id sid tid token lemma pos ap-7718 1 1 acta acta PROPN ap-7718 1 2 polytechnica polytechnica PROPN ap-7718 1 3 https://doi.org/10.14311/ap.2022.62.0211 https://doi.org/10.14311/ap.2022.62.0211 PROPN ap-7718 1 4 acta acta PROPN ap-7718 1 5 polytechnica polytechnica PROPN ap-7718 1 6 62(1):211–221 62(1):211–221 PROPN ap-7718 1 7 , , PUNCT ap-7718 1 8 2022 2022 NUM ap-7718 1 9 © © ADP ap-7718 1 10 2022 2022 NUM ap-7718 1 11 the the DET ap-7718 1 12 author(s author(s NOUN ap-7718 1 13 ) ) PUNCT ap-7718 1 14 . . PUNCT ap-7718 2 1 licensed license VERB ap-7718 2 2 under under ADP ap-7718 2 3 a a DET ap-7718 2 4 cc cc NOUN ap-7718 2 5 - - PUNCT ap-7718 2 6 by by ADP ap-7718 2 7 4.0 4.0 NUM ap-7718 2 8 licence licence NOUN ap-7718 2 9 published publish VERB ap-7718 2 10 by by ADP ap-7718 2 11 the the DET ap-7718 2 12 czech czech PROPN ap-7718 2 13 technical technical PROPN ap-7718 2 14 university university PROPN ap-7718 2 15 in in ADP ap-7718 2 16 prague prague PROPN ap-7718 2 17 time time NOUN ap-7718 2 18 - - PUNCT ap-7718 2 19 dependent dependent ADJ ap-7718 2 20 mass mass NOUN ap-7718 2 21 oscillators oscillator NOUN ap-7718 2 22 : : PUNCT ap-7718 2 23 constants constant NOUN ap-7718 2 24 of of ADP ap-7718 2 25 motion motion NOUN ap-7718 2 26 and and CCONJ ap-7718 2 27 semiclasical semiclasical ADJ ap-7718 2 28 states state NOUN ap-7718 2 29 kevin kevin PROPN ap-7718 2 30 zelaya zelaya PROPN ap-7718 2 31 czech czech PROPN ap-7718 2 32 academy academy PROPN ap-7718 2 33 of of ADP ap-7718 2 34 science science PROPN ap-7718 2 35 , , PUNCT ap-7718 2 36 nuclear nuclear PROPN ap-7718 2 37 physics physics PROPN ap-7718 2 38 institute institute PROPN ap-7718 2 39 , , PUNCT ap-7718 2 40 250 250 NUM ap-7718 2 41 68 68 NUM ap-7718 2 42 řež řež NOUN ap-7718 2 43 , , PUNCT ap-7718 2 44 czech czech PROPN ap-7718 2 45 republic republic NOUN ap-7718 2 46 correspondence correspondence NOUN ap-7718 2 47 : : PUNCT ap-7718 2 48 kdzelaya@fis.cinvestav.mx kdzelaya@fis.cinvestav.mx PROPN ap-7718 2 49 abstract abstract NOUN ap-7718 2 50 . . PUNCT ap-7718 3 1 this this DET ap-7718 3 2 work work NOUN ap-7718 3 3 reports report VERB ap-7718 3 4 the the DET ap-7718 3 5 construction construction NOUN ap-7718 3 6 of of ADP ap-7718 3 7 constants constant NOUN ap-7718 3 8 of of ADP ap-7718 3 9 motion motion NOUN ap-7718 3 10 for for ADP ap-7718 3 11 a a DET ap-7718 3 12 family family NOUN ap-7718 3 13 of of ADP ap-7718 3 14 time time NOUN ap-7718 3 15 - - PUNCT ap-7718 3 16 dependent dependent ADJ ap-7718 3 17 mass mass NOUN ap-7718 3 18 oscillators oscillator NOUN ap-7718 3 19 , , PUNCT ap-7718 3 20 achieved achieve VERB ap-7718 3 21 by by ADP ap-7718 3 22 implementing implement VERB ap-7718 3 23 the the DET ap-7718 3 24 formalism formalism NOUN ap-7718 3 25 of of ADP ap-7718 3 26 form form NOUN ap-7718 3 27 - - PUNCT ap-7718 3 28 preserving preserve VERB ap-7718 3 29 point point NOUN ap-7718 3 30 transformations transformation NOUN ap-7718 3 31 . . PUNCT ap-7718 4 1 the the DET ap-7718 4 2 latter latter ADJ ap-7718 4 3 allows allow VERB ap-7718 4 4 obtaining obtain VERB ap-7718 4 5 a a DET ap-7718 4 6 spectral spectral ADJ ap-7718 4 7 problem problem NOUN ap-7718 4 8 for for ADP ap-7718 4 9 each each DET ap-7718 4 10 constant constant ADJ ap-7718 4 11 of of ADP ap-7718 4 12 motion motion NOUN ap-7718 4 13 , , PUNCT ap-7718 4 14 one one NUM ap-7718 4 15 of of ADP ap-7718 4 16 which which PRON ap-7718 4 17 leads lead VERB ap-7718 4 18 to to ADP ap-7718 4 19 a a DET ap-7718 4 20 non non ADJ ap-7718 4 21 - - ADJ ap-7718 4 22 orthogonal orthogonal ADJ ap-7718 4 23 set set NOUN ap-7718 4 24 of of ADP ap-7718 4 25 eigensolutions eigensolution NOUN ap-7718 4 26 that that PRON ap-7718 4 27 are be AUX ap-7718 4 28 , , PUNCT ap-7718 4 29 in in ADP ap-7718 4 30 turn turn NOUN ap-7718 4 31 , , PUNCT ap-7718 4 32 coherent coherent ADJ ap-7718 4 33 states state NOUN ap-7718 4 34 . . PUNCT ap-7718 5 1 that that PRON ap-7718 5 2 is be AUX ap-7718 5 3 , , PUNCT ap-7718 5 4 eigensolutions eigensolution NOUN ap-7718 5 5 whose whose DET ap-7718 5 6 wavepacket wavepacket NOUN ap-7718 5 7 follows follow VERB ap-7718 5 8 a a DET ap-7718 5 9 classical classical ADJ ap-7718 5 10 trajectory trajectory NOUN ap-7718 5 11 and and CCONJ ap-7718 5 12 saturate saturate VERB ap-7718 5 13 , , PUNCT ap-7718 5 14 in in ADP ap-7718 5 15 this this DET ap-7718 5 16 case case NOUN ap-7718 5 17 , , PUNCT ap-7718 5 18 the the DET ap-7718 5 19 schrödinger schrödinger ADJ ap-7718 5 20 - - PUNCT ap-7718 5 21 robertson robertson PROPN ap-7718 5 22 uncertainty uncertainty PROPN ap-7718 5 23 relationship relationship NOUN ap-7718 5 24 . . PUNCT ap-7718 6 1 results result NOUN ap-7718 6 2 obtained obtain VERB ap-7718 6 3 in in ADP ap-7718 6 4 this this DET ap-7718 6 5 form form NOUN ap-7718 6 6 are be AUX ap-7718 6 7 relatively relatively ADV ap-7718 6 8 general general ADJ ap-7718 6 9 , , PUNCT ap-7718 6 10 and and CCONJ ap-7718 6 11 some some DET ap-7718 6 12 particular particular ADJ ap-7718 6 13 examples example NOUN ap-7718 6 14 are be AUX ap-7718 6 15 considered consider VERB ap-7718 6 16 to to PART ap-7718 6 17 illustrate illustrate VERB ap-7718 6 18 the the DET ap-7718 6 19 results result NOUN ap-7718 6 20 further far ADV ap-7718 6 21 . . PUNCT ap-7718 7 1 notably notably ADV ap-7718 7 2 , , PUNCT ap-7718 7 3 a a DET ap-7718 7 4 regularized regularize VERB ap-7718 7 5 caldirola caldirola PROPN ap-7718 7 6 - - PUNCT ap-7718 7 7 kanai kanai PROPN ap-7718 7 8 mass mass PROPN ap-7718 7 9 term term NOUN ap-7718 7 10 is be AUX ap-7718 7 11 introduced introduce VERB ap-7718 7 12 in in ADP ap-7718 7 13 an an DET ap-7718 7 14 attempt attempt NOUN ap-7718 7 15 to to PART ap-7718 7 16 amend amend VERB ap-7718 7 17 some some PRON ap-7718 7 18 of of ADP ap-7718 7 19 the the DET ap-7718 7 20 unusual unusual ADJ ap-7718 7 21 features feature NOUN ap-7718 7 22 found find VERB ap-7718 7 23 in in ADP ap-7718 7 24 the the DET ap-7718 7 25 conventional conventional ADJ ap-7718 7 26 caldirola caldirola PROPN ap-7718 7 27 - - PUNCT ap-7718 7 28 kanai kanai PROPN ap-7718 7 29 case case NOUN ap-7718 7 30 . . PUNCT ap-7718 8 1 keywords keyword NOUN ap-7718 8 2 : : PUNCT ap-7718 8 3 time time NOUN ap-7718 8 4 - - PUNCT ap-7718 8 5 dependent dependent ADJ ap-7718 8 6 mass mass NOUN ap-7718 8 7 oscillators oscillator NOUN ap-7718 8 8 , , PUNCT ap-7718 8 9 caldirola caldirola PROPN ap-7718 8 10 - - PUNCT ap-7718 8 11 kanai kanai PROPN ap-7718 8 12 oscillator oscillator PROPN ap-7718 8 13 , , PUNCT ap-7718 8 14 quantum quantum ADJ ap-7718 8 15 invariants invariant NOUN ap-7718 8 16 , , PUNCT ap-7718 8 17 coherent coherent ADJ ap-7718 8 18 states state NOUN ap-7718 8 19 , , PUNCT ap-7718 8 20 semiclassical semiclassical ADJ ap-7718 8 21 dynamics dynamic NOUN ap-7718 8 22 . . PUNCT ap-7718 9 1 1 1 X ap-7718 9 2 . . X ap-7718 9 3 introduction introduction NOUN ap-7718 9 4 the the DET ap-7718 9 5 search search NOUN ap-7718 9 6 for for ADP ap-7718 9 7 exact exact ADJ ap-7718 9 8 solutions solution NOUN ap-7718 9 9 for for ADP ap-7718 9 10 time time NOUN ap-7718 9 11 - - PUNCT ap-7718 9 12 dependent dependent ADJ ap-7718 9 13 ( ( PUNCT ap-7718 9 14 nonstationary nonstationary ADJ ap-7718 9 15 ) ) PUNCT ap-7718 9 16 quantum quantum NOUN ap-7718 9 17 models model NOUN ap-7718 9 18 is be AUX ap-7718 9 19 challenging challenge VERB ap-7718 9 20 task task NOUN ap-7718 9 21 as as SCONJ ap-7718 9 22 compared compare VERB ap-7718 9 23 to to ADP ap-7718 9 24 the the DET ap-7718 9 25 stationary stationary ADJ ap-7718 9 26 ( ( PUNCT ap-7718 9 27 time time NOUN ap-7718 9 28 - - PUNCT ap-7718 9 29 independent independent ADJ ap-7718 9 30 ) ) PUNCT ap-7718 9 31 counterpart counterpart NOUN ap-7718 9 32 . . PUNCT ap-7718 10 1 in in ADP ap-7718 10 2 the the DET ap-7718 10 3 stationary stationary ADJ ap-7718 10 4 case case NOUN ap-7718 10 5 , , PUNCT ap-7718 10 6 the the DET ap-7718 10 7 dynamical dynamical ADJ ap-7718 10 8 law law NOUN ap-7718 10 9 ( ( PUNCT ap-7718 10 10 schrödinger schrödinger ADJ ap-7718 10 11 equation equation NOUN ap-7718 10 12 ) ) PUNCT ap-7718 10 13 reduces reduce VERB ap-7718 10 14 to to ADP ap-7718 10 15 an an DET ap-7718 10 16 eigenvalue eigenvalue ADJ ap-7718 10 17 equation equation NOUN ap-7718 10 18 associated associate VERB ap-7718 10 19 with with ADP ap-7718 10 20 the the DET ap-7718 10 21 energy energy NOUN ap-7718 10 22 observable observable ADJ ap-7718 10 23 , , PUNCT ap-7718 10 24 the the DET ap-7718 10 25 hamiltonian hamiltonian NOUN ap-7718 10 26 , , PUNCT ap-7718 10 27 for for ADP ap-7718 10 28 which which PRON ap-7718 10 29 several several ADJ ap-7718 10 30 methods method NOUN ap-7718 10 31 can can AUX ap-7718 10 32 be be AUX ap-7718 10 33 implemented implement VERB ap-7718 10 34 to to PART ap-7718 10 35 obtain obtain VERB ap-7718 10 36 exact exact ADJ ap-7718 10 37 solutions solution NOUN ap-7718 10 38 . . PUNCT ap-7718 11 1 particularly particularly ADV ap-7718 11 2 , , PUNCT ap-7718 11 3 new new ADJ ap-7718 11 4 exactly exactly ADV ap-7718 11 5 solvable solvable ADJ ap-7718 11 6 models model NOUN ap-7718 11 7 can can AUX ap-7718 11 8 be be AUX ap-7718 11 9 constructed construct VERB ap-7718 11 10 from from ADP ap-7718 11 11 previously previously ADV ap-7718 11 12 known know VERB ap-7718 11 13 ones one NOUN ap-7718 11 14 through through ADP ap-7718 11 15 darboux darboux ADJ ap-7718 11 16 transformations transformation NOUN ap-7718 11 17 [ [ X ap-7718 11 18 1 1 NUM ap-7718 11 19 ] ] PUNCT ap-7718 11 20 ( ( PUNCT ap-7718 11 21 also also ADV ap-7718 11 22 known know VERB ap-7718 11 23 as as ADP ap-7718 11 24 susy susy NOUN ap-7718 11 25 - - PUNCT ap-7718 11 26 qm qm PROPN ap-7718 11 27 ) ) PUNCT ap-7718 11 28 . . PUNCT ap-7718 12 1 in in ADP ap-7718 12 2 the the DET ap-7718 12 3 nonstationary nonstationary ADJ ap-7718 12 4 case case NOUN ap-7718 12 5 , , PUNCT ap-7718 12 6 it it PRON ap-7718 12 7 is be AUX ap-7718 12 8 still still ADV ap-7718 12 9 possible possible ADJ ap-7718 12 10 to to PART ap-7718 12 11 recover recover VERB ap-7718 12 12 an an DET ap-7718 12 13 eigenvalue eigenvalue ADJ ap-7718 12 14 problem problem NOUN ap-7718 12 15 for for ADP ap-7718 12 16 the the DET ap-7718 12 17 hamiltonian hamiltonian NOUN ap-7718 12 18 if if SCONJ ap-7718 12 19 one one PRON ap-7718 12 20 restricts restrict VERB ap-7718 12 21 to to ADP ap-7718 12 22 the the DET ap-7718 12 23 adiabatic adiabatic ADJ ap-7718 12 24 approximation approximation NOUN ap-7718 12 25 [ [ X ap-7718 12 26 2 2 NUM ap-7718 12 27 , , PUNCT ap-7718 12 28 3 3 NUM ap-7718 12 29 ] ] PUNCT ap-7718 12 30 . . PUNCT ap-7718 13 1 however however ADV ap-7718 13 2 , , PUNCT ap-7718 13 3 in in ADP ap-7718 13 4 general general ADJ ap-7718 13 5 , , PUNCT ap-7718 13 6 the the DET ap-7718 13 7 latter latter ADJ ap-7718 13 8 is be AUX ap-7718 13 9 not not PART ap-7718 13 10 feasible feasible ADJ ap-7718 13 11 , , PUNCT ap-7718 13 12 and and CCONJ ap-7718 13 13 other other ADJ ap-7718 13 14 workarounds workaround NOUN ap-7718 13 15 should should AUX ap-7718 13 16 be be AUX ap-7718 13 17 implemented implement VERB ap-7718 13 18 . . PUNCT ap-7718 14 1 despite despite SCONJ ap-7718 14 2 all all DET ap-7718 14 3 these these DET ap-7718 14 4 challenges challenge NOUN ap-7718 14 5 , , PUNCT ap-7718 14 6 time time NOUN ap-7718 14 7 - - PUNCT ap-7718 14 8 dependent dependent ADJ ap-7718 14 9 phenomena phenomenon NOUN ap-7718 14 10 find find VERB ap-7718 14 11 exciting exciting ADJ ap-7718 14 12 applications application NOUN ap-7718 14 13 in in ADP ap-7718 14 14 physical physical ADJ ap-7718 14 15 systems system NOUN ap-7718 14 16 such such ADJ ap-7718 14 17 as as ADP ap-7718 14 18 electromagnetic electromagnetic ADJ ap-7718 14 19 traps trap NOUN ap-7718 14 20 of of ADP ap-7718 14 21 charged charge VERB ap-7718 14 22 particles particle NOUN ap-7718 14 23 and and CCONJ ap-7718 14 24 plasma plasma NOUN ap-7718 14 25 physics physics NOUN ap-7718 15 1 [ [ X ap-7718 15 2 4–8 4–8 NOUN ap-7718 15 3 ] ] X ap-7718 15 4 . . PUNCT ap-7718 16 1 the the DET ap-7718 16 2 parametric parametric ADJ ap-7718 16 3 oscillator oscillator NOUN ap-7718 16 4 is be AUX ap-7718 16 5 perhaps perhaps ADV ap-7718 16 6 the the DET ap-7718 16 7 most most ADV ap-7718 16 8 wellknown wellknown ADJ ap-7718 16 9 exactly exactly ADV ap-7718 16 10 solvable solvable ADJ ap-7718 16 11 nonstationary nonstationary ADJ ap-7718 16 12 model model NOUN ap-7718 16 13 in in ADP ap-7718 16 14 quantum quantum ADJ ap-7718 16 15 mechanics mechanic NOUN ap-7718 16 16 . . PUNCT ap-7718 17 1 a a DET ap-7718 17 2 straightforward straightforward ADJ ap-7718 17 3 method method NOUN ap-7718 17 4 to to PART ap-7718 17 5 solve solve VERB ap-7718 17 6 such such DET ap-7718 17 7 a a DET ap-7718 17 8 problem problem NOUN ap-7718 17 9 was be AUX ap-7718 17 10 introduced introduce VERB ap-7718 17 11 by by ADP ap-7718 17 12 lews lew NOUN ap-7718 17 13 and and CCONJ ap-7718 17 14 riesenfeld riesenfeld VERB ap-7718 17 15 [ [ X ap-7718 17 16 9 9 NUM ap-7718 17 17 ] ] PUNCT ap-7718 17 18 by by ADP ap-7718 17 19 noticing notice VERB ap-7718 17 20 that that SCONJ ap-7718 17 21 the the DET ap-7718 17 22 appropriate appropriate ADJ ap-7718 17 23 constant constant NOUN ap-7718 17 24 of of ADP ap-7718 17 25 motion motion NOUN ap-7718 17 26 ( ( PUNCT ap-7718 17 27 quantum quantum NOUN ap-7718 17 28 invariant invariant NOUN ap-7718 17 29 ) ) PUNCT ap-7718 17 30 admits admit VERB ap-7718 17 31 a a DET ap-7718 17 32 nonstationary nonstationary ADJ ap-7718 17 33 eigenvalue eigenvalue ADJ ap-7718 17 34 equation equation NOUN ap-7718 17 35 with with ADP ap-7718 17 36 time time NOUN ap-7718 17 37 - - PUNCT ap-7718 17 38 dependent dependent ADJ ap-7718 17 39 solutions solution NOUN ap-7718 17 40 and and CCONJ ap-7718 17 41 constant constant ADJ ap-7718 17 42 eigenvalues eigenvalue NOUN ap-7718 17 43 . . PUNCT ap-7718 18 1 in in ADP ap-7718 18 2 this this DET ap-7718 18 3 form form NOUN ap-7718 18 4 , , PUNCT ap-7718 18 5 nonstationary nonstationary ADJ ap-7718 18 6 models model NOUN ap-7718 18 7 can can AUX ap-7718 18 8 be be AUX ap-7718 18 9 addressed address VERB ap-7718 18 10 similarly similarly ADV ap-7718 18 11 to to ADP ap-7718 18 12 their their PRON ap-7718 18 13 stationary stationary ADJ ap-7718 18 14 counterparts counterpart NOUN ap-7718 18 15 . . PUNCT ap-7718 19 1 this this PRON ap-7718 19 2 paved pave VERB ap-7718 19 3 the the DET ap-7718 19 4 way way NOUN ap-7718 19 5 to to PART ap-7718 19 6 solve solve VERB ap-7718 19 7 other other ADJ ap-7718 19 8 time time NOUN ap-7718 19 9 - - PUNCT ap-7718 19 10 dependent dependent ADJ ap-7718 19 11 problems problem NOUN ap-7718 19 12 [ [ X ap-7718 19 13 10–14 10–14 NUM ap-7718 19 14 ] ] PUNCT ap-7718 19 15 . . PUNCT ap-7718 20 1 recently recently ADV ap-7718 20 2 , , PUNCT ap-7718 20 3 the the DET ap-7718 20 4 darboux darboux ADJ ap-7718 20 5 transformation transformation NOUN ap-7718 20 6 has have AUX ap-7718 20 7 been be AUX ap-7718 20 8 adapted adapt VERB ap-7718 20 9 into into ADP ap-7718 20 10 the the DET ap-7718 20 11 quantum quantum ADJ ap-7718 20 12 invariant invariant ADJ ap-7718 20 13 scheme scheme NOUN ap-7718 20 14 to to PART ap-7718 20 15 construct construct VERB ap-7718 20 16 new new ADJ ap-7718 20 17 time time NOUN ap-7718 20 18 - - PUNCT ap-7718 20 19 dependent dependent ADJ ap-7718 20 20 hamiltonians hamiltonian NOUN ap-7718 20 21 , , PUNCT ap-7718 20 22 together together ADV ap-7718 20 23 with with ADP ap-7718 20 24 the the DET ap-7718 20 25 corresponding correspond VERB ap-7718 20 26 quantum quantum ADJ ap-7718 20 27 invariant invariant NOUN ap-7718 20 28 and and CCONJ ap-7718 20 29 the the DET ap-7718 20 30 set set NOUN ap-7718 20 31 of of ADP ap-7718 20 32 solutions solution NOUN ap-7718 20 33 [ [ X ap-7718 20 34 15–17 15–17 NUM ap-7718 20 35 ] ] PUNCT ap-7718 20 36 . . PUNCT ap-7718 21 1 alternatively alternatively ADV ap-7718 21 2 , , PUNCT ap-7718 21 3 other other ADJ ap-7718 21 4 methods method NOUN ap-7718 21 5 exist exist VERB ap-7718 21 6 to to PART ap-7718 21 7 build build VERB ap-7718 21 8 new new ADJ ap-7718 21 9 time time NOUN ap-7718 21 10 - - PUNCT ap-7718 21 11 dependent dependent ADJ ap-7718 21 12 models model NOUN ap-7718 21 13 , , PUNCT ap-7718 21 14 such such ADJ ap-7718 21 15 as as ADP ap-7718 21 16 the the DET ap-7718 21 17 modified modify VERB ap-7718 21 18 darboux darboux NOUN ap-7718 21 19 transfomation transfomation NOUN ap-7718 21 20 introduced introduce VERB ap-7718 21 21 by by ADP ap-7718 21 22 bagrov bagrov NOUN ap-7718 21 23 et et PROPN ap-7718 21 24 al al PROPN ap-7718 21 25 . . PUNCT ap-7718 22 1 [ [ X ap-7718 22 2 18 18 NUM ap-7718 22 3 ] ] PUNCT ap-7718 22 4 , , PUNCT ap-7718 22 5 which which PRON ap-7718 22 6 relies rely VERB ap-7718 22 7 on on ADP ap-7718 22 8 a a DET ap-7718 22 9 differential differential ADJ ap-7718 22 10 operator operator NOUN ap-7718 22 11 that that PRON ap-7718 22 12 intertwines intertwine VERB ap-7718 22 13 a a DET ap-7718 22 14 known know VERB ap-7718 22 15 schrödinger schrödinger ADJ ap-7718 22 16 equation equation NOUN ap-7718 22 17 with with ADP ap-7718 22 18 an an DET ap-7718 22 19 unknown unknown ADJ ap-7718 22 20 one one NOUN ap-7718 22 21 . . PUNCT ap-7718 23 1 this this PRON ap-7718 23 2 has have AUX ap-7718 23 3 led lead VERB ap-7718 23 4 to to ADP ap-7718 23 5 new new ADJ ap-7718 23 6 results result NOUN ap-7718 23 7 in in ADP ap-7718 23 8 the the DET ap-7718 23 9 nonstationary nonstationary ADJ ap-7718 23 10 hermitian hermitian ADJ ap-7718 23 11 regime regime NOUN ap-7718 23 12 [ [ X ap-7718 23 13 19–21 19–21 NUM ap-7718 23 14 ] ] PUNCT ap-7718 23 15 . . PUNCT ap-7718 24 1 a a DET ap-7718 24 2 non non ADJ ap-7718 24 3 - - ADJ ap-7718 24 4 hermitian hermitian ADJ ap-7718 24 5 pt pt ADJ ap-7718 24 6 - - ADJ ap-7718 24 7 symmetric symmetric ADJ ap-7718 24 8 extension extension NOUN ap-7718 24 9 has have AUX ap-7718 24 10 been be AUX ap-7718 24 11 discussed discuss VERB ap-7718 24 12 in in ADP ap-7718 24 13 [ [ X ap-7718 24 14 22 22 NUM ap-7718 24 15 ] ] PUNCT ap-7718 24 16 , , PUNCT ap-7718 24 17 and and CCONJ ap-7718 24 18 some some DET ap-7718 24 19 further further ADJ ap-7718 24 20 models model NOUN ap-7718 24 21 were be AUX ap-7718 24 22 reported report VERB ap-7718 24 23 in in ADP ap-7718 24 24 [ [ X ap-7718 24 25 23 23 NUM ap-7718 24 26 , , PUNCT ap-7718 24 27 24 24 NUM ap-7718 24 28 ] ] PUNCT ap-7718 24 29 . . PUNCT ap-7718 25 1 on on ADP ap-7718 25 2 the the DET ap-7718 25 3 other other ADJ ap-7718 25 4 hand hand NOUN ap-7718 25 5 , , PUNCT ap-7718 25 6 the the DET ap-7718 25 7 point point NOUN ap-7718 25 8 transformations transformation NOUN ap-7718 25 9 formalism formalism NOUN ap-7718 25 10 [ [ X ap-7718 25 11 25 25 NUM ap-7718 25 12 ] ] PUNCT ap-7718 25 13 has have AUX ap-7718 25 14 been be AUX ap-7718 25 15 proved prove VERB ap-7718 25 16 useful useful ADJ ap-7718 25 17 to to PART ap-7718 25 18 construct construct VERB ap-7718 25 19 and and CCONJ ap-7718 25 20 solve solve VERB ap-7718 25 21 time time NOUN ap-7718 25 22 - - PUNCT ap-7718 25 23 dependent dependent ADJ ap-7718 25 24 oscillators oscillator NOUN ap-7718 25 25 . . PUNCT ap-7718 26 1 this this PRON ap-7718 26 2 was be AUX ap-7718 26 3 achieved achieve VERB ap-7718 26 4 by by ADP ap-7718 26 5 implementing implement VERB ap-7718 26 6 a a DET ap-7718 26 7 geometrical geometrical ADJ ap-7718 26 8 deformation deformation NOUN ap-7718 26 9 that that PRON ap-7718 26 10 transforms transform VERB ap-7718 26 11 the the DET ap-7718 26 12 stationary stationary ADJ ap-7718 26 13 oscillator oscillator NOUN ap-7718 26 14 schrödinger schrödinger ADJ ap-7718 26 15 equation equation NOUN ap-7718 26 16 into into ADP ap-7718 26 17 one one NUM ap-7718 26 18 with with ADP ap-7718 26 19 time time NOUN ap-7718 26 20 - - PUNCT ap-7718 26 21 dependent dependent ADJ ap-7718 26 22 frequency frequency NOUN ap-7718 26 23 and and CCONJ ap-7718 26 24 mass mass NOUN ap-7718 27 1 [ [ X ap-7718 27 2 26 26 NUM ap-7718 27 3 , , PUNCT ap-7718 27 4 27 27 NUM ap-7718 27 5 ] ] PUNCT ap-7718 27 6 . . PUNCT ap-7718 28 1 this this PRON ap-7718 28 2 allows allow VERB ap-7718 28 3 obtaining obtain VERB ap-7718 28 4 further further ADJ ap-7718 28 5 information information NOUN ap-7718 28 6 such such ADJ ap-7718 28 7 as as ADP ap-7718 28 8 the the DET ap-7718 28 9 constants constant NOUN ap-7718 28 10 of of ADP ap-7718 28 11 motion motion NOUN ap-7718 28 12 , , PUNCT ap-7718 28 13 which which PRON ap-7718 28 14 are be AUX ap-7718 28 15 preserved preserve VERB ap-7718 28 16 throughout throughout ADP ap-7718 28 17 the the DET ap-7718 28 18 point point NOUN ap-7718 28 19 transformation transformation NOUN ap-7718 29 1 [ [ X ap-7718 29 2 25 25 NUM ap-7718 29 3 ] ] PUNCT ap-7718 29 4 , , PUNCT ap-7718 29 5 leading lead VERB ap-7718 29 6 to to ADP ap-7718 29 7 a a DET ap-7718 29 8 straightforward straightforward ADJ ap-7718 29 9 way way NOUN ap-7718 29 10 to to PART ap-7718 29 11 get get VERB ap-7718 29 12 such such ADJ ap-7718 29 13 constants constant NOUN ap-7718 29 14 of of ADP ap-7718 29 15 motion motion NOUN ap-7718 29 16 without without ADP ap-7718 29 17 imposing impose VERB ap-7718 29 18 any any DET ap-7718 29 19 ansatz ansatz NOUN ap-7718 29 20 . . PUNCT ap-7718 30 1 a a DET ap-7718 30 2 further further ADJ ap-7718 30 3 extension extension NOUN ap-7718 30 4 for for ADP ap-7718 30 5 non non ADJ ap-7718 30 6 - - ADJ ap-7718 30 7 hermitian hermitian ADJ ap-7718 30 8 systems system NOUN ap-7718 30 9 was be AUX ap-7718 30 10 introduced introduce VERB ap-7718 30 11 in in ADP ap-7718 30 12 [ [ X ap-7718 30 13 28 28 NUM ap-7718 30 14 ] ] PUNCT ap-7718 30 15 , , PUNCT ap-7718 30 16 whereas whereas SCONJ ap-7718 30 17 a a DET ap-7718 30 18 non non ADJ ap-7718 30 19 - - ADJ ap-7718 30 20 hermitian hermitian ADJ ap-7718 30 21 extension extension NOUN ap-7718 30 22 of of ADP ap-7718 30 23 the the DET ap-7718 30 24 generalized generalized ADJ ap-7718 30 25 caldirola caldirola PROPN ap-7718 30 26 - - PUNCT ap-7718 30 27 kanai kanai PROPN ap-7718 30 28 oscillator oscillator PROPN ap-7718 30 29 was be AUX ap-7718 30 30 discussed discuss VERB ap-7718 30 31 in in ADP ap-7718 30 32 [ [ X ap-7718 30 33 29 29 NUM ap-7718 30 34 ] ] PUNCT ap-7718 30 35 . . PUNCT ap-7718 31 1 in in ADP ap-7718 31 2 this this DET ap-7718 31 3 work work NOUN ap-7718 31 4 , , PUNCT ap-7718 31 5 the the DET ap-7718 31 6 point point NOUN ap-7718 31 7 transformation transformation NOUN ap-7718 31 8 formalism formalism NOUN ap-7718 31 9 is be AUX ap-7718 31 10 exploited exploit VERB ap-7718 31 11 to to PART ap-7718 31 12 construct construct VERB ap-7718 31 13 and and CCONJ ap-7718 31 14 study study VERB ap-7718 31 15 the the DET ap-7718 31 16 dynamics dynamic NOUN ap-7718 31 17 of of ADP ap-7718 31 18 semiclassical semiclassical ADJ ap-7718 31 19 states state NOUN ap-7718 31 20 associated associate VERB ap-7718 31 21 with with ADP ap-7718 31 22 time time NOUN ap-7718 31 23 - - PUNCT ap-7718 31 24 dependent dependent ADJ ap-7718 31 25 mass mass NOUN ap-7718 31 26 oscillators oscillator NOUN ap-7718 31 27 . . PUNCT ap-7718 32 1 this this PRON ap-7718 32 2 is be AUX ap-7718 32 3 achieved achieve VERB ap-7718 32 4 by by ADP ap-7718 32 5 using use VERB ap-7718 32 6 the the DET ap-7718 32 7 aforementioned aforementioned ADJ ap-7718 32 8 preservation preservation NOUN ap-7718 32 9 of of ADP ap-7718 32 10 constants constant NOUN ap-7718 32 11 of of ADP ap-7718 32 12 motion motion NOUN ap-7718 32 13 and and CCONJ ap-7718 32 14 identifying identify VERB ap-7718 32 15 their their PRON ap-7718 32 16 corresponding corresponding ADJ ap-7718 32 17 spectral spectral ADJ ap-7718 32 18 problem problem NOUN ap-7718 32 19 . . PUNCT ap-7718 33 1 notably notably ADV ap-7718 33 2 , , PUNCT ap-7718 33 3 it it PRON ap-7718 33 4 is be AUX ap-7718 33 5 shown show VERB ap-7718 33 6 that that SCONJ ap-7718 33 7 one one NUM ap-7718 33 8 constant constant ADJ ap-7718 33 9 of of ADP ap-7718 33 10 motion motion NOUN ap-7718 33 11 leads lead VERB ap-7718 33 12 to to ADP ap-7718 33 13 an an DET ap-7718 33 14 orthogonal orthogonal ADJ ap-7718 33 15 set set NOUN ap-7718 33 16 of of ADP ap-7718 33 17 solutions solution NOUN ap-7718 33 18 , , PUNCT ap-7718 33 19 whereas whereas SCONJ ap-7718 33 20 a a DET ap-7718 33 21 different different ADJ ap-7718 33 22 one one NOUN ap-7718 33 23 leads lead VERB ap-7718 33 24 to to ADP ap-7718 33 25 nonorthogonal nonorthogonal ADJ ap-7718 33 26 solutions solution NOUN ap-7718 33 27 that that PRON ap-7718 33 28 behave behave VERB ap-7718 33 29 like like ADP ap-7718 33 30 semiclassical semiclassical ADJ ap-7718 33 31 states state NOUN ap-7718 33 32 . . PUNCT ap-7718 34 1 that that PRON ap-7718 34 2 is be AUX ap-7718 34 3 , , PUNCT ap-7718 34 4 gaussian gaussian ADJ ap-7718 34 5 wavepackets wavepacket NOUN ap-7718 34 6 whose whose DET ap-7718 34 7 maximum maximum ADJ ap-7718 34 8 point point NOUN ap-7718 34 9 follows follow VERB ap-7718 34 10 the the DET ap-7718 34 11 corresponding corresponding ADJ ap-7718 34 12 classical classical ADJ ap-7718 34 13 trajectory trajectory NOUN ap-7718 34 14 and and CCONJ ap-7718 34 15 minimize minimize VERB ap-7718 34 16 , , PUNCT ap-7718 34 17 in in ADP ap-7718 34 18 this this DET ap-7718 34 19 case case NOUN ap-7718 34 20 , , PUNCT ap-7718 34 21 the the DET ap-7718 34 22 211 211 NUM ap-7718 34 23 https://doi.org/10.14311/ap.2022.62.0211 https://doi.org/10.14311/ap.2022.62.0211 NOUN ap-7718 34 24 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN ap-7718 34 25 https://www.cvut.cz/en https://www.cvut.cz/en PROPN ap-7718 34 26 kevin kevin PROPN ap-7718 34 27 zelaya zelaya PROPN ap-7718 34 28 acta acta PROPN ap-7718 34 29 polytechnica polytechnica PROPN ap-7718 34 30 schrödinger schrödinger PROPN ap-7718 34 31 - - PUNCT ap-7718 34 32 robertson robertson PROPN ap-7718 34 33 uncertainty uncertainty NOUN ap-7718 34 34 principle principle NOUN ap-7718 34 35 . . PUNCT ap-7718 35 1 two two NUM ap-7718 35 2 particular particular ADJ ap-7718 35 3 examples example NOUN ap-7718 35 4 are be AUX ap-7718 35 5 considered consider VERB ap-7718 35 6 to to PART ap-7718 35 7 illustrate illustrate VERB ap-7718 35 8 the the DET ap-7718 35 9 usefulness usefulness NOUN ap-7718 35 10 of of ADP ap-7718 35 11 the the DET ap-7718 35 12 approach approach NOUN ap-7718 35 13 further far ADV ap-7718 35 14 . . PUNCT ap-7718 36 1 2 2 X ap-7718 36 2 . . X ap-7718 36 3 materials material NOUN ap-7718 36 4 and and CCONJ ap-7718 36 5 methods method NOUN ap-7718 36 6 throughout throughout ADP ap-7718 36 7 this this DET ap-7718 36 8 manuscript manuscript NOUN ap-7718 36 9 , , PUNCT ap-7718 36 10 the the DET ap-7718 36 11 time time NOUN ap-7718 36 12 - - PUNCT ap-7718 36 13 dependent dependent ADJ ap-7718 36 14 mass mass ADJ ap-7718 36 15 m(t m(t NOUN ap-7718 36 16 ) ) PUNCT ap-7718 36 17 and and CCONJ ap-7718 36 18 frequency frequency NOUN ap-7718 36 19 ω2(t ω2(t SYM ap-7718 36 20 ) ) PUNCT ap-7718 36 21 oscillator oscillator NOUN ap-7718 36 22 subjected subject VERB ap-7718 36 23 to to ADP ap-7718 36 24 an an DET ap-7718 36 25 external external ADJ ap-7718 36 26 driving driving NOUN ap-7718 36 27 force force NOUN ap-7718 36 28 f f PROPN ap-7718 36 29 ( ( PUNCT ap-7718 36 30 t t PROPN ap-7718 36 31 ) ) PUNCT ap-7718 36 32 is be AUX ap-7718 36 33 considered consider VERB ap-7718 36 34 . . PUNCT ap-7718 37 1 such such DET ap-7718 37 2 a a DET ap-7718 37 3 model model NOUN ap-7718 37 4 is be AUX ap-7718 37 5 characterized characterize VERB ap-7718 37 6 by by ADP ap-7718 37 7 the the DET ap-7718 37 8 time time NOUN ap-7718 37 9 - - PUNCT ap-7718 37 10 dependent dependent ADJ ap-7718 37 11 hamiltonian hamiltonian ADJ ap-7718 37 12 ĥck(t ĥck(t NOUN ap-7718 37 13 ) ) PUNCT ap-7718 38 1 = = SYM ap-7718 38 2 p̂2 p̂2 ADP ap-7718 38 3 2m(t 2m(t PROPN ap-7718 38 4 ) ) PUNCT ap-7718 39 1 + + CCONJ ap-7718 39 2 m(t)ω2(t m(t)ω2(t NOUN ap-7718 39 3 ) ) PUNCT ap-7718 39 4 2 2 NUM ap-7718 39 5 x̂2 x̂2 NOUN ap-7718 40 1 + + CCONJ ap-7718 40 2 f f X ap-7718 40 3 ( ( PUNCT ap-7718 40 4 t)x̂ t)x̂ PROPN ap-7718 40 5 , , PUNCT ap-7718 40 6 ( ( PUNCT ap-7718 40 7 1 1 X ap-7718 40 8 ) ) PUNCT ap-7718 40 9 with with ADP ap-7718 40 10 x̂ x̂ NOUN ap-7718 40 11 and and CCONJ ap-7718 40 12 p̂x p̂x DET ap-7718 40 13 the the DET ap-7718 40 14 canonical canonical ADJ ap-7718 40 15 position position NOUN ap-7718 40 16 and and CCONJ ap-7718 40 17 momentum momentum NOUN ap-7718 40 18 operators operator NOUN ap-7718 40 19 , , PUNCT ap-7718 40 20 respectively respectively ADV ap-7718 40 21 , , PUNCT ap-7718 40 22 with with ADP ap-7718 40 23 [ [ X ap-7718 40 24 x̂ x̂ NOUN ap-7718 40 25 , , PUNCT ap-7718 40 26 p̂x p̂x NOUN ap-7718 40 27 ] ] X ap-7718 40 28 = = SYM ap-7718 40 29 iℏi iℏi NOUN ap-7718 40 30 . . PUNCT ap-7718 41 1 henceforth henceforth ADV ap-7718 41 2 , , PUNCT ap-7718 41 3 the the DET ap-7718 41 4 identity identity NOUN ap-7718 41 5 operator operator NOUN ap-7718 41 6 i i PRON ap-7718 41 7 is be AUX ap-7718 41 8 omitted omit VERB ap-7718 41 9 each each DET ap-7718 41 10 time time NOUN ap-7718 41 11 it it PRON ap-7718 41 12 multiplies multiply VERB ap-7718 41 13 a a DET ap-7718 41 14 constant constant ADJ ap-7718 41 15 or or CCONJ ap-7718 41 16 a a DET ap-7718 41 17 function function NOUN ap-7718 41 18 . . PUNCT ap-7718 42 1 the the DET ap-7718 42 2 corresponding correspond VERB ap-7718 42 3 schrödinger schrödinger ADJ ap-7718 42 4 equation equation NOUN ap-7718 42 5 iℏ iℏ ADP ap-7718 42 6 ∂ψ ∂ψ PROPN ap-7718 42 7 ∂t ∂t PROPN ap-7718 42 8 = = SYM ap-7718 42 9 − − PROPN ap-7718 42 10 ℏ2 ℏ2 NOUN ap-7718 42 11 2m(t 2m(t NUM ap-7718 42 12 ) ) PUNCT ap-7718 43 1 ∂2ψ ∂2ψ PROPN ap-7718 43 2 ∂x2 ∂x2 NOUN ap-7718 43 3 + + NOUN ap-7718 43 4 m(t)ω2(t)x2 m(t)ω2(t)x2 NOUN ap-7718 43 5 2 2 NUM ap-7718 43 6 ψ+f ψ+f SYM ap-7718 43 7 ( ( PUNCT ap-7718 43 8 t)xψ t)xψ NOUN ap-7718 43 9 , , PUNCT ap-7718 43 10 ( ( PUNCT ap-7718 43 11 2 2 X ap-7718 43 12 ) ) PUNCT ap-7718 43 13 is be AUX ap-7718 43 14 recovered recover VERB ap-7718 43 15 by by ADP ap-7718 43 16 using use VERB ap-7718 43 17 the the DET ap-7718 43 18 coordinate coordinate NOUN ap-7718 43 19 representation representation NOUN ap-7718 43 20 px px PROPN ap-7718 43 21 ≡ ≡ PROPN ap-7718 43 22 −iℏ −iℏ PROPN ap-7718 43 23 ∂ ∂ NOUN ap-7718 43 24 ∂x ∂x PROPN ap-7718 43 25 and and CCONJ ap-7718 43 26 x̂ x̂ NUM ap-7718 43 27 ≡ ≡ PROPN ap-7718 43 28 x x PUNCT ap-7718 43 29 ∈ ∈ PROPN ap-7718 43 30 r. r. NOUN ap-7718 43 31 the the DET ap-7718 43 32 solutions solution NOUN ap-7718 43 33 of of ADP ap-7718 43 34 eq eq PROPN ap-7718 43 35 . . PUNCT ap-7718 44 1 ( ( PUNCT ap-7718 44 2 2 2 X ap-7718 44 3 ) ) PUNCT ap-7718 44 4 have have AUX ap-7718 44 5 been be AUX ap-7718 44 6 discussed discuss VERB ap-7718 44 7 by by ADP ap-7718 44 8 several several ADJ ap-7718 44 9 authors author NOUN ap-7718 44 10 , , PUNCT ap-7718 44 11 see see VERB ap-7718 44 12 [ [ X ap-7718 44 13 27 27 NUM ap-7718 44 14 , , PUNCT ap-7718 44 15 30–32 30–32 NUM ap-7718 44 16 ] ] PUNCT ap-7718 44 17 . . PUNCT ap-7718 45 1 here here ADV ap-7718 45 2 , , PUNCT ap-7718 45 3 a a DET ap-7718 45 4 brief brief ADJ ap-7718 45 5 summary summary NOUN ap-7718 45 6 of of ADP ap-7718 45 7 the the DET ap-7718 45 8 point point NOUN ap-7718 45 9 transformation transformation NOUN ap-7718 45 10 approach approach NOUN ap-7718 45 11 discussed discuss VERB ap-7718 45 12 in in ADP ap-7718 45 13 [ [ X ap-7718 45 14 26 26 NUM ap-7718 45 15 , , PUNCT ap-7718 45 16 27 27 NUM ap-7718 45 17 ] ] PUNCT ap-7718 45 18 is be AUX ap-7718 45 19 provided provide VERB ap-7718 45 20 . . PUNCT ap-7718 46 1 this this PRON ap-7718 46 2 eases ease VERB ap-7718 46 3 the the DET ap-7718 46 4 discussion discussion NOUN ap-7718 46 5 of of ADP ap-7718 46 6 semiclassical semiclassical ADJ ap-7718 46 7 states state NOUN ap-7718 46 8 and and CCONJ ap-7718 46 9 dynamics dynamic NOUN ap-7718 46 10 to to PART ap-7718 46 11 be be AUX ap-7718 46 12 presented present VERB ap-7718 46 13 later later ADV ap-7718 46 14 in in ADP ap-7718 46 15 section section NOUN ap-7718 46 16 3 3 NUM ap-7718 46 17 . . NOUN ap-7718 46 18 2.1 2.1 NUM ap-7718 46 19 . . PUNCT ap-7718 47 1 point point NOUN ap-7718 47 2 transformations transformation NOUN ap-7718 47 3 in in ADP ap-7718 47 4 general general ADJ ap-7718 47 5 , , PUNCT ap-7718 47 6 the the DET ap-7718 47 7 method method NOUN ap-7718 47 8 of of ADP ap-7718 47 9 form form NOUN ap-7718 47 10 - - PUNCT ap-7718 47 11 preserving preserve VERB ap-7718 47 12 point point NOUN ap-7718 47 13 transformations transformation NOUN ap-7718 47 14 relies rely VERB ap-7718 47 15 on on ADP ap-7718 47 16 a a DET ap-7718 47 17 geometrical geometrical ADJ ap-7718 47 18 deformation deformation NOUN ap-7718 47 19 that that PRON ap-7718 47 20 maps map VERB ap-7718 47 21 an an DET ap-7718 47 22 initial initial ADJ ap-7718 47 23 differential differential ADJ ap-7718 47 24 equation equation NOUN ap-7718 47 25 with with ADP ap-7718 47 26 variable variable ADJ ap-7718 47 27 coefficients coefficient NOUN ap-7718 47 28 into into ADP ap-7718 47 29 another another DET ap-7718 47 30 one one NUM ap-7718 47 31 of of ADP ap-7718 47 32 the the DET ap-7718 47 33 same same ADJ ap-7718 47 34 form form NOUN ap-7718 47 35 but but CCONJ ap-7718 47 36 with with ADP ap-7718 47 37 different different ADJ ap-7718 47 38 coefficients coefficient NOUN ap-7718 47 39 . . PUNCT ap-7718 48 1 to to PART ap-7718 48 2 illustrate illustrate VERB ap-7718 48 3 this this PRON ap-7718 48 4 , , PUNCT ap-7718 48 5 be be AUX ap-7718 48 6 the the DET ap-7718 48 7 stationary stationary ADJ ap-7718 48 8 oscillator oscillator NOUN ap-7718 48 9 hamiltonian hamiltonian NOUN ap-7718 48 10 ĥosc ĥosc NOUN ap-7718 48 11 = = PUNCT ap-7718 48 12 p̂2 p̂2 NOUN ap-7718 48 13 y y PROPN ap-7718 48 14 2m0 2m0 NUM ap-7718 48 15 + + CCONJ ap-7718 48 16 m0w m0w X ap-7718 48 17 2 2 NUM ap-7718 48 18 0 0 NUM ap-7718 48 19 ŷ ŷ NUM ap-7718 48 20 2 2 NUM ap-7718 48 21 2 2 NUM ap-7718 48 22 , , PUNCT ap-7718 48 23 [ [ X ap-7718 48 24 ŷ ŷ NUM ap-7718 48 25 , , PUNCT ap-7718 48 26 p̂y p̂y NOUN ap-7718 48 27 ] ] X ap-7718 48 28 = = SYM ap-7718 48 29 iℏ iℏ NOUN ap-7718 48 30 , , PUNCT ap-7718 48 31 ( ( PUNCT ap-7718 48 32 3 3 NUM ap-7718 48 33 ) ) PUNCT ap-7718 48 34 with with ADP ap-7718 48 35 ŷ ŷ NUM ap-7718 48 36 and and CCONJ ap-7718 48 37 p̂y p̂y X ap-7718 48 38 another another DET ap-7718 48 39 couple couple NOUN ap-7718 48 40 of of ADP ap-7718 48 41 canonical canonical ADJ ap-7718 48 42 position position NOUN ap-7718 48 43 and and CCONJ ap-7718 48 44 momentum momentum NOUN ap-7718 48 45 observables observable NOUN ap-7718 48 46 , , PUNCT ap-7718 48 47 respectively respectively ADV ap-7718 48 48 . . PUNCT ap-7718 49 1 the the DET ap-7718 49 2 corresponding correspond VERB ap-7718 49 3 schrödinger schrödinger ADJ ap-7718 49 4 equation equation NOUN ap-7718 49 5 iℏ iℏ ADP ap-7718 49 6 ∂ψ ∂ψ PROPN ap-7718 49 7 ∂τ ∂τ PROPN ap-7718 49 8 = = SYM ap-7718 49 9 − − PROPN ap-7718 49 10 ℏ2 ℏ2 NOUN ap-7718 49 11 2m0 2m0 NUM ap-7718 49 12 ∂2ψ ∂2ψ PROPN ap-7718 49 13 ∂y2 ∂y2 PROPN ap-7718 49 14 + + NUM ap-7718 49 15 m0w m0w X ap-7718 49 16 2 2 NUM ap-7718 49 17 0y 0y NOUN ap-7718 49 18 2 2 NUM ap-7718 49 19 2 2 NUM ap-7718 49 20 ψ ψ NOUN ap-7718 49 21 , , PUNCT ap-7718 49 22 ( ( PUNCT ap-7718 49 23 4 4 X ap-7718 49 24 ) ) PUNCT ap-7718 49 25 admits admit VERB ap-7718 49 26 the the DET ap-7718 49 27 well well ADV ap-7718 49 28 - - PUNCT ap-7718 49 29 known know VERB ap-7718 49 30 solutions solution NOUN ap-7718 49 31 [ [ X ap-7718 49 32 2 2 NUM ap-7718 49 33 ] ] PUNCT ap-7718 49 34 ψn(y ψn(y NUM ap-7718 49 35 , , PUNCT ap-7718 49 36 τ τ X ap-7718 49 37 ) ) PUNCT ap-7718 49 38 = = PUNCT ap-7718 50 1 e−iw0(n+ e−iw0(n+ PROPN ap-7718 50 2 1 1 NUM ap-7718 50 3 2 2 NUM ap-7718 50 4 ) ) PUNCT ap-7718 50 5 τ τ PROPN ap-7718 50 6 φn(y φn(y NUM ap-7718 50 7 ) ) PUNCT ap-7718 50 8 , , PUNCT ap-7718 50 9 ( ( PUNCT ap-7718 50 10 5 5 X ap-7718 50 11 ) ) PUNCT ap-7718 50 12 where where SCONJ ap-7718 50 13 φn(y φn(y NUM ap-7718 50 14 ) ) PUNCT ap-7718 50 15 = = SYM ap-7718 50 16 √ √ NUM ap-7718 50 17 1 1 NUM ap-7718 50 18 2nn 2nn NOUN ap-7718 50 19 ! ! PUNCT ap-7718 51 1 √ √ NUM ap-7718 52 1 m0w0 m0w0 PUNCT ap-7718 52 2 πℏ πℏ NOUN ap-7718 52 3 e− e− PROPN ap-7718 52 4 m0w0 m0w0 PROPN ap-7718 52 5 2ℏ 2ℏ NUM ap-7718 53 1 y2 y2 INTJ ap-7718 53 2 hn hn PROPN ap-7718 53 3 ( ( PUNCT ap-7718 53 4 √ √ PROPN ap-7718 53 5 m0w0 m0w0 PROPN ap-7718 53 6 ℏ ℏ PROPN ap-7718 53 7 y y PROPN ap-7718 53 8 ) ) PUNCT ap-7718 53 9 , , PUNCT ap-7718 53 10 ( ( PUNCT ap-7718 53 11 6 6 NUM ap-7718 53 12 ) ) PUNCT ap-7718 53 13 with with ADP ap-7718 53 14 hn(z hn(z NOUN ap-7718 53 15 ) ) PUNCT ap-7718 53 16 the the DET ap-7718 53 17 hermite hermite ADJ ap-7718 53 18 polynomials polynomial VERB ap-7718 53 19 [ [ X ap-7718 53 20 33 33 NUM ap-7718 53 21 ] ] PUNCT ap-7718 53 22 , , PUNCT ap-7718 53 23 fulfills fulfill VERB ap-7718 53 24 the the DET ap-7718 53 25 stationary stationary ADJ ap-7718 53 26 eigenvalue eigenvalue PROPN ap-7718 53 27 problem problem NOUN ap-7718 53 28 hoscφn(y hoscφn(y NOUN ap-7718 53 29 ) ) PUNCT ap-7718 53 30 = = SYM ap-7718 53 31 e(osc e(osc X ap-7718 53 32 ) ) PUNCT ap-7718 53 33 n n NOUN ap-7718 53 34 φn(y φn(y NUM ap-7718 53 35 ) ) PUNCT ap-7718 53 36 , , PUNCT ap-7718 53 37 e(osc e(osc PROPN ap-7718 53 38 ) ) PUNCT ap-7718 53 39 n n NOUN ap-7718 53 40 = = PUNCT ap-7718 53 41 ℏw0 ℏw0 X ap-7718 53 42 ( ( PUNCT ap-7718 53 43 n+ n+ NUM ap-7718 53 44 1 1 NUM ap-7718 53 45 2 2 NUM ap-7718 53 46 ) ) PUNCT ap-7718 53 47 , , PUNCT ap-7718 53 48 ( ( PUNCT ap-7718 53 49 7 7 X ap-7718 53 50 ) ) PUNCT ap-7718 53 51 with with ADP ap-7718 53 52 hosc hosc NOUN ap-7718 53 53 the the DET ap-7718 53 54 coordinate coordinate NOUN ap-7718 53 55 representation representation NOUN ap-7718 53 56 of of ADP ap-7718 53 57 ĥosc ĥosc ADJ ap-7718 53 58 , , PUNCT ap-7718 53 59 i.e. i.e. X ap-7718 53 60 , , PUNCT ap-7718 53 61 a a DET ap-7718 53 62 second second ADJ ap-7718 53 63 - - PUNCT ap-7718 53 64 order order NOUN ap-7718 53 65 differential differential NOUN ap-7718 53 66 operator operator NOUN ap-7718 53 67 that that PRON ap-7718 53 68 admits admit VERB ap-7718 53 69 a a DET ap-7718 53 70 sturm sturm PROPN ap-7718 53 71 - - PUNCT ap-7718 53 72 liouville liouville NOUN ap-7718 53 73 problem problem NOUN ap-7718 53 74 . . PUNCT ap-7718 54 1 to to PART ap-7718 54 2 implement implement VERB ap-7718 54 3 the the DET ap-7718 54 4 point point NOUN ap-7718 54 5 transformation transformation NOUN ap-7718 54 6 , , PUNCT ap-7718 54 7 one one PRON ap-7718 54 8 imposes impose VERB ap-7718 54 9 a a DET ap-7718 54 10 set set NOUN ap-7718 54 11 of of ADP ap-7718 54 12 relationships relationship NOUN ap-7718 54 13 between between ADP ap-7718 54 14 the the DET ap-7718 54 15 coordinates coordinate NOUN ap-7718 54 16 , , PUNCT ap-7718 54 17 time time NOUN ap-7718 54 18 paramaters paramater NOUN ap-7718 54 19 , , PUNCT ap-7718 54 20 and and CCONJ ap-7718 54 21 solutions solution NOUN ap-7718 54 22 of of ADP ap-7718 54 23 both both DET ap-7718 54 24 systems system NOUN ap-7718 54 25 in in ADP ap-7718 54 26 consideration consideration NOUN ap-7718 54 27 [ [ X ap-7718 54 28 25 25 NUM ap-7718 54 29 ] ] PUNCT ap-7718 54 30 . . PUNCT ap-7718 55 1 in in ADP ap-7718 55 2 general general ADJ ap-7718 55 3 one one NUM ap-7718 55 4 has have VERB ap-7718 55 5 y(x y(x PROPN ap-7718 55 6 , , PUNCT ap-7718 55 7 t t PROPN ap-7718 55 8 ) ) PUNCT ap-7718 55 9 , , PUNCT ap-7718 55 10 τ(x τ(x PROPN ap-7718 55 11 , , PUNCT ap-7718 55 12 t t PROPN ap-7718 55 13 ) ) PUNCT ap-7718 55 14 , , PUNCT ap-7718 55 15 ψ(y(x ψ(y(x PROPN ap-7718 55 16 , , PUNCT ap-7718 55 17 t t PROPN ap-7718 55 18 ) ) PUNCT ap-7718 55 19 , , PUNCT ap-7718 55 20 τ(x τ(x PROPN ap-7718 55 21 , , PUNCT ap-7718 55 22 t t PROPN ap-7718 55 23 ) ) PUNCT ap-7718 55 24 ) ) PUNCT ap-7718 56 1 ≡ ≡ PROPN ap-7718 56 2 g(x g(x PROPN ap-7718 56 3 , , PUNCT ap-7718 56 4 t;ψ t;ψ NUM ap-7718 56 5 ) ) PUNCT ap-7718 56 6 , , PUNCT ap-7718 56 7 ( ( PUNCT ap-7718 56 8 8) 8) NUM ap-7718 56 9 where where SCONJ ap-7718 56 10 g(x g(x NOUN ap-7718 56 11 , , PUNCT ap-7718 56 12 t;ψ t;ψ NUM ap-7718 56 13 ) ) PUNCT ap-7718 56 14 is be AUX ap-7718 56 15 a a DET ap-7718 56 16 reparametrization reparametrization NOUN ap-7718 56 17 of of ADP ap-7718 56 18 ψ ψ PRON ap-7718 56 19 as as ADP ap-7718 56 20 an an DET ap-7718 56 21 explicit explicit ADJ ap-7718 56 22 function function NOUN ap-7718 56 23 of of ADP ap-7718 56 24 x x PROPN ap-7718 56 25 , , PUNCT ap-7718 56 26 t t PROPN ap-7718 56 27 , , PUNCT ap-7718 56 28 and and CCONJ ap-7718 56 29 ψ ψ X ap-7718 56 30 . . PUNCT ap-7718 57 1 in in ADP ap-7718 57 2 the the DET ap-7718 57 3 case case NOUN ap-7718 57 4 under under ADP ap-7718 57 5 consideration consideration NOUN ap-7718 57 6 , , PUNCT ap-7718 57 7 some some DET ap-7718 57 8 further further ADJ ap-7718 57 9 conditions condition NOUN ap-7718 57 10 are be AUX ap-7718 57 11 required require VERB ap-7718 57 12 to to PART ap-7718 57 13 preserve preserve VERB ap-7718 57 14 the the DET ap-7718 57 15 linearity linearity NOUN ap-7718 57 16 and and CCONJ ap-7718 57 17 the the DET ap-7718 57 18 hermiticity hermiticity NOUN ap-7718 57 19 of of ADP ap-7718 57 20 ĥosc ĥosc ADJ ap-7718 57 21 and and CCONJ ap-7718 57 22 ĥck(t ĥck(t NOUN ap-7718 57 23 ) ) PUNCT ap-7718 57 24 . . PUNCT ap-7718 58 1 a a DET ap-7718 58 2 detailed detailed ADJ ap-7718 58 3 discussion discussion NOUN ap-7718 58 4 on on ADP ap-7718 58 5 the the DET ap-7718 58 6 matter matter NOUN ap-7718 58 7 can can AUX ap-7718 58 8 be be AUX ap-7718 58 9 found find VERB ap-7718 58 10 in in ADP ap-7718 58 11 [ [ X ap-7718 58 12 27 27 NUM ap-7718 58 13 ] ] PUNCT ap-7718 58 14 . . PUNCT ap-7718 59 1 here here ADV ap-7718 59 2 , , PUNCT ap-7718 59 3 the the DET ap-7718 59 4 final final ADJ ap-7718 59 5 form form NOUN ap-7718 59 6 of of ADP ap-7718 59 7 the the DET ap-7718 59 8 point point NOUN ap-7718 59 9 transformation transformation NOUN ap-7718 59 10 is be AUX ap-7718 59 11 used use VERB ap-7718 59 12 , , PUNCT ap-7718 59 13 leading lead VERB ap-7718 59 14 to to ADP ap-7718 59 15 y(x y(x PROPN ap-7718 59 16 , , PUNCT ap-7718 59 17 t t PROPN ap-7718 59 18 ) ) PUNCT ap-7718 59 19 = = SYM ap-7718 59 20 µ(t)x+ µ(t)x+ X ap-7718 59 21 γ(t γ(t NOUN ap-7718 59 22 ) ) PUNCT ap-7718 59 23 σ(t σ(t PROPN ap-7718 59 24 ) ) PUNCT ap-7718 59 25 , , PUNCT ap-7718 59 26 τ(t τ(t ADP ap-7718 59 27 ) ) PUNCT ap-7718 59 28 = = SYM ap-7718 59 29 ∫ ∫ PROPN ap-7718 59 30 t t PROPN ap-7718 59 31 dt′ dt′ NOUN ap-7718 59 32 σ2(t′ σ2(t′ NUM ap-7718 59 33 ) ) PUNCT ap-7718 59 34 , , PUNCT ap-7718 59 35 ( ( PUNCT ap-7718 59 36 9 9 NUM ap-7718 59 37 ) ) PUNCT ap-7718 59 38 and and CCONJ ap-7718 59 39 ψ(y(x ψ(y(x PROPN ap-7718 59 40 , , PUNCT ap-7718 59 41 t t PROPN ap-7718 59 42 ) ) PUNCT ap-7718 59 43 , , PUNCT ap-7718 59 44 τ(t τ(t NOUN ap-7718 59 45 ) ) PUNCT ap-7718 59 46 ) ) PUNCT ap-7718 60 1 ≡ ≡ PROPN ap-7718 60 2 g(x g(x NOUN ap-7718 60 3 , , PUNCT ap-7718 60 4 t;ψ t;ψ NUM ap-7718 60 5 ) ) PUNCT ap-7718 60 6 = = SYM ap-7718 60 7 a(x a(x NOUN ap-7718 60 8 , , PUNCT ap-7718 60 9 t)ψ(x t)ψ(x NOUN ap-7718 60 10 , , PUNCT ap-7718 60 11 t t PROPN ap-7718 60 12 ) ) PUNCT ap-7718 60 13 , , PUNCT ap-7718 60 14 ( ( PUNCT ap-7718 60 15 10 10 NUM ap-7718 60 16 ) ) PUNCT ap-7718 60 17 with with ADP ap-7718 60 18 m(t m(t NOUN ap-7718 60 19 ) ) PUNCT ap-7718 60 20 = = SYM ap-7718 60 21 µ2(t µ2(t PROPN ap-7718 60 22 ) ) PUNCT ap-7718 60 23 , , PUNCT ap-7718 60 24 together together ADV ap-7718 60 25 with with ADP ap-7718 60 26 σ(t σ(t PROPN ap-7718 60 27 ) ) PUNCT ap-7718 60 28 and and CCONJ ap-7718 60 29 γ(t γ(t NOUN ap-7718 60 30 ) ) PUNCT ap-7718 60 31 some some DET ap-7718 60 32 real real ADV ap-7718 60 33 - - PUNCT ap-7718 60 34 valued value VERB ap-7718 60 35 functions function NOUN ap-7718 60 36 to to PART ap-7718 60 37 be be AUX ap-7718 60 38 determined determine VERB ap-7718 60 39 . . PUNCT ap-7718 61 1 by by ADP ap-7718 61 2 substituting substitute VERB ap-7718 61 3 ( ( PUNCT ap-7718 61 4 9 9 NUM ap-7718 61 5 ) ) PUNCT ap-7718 61 6 into into ADP ap-7718 61 7 the the DET ap-7718 61 8 schrödinger schrödinger ADJ ap-7718 61 9 equation equation NOUN ap-7718 61 10 ( ( PUNCT ap-7718 61 11 4 4 NUM ap-7718 61 12 ) ) PUNCT ap-7718 61 13 , , PUNCT ap-7718 61 14 and and CCONJ ap-7718 61 15 after after ADP ap-7718 61 16 some some DET ap-7718 61 17 calculations calculation NOUN ap-7718 61 18 , , PUNCT ap-7718 61 19 one one NUM ap-7718 61 20 arrives arrive VERB ap-7718 61 21 to to ADP ap-7718 61 22 a a DET ap-7718 61 23 new new ADJ ap-7718 61 24 partial partial ADJ ap-7718 61 25 differential differential NOUN ap-7718 61 26 equation equation NOUN ap-7718 61 27 for for ADP ap-7718 61 28 ψ(x ψ(x PROPN ap-7718 61 29 , , PUNCT ap-7718 61 30 t t PROPN ap-7718 61 31 ) ) PUNCT ap-7718 61 32 that that PRON ap-7718 61 33 takes take VERB ap-7718 61 34 the the DET ap-7718 61 35 exact exact ADJ ap-7718 61 36 form form NOUN ap-7718 61 37 in in ADP ap-7718 61 38 ( ( PUNCT ap-7718 61 39 2 2 NUM ap-7718 61 40 ) ) PUNCT ap-7718 61 41 . . PUNCT ap-7718 62 1 the the DET ap-7718 62 2 latter latter ADJ ap-7718 62 3 allows allow VERB ap-7718 62 4 obtaining obtain VERB ap-7718 62 5 a(x a(x NOUN ap-7718 62 6 , , PUNCT ap-7718 62 7 t t NOUN ap-7718 62 8 ) ) PUNCT ap-7718 62 9 = = SYM ap-7718 62 10 √ √ PROPN ap-7718 62 11 σ σ NUM ap-7718 62 12 µ µ X ap-7718 62 13 exp exp NOUN ap-7718 62 14 a(x a(x PROPN ap-7718 62 15 , , PUNCT ap-7718 62 16 t t PROPN ap-7718 62 17 ) ) PUNCT ap-7718 62 18 , , PUNCT ap-7718 62 19 a(x a(x PROPN ap-7718 62 20 , , PUNCT ap-7718 62 21 t t PROPN ap-7718 62 22 ) ) PUNCT ap-7718 62 23 : : PUNCT ap-7718 63 1 = = PUNCT ap-7718 63 2 ( ( PUNCT ap-7718 63 3 i i NOUN ap-7718 63 4 m0w0 m0w0 PROPN ap-7718 63 5 ℏ ℏ PROPN ap-7718 63 6 µ µ PROPN ap-7718 63 7 σ σ X ap-7718 63 8 ( ( PUNCT ap-7718 63 9 wµ wµ ADV ap-7718 63 10 2 2 NUM ap-7718 63 11 x2 x2 NOUN ap-7718 64 1 + + NOUN ap-7718 64 2 wγx wγx ADV ap-7718 64 3 ) ) PUNCT ap-7718 65 1 + + CCONJ ap-7718 65 2 iη iη ADJ ap-7718 65 3 ) ) PUNCT ap-7718 65 4 , , PUNCT ap-7718 65 5 ( ( PUNCT ap-7718 65 6 11 11 NUM ap-7718 65 7 ) ) PUNCT ap-7718 65 8 where where SCONJ ap-7718 65 9 a(x a(x NOUN ap-7718 65 10 , , PUNCT ap-7718 65 11 t t PROPN ap-7718 65 12 ) ) PUNCT ap-7718 65 13 is be AUX ap-7718 65 14 a a DET ap-7718 65 15 local local ADJ ap-7718 65 16 time time NOUN ap-7718 65 17 - - PUNCT ap-7718 65 18 dependent dependent ADJ ap-7718 65 19 complex complex ADJ ap-7718 65 20 - - PUNCT ap-7718 65 21 phase phase NOUN ap-7718 65 22 and and CCONJ ap-7718 65 23 [ [ X ap-7718 65 24 27 27 NUM ap-7718 65 25 ] ] SYM ap-7718 65 26 η(t η(t NOUN ap-7718 65 27 ) ) PUNCT ap-7718 65 28 : : PUNCT ap-7718 66 1 = = SYM ap-7718 66 2 m0 m0 PROPN ap-7718 66 3 2ℏ 2ℏ NUM ap-7718 66 4 γ(t)wγ(t γ(t)wγ(t NOUN ap-7718 66 5 ) ) PUNCT ap-7718 66 6 σ(t σ(t PROPN ap-7718 66 7 ) ) PUNCT ap-7718 67 1 − − PROPN ap-7718 68 1 1 1 NUM ap-7718 68 2 2ℏ 2ℏ NUM ap-7718 68 3 ∫ ∫ PROPN ap-7718 68 4 t t PROPN ap-7718 68 5 dt′ dt′ PROPN ap-7718 68 6 f f PROPN ap-7718 68 7 ( ( PUNCT ap-7718 68 8 t′ t′ NUM ap-7718 68 9 ) ) PUNCT ap-7718 68 10 µ(t′ µ(t′ PROPN ap-7718 68 11 ) ) PUNCT ap-7718 68 12 , , PUNCT ap-7718 68 13 wµ(t wµ(t PUNCT ap-7718 68 14 ) ) PUNCT ap-7718 69 1 = = SYM ap-7718 69 2 σ(t)µ̇(t σ(t)µ̇(t ADJ ap-7718 69 3 ) ) PUNCT ap-7718 69 4 − − PROPN ap-7718 69 5 σ̇(t)µ(t σ̇(t)µ(t PROPN ap-7718 69 6 ) ) PUNCT ap-7718 69 7 , , PUNCT ap-7718 69 8 wγ(t wγ(t X ap-7718 69 9 ) ) PUNCT ap-7718 69 10 = = SYM ap-7718 69 11 σ(t)γ̇(t σ(t)γ̇(t X ap-7718 69 12 ) ) PUNCT ap-7718 69 13 − − PROPN ap-7718 69 14 σ̇(t)γ(t σ̇(t)γ(t NOUN ap-7718 69 15 ) ) PUNCT ap-7718 69 16 , , PUNCT ap-7718 69 17 ( ( PUNCT ap-7718 69 18 12 12 NUM ap-7718 69 19 ) ) PUNCT ap-7718 69 20 with with ADP ap-7718 69 21 ḟ(t ḟ(t ADJ ap-7718 69 22 ) ) PUNCT ap-7718 69 23 ≡ ≡ PROPN ap-7718 69 24 df(t df(t NOUN ap-7718 69 25 ) ) PUNCT ap-7718 69 26 dt dt ADP ap-7718 69 27 a a DET ap-7718 69 28 short short ADJ ap-7718 69 29 - - PUNCT ap-7718 69 30 hand hand NOUN ap-7718 69 31 notation notation NOUN ap-7718 69 32 for for ADP ap-7718 69 33 the the DET ap-7718 69 34 time time NOUN ap-7718 69 35 derivative derivative ADJ ap-7718 69 36 . . PUNCT ap-7718 70 1 in in ADP ap-7718 70 2 the the DET ap-7718 70 3 latter latter ADJ ap-7718 70 4 , , PUNCT ap-7718 70 5 σ(t σ(t PROPN ap-7718 70 6 ) ) PUNCT ap-7718 70 7 and and CCONJ ap-7718 70 8 γ(t γ(t NOUN ap-7718 70 9 ) ) PUNCT ap-7718 70 10 fulfill fulfill VERB ap-7718 70 11 the the DET ap-7718 70 12 nonlinear nonlinear ADJ ap-7718 70 13 ermakov ermakov ADJ ap-7718 70 14 equation equation NOUN ap-7718 70 15 σ̈(t σ̈(t NUM ap-7718 70 16 ) ) PUNCT ap-7718 71 1 + + CCONJ ap-7718 71 2 ( ( PUNCT ap-7718 71 3 ω2(t ω2(t NOUN ap-7718 71 4 ) ) PUNCT ap-7718 71 5 − − NOUN ap-7718 71 6 µ̈(t µ̈(t PROPN ap-7718 71 7 ) ) PUNCT ap-7718 71 8 µ(t µ(t ADJ ap-7718 71 9 ) ) PUNCT ap-7718 71 10 ) ) PUNCT ap-7718 72 1 σ(t σ(t PROPN ap-7718 72 2 ) ) PUNCT ap-7718 73 1 = = SYM ap-7718 73 2 w2 w2 NOUN ap-7718 73 3 0 0 NUM ap-7718 73 4 σ3(t σ3(t PROPN ap-7718 73 5 ) ) PUNCT ap-7718 73 6 , , PUNCT ap-7718 73 7 ( ( PUNCT ap-7718 73 8 13 13 NUM ap-7718 73 9 ) ) PUNCT ap-7718 73 10 and and CCONJ ap-7718 73 11 non non ADJ ap-7718 73 12 - - ADJ ap-7718 73 13 homogeneous homogeneous ADJ ap-7718 73 14 equation equation NOUN ap-7718 73 15 γ̈(t γ̈(t PROPN ap-7718 73 16 ) ) PUNCT ap-7718 74 1 + + CCONJ ap-7718 74 2 ( ( PUNCT ap-7718 74 3 ω2(t ω2(t NOUN ap-7718 74 4 ) ) PUNCT ap-7718 74 5 − − NOUN ap-7718 74 6 µ̈(t µ̈(t PROPN ap-7718 74 7 ) ) PUNCT ap-7718 74 8 µ(t µ(t ADJ ap-7718 74 9 ) ) PUNCT ap-7718 74 10 ) ) PUNCT ap-7718 75 1 γ(t γ(t NOUN ap-7718 75 2 ) ) PUNCT ap-7718 75 3 = = SYM ap-7718 75 4 f f PROPN ap-7718 75 5 ( ( PUNCT ap-7718 75 6 t t PROPN ap-7718 75 7 ) ) PUNCT ap-7718 75 8 m0µ(t m0µ(t PROPN ap-7718 75 9 ) ) PUNCT ap-7718 75 10 , , PUNCT ap-7718 75 11 ( ( PUNCT ap-7718 75 12 14 14 NUM ap-7718 75 13 ) ) PUNCT ap-7718 75 14 the the DET ap-7718 75 15 solutions solution NOUN ap-7718 75 16 of of ADP ap-7718 75 17 the the DET ap-7718 75 18 ermakov ermakov ADJ ap-7718 75 19 equation equation NOUN ap-7718 75 20 are be AUX ap-7718 75 21 wellknown wellknown ADJ ap-7718 76 1 [ [ X ap-7718 76 2 34–36 34–36 NUM ap-7718 76 3 ] ] PUNCT ap-7718 76 4 and and CCONJ ap-7718 76 5 computed compute VERB ap-7718 76 6 from from ADP ap-7718 76 7 two two NUM ap-7718 76 8 linearly linearly ADV ap-7718 76 9 independent independent ADJ ap-7718 76 10 solutions solution NOUN ap-7718 76 11 of of ADP ap-7718 76 12 the the DET ap-7718 76 13 associated associated ADJ ap-7718 76 14 linear linear PROPN ap-7718 76 15 equation equation NOUN ap-7718 76 16 q̈j(t q̈j(t PROPN ap-7718 76 17 ) ) PUNCT ap-7718 77 1 + + CCONJ ap-7718 77 2 ( ( PUNCT ap-7718 77 3 ω2(t ω2(t NOUN ap-7718 77 4 ) ) PUNCT ap-7718 77 5 − − NOUN ap-7718 77 6 µ̈(t µ̈(t PROPN ap-7718 77 7 ) ) PUNCT ap-7718 77 8 µ(t µ(t ADJ ap-7718 77 9 ) ) PUNCT ap-7718 77 10 ) ) PUNCT ap-7718 78 1 qj(t qj(t X ap-7718 78 2 ) ) PUNCT ap-7718 78 3 = = SYM ap-7718 79 1 0 0 NUM ap-7718 79 2 , , PUNCT ap-7718 79 3 j j PROPN ap-7718 79 4 = = SYM ap-7718 79 5 1 1 NUM ap-7718 79 6 , , PUNCT ap-7718 79 7 2 2 NUM ap-7718 79 8 , , PUNCT ap-7718 79 9 ( ( PUNCT ap-7718 79 10 15 15 NUM ap-7718 79 11 ) ) SYM ap-7718 79 12 212 212 NUM ap-7718 79 13 vol vol NOUN ap-7718 79 14 . . PUNCT ap-7718 80 1 62 62 NUM ap-7718 80 2 no no INTJ ap-7718 80 3 . . PUNCT ap-7718 81 1 1/2022 1/2022 NUM ap-7718 81 2 td td NOUN ap-7718 81 3 mass mass NOUN ap-7718 81 4 oscillators oscillator NOUN ap-7718 81 5 : : PUNCT ap-7718 81 6 constants constant NOUN ap-7718 81 7 of of ADP ap-7718 81 8 motion motion NOUN ap-7718 81 9 and and CCONJ ap-7718 81 10 semiclasical semiclasical ADJ ap-7718 81 11 states state NOUN ap-7718 81 12 through through ADP ap-7718 81 13 the the DET ap-7718 81 14 nonlinear nonlinear ADJ ap-7718 81 15 combination combination NOUN ap-7718 81 16 σ(t σ(t NOUN ap-7718 81 17 ) ) PUNCT ap-7718 82 1 = = NOUN ap-7718 82 2 [ [ PUNCT ap-7718 82 3 aq2 aq2 NOUN ap-7718 82 4 1(t 1(t NUM ap-7718 82 5 ) ) PUNCT ap-7718 82 6 + + CCONJ ap-7718 82 7 bq1(t)q2(t bq1(t)q2(t VERB ap-7718 82 8 ) ) PUNCT ap-7718 82 9 + + CCONJ ap-7718 82 10 cq2 cq2 NOUN ap-7718 82 11 2(t 2(t NUM ap-7718 82 12 ) ) PUNCT ap-7718 82 13 ] ] PUNCT ap-7718 82 14 1 1 NUM ap-7718 82 15 2 2 NUM ap-7718 82 16 , , PUNCT ap-7718 82 17 ( ( PUNCT ap-7718 82 18 16 16 NUM ap-7718 82 19 ) ) PUNCT ap-7718 82 20 with with ADP ap-7718 82 21 b2 b2 NOUN ap-7718 82 22 −4ac −4ac NOUN ap-7718 82 23 = = SYM ap-7718 82 24 −4 −4 PROPN ap-7718 82 25 w2 w2 PROPN ap-7718 82 26 0 0 NUM ap-7718 82 27 w2 w2 PROPN ap-7718 82 28 0 0 NUM ap-7718 82 29 and and CCONJ ap-7718 82 30 w0 w0 PROPN ap-7718 82 31 = = SYM ap-7718 82 32 wr(q1(t wr(q1(t PROPN ap-7718 82 33 ) ) PUNCT ap-7718 82 34 , , PUNCT ap-7718 82 35 q2(t q2(t PROPN ap-7718 82 36 ) ) PUNCT ap-7718 82 37 ) ) PUNCT ap-7718 83 1 ̸= ̸= NOUN ap-7718 83 2 0 0 NUM ap-7718 83 3 the the DET ap-7718 83 4 wronskian wronskian NOUN ap-7718 83 5 of of ADP ap-7718 83 6 two two NUM ap-7718 83 7 linearly linearly ADV ap-7718 83 8 independent independent ADJ ap-7718 83 9 solutions solution NOUN ap-7718 83 10 of of ADP ap-7718 83 11 ( ( PUNCT ap-7718 83 12 15 15 NUM ap-7718 83 13 ) ) PUNCT ap-7718 83 14 , , PUNCT ap-7718 83 15 which which PRON ap-7718 83 16 is be AUX ap-7718 83 17 in in ADP ap-7718 83 18 general general ADJ ap-7718 83 19 a a DET ap-7718 83 20 time time NOUN ap-7718 83 21 - - PUNCT ap-7718 83 22 independent independent ADJ ap-7718 83 23 complex complex ADJ ap-7718 83 24 constant constant ADJ ap-7718 83 25 . . PUNCT ap-7718 84 1 the the DET ap-7718 84 2 previous previous ADJ ap-7718 84 3 constraint constraint NOUN ap-7718 84 4 on on ADP ap-7718 84 5 a a DET ap-7718 84 6 , , PUNCT ap-7718 84 7 b b NOUN ap-7718 84 8 , , PUNCT ap-7718 84 9 and and CCONJ ap-7718 84 10 c c NOUN ap-7718 84 11 guarantees guarantee VERB ap-7718 84 12 that that SCONJ ap-7718 84 13 σ(t σ(t PROPN ap-7718 84 14 ) ) PUNCT ap-7718 84 15 is be AUX ap-7718 84 16 different different ADJ ap-7718 84 17 from from ADP ap-7718 84 18 zero zero NUM ap-7718 84 19 [ [ X ap-7718 84 20 26 26 NUM ap-7718 84 21 ] ] PUNCT ap-7718 84 22 for for ADP ap-7718 84 23 t t PROPN ap-7718 84 24 ∈ ∈ PROPN ap-7718 84 25 r. r. PROPN ap-7718 84 26 in in ADP ap-7718 84 27 this this DET ap-7718 84 28 form form NOUN ap-7718 84 29 , , PUNCT ap-7718 84 30 one one PRON ap-7718 84 31 obtains obtain VERB ap-7718 84 32 a a DET ap-7718 84 33 set set NOUN ap-7718 84 34 of of ADP ap-7718 84 35 solutions solution NOUN ap-7718 84 36 { { PUNCT ap-7718 84 37 ψn(x ψn(x ADP ap-7718 84 38 , , PUNCT ap-7718 84 39 t)}∞ t)}∞ ADV ap-7718 84 40 n=0 n=0 NUM ap-7718 84 41 to to ADP ap-7718 84 42 the the DET ap-7718 84 43 schrödinger schrödinger ADJ ap-7718 84 44 equation equation NOUN ap-7718 84 45 ( ( PUNCT ap-7718 84 46 2 2 NUM ap-7718 84 47 ) ) PUNCT ap-7718 84 48 , , PUNCT ap-7718 84 49 where where SCONJ ap-7718 84 50 ψn(x ψn(x ADP ap-7718 84 51 , , PUNCT ap-7718 84 52 t t PROPN ap-7718 84 53 ) ) PUNCT ap-7718 84 54 = = SYM ap-7718 84 55 √ √ ADP ap-7718 84 56 µ(t µ(t ADJ ap-7718 84 57 ) ) PUNCT ap-7718 84 58 σ(t σ(t PROPN ap-7718 84 59 ) ) PUNCT ap-7718 85 1 [ [ X ap-7718 85 2 a(x a(x NOUN ap-7718 85 3 , , PUNCT ap-7718 85 4 t)]−1e−iw0(n+ t)]−1e−iw0(n+ NOUN ap-7718 85 5 1 1 NUM ap-7718 85 6 2 2 NUM ap-7718 85 7 ) ) PUNCT ap-7718 85 8 τ(t τ(t NOUN ap-7718 85 9 ) ) PUNCT ap-7718 85 10 × × NOUN ap-7718 85 11 e e NOUN ap-7718 85 12 − − PROPN ap-7718 85 13 m0w0 m0w0 PROPN ap-7718 85 14 2ℏ 2ℏ NUM ap-7718 85 15 ( ( PUNCT ap-7718 85 16 µ(t)x+γ(t µ(t)x+γ(t X ap-7718 85 17 ) ) PUNCT ap-7718 85 18 σ(t σ(t PROPN ap-7718 85 19 ) ) PUNCT ap-7718 85 20 ) ) PUNCT ap-7718 85 21 2 2 NUM ap-7718 85 22 hn hn PROPN ap-7718 85 23 ( ( PUNCT ap-7718 85 24 √ √ PROPN ap-7718 85 25 m0w0 m0w0 NUM ap-7718 85 26 ℏ ℏ PROPN ap-7718 85 27 µ(t)x+ µ(t)x+ X ap-7718 85 28 γ(t γ(t NOUN ap-7718 85 29 ) ) PUNCT ap-7718 85 30 σ(t σ(t PROPN ap-7718 85 31 ) ) PUNCT ap-7718 85 32 ) ) PUNCT ap-7718 85 33 . . PUNCT ap-7718 86 1 ( ( PUNCT ap-7718 86 2 17 17 NUM ap-7718 86 3 ) ) PUNCT ap-7718 86 4 from from ADP ap-7718 86 5 ( ( PUNCT ap-7718 86 6 10)-(11 10)-(11 NUM ap-7718 86 7 ) ) PUNCT ap-7718 86 8 it it PRON ap-7718 86 9 follows follow VERB ap-7718 86 10 that that SCONJ ap-7718 86 11 ( ( PUNCT ap-7718 86 12 ψm ψm INTJ ap-7718 86 13 , , PUNCT ap-7718 86 14 ψn ψn ADV ap-7718 86 15 ) ) PUNCT ap-7718 86 16 : : PUNCT ap-7718 87 1 = = SYM ap-7718 87 2 ∫ ∫ PROPN ap-7718 87 3 r r NOUN ap-7718 87 4 dxψ∗ dxψ∗ NOUN ap-7718 87 5 m(x m(x PROPN ap-7718 87 6 , , PUNCT ap-7718 87 7 t)ψn(x t)ψn(x PROPN ap-7718 87 8 , , PUNCT ap-7718 87 9 t t PROPN ap-7718 87 10 ) ) PUNCT ap-7718 87 11 = = SYM ap-7718 88 1 ∫ ∫ PROPN ap-7718 88 2 r r PROPN ap-7718 88 3 dyψ∗ dyψ∗ PROPN ap-7718 88 4 m(y m(y PROPN ap-7718 88 5 , , PUNCT ap-7718 88 6 τ)ψn(y τ)ψn(y PROPN ap-7718 88 7 , , PUNCT ap-7718 88 8 τ τ X ap-7718 88 9 ) ) PUNCT ap-7718 88 10 = = SYM ap-7718 88 11 δn δn PROPN ap-7718 88 12 , , PUNCT ap-7718 88 13 m m PRON ap-7718 88 14 , , PUNCT ap-7718 88 15 ( ( PUNCT ap-7718 88 16 18 18 NUM ap-7718 88 17 ) ) PUNCT ap-7718 88 18 with with ADP ap-7718 88 19 z∗ z∗ PROPN ap-7718 88 20 the the DET ap-7718 88 21 complex complex ADJ ap-7718 88 22 conjugate conjugate NOUN ap-7718 88 23 of of ADP ap-7718 88 24 z. z. PROPN ap-7718 88 25 that that PRON ap-7718 88 26 is be AUX ap-7718 88 27 , , PUNCT ap-7718 88 28 the the DET ap-7718 88 29 inner inner ADJ ap-7718 88 30 product product NOUN ap-7718 88 31 is be AUX ap-7718 88 32 preserved preserve VERB ap-7718 88 33 and and CCONJ ap-7718 88 34 thus thus ADV ap-7718 88 35 the the DET ap-7718 88 36 set set NOUN ap-7718 88 37 { { PUNCT ap-7718 88 38 ψn(x ψn(x ADP ap-7718 88 39 , , PUNCT ap-7718 88 40 t)}∞ t)}∞ ADV ap-7718 88 41 n=0 n=0 NUM ap-7718 88 42 is be AUX ap-7718 88 43 orthonormal orthonormal ADJ ap-7718 88 44 in in ADP ap-7718 88 45 l2(r l2(r PROPN ap-7718 88 46 , , PUNCT ap-7718 88 47 dx dx PROPN ap-7718 88 48 ) ) PUNCT ap-7718 88 49 . . PUNCT ap-7718 89 1 the the DET ap-7718 89 2 expressions expression NOUN ap-7718 89 3 presented present VERB ap-7718 89 4 so so ADV ap-7718 89 5 far far ADV ap-7718 89 6 are be AUX ap-7718 89 7 general general ADJ ap-7718 89 8 , , PUNCT ap-7718 89 9 and and CCONJ ap-7718 89 10 specific specific ADJ ap-7718 89 11 result result NOUN ap-7718 89 12 may may AUX ap-7718 89 13 be be AUX ap-7718 89 14 obtained obtain VERB ap-7718 89 15 once once ADV ap-7718 89 16 the the DET ap-7718 89 17 timedependent timedependent NOUN ap-7718 89 18 mass mass PROPN ap-7718 89 19 and and CCONJ ap-7718 89 20 frequency frequency NOUN ap-7718 89 21 terms term NOUN ap-7718 89 22 are be AUX ap-7718 89 23 specified specify VERB ap-7718 89 24 . . PUNCT ap-7718 90 1 this this PRON ap-7718 90 2 is be AUX ap-7718 90 3 discussed discuss VERB ap-7718 90 4 in in ADP ap-7718 90 5 the the DET ap-7718 90 6 following follow VERB ap-7718 90 7 sections section NOUN ap-7718 90 8 . . PUNCT ap-7718 91 1 before before ADP ap-7718 91 2 concluding conclude VERB ap-7718 91 3 , , PUNCT ap-7718 91 4 an an DET ap-7718 91 5 explicit explicit ADJ ap-7718 91 6 expression expression NOUN ap-7718 91 7 for for ADP ap-7718 91 8 τ(t τ(t NOUN ap-7718 91 9 ) ) PUNCT ap-7718 91 10 can can AUX ap-7718 91 11 be be AUX ap-7718 91 12 determined determine VERB ap-7718 91 13 in in ADP ap-7718 91 14 terms term NOUN ap-7718 91 15 of of ADP ap-7718 91 16 the the DET ap-7718 91 17 two two NUM ap-7718 91 18 linearly linearly ADV ap-7718 91 19 independent independent ADJ ap-7718 91 20 solutions solution NOUN ap-7718 91 21 q1(t q1(t PART ap-7718 91 22 ) ) PUNCT ap-7718 91 23 and and CCONJ ap-7718 91 24 q2(t q2(t PROPN ap-7718 91 25 ) ) PUNCT ap-7718 91 26 as as ADV ap-7718 91 27 well well ADV ap-7718 91 28 . . PUNCT ap-7718 92 1 one one PRON ap-7718 92 2 gets get VERB ap-7718 92 3 τ(t τ(t NOUN ap-7718 92 4 ) ) PUNCT ap-7718 92 5 = = SYM ap-7718 92 6 w−1 w−1 PROPN ap-7718 92 7 0 0 NUM ap-7718 92 8 arctan arctan PROPN ap-7718 92 9 [ [ PUNCT ap-7718 92 10 w0 w0 PROPN ap-7718 92 11 2w0 2w0 NUM ap-7718 92 12 ( ( PUNCT ap-7718 92 13 b+ b+ X ap-7718 92 14 2cq2(t 2cq2(t NUM ap-7718 92 15 ) ) PUNCT ap-7718 92 16 q1(t q1(t NUM ap-7718 92 17 ) ) PUNCT ap-7718 92 18 ) ) PUNCT ap-7718 92 19 ] ] PUNCT ap-7718 92 20 . . PUNCT ap-7718 93 1 ( ( PUNCT ap-7718 93 2 19 19 NUM ap-7718 93 3 ) ) PUNCT ap-7718 93 4 3 3 NUM ap-7718 93 5 . . X ap-7718 93 6 results result NOUN ap-7718 93 7 : : PUNCT ap-7718 93 8 constants constant NOUN ap-7718 93 9 of of ADP ap-7718 93 10 motion motion NOUN ap-7718 93 11 and and CCONJ ap-7718 93 12 semiclassical semiclassical ADJ ap-7718 93 13 states state NOUN ap-7718 93 14 additional additional ADJ ap-7718 93 15 information information NOUN ap-7718 93 16 can can AUX ap-7718 93 17 be be AUX ap-7718 93 18 extracted extract VERB ap-7718 93 19 from from ADP ap-7718 93 20 the the DET ap-7718 93 21 stationary stationary ADJ ap-7718 93 22 oscillator oscillator NOUN ap-7718 93 23 into into ADP ap-7718 93 24 the the DET ap-7718 93 25 time time NOUN ap-7718 93 26 - - PUNCT ap-7718 93 27 dependent dependent ADJ ap-7718 93 28 model model NOUN ap-7718 93 29 . . PUNCT ap-7718 94 1 particularly particularly ADV ap-7718 94 2 , , PUNCT ap-7718 94 3 point point NOUN ap-7718 94 4 transformations transformation NOUN ap-7718 94 5 preserve preserve VERB ap-7718 94 6 first first ADJ ap-7718 94 7 - - PUNCT ap-7718 94 8 integrals integral NOUN ap-7718 94 9 of of ADP ap-7718 94 10 the the DET ap-7718 94 11 initial initial ADJ ap-7718 94 12 equation equation NOUN ap-7718 94 13 [ [ X ap-7718 94 14 25 25 NUM ap-7718 94 15 ] ] PUNCT ap-7718 94 16 . . PUNCT ap-7718 95 1 in in ADP ap-7718 95 2 the the DET ap-7718 95 3 context context NOUN ap-7718 95 4 of of ADP ap-7718 95 5 the the DET ap-7718 95 6 schrödinger schrödinger ADJ ap-7718 95 7 equation equation NOUN ap-7718 95 8 , , PUNCT ap-7718 95 9 such such ADJ ap-7718 95 10 first first ADJ ap-7718 95 11 - - PUNCT ap-7718 95 12 integrals integral NOUN ap-7718 95 13 correspond correspond VERB ap-7718 95 14 to to ADP ap-7718 95 15 constants constant NOUN ap-7718 95 16 of of ADP ap-7718 95 17 motion motion NOUN ap-7718 95 18 , , PUNCT ap-7718 95 19 also also ADV ap-7718 95 20 known know VERB ap-7718 95 21 as as ADP ap-7718 95 22 quantum quantum NOUN ap-7718 95 23 invariant invariant NOUN ap-7718 95 24 , , PUNCT ap-7718 95 25 associated associate VERB ap-7718 95 26 with with ADP ap-7718 95 27 the the DET ap-7718 95 28 physical physical ADJ ap-7718 95 29 models model NOUN ap-7718 95 30 under under ADP ap-7718 95 31 consideration consideration NOUN ap-7718 95 32 . . PUNCT ap-7718 96 1 from from ADP ap-7718 96 2 the the DET ap-7718 96 3 stationary stationary ADJ ap-7718 96 4 oscillator oscillator NOUN ap-7718 96 5 , , PUNCT ap-7718 96 6 it it PRON ap-7718 96 7 is be AUX ap-7718 96 8 straightforward straightforward ADJ ap-7718 96 9 to to PART ap-7718 96 10 realize realize VERB ap-7718 96 11 that that SCONJ ap-7718 96 12 the the DET ap-7718 96 13 hamiltonian hamiltonian NOUN ap-7718 96 14 ĥosc ĥosc NOUN ap-7718 96 15 is be AUX ap-7718 96 16 a a DET ap-7718 96 17 constant constant ADJ ap-7718 96 18 of of ADP ap-7718 96 19 motion motion NOUN ap-7718 96 20 that that PRON ap-7718 96 21 characterize characterize VERB ap-7718 96 22 the the DET ap-7718 96 23 energy energy NOUN ap-7718 96 24 observable observable ADJ ap-7718 96 25 . . PUNCT ap-7718 97 1 in in ADP ap-7718 97 2 the the DET ap-7718 97 3 time time NOUN ap-7718 97 4 - - PUNCT ap-7718 97 5 dependent dependent ADJ ap-7718 97 6 case case NOUN ap-7718 97 7 , , PUNCT ap-7718 97 8 ĥck(t ĥck(t NOUN ap-7718 97 9 ) ) PUNCT ap-7718 97 10 is be AUX ap-7718 97 11 no no ADV ap-7718 97 12 longer long ADV ap-7718 97 13 a a DET ap-7718 97 14 constant constant NOUN ap-7718 97 15 of of ADP ap-7718 97 16 motion motion NOUN ap-7718 97 17 , , PUNCT ap-7718 97 18 as as ADP ap-7718 97 19 dĥck(t dĥck(t X ap-7718 97 20 ) ) PUNCT ap-7718 97 21 dt dt PUNCT ap-7718 98 1 = = NOUN ap-7718 98 2 ̸ ̸ NUM ap-7718 98 3 0 0 NUM ap-7718 98 4 . . PUNCT ap-7718 99 1 this this PRON ap-7718 99 2 implies imply VERB ap-7718 99 3 that that SCONJ ap-7718 99 4 an an DET ap-7718 99 5 eigenvalue eigenvalue NOUN ap-7718 99 6 problem problem NOUN ap-7718 99 7 associated associate VERB ap-7718 99 8 with with ADP ap-7718 99 9 ĥck ĥck NOUN ap-7718 99 10 is be AUX ap-7718 99 11 not not PART ap-7718 99 12 possible1 possible1 ADJ ap-7718 99 13 . . PUNCT ap-7718 100 1 1one 1one NOUN ap-7718 100 2 can can AUX ap-7718 100 3 still still ADV ap-7718 100 4 link link VERB ap-7718 100 5 an an DET ap-7718 100 6 eigenvalue eigenvalue NOUN ap-7718 100 7 problem problem NOUN ap-7718 100 8 with with ADP ap-7718 100 9 ĥck(t ĥck(t NOUN ap-7718 100 10 ) ) PUNCT ap-7718 100 11 under under ADP ap-7718 100 12 the the DET ap-7718 100 13 adiabatic adiabatic ADJ ap-7718 100 14 approximation approximation NOUN ap-7718 100 15 [ [ X ap-7718 100 16 3 3 NUM ap-7718 100 17 ] ] PUNCT ap-7718 100 18 . . PUNCT ap-7718 101 1 this this DET ap-7718 101 2 work work NOUN ap-7718 101 3 focuses focus VERB ap-7718 101 4 on on ADP ap-7718 101 5 exact exact ADJ ap-7718 101 6 solutions solution NOUN ap-7718 101 7 and and CCONJ ap-7718 101 8 such such DET ap-7718 101 9 an an DET ap-7718 101 10 approach approach NOUN ap-7718 101 11 will will AUX ap-7718 101 12 be be AUX ap-7718 101 13 disregarded disregard VERB ap-7718 101 14 . . PUNCT ap-7718 102 1 on on ADP ap-7718 102 2 the the DET ap-7718 102 3 other other ADJ ap-7718 102 4 hand hand NOUN ap-7718 102 5 , , PUNCT ap-7718 102 6 an an DET ap-7718 102 7 orthonormal orthonormal ADJ ap-7718 102 8 set set NOUN ap-7718 102 9 of of ADP ap-7718 102 10 solutions solution NOUN ap-7718 102 11 { { PUNCT ap-7718 102 12 ψn(x ψn(x ADP ap-7718 102 13 , , PUNCT ap-7718 102 14 t)}∞ t)}∞ ADV ap-7718 102 15 n=0 n=0 PRON ap-7718 102 16 has have AUX ap-7718 102 17 been be AUX ap-7718 102 18 already already ADV ap-7718 102 19 identified identify VERB ap-7718 102 20 , , PUNCT ap-7718 102 21 and and CCONJ ap-7718 102 22 it it PRON ap-7718 102 23 is be AUX ap-7718 102 24 still still ADV ap-7718 102 25 unclear unclear ADJ ap-7718 102 26 the the DET ap-7718 102 27 eigenvalue eigenvalue PROPN ap-7718 102 28 problem problem NOUN ap-7718 102 29 that that PRON ap-7718 102 30 such such DET ap-7718 102 31 a a DET ap-7718 102 32 set set VERB ap-7718 102 33 solves solve NOUN ap-7718 102 34 . . PUNCT ap-7718 103 1 this this DET ap-7718 103 2 problem problem NOUN ap-7718 103 3 was be AUX ap-7718 103 4 addressed address VERB ap-7718 103 5 by by ADP ap-7718 103 6 lewis lewis PROPN ap-7718 103 7 - - PUNCT ap-7718 103 8 riesenfeld riesenfeld NOUN ap-7718 103 9 [ [ X ap-7718 103 10 9 9 NUM ap-7718 103 11 ] ] PUNCT ap-7718 103 12 while while SCONJ ap-7718 103 13 solving solve VERB ap-7718 103 14 the the DET ap-7718 103 15 dynamics dynamic NOUN ap-7718 103 16 of of ADP ap-7718 103 17 the the DET ap-7718 103 18 parametric parametric ADJ ap-7718 103 19 oscillator oscillator NOUN ap-7718 103 20 . . PUNCT ap-7718 104 1 they they PRON ap-7718 104 2 notice notice VERB ap-7718 104 3 that that SCONJ ap-7718 104 4 even even ADV ap-7718 104 5 in in ADP ap-7718 104 6 the the DET ap-7718 104 7 time time NOUN ap-7718 104 8 - - PUNCT ap-7718 104 9 dependent dependent ADJ ap-7718 104 10 regime regime NOUN ap-7718 104 11 , , PUNCT ap-7718 104 12 there there PRON ap-7718 104 13 may may AUX ap-7718 104 14 be be AUX ap-7718 104 15 a a DET ap-7718 104 16 constant constant NOUN ap-7718 104 17 of of ADP ap-7718 104 18 motion motion NOUN ap-7718 104 19 î0(t î0(t VERB ap-7718 104 20 ) ) PUNCT ap-7718 104 21 that that PRON ap-7718 104 22 admits admit VERB ap-7718 104 23 a a DET ap-7718 104 24 spectral spectral ADJ ap-7718 104 25 problem problem NOUN ap-7718 104 26 î0(t)ϕ(x î0(t)ϕ(x PRON ap-7718 104 27 , , PUNCT ap-7718 104 28 t t PROPN ap-7718 104 29 ) ) PUNCT ap-7718 104 30 = = PUNCT ap-7718 104 31 λϕ(x λϕ(x PROPN ap-7718 104 32 , , PUNCT ap-7718 104 33 t t PROPN ap-7718 104 34 ) ) PUNCT ap-7718 104 35 , , PUNCT ap-7718 104 36 ( ( PUNCT ap-7718 104 37 20 20 NUM ap-7718 104 38 ) ) PUNCT ap-7718 104 39 where where SCONJ ap-7718 104 40 the the DET ap-7718 104 41 eigenvalues eigenvalues PROPN ap-7718 104 42 λ λ NOUN ap-7718 104 43 are be AUX ap-7718 104 44 time time NOUN ap-7718 104 45 - - PUNCT ap-7718 104 46 independent independent ADJ ap-7718 104 47 . . PUNCT ap-7718 105 1 the the DET ap-7718 105 2 existence existence NOUN ap-7718 105 3 and and CCONJ ap-7718 105 4 uniqueness uniqueness NOUN ap-7718 105 5 of of ADP ap-7718 105 6 such such DET ap-7718 105 7 a a DET ap-7718 105 8 quantum quantum ADJ ap-7718 105 9 invariant invariant NOUN ap-7718 105 10 is be AUX ap-7718 105 11 not not PART ap-7718 105 12 necessarily necessarily ADV ap-7718 105 13 ensured ensure VERB ap-7718 105 14 . . PUNCT ap-7718 106 1 still still ADV ap-7718 106 2 , , PUNCT ap-7718 106 3 for for ADP ap-7718 106 4 the the DET ap-7718 106 5 parametric parametric PROPN ap-7718 106 6 oscillator oscillator PROPN ap-7718 106 7 , , PUNCT ap-7718 106 8 lewis lewis PROPN ap-7718 106 9 and and CCONJ ap-7718 106 10 riesenfeld riesenfeld PROPN ap-7718 106 11 managed manage VERB ap-7718 106 12 to to PART ap-7718 106 13 find find VERB ap-7718 106 14 the the DET ap-7718 106 15 quantum quantum NOUN ap-7718 106 16 invariant invariant NOUN ap-7718 106 17 and and CCONJ ap-7718 106 18 solve solve VERB ap-7718 106 19 the the DET ap-7718 106 20 related relate VERB ap-7718 106 21 spectral spectral ADJ ap-7718 106 22 problem problem NOUN ap-7718 106 23 . . PUNCT ap-7718 107 1 here here ADV ap-7718 107 2 , , PUNCT ap-7718 107 3 some some DET ap-7718 107 4 quantum quantum NOUN ap-7718 107 5 invariants invariant NOUN ap-7718 107 6 associated associate VERB ap-7718 107 7 with with ADP ap-7718 107 8 ĥck ĥck NOUN ap-7718 107 9 can can AUX ap-7718 107 10 be be AUX ap-7718 107 11 found find VERB ap-7718 107 12 through through ADP ap-7718 107 13 point point NOUN ap-7718 107 14 transformations transformation NOUN ap-7718 107 15 . . PUNCT ap-7718 108 1 first first ADV ap-7718 108 2 , , PUNCT ap-7718 108 3 notice notice VERB ap-7718 108 4 that that SCONJ ap-7718 108 5 the the DET ap-7718 108 6 point point NOUN ap-7718 108 7 transformation transformation NOUN ap-7718 108 8 was be AUX ap-7718 108 9 implemented implement VERB ap-7718 108 10 in in ADP ap-7718 108 11 the the DET ap-7718 108 12 schrödinger schrödinger ADJ ap-7718 108 13 equation equation NOUN ap-7718 108 14 to to PART ap-7718 108 15 get get VERB ap-7718 108 16 the the DET ap-7718 108 17 time time NOUN ap-7718 108 18 - - PUNCT ap-7718 108 19 dependent dependent ADJ ap-7718 108 20 counterpart counterpart NOUN ap-7718 108 21 . . PUNCT ap-7718 109 1 the the DET ap-7718 109 2 same same ADJ ap-7718 109 3 transformation transformation NOUN ap-7718 109 4 can can AUX ap-7718 109 5 be be AUX ap-7718 109 6 applied apply VERB ap-7718 109 7 to to ADP ap-7718 109 8 a a DET ap-7718 109 9 constant constant NOUN ap-7718 109 10 of of ADP ap-7718 109 11 motion motion NOUN ap-7718 109 12 of of ADP ap-7718 109 13 the the DET ap-7718 109 14 harmonic harmonic ADJ ap-7718 109 15 oscillator oscillator NOUN ap-7718 109 16 to to PART ap-7718 109 17 get get VERB ap-7718 109 18 the the DET ap-7718 109 19 corresponding correspond VERB ap-7718 109 20 one one NUM ap-7718 109 21 on on ADP ap-7718 109 22 the the DET ap-7718 109 23 time time NOUN ap-7718 109 24 - - PUNCT ap-7718 109 25 dependent dependent ADJ ap-7718 109 26 model model NOUN ap-7718 109 27 . . PUNCT ap-7718 110 1 particularly particularly ADV ap-7718 110 2 , , PUNCT ap-7718 110 3 by by AUX ap-7718 110 4 consider consider VERB ap-7718 110 5 the the DET ap-7718 110 6 eigenvalue eigenvalue PROPN ap-7718 110 7 problem problem NOUN ap-7718 110 8 ( ( PUNCT ap-7718 110 9 7 7 NUM ap-7718 110 10 ) ) PUNCT ap-7718 110 11 , , PUNCT ap-7718 110 12 and and CCONJ ap-7718 110 13 after after ADP ap-7718 110 14 some some DET ap-7718 110 15 calculations calculation NOUN ap-7718 110 16 , , PUNCT ap-7718 110 17 one one PRON ap-7718 110 18 gets get VERB ap-7718 110 19 a a DET ap-7718 110 20 first first ADJ ap-7718 110 21 quantum quantum ADJ ap-7718 110 22 invariant invariant NOUN ap-7718 110 23 of of ADP ap-7718 110 24 the the DET ap-7718 110 25 form form NOUN ap-7718 110 26 î1(t î1(t PROPN ap-7718 110 27 ) ) PUNCT ap-7718 110 28 : : PUNCT ap-7718 110 29 = = SYM ap-7718 110 30 σ2(t σ2(t PROPN ap-7718 110 31 ) ) PUNCT ap-7718 110 32 2m0µ2(t 2m0µ2(t NOUN ap-7718 110 33 ) ) PUNCT ap-7718 110 34 p̂ p̂ X ap-7718 110 35 2 2 NUM ap-7718 110 36 x x SYM ap-7718 110 37 + + CCONJ ap-7718 110 38 m0 m0 NOUN ap-7718 110 39 2 2 NUM ap-7718 110 40 ( ( PUNCT ap-7718 110 41 w w PROPN ap-7718 110 42 2 2 NUM ap-7718 110 43 µ(t µ(t ADJ ap-7718 110 44 ) ) PUNCT ap-7718 111 1 + + NUM ap-7718 111 2 w2 w2 NOUN ap-7718 111 3 0 0 NUM ap-7718 111 4 µ2(t µ2(t NOUN ap-7718 111 5 ) ) PUNCT ap-7718 111 6 σ2(t σ2(t PROPN ap-7718 111 7 ) ) PUNCT ap-7718 111 8 ) ) PUNCT ap-7718 111 9 x̂2 x̂2 PUNCT ap-7718 112 1 + + CCONJ ap-7718 112 2 σwµ(t σwµ(t NOUN ap-7718 112 3 ) ) PUNCT ap-7718 112 4 2µ(t 2µ(t NOUN ap-7718 112 5 ) ) PUNCT ap-7718 112 6 ( ( PUNCT ap-7718 112 7 x̂p̂x x̂p̂x X ap-7718 112 8 + + CCONJ ap-7718 112 9 p̂xx̂ p̂xx̂ PROPN ap-7718 112 10 ) ) PUNCT ap-7718 112 11 + + NUM ap-7718 112 12 σwγ(t σwγ(t NOUN ap-7718 112 13 ) ) PUNCT ap-7718 112 14 µ(t µ(t ADJ ap-7718 112 15 ) ) PUNCT ap-7718 112 16 p̂x p̂x NOUN ap-7718 113 1 + + NOUN ap-7718 113 2 m0 m0 NOUN ap-7718 113 3 ( ( PUNCT ap-7718 113 4 wγ(t)wµ(t wγ(t)wµ(t NOUN ap-7718 113 5 ) ) PUNCT ap-7718 113 6 + + NUM ap-7718 113 7 w2 w2 NOUN ap-7718 113 8 0 0 NUM ap-7718 113 9 µ(t)γ(t µ(t)γ(t ADV ap-7718 113 10 ) ) PUNCT ap-7718 113 11 σ2(t σ2(t NOUN ap-7718 113 12 ) ) PUNCT ap-7718 113 13 ) ) PUNCT ap-7718 113 14 x̂ x̂ PUNCT ap-7718 114 1 + + CCONJ ap-7718 114 2 ( ( PUNCT ap-7718 114 3 m0 m0 PROPN ap-7718 114 4 2 2 NUM ap-7718 114 5 w w PROPN ap-7718 114 6 2 2 NUM ap-7718 114 7 γ γ X ap-7718 114 8 ( ( PUNCT ap-7718 114 9 t t PROPN ap-7718 114 10 ) ) PUNCT ap-7718 114 11 + + SYM ap-7718 114 12 γ2(t γ2(t SYM ap-7718 114 13 ) ) PUNCT ap-7718 114 14 σ2(t σ2(t PROPN ap-7718 114 15 ) ) PUNCT ap-7718 114 16 ) ) PUNCT ap-7718 114 17 . . PUNCT ap-7718 115 1 ( ( PUNCT ap-7718 115 2 21 21 NUM ap-7718 115 3 ) ) PUNCT ap-7718 115 4 it it PRON ap-7718 115 5 is be AUX ap-7718 115 6 straightforward straightforward ADJ ap-7718 115 7 to to PART ap-7718 115 8 show show VERB ap-7718 115 9 that that SCONJ ap-7718 115 10 î1(t î1(t NOUN ap-7718 115 11 ) ) PUNCT ap-7718 115 12 is be AUX ap-7718 115 13 indeed indeed ADV ap-7718 115 14 a a DET ap-7718 115 15 quantum quantum NOUN ap-7718 115 16 invariant invariant NOUN ap-7718 115 17 , , PUNCT ap-7718 115 18 i i PRON ap-7718 115 19 ℏ ℏ PROPN ap-7718 116 1 [ [ X ap-7718 116 2 ĥck ĥck X ap-7718 116 3 , , PUNCT ap-7718 116 4 î1(t î1(t NOUN ap-7718 116 5 ) ) PUNCT ap-7718 116 6 ] ] PUNCT ap-7718 117 1 + + CCONJ ap-7718 117 2 ∂î1(t ∂î1(t PROPN ap-7718 117 3 ) ) PUNCT ap-7718 117 4 ∂t ∂t PROPN ap-7718 117 5 = = SYM ap-7718 117 6 0 0 PROPN ap-7718 117 7 . . PUNCT ap-7718 118 1 ( ( PUNCT ap-7718 118 2 22 22 NUM ap-7718 118 3 ) ) PUNCT ap-7718 118 4 moreover moreover ADV ap-7718 118 5 , , PUNCT ap-7718 118 6 i1(t i1(t PROPN ap-7718 118 7 ) ) PUNCT ap-7718 118 8 , , PUNCT ap-7718 118 9 the the DET ap-7718 118 10 coordinate coordinate NOUN ap-7718 118 11 representation representation NOUN ap-7718 118 12 of of ADP ap-7718 118 13 î1(t î1(t PROPN ap-7718 118 14 ) ) PUNCT ap-7718 118 15 , , PUNCT ap-7718 118 16 defines define VERB ap-7718 118 17 a a DET ap-7718 118 18 sturm sturm NOUN ap-7718 118 19 - - PUNCT ap-7718 118 20 liouville liouville NOUN ap-7718 118 21 problem problem NOUN ap-7718 118 22 with with ADP ap-7718 118 23 timedependent timedependent NOUN ap-7718 118 24 coefficients coefficient NOUN ap-7718 118 25 , , PUNCT ap-7718 118 26 i1(t)ψn(x i1(t)ψn(x PROPN ap-7718 118 27 , , PUNCT ap-7718 118 28 t t PROPN ap-7718 118 29 ) ) PUNCT ap-7718 118 30 = = PUNCT ap-7718 119 1 ℏw0 ℏw0 X ap-7718 119 2 ( ( PUNCT ap-7718 119 3 n+ n+ NUM ap-7718 119 4 1 1 NUM ap-7718 119 5 2 2 NUM ap-7718 119 6 ) ) PUNCT ap-7718 119 7 ψn(x ψn(x ADP ap-7718 119 8 , , PUNCT ap-7718 119 9 t t PROPN ap-7718 119 10 ) ) PUNCT ap-7718 119 11 , , PUNCT ap-7718 119 12 ( ( PUNCT ap-7718 119 13 23 23 NUM ap-7718 119 14 ) ) PUNCT ap-7718 119 15 which which PRON ap-7718 119 16 justifies justify VERB ap-7718 119 17 the the DET ap-7718 119 18 existence existence NOUN ap-7718 119 19 of of ADP ap-7718 119 20 the the DET ap-7718 119 21 orthogonal orthogonal ADJ ap-7718 119 22 set set NOUN ap-7718 119 23 of of ADP ap-7718 119 24 solutions solution NOUN ap-7718 119 25 found find VERB ap-7718 119 26 in in ADP ap-7718 119 27 section section NOUN ap-7718 119 28 2 2 NUM ap-7718 119 29 . . PUNCT ap-7718 119 30 note note VERB ap-7718 119 31 that that SCONJ ap-7718 119 32 orthogonality orthogonality NOUN ap-7718 119 33 has have AUX ap-7718 119 34 been be AUX ap-7718 119 35 alternatively alternatively ADV ap-7718 119 36 proved prove VERB ap-7718 119 37 in in ADP ap-7718 119 38 ( ( PUNCT ap-7718 119 39 18 18 NUM ap-7718 119 40 ) ) PUNCT ap-7718 119 41 using use VERB ap-7718 119 42 the the DET ap-7718 119 43 preservation preservation NOUN ap-7718 119 44 of of ADP ap-7718 119 45 the the DET ap-7718 119 46 inner inner ADJ ap-7718 119 47 product product NOUN ap-7718 119 48 . . PUNCT ap-7718 120 1 remarkably remarkably ADV ap-7718 120 2 , , PUNCT ap-7718 120 3 there there PRON ap-7718 120 4 are be VERB ap-7718 120 5 still still ADV ap-7718 120 6 more more ADJ ap-7718 120 7 quantum quantum ADJ ap-7718 120 8 invariants invariant NOUN ap-7718 120 9 to to PART ap-7718 120 10 be be AUX ap-7718 120 11 exploited exploit VERB ap-7718 120 12 . . PUNCT ap-7718 121 1 to to PART ap-7718 121 2 see see VERB ap-7718 121 3 this this PRON ap-7718 121 4 , , PUNCT ap-7718 121 5 let let VERB ap-7718 121 6 us we PRON ap-7718 121 7 consider consider VERB ap-7718 121 8 the the DET ap-7718 121 9 operators operator NOUN ap-7718 121 10 â â PUNCT ap-7718 121 11 = = SYM ap-7718 122 1 √ √ PROPN ap-7718 122 2 m0w0 m0w0 PROPN ap-7718 122 3 2ℏ 2ℏ NUM ap-7718 122 4 ŷ ŷ NUM ap-7718 123 1 + + CCONJ ap-7718 123 2 i i PRON ap-7718 123 3 p̂y√ p̂y√ VERB ap-7718 123 4 2m0ℏw0 2m0ℏw0 NUM ap-7718 123 5 , , PUNCT ap-7718 124 1 ↠↠ADJ ap-7718 124 2 = = NOUN ap-7718 124 3 √ √ NUM ap-7718 124 4 m0w0 m0w0 PROPN ap-7718 124 5 2ℏ 2ℏ NUM ap-7718 125 1 ŷ ŷ NUM ap-7718 125 2 − − PROPN ap-7718 126 1 i i PRON ap-7718 126 2 p̂y√ p̂y√ VERB ap-7718 126 3 2m0ℏw0 2m0ℏw0 NUM ap-7718 126 4 , , PUNCT ap-7718 126 5 ( ( PUNCT ap-7718 126 6 24 24 NUM ap-7718 126 7 ) ) PUNCT ap-7718 126 8 213 213 NUM ap-7718 126 9 kevin kevin PROPN ap-7718 126 10 zelaya zelaya PROPN ap-7718 126 11 acta acta PROPN ap-7718 126 12 polytechnica polytechnica PROPN ap-7718 126 13 which which PRON ap-7718 126 14 factorize factorize VERB ap-7718 126 15 the the DET ap-7718 126 16 stationary stationary ADJ ap-7718 126 17 oscillator oscillator NOUN ap-7718 126 18 hamiltonian hamiltonian NOUN ap-7718 126 19 as as ADP ap-7718 126 20 ĥosc ĥosc ADV ap-7718 126 21 = = SYM ap-7718 126 22 ℏw0(â†â+ ℏw0(â†â+ NOUN ap-7718 126 23 1 1 NUM ap-7718 126 24 2 2 NUM ap-7718 126 25 ) ) PUNCT ap-7718 126 26 and and CCONJ ap-7718 126 27 fulfill fulfill VERB ap-7718 126 28 the the DET ap-7718 126 29 commutation commutation NOUN ap-7718 126 30 relationship relationship NOUN ap-7718 126 31 [ [ X ap-7718 126 32 â â X ap-7718 126 33 , , PUNCT ap-7718 126 34 ↠↠ADJ ap-7718 126 35 ] ] X ap-7718 126 36 = = SYM ap-7718 126 37 1 1 X ap-7718 126 38 . . PUNCT ap-7718 127 1 although although SCONJ ap-7718 127 2 â â PRON ap-7718 127 3 and and CCONJ ap-7718 127 4 ↠↠ADJ ap-7718 127 5 are be AUX ap-7718 127 6 not not PART ap-7718 127 7 constants constant NOUN ap-7718 127 8 of of ADP ap-7718 127 9 motion motion NOUN ap-7718 127 10 of of ADP ap-7718 127 11 ĥosc ĥosc ADV ap-7718 127 12 , , PUNCT ap-7718 127 13 one one PRON ap-7718 127 14 can can AUX ap-7718 127 15 introduce introduce VERB ap-7718 127 16 a a DET ap-7718 127 17 new new ADJ ap-7718 127 18 pair pair NOUN ap-7718 127 19 of of ADP ap-7718 127 20 operators operator NOUN ap-7718 127 21 â â ADP ap-7718 127 22 : : PUNCT ap-7718 127 23 = = SYM ap-7718 127 24 eiw0τ eiw0τ PROPN ap-7718 127 25 â â ADJ ap-7718 127 26 , , PUNCT ap-7718 127 27 ↠↠ADJ ap-7718 127 28 : : PUNCT ap-7718 127 29 = = PUNCT ap-7718 127 30 e−iw0τ e−iw0τ PROPN ap-7718 127 31 ↠↠PROPN ap-7718 127 32 , , PUNCT ap-7718 127 33 ( ( PUNCT ap-7718 127 34 25 25 NUM ap-7718 127 35 ) ) PUNCT ap-7718 127 36 where where SCONJ ap-7718 127 37 the the DET ap-7718 127 38 straightforward straightforward ADJ ap-7718 127 39 calculations calculation NOUN ap-7718 127 40 show show VERB ap-7718 127 41 that that SCONJ ap-7718 128 1 i i PRON ap-7718 128 2 ℏ ℏ VERB ap-7718 129 1 [ [ X ap-7718 129 2 ĥosc ĥosc ADV ap-7718 129 3 , , PUNCT ap-7718 129 4 â â ADP ap-7718 129 5 ] ] X ap-7718 129 6 + + CCONJ ap-7718 129 7 ∂â ∂â NOUN ap-7718 129 8 ∂τ ∂τ PROPN ap-7718 129 9 = = SYM ap-7718 129 10 0 0 NUM ap-7718 129 11 , , PUNCT ap-7718 129 12 and and CCONJ ap-7718 129 13 similarly similarly ADV ap-7718 129 14 for for ADP ap-7718 129 15 â†. â†. NOUN ap-7718 129 16 that that PRON ap-7718 129 17 is be AUX ap-7718 129 18 , , PUNCT ap-7718 129 19 a a PRON ap-7718 129 20 and and CCONJ ap-7718 129 21 a† a† NOUN ap-7718 129 22 are be AUX ap-7718 129 23 quantum quantum ADJ ap-7718 129 24 invariants invariant NOUN ap-7718 129 25 of of ADP ap-7718 129 26 ĥosc ĥosc PROPN ap-7718 129 27 . . PUNCT ap-7718 130 1 the the DET ap-7718 130 2 latter latter ADJ ap-7718 130 3 can can AUX ap-7718 130 4 now now ADV ap-7718 130 5 be be AUX ap-7718 130 6 mapped map VERB ap-7718 130 7 into into ADP ap-7718 130 8 the the DET ap-7718 130 9 timedependent timedependent NOUN ap-7718 130 10 model model NOUN ap-7718 130 11 , , PUNCT ap-7718 130 12 leading lead VERB ap-7718 130 13 straightforwardly straightforwardly ADV ap-7718 130 14 to to ADP ap-7718 130 15 new new ADJ ap-7718 130 16 quantum quantum NOUN ap-7718 130 17 invariants invariant NOUN ap-7718 130 18 of of ADP ap-7718 130 19 ĥck(t ĥck(t NOUN ap-7718 130 20 ) ) PUNCT ap-7718 130 21 of of ADP ap-7718 130 22 the the DET ap-7718 130 23 form form NOUN ap-7718 130 24 îa(t îa(t PROPN ap-7718 130 25 ) ) PUNCT ap-7718 130 26 = = PUNCT ap-7718 131 1 eiw0τ(t eiw0τ(t X ap-7718 131 2 ) ) PUNCT ap-7718 131 3 [ [ PUNCT ap-7718 131 4 i√ i√ PROPN ap-7718 131 5 2m0ℏw0 2m0ℏw0 NUM ap-7718 131 6 σ(t σ(t NOUN ap-7718 131 7 ) ) PUNCT ap-7718 131 8 µ(t µ(t ADJ ap-7718 131 9 ) ) PUNCT ap-7718 131 10 p̂x p̂x NOUN ap-7718 132 1 + + CCONJ ap-7718 132 2 ( ( PUNCT ap-7718 132 3 √ √ NUM ap-7718 132 4 m0w0 m0w0 PROPN ap-7718 132 5 2ℏ 2ℏ NUM ap-7718 132 6 µ(t µ(t ADJ ap-7718 132 7 ) ) PUNCT ap-7718 132 8 σ(t σ(t PROPN ap-7718 132 9 ) ) PUNCT ap-7718 133 1 + + CCONJ ap-7718 133 2 i i PRON ap-7718 133 3 √ √ VERB ap-7718 133 4 m0 m0 PROPN ap-7718 133 5 2ℏw0 2ℏw0 NUM ap-7718 133 6 wµ(t wµ(t NUM ap-7718 133 7 ) ) PUNCT ap-7718 133 8 ) ) PUNCT ap-7718 133 9 x̂ x̂ PUNCT ap-7718 134 1 + + CCONJ ap-7718 134 2 ( ( PUNCT ap-7718 134 3 √ √ NUM ap-7718 134 4 m0w0 m0w0 PROPN ap-7718 134 5 2ℏ 2ℏ NUM ap-7718 134 6 γ(t γ(t NOUN ap-7718 134 7 ) ) PUNCT ap-7718 134 8 σ(t σ(t PROPN ap-7718 134 9 ) ) PUNCT ap-7718 135 1 + + CCONJ ap-7718 135 2 i i PRON ap-7718 135 3 √ √ VERB ap-7718 135 4 m0 m0 PROPN ap-7718 135 5 2ℏw0 2ℏw0 NUM ap-7718 135 6 wγ(t wγ(t NOUN ap-7718 135 7 ) ) PUNCT ap-7718 135 8 ) ) PUNCT ap-7718 135 9 ] ] PUNCT ap-7718 135 10 , , PUNCT ap-7718 135 11 ( ( PUNCT ap-7718 135 12 26 26 NUM ap-7718 135 13 ) ) PUNCT ap-7718 135 14 and and CCONJ ap-7718 135 15 its its PRON ap-7718 135 16 adjoint adjoint NOUN ap-7718 135 17 î† î† PROPN ap-7718 135 18 a(t a(t NOUN ap-7718 135 19 ) ) PUNCT ap-7718 135 20 . . PUNCT ap-7718 136 1 before before ADP ap-7718 136 2 proceeding proceeding NOUN ap-7718 136 3 , , PUNCT ap-7718 136 4 it it PRON ap-7718 136 5 is be AUX ap-7718 136 6 worth worth ADJ ap-7718 136 7 to to ADP ap-7718 136 8 recalling recall VERB ap-7718 136 9 that that SCONJ ap-7718 136 10 two two NUM ap-7718 136 11 arbitrary arbitrary ADJ ap-7718 136 12 quantum quantum NOUN ap-7718 136 13 invariants invariant NOUN ap-7718 136 14 î(t î(t PRON ap-7718 136 15 ) ) PUNCT ap-7718 136 16 and and CCONJ ap-7718 136 17 ˆ̃ ˆ̃ PROPN ap-7718 136 18 i(t i(t PROPN ap-7718 136 19 ) ) PUNCT ap-7718 136 20 of of ADP ap-7718 136 21 a a DET ap-7718 136 22 given give VERB ap-7718 136 23 hamiltonian hamiltonian NOUN ap-7718 136 24 ĥ(t ĥ(t NOUN ap-7718 136 25 ) ) PUNCT ap-7718 136 26 can can AUX ap-7718 136 27 be be AUX ap-7718 136 28 used use VERB ap-7718 136 29 to to PART ap-7718 136 30 construct construct VERB ap-7718 136 31 further further ADJ ap-7718 136 32 invariants invariant NOUN ap-7718 136 33 . . PUNCT ap-7718 137 1 this this PRON ap-7718 137 2 follows follow VERB ap-7718 137 3 from from ADP ap-7718 137 4 the the DET ap-7718 137 5 fact fact NOUN ap-7718 137 6 that that SCONJ ap-7718 137 7 the the DET ap-7718 137 8 linear linear ADJ ap-7718 137 9 combination combination NOUN ap-7718 137 10 ℓî(t ℓî(t NOUN ap-7718 137 11 ) ) PUNCT ap-7718 137 12 + + NUM ap-7718 138 1 ℓ̃ ℓ̃ PROPN ap-7718 138 2 ˆ̃ ˆ̃ PROPN ap-7718 138 3 i(t i(t PROPN ap-7718 138 4 ) ) PUNCT ap-7718 138 5 and and CCONJ ap-7718 138 6 the the DET ap-7718 138 7 product product NOUN ap-7718 138 8 ℓî(t)ˆ̃i(t ℓî(t)ˆ̃i(t NOUN ap-7718 138 9 ) ) PUNCT ap-7718 138 10 of of ADP ap-7718 138 11 quantum quantum NOUN ap-7718 138 12 invariants invariant NOUN ap-7718 138 13 are be AUX ap-7718 138 14 also also ADV ap-7718 138 15 quantum quantum ADJ ap-7718 138 16 invariants invariant NOUN ap-7718 138 17 of of ADP ap-7718 138 18 the the DET ap-7718 138 19 same same ADJ ap-7718 138 20 hamiltonian hamiltonian NOUN ap-7718 138 21 ĥ(t ĥ(t NOUN ap-7718 138 22 ) ) PUNCT ap-7718 138 23 , , PUNCT ap-7718 138 24 for for ADP ap-7718 138 25 ℓ ℓ NOUN ap-7718 138 26 , , PUNCT ap-7718 138 27 ℓ̃ ℓ̃ PROPN ap-7718 138 28 , , PUNCT ap-7718 138 29 and and CCONJ ap-7718 138 30 ℓ ℓ NOUN ap-7718 138 31 timeindependent timeindependent NOUN ap-7718 138 32 coefficients coefficient NOUN ap-7718 138 33 . . PUNCT ap-7718 139 1 in in ADP ap-7718 139 2 this this DET ap-7718 139 3 form form NOUN ap-7718 139 4 , , PUNCT ap-7718 139 5 îa(t îa(t PROPN ap-7718 139 6 ) ) PUNCT ap-7718 139 7 and and CCONJ ap-7718 139 8 î† î† PROPN ap-7718 139 9 a(t a(t NOUN ap-7718 139 10 ) ) PUNCT ap-7718 139 11 generate generate VERB ap-7718 139 12 î1(t î1(t NOUN ap-7718 139 13 ) ) PUNCT ap-7718 139 14 through through ADP ap-7718 139 15 î1(t î1(t PROPN ap-7718 139 16 ) ) PUNCT ap-7718 139 17 = = PUNCT ap-7718 140 1 ℏw0 ℏw0 INTJ ap-7718 140 2 ( ( PUNCT ap-7718 140 3 î† î† PROPN ap-7718 140 4 a(t)îa(t a(t)îa(t PROPN ap-7718 140 5 ) ) PUNCT ap-7718 141 1 + + CCONJ ap-7718 141 2 1 1 NUM ap-7718 141 3 2 2 NUM ap-7718 141 4 ) ) PUNCT ap-7718 141 5 , , PUNCT ap-7718 141 6 ( ( PUNCT ap-7718 141 7 27 27 NUM ap-7718 141 8 ) ) PUNCT ap-7718 141 9 which which PRON ap-7718 141 10 is be AUX ap-7718 141 11 analogous analogous ADJ ap-7718 141 12 to to ADP ap-7718 141 13 the the DET ap-7718 141 14 factorization factorization NOUN ap-7718 141 15 of of ADP ap-7718 141 16 the the DET ap-7718 141 17 stationary stationary ADJ ap-7718 141 18 oscillator oscillator NOUN ap-7718 141 19 . . PUNCT ap-7718 142 1 similarly similarly ADV ap-7718 142 2 , , PUNCT ap-7718 142 3 the the DET ap-7718 142 4 commutation commutation NOUN ap-7718 142 5 relationship relationship NOUN ap-7718 142 6 [ [ X ap-7718 142 7 â â X ap-7718 142 8 , , PUNCT ap-7718 142 9 ↠↠ADJ ap-7718 142 10 ] ] X ap-7718 142 11 = = SYM ap-7718 142 12 1 1 NUM ap-7718 142 13 of of ADP ap-7718 142 14 the the DET ap-7718 142 15 stationary stationary ADJ ap-7718 142 16 oscillator oscillator NOUN ap-7718 142 17 is be AUX ap-7718 142 18 preserved preserve VERB ap-7718 142 19 . . PUNCT ap-7718 143 1 one one NUM ap-7718 143 2 thus thus ADV ap-7718 143 3 get get VERB ap-7718 143 4 [ [ X ap-7718 143 5 îa(t îa(t PROPN ap-7718 143 6 ) ) PUNCT ap-7718 143 7 , , PUNCT ap-7718 143 8 î† î† PROPN ap-7718 143 9 a(t a(t PROPN ap-7718 143 10 ) ) PUNCT ap-7718 143 11 ] ] PUNCT ap-7718 144 1 = = PUNCT ap-7718 144 2 1 1 NUM ap-7718 144 3 together together ADV ap-7718 144 4 with with ADP ap-7718 144 5 [ [ X ap-7718 144 6 î1(t î1(t NOUN ap-7718 144 7 ) ) PUNCT ap-7718 144 8 , , PUNCT ap-7718 144 9 îa(t îa(t PROPN ap-7718 144 10 ) ) PUNCT ap-7718 144 11 ] ] PUNCT ap-7718 145 1 = = SYM ap-7718 145 2 −ℏw0îa(t −ℏw0îa(t PROPN ap-7718 145 3 ) ) PUNCT ap-7718 145 4 , , PUNCT ap-7718 146 1 [ [ X ap-7718 146 2 î1(t î1(t NOUN ap-7718 146 3 ) ) PUNCT ap-7718 146 4 , , PUNCT ap-7718 146 5 î† î† PROPN ap-7718 146 6 a(t a(t PROPN ap-7718 146 7 ) ) PUNCT ap-7718 146 8 ] ] PUNCT ap-7718 147 1 = = PUNCT ap-7718 147 2 ℏw0î ℏw0î PUNCT ap-7718 147 3 † † X ap-7718 147 4 a(t a(t PROPN ap-7718 147 5 ) ) PUNCT ap-7718 147 6 , , PUNCT ap-7718 147 7 ( ( PUNCT ap-7718 147 8 28 28 NUM ap-7718 147 9 ) ) PUNCT ap-7718 147 10 which which PRON ap-7718 147 11 means mean VERB ap-7718 147 12 that that SCONJ ap-7718 147 13 îa(t îa(t PROPN ap-7718 147 14 ) ) PUNCT ap-7718 147 15 and and CCONJ ap-7718 147 16 î† î† PROPN ap-7718 147 17 a(t a(t PROPN ap-7718 147 18 ) ) PUNCT ap-7718 147 19 are be AUX ap-7718 147 20 annihilation annihilation NOUN ap-7718 147 21 and and CCONJ ap-7718 147 22 creation creation NOUN ap-7718 147 23 operators operator NOUN ap-7718 147 24 , , PUNCT ap-7718 147 25 respectively respectively ADV ap-7718 147 26 , , PUNCT ap-7718 147 27 for for ADP ap-7718 147 28 the the DET ap-7718 147 29 eigensolutions eigensolution NOUN ap-7718 147 30 of of ADP ap-7718 147 31 î1(t î1(t NOUN ap-7718 147 32 ) ) PUNCT ap-7718 147 33 . . PUNCT ap-7718 148 1 the the DET ap-7718 148 2 latter latter ADJ ap-7718 148 3 leads lead VERB ap-7718 148 4 to to PART ap-7718 148 5 îa(t)ψn+1(x îa(t)ψn+1(x VERB ap-7718 148 6 , , PUNCT ap-7718 148 7 t t PROPN ap-7718 148 8 ) ) PUNCT ap-7718 148 9 = = SYM ap-7718 149 1 √ √ ADP ap-7718 149 2 ℏw0(n+ ℏw0(n+ PRON ap-7718 149 3 1)ψn(x 1)ψn(x NUM ap-7718 149 4 , , PUNCT ap-7718 149 5 t t PROPN ap-7718 149 6 ) ) PUNCT ap-7718 149 7 , , PUNCT ap-7718 149 8 î† î† PROPN ap-7718 149 9 a(t)ψn(x a(t)ψn(x PROPN ap-7718 149 10 , , PUNCT ap-7718 149 11 t t PROPN ap-7718 149 12 ) ) PUNCT ap-7718 149 13 = = SYM ap-7718 150 1 √ √ ADP ap-7718 150 2 ℏw0(n+ ℏw0(n+ PRON ap-7718 150 3 1)ψn+1(x 1)ψn+1(x PROPN ap-7718 150 4 , , PUNCT ap-7718 150 5 t t PROPN ap-7718 150 6 ) ) PUNCT ap-7718 150 7 , , PUNCT ap-7718 150 8 ( ( PUNCT ap-7718 150 9 29 29 NUM ap-7718 150 10 ) ) PUNCT ap-7718 150 11 for for ADP ap-7718 150 12 n n NOUN ap-7718 150 13 = = SYM ap-7718 150 14 0 0 NUM ap-7718 150 15 , , PUNCT ap-7718 150 16 . . PUNCT ap-7718 150 17 . . PUNCT ap-7718 150 18 . . PUNCT ap-7718 151 1 . . PUNCT ap-7718 152 1 on on ADP ap-7718 152 2 the the DET ap-7718 152 3 other other ADJ ap-7718 152 4 hand hand NOUN ap-7718 152 5 , , PUNCT ap-7718 152 6 the the DET ap-7718 152 7 orthonormal orthonormal ADJ ap-7718 152 8 set set NOUN ap-7718 152 9 { { PUNCT ap-7718 152 10 ψn(x ψn(x NOUN ap-7718 152 11 , , PUNCT ap-7718 152 12 t)}∞ t)}∞ ADV ap-7718 152 13 n=0 n=0 NUM ap-7718 152 14 can can AUX ap-7718 152 15 be be AUX ap-7718 152 16 used use VERB ap-7718 152 17 as as ADP ap-7718 152 18 a a DET ap-7718 152 19 basis basis NOUN ap-7718 152 20 to to PART ap-7718 152 21 expand expand VERB ap-7718 152 22 any any DET ap-7718 152 23 arbitrary arbitrary ADJ ap-7718 152 24 solution solution NOUN ap-7718 152 25 ψ(x ψ(x PROPN ap-7718 152 26 , , PUNCT ap-7718 152 27 t t PROPN ap-7718 152 28 ) ) PUNCT ap-7718 152 29 of of ADP ap-7718 152 30 ( ( PUNCT ap-7718 152 31 2 2 NUM ap-7718 152 32 ) ) PUNCT ap-7718 152 33 through through ADP ap-7718 152 34 ψ(x ψ(x PROPN ap-7718 152 35 , , PUNCT ap-7718 152 36 t t PROPN ap-7718 152 37 ) ) PUNCT ap-7718 152 38 = = PUNCT ap-7718 153 1 ∞∑ ∞∑ PRON ap-7718 153 2 n=0 n=0 PROPN ap-7718 153 3 cnψn(x cnψn(x PROPN ap-7718 153 4 , , PUNCT ap-7718 153 5 t t PROPN ap-7718 153 6 ) ) PUNCT ap-7718 153 7 , , PUNCT ap-7718 153 8 cn cn PROPN ap-7718 153 9 : : PUNCT ap-7718 153 10 = = SYM ap-7718 153 11 ( ( PUNCT ap-7718 153 12 ψn(x ψn(x X ap-7718 153 13 , , PUNCT ap-7718 153 14 t t PROPN ap-7718 153 15 ) ) PUNCT ap-7718 153 16 , , PUNCT ap-7718 153 17 ψ(x ψ(x PROPN ap-7718 153 18 , , PUNCT ap-7718 153 19 t t PROPN ap-7718 153 20 ) ) PUNCT ap-7718 153 21 ) ) PUNCT ap-7718 153 22 . . PUNCT ap-7718 154 1 ( ( PUNCT ap-7718 154 2 30 30 NUM ap-7718 154 3 ) ) PUNCT ap-7718 154 4 now now ADV ap-7718 154 5 , , PUNCT ap-7718 154 6 from from ADP ap-7718 154 7 the the DET ap-7718 154 8 above above ADJ ap-7718 154 9 results result NOUN ap-7718 154 10 , , PUNCT ap-7718 154 11 one one PRON ap-7718 154 12 may may AUX ap-7718 154 13 investigate investigate VERB ap-7718 154 14 the the DET ap-7718 154 15 spectral spectral ADJ ap-7718 154 16 problem problem NOUN ap-7718 154 17 related relate VERB ap-7718 154 18 to to ADP ap-7718 154 19 the the DET ap-7718 154 20 remaining remain VERB ap-7718 154 21 quantum quantum NOUN ap-7718 154 22 invariants invariant NOUN ap-7718 154 23 îa(t îa(t PROPN ap-7718 154 24 ) ) PUNCT ap-7718 154 25 and and CCONJ ap-7718 154 26 î† î† PROPN ap-7718 154 27 a(t a(t NOUN ap-7718 154 28 ) ) PUNCT ap-7718 154 29 . . PUNCT ap-7718 155 1 by by ADP ap-7718 155 2 considering consider VERB ap-7718 155 3 the the DET ap-7718 155 4 annihilation annihilation NOUN ap-7718 155 5 operator operator NOUN ap-7718 155 6 îa(t îa(t PROPN ap-7718 155 7 ) ) PUNCT ap-7718 155 8 , , PUNCT ap-7718 155 9 one one PRON ap-7718 155 10 obtains obtain VERB ap-7718 155 11 the the DET ap-7718 155 12 eigenvalue eigenvalue PROPN ap-7718 155 13 problem problem NOUN ap-7718 155 14 îa(t)ξα(x îa(t)ξα(x PROPN ap-7718 155 15 , , PUNCT ap-7718 155 16 t t PROPN ap-7718 155 17 ) ) PUNCT ap-7718 155 18 = = SYM ap-7718 155 19 αξα(x αξα(x PROPN ap-7718 155 20 , , PUNCT ap-7718 155 21 t t PROPN ap-7718 155 22 ) ) PUNCT ap-7718 155 23 , , PUNCT ap-7718 155 24 ( ( PUNCT ap-7718 155 25 31 31 NUM ap-7718 155 26 ) ) PUNCT ap-7718 155 27 where where SCONJ ap-7718 155 28 the the DET ap-7718 155 29 eigensolution eigensolution NOUN ap-7718 155 30 ξα(x ξα(x NOUN ap-7718 155 31 , , PUNCT ap-7718 155 32 t t PROPN ap-7718 155 33 ) ) PUNCT ap-7718 155 34 can can AUX ap-7718 155 35 be be AUX ap-7718 155 36 expanded expand VERB ap-7718 155 37 as as ADP ap-7718 155 38 the the DET ap-7718 155 39 linear linear ADJ ap-7718 155 40 combination combination NOUN ap-7718 155 41 ξα(x ξα(x NOUN ap-7718 155 42 , , PUNCT ap-7718 155 43 t t PROPN ap-7718 155 44 ) ) PUNCT ap-7718 155 45 = = PUNCT ap-7718 156 1 ∞∑ ∞∑ ADJ ap-7718 156 2 n=0 n=0 NUM ap-7718 156 3 c̃n(α)ψn(x c̃n(α)ψn(x PROPN ap-7718 156 4 , , PUNCT ap-7718 156 5 t t PROPN ap-7718 156 6 ) ) PUNCT ap-7718 156 7 , , PUNCT ap-7718 156 8 α α PROPN ap-7718 156 9 ∈ ∈ PROPN ap-7718 156 10 c. c. NOUN ap-7718 156 11 ( ( PUNCT ap-7718 156 12 32 32 NUM ap-7718 156 13 ) ) PUNCT ap-7718 156 14 this this PRON ap-7718 156 15 corresponds correspond VERB ap-7718 156 16 to to ADP ap-7718 156 17 the the DET ap-7718 156 18 construction construction NOUN ap-7718 156 19 of of ADP ap-7718 156 20 coherent coherent ADJ ap-7718 156 21 states state NOUN ap-7718 156 22 using use VERB ap-7718 156 23 the the DET ap-7718 156 24 barut barut NOUN ap-7718 156 25 - - PUNCT ap-7718 156 26 girardelo girardelo NOUN ap-7718 156 27 approach approach NOUN ap-7718 156 28 [ [ X ap-7718 156 29 37 37 NUM ap-7718 156 30 ] ] PUNCT ap-7718 156 31 . . PUNCT ap-7718 157 1 the the DET ap-7718 157 2 complex complex ADJ ap-7718 157 3 coefficients coefficient NOUN ap-7718 157 4 c̃n(α c̃n(α PRON ap-7718 157 5 ) ) PUNCT ap-7718 157 6 are be AUX ap-7718 157 7 determined determine VERB ap-7718 157 8 by by ADP ap-7718 157 9 using use VERB ap-7718 157 10 the the DET ap-7718 157 11 action action NOUN ap-7718 157 12 of of ADP ap-7718 157 13 the the DET ap-7718 157 14 ladder ladder NOUN ap-7718 157 15 operators operator NOUN ap-7718 157 16 ( ( PUNCT ap-7718 157 17 29 29 NUM ap-7718 157 18 ) ) PUNCT ap-7718 157 19 and and CCONJ ap-7718 157 20 exploiting exploit VERB ap-7718 157 21 the the DET ap-7718 157 22 orthonormality orthonormality NOUN ap-7718 157 23 of of ADP ap-7718 157 24 the the DET ap-7718 157 25 set set NOUN ap-7718 157 26 { { PUNCT ap-7718 157 27 ψn(x ψn(x NOUN ap-7718 157 28 , , PUNCT ap-7718 157 29 t)}∞ t)}∞ ADV ap-7718 157 30 n=0 n=0 NUM ap-7718 157 31 . . PUNCT ap-7718 158 1 after after ADP ap-7718 158 2 substituting substitute VERB ap-7718 158 3 the the DET ap-7718 158 4 linear linear ADJ ap-7718 158 5 combination combination NOUN ap-7718 158 6 ξα(x ξα(x NOUN ap-7718 158 7 , , PUNCT ap-7718 158 8 t t PROPN ap-7718 158 9 ) ) PUNCT ap-7718 158 10 into into ADP ap-7718 158 11 the the DET ap-7718 158 12 corresponding corresponding ADJ ap-7718 158 13 eigenvalue eigenvalue PROPN ap-7718 158 14 problem problem NOUN ap-7718 158 15 in in ADP ap-7718 158 16 ( ( PUNCT ap-7718 158 17 31 31 NUM ap-7718 158 18 ) ) PUNCT ap-7718 158 19 , , PUNCT ap-7718 158 20 one one PRON ap-7718 158 21 obtains obtain VERB ap-7718 158 22 the the DET ap-7718 158 23 one one NUM ap-7718 158 24 - - PUNCT ap-7718 158 25 parameter parameter NOUN ap-7718 158 26 normalized normalize VERB ap-7718 158 27 eigensolutions eigensolution NOUN ap-7718 158 28 ξα(x ξα(x NOUN ap-7718 158 29 , , PUNCT ap-7718 158 30 t t PROPN ap-7718 158 31 ) ) PUNCT ap-7718 158 32 = = NOUN ap-7718 158 33 exp exp NOUN ap-7718 158 34 ( ( PUNCT ap-7718 158 35 − − PROPN ap-7718 158 36 |α|2 |α|2 NOUN ap-7718 158 37 2ℏw0 2ℏw0 NUM ap-7718 158 38 ) ) PUNCT ap-7718 159 1 ∞∑ ∞∑ NUM ap-7718 159 2 n=0 n=0 NUM ap-7718 159 3 ( ( PUNCT ap-7718 159 4 α√ α√ PROPN ap-7718 159 5 ℏw0 ℏw0 ADJ ap-7718 159 6 ) ) PUNCT ap-7718 159 7 n n PRON ap-7718 159 8 ψn(x ψn(x ADP ap-7718 159 9 , , PUNCT ap-7718 159 10 t)√ t)√ NOUN ap-7718 159 11 n n NUM ap-7718 159 12 ! ! PUNCT ap-7718 159 13 . . PUNCT ap-7718 160 1 ( ( PUNCT ap-7718 160 2 33 33 NUM ap-7718 160 3 ) ) PUNCT ap-7718 160 4 henceforth henceforth ADV ap-7718 160 5 , , PUNCT ap-7718 160 6 the the DET ap-7718 160 7 latter latter ADJ ap-7718 160 8 are be AUX ap-7718 160 9 called call VERB ap-7718 160 10 time time NOUN ap-7718 160 11 - - PUNCT ap-7718 160 12 dependent dependent ADJ ap-7718 160 13 coherent coherent ADJ ap-7718 160 14 states state NOUN ap-7718 160 15 or or CCONJ ap-7718 160 16 semiclassical semiclassical ADJ ap-7718 160 17 states state NOUN ap-7718 160 18 interchangeably interchangeably ADV ap-7718 160 19 . . PUNCT ap-7718 161 1 similar similar ADJ ap-7718 161 2 to to ADP ap-7718 161 3 glauber glauber PROPN ap-7718 161 4 coherent coherent ADJ ap-7718 161 5 states state NOUN ap-7718 161 6 [ [ X ap-7718 161 7 38 38 NUM ap-7718 161 8 ] ] PUNCT ap-7718 161 9 , , PUNCT ap-7718 161 10 the the DET ap-7718 161 11 eigensolutions eigensolution NOUN ap-7718 161 12 of of ADP ap-7718 161 13 the the DET ap-7718 161 14 annihilation annihilation NOUN ap-7718 161 15 operator operator NOUN ap-7718 161 16 îa îa NOUN ap-7718 161 17 are be AUX ap-7718 161 18 not not PART ap-7718 161 19 orthogonal orthogonal ADJ ap-7718 161 20 among among ADP ap-7718 161 21 themselves themselves PRON ap-7718 161 22 . . PUNCT ap-7718 162 1 this this PRON ap-7718 162 2 follows follow VERB ap-7718 162 3 from from ADP ap-7718 162 4 the the DET ap-7718 162 5 overlap overlap NOUN ap-7718 162 6 between between ADP ap-7718 162 7 two two NUM ap-7718 162 8 solutions solution NOUN ap-7718 162 9 with with ADP ap-7718 162 10 different different ADJ ap-7718 162 11 eigenvalues eigenvalue NOUN ap-7718 162 12 , , PUNCT ap-7718 162 13 let let VERB ap-7718 162 14 say say VERB ap-7718 162 15 α α PRON ap-7718 162 16 and and CCONJ ap-7718 162 17 β β NOUN ap-7718 162 18 , , PUNCT ap-7718 162 19 leading lead VERB ap-7718 162 20 to to ADP ap-7718 162 21 |(ξβ |(ξβ NUM ap-7718 162 22 , , PUNCT ap-7718 162 23 ξα)|2 ξα)|2 PROPN ap-7718 162 24 = = PUNCT ap-7718 162 25 exp exp NOUN ap-7718 162 26 ( ( PUNCT ap-7718 162 27 −|α|2 −|α|2 PROPN ap-7718 162 28 + + CCONJ ap-7718 162 29 |β|2 |β|2 ADJ ap-7718 162 30 − − NOUN ap-7718 162 31 2 2 NUM ap-7718 162 32 re(α∗β re(α∗β NOUN ap-7718 162 33 ) ) PUNCT ap-7718 162 34 ℏw0 ℏw0 CCONJ ap-7718 162 35 ) ) PUNCT ap-7718 162 36 , , PUNCT ap-7718 162 37 ( ( PUNCT ap-7718 162 38 34 34 NUM ap-7718 162 39 ) ) PUNCT ap-7718 162 40 which which PRON ap-7718 162 41 is be AUX ap-7718 162 42 different different ADJ ap-7718 162 43 from from ADP ap-7718 162 44 zero zero NUM ap-7718 162 45 for for ADP ap-7718 162 46 every every DET ap-7718 162 47 α α NOUN ap-7718 162 48 , , PUNCT ap-7718 162 49 β β X ap-7718 162 50 ∈ ∈ PROPN ap-7718 162 51 c c X ap-7718 162 52 , , PUNCT ap-7718 162 53 with with ADP ap-7718 162 54 the the DET ap-7718 162 55 inner inner ADJ ap-7718 162 56 product product NOUN ap-7718 162 57 defined define VERB ap-7718 162 58 in in ADP ap-7718 162 59 ( ( PUNCT ap-7718 162 60 18 18 NUM ap-7718 162 61 ) ) PUNCT ap-7718 162 62 . . PUNCT ap-7718 163 1 interestingly interestingly ADV ap-7718 163 2 , , PUNCT ap-7718 163 3 the the DET ap-7718 163 4 eigensolution eigensolution NOUN ap-7718 163 5 ξα(x ξα(x NOUN ap-7718 163 6 , , PUNCT ap-7718 163 7 t t PROPN ap-7718 163 8 ) ) PUNCT ap-7718 163 9 can can AUX ap-7718 163 10 be be AUX ap-7718 163 11 brought bring VERB ap-7718 163 12 into into ADP ap-7718 163 13 an an DET ap-7718 163 14 alternative alternative ADJ ap-7718 163 15 and and CCONJ ap-7718 163 16 handy handy ADJ ap-7718 163 17 expression expression NOUN ap-7718 163 18 by by ADP ap-7718 163 19 using use VERB ap-7718 163 20 the the DET ap-7718 163 21 explicit explicit ADJ ap-7718 163 22 form form NOUN ap-7718 163 23 of of ADP ap-7718 163 24 ψn(x ψn(x ADP ap-7718 163 25 , , PUNCT ap-7718 163 26 t t PROPN ap-7718 163 27 ) ) PUNCT ap-7718 163 28 given give VERB ap-7718 163 29 in in ADP ap-7718 163 30 ( ( PUNCT ap-7718 163 31 17 17 NUM ap-7718 163 32 ) ) PUNCT ap-7718 163 33 , , PUNCT ap-7718 163 34 together together ADV ap-7718 163 35 with with ADP ap-7718 163 36 the the DET ap-7718 163 37 well well ADV ap-7718 163 38 - - PUNCT ap-7718 163 39 known know VERB ap-7718 163 40 summation summation NOUN ap-7718 163 41 rules rule NOUN ap-7718 163 42 for for ADP ap-7718 163 43 the the DET ap-7718 163 44 hermite hermite ADJ ap-7718 163 45 polynomials polynomial NOUN ap-7718 163 46 . . PUNCT ap-7718 164 1 by by ADP ap-7718 164 2 doing do VERB ap-7718 164 3 so so ADV ap-7718 164 4 one one PRON ap-7718 164 5 gets get VERB ap-7718 164 6 ξα(x ξα(x NOUN ap-7718 164 7 , , PUNCT ap-7718 164 8 t t PROPN ap-7718 164 9 ) ) PUNCT ap-7718 164 10 ≡ ≡ PROPN ap-7718 164 11 √ √ ADP ap-7718 164 12 µ(t µ(t ADJ ap-7718 164 13 ) ) PUNCT ap-7718 164 14 σ(t σ(t PROPN ap-7718 164 15 ) ) PUNCT ap-7718 164 16 √ √ NUM ap-7718 164 17 m0w0 m0w0 PUNCT ap-7718 164 18 πℏ πℏ NOUN ap-7718 165 1 [ [ X ap-7718 165 2 a(x a(x NOUN ap-7718 165 3 , , PUNCT ap-7718 165 4 t)]−1e−i t)]−1e−i PRON ap-7718 165 5 w0τ(t w0τ(t PROPN ap-7718 165 6 ) ) PUNCT ap-7718 165 7 2 2 NUM ap-7718 165 8 × × NOUN ap-7718 165 9 exp exp NOUN ap-7718 165 10 [ [ PUNCT ap-7718 165 11 i i PRON ap-7718 165 12 √ √ PROPN ap-7718 165 13 2m0w0 2m0w0 NUM ap-7718 165 14 ℏ ℏ PROPN ap-7718 165 15 ( ( PUNCT ap-7718 165 16 µ(t)x+ µ(t)x+ X ap-7718 165 17 γ(t γ(t NOUN ap-7718 165 18 ) ) PUNCT ap-7718 165 19 σ(t σ(t PROPN ap-7718 165 20 ) ) PUNCT ap-7718 165 21 ) ) PUNCT ap-7718 166 1 i i PRON ap-7718 166 2 m m VERB ap-7718 166 3 α̃(t α̃(t PROPN ap-7718 166 4 ) ) PUNCT ap-7718 166 5 ] ] PUNCT ap-7718 167 1 ×exp ×exp PROPN ap-7718 167 2 [ [ PUNCT ap-7718 167 3 −m0w0 −m0w0 PROPN ap-7718 167 4 2ℏ 2ℏ NUM ap-7718 167 5 ( ( PUNCT ap-7718 167 6 µ(t)x+ µ(t)x+ X ap-7718 167 7 γ(t γ(t NOUN ap-7718 167 8 ) ) PUNCT ap-7718 167 9 σ(t σ(t PROPN ap-7718 167 10 ) ) PUNCT ap-7718 167 11 − − NOUN ap-7718 168 1 √ √ NUM ap-7718 168 2 2ℏ 2ℏ NUM ap-7718 168 3 m0w0 m0w0 PROPN ap-7718 168 4 re re ADP ap-7718 168 5 α̃(t α̃(t PROPN ap-7718 168 6 ) ) PUNCT ap-7718 168 7 ) ) PUNCT ap-7718 169 1 2 2 NUM ap-7718 169 2 ] ] PUNCT ap-7718 169 3 , , PUNCT ap-7718 169 4 ( ( PUNCT ap-7718 169 5 35 35 NUM ap-7718 169 6 ) ) PUNCT ap-7718 169 7 with with ADP ap-7718 169 8 α̃(t α̃(t NOUN ap-7718 169 9 ) ) PUNCT ap-7718 169 10 = = SYM ap-7718 169 11 αe−iw0τ(t αe−iw0τ(t NOUN ap-7718 169 12 ) ) PUNCT ap-7718 169 13 . . PUNCT ap-7718 170 1 thus thus ADV ap-7718 170 2 , , PUNCT ap-7718 170 3 ξα(x ξα(x NUM ap-7718 170 4 , , PUNCT ap-7718 170 5 t t PROPN ap-7718 170 6 ) ) PUNCT ap-7718 170 7 is be AUX ap-7718 170 8 a a DET ap-7718 170 9 normalized normalize VERB ap-7718 170 10 gaussian gaussian ADJ ap-7718 170 11 wavepacket wavepacket NOUN ap-7718 170 12 with with ADP ap-7718 170 13 time time NOUN ap-7718 170 14 - - PUNCT ap-7718 170 15 dependent dependent ADJ ap-7718 170 16 width width NOUN ap-7718 170 17 . . PUNCT ap-7718 171 1 the the DET ap-7718 171 2 complex complex ADJ ap-7718 171 3 constants constant NOUN ap-7718 171 4 α α PRON ap-7718 171 5 plays play VERB ap-7718 171 6 the the DET ap-7718 171 7 role role NOUN ap-7718 171 8 of of ADP ap-7718 171 9 the the DET ap-7718 171 10 initial initial ADJ ap-7718 171 11 conditions condition NOUN ap-7718 171 12 of of ADP ap-7718 171 13 the the DET ap-7718 171 14 wavepacket wavepacket NOUN ap-7718 171 15 at at ADP ap-7718 171 16 a a DET ap-7718 171 17 given give VERB ap-7718 171 18 time time NOUN ap-7718 171 19 t0 t0 PROPN ap-7718 171 20 . . PUNCT ap-7718 172 1 see see VERB ap-7718 172 2 the the DET ap-7718 172 3 discussion discussion NOUN ap-7718 172 4 in in ADP ap-7718 172 5 the the DET ap-7718 172 6 following follow VERB ap-7718 172 7 section section NOUN ap-7718 172 8 . . PUNCT ap-7718 173 1 despite despite SCONJ ap-7718 173 2 the the DET ap-7718 173 3 lack lack NOUN ap-7718 173 4 of of ADP ap-7718 173 5 orthogonality orthogonality NOUN ap-7718 173 6 in in ADP ap-7718 173 7 the the DET ap-7718 173 8 elements element NOUN ap-7718 173 9 of of ADP ap-7718 173 10 the the DET ap-7718 173 11 set set NOUN ap-7718 173 12 { { PUNCT ap-7718 173 13 ξα(x ξα(x NUM ap-7718 173 14 , , PUNCT ap-7718 173 15 t)}α∈c t)}α∈c PROPN ap-7718 173 16 , , PUNCT ap-7718 173 17 they they PRON ap-7718 173 18 can can AUX ap-7718 173 19 be be AUX ap-7718 173 20 still still ADV ap-7718 173 21 used use VERB ap-7718 173 22 as as ADP ap-7718 173 23 a a DET ap-7718 173 24 nonorthogonal nonorthogonal ADJ ap-7718 173 25 basis basis NOUN ap-7718 173 26 so so SCONJ ap-7718 173 27 that that SCONJ ap-7718 173 28 any any DET ap-7718 173 29 arbitrary arbitrary ADJ ap-7718 173 30 solution solution NOUN ap-7718 173 31 of of ADP ap-7718 173 32 ( ( PUNCT ap-7718 173 33 2 2 X ap-7718 173 34 ) ) PUNCT ap-7718 173 35 can can AUX ap-7718 173 36 be be AUX ap-7718 173 37 constructed construct VERB ap-7718 173 38 through through ADP ap-7718 173 39 the the DET ap-7718 173 40 appropriate appropriate ADJ ap-7718 173 41 linear linear ADJ ap-7718 173 42 superposition superposition NOUN ap-7718 173 43 . . PUNCT ap-7718 174 1 that that PRON ap-7718 174 2 is is ADV ap-7718 174 3 , , PUNCT ap-7718 174 4 a a DET ap-7718 174 5 given give VERB ap-7718 174 6 solution solution NOUN ap-7718 174 7 ψ(x ψ(x PROPN ap-7718 174 8 , , PUNCT ap-7718 174 9 t t PROPN ap-7718 174 10 ) ) PUNCT ap-7718 174 11 of of ADP ap-7718 174 12 ( ( PUNCT ap-7718 174 13 2 2 X ap-7718 174 14 ) ) PUNCT ap-7718 174 15 expands expand VERB ap-7718 174 16 as as ADP ap-7718 174 17 ψ(x ψ(x PROPN ap-7718 174 18 , , PUNCT ap-7718 174 19 t t PROPN ap-7718 174 20 ) ) PUNCT ap-7718 174 21 = = SYM ap-7718 175 1 ∫ ∫ PROPN ap-7718 175 2 c c PROPN ap-7718 175 3 d2α d2α NOUN ap-7718 175 4 πℏ πℏ NOUN ap-7718 175 5 c(α)ξα(x c(α)ξα(x PROPN ap-7718 175 6 , , PUNCT ap-7718 175 7 t t PROPN ap-7718 175 8 ) ) PUNCT ap-7718 175 9 , , PUNCT ap-7718 175 10 ( ( PUNCT ap-7718 175 11 36 36 NUM ap-7718 175 12 ) ) PUNCT ap-7718 175 13 214 214 NUM ap-7718 175 14 vol vol NOUN ap-7718 175 15 . . PUNCT ap-7718 176 1 62 62 NUM ap-7718 176 2 no no INTJ ap-7718 176 3 . . PUNCT ap-7718 177 1 1/2022 1/2022 NUM ap-7718 177 2 td td NOUN ap-7718 177 3 mass mass NOUN ap-7718 177 4 oscillators oscillator NOUN ap-7718 177 5 : : PUNCT ap-7718 177 6 constants constant NOUN ap-7718 177 7 of of ADP ap-7718 177 8 motion motion NOUN ap-7718 177 9 and and CCONJ ap-7718 177 10 semiclasical semiclasical ADJ ap-7718 177 11 states state NOUN ap-7718 177 12 where where SCONJ ap-7718 177 13 c(α c(α NOUN ap-7718 177 14 ) ) PUNCT ap-7718 177 15 = = SYM ap-7718 177 16 ( ( PUNCT ap-7718 177 17 ξα(x ξα(x X ap-7718 177 18 , , PUNCT ap-7718 177 19 t t PROPN ap-7718 177 20 ) ) PUNCT ap-7718 177 21 , , PUNCT ap-7718 177 22 ψ(x ψ(x PROPN ap-7718 177 23 , , PUNCT ap-7718 177 24 t t PROPN ap-7718 177 25 ) ) PUNCT ap-7718 177 26 ) ) PUNCT ap-7718 177 27 . . PUNCT ap-7718 178 1 so so ADV ap-7718 178 2 far far ADV ap-7718 178 3 , , PUNCT ap-7718 178 4 the the DET ap-7718 178 5 spectral spectral ADJ ap-7718 178 6 problem problem NOUN ap-7718 178 7 related relate VERB ap-7718 178 8 to to ADP ap-7718 178 9 the the DET ap-7718 178 10 quantum quantum NOUN ap-7718 178 11 invariants invariant NOUN ap-7718 178 12 î1(t î1(t PROPN ap-7718 178 13 ) ) PUNCT ap-7718 178 14 and and CCONJ ap-7718 178 15 îa(t îa(t PROPN ap-7718 178 16 ) ) PUNCT ap-7718 178 17 has have AUX ap-7718 178 18 led lead VERB ap-7718 178 19 to to ADP ap-7718 178 20 a a DET ap-7718 178 21 discrete discrete NOUN ap-7718 178 22 and and CCONJ ap-7718 178 23 a a DET ap-7718 178 24 continuous continuous ADJ ap-7718 178 25 representation representation NOUN ap-7718 178 26 , , PUNCT ap-7718 178 27 respectively respectively ADV ap-7718 178 28 , , PUNCT ap-7718 178 29 in in ADP ap-7718 178 30 which which PRON ap-7718 178 31 any any DET ap-7718 178 32 solution solution NOUN ap-7718 178 33 of of ADP ap-7718 178 34 ( ( PUNCT ap-7718 178 35 2 2 X ap-7718 178 36 ) ) PUNCT ap-7718 178 37 can can AUX ap-7718 178 38 be be AUX ap-7718 178 39 expanded expand VERB ap-7718 178 40 . . PUNCT ap-7718 179 1 although although SCONJ ap-7718 179 2 the the DET ap-7718 179 3 eigenvalue eigenvalue PROPN ap-7718 179 4 problem problem NOUN ap-7718 179 5 related relate VERB ap-7718 179 6 to to ADP ap-7718 179 7 the the DET ap-7718 179 8 quantum quantum ADJ ap-7718 179 9 invariant invariant NOUN ap-7718 179 10 î† î† PROPN ap-7718 179 11 a(t a(t PROPN ap-7718 179 12 ) ) PUNCT ap-7718 179 13 can can AUX ap-7718 179 14 be be AUX ap-7718 179 15 established establish VERB ap-7718 179 16 , , PUNCT ap-7718 179 17 it it PRON ap-7718 179 18 leads lead VERB ap-7718 179 19 to to ADP ap-7718 179 20 non non ADJ ap-7718 179 21 - - ADJ ap-7718 179 22 finite finite ADJ ap-7718 179 23 norm norm NOUN ap-7718 179 24 solutions solution NOUN ap-7718 179 25 and and CCONJ ap-7718 179 26 is be AUX ap-7718 179 27 thus thus ADV ap-7718 179 28 discarded discard VERB ap-7718 179 29 . . PUNCT ap-7718 180 1 3.1 3.1 NUM ap-7718 180 2 . . PUNCT ap-7718 180 3 semiclassical semiclassical ADJ ap-7718 180 4 dynamics dynamic NOUN ap-7718 180 5 with with ADP ap-7718 180 6 the the DET ap-7718 180 7 time time NOUN ap-7718 180 8 - - PUNCT ap-7718 180 9 dependent dependent ADJ ap-7718 180 10 coherent coherent ADJ ap-7718 180 11 states state NOUN ap-7718 180 12 already already ADV ap-7718 180 13 constructed construct VERB ap-7718 180 14 , , PUNCT ap-7718 180 15 one one PRON ap-7718 180 16 can can AUX ap-7718 180 17 now now ADV ap-7718 180 18 study study VERB ap-7718 180 19 the the DET ap-7718 180 20 evolution evolution NOUN ap-7718 180 21 on on ADP ap-7718 180 22 time time NOUN ap-7718 180 23 of of ADP ap-7718 180 24 such such ADJ ap-7718 180 25 state state NOUN ap-7718 180 26 and and CCONJ ap-7718 180 27 its its PRON ap-7718 180 28 relation relation NOUN ap-7718 180 29 with with ADP ap-7718 180 30 physical physical ADJ ap-7718 180 31 observables observable NOUN ap-7718 180 32 such such ADJ ap-7718 180 33 as as ADP ap-7718 180 34 position position NOUN ap-7718 180 35 and and CCONJ ap-7718 180 36 momentum momentum NOUN ap-7718 180 37 x̂ x̂ NOUN ap-7718 180 38 and and CCONJ ap-7718 180 39 p̂x p̂x NOUN ap-7718 180 40 , , PUNCT ap-7718 180 41 respectively respectively ADV ap-7718 180 42 . . PUNCT ap-7718 181 1 to to ADP ap-7718 181 2 this this DET ap-7718 181 3 end end NOUN ap-7718 181 4 , , PUNCT ap-7718 181 5 note note VERB ap-7718 181 6 that that SCONJ ap-7718 181 7 the the DET ap-7718 181 8 quantum quantum NOUN ap-7718 181 9 invariants invariant NOUN ap-7718 181 10 obtained obtain VERB ap-7718 181 11 through through ADP ap-7718 181 12 point point NOUN ap-7718 181 13 transformations transformation NOUN ap-7718 181 14 preserve preserve VERB ap-7718 181 15 the the DET ap-7718 181 16 commutation commutation NOUN ap-7718 181 17 relation relation NOUN ap-7718 181 18 ( ( PUNCT ap-7718 181 19 28 28 NUM ap-7718 181 20 ) ) PUNCT ap-7718 181 21 of of ADP ap-7718 181 22 the the DET ap-7718 181 23 corresponding correspond VERB ap-7718 181 24 operators operator NOUN ap-7718 181 25 of of ADP ap-7718 181 26 the the DET ap-7718 181 27 stationary stationary ADJ ap-7718 181 28 oscillator oscillator NOUN ap-7718 181 29 . . PUNCT ap-7718 182 1 that that PRON ap-7718 182 2 is is ADV ap-7718 182 3 , , PUNCT ap-7718 182 4 the the DET ap-7718 182 5 set set NOUN ap-7718 182 6 { { PUNCT ap-7718 182 7 îa(t îa(t PROPN ap-7718 182 8 ) ) PUNCT ap-7718 182 9 , , PUNCT ap-7718 182 10 î† î† PROPN ap-7718 182 11 a(t a(t PROPN ap-7718 182 12 ) ) PUNCT ap-7718 182 13 , , PUNCT ap-7718 182 14 î† î† PROPN ap-7718 182 15 a(t)îa(t a(t)îa(t PROPN ap-7718 182 16 ) ) PUNCT ap-7718 182 17 } } PUNCT ap-7718 182 18 fulfill fulfill VERB ap-7718 182 19 the the DET ap-7718 182 20 weyl weyl PROPN ap-7718 182 21 - - PUNCT ap-7718 182 22 heisenberg heisenberg PROPN ap-7718 182 23 algebra algebra PROPN ap-7718 182 24 [ [ X ap-7718 182 25 39 39 NUM ap-7718 182 26 ] ] PUNCT ap-7718 182 27 . . PUNCT ap-7718 183 1 this this PRON ap-7718 183 2 allows allow VERB ap-7718 183 3 the the DET ap-7718 183 4 construction construction NOUN ap-7718 183 5 of of ADP ap-7718 183 6 a a DET ap-7718 183 7 unitary unitary ADJ ap-7718 183 8 displacement displacement ADJ ap-7718 183 9 operator operator NOUN ap-7718 183 10 of of ADP ap-7718 183 11 the the DET ap-7718 183 12 form form NOUN ap-7718 183 13 [ [ X ap-7718 183 14 39 39 NUM ap-7718 183 15 ] ] PUNCT ap-7718 183 16 d(α d(α PROPN ap-7718 183 17 ; ; PUNCT ap-7718 183 18 t t X ap-7718 183 19 ) ) PUNCT ap-7718 183 20 = = SYM ap-7718 183 21 eαî† eαî† X ap-7718 183 22 a(t)−α∗îa(t a(t)−α∗îa(t PROPN ap-7718 183 23 ) ) PUNCT ap-7718 183 24 = = PUNCT ap-7718 183 25 e− e− PROPN ap-7718 183 26 |α| |α| PROPN ap-7718 183 27 2 2 NUM ap-7718 183 28 eαî† eαî† NOUN ap-7718 183 29 ae−α∗îa ae−α∗îa PROPN ap-7718 183 30 , , PUNCT ap-7718 183 31 α α PROPN ap-7718 183 32 ∈ ∈ PROPN ap-7718 183 33 c c X ap-7718 183 34 , , PUNCT ap-7718 183 35 ( ( PUNCT ap-7718 183 36 37 37 NUM ap-7718 183 37 ) ) PUNCT ap-7718 183 38 so so SCONJ ap-7718 183 39 that that PRON ap-7718 183 40 d†(α d†(α VERB ap-7718 183 41 , , PUNCT ap-7718 183 42 t)îa(t)d(α t)îa(t)d(α PROPN ap-7718 183 43 , , PUNCT ap-7718 183 44 t t PROPN ap-7718 183 45 ) ) PUNCT ap-7718 183 46 = = SYM ap-7718 183 47 îa(t îa(t PROPN ap-7718 183 48 ) ) PUNCT ap-7718 184 1 + + NUM ap-7718 184 2 α α NOUN ap-7718 184 3 , , PUNCT ap-7718 184 4 d†(α d†(α VERB ap-7718 184 5 ; ; PUNCT ap-7718 184 6 t)î† t)î† NUM ap-7718 184 7 a(t)d(α a(t)d(α NOUN ap-7718 184 8 ; ; PUNCT ap-7718 184 9 t t X ap-7718 184 10 ) ) PUNCT ap-7718 184 11 = = PUNCT ap-7718 185 1 î† î† PROPN ap-7718 185 2 a(t a(t NOUN ap-7718 185 3 ) ) PUNCT ap-7718 186 1 + + NUM ap-7718 186 2 α∗. α∗. NOUN ap-7718 186 3 ( ( PUNCT ap-7718 186 4 38 38 NUM ap-7718 186 5 ) ) PUNCT ap-7718 186 6 it it PRON ap-7718 186 7 follows follow VERB ap-7718 186 8 that that SCONJ ap-7718 186 9 the the DET ap-7718 186 10 action action NOUN ap-7718 186 11 of of ADP ap-7718 186 12 the the DET ap-7718 186 13 first first ADJ ap-7718 186 14 relationship relationship NOUN ap-7718 186 15 acted act VERB ap-7718 186 16 on on ADP ap-7718 186 17 ψ0(x ψ0(x SYM ap-7718 186 18 , , PUNCT ap-7718 186 19 t t PROPN ap-7718 186 20 ) ) PUNCT ap-7718 186 21 leads lead VERB ap-7718 186 22 to to ADP ap-7718 186 23 îa(t)d(α îa(t)d(α NOUN ap-7718 186 24 , , PUNCT ap-7718 186 25 t)ψ0(x t)ψ0(x NOUN ap-7718 186 26 , , PUNCT ap-7718 186 27 t t PROPN ap-7718 186 28 ) ) PUNCT ap-7718 186 29 = = SYM ap-7718 187 1 αd(α αd(α NOUN ap-7718 187 2 , , PUNCT ap-7718 187 3 t)ψ0(x t)ψ0(x NOUN ap-7718 187 4 , , PUNCT ap-7718 187 5 t t PROPN ap-7718 187 6 ) ) PUNCT ap-7718 187 7 , , PUNCT ap-7718 187 8 from from ADP ap-7718 187 9 which which PRON ap-7718 187 10 one one PRON ap-7718 187 11 recovers recover VERB ap-7718 187 12 the the DET ap-7718 187 13 eigenvalue eigenvalue ADJ ap-7718 187 14 equation equation NOUN ap-7718 187 15 previously previously ADV ap-7718 187 16 analyzed analyze VERB ap-7718 187 17 in in ADP ap-7718 187 18 ( ( PUNCT ap-7718 187 19 31 31 NUM ap-7718 187 20 ) ) PUNCT ap-7718 187 21 by by ADP ap-7718 187 22 identifying identify VERB ap-7718 187 23 ξα(x ξα(x NOUN ap-7718 187 24 , , PUNCT ap-7718 187 25 t t PROPN ap-7718 187 26 ) ) PUNCT ap-7718 187 27 = = PUNCT ap-7718 188 1 d(α d(α NOUN ap-7718 188 2 , , PUNCT ap-7718 188 3 t)ψ0(x t)ψ0(x NOUN ap-7718 188 4 , , PUNCT ap-7718 188 5 t t PROPN ap-7718 188 6 ) ) PUNCT ap-7718 188 7 . . PUNCT ap-7718 189 1 this this PRON ap-7718 189 2 corresponds correspond VERB ap-7718 189 3 to to ADP ap-7718 189 4 the the DET ap-7718 189 5 coherent coherent ADJ ap-7718 189 6 states state NOUN ap-7718 189 7 construction construction NOUN ap-7718 189 8 of of ADP ap-7718 189 9 perelomov perelomov NOUN ap-7718 189 10 [ [ X ap-7718 189 11 39 39 NUM ap-7718 189 12 ] ] PUNCT ap-7718 189 13 . . PUNCT ap-7718 190 1 so so ADV ap-7718 190 2 far far ADV ap-7718 190 3 , , PUNCT ap-7718 190 4 two two NUM ap-7718 190 5 different different ADJ ap-7718 190 6 and and CCONJ ap-7718 190 7 equivalent equivalent ADJ ap-7718 190 8 ways way NOUN ap-7718 190 9 to to PART ap-7718 190 10 construct construct VERB ap-7718 190 11 the the DET ap-7718 190 12 solutions solution NOUN ap-7718 190 13 ξα(x ξα(x NOUN ap-7718 190 14 , , PUNCT ap-7718 190 15 t t PROPN ap-7718 190 16 ) ) PUNCT ap-7718 190 17 have have AUX ap-7718 190 18 been be AUX ap-7718 190 19 identified identify VERB ap-7718 190 20 , , PUNCT ap-7718 190 21 a a DET ap-7718 190 22 property property NOUN ap-7718 190 23 akin akin ADJ ap-7718 190 24 to to ADP ap-7718 190 25 glauber glauber PROPN ap-7718 190 26 coherent coherent ADJ ap-7718 190 27 states state NOUN ap-7718 190 28 . . PUNCT ap-7718 191 1 to to PART ap-7718 191 2 further far ADV ap-7718 191 3 explore explore VERB ap-7718 191 4 the the DET ap-7718 191 5 time time NOUN ap-7718 191 6 - - PUNCT ap-7718 191 7 dependent dependent ADJ ap-7718 191 8 coherent coherent ADJ ap-7718 191 9 states state NOUN ap-7718 191 10 , , PUNCT ap-7718 191 11 one one PRON ap-7718 191 12 can can AUX ap-7718 191 13 take take VERB ap-7718 191 14 the the DET ap-7718 191 15 unitary unitary ADJ ap-7718 191 16 transformations transformation NOUN ap-7718 191 17 ( ( PUNCT ap-7718 191 18 38 38 NUM ap-7718 191 19 ) ) PUNCT ap-7718 191 20 and and CCONJ ap-7718 191 21 combine combine VERB ap-7718 191 22 them they PRON ap-7718 191 23 with with ADP ap-7718 191 24 the the DET ap-7718 191 25 relationship relationship NOUN ap-7718 191 26 between between ADP ap-7718 191 27 the the DET ap-7718 191 28 ladder ladder NOUN ap-7718 191 29 operators operator NOUN ap-7718 191 30 and and CCONJ ap-7718 191 31 the the DET ap-7718 191 32 physical physical ADJ ap-7718 191 33 position position NOUN ap-7718 191 34 x̂ x̂ NOUN ap-7718 191 35 and and CCONJ ap-7718 191 36 momentum momentum NOUN ap-7718 191 37 p̂x p̂x NOUN ap-7718 191 38 observables observable NOUN ap-7718 191 39 presented present VERB ap-7718 191 40 in in ADP ap-7718 191 41 ( ( PUNCT ap-7718 191 42 26 26 NUM ap-7718 191 43 ) ) PUNCT ap-7718 191 44 . . PUNCT ap-7718 192 1 after after ADP ap-7718 192 2 some some DET ap-7718 192 3 calculations calculation NOUN ap-7718 192 4 one one PRON ap-7718 192 5 obtains obtain VERB ap-7718 192 6 ⟨x̂⟩α(t ⟨x̂⟩α(t NOUN ap-7718 192 7 ) ) PUNCT ap-7718 192 8 = = PUNCT ap-7718 193 1 √ √ PROPN ap-7718 193 2 2ℏ 2ℏ NUM ap-7718 193 3 m0w0 m0w0 PROPN ap-7718 193 4 σ(t σ(t PROPN ap-7718 193 5 ) ) PUNCT ap-7718 194 1 µ(t)r µ(t)r PROPN ap-7718 194 2 cos cos INTJ ap-7718 194 3 ( ( PUNCT ap-7718 194 4 w0τ(t w0τ(t PROPN ap-7718 194 5 ) ) PUNCT ap-7718 194 6 − − NUM ap-7718 194 7 θ θ X ap-7718 194 8 ) ) PUNCT ap-7718 194 9 − − NOUN ap-7718 194 10 γ(t γ(t NOUN ap-7718 194 11 ) ) PUNCT ap-7718 194 12 µ(t µ(t ADJ ap-7718 194 13 ) ) PUNCT ap-7718 194 14 , , PUNCT ap-7718 194 15 ( ( PUNCT ap-7718 194 16 39 39 NUM ap-7718 194 17 ) ) PUNCT ap-7718 194 18 where where SCONJ ap-7718 194 19 α α PROPN ap-7718 194 20 = = SYM ap-7718 194 21 reiθ reiθ PROPN ap-7718 194 22 . . PUNCT ap-7718 195 1 by by ADP ap-7718 195 2 using use VERB ap-7718 195 3 ( ( PUNCT ap-7718 195 4 19 19 NUM ap-7718 195 5 ) ) PUNCT ap-7718 195 6 and and CCONJ ap-7718 195 7 some some DET ap-7718 195 8 elementary elementary ADJ ap-7718 195 9 trigonometric trigonometric ADJ ap-7718 195 10 identities identity NOUN ap-7718 195 11 , , PUNCT ap-7718 195 12 one one PRON ap-7718 195 13 recovers recover VERB ap-7718 195 14 an an DET ap-7718 195 15 explicit explicit ADJ ap-7718 195 16 expression expression NOUN ap-7718 195 17 in in ADP ap-7718 195 18 terms term NOUN ap-7718 195 19 of of ADP ap-7718 195 20 q1(t q1(t NOUN ap-7718 195 21 ) ) PUNCT ap-7718 195 22 and and CCONJ ap-7718 195 23 q2(t q2(t NOUN ap-7718 195 24 ) ) PUNCT ap-7718 195 25 as as ADP ap-7718 195 26 ⟨x̂⟩α(t ⟨x̂⟩α(t VERB ap-7718 195 27 ) ) PUNCT ap-7718 195 28 = = SYM ap-7718 195 29 −γ(t −γ(t NOUN ap-7718 195 30 ) ) PUNCT ap-7718 195 31 µ(t µ(t ADJ ap-7718 195 32 ) ) PUNCT ap-7718 196 1 + + CCONJ ap-7718 196 2 √ √ NUM ap-7718 196 3 2ℏw0 2ℏw0 NUM ap-7718 196 4 m0c m0c PROPN ap-7718 196 5 r r NOUN ap-7718 196 6 w0 w0 PROPN ap-7718 196 7 [ [ X ap-7718 196 8 ( ( PUNCT ap-7718 196 9 cos cos PROPN ap-7718 196 10 θ θ PROPN ap-7718 196 11 + + PROPN ap-7718 196 12 w0 w0 PROPN ap-7718 196 13 2w0 2w0 PROPN ap-7718 196 14 c c NOUN ap-7718 196 15 sin sin PROPN ap-7718 196 16 θ θ PROPN ap-7718 196 17 ) ) PUNCT ap-7718 197 1 q1(t q1(t X ap-7718 197 2 ) ) PUNCT ap-7718 197 3 µ(t µ(t ADJ ap-7718 197 4 ) ) PUNCT ap-7718 197 5 + + NUM ap-7718 197 6 w0 w0 PROPN ap-7718 197 7 w0 w0 PROPN ap-7718 197 8 c c PROPN ap-7718 197 9 sin sin NOUN ap-7718 197 10 θ θ PROPN ap-7718 197 11 q2(t q2(t NOUN ap-7718 197 12 ) ) PUNCT ap-7718 197 13 µ(t µ(t ADJ ap-7718 197 14 ) ) PUNCT ap-7718 197 15 ] ] PUNCT ap-7718 197 16 . . PUNCT ap-7718 198 1 ( ( PUNCT ap-7718 198 2 40 40 NUM ap-7718 198 3 ) ) PUNCT ap-7718 198 4 similarly similarly ADV ap-7718 198 5 , , PUNCT ap-7718 198 6 the the DET ap-7718 198 7 calculations calculation NOUN ap-7718 198 8 for for ADP ap-7718 198 9 the the DET ap-7718 198 10 momentum momentum NOUN ap-7718 198 11 observable observable ADJ ap-7718 198 12 leads lead VERB ap-7718 198 13 to to PART ap-7718 198 14 ⟨p̂x⟩α(t ⟨p̂x⟩α(t VERB ap-7718 198 15 ) ) PUNCT ap-7718 198 16 = = SYM ap-7718 198 17 −m0 −m0 VERB ap-7718 198 18 µ(t µ(t ADJ ap-7718 198 19 ) ) PUNCT ap-7718 198 20 σ(t σ(t PROPN ap-7718 198 21 ) ) PUNCT ap-7718 198 22 ( ( PUNCT ap-7718 198 23 wµ(t)⟨x̂⟩α(t wµ(t)⟨x̂⟩α(t VERB ap-7718 198 24 ) ) PUNCT ap-7718 198 25 + + NOUN ap-7718 198 26 wγ(t wγ(t NOUN ap-7718 198 27 ) ) PUNCT ap-7718 198 28 ) ) PUNCT ap-7718 199 1 − − ADP ap-7718 200 1 √ √ NUM ap-7718 200 2 2m0ℏw0 2m0ℏw0 NUM ap-7718 200 3 µ(t µ(t ADJ ap-7718 200 4 ) ) PUNCT ap-7718 200 5 σ(t)r σ(t)r PROPN ap-7718 200 6 sin sin NOUN ap-7718 200 7 ( ( PUNCT ap-7718 200 8 w0τ(t w0τ(t PROPN ap-7718 200 9 ) ) PUNCT ap-7718 200 10 − − NUM ap-7718 200 11 θ θ NOUN ap-7718 200 12 ) ) PUNCT ap-7718 200 13 ) ) PUNCT ap-7718 200 14 . . PUNCT ap-7718 201 1 ( ( PUNCT ap-7718 201 2 41 41 NUM ap-7718 201 3 ) ) PUNCT ap-7718 201 4 in in ADP ap-7718 201 5 the the DET ap-7718 201 6 latter latter ADJ ap-7718 201 7 , , PUNCT ap-7718 201 8 ⟨ô⟩α(t ⟨ô⟩α(t ADJ ap-7718 201 9 ) ) PUNCT ap-7718 201 10 ≡ ≡ PROPN ap-7718 201 11 ( ( PUNCT ap-7718 201 12 ξα(x ξα(x PROPN ap-7718 201 13 , , PUNCT ap-7718 201 14 t t PROPN ap-7718 201 15 ) ) PUNCT ap-7718 201 16 , , PUNCT ap-7718 201 17 ôξα(x ôξα(x NUM ap-7718 201 18 , , PUNCT ap-7718 201 19 t t PROPN ap-7718 201 20 ) ) PUNCT ap-7718 201 21 ) ) PUNCT ap-7718 201 22 stands stand VERB ap-7718 201 23 for for ADP ap-7718 201 24 the the DET ap-7718 201 25 average average ADJ ap-7718 201 26 value value NOUN ap-7718 201 27 of of ADP ap-7718 201 28 the the DET ap-7718 201 29 observable observable ADJ ap-7718 201 30 ô ô NUM ap-7718 201 31 computed compute VERB ap-7718 201 32 through through ADP ap-7718 201 33 the the DET ap-7718 201 34 time time NOUN ap-7718 201 35 - - PUNCT ap-7718 201 36 dependent dependent ADJ ap-7718 201 37 coherent coherent ADJ ap-7718 201 38 state state NOUN ap-7718 201 39 ξα(x ξα(x PROPN ap-7718 201 40 , , PUNCT ap-7718 201 41 t t PROPN ap-7718 201 42 ) ) PUNCT ap-7718 201 43 . . PUNCT ap-7718 202 1 the the DET ap-7718 202 2 expectation expectation NOUN ap-7718 202 3 value value NOUN ap-7718 202 4 of of ADP ap-7718 202 5 the the DET ap-7718 202 6 momentum momentum NOUN ap-7718 202 7 ( ( PUNCT ap-7718 202 8 41 41 NUM ap-7718 202 9 ) ) PUNCT ap-7718 202 10 can can AUX ap-7718 202 11 be be AUX ap-7718 202 12 further far ADV ap-7718 202 13 simplified simplify VERB ap-7718 202 14 so so SCONJ ap-7718 202 15 that that SCONJ ap-7718 202 16 it it PRON ap-7718 202 17 simply simply ADV ap-7718 202 18 rewrites rewrite VERB ap-7718 202 19 as as ADP ap-7718 202 20 ⟨p̂x⟩α(t ⟨p̂x⟩α(t ADJ ap-7718 202 21 ) ) PUNCT ap-7718 202 22 = = SYM ap-7718 202 23 m(t m(t NOUN ap-7718 202 24 ) ) PUNCT ap-7718 203 1 d d NOUN ap-7718 203 2 dt dt PUNCT ap-7718 203 3 ⟨x̂⟩α(t ⟨x̂⟩α(t PROPN ap-7718 203 4 ) ) PUNCT ap-7718 203 5 , , PUNCT ap-7718 203 6 m(t m(t NOUN ap-7718 203 7 ) ) PUNCT ap-7718 203 8 = = PUNCT ap-7718 203 9 m0µ m0µ PROPN ap-7718 203 10 2(t 2(t NUM ap-7718 203 11 ) ) PUNCT ap-7718 203 12 , , PUNCT ap-7718 203 13 ( ( PUNCT ap-7718 203 14 42 42 NUM ap-7718 203 15 ) ) PUNCT ap-7718 203 16 which which PRON ap-7718 203 17 is be AUX ap-7718 203 18 an an DET ap-7718 203 19 analogous analogous ADJ ap-7718 203 20 relation relation NOUN ap-7718 203 21 to to ADP ap-7718 203 22 that that PRON ap-7718 203 23 obtained obtain VERB ap-7718 203 24 from from ADP ap-7718 203 25 the the DET ap-7718 203 26 canonical canonical ADJ ap-7718 203 27 equations equation NOUN ap-7718 203 28 of of ADP ap-7718 203 29 motion motion NOUN ap-7718 203 30 of of ADP ap-7718 203 31 the the DET ap-7718 203 32 corresponding corresponding ADJ ap-7718 203 33 classical classical ADJ ap-7718 203 34 hamiltonian hamiltonian NOUN ap-7718 203 35 . . PUNCT ap-7718 204 1 this this PRON ap-7718 204 2 is be AUX ap-7718 204 3 also also ADV ap-7718 204 4 consequence consequence NOUN ap-7718 204 5 of of ADP ap-7718 204 6 the the DET ap-7718 204 7 quadratic quadratic ADJ ap-7718 204 8 nature nature NOUN ap-7718 204 9 of of ADP ap-7718 204 10 the the DET ap-7718 204 11 time time NOUN ap-7718 204 12 - - PUNCT ap-7718 204 13 dependent dependent ADJ ap-7718 204 14 hamiltonian hamiltonian ADJ ap-7718 204 15 ĥck(t ĥck(t NOUN ap-7718 204 16 ) ) PUNCT ap-7718 204 17 and and CCONJ ap-7718 204 18 the the DET ap-7718 204 19 ehrenfest ehrenf ADJ ap-7718 204 20 theorem theorem VERB ap-7718 204 21 . . PROPN ap-7718 205 1 from from ADP ap-7718 205 2 the the DET ap-7718 205 3 expectation expectation NOUN ap-7718 205 4 values value NOUN ap-7718 205 5 obtained obtain VERB ap-7718 205 6 in in ADP ap-7718 205 7 ( ( PUNCT ap-7718 205 8 39)-(42 39)-(42 NUM ap-7718 205 9 ) ) PUNCT ap-7718 205 10 , , PUNCT ap-7718 205 11 a a DET ap-7718 205 12 relationship relationship NOUN ap-7718 205 13 between between ADP ap-7718 205 14 the the DET ap-7718 205 15 complex complex ADJ ap-7718 205 16 parameter parameter NOUN ap-7718 205 17 α α NOUN ap-7718 205 18 = = PUNCT ap-7718 205 19 reiθ reiθ PROPN ap-7718 205 20 and and CCONJ ap-7718 205 21 the the DET ap-7718 205 22 expectation expectation NOUN ap-7718 205 23 values value NOUN ap-7718 205 24 at at ADP ap-7718 205 25 a a DET ap-7718 205 26 given give VERB ap-7718 205 27 initial initial ADJ ap-7718 205 28 time time NOUN ap-7718 205 29 t t PROPN ap-7718 205 30 = = SYM ap-7718 205 31 t0 t0 PROPN ap-7718 205 32 can can AUX ap-7718 205 33 be be AUX ap-7718 205 34 established establish VERB ap-7718 205 35 . . PUNCT ap-7718 206 1 the the DET ap-7718 206 2 straightforward straightforward ADJ ap-7718 206 3 calculations calculation NOUN ap-7718 206 4 lead lead VERB ap-7718 206 5 to to ADP ap-7718 206 6 ( ( PUNCT ap-7718 206 7 re re VERB ap-7718 206 8 α̃t0 α̃t0 PROPN ap-7718 206 9 i i PROPN ap-7718 206 10 m m PROPN ap-7718 206 11 α̃t0 α̃t0 ADJ ap-7718 206 12 ) ) PUNCT ap-7718 207 1 = = SYM ap-7718 207 2 ( ( PUNCT ap-7718 207 3 √m0w0 √m0w0 PROPN ap-7718 207 4 2ℏ 2ℏ NUM ap-7718 207 5 γt0 γt0 NOUN ap-7718 207 6 σt0√ σt0√ PROPN ap-7718 207 7 m0 m0 PROPN ap-7718 207 8 2ℏw0 2ℏw0 NUM ap-7718 207 9 wγt0 wγt0 PROPN ap-7718 207 10 ) ) PUNCT ap-7718 208 1 + + CCONJ ap-7718 208 2 ( ( PUNCT ap-7718 208 3 √m0w0 √m0w0 PROPN ap-7718 208 4 2ℏ 2ℏ PROPN ap-7718 208 5 µt0 µt0 PROPN ap-7718 208 6 σt0 σt0 PROPN ap-7718 208 7 0√ 0√ PROPN ap-7718 208 8 m0 m0 PROPN ap-7718 208 9 2ℏw0 2ℏw0 NUM ap-7718 208 10 wµt0 wµt0 PROPN ap-7718 208 11 1√ 1√ PROPN ap-7718 208 12 2m0ℏw0 2m0ℏw0 NUM ap-7718 208 13 σt0 σt0 NOUN ap-7718 208 14 µt0 µt0 PROPN ap-7718 208 15 ) ) PUNCT ap-7718 208 16 ( ( PUNCT ap-7718 208 17 ⟨x̂⟩t0 ⟨x̂⟩t0 NOUN ap-7718 208 18 ⟨p̂x⟩t0 ⟨p̂x⟩t0 NUM ap-7718 208 19 ) ) PUNCT ap-7718 208 20 , , PUNCT ap-7718 208 21 ( ( PUNCT ap-7718 208 22 43 43 NUM ap-7718 208 23 ) ) PUNCT ap-7718 208 24 with with ADP ap-7718 208 25 α̃t0 α̃t0 ADJ ap-7718 208 26 = = SYM ap-7718 208 27 αe−iw0τt0 αe−iw0τt0 NOUN ap-7718 208 28 = = NOUN ap-7718 208 29 rei(θ−w0τt0 rei(θ−w0τt0 NOUN ap-7718 208 30 ) ) PUNCT ap-7718 208 31 , , PUNCT ap-7718 208 32 τt0 τt0 NOUN ap-7718 208 33 = = PUNCT ap-7718 208 34 τ(t0 τ(t0 NOUN ap-7718 208 35 ) ) PUNCT ap-7718 208 36 , , PUNCT ap-7718 208 37 σt0 σt0 NOUN ap-7718 208 38 = = SYM ap-7718 208 39 σ(t0 σ(t0 NOUN ap-7718 208 40 ) ) PUNCT ap-7718 208 41 , , PUNCT ap-7718 208 42 γt0 γt0 X ap-7718 208 43 = = SYM ap-7718 208 44 γ(t0 γ(t0 PROPN ap-7718 208 45 ) ) PUNCT ap-7718 208 46 , , PUNCT ap-7718 208 47 wγt0 wγt0 NOUN ap-7718 208 48 = = SYM ap-7718 208 49 wγ(t0 wγ(t0 PROPN ap-7718 208 50 ) ) PUNCT ap-7718 208 51 , , PUNCT ap-7718 208 52 wµt0 wµt0 PROPN ap-7718 208 53 = = SYM ap-7718 208 54 wµ(t0 wµ(t0 PROPN ap-7718 208 55 ) ) PUNCT ap-7718 208 56 , , PUNCT ap-7718 208 57 ⟨x̂⟩t0 ⟨x̂⟩t0 NOUN ap-7718 208 58 = = SYM ap-7718 208 59 ⟨x̂⟩α(t0 ⟨x̂⟩α(t0 PROPN ap-7718 208 60 ) ) PUNCT ap-7718 208 61 , , PUNCT ap-7718 208 62 and and CCONJ ap-7718 208 63 ⟨p̂x⟩t0 ⟨p̂x⟩t0 X ap-7718 208 64 = = SYM ap-7718 208 65 ⟨p̂x⟩α(t0 ⟨p̂x⟩α(t0 NOUN ap-7718 208 66 ) ) PUNCT ap-7718 208 67 . . PUNCT ap-7718 209 1 on on ADP ap-7718 209 2 the the DET ap-7718 209 3 other other ADJ ap-7718 209 4 hand hand NOUN ap-7718 209 5 , , PUNCT ap-7718 209 6 one one PRON ap-7718 209 7 can can AUX ap-7718 209 8 write write VERB ap-7718 209 9 the the DET ap-7718 209 10 probability probability NOUN ap-7718 209 11 density density NOUN ap-7718 209 12 associated associate VERB ap-7718 209 13 with with ADP ap-7718 209 14 the the DET ap-7718 209 15 time time NOUN ap-7718 209 16 - - PUNCT ap-7718 209 17 dependent dependent ADJ ap-7718 209 18 coherent coherent ADJ ap-7718 209 19 state state NOUN ap-7718 209 20 in in ADP ap-7718 209 21 terms term NOUN ap-7718 209 22 of of ADP ap-7718 209 23 ⟨x̂⟩α(t ⟨x̂⟩α(t NOUN ap-7718 209 24 ) ) PUNCT ap-7718 209 25 through through ADP ap-7718 209 26 pα(x pα(x NUM ap-7718 209 27 , , PUNCT ap-7718 209 28 t t PROPN ap-7718 209 29 ) ) PUNCT ap-7718 209 30 : : PUNCT ap-7718 210 1 = = SYM ap-7718 210 2 |ξα(x |ξα(x PROPN ap-7718 210 3 , , PUNCT ap-7718 210 4 t)|2 t)|2 NOUN ap-7718 210 5 = = SYM ap-7718 210 6 √ √ PROPN ap-7718 210 7 m0w0 m0w0 PROPN ap-7718 210 8 πℏ πℏ NOUN ap-7718 210 9 µ(t µ(t ADJ ap-7718 210 10 ) ) PUNCT ap-7718 210 11 σ(t σ(t PROPN ap-7718 210 12 ) ) PUNCT ap-7718 210 13 × × NOUN ap-7718 210 14 exp exp NOUN ap-7718 210 15 [ [ PUNCT ap-7718 210 16 −m0w0 −m0w0 PROPN ap-7718 210 17 2ℏ 2ℏ NUM ap-7718 210 18 µ2(t µ2(t NOUN ap-7718 210 19 ) ) PUNCT ap-7718 210 20 σ2(t σ2(t PROPN ap-7718 210 21 ) ) PUNCT ap-7718 210 22 ( ( PUNCT ap-7718 210 23 x− x− PROPN ap-7718 210 24 ⟨x̂⟩α(t))2 ⟨x̂⟩α(t))2 PROPN ap-7718 210 25 ] ] PUNCT ap-7718 210 26 , , PUNCT ap-7718 210 27 ( ( PUNCT ap-7718 210 28 44 44 NUM ap-7718 210 29 ) ) PUNCT ap-7718 210 30 which which PRON ap-7718 210 31 is be AUX ap-7718 210 32 a a DET ap-7718 210 33 gaussian gaussian ADJ ap-7718 210 34 wavepacket wavepacket NOUN ap-7718 210 35 whose whose DET ap-7718 210 36 maximum maximum NOUN ap-7718 210 37 follows follow VERB ap-7718 210 38 the the DET ap-7718 210 39 classical classical ADJ ap-7718 210 40 trajectory trajectory NOUN ap-7718 210 41 . . PUNCT ap-7718 211 1 that that PRON ap-7718 211 2 is is ADV ap-7718 211 3 , , PUNCT ap-7718 211 4 the the DET ap-7718 211 5 timedependent timedependent NOUN ap-7718 211 6 coherent coherent ADJ ap-7718 211 7 state state NOUN ap-7718 211 8 is be AUX ap-7718 211 9 considered consider VERB ap-7718 211 10 as as ADP ap-7718 211 11 a a DET ap-7718 211 12 semiclassical semiclassical ADJ ap-7718 211 13 state state NOUN ap-7718 211 14 . . PUNCT ap-7718 212 1 before before ADP ap-7718 212 2 concluding conclude VERB ap-7718 212 3 this this DET ap-7718 212 4 section section NOUN ap-7718 212 5 , , PUNCT ap-7718 212 6 it it PRON ap-7718 212 7 is be AUX ap-7718 212 8 worth worth ADJ ap-7718 212 9 exploring explore VERB ap-7718 212 10 the the DET ap-7718 212 11 corresponding correspond VERB ap-7718 212 12 uncertainty uncertainty NOUN ap-7718 212 13 relations relation NOUN ap-7718 212 14 associated associate VERB ap-7718 212 15 with with ADP ap-7718 212 16 the the DET ap-7718 212 17 canonical canonical ADJ ap-7718 212 18 observables observable NOUN ap-7718 212 19 , , PUNCT ap-7718 212 20 which which PRON ap-7718 212 21 can can AUX ap-7718 212 22 be be AUX ap-7718 212 23 computed compute VERB ap-7718 212 24 by by ADP ap-7718 212 25 using use VERB ap-7718 212 26 ( ( PUNCT ap-7718 212 27 26 26 NUM ap-7718 212 28 ) ) PUNCT ap-7718 212 29 , , PUNCT ap-7718 212 30 ( ( PUNCT ap-7718 212 31 31 31 NUM ap-7718 212 32 ) ) PUNCT ap-7718 212 33 , , PUNCT ap-7718 212 34 and and CCONJ ap-7718 212 35 ( ( PUNCT ap-7718 212 36 38 38 NUM ap-7718 212 37 ) ) PUNCT ap-7718 212 38 . . PUNCT ap-7718 213 1 after after ADP ap-7718 213 2 some some DET ap-7718 213 3 calculations calculation NOUN ap-7718 213 4 one one PRON ap-7718 213 5 gets get VERB ap-7718 213 6 ( ( PUNCT ap-7718 213 7 ∆x̂)2 ∆x̂)2 X ap-7718 213 8 α α NOUN ap-7718 213 9 = = SYM ap-7718 213 10 ℏ ℏ PROPN ap-7718 213 11 2m0w0 2m0w0 NUM ap-7718 213 12 σ2(t σ2(t NOUN ap-7718 213 13 ) ) PUNCT ap-7718 213 14 µ2(t µ2(t PROPN ap-7718 213 15 ) ) PUNCT ap-7718 213 16 , , PUNCT ap-7718 213 17 ( ( PUNCT ap-7718 213 18 ∆p̂x)2 ∆p̂x)2 VERB ap-7718 213 19 α α X ap-7718 213 20 = = PUNCT ap-7718 213 21 m0ℏw0 m0ℏw0 VERB ap-7718 213 22 2 2 NUM ap-7718 213 23 µ2(t µ2(t NOUN ap-7718 213 24 ) ) PUNCT ap-7718 213 25 σ2(t σ2(t PROPN ap-7718 213 26 ) ) PUNCT ap-7718 213 27 ( ( PUNCT ap-7718 213 28 1 1 NUM ap-7718 213 29 + + NUM ap-7718 213 30 σ2(t)w σ2(t)w NOUN ap-7718 213 31 2 2 NUM ap-7718 213 32 µ(t µ(t ADJ ap-7718 213 33 ) ) PUNCT ap-7718 213 34 w2 w2 NOUN ap-7718 213 35 0µ 0µ NOUN ap-7718 213 36 2(t 2(t NUM ap-7718 213 37 ) ) PUNCT ap-7718 213 38 ) ) PUNCT ap-7718 213 39 , , PUNCT ap-7718 213 40 ( ( PUNCT ap-7718 213 41 45 45 NUM ap-7718 213 42 ) ) PUNCT ap-7718 213 43 215 215 NUM ap-7718 213 44 kevin kevin PROPN ap-7718 213 45 zelaya zelaya PROPN ap-7718 213 46 acta acta PROPN ap-7718 213 47 polytechnica polytechnica PROPN ap-7718 213 48 from from ADP ap-7718 213 49 which which PRON ap-7718 213 50 the the DET ap-7718 213 51 uncertainty uncertainty NOUN ap-7718 213 52 relation relation NOUN ap-7718 213 53 reduces reduce VERB ap-7718 213 54 to to ADP ap-7718 213 55 ( ( PUNCT ap-7718 213 56 ∆x̂)2 ∆x̂)2 ADP ap-7718 213 57 α α PROPN ap-7718 213 58 ( ( PUNCT ap-7718 213 59 ∆p̂x)2 ∆p̂x)2 NOUN ap-7718 213 60 α α NOUN ap-7718 214 1 = = PUNCT ap-7718 214 2 ℏ2 ℏ2 NOUN ap-7718 214 3 4 4 NUM ap-7718 214 4 ( ( PUNCT ap-7718 214 5 1 1 NUM ap-7718 214 6 + + NUM ap-7718 214 7 σ2(t)w σ2(t)w NOUN ap-7718 214 8 2 2 NUM ap-7718 214 9 µ(t µ(t ADJ ap-7718 214 10 ) ) PUNCT ap-7718 214 11 w2 w2 NOUN ap-7718 214 12 0µ 0µ NOUN ap-7718 214 13 2(t 2(t NUM ap-7718 214 14 ) ) PUNCT ap-7718 214 15 ) ) PUNCT ap-7718 214 16 , , PUNCT ap-7718 214 17 ( ( PUNCT ap-7718 214 18 46 46 NUM ap-7718 214 19 ) ) PUNCT ap-7718 214 20 where where SCONJ ap-7718 214 21 it it PRON ap-7718 214 22 is be AUX ap-7718 214 23 clear clear ADJ ap-7718 214 24 that that SCONJ ap-7718 214 25 , , PUNCT ap-7718 214 26 in in ADP ap-7718 214 27 general general ADJ ap-7718 214 28 , , PUNCT ap-7718 214 29 ξα(x ξα(x NUM ap-7718 214 30 , , PUNCT ap-7718 214 31 t t PROPN ap-7718 214 32 ) ) PUNCT ap-7718 214 33 does do AUX ap-7718 214 34 not not PART ap-7718 214 35 minimize minimize VERB ap-7718 214 36 the the DET ap-7718 214 37 heisenberg heisenberg PROPN ap-7718 214 38 uncertainty uncertainty PROPN ap-7718 214 39 relationship relationship NOUN ap-7718 214 40 , , PUNCT ap-7718 214 41 except except SCONJ ap-7718 214 42 for for ADP ap-7718 214 43 those those DET ap-7718 214 44 times time NOUN ap-7718 214 45 t′ t′ NUM ap-7718 214 46 at at ADP ap-7718 214 47 which which PRON ap-7718 214 48 wµ(t′ wµ(t′ NOUN ap-7718 214 49 ) ) PUNCT ap-7718 214 50 = = SYM ap-7718 214 51 0 0 X ap-7718 214 52 . . PUNCT ap-7718 215 1 the the DET ap-7718 215 2 latter latter NOUN ap-7718 215 3 follows follow VERB ap-7718 215 4 from from ADP ap-7718 215 5 the the DET ap-7718 215 6 fact fact NOUN ap-7718 215 7 that that SCONJ ap-7718 215 8 σ σ NOUN ap-7718 215 9 ̸= ̸= PROPN ap-7718 215 10 0 0 NUM ap-7718 215 11 for for ADP ap-7718 215 12 t t PROPN ap-7718 215 13 ∈ ∈ PROPN ap-7718 215 14 r. r. PROPN ap-7718 215 15 still still ADV ap-7718 215 16 , , PUNCT ap-7718 215 17 there there PRON ap-7718 215 18 are be VERB ap-7718 215 19 two two NUM ap-7718 215 20 special special ADJ ap-7718 215 21 cases case NOUN ap-7718 215 22 for for ADP ap-7718 215 23 which which PRON ap-7718 215 24 eq eq ADP ap-7718 215 25 . . PUNCT ap-7718 216 1 ( ( PUNCT ap-7718 216 2 46 46 NUM ap-7718 216 3 ) ) PUNCT ap-7718 216 4 minimizes minimize VERB ap-7718 216 5 at at ADP ap-7718 216 6 all all DET ap-7718 216 7 times time NOUN ap-7718 216 8 . . PUNCT ap-7718 217 1 • • NOUN ap-7718 217 2 for for ADP ap-7718 217 3 µ(t µ(t ADJ ap-7718 217 4 ) ) PUNCT ap-7718 217 5 = = SYM ap-7718 217 6 µ0 µ0 NOUN ap-7718 217 7 and and CCONJ ap-7718 217 8 ω(t ω(t NOUN ap-7718 217 9 ) ) PUNCT ap-7718 217 10 = = SYM ap-7718 217 11 w1 w1 NOUN ap-7718 218 1 , , PUNCT ap-7718 218 2 one one PRON ap-7718 218 3 can can AUX ap-7718 218 4 always always ADV ap-7718 218 5 find find VERB ap-7718 218 6 a a DET ap-7718 218 7 constant constant ADJ ap-7718 218 8 solution solution NOUN ap-7718 218 9 σ4(t σ4(t NUM ap-7718 218 10 ) ) PUNCT ap-7718 218 11 = = PROPN ap-7718 218 12 w2 w2 NOUN ap-7718 218 13 0 0 NUM ap-7718 218 14 / / SYM ap-7718 218 15 w w PROPN ap-7718 218 16 2 2 NUM ap-7718 218 17 1 1 NUM ap-7718 218 18 so so SCONJ ap-7718 218 19 that that SCONJ ap-7718 218 20 wµ wµ ADP ap-7718 218 21 = = SYM ap-7718 218 22 0 0 PROPN ap-7718 218 23 . . PUNCT ap-7718 219 1 the the DET ap-7718 219 2 uncertainty uncertainty NOUN ap-7718 219 3 relationship relationship NOUN ap-7718 219 4 ( ( PUNCT ap-7718 219 5 46 46 NUM ap-7718 219 6 ) ) PUNCT ap-7718 219 7 is be AUX ap-7718 219 8 minimized minimize VERB ap-7718 219 9 , , PUNCT ap-7718 219 10 and and CCONJ ap-7718 219 11 the the DET ap-7718 219 12 time time NOUN ap-7718 219 13 - - PUNCT ap-7718 219 14 dependent dependent ADJ ap-7718 219 15 hamiltonian hamiltonian NOUN ap-7718 219 16 becomes become VERB ap-7718 219 17 ĥck(t ĥck(t NOUN ap-7718 219 18 ) ) PUNCT ap-7718 219 19 = = PUNCT ap-7718 220 1 p̂2 p̂2 NOUN ap-7718 220 2 x x X ap-7718 220 3 2m0µ2 2m0µ2 NUM ap-7718 220 4 0 0 NUM ap-7718 221 1 + + CCONJ ap-7718 221 2 m0µ m0µ PROPN ap-7718 221 3 2 2 NUM ap-7718 221 4 0w 0w NOUN ap-7718 221 5 2 2 NUM ap-7718 221 6 1 1 NUM ap-7718 221 7 2 2 NUM ap-7718 221 8 x̂2 x̂2 NOUN ap-7718 222 1 + + CCONJ ap-7718 222 2 f f X ap-7718 222 3 ( ( PUNCT ap-7718 222 4 t)x̂ t)x̂ PROPN ap-7718 222 5 , , PUNCT ap-7718 222 6 ( ( PUNCT ap-7718 222 7 47 47 NUM ap-7718 222 8 ) ) PUNCT ap-7718 222 9 which which PRON ap-7718 222 10 is be AUX ap-7718 222 11 nothing nothing PRON ap-7718 222 12 but but SCONJ ap-7718 222 13 a a DET ap-7718 222 14 stationary stationary ADJ ap-7718 222 15 oscillator oscillator NOUN ap-7718 222 16 with with ADP ap-7718 222 17 an an DET ap-7718 222 18 external external ADJ ap-7718 222 19 time time NOUN ap-7718 222 20 - - PUNCT ap-7718 222 21 dependent dependent ADJ ap-7718 222 22 driving driving NOUN ap-7718 222 23 force force NOUN ap-7718 222 24 f f PROPN ap-7718 222 25 ( ( PUNCT ap-7718 222 26 t)2 t)2 PROPN ap-7718 222 27 . . PUNCT ap-7718 222 28 thus thus ADV ap-7718 222 29 , , PUNCT ap-7718 222 30 the the DET ap-7718 222 31 uncertainty uncertainty NOUN ap-7718 222 32 relation relation NOUN ap-7718 222 33 gets get AUX ap-7718 222 34 minimized minimize VERB ap-7718 222 35 in in ADP ap-7718 222 36 the the DET ap-7718 222 37 stationary stationary ADJ ap-7718 222 38 limit limit NOUN ap-7718 222 39 , , PUNCT ap-7718 222 40 as as SCONJ ap-7718 222 41 expected expect VERB ap-7718 222 42 . . PUNCT ap-7718 223 1 • • INTJ ap-7718 223 2 for for ADP ap-7718 223 3 ω2(t ω2(t SYM ap-7718 223 4 ) ) PUNCT ap-7718 223 5 = = NOUN ap-7718 223 6 w2 w2 NOUN ap-7718 223 7 0µ 0µ NOUN ap-7718 223 8 −4(t −4(t VERB ap-7718 223 9 ) ) PUNCT ap-7718 223 10 , , PUNCT ap-7718 223 11 there there PRON ap-7718 223 12 is be VERB ap-7718 223 13 a a DET ap-7718 223 14 solution solution NOUN ap-7718 223 15 σ(t σ(t NOUN ap-7718 223 16 ) ) PUNCT ap-7718 224 1 = = SYM ap-7718 224 2 µ(t µ(t ADJ ap-7718 224 3 ) ) PUNCT ap-7718 224 4 for for ADP ap-7718 224 5 which which PRON ap-7718 224 6 wµ wµ ADP ap-7718 224 7 = = NOUN ap-7718 224 8 0 0 PROPN ap-7718 224 9 . . PUNCT ap-7718 225 1 this this PRON ap-7718 225 2 leads lead VERB ap-7718 225 3 to to ADP ap-7718 225 4 a a DET ap-7718 225 5 hamiltonian hamiltonian NOUN ap-7718 225 6 of of ADP ap-7718 225 7 the the DET ap-7718 225 8 form form NOUN ap-7718 225 9 ĥck(t ĥck(t NOUN ap-7718 225 10 ) ) PUNCT ap-7718 225 11 = = SYM ap-7718 225 12 1 1 NUM ap-7718 225 13 µ2(t µ2(t NOUN ap-7718 225 14 ) ) PUNCT ap-7718 225 15 ( ( PUNCT ap-7718 225 16 p̂2 p̂2 NOUN ap-7718 225 17 x x X ap-7718 225 18 2m0 2m0 NUM ap-7718 226 1 + + CCONJ ap-7718 226 2 m0w m0w X ap-7718 226 3 2 2 NUM ap-7718 226 4 0 0 NUM ap-7718 226 5 2 2 NUM ap-7718 226 6 x̂2 x̂2 NOUN ap-7718 226 7 + + CCONJ ap-7718 226 8 µ2(t)f µ2(t)f SYM ap-7718 226 9 ( ( PUNCT ap-7718 226 10 t)x̂ t)x̂ PROPN ap-7718 226 11 ) ) PUNCT ap-7718 226 12 . . PUNCT ap-7718 227 1 ( ( PUNCT ap-7718 227 2 48 48 NUM ap-7718 227 3 ) ) PUNCT ap-7718 227 4 although although SCONJ ap-7718 227 5 the the DET ap-7718 227 6 solutions solution NOUN ap-7718 227 7 ξα(x ξα(x NOUN ap-7718 227 8 , , PUNCT ap-7718 227 9 t t PROPN ap-7718 227 10 ) ) PUNCT ap-7718 227 11 minimize minimize VERB ap-7718 227 12 the the DET ap-7718 227 13 heisenberg heisenberg PROPN ap-7718 227 14 uncertainty uncertainty NOUN ap-7718 227 15 relation relation NOUN ap-7718 227 16 only only ADV ap-7718 227 17 on on ADP ap-7718 227 18 some some DET ap-7718 227 19 restricted restricted ADJ ap-7718 227 20 cases case NOUN ap-7718 227 21 , , PUNCT ap-7718 227 22 one one PRON ap-7718 227 23 can can AUX ap-7718 227 24 still still ADV ap-7718 227 25 explore explore VERB ap-7718 227 26 the the DET ap-7718 227 27 schrödinger schrödinger ADJ ap-7718 227 28 - - PUNCT ap-7718 227 29 robertson robertson PROPN ap-7718 227 30 inequality inequality NOUN ap-7718 227 31 [ [ X ap-7718 227 32 40 40 NUM ap-7718 227 33 , , PUNCT ap-7718 227 34 41 41 NUM ap-7718 227 35 ] ] PUNCT ap-7718 227 36 . . PUNCT ap-7718 228 1 this this PRON ap-7718 228 2 is be AUX ap-7718 228 3 defined define VERB ap-7718 228 4 for for ADP ap-7718 228 5 a a DET ap-7718 228 6 pair pair NOUN ap-7718 228 7 of of ADP ap-7718 228 8 observables observable NOUN ap-7718 228 9 â â VERB ap-7718 228 10 and and CCONJ ap-7718 228 11 b̂ b̂ ADV ap-7718 228 12 through through ADV ap-7718 228 13 ( ( PUNCT ap-7718 228 14 ∆â ∆â NOUN ap-7718 228 15 ) ) SYM ap-7718 228 16 2 2 NUM ap-7718 228 17 ( ( PUNCT ap-7718 228 18 ∆b̂ ∆b̂ NOUN ap-7718 228 19 ) ) PUNCT ap-7718 228 20 2 2 NUM ap-7718 228 21 ≥ ≥ NOUN ap-7718 228 22 |⟨[â |⟨[â NOUN ap-7718 228 23 , , PUNCT ap-7718 228 24 b̂]⟩|2 b̂]⟩|2 NOUN ap-7718 228 25 4 4 NUM ap-7718 228 26 + + NOUN ap-7718 229 1 σ2 σ2 PROPN ap-7718 229 2 a a DET ap-7718 229 3 , , PUNCT ap-7718 229 4 b b NOUN ap-7718 229 5 , , PUNCT ap-7718 229 6 ( ( PUNCT ap-7718 229 7 49 49 NUM ap-7718 229 8 ) ) PUNCT ap-7718 229 9 where where SCONJ ap-7718 229 10 σa σa PROPN ap-7718 229 11 , , PUNCT ap-7718 229 12 b b NOUN ap-7718 229 13 : : PUNCT ap-7718 229 14 = = SYM ap-7718 229 15 1 1 NUM ap-7718 229 16 2 2 NUM ap-7718 229 17 ⟨âb̂ ⟨âb̂ NOUN ap-7718 230 1 + + CCONJ ap-7718 230 2 b̂â⟩ b̂â⟩ PROPN ap-7718 230 3 − − PROPN ap-7718 230 4 ⟨â⟩⟨b̂⟩ ⟨â⟩⟨b̂⟩ PROPN ap-7718 230 5 stands stand VERB ap-7718 230 6 for for ADP ap-7718 230 7 the the DET ap-7718 230 8 correlation correlation NOUN ap-7718 230 9 function function NOUN ap-7718 230 10 . . PUNCT ap-7718 231 1 for for ADP ap-7718 231 2 the the DET ap-7718 231 3 canonical canonical ADJ ap-7718 231 4 position position NOUN ap-7718 231 5 x̂ x̂ NOUN ap-7718 231 6 and and CCONJ ap-7718 231 7 momentum momentum NOUN ap-7718 231 8 p̂x p̂x NOUN ap-7718 231 9 observables observable VERB ap-7718 231 10 one one NUM ap-7718 231 11 gets get VERB ap-7718 231 12 σ2 σ2 NOUN ap-7718 231 13 x x X ap-7718 231 14 , , PUNCT ap-7718 231 15 px px PROPN ap-7718 231 16 = = PUNCT ap-7718 231 17 ℏ2 ℏ2 NOUN ap-7718 231 18 4w2 4w2 NOUN ap-7718 231 19 0 0 NUM ap-7718 232 1 σ2(t)w σ2(t)w SYM ap-7718 232 2 2 2 NUM ap-7718 232 3 µ(t µ(t ADJ ap-7718 232 4 ) ) PUNCT ap-7718 232 5 µ2(t µ2(t NOUN ap-7718 232 6 ) ) PUNCT ap-7718 232 7 , , PUNCT ap-7718 232 8 ( ( PUNCT ap-7718 232 9 50 50 NUM ap-7718 232 10 ) ) PUNCT ap-7718 232 11 when when SCONJ ap-7718 232 12 computed compute VERB ap-7718 232 13 through through ADP ap-7718 232 14 ξα(x ξα(x NUM ap-7718 232 15 , , PUNCT ap-7718 232 16 t t PROPN ap-7718 232 17 ) ) PUNCT ap-7718 232 18 . . PUNCT ap-7718 233 1 thus thus ADV ap-7718 233 2 , , PUNCT ap-7718 233 3 the the DET ap-7718 233 4 semiclassical semiclassical ADJ ap-7718 233 5 states state NOUN ap-7718 233 6 ξα(x ξα(x NOUN ap-7718 233 7 , , PUNCT ap-7718 233 8 t t PROPN ap-7718 233 9 ) ) PUNCT ap-7718 233 10 minimize minimize VERB ap-7718 233 11 the the DET ap-7718 233 12 schrödinger schrödinger ADJ ap-7718 233 13 - - PUNCT ap-7718 233 14 robertson robertson PROPN ap-7718 233 15 relationship relationship NOUN ap-7718 233 16 for for ADP ap-7718 233 17 t t PROPN ap-7718 233 18 ∈ ∈ PROPN ap-7718 233 19 r. r. PROPN ap-7718 233 20 4 4 PROPN ap-7718 233 21 . . PUNCT ap-7718 234 1 discussion discussion NOUN ap-7718 234 2 : : PUNCT ap-7718 234 3 conventional conventional ADJ ap-7718 234 4 and and CCONJ ap-7718 234 5 regularized regularize VERB ap-7718 234 6 caldirola caldirola PROPN ap-7718 234 7 - - PUNCT ap-7718 234 8 kanai kanai PROPN ap-7718 234 9 oscillators oscillator NOUN ap-7718 234 10 so so ADV ap-7718 234 11 far far ADV ap-7718 234 12 , , PUNCT ap-7718 234 13 the the DET ap-7718 234 14 most most ADV ap-7718 234 15 general general ADJ ap-7718 234 16 setup setup NOUN ap-7718 234 17 has have AUX ap-7718 234 18 been be AUX ap-7718 234 19 addressed address VERB ap-7718 234 20 for for ADP ap-7718 234 21 a a DET ap-7718 234 22 time time NOUN ap-7718 234 23 - - PUNCT ap-7718 234 24 dependent dependent ADJ ap-7718 234 25 mass mass NOUN ap-7718 234 26 oscillator oscillator NOUN ap-7718 234 27 . . PUNCT ap-7718 235 1 two two NUM ap-7718 235 2 particular particular ADJ ap-7718 235 3 2the 2the PROPN ap-7718 235 4 hamiltonian hamiltonian NOUN ap-7718 235 5 ( ( PUNCT ap-7718 235 6 47 47 NUM ap-7718 235 7 ) ) PUNCT ap-7718 235 8 is be AUX ap-7718 235 9 essentially essentially ADV ap-7718 235 10 stationary stationary ADJ ap-7718 235 11 , , PUNCT ap-7718 235 12 for for ADP ap-7718 235 13 the the DET ap-7718 235 14 term term NOUN ap-7718 235 15 f f PROPN ap-7718 235 16 ( ( PUNCT ap-7718 235 17 t t PROPN ap-7718 235 18 ) ) PUNCT ap-7718 235 19 can can AUX ap-7718 235 20 be be AUX ap-7718 235 21 absorbed absorb VERB ap-7718 235 22 through through ADP ap-7718 235 23 an an DET ap-7718 235 24 appropriate appropriate ADJ ap-7718 235 25 reparametrization reparametrization NOUN ap-7718 235 26 of of ADP ap-7718 235 27 the the DET ap-7718 235 28 canonical canonical ADJ ap-7718 235 29 coordinate coordinate NOUN ap-7718 235 30 . . PUNCT ap-7718 236 1 examples example NOUN ap-7718 236 2 are be AUX ap-7718 236 3 considered consider VERB ap-7718 236 4 in in ADP ap-7718 236 5 this this DET ap-7718 236 6 section section NOUN ap-7718 236 7 to to PART ap-7718 236 8 further far ADV ap-7718 236 9 illustrate illustrate VERB ap-7718 236 10 the the DET ap-7718 236 11 usefulness usefulness NOUN ap-7718 236 12 and and CCONJ ap-7718 236 13 behavior behavior NOUN ap-7718 236 14 of of ADP ap-7718 236 15 the the DET ap-7718 236 16 so so ADV ap-7718 236 17 - - PUNCT ap-7718 236 18 constructed construct VERB ap-7718 236 19 solutions solution NOUN ap-7718 236 20 and and CCONJ ap-7718 236 21 coherent coherent ADJ ap-7718 236 22 states state NOUN ap-7718 236 23 . . PUNCT ap-7718 237 1 henceforth henceforth ADV ap-7718 237 2 , , PUNCT ap-7718 237 3 all all DET ap-7718 237 4 calculations calculation NOUN ap-7718 237 5 are be AUX ap-7718 237 6 carried carry VERB ap-7718 237 7 on on ADP ap-7718 237 8 by by ADP ap-7718 237 9 working work VERB ap-7718 237 10 in in ADP ap-7718 237 11 units unit NOUN ap-7718 237 12 of of ADP ap-7718 237 13 ℏ ℏ NOUN ap-7718 237 14 = = SYM ap-7718 237 15 1 1 NUM ap-7718 237 16 to to PART ap-7718 237 17 simplify simplify VERB ap-7718 237 18 the the DET ap-7718 237 19 ongoing ongoing ADJ ap-7718 237 20 discussion discussion NOUN ap-7718 237 21 . . PUNCT ap-7718 238 1 throughout throughout ADP ap-7718 238 2 the the DET ap-7718 238 3 rest rest NOUN ap-7718 238 4 of of ADP ap-7718 238 5 this this DET ap-7718 238 6 manuscript manuscript NOUN ap-7718 238 7 , , PUNCT ap-7718 238 8 the the DET ap-7718 238 9 following follow VERB ap-7718 238 10 two two NUM ap-7718 238 11 time time NOUN ap-7718 238 12 - - PUNCT ap-7718 238 13 dependent dependent ADJ ap-7718 238 14 masses masse NOUN ap-7718 238 15 are be AUX ap-7718 238 16 considered consider VERB ap-7718 238 17 : : PUNCT ap-7718 238 18 µck(t µck(t NUM ap-7718 238 19 ) ) PUNCT ap-7718 238 20 = = SYM ap-7718 238 21 e−κt e−κt PROPN ap-7718 238 22 , , PUNCT ap-7718 238 23 κ κ X ap-7718 238 24 ≥ ≥ NOUN ap-7718 238 25 0 0 NUM ap-7718 238 26 , , PUNCT ap-7718 238 27 ( ( PUNCT ap-7718 238 28 51a 51a NOUN ap-7718 238 29 ) ) PUNCT ap-7718 238 30 µrck(t µrck(t NOUN ap-7718 238 31 ) ) PUNCT ap-7718 238 32 = = SYM ap-7718 238 33 e−κt e−κt NOUN ap-7718 238 34 + + CCONJ ap-7718 238 35 µ0 µ0 PROPN ap-7718 238 36 , , PUNCT ap-7718 238 37 κ κ NOUN ap-7718 238 38 , , PUNCT ap-7718 238 39 µ0 µ0 NOUN ap-7718 238 40 ≥ ≥ NOUN ap-7718 238 41 0 0 NUM ap-7718 238 42 . . PUNCT ap-7718 238 43 ( ( PUNCT ap-7718 238 44 51b 51b NOUN ap-7718 238 45 ) ) PUNCT ap-7718 238 46 the the DET ap-7718 238 47 first first ADJ ap-7718 238 48 one one NUM ap-7718 238 49 corresponds correspond NOUN ap-7718 238 50 to to ADP ap-7718 238 51 the the DET ap-7718 238 52 well well ADV ap-7718 238 53 - - PUNCT ap-7718 238 54 known know VERB ap-7718 238 55 caldirola caldirola PROPN ap-7718 238 56 - - PUNCT ap-7718 238 57 kanai kanai PROPN ap-7718 238 58 oscillator oscillator PROPN ap-7718 238 59 [ [ X ap-7718 238 60 42 42 NUM ap-7718 238 61 , , PUNCT ap-7718 238 62 43 43 NUM ap-7718 238 63 ] ] PUNCT ap-7718 238 64 , , PUNCT ap-7718 238 65 which which PRON ap-7718 238 66 contains contain VERB ap-7718 238 67 a a DET ap-7718 238 68 mass mass ADJ ap-7718 238 69 - - PUNCT ap-7718 238 70 term term NOUN ap-7718 238 71 that that PRON ap-7718 238 72 asymptotically asymptotically ADV ap-7718 238 73 approaches approach VERB ap-7718 238 74 to to ADP ap-7718 238 75 zero zero NUM ap-7718 238 76 . . PUNCT ap-7718 239 1 this this PRON ap-7718 239 2 is be AUX ap-7718 239 3 a a DET ap-7718 239 4 rather rather ADV ap-7718 239 5 unrealistic unrealistic ADJ ap-7718 239 6 scenario scenario NOUN ap-7718 239 7 in in ADP ap-7718 239 8 the the DET ap-7718 239 9 context context NOUN ap-7718 239 10 of of ADP ap-7718 239 11 the the DET ap-7718 239 12 schrödinger schrödinger ADJ ap-7718 239 13 equation equation NOUN ap-7718 239 14 . . PUNCT ap-7718 240 1 still still ADV ap-7718 240 2 , , PUNCT ap-7718 240 3 one one PRON ap-7718 240 4 can can AUX ap-7718 240 5 study study VERB ap-7718 240 6 the the DET ap-7718 240 7 dynamics dynamic NOUN ap-7718 240 8 on on ADP ap-7718 240 9 a a DET ap-7718 240 10 given give VERB ap-7718 240 11 time time NOUN ap-7718 240 12 range range NOUN ap-7718 240 13 , , PUNCT ap-7718 240 14 let let VERB ap-7718 240 15 say say VERB ap-7718 240 16 t t PROPN ap-7718 240 17 ∈ ∈ PROPN ap-7718 241 1 [ [ X ap-7718 241 2 0 0 NUM ap-7718 241 3 , , PUNCT ap-7718 241 4 t t X ap-7718 241 5 ] ] PUNCT ap-7718 241 6 , , PUNCT ap-7718 241 7 where where SCONJ ap-7718 241 8 t t PROPN ap-7718 241 9 denotes denote VERB ap-7718 241 10 the the DET ap-7718 241 11 time time NOUN ap-7718 241 12 spent spend VERB ap-7718 241 13 by by ADP ap-7718 241 14 the the DET ap-7718 241 15 mass mass NOUN ap-7718 241 16 to to PART ap-7718 241 17 reduce reduce VERB ap-7718 241 18 its its PRON ap-7718 241 19 initial initial ADJ ap-7718 241 20 value value NOUN ap-7718 241 21 in in ADP ap-7718 241 22 a a DET ap-7718 241 23 factor factor NOUN ap-7718 241 24 e−1 e−1 PROPN ap-7718 241 25 . . PUNCT ap-7718 242 1 in in ADP ap-7718 242 2 other other ADJ ap-7718 242 3 words word NOUN ap-7718 242 4 , , PUNCT ap-7718 242 5 t t PROPN ap-7718 242 6 = = SYM ap-7718 242 7 κ−1 κ−1 PROPN ap-7718 242 8 is be AUX ap-7718 242 9 equivalent equivalent ADJ ap-7718 242 10 to to ADP ap-7718 242 11 the the DET ap-7718 242 12 lifetime lifetime NOUN ap-7718 242 13 of of ADP ap-7718 242 14 a a DET ap-7718 242 15 decaying decay VERB ap-7718 242 16 system system NOUN ap-7718 242 17 . . PUNCT ap-7718 243 1 one one NUM ap-7718 243 2 thus thus ADV ap-7718 243 3 may may AUX ap-7718 243 4 disregard disregard VERB ap-7718 243 5 the the DET ap-7718 243 6 dynamics dynamic NOUN ap-7718 243 7 for for ADP ap-7718 243 8 t t PROPN ap-7718 243 9 > > X ap-7718 243 10 t t PROPN ap-7718 243 11 . . PUNCT ap-7718 244 1 to to PART ap-7718 244 2 amend amend VERB ap-7718 244 3 such such ADJ ap-7718 244 4 issue issue NOUN ap-7718 244 5 , , PUNCT ap-7718 244 6 the the DET ap-7718 244 7 second second ADJ ap-7718 244 8 mass mass ADJ ap-7718 244 9 term term NOUN ap-7718 244 10 µrck(t µrck(t NOUN ap-7718 244 11 ) ) PUNCT ap-7718 244 12 has have AUX ap-7718 244 13 been be AUX ap-7718 244 14 introduced introduce VERB ap-7718 244 15 , , PUNCT ap-7718 244 16 which which PRON ap-7718 244 17 transits transit VERB ap-7718 244 18 from from ADP ap-7718 244 19 µrck(0 µrck(0 NOUN ap-7718 244 20 ) ) PUNCT ap-7718 244 21 = = SYM ap-7718 244 22 1 1 NUM ap-7718 244 23 to to ADP ap-7718 244 24 µck(t µck(t NOUN ap-7718 244 25 → → SYM ap-7718 244 26 ∞ ∞ NUM ap-7718 244 27 ) ) PUNCT ap-7718 244 28 = = SYM ap-7718 244 29 µ0 µ0 NOUN ap-7718 244 30 . . PUNCT ap-7718 245 1 thus thus ADV ap-7718 245 2 , , PUNCT ap-7718 245 3 there there PRON ap-7718 245 4 is be VERB ap-7718 245 5 no no DET ap-7718 245 6 need need NOUN ap-7718 245 7 to to PART ap-7718 245 8 introduce introduce VERB ap-7718 245 9 any any DET ap-7718 245 10 artificial artificial ADJ ap-7718 245 11 truncation truncation NOUN ap-7718 245 12 on on ADP ap-7718 245 13 the the DET ap-7718 245 14 time time NOUN ap-7718 245 15 domain domain NOUN ap-7718 245 16 . . PUNCT ap-7718 246 1 the the DET ap-7718 246 2 hamiltonian hamiltonian NOUN ap-7718 246 3 associated associate VERB ap-7718 246 4 with with ADP ap-7718 246 5 this this DET ap-7718 246 6 mass mass ADJ ap-7718 246 7 - - PUNCT ap-7718 246 8 term term NOUN ap-7718 246 9 will will AUX ap-7718 246 10 be be AUX ap-7718 246 11 called call VERB ap-7718 246 12 regularized regularize VERB ap-7718 246 13 caldirola caldirola PROPN ap-7718 246 14 - - PUNCT ap-7718 246 15 kanai kanai PROPN ap-7718 246 16 oscillator oscillator PROPN ap-7718 246 17 . . PUNCT ap-7718 247 1 despite despite SCONJ ap-7718 247 2 the the DET ap-7718 247 3 apparent apparent ADJ ap-7718 247 4 advantages advantage NOUN ap-7718 247 5 of of ADP ap-7718 247 6 the the DET ap-7718 247 7 regularized regularize VERB ap-7718 247 8 system system NOUN ap-7718 247 9 , , PUNCT ap-7718 247 10 analytic analytic ADJ ap-7718 247 11 expressions expression NOUN ap-7718 247 12 for for ADP ap-7718 247 13 σ(t σ(t NOUN ap-7718 247 14 ) ) PUNCT ap-7718 247 15 are be AUX ap-7718 247 16 significantly significantly ADV ap-7718 247 17 more more ADV ap-7718 247 18 complicated complicated ADJ ap-7718 247 19 with with ADP ap-7718 247 20 respect respect NOUN ap-7718 247 21 to to ADP ap-7718 247 22 those those PRON ap-7718 247 23 obtained obtain VERB ap-7718 247 24 from from ADP ap-7718 247 25 µck(t µck(t PROPN ap-7718 247 26 ) ) PUNCT ap-7718 247 27 . . PUNCT ap-7718 248 1 still still ADV ap-7718 248 2 , , PUNCT ap-7718 248 3 exact exact ADJ ap-7718 248 4 result result NOUN ap-7718 248 5 can can AUX ap-7718 248 6 be be AUX ap-7718 248 7 obtained obtain VERB ap-7718 248 8 . . PUNCT ap-7718 249 1 the the DET ap-7718 249 2 discussion discussion NOUN ap-7718 249 3 is be AUX ap-7718 249 4 thus thus ADV ap-7718 249 5 divided divide VERB ap-7718 249 6 for for ADP ap-7718 249 7 each each DET ap-7718 249 8 case case NOUN ap-7718 249 9 separately separately ADV ap-7718 249 10 . . PUNCT ap-7718 250 1 4.1 4.1 NUM ap-7718 250 2 . . PUNCT ap-7718 250 3 caldirola caldirola PROPN ap-7718 250 4 - - PUNCT ap-7718 250 5 kanai kanai PROPN ap-7718 250 6 case case NOUN ap-7718 250 7 the the DET ap-7718 250 8 so so ADV ap-7718 250 9 - - PUNCT ap-7718 250 10 called call VERB ap-7718 250 11 caldirola caldirola PROPN ap-7718 250 12 - - PUNCT ap-7718 250 13 kanai kanai PROPN ap-7718 250 14 system system NOUN ap-7718 250 15 is be AUX ap-7718 250 16 another another DET ap-7718 250 17 wellknown wellknown ADJ ap-7718 250 18 nonstationary nonstationary ADJ ap-7718 250 19 problem problem NOUN ap-7718 250 20 , , PUNCT ap-7718 250 21 characterized characterize VERB ap-7718 250 22 by by ADP ap-7718 250 23 timedependent timedependent NOUN ap-7718 250 24 mass mass PROPN ap-7718 250 25 decaying decay VERB ap-7718 250 26 exponentially exponentially ADV ap-7718 250 27 on on ADP ap-7718 250 28 time time NOUN ap-7718 250 29 . . PUNCT ap-7718 251 1 it it PRON ap-7718 251 2 was be AUX ap-7718 251 3 independently independently ADV ap-7718 251 4 introduced introduce VERB ap-7718 251 5 by by ADP ap-7718 251 6 caldirola caldirola PROPN ap-7718 251 7 [ [ X ap-7718 251 8 42 42 NUM ap-7718 251 9 ] ] PUNCT ap-7718 251 10 and and CCONJ ap-7718 251 11 kanai kanai PROPN ap-7718 252 1 [ [ X ap-7718 252 2 43 43 NUM ap-7718 252 3 ] ] PUNCT ap-7718 252 4 in in ADP ap-7718 252 5 an an DET ap-7718 252 6 attempt attempt NOUN ap-7718 252 7 to to PART ap-7718 252 8 describe describe VERB ap-7718 252 9 the the DET ap-7718 252 10 quantum quantum ADJ ap-7718 252 11 counterpart counterpart NOUN ap-7718 252 12 of of ADP ap-7718 252 13 a a DET ap-7718 252 14 damped damped ADJ ap-7718 252 15 oscillator oscillator NOUN ap-7718 252 16 . . PUNCT ap-7718 253 1 this this DET ap-7718 253 2 model model NOUN ap-7718 253 3 has have AUX ap-7718 253 4 been be AUX ap-7718 253 5 addressed address VERB ap-7718 253 6 by by ADP ap-7718 253 7 different different ADJ ap-7718 253 8 means mean NOUN ap-7718 253 9 , , PUNCT ap-7718 253 10 such such ADJ ap-7718 253 11 using use VERB ap-7718 253 12 a a DET ap-7718 253 13 fourier fourier NOUN ap-7718 253 14 transform transform NOUN ap-7718 253 15 to to PART ap-7718 253 16 map map VERB ap-7718 253 17 the the DET ap-7718 253 18 map map NOUN ap-7718 253 19 it it PRON ap-7718 253 20 into into ADP ap-7718 253 21 a a DET ap-7718 253 22 parametric parametric ADJ ap-7718 253 23 oscillator oscillator NOUN ap-7718 253 24 [ [ X ap-7718 253 25 32 32 NUM ap-7718 253 26 ] ] PUNCT ap-7718 253 27 , , PUNCT ap-7718 253 28 and and CCONJ ap-7718 253 29 using use VERB ap-7718 253 30 the the DET ap-7718 253 31 quantum quantum NOUN ap-7718 253 32 arnold arnold PROPN ap-7718 253 33 transformation transformation NOUN ap-7718 254 1 [ [ X ap-7718 254 2 30 30 NUM ap-7718 254 3 ] ] PUNCT ap-7718 254 4 . . PUNCT ap-7718 255 1 for for ADP ap-7718 255 2 this this DET ap-7718 255 3 particular particular ADJ ap-7718 255 4 case case NOUN ap-7718 255 5 , , PUNCT ap-7718 255 6 a a DET ap-7718 255 7 constant constant ADJ ap-7718 255 8 frequency frequency NOUN ap-7718 255 9 ω2(t ω2(t NOUN ap-7718 255 10 ) ) PUNCT ap-7718 255 11 = = SYM ap-7718 255 12 w2 w2 NOUN ap-7718 255 13 1 1 NUM ap-7718 255 14 and and CCONJ ap-7718 255 15 a a DET ap-7718 255 16 driven drive VERB ap-7718 255 17 force force NOUN ap-7718 255 18 f f PROPN ap-7718 255 19 ( ( PUNCT ap-7718 255 20 t t PROPN ap-7718 255 21 ) ) PUNCT ap-7718 255 22 = = PROPN ap-7718 255 23 a0 a0 PROPN ap-7718 255 24 cos(νt cos(νt NOUN ap-7718 255 25 ) ) PUNCT ap-7718 255 26 , , PUNCT ap-7718 255 27 for for ADP ap-7718 255 28 ν ν X ap-7718 255 29 , , PUNCT ap-7718 255 30 a0 a0 PROPN ap-7718 255 31 ∈ ∈ PROPN ap-7718 255 32 r r NOUN ap-7718 255 33 , , PUNCT ap-7718 255 34 are be AUX ap-7718 255 35 considered consider VERB ap-7718 255 36 . . PUNCT ap-7718 256 1 this this PRON ap-7718 256 2 leads lead VERB ap-7718 256 3 to to ADP ap-7718 256 4 a a DET ap-7718 256 5 forced force VERB ap-7718 256 6 caldirola caldirola PROPN ap-7718 256 7 - - PUNCT ap-7718 256 8 kanai kanai PROPN ap-7718 256 9 oscillator oscillator PROPN ap-7718 256 10 hamiltonian hamiltonian NOUN ap-7718 257 1 [ [ X ap-7718 257 2 10 10 NUM ap-7718 257 3 , , PUNCT ap-7718 257 4 44 44 NUM ap-7718 257 5 ] ] PUNCT ap-7718 257 6 of of ADP ap-7718 257 7 the the DET ap-7718 257 8 form form NOUN ap-7718 257 9 ĥck(t ĥck(t NOUN ap-7718 257 10 ) ) PUNCT ap-7718 257 11 = = PUNCT ap-7718 257 12 e2κt e2κt NOUN ap-7718 257 13 2m0 2m0 NUM ap-7718 258 1 p̂2 p̂2 NOUN ap-7718 258 2 x x PUNCT ap-7718 259 1 + + NOUN ap-7718 259 2 e−2κtm0w e−2κtm0w AUX ap-7718 259 3 2 2 NUM ap-7718 259 4 1 1 NUM ap-7718 259 5 2 2 NUM ap-7718 259 6 x̂2 x̂2 NOUN ap-7718 259 7 + + ADJ ap-7718 259 8 a0 a0 NOUN ap-7718 259 9 cos(νt)x̂. cos(νt)x̂. NOUN ap-7718 259 10 ( ( PUNCT ap-7718 259 11 52 52 NUM ap-7718 259 12 ) ) PUNCT ap-7718 259 13 from from ADP ap-7718 259 14 the the DET ap-7718 259 15 results result NOUN ap-7718 259 16 obtained obtain VERB ap-7718 259 17 in in ADP ap-7718 259 18 previous previous ADJ ap-7718 259 19 sections section NOUN ap-7718 259 20 , , PUNCT ap-7718 259 21 one one PRON ap-7718 259 22 gets get VERB ap-7718 259 23 the the DET ap-7718 259 24 solutions solution NOUN ap-7718 259 25 to to ADP ap-7718 259 26 the the DET ap-7718 259 27 ermakov ermakov ADJ ap-7718 259 28 and and CCONJ ap-7718 259 29 nonhomogeneous nonhomogeneous ADJ ap-7718 259 30 equations equation NOUN ap-7718 259 31 as as ADP ap-7718 259 32 σ(t σ(t PROPN ap-7718 259 33 ) ) PUNCT ap-7718 260 1 = = PRON ap-7718 260 2 ( ( PUNCT ap-7718 260 3 aq2 aq2 PROPN ap-7718 260 4 1(t 1(t NUM ap-7718 260 5 ) ) PUNCT ap-7718 260 6 + + CCONJ ap-7718 260 7 bq1(t)q2(t bq1(t)q2(t VERB ap-7718 260 8 ) ) PUNCT ap-7718 260 9 + + CCONJ ap-7718 260 10 cq2 cq2 NOUN ap-7718 260 11 2(t 2(t NUM ap-7718 260 12 ) ) PUNCT ap-7718 260 13 ) ) PUNCT ap-7718 260 14 1 1 NUM ap-7718 260 15 2 2 NUM ap-7718 260 16 , , PUNCT ap-7718 260 17 γ(t γ(t NOUN ap-7718 260 18 ) ) PUNCT ap-7718 260 19 = = SYM ap-7718 260 20 γ1q1(t γ1q1(t X ap-7718 260 21 ) ) PUNCT ap-7718 260 22 + + CCONJ ap-7718 260 23 γ2q2(t γ2q2(t NOUN ap-7718 260 24 ) ) PUNCT ap-7718 260 25 + + NUM ap-7718 260 26 γp(t γp(t NOUN ap-7718 260 27 ) ) PUNCT ap-7718 260 28 , , PUNCT ap-7718 260 29 ( ( PUNCT ap-7718 260 30 53 53 NUM ap-7718 260 31 ) ) PUNCT ap-7718 260 32 216 216 NUM ap-7718 260 33 vol vol NOUN ap-7718 260 34 . . PUNCT ap-7718 261 1 62 62 NUM ap-7718 261 2 no no INTJ ap-7718 261 3 . . PUNCT ap-7718 262 1 1/2022 1/2022 NUM ap-7718 262 2 td td NOUN ap-7718 262 3 mass mass NOUN ap-7718 262 4 oscillators oscillator NOUN ap-7718 262 5 : : PUNCT ap-7718 262 6 constants constant NOUN ap-7718 262 7 of of ADP ap-7718 262 8 motion motion NOUN ap-7718 262 9 and and CCONJ ap-7718 262 10 semiclasical semiclasical ADJ ap-7718 262 11 states state NOUN ap-7718 262 12 ( ( PUNCT ap-7718 262 13 a a NOUN ap-7718 262 14 ) ) PUNCT ap-7718 262 15 . . PUNCT ap-7718 262 16 wµ(t wµ(t PUNCT ap-7718 262 17 ) ) PUNCT ap-7718 262 18 ( ( PUNCT ap-7718 262 19 b b NOUN ap-7718 262 20 ) ) PUNCT ap-7718 262 21 . . PUNCT ap-7718 263 1 figure figure NOUN ap-7718 263 2 1 1 NUM ap-7718 263 3 . . PUNCT ap-7718 264 1 ( ( PUNCT ap-7718 264 2 a a NOUN ap-7718 264 3 ) ) PUNCT ap-7718 264 4 wµ(t wµ(t PUNCT ap-7718 264 5 ) ) PUNCT ap-7718 265 1 = = SYM ap-7718 265 2 σ(t)µ̇(t σ(t)µ̇(t ADJ ap-7718 265 3 ) ) PUNCT ap-7718 265 4 − − PROPN ap-7718 265 5 σ̇(t)µ(t σ̇(t)µ(t PROPN ap-7718 265 6 ) ) PUNCT ap-7718 265 7 for for ADP ap-7718 265 8 the the DET ap-7718 265 9 caldirola caldirola PROPN ap-7718 265 10 - - PUNCT ap-7718 265 11 kanai kanai PROPN ap-7718 265 12 mass mass PROPN ap-7718 265 13 term term NOUN ap-7718 265 14 µck(t µck(t PROPN ap-7718 265 15 ) ) PUNCT ap-7718 265 16 . . PUNCT ap-7718 266 1 ( ( PUNCT ap-7718 266 2 b b X ap-7718 266 3 ) ) PUNCT ap-7718 266 4 variances variance NOUN ap-7718 266 5 ( ( PUNCT ap-7718 266 6 ∆x̂)2 ∆x̂)2 PROPN ap-7718 266 7 α α PROPN ap-7718 266 8 ( ( PUNCT ap-7718 266 9 solidblue solidblue NOUN ap-7718 266 10 ) ) PUNCT ap-7718 266 11 , , PUNCT ap-7718 266 12 ( ( PUNCT ap-7718 266 13 ∆p̂x)2 ∆p̂x)2 NOUN ap-7718 266 14 α α PROPN ap-7718 266 15 ( ( PUNCT ap-7718 266 16 dashed dash VERB ap-7718 266 17 - - PUNCT ap-7718 266 18 red red VERB ap-7718 266 19 ) ) PUNCT ap-7718 266 20 , , PUNCT ap-7718 266 21 the the DET ap-7718 266 22 schrödinger schrödinger ADJ ap-7718 266 23 - - PUNCT ap-7718 266 24 robertson robertson PROPN ap-7718 266 25 uncertainty uncertainty NOUN ap-7718 266 26 minimum minimum NOUN ap-7718 266 27 ( ( PUNCT ap-7718 266 28 dotted dot VERB ap-7718 266 29 - - PUNCT ap-7718 266 30 green green NOUN ap-7718 266 31 ) ) PUNCT ap-7718 266 32 , , PUNCT ap-7718 266 33 and and CCONJ ap-7718 266 34 the the DET ap-7718 266 35 heisenberg heisenberg PROPN ap-7718 266 36 uncertainty uncertainty PROPN ap-7718 266 37 minimum minimum ADJ ap-7718 266 38 ( ( PUNCT ap-7718 266 39 thick thick ADJ ap-7718 266 40 - - PUNCT ap-7718 266 41 solid solid ADJ ap-7718 266 42 - - PUNCT ap-7718 266 43 black black NOUN ap-7718 266 44 ) ) PUNCT ap-7718 266 45 associated associate VERB ap-7718 266 46 with with ADP ap-7718 266 47 the the DET ap-7718 266 48 coherent coherent ADJ ap-7718 266 49 states state NOUN ap-7718 266 50 ξα(x ξα(x NOUN ap-7718 266 51 , , PUNCT ap-7718 266 52 t t PROPN ap-7718 266 53 ) ) PUNCT ap-7718 266 54 and and CCONJ ap-7718 266 55 the the DET ap-7718 266 56 mass mass ADJ ap-7718 266 57 term term NOUN ap-7718 266 58 µck(t µck(t PROPN ap-7718 266 59 ) ) PUNCT ap-7718 266 60 . . PUNCT ap-7718 267 1 the the DET ap-7718 267 2 parameters parameter NOUN ap-7718 267 3 have have AUX ap-7718 267 4 been be AUX ap-7718 267 5 fixed fix VERB ap-7718 267 6 as as ADP ap-7718 267 7 a a DET ap-7718 267 8 = = SYM ap-7718 267 9 c c NOUN ap-7718 267 10 = = SYM ap-7718 267 11 w0 w0 PROPN ap-7718 267 12 = = SYM ap-7718 267 13 1 1 NUM ap-7718 267 14 , , PUNCT ap-7718 267 15 w1 w1 NOUN ap-7718 267 16 = = SYM ap-7718 267 17 2 2 NUM ap-7718 267 18 , , PUNCT ap-7718 267 19 and and CCONJ ap-7718 267 20 κ κ X ap-7718 267 21 = = SYM ap-7718 267 22 0.5 0.5 NUM ap-7718 267 23 . . PUNCT ap-7718 268 1 ( ( PUNCT ap-7718 268 2 a a NOUN ap-7718 268 3 ) ) PUNCT ap-7718 268 4 . . PUNCT ap-7718 269 1 n n NOUN ap-7718 269 2 = = SYM ap-7718 269 3 0 0 PUNCT ap-7718 269 4 ( ( PUNCT ap-7718 269 5 b b NOUN ap-7718 269 6 ) ) PUNCT ap-7718 269 7 . . PUNCT ap-7718 270 1 n n NOUN ap-7718 270 2 = = SYM ap-7718 270 3 1 1 NUM ap-7718 270 4 ( ( PUNCT ap-7718 270 5 c c NOUN ap-7718 270 6 ) ) PUNCT ap-7718 270 7 . . PUNCT ap-7718 271 1 figure figure NOUN ap-7718 271 2 2 2 NUM ap-7718 271 3 . . NOUN ap-7718 271 4 probability probability NOUN ap-7718 271 5 density density NOUN ap-7718 271 6 pn pn NOUN ap-7718 271 7 = = SYM ap-7718 271 8 |ψn(x |ψn(x PROPN ap-7718 271 9 , , PUNCT ap-7718 271 10 t)|2 t)|2 NOUN ap-7718 271 11 for for ADP ap-7718 271 12 n n NOUN ap-7718 271 13 = = SYM ap-7718 271 14 0 0 PUNCT ap-7718 271 15 ( ( PUNCT ap-7718 271 16 a a NOUN ap-7718 271 17 ) ) PUNCT ap-7718 271 18 , , PUNCT ap-7718 271 19 n n NOUN ap-7718 271 20 = = SYM ap-7718 271 21 1 1 NUM ap-7718 271 22 ( ( PUNCT ap-7718 271 23 b b NOUN ap-7718 271 24 ) ) PUNCT ap-7718 271 25 , , PUNCT ap-7718 271 26 and and CCONJ ap-7718 271 27 pα pα VERB ap-7718 271 28 = = SYM ap-7718 271 29 |ψn(x |ψn(x PROPN ap-7718 271 30 , , PUNCT ap-7718 271 31 t)|2 t)|2 NOUN ap-7718 271 32 ( ( PUNCT ap-7718 271 33 c c NOUN ap-7718 271 34 ) ) PUNCT ap-7718 271 35 associated associate VERB ap-7718 271 36 with with ADP ap-7718 271 37 the the DET ap-7718 271 38 caldirola caldirola PROPN ap-7718 271 39 - - PUNCT ap-7718 271 40 kanai kanai PROPN ap-7718 271 41 mass mass PROPN ap-7718 271 42 term term NOUN ap-7718 271 43 µck(t µck(t PROPN ap-7718 271 44 ) ) PUNCT ap-7718 271 45 . . PUNCT ap-7718 272 1 for for ADP ap-7718 272 2 simplicity simplicity NOUN ap-7718 272 3 , , PUNCT ap-7718 272 4 the the DET ap-7718 272 5 external external ADJ ap-7718 272 6 force force NOUN ap-7718 272 7 f f PROPN ap-7718 272 8 ( ( PUNCT ap-7718 272 9 t t PROPN ap-7718 272 10 ) ) PUNCT ap-7718 272 11 and and CCONJ ap-7718 272 12 γ(t γ(t NOUN ap-7718 272 13 ) ) PUNCT ap-7718 272 14 have have AUX ap-7718 272 15 been be AUX ap-7718 272 16 fixed fix VERB ap-7718 272 17 to to ADP ap-7718 272 18 zero zero NUM ap-7718 272 19 . . PUNCT ap-7718 273 1 the the DET ap-7718 273 2 rest rest NOUN ap-7718 273 3 of of ADP ap-7718 273 4 parameters parameter NOUN ap-7718 273 5 have have AUX ap-7718 273 6 been be AUX ap-7718 273 7 fixed fix VERB ap-7718 273 8 as as ADP ap-7718 273 9 w0 w0 PROPN ap-7718 273 10 = = PROPN ap-7718 273 11 a a DET ap-7718 273 12 = = SYM ap-7718 273 13 b b NOUN ap-7718 273 14 = = SYM ap-7718 273 15 1 1 NUM ap-7718 273 16 , , PUNCT ap-7718 273 17 w1 w1 NOUN ap-7718 273 18 = = SYM ap-7718 273 19 2 2 NUM ap-7718 273 20 , , PUNCT ap-7718 273 21 and and CCONJ ap-7718 273 22 κ κ X ap-7718 273 23 = = SYM ap-7718 273 24 0.5 0.5 NUM ap-7718 273 25 . . PUNCT ap-7718 273 26 respectively respectively ADV ap-7718 273 27 , , PUNCT ap-7718 273 28 with with ADP ap-7718 273 29 γ1 γ1 NOUN ap-7718 273 30 and and CCONJ ap-7718 273 31 γ2 γ2 ADJ ap-7718 273 32 arbitrary arbitrary ADJ ap-7718 273 33 real real ADJ ap-7718 273 34 constants constant NOUN ap-7718 273 35 , , PUNCT ap-7718 273 36 b2 b2 NOUN ap-7718 273 37 − − NOUN ap-7718 273 38 4ac 4ac NOUN ap-7718 273 39 = = SYM ap-7718 273 40 −16 −16 PROPN ap-7718 273 41 w2 w2 NOUN ap-7718 273 42 0 0 NUM ap-7718 273 43 w2 w2 PROPN ap-7718 273 44 1−κ2 1−κ2 NUM ap-7718 273 45 , , PUNCT ap-7718 273 46 and and CCONJ ap-7718 273 47 q1(t q1(t X ap-7718 273 48 ) ) PUNCT ap-7718 273 49 = = SYM ap-7718 273 50 cos cos PROPN ap-7718 273 51 ( ( PUNCT ap-7718 273 52 √ √ PROPN ap-7718 273 53 w2 w2 NOUN ap-7718 273 54 − − PROPN ap-7718 273 55 κ2 κ2 PROPN ap-7718 273 56 t t PROPN ap-7718 273 57 ) ) PUNCT ap-7718 273 58 , , PUNCT ap-7718 273 59 q2(t q2(t PROPN ap-7718 273 60 ) ) PUNCT ap-7718 273 61 = = PUNCT ap-7718 273 62 sin sin NOUN ap-7718 273 63 ( ( PUNCT ap-7718 273 64 √ √ PROPN ap-7718 273 65 w2 w2 NOUN ap-7718 273 66 − − PROPN ap-7718 273 67 κ2 κ2 PROPN ap-7718 273 68 t t PROPN ap-7718 273 69 ) ) PUNCT ap-7718 273 70 , , PUNCT ap-7718 273 71 γp(t γp(t PUNCT ap-7718 273 72 ) ) PUNCT ap-7718 274 1 = = PUNCT ap-7718 274 2 a0e a0e PRON ap-7718 274 3 −kt −kt PROPN ap-7718 274 4 ( ( PUNCT ap-7718 274 5 w2 w2 NOUN ap-7718 274 6 1 1 NUM ap-7718 274 7 − − PROPN ap-7718 274 8 ν2 ν2 PROPN ap-7718 274 9 ) ) PUNCT ap-7718 274 10 cos(νt cos(νt PROPN ap-7718 274 11 ) ) PUNCT ap-7718 274 12 − − PROPN ap-7718 274 13 2κν 2κν NOUN ap-7718 274 14 sin(νt sin(νt PROPN ap-7718 274 15 ) ) PUNCT ap-7718 274 16 ( ( PUNCT ap-7718 274 17 w2 w2 NOUN ap-7718 274 18 1 1 NUM ap-7718 274 19 + + CCONJ ap-7718 274 20 ν2)2 ν2)2 CCONJ ap-7718 274 21 − − VERB ap-7718 274 22 4ν2(w2 4ν2(w2 NUM ap-7718 274 23 1 1 NUM ap-7718 274 24 − − PROPN ap-7718 274 25 κ2 κ2 NOUN ap-7718 274 26 ) ) PUNCT ap-7718 274 27 . . PUNCT ap-7718 275 1 ( ( PUNCT ap-7718 275 2 54 54 NUM ap-7718 275 3 ) ) PUNCT ap-7718 275 4 in in ADP ap-7718 275 5 the the DET ap-7718 275 6 sequel sequel NOUN ap-7718 275 7 , , PUNCT ap-7718 275 8 κ κ X ap-7718 275 9 = = SYM ap-7718 275 10 0.5 0.5 NUM ap-7718 275 11 is be AUX ap-7718 275 12 consider consider VERB ap-7718 275 13 so so SCONJ ap-7718 275 14 that that SCONJ ap-7718 275 15 the the DET ap-7718 275 16 caldirola caldirola PROPN ap-7718 275 17 - - PUNCT ap-7718 275 18 kanai kanai PROPN ap-7718 275 19 oscillator oscillator PROPN ap-7718 275 20 is be AUX ap-7718 275 21 constrained constrain VERB ap-7718 275 22 to to ADP ap-7718 275 23 the the DET ap-7718 275 24 time time NOUN ap-7718 275 25 interval interval NOUN ap-7718 275 26 t t PROPN ap-7718 275 27 ∈ ∈ PROPN ap-7718 276 1 [ [ X ap-7718 276 2 0 0 NUM ap-7718 276 3 , , PUNCT ap-7718 276 4 2 2 NUM ap-7718 276 5 ] ] PUNCT ap-7718 276 6 . . PUNCT ap-7718 277 1 further further ADJ ap-7718 277 2 discussions discussion NOUN ap-7718 277 3 concerning concern VERB ap-7718 277 4 the the DET ap-7718 277 5 dynamics dynamic NOUN ap-7718 277 6 will will AUX ap-7718 277 7 be be AUX ap-7718 277 8 restricted restrict VERB ap-7718 277 9 to to ADP ap-7718 277 10 such such DET ap-7718 277 11 a a DET ap-7718 277 12 time time NOUN ap-7718 277 13 interval interval NOUN ap-7718 277 14 . . PUNCT ap-7718 278 1 it it PRON ap-7718 278 2 is be AUX ap-7718 278 3 worth worth ADJ ap-7718 278 4 recalling recall VERB ap-7718 278 5 that that SCONJ ap-7718 278 6 the the DET ap-7718 278 7 zeros zero NOUN ap-7718 278 8 of of ADP ap-7718 278 9 wµ(t wµ(t PROPN ap-7718 278 10 ) ) PUNCT ap-7718 278 11 correspond correspond VERB ap-7718 278 12 to to ADP ap-7718 278 13 the the DET ap-7718 278 14 times time NOUN ap-7718 278 15 for for ADP ap-7718 278 16 which which PRON ap-7718 278 17 the the DET ap-7718 278 18 heisenberg heisenberg PROPN ap-7718 278 19 uncertainty uncertainty NOUN ap-7718 278 20 relationship relationship NOUN ap-7718 278 21 saturates saturate VERB ap-7718 278 22 . . PUNCT ap-7718 279 1 although although SCONJ ap-7718 279 2 the the DET ap-7718 279 3 expression expression NOUN ap-7718 279 4 for for ADP ap-7718 279 5 wµ(t wµ(t NUM ap-7718 279 6 ) ) PUNCT ap-7718 279 7 is be AUX ap-7718 279 8 rather rather ADV ap-7718 279 9 simple simple ADJ ap-7718 279 10 in in ADP ap-7718 279 11 this this DET ap-7718 279 12 case case NOUN ap-7718 279 13 , , PUNCT ap-7718 279 14 determining determine VERB ap-7718 279 15 the the DET ap-7718 279 16 zeros zero NOUN ap-7718 279 17 consist consist NOUN ap-7718 279 18 of of ADP ap-7718 279 19 solving solve VERB ap-7718 279 20 a a DET ap-7718 279 21 transcendental transcendental ADJ ap-7718 279 22 equation equation NOUN ap-7718 279 23 . . PUNCT ap-7718 280 1 thus thus ADV ap-7718 280 2 , , PUNCT ap-7718 280 3 to to PART ap-7718 280 4 get get VERB ap-7718 280 5 further further ADJ ap-7718 280 6 insight insight NOUN ap-7718 280 7 , , PUNCT ap-7718 280 8 one one PRON ap-7718 280 9 may may AUX ap-7718 280 10 analyze analyze VERB ap-7718 280 11 figure figure VERB ap-7718 280 12 1a 1a NOUN ap-7718 280 13 , , PUNCT ap-7718 280 14 which which PRON ap-7718 280 15 depicts depict VERB ap-7718 280 16 the the DET ap-7718 280 17 behavior behavior NOUN ap-7718 280 18 of of ADP ap-7718 280 19 such such DET ap-7718 280 20 a a DET ap-7718 280 21 function function NOUN ap-7718 280 22 for for ADP ap-7718 280 23 µck(t µck(t NOUN ap-7718 280 24 ) ) PUNCT ap-7718 280 25 ( ( PUNCT ap-7718 280 26 solid solid ADJ ap-7718 280 27 - - PUNCT ap-7718 280 28 blue blue ADJ ap-7718 280 29 ) ) PUNCT ap-7718 280 30 . . PUNCT ap-7718 281 1 from from ADP ap-7718 281 2 the the DET ap-7718 281 3 latter latter ADJ ap-7718 281 4 , , PUNCT ap-7718 281 5 one one PRON ap-7718 281 6 can can AUX ap-7718 281 7 see see VERB ap-7718 281 8 that that DET ap-7718 281 9 zeroes zero NOUN ap-7718 281 10 do do AUX ap-7718 281 11 exist exist VERB ap-7718 281 12 indeed indeed ADV ap-7718 281 13 , , PUNCT ap-7718 281 14 and and CCONJ ap-7718 281 15 thus thus ADV ap-7718 281 16 one one NUM ap-7718 281 17 should should AUX ap-7718 281 18 expect expect VERB ap-7718 281 19 points point NOUN ap-7718 281 20 in in ADP ap-7718 281 21 time time NOUN ap-7718 281 22 for for ADP ap-7718 281 23 which which PRON ap-7718 281 24 the the DET ap-7718 281 25 heisenberg heisenberg PROPN ap-7718 281 26 inequality inequality PROPN ap-7718 281 27 saturates saturate VERB ap-7718 281 28 . . PUNCT ap-7718 282 1 despite despite SCONJ ap-7718 282 2 the the DET ap-7718 282 3 latter latter ADJ ap-7718 282 4 , , PUNCT ap-7718 282 5 the the DET ap-7718 282 6 schrödinger schrödinger ADJ ap-7718 282 7 - - PUNCT ap-7718 282 8 robertson robertson PROPN ap-7718 282 9 inequality inequality NOUN ap-7718 282 10 saturates saturate VERB ap-7718 282 11 at at ADP ap-7718 282 12 all all DET ap-7718 282 13 times time NOUN ap-7718 282 14 . . PUNCT ap-7718 283 1 in in ADP ap-7718 283 2 figure figure NOUN ap-7718 283 3 1b 1b NUM ap-7718 283 4 , , PUNCT ap-7718 283 5 one one PRON ap-7718 283 6 can can AUX ap-7718 283 7 see see VERB ap-7718 283 8 the the DET ap-7718 283 9 behavior behavior NOUN ap-7718 283 10 of of ADP ap-7718 283 11 the the DET ap-7718 283 12 variances variance NOUN ap-7718 283 13 , , PUNCT ap-7718 283 14 from from ADP ap-7718 283 15 which which PRON ap-7718 283 16 it it PRON ap-7718 283 17 is be AUX ap-7718 283 18 clear clear ADJ ap-7718 283 19 that that SCONJ ap-7718 283 20 the the DET ap-7718 283 21 variance variance NOUN ap-7718 283 22 in in ADP ap-7718 283 23 the the DET ap-7718 283 24 position position NOUN ap-7718 283 25 blows blow VERB ap-7718 283 26 up up ADP ap-7718 283 27 at at ADP ap-7718 283 28 time time NOUN ap-7718 283 29 pass pass VERB ap-7718 283 30 by by ADP ap-7718 283 31 , , PUNCT ap-7718 283 32 whereas whereas SCONJ ap-7718 283 33 the the DET ap-7718 283 34 momentum momentum NOUN ap-7718 283 35 variance variance NOUN ap-7718 283 36 squeezes squeeze VERB ap-7718 283 37 indefinitely indefinitely ADV ap-7718 283 38 , , PUNCT ap-7718 283 39 approaching approach VERB ap-7718 283 40 asymptotically asymptotically ADV ap-7718 283 41 to to ADP ap-7718 283 42 zero zero NUM ap-7718 283 43 . . PUNCT ap-7718 284 1 this this DET ap-7718 284 2 odd odd ADJ ap-7718 284 3 behavior behavior NOUN ap-7718 284 4 results result VERB ap-7718 284 5 from from ADP ap-7718 284 6 a a DET ap-7718 284 7 mass mass ADJ ap-7718 284 8 term term NOUN ap-7718 284 9 that that PRON ap-7718 284 10 quickly quickly ADV ap-7718 284 11 decays decay VERB ap-7718 284 12 , , PUNCT ap-7718 284 13 approaching approach VERB ap-7718 284 14 zero zero NUM ap-7718 284 15 but but CCONJ ap-7718 284 16 never never ADV ap-7718 284 17 converging converge VERB ap-7718 284 18 to to ADP ap-7718 284 19 it it PRON ap-7718 284 20 . . PUNCT ap-7718 285 1 for for ADP ap-7718 285 2 those those DET ap-7718 285 3 reasons reason NOUN ap-7718 285 4 , , PUNCT ap-7718 285 5 a a DET ap-7718 285 6 truncation truncation NOUN ap-7718 285 7 on on ADP ap-7718 285 8 the the DET ap-7718 285 9 time time NOUN ap-7718 285 10 interval interval NOUN ap-7718 285 11 was be AUX ap-7718 285 12 previously previously ADV ap-7718 285 13 introduced introduce VERB ap-7718 285 14 in in ADP ap-7718 285 15 the the DET ap-7718 285 16 form form NOUN ap-7718 285 17 of of ADP ap-7718 285 18 a a DET ap-7718 285 19 mean mean ADJ ap-7718 285 20 lifetime lifetime NOUN ap-7718 285 21 , , PUNCT ap-7718 285 22 which which PRON ap-7718 285 23 in in ADP ap-7718 285 24 this this DET ap-7718 285 25 case case NOUN ap-7718 285 26 becomes become VERB ap-7718 285 27 t t NOUN ap-7718 285 28 = = SYM ap-7718 286 1 κ−1 κ−1 PROPN ap-7718 286 2 = = SYM ap-7718 286 3 2 2 X ap-7718 286 4 . . PUNCT ap-7718 287 1 in in ADP ap-7718 287 2 this this DET ap-7718 287 3 form form NOUN ap-7718 287 4 , , PUNCT ap-7718 287 5 one one PRON ap-7718 287 6 still still ADV ap-7718 287 7 has have VERB ap-7718 287 8 a a DET ap-7718 287 9 realistic realistic ADJ ap-7718 287 10 behavior behavior NOUN ap-7718 287 11 for for ADP ap-7718 287 12 t t PROPN ap-7718 287 13 ∈ ∈ PROPN ap-7718 287 14 ( ( PUNCT ap-7718 287 15 0 0 NUM ap-7718 287 16 , , PUNCT ap-7718 287 17 2 2 NUM ap-7718 287 18 ) ) PUNCT ap-7718 287 19 . . PUNCT ap-7718 288 1 the the DET ap-7718 288 2 previous previous ADJ ap-7718 288 3 results result NOUN ap-7718 288 4 can can AUX ap-7718 288 5 be be AUX ap-7718 288 6 verified verify VERB ap-7718 288 7 by by ADP ap-7718 288 8 looking look VERB ap-7718 288 9 at at ADP ap-7718 288 10 the the DET ap-7718 288 11 probability probability NOUN ap-7718 288 12 density density NOUN ap-7718 288 13 associated associate VERB ap-7718 288 14 with with ADP ap-7718 288 15 the the DET ap-7718 288 16 solutions solution NOUN ap-7718 288 17 ψn(x ψn(x ADP ap-7718 288 18 , , PUNCT ap-7718 288 19 t t PROPN ap-7718 288 20 ) ) PUNCT ap-7718 288 21 and and CCONJ ap-7718 288 22 the the DET ap-7718 288 23 coherent coherent ADJ ap-7718 288 24 state state NOUN ap-7718 288 25 ξα(x ξα(x PROPN ap-7718 288 26 , , PUNCT ap-7718 288 27 t t PROPN ap-7718 288 28 ) ) PUNCT ap-7718 288 29 , , PUNCT ap-7718 288 30 which which PRON ap-7718 288 31 is be AUX ap-7718 288 32 depicted depict VERB ap-7718 288 33 in in ADP ap-7718 288 34 figure figure NOUN ap-7718 288 35 2 2 NUM ap-7718 288 36 . . PUNCT ap-7718 289 1 from from ADP ap-7718 289 2 those those DET ap-7718 289 3 probability probability NOUN ap-7718 289 4 densities density NOUN ap-7718 289 5 , , PUNCT ap-7718 289 6 one one PRON ap-7718 289 7 may may AUX ap-7718 289 8 see see VERB ap-7718 289 9 the the DET ap-7718 289 10 increase increase NOUN ap-7718 289 11 on on ADP ap-7718 289 12 the the DET ap-7718 289 13 position position NOUN ap-7718 289 14 variance variance NOUN ap-7718 289 15 ( ( PUNCT ap-7718 289 16 ∆x̂)2 ∆x̂)2 PROPN ap-7718 289 17 α α NUM ap-7718 289 18 , , PUNCT ap-7718 289 19 for for SCONJ ap-7718 289 20 the the DET ap-7718 289 21 wavepacket wavepacket NOUN ap-7718 289 22 spreads spread VERB ap-7718 289 23 rapidly rapidly ADV ap-7718 289 24 on on ADP ap-7718 289 25 time time NOUN ap-7718 289 26 , , PUNCT ap-7718 289 27 217 217 NUM ap-7718 289 28 kevin kevin PROPN ap-7718 289 29 zelaya zelaya PROPN ap-7718 289 30 acta acta PROPN ap-7718 289 31 polytechnica polytechnica PROPN ap-7718 289 32 ( ( PUNCT ap-7718 289 33 a a NOUN ap-7718 289 34 ) ) PUNCT ap-7718 289 35 . . PUNCT ap-7718 290 1 wµ(t wµ(t PUNCT ap-7718 290 2 ) ) PUNCT ap-7718 291 1 ( ( PUNCT ap-7718 291 2 b b NOUN ap-7718 291 3 ) ) PUNCT ap-7718 291 4 . . PUNCT ap-7718 292 1 figure figure VERB ap-7718 292 2 3 3 NUM ap-7718 292 3 . . PUNCT ap-7718 293 1 ( ( PUNCT ap-7718 293 2 a a NOUN ap-7718 293 3 ) ) PUNCT ap-7718 293 4 wµ(t wµ(t PUNCT ap-7718 293 5 ) ) PUNCT ap-7718 294 1 = = SYM ap-7718 294 2 σ(t)µ̇(t σ(t)µ̇(t ADJ ap-7718 294 3 ) ) PUNCT ap-7718 294 4 − − PROPN ap-7718 294 5 σ̇(t)µ(t σ̇(t)µ(t PROPN ap-7718 294 6 ) ) PUNCT ap-7718 294 7 for for ADP ap-7718 294 8 the the DET ap-7718 294 9 regularized regularize VERB ap-7718 294 10 caldirola caldirola PROPN ap-7718 294 11 - - PUNCT ap-7718 294 12 kanai kanai PROPN ap-7718 294 13 mass mass PROPN ap-7718 294 14 term term PROPN ap-7718 294 15 µrck(t µrck(t PROPN ap-7718 294 16 ) ) PUNCT ap-7718 294 17 . . PUNCT ap-7718 295 1 ( ( PUNCT ap-7718 295 2 b b X ap-7718 295 3 ) ) PUNCT ap-7718 295 4 variances variance NOUN ap-7718 295 5 ( ( PUNCT ap-7718 295 6 ∆x̂)2 ∆x̂)2 PROPN ap-7718 295 7 α α X ap-7718 295 8 ( ( PUNCT ap-7718 295 9 solid solid ADJ ap-7718 295 10 - - PUNCT ap-7718 295 11 blue blue ADJ ap-7718 295 12 ) ) PUNCT ap-7718 295 13 , , PUNCT ap-7718 295 14 ( ( PUNCT ap-7718 295 15 ∆p̂x)2 ∆p̂x)2 NOUN ap-7718 295 16 α α PROPN ap-7718 295 17 ( ( PUNCT ap-7718 295 18 dashed dash VERB ap-7718 295 19 - - PUNCT ap-7718 295 20 red red VERB ap-7718 295 21 ) ) PUNCT ap-7718 295 22 , , PUNCT ap-7718 295 23 the the DET ap-7718 295 24 schrödinger schrödinger ADJ ap-7718 295 25 - - PUNCT ap-7718 295 26 robertson robertson PROPN ap-7718 295 27 uncertainty uncertainty NOUN ap-7718 295 28 minimum minimum NOUN ap-7718 295 29 ( ( PUNCT ap-7718 295 30 dotted dot VERB ap-7718 295 31 - - PUNCT ap-7718 295 32 green green NOUN ap-7718 295 33 ) ) PUNCT ap-7718 295 34 , , PUNCT ap-7718 295 35 and and CCONJ ap-7718 295 36 the the DET ap-7718 295 37 heisenberg heisenberg PROPN ap-7718 295 38 uncertainty uncertainty PROPN ap-7718 295 39 minimum minimum ADJ ap-7718 295 40 ( ( PUNCT ap-7718 295 41 thick thick ADJ ap-7718 295 42 - - PUNCT ap-7718 295 43 solid solid ADJ ap-7718 295 44 - - PUNCT ap-7718 295 45 black black NOUN ap-7718 295 46 ) ) PUNCT ap-7718 295 47 associated associate VERB ap-7718 295 48 with with ADP ap-7718 295 49 the the DET ap-7718 295 50 coherent coherent ADJ ap-7718 295 51 states state NOUN ap-7718 295 52 ξα(x ξα(x NOUN ap-7718 295 53 , , PUNCT ap-7718 295 54 t t PROPN ap-7718 295 55 ) ) PUNCT ap-7718 295 56 and and CCONJ ap-7718 295 57 the the DET ap-7718 295 58 mass mass ADJ ap-7718 295 59 term term NOUN ap-7718 295 60 µrck(t µrck(t PROPN ap-7718 295 61 ) ) PUNCT ap-7718 295 62 . . PUNCT ap-7718 296 1 the the DET ap-7718 296 2 parameters parameter NOUN ap-7718 296 3 have have AUX ap-7718 296 4 been be AUX ap-7718 296 5 fixed fix VERB ap-7718 296 6 as as ADP ap-7718 296 7 a a DET ap-7718 296 8 = = SYM ap-7718 296 9 c c NOUN ap-7718 296 10 = = SYM ap-7718 296 11 w0 w0 PROPN ap-7718 296 12 = = SYM ap-7718 296 13 1 1 NUM ap-7718 296 14 , , PUNCT ap-7718 296 15 w1 w1 NOUN ap-7718 296 16 = = SYM ap-7718 296 17 2 2 NUM ap-7718 296 18 , , PUNCT ap-7718 296 19 µ0 µ0 NOUN ap-7718 296 20 = = SYM ap-7718 296 21 0.3 0.3 NUM ap-7718 296 22 , , PUNCT ap-7718 296 23 and and CCONJ ap-7718 296 24 κ κ X ap-7718 296 25 = = NOUN ap-7718 296 26 0.5 0.5 NUM ap-7718 296 27 . . PUNCT ap-7718 296 28 to to ADP ap-7718 296 29 the the DET ap-7718 296 30 point point NOUN ap-7718 296 31 that that SCONJ ap-7718 296 32 , , PUNCT ap-7718 296 33 for for ADP ap-7718 296 34 times time NOUN ap-7718 296 35 t t PROPN ap-7718 296 36 > > X ap-7718 296 37 4 4 NUM ap-7718 296 38 is be AUX ap-7718 296 39 almost almost ADV ap-7718 296 40 indistinguishable indistinguishable ADJ ap-7718 296 41 . . PUNCT ap-7718 297 1 for for ADP ap-7718 297 2 completeness completeness NOUN ap-7718 297 3 , , PUNCT ap-7718 297 4 the the DET ap-7718 297 5 classical classical ADJ ap-7718 297 6 trajectory trajectory NOUN ap-7718 297 7 is be AUX ap-7718 297 8 depicted depict VERB ap-7718 297 9 as as ADP ap-7718 297 10 a a DET ap-7718 297 11 dashed dash VERB ap-7718 297 12 - - PUNCT ap-7718 297 13 black black ADJ ap-7718 297 14 curve curve NOUN ap-7718 297 15 in in ADP ap-7718 297 16 figure figure NOUN ap-7718 297 17 2c 2c NOUN ap-7718 297 18 , , PUNCT ap-7718 297 19 where where SCONJ ap-7718 297 20 the the DET ap-7718 297 21 initial initial ADJ ap-7718 297 22 conditions condition NOUN ap-7718 297 23 ⟨x̂⟩t0 ⟨x̂⟩t0 NOUN ap-7718 297 24 = = SYM ap-7718 297 25 2 2 NUM ap-7718 297 26 and and CCONJ ap-7718 297 27 ⟨p̂x⟩t0 ⟨p̂x⟩t0 SYM ap-7718 297 28 = = SYM ap-7718 297 29 0 0 NUM ap-7718 297 30 have have AUX ap-7718 297 31 been be AUX ap-7718 297 32 used use VERB ap-7718 297 33 . . PUNCT ap-7718 298 1 4.2 4.2 NUM ap-7718 298 2 . . PUNCT ap-7718 298 3 regularized regularize VERB ap-7718 298 4 caldirola caldirola PROPN ap-7718 298 5 - - PUNCT ap-7718 298 6 kanai kanai PROPN ap-7718 298 7 in in ADP ap-7718 298 8 this this DET ap-7718 298 9 section section NOUN ap-7718 298 10 , , PUNCT ap-7718 298 11 the the DET ap-7718 298 12 regularized regularize VERB ap-7718 298 13 caldirola caldirola PROPN ap-7718 298 14 - - PUNCT ap-7718 298 15 kanai kanai PROPN ap-7718 298 16 oscillator oscillator PROPN ap-7718 298 17 is be AUX ap-7718 298 18 introduced introduce VERB ap-7718 298 19 so so SCONJ ap-7718 298 20 that that SCONJ ap-7718 298 21 it it PRON ap-7718 298 22 amends amend VERB ap-7718 298 23 the the DET ap-7718 298 24 difficulties difficulty NOUN ap-7718 298 25 found find VERB ap-7718 298 26 in in ADP ap-7718 298 27 the the DET ap-7718 298 28 caldirola caldirola PROPN ap-7718 298 29 - - PUNCT ap-7718 298 30 kanai kanai PROPN ap-7718 298 31 for for ADP ap-7718 298 32 t t PROPN ap-7718 298 33 > > PUNCT ap-7718 298 34 > > X ap-7718 298 35 t t PROPN ap-7718 298 36 . . PUNCT ap-7718 299 1 this this DET ap-7718 299 2 model model NOUN ap-7718 299 3 is be AUX ap-7718 299 4 characterized characterize VERB ap-7718 299 5 by by ADP ap-7718 299 6 a a DET ap-7718 299 7 constant constant ADJ ap-7718 299 8 frequency frequency NOUN ap-7718 299 9 ω2(t ω2(t NOUN ap-7718 299 10 ) ) PUNCT ap-7718 299 11 = = SYM ap-7718 299 12 w2 w2 NOUN ap-7718 299 13 1 1 NUM ap-7718 299 14 and and CCONJ ap-7718 299 15 a a DET ap-7718 299 16 mass mass ADJ ap-7718 299 17 term term NOUN ap-7718 299 18 µrck(t µrck(t NOUN ap-7718 299 19 ) ) PUNCT ap-7718 299 20 = = SYM ap-7718 300 1 µ0e µ0e PROPN ap-7718 300 2 −κt −κt PROPN ap-7718 300 3 + + CCONJ ap-7718 300 4 µ1 µ1 PROPN ap-7718 300 5 , , PUNCT ap-7718 300 6 with with ADP ap-7718 300 7 w1 w1 NOUN ap-7718 300 8 , , PUNCT ap-7718 300 9 µ0 µ0 PROPN ap-7718 300 10 , , PUNCT ap-7718 300 11 µ1 µ1 PROPN ap-7718 300 12 , , PUNCT ap-7718 300 13 κ κ X ap-7718 300 14 > > X ap-7718 300 15 0 0 PROPN ap-7718 300 16 . . PUNCT ap-7718 301 1 the the DET ap-7718 301 2 mass mass ADJ ap-7718 301 3 term term NOUN ap-7718 301 4 will will AUX ap-7718 301 5 converge converge VERB ap-7718 301 6 at at ADP ap-7718 301 7 a a DET ap-7718 301 8 constant constant ADJ ap-7718 301 9 value value NOUN ap-7718 301 10 ( ( PUNCT ap-7718 301 11 different different ADJ ap-7718 301 12 from from ADP ap-7718 301 13 zero zero NUM ap-7718 301 14 ) ) PUNCT ap-7718 301 15 and and CCONJ ap-7718 301 16 the the DET ap-7718 301 17 anomalies anomaly NOUN ap-7718 301 18 found find VERB ap-7718 301 19 in in ADP ap-7718 301 20 the the DET ap-7718 301 21 conventional conventional ADJ ap-7718 301 22 caldirola caldirola PROPN ap-7718 301 23 - - PUNCT ap-7718 301 24 kanai kanai PROPN ap-7718 301 25 case case NOUN ap-7718 301 26 will will AUX ap-7718 301 27 be be AUX ap-7718 301 28 fixed fix VERB ap-7718 301 29 . . PUNCT ap-7718 302 1 the the DET ap-7718 302 2 main main ADJ ap-7718 302 3 consequence consequence NOUN ap-7718 302 4 of of ADP ap-7718 302 5 the the DET ap-7718 302 6 mass mass ADJ ap-7718 302 7 regularization regularization NOUN ap-7718 302 8 is be AUX ap-7718 302 9 that that SCONJ ap-7718 302 10 the the DET ap-7718 302 11 classical classical ADJ ap-7718 302 12 equation equation NOUN ap-7718 302 13 of of ADP ap-7718 302 14 motion motion NOUN ap-7718 302 15 is be AUX ap-7718 302 16 not not PART ap-7718 302 17 as as ADV ap-7718 302 18 trivial trivial ADJ ap-7718 302 19 as as ADP ap-7718 302 20 in in ADP ap-7718 302 21 section section NOUN ap-7718 302 22 4 4 NUM ap-7718 302 23 . . PUNCT ap-7718 303 1 in in ADP ap-7718 303 2 turn turn NOUN ap-7718 303 3 one one NUM ap-7718 303 4 has have VERB ap-7718 303 5 q̈(t q̈(t NOUN ap-7718 303 6 ) ) PUNCT ap-7718 304 1 + + CCONJ ap-7718 304 2 ( ( PUNCT ap-7718 304 3 w2 w2 NOUN ap-7718 304 4 1 1 NUM ap-7718 304 5 − − PROPN ap-7718 304 6 κ2 κ2 NOUN ap-7718 304 7 1 1 NUM ap-7718 304 8 + + CCONJ ap-7718 304 9 µ0eκt µ0eκt NUM ap-7718 304 10 ) ) PUNCT ap-7718 304 11 q(t q(t PROPN ap-7718 304 12 ) ) PUNCT ap-7718 304 13 = = SYM ap-7718 304 14 0 0 X ap-7718 304 15 . . PUNCT ap-7718 305 1 ( ( PUNCT ap-7718 305 2 55 55 NUM ap-7718 305 3 ) ) PUNCT ap-7718 305 4 two two NUM ap-7718 305 5 linearly linearly ADV ap-7718 305 6 independent independent ADJ ap-7718 305 7 solutions solution NOUN ap-7718 305 8 to to ADP ap-7718 305 9 the the DET ap-7718 305 10 corresponding corresponding ADJ ap-7718 305 11 linear linear ADJ ap-7718 305 12 equation equation NOUN ap-7718 305 13 ( ( PUNCT ap-7718 305 14 15 15 NUM ap-7718 305 15 ) ) PUNCT ap-7718 305 16 can can AUX ap-7718 305 17 be be AUX ap-7718 305 18 found find VERB ap-7718 305 19 as as ADP ap-7718 305 20 q1(t q1(t PROPN ap-7718 305 21 ) ) PUNCT ap-7718 305 22 = = SYM ap-7718 305 23 z(t)i z(t)i PROPN ap-7718 305 24 w1 w1 NOUN ap-7718 305 25 k k PROPN ap-7718 305 26 2f1 2f1 NUM ap-7718 305 27 a1 a1 PROPN ap-7718 305 28 , , PUNCT ap-7718 305 29 a2 a2 PROPN ap-7718 305 30 1 1 NUM ap-7718 305 31 − − PROPN ap-7718 305 32 2iw1 2iw1 NUM ap-7718 306 1 k k PROPN ap-7718 306 2 ∣∣∣∣∣∣ ∣∣∣∣∣∣ ADP ap-7718 306 3 −1 −1 NOUN ap-7718 306 4 z(t z(t NOUN ap-7718 306 5 ) ) PUNCT ap-7718 307 1   PROPN ap-7718 307 2 , , PUNCT ap-7718 307 3 q2(t q2(t NOUN ap-7718 307 4 ) ) PUNCT ap-7718 307 5 = = SYM ap-7718 307 6 q∗ q∗ NOUN ap-7718 307 7 1(t 1(t NUM ap-7718 307 8 ) ) PUNCT ap-7718 307 9 , , PUNCT ap-7718 307 10 ( ( PUNCT ap-7718 307 11 56 56 NUM ap-7718 307 12 ) ) PUNCT ap-7718 307 13 where where SCONJ ap-7718 307 14 z(t z(t NOUN ap-7718 307 15 ) ) PUNCT ap-7718 307 16 = = SYM ap-7718 307 17 µ0e µ0e PROPN ap-7718 307 18 κt κt NOUN ap-7718 307 19 , , PUNCT ap-7718 307 20 a1 a1 NOUN ap-7718 307 21 = = SYM ap-7718 307 22 −iw1 −iw1 X ap-7718 307 23 k k PROPN ap-7718 307 24 −i −i PROPN ap-7718 307 25 √ √ PROPN ap-7718 307 26 w2 w2 PROPN ap-7718 307 27 1 1 NUM ap-7718 307 28 k2 k2 NOUN ap-7718 307 29 − − PROPN ap-7718 307 30 1 1 NUM ap-7718 307 31 , , PUNCT ap-7718 307 32 and and CCONJ ap-7718 307 33 a2 a2 PROPN ap-7718 307 34 = = SYM ap-7718 308 1 −iw1 −iw1 X ap-7718 309 1 k k PROPN ap-7718 310 1 + + CCONJ ap-7718 310 2 i i PRON ap-7718 310 3 √ √ VERB ap-7718 310 4 w2 w2 NOUN ap-7718 310 5 1 1 NUM ap-7718 310 6 k2 k2 NOUN ap-7718 310 7 − − PROPN ap-7718 310 8 1 1 NUM ap-7718 310 9 . . PUNCT ap-7718 311 1 on on ADP ap-7718 311 2 the the DET ap-7718 311 3 other other ADJ ap-7718 311 4 hand hand NOUN ap-7718 311 5 , , PUNCT ap-7718 311 6 2f1(a 2f1(a NUM ap-7718 311 7 , , PUNCT ap-7718 311 8 b b X ap-7718 311 9 ; ; PUNCT ap-7718 311 10 c;z c;z PROPN ap-7718 311 11 ) ) PUNCT ap-7718 311 12 stands stand VERB ap-7718 311 13 for for ADP ap-7718 311 14 the the DET ap-7718 311 15 hypergeometric hypergeometric ADJ ap-7718 311 16 function function NOUN ap-7718 311 17 [ [ X ap-7718 311 18 33 33 NUM ap-7718 311 19 ] ] PUNCT ap-7718 311 20 , , PUNCT ap-7718 311 21 which which PRON ap-7718 311 22 converges converge VERB ap-7718 311 23 in in ADP ap-7718 311 24 the the DET ap-7718 311 25 complex complex ADJ ap-7718 311 26 unit unit NOUN ap-7718 311 27 - - PUNCT ap-7718 311 28 disk disk NOUN ap-7718 311 29 |z| |z| NOUN ap-7718 311 30 < < X ap-7718 311 31 1 1 NUM ap-7718 311 32 . . PUNCT ap-7718 311 33 given give VERB ap-7718 311 34 that that PRON ap-7718 311 35 z(t z(t NOUN ap-7718 311 36 ) ) PUNCT ap-7718 311 37 : : PUNCT ap-7718 312 1 r r NOUN ap-7718 312 2 → → SYM ap-7718 312 3 ( ( PUNCT ap-7718 312 4 1,∞ 1,∞ NUM ap-7718 312 5 ) ) PUNCT ap-7718 312 6 , , PUNCT ap-7718 312 7 the the DET ap-7718 312 8 solutions solution NOUN ap-7718 312 9 q1,2(t q1,2(t PROPN ap-7718 312 10 ) ) PUNCT ap-7718 312 11 in in ADP ap-7718 312 12 ( ( PUNCT ap-7718 312 13 56 56 NUM ap-7718 312 14 ) ) PUNCT ap-7718 312 15 converge converge NOUN ap-7718 312 16 for for ADP ap-7718 312 17 t t PROPN ap-7718 312 18 ∈ ∈ PROPN ap-7718 312 19 r. r. PROPN ap-7718 312 20 since since SCONJ ap-7718 312 21 both both DET ap-7718 312 22 solutions solution NOUN ap-7718 312 23 in in ADP ap-7718 312 24 ( ( PUNCT ap-7718 312 25 56 56 NUM ap-7718 312 26 ) ) PUNCT ap-7718 312 27 are be AUX ap-7718 312 28 complex complex ADV ap-7718 312 29 - - PUNCT ap-7718 312 30 valued value VERB ap-7718 312 31 , , PUNCT ap-7718 312 32 with with ADP ap-7718 312 33 q2 q2 NOUN ap-7718 312 34 = = SYM ap-7718 312 35 q∗ q∗ PROPN ap-7718 312 36 1 1 NUM ap-7718 312 37 , , PUNCT ap-7718 312 38 one one PRON ap-7718 312 39 can can AUX ap-7718 312 40 construct construct VERB ap-7718 312 41 a a DET ap-7718 312 42 real real ADV ap-7718 312 43 - - PUNCT ap-7718 312 44 valued value VERB ap-7718 312 45 solution solution NOUN ap-7718 312 46 to to ADP ap-7718 312 47 the the DET ap-7718 312 48 ermakov ermakov ADJ ap-7718 312 49 equation equation NOUN ap-7718 312 50 by by ADP ap-7718 312 51 taking take VERB ap-7718 312 52 q1 q1 NOUN ap-7718 312 53 = = NOUN ap-7718 312 54 re(q1 re(q1 NOUN ap-7718 312 55 ) ) PUNCT ap-7718 312 56 and and CCONJ ap-7718 312 57 q2 q2 NOUN ap-7718 312 58 = = NOUN ap-7718 312 59 im(q1 im(q1 PROPN ap-7718 312 60 ) ) PUNCT ap-7718 312 61 . . PUNCT ap-7718 313 1 to to PART ap-7718 313 2 simplify simplify VERB ap-7718 313 3 the the DET ap-7718 313 4 ongoing ongoing ADJ ap-7718 313 5 discussion discussion NOUN ap-7718 313 6 , , PUNCT ap-7718 313 7 the the DET ap-7718 313 8 external external ADJ ap-7718 313 9 force force NOUN ap-7718 313 10 is be AUX ap-7718 313 11 considered consider VERB ap-7718 313 12 null null ADJ ap-7718 313 13 , , PUNCT ap-7718 313 14 f f PROPN ap-7718 313 15 ( ( PUNCT ap-7718 313 16 t t PROPN ap-7718 313 17 ) ) PUNCT ap-7718 313 18 = = NOUN ap-7718 314 1 0 0 X ap-7718 314 2 . . X ap-7718 315 1 one one NOUN ap-7718 315 2 thus thus ADV ap-7718 315 3 obtains obtain VERB ap-7718 315 4 σ2(t σ2(t PRON ap-7718 315 5 ) ) PUNCT ap-7718 315 6 = = PUNCT ap-7718 315 7 a a DET ap-7718 315 8 re[q1(t)]2 re[q1(t)]2 PROPN ap-7718 315 9 + + PROPN ap-7718 315 10 b b X ap-7718 315 11 re[q1(t re[q1(t PROPN ap-7718 315 12 ) ) PUNCT ap-7718 315 13 ] ] PUNCT ap-7718 315 14 im[q1(t im[q1(t ADV ap-7718 315 15 ) ) PUNCT ap-7718 315 16 ] ] PUNCT ap-7718 316 1 + + CCONJ ap-7718 316 2 c c X ap-7718 316 3 im[q1(t)]2 im[q1(t)]2 PROPN ap-7718 316 4 , , PUNCT ap-7718 316 5 ( ( PUNCT ap-7718 316 6 57 57 NUM ap-7718 316 7 ) ) PUNCT ap-7718 316 8 γ(t γ(t NOUN ap-7718 316 9 ) ) PUNCT ap-7718 316 10 = = SYM ap-7718 316 11 γ1 γ1 PROPN ap-7718 316 12 re[q1(t re[q1(t PROPN ap-7718 316 13 ) ) PUNCT ap-7718 316 14 ] ] PUNCT ap-7718 317 1 + + CCONJ ap-7718 317 2 γ2 γ2 PROPN ap-7718 317 3 im[q1(t im[q1(t ADV ap-7718 317 4 ) ) PUNCT ap-7718 317 5 ] ] PUNCT ap-7718 317 6 , , PUNCT ap-7718 317 7 ( ( PUNCT ap-7718 317 8 58 58 NUM ap-7718 317 9 ) ) PUNCT ap-7718 317 10 where where SCONJ ap-7718 317 11 the the DET ap-7718 317 12 wronskian wronskian NOUN ap-7718 317 13 of of ADP ap-7718 317 14 the the DET ap-7718 317 15 two two NUM ap-7718 317 16 linearly linearly ADV ap-7718 317 17 independent independent ADJ ap-7718 317 18 solutions solution NOUN ap-7718 317 19 re re NOUN ap-7718 317 20 q1 q1 PROPN ap-7718 317 21 and and CCONJ ap-7718 317 22 i i PROPN ap-7718 317 23 m m PROPN ap-7718 317 24 q1 q1 PROPN ap-7718 317 25 becomes become VERB ap-7718 317 26 w0 w0 PROPN ap-7718 317 27 = = PROPN ap-7718 317 28 w1 w1 NOUN ap-7718 317 29 , , PUNCT ap-7718 317 30 leading lead VERB ap-7718 317 31 to to ADP ap-7718 317 32 the the DET ap-7718 317 33 constraint constraint NOUN ap-7718 317 34 b2 b2 NOUN ap-7718 317 35 − − NOUN ap-7718 317 36 4ac 4ac NOUN ap-7718 317 37 = = SYM ap-7718 318 1 −4 −4 PROPN ap-7718 318 2 w2 w2 PROPN ap-7718 318 3 0 0 PROPN ap-7718 318 4 w2 w2 PROPN ap-7718 318 5 1 1 NUM ap-7718 318 6 . . PUNCT ap-7718 319 1 similarly similarly ADV ap-7718 319 2 to to ADP ap-7718 319 3 the the DET ap-7718 319 4 caldirola caldirola PROPN ap-7718 319 5 - - PUNCT ap-7718 319 6 kanai kanai PROPN ap-7718 319 7 case case NOUN ap-7718 319 8 , , PUNCT ap-7718 319 9 the the DET ap-7718 319 10 heisenberg heisenberg PROPN ap-7718 319 11 uncertainty uncertainty NOUN ap-7718 319 12 relation relation NOUN ap-7718 319 13 saturates saturate VERB ap-7718 319 14 for for ADP ap-7718 319 15 times time NOUN ap-7718 319 16 tm tm PROPN ap-7718 319 17 such such ADJ ap-7718 319 18 that that SCONJ ap-7718 319 19 wµ(tm wµ(tm NOUN ap-7718 319 20 ) ) PUNCT ap-7718 320 1 = = PUNCT ap-7718 320 2 0 0 X ap-7718 320 3 . . PUNCT ap-7718 321 1 in in ADP ap-7718 321 2 this this DET ap-7718 321 3 case case NOUN ap-7718 321 4 , , PUNCT ap-7718 321 5 an an DET ap-7718 321 6 analytic analytic ADJ ap-7718 321 7 expression expression NOUN ap-7718 321 8 for for ADP ap-7718 321 9 such such ADJ ap-7718 321 10 points point NOUN ap-7718 321 11 is be AUX ap-7718 321 12 fairly fairly ADV ap-7718 321 13 complicated complicated ADJ ap-7718 321 14 . . PUNCT ap-7718 322 1 instead instead ADV ap-7718 322 2 , , PUNCT ap-7718 322 3 one one PRON ap-7718 322 4 may may AUX ap-7718 322 5 look look VERB ap-7718 322 6 at at ADP ap-7718 322 7 the the DET ap-7718 322 8 behavior behavior NOUN ap-7718 322 9 of of ADP ap-7718 322 10 wµ(t wµ(t NOUN ap-7718 322 11 ) ) PUNCT ap-7718 322 12 depicted depict VERB ap-7718 322 13 in in ADP ap-7718 322 14 figure figure NOUN ap-7718 322 15 3a 3a NUM ap-7718 322 16 , , PUNCT ap-7718 322 17 from from ADP ap-7718 322 18 which which PRON ap-7718 322 19 it it PRON ap-7718 322 20 is be AUX ap-7718 322 21 clear clear ADJ ap-7718 322 22 that that SCONJ ap-7718 322 23 such such ADJ ap-7718 322 24 points point NOUN ap-7718 322 25 exist exist VERB ap-7718 322 26 . . PUNCT ap-7718 323 1 on on ADP ap-7718 323 2 the the DET ap-7718 323 3 other other ADJ ap-7718 323 4 hand hand NOUN ap-7718 323 5 , , PUNCT ap-7718 323 6 figure figure VERB ap-7718 323 7 3b 3b NOUN ap-7718 323 8 reveals reveal VERB ap-7718 323 9 that that SCONJ ap-7718 323 10 , , PUNCT ap-7718 323 11 in in ADP ap-7718 323 12 contradistinction contradistinction NOUN ap-7718 323 13 to to ADP ap-7718 323 14 the the DET ap-7718 323 15 caldirola caldirola PROPN ap-7718 323 16 - - PUNCT ap-7718 323 17 kanai kanai PROPN ap-7718 323 18 case case NOUN ap-7718 323 19 , , PUNCT ap-7718 323 20 the the DET ap-7718 323 21 position position NOUN ap-7718 323 22 variance variance NOUN ap-7718 323 23 does do AUX ap-7718 323 24 not not PART ap-7718 323 25 grow grow VERB ap-7718 323 26 indefinitely indefinitely ADV ap-7718 323 27 in in ADP ap-7718 323 28 time time NOUN ap-7718 323 29 . . PUNCT ap-7718 324 1 this this PRON ap-7718 324 2 is be AUX ap-7718 324 3 rather rather ADV ap-7718 324 4 expected expect VERB ap-7718 324 5 as as ADP ap-7718 324 6 , , PUNCT ap-7718 324 7 for for ADP ap-7718 324 8 asymptotic asymptotic ADJ ap-7718 324 9 times time NOUN ap-7718 324 10 t t PROPN ap-7718 324 11 > > X ap-7718 324 12 > > X ap-7718 324 13 1 1 NUM ap-7718 324 14 , , PUNCT ap-7718 324 15 the the DET ap-7718 324 16 mass mass ADJ ap-7718 324 17 term term NOUN ap-7718 324 18 converges converge VERB ap-7718 324 19 to to ADP ap-7718 324 20 a a DET ap-7718 324 21 finite finite ADJ ap-7718 324 22 value value NOUN ap-7718 324 23 different different ADJ ap-7718 324 24 from from ADP ap-7718 324 25 zero zero NUM ap-7718 324 26 . . PUNCT ap-7718 325 1 that that PRON ap-7718 325 2 is is ADV ap-7718 325 3 , , PUNCT ap-7718 325 4 the the DET ap-7718 325 5 hamiltonian hamiltonian NOUN ap-7718 325 6 becomes become VERB ap-7718 325 7 stationary stationary ADJ ap-7718 325 8 for for ADP ap-7718 325 9 asymptotic asymptotic ADJ ap-7718 325 10 values value NOUN ap-7718 325 11 . . PUNCT ap-7718 326 1 before before ADP ap-7718 326 2 conclude conclude NOUN ap-7718 326 3 , , PUNCT ap-7718 326 4 the the DET ap-7718 326 5 probability probability NOUN ap-7718 326 6 density density NOUN ap-7718 326 7 for for ADP ap-7718 326 8 ψn(x ψn(x ADP ap-7718 326 9 , , PUNCT ap-7718 326 10 t t PROPN ap-7718 326 11 ) ) PUNCT ap-7718 326 12 and and CCONJ ap-7718 326 13 ξα(x ξα(x NUM ap-7718 326 14 , , PUNCT ap-7718 326 15 t t PROPN ap-7718 326 16 ) ) PUNCT ap-7718 326 17 are be AUX ap-7718 326 18 shown show VERB ap-7718 326 19 in in ADP ap-7718 326 20 figure figure NOUN ap-7718 326 21 4 4 NUM ap-7718 326 22 . . PUNCT ap-7718 327 1 in in ADP ap-7718 327 2 the the DET ap-7718 327 3 latter latter ADJ ap-7718 327 4 , , PUNCT ap-7718 327 5 it it PRON ap-7718 327 6 can can AUX ap-7718 327 7 be be AUX ap-7718 327 8 verified verify VERB ap-7718 327 9 that that SCONJ ap-7718 327 10 the the DET ap-7718 327 11 width width NOUN ap-7718 327 12 of of ADP ap-7718 327 13 the the DET ap-7718 327 14 wavepackets wavepacket NOUN ap-7718 327 15 oscillates oscillate NOUN ap-7718 327 16 in in ADP ap-7718 327 17 a a DET ap-7718 327 18 bounded bounded ADJ ap-7718 327 19 way way NOUN ap-7718 327 20 for for ADP ap-7718 327 21 times time NOUN ap-7718 327 22 t t PROPN ap-7718 327 23 > > X ap-7718 327 24 2 2 X ap-7718 327 25 . . PUNCT ap-7718 328 1 particularly particularly ADV ap-7718 328 2 , , PUNCT ap-7718 328 3 for for ADP ap-7718 328 4 the the DET ap-7718 328 5 coherent coherent ADJ ap-7718 328 6 state state NOUN ap-7718 328 7 case case NOUN ap-7718 328 8 of of ADP ap-7718 328 9 figure figure NOUN ap-7718 328 10 4c 4c NOUN ap-7718 328 11 , , PUNCT ap-7718 328 12 the the DET ap-7718 328 13 dynamics dynamic NOUN ap-7718 328 14 of of ADP ap-7718 328 15 the the DET ap-7718 328 16 wavepacket wavepacket NOUN ap-7718 328 17 can can AUX ap-7718 328 18 be be AUX ap-7718 328 19 identified identify VERB ap-7718 328 20 clearly clearly ADV ap-7718 328 21 , , PUNCT ap-7718 328 22 where where SCONJ ap-7718 328 23 the the DET ap-7718 328 24 maximum maximum ADJ ap-7718 328 25 point point NOUN ap-7718 328 26 follows follow VERB ap-7718 328 27 the the DET ap-7718 328 28 corresponding corresponding ADJ ap-7718 328 29 classical classical ADJ ap-7718 328 30 trajectory trajectory NOUN ap-7718 328 31 ( ( PUNCT ap-7718 328 32 dashed dash VERB ap-7718 328 33 - - PUNCT ap-7718 328 34 black black NOUN ap-7718 328 35 ) ) PUNCT ap-7718 328 36 . . PUNCT ap-7718 329 1 therefore therefore ADV ap-7718 329 2 , , PUNCT ap-7718 329 3 there there PRON ap-7718 329 4 is be VERB ap-7718 329 5 no no DET ap-7718 329 6 need need NOUN ap-7718 329 7 to to PART ap-7718 329 8 introduce introduce VERB ap-7718 329 9 a a DET ap-7718 329 10 truncation truncation NOUN ap-7718 329 11 time time NOUN ap-7718 329 12 t t PROPN ap-7718 329 13 , , PUNCT ap-7718 329 14 for for SCONJ ap-7718 329 15 the the DET ap-7718 329 16 mass mass NOUN ap-7718 329 17 converges converge NOUN ap-7718 329 18 to to ADP ap-7718 329 19 a a DET ap-7718 329 20 constant constant ADJ ap-7718 329 21 value value NOUN ap-7718 329 22 different different ADJ ap-7718 329 23 from from ADP ap-7718 329 24 zero zero NUM ap-7718 329 25 , , PUNCT ap-7718 329 26 remaining remain VERB ap-7718 329 27 physically physically ADV ap-7718 329 28 reasonable reasonable ADJ ap-7718 329 29 at at ADP ap-7718 329 30 all all DET ap-7718 329 31 times time NOUN ap-7718 329 32 . . PUNCT ap-7718 330 1 5 5 NUM ap-7718 330 2 . . X ap-7718 330 3 conclusions conclusion NOUN ap-7718 330 4 in in ADP ap-7718 330 5 this this DET ap-7718 330 6 work work NOUN ap-7718 330 7 , , PUNCT ap-7718 330 8 the the DET ap-7718 330 9 class class NOUN ap-7718 330 10 of of ADP ap-7718 330 11 form form NOUN ap-7718 330 12 - - PUNCT ap-7718 330 13 preserving preserve VERB ap-7718 330 14 point point NOUN ap-7718 330 15 transformations transformation NOUN ap-7718 330 16 has have AUX ap-7718 330 17 been be AUX ap-7718 330 18 used use VERB ap-7718 330 19 to to PART ap-7718 330 20 construct construct VERB ap-7718 330 21 the the DET ap-7718 330 22 constants constant NOUN ap-7718 330 23 of of ADP ap-7718 330 24 218 218 NUM ap-7718 330 25 vol vol NOUN ap-7718 330 26 . . PUNCT ap-7718 331 1 62 62 NUM ap-7718 331 2 no no INTJ ap-7718 331 3 . . PUNCT ap-7718 332 1 1/2022 1/2022 NUM ap-7718 332 2 td td NOUN ap-7718 332 3 mass mass NOUN ap-7718 332 4 oscillators oscillator NOUN ap-7718 332 5 : : PUNCT ap-7718 332 6 constants constant NOUN ap-7718 332 7 of of ADP ap-7718 332 8 motion motion NOUN ap-7718 332 9 and and CCONJ ap-7718 332 10 semiclasical semiclasical ADJ ap-7718 332 11 states state NOUN ap-7718 332 12 ( ( PUNCT ap-7718 332 13 a a NOUN ap-7718 332 14 ) ) PUNCT ap-7718 332 15 . . PUNCT ap-7718 333 1 n n NOUN ap-7718 333 2 = = SYM ap-7718 333 3 0 0 PUNCT ap-7718 333 4 ( ( PUNCT ap-7718 333 5 b b NOUN ap-7718 333 6 ) ) PUNCT ap-7718 333 7 . . PUNCT ap-7718 334 1 n n NOUN ap-7718 334 2 = = SYM ap-7718 334 3 1 1 NUM ap-7718 334 4 ( ( PUNCT ap-7718 334 5 c c NOUN ap-7718 334 6 ) ) PUNCT ap-7718 334 7 . . PUNCT ap-7718 335 1 figure figure VERB ap-7718 335 2 4 4 NUM ap-7718 335 3 . . NOUN ap-7718 335 4 probability probability NOUN ap-7718 335 5 density density NOUN ap-7718 335 6 pn pn NOUN ap-7718 335 7 = = SYM ap-7718 335 8 |ψn(x |ψn(x PROPN ap-7718 335 9 , , PUNCT ap-7718 335 10 t)|2 t)|2 NOUN ap-7718 335 11 for for ADP ap-7718 335 12 n n NOUN ap-7718 335 13 = = SYM ap-7718 335 14 0 0 PUNCT ap-7718 335 15 ( ( PUNCT ap-7718 335 16 a a NOUN ap-7718 335 17 ) ) PUNCT ap-7718 335 18 , , PUNCT ap-7718 335 19 n n NOUN ap-7718 335 20 = = SYM ap-7718 335 21 1 1 NUM ap-7718 335 22 ( ( PUNCT ap-7718 335 23 b b NOUN ap-7718 335 24 ) ) PUNCT ap-7718 335 25 , , PUNCT ap-7718 335 26 and and CCONJ ap-7718 335 27 pα pα VERB ap-7718 335 28 = = SYM ap-7718 335 29 |ψn(x |ψn(x PROPN ap-7718 335 30 , , PUNCT ap-7718 335 31 t)|2 t)|2 NOUN ap-7718 335 32 ( ( PUNCT ap-7718 335 33 c c NOUN ap-7718 335 34 ) ) PUNCT ap-7718 335 35 associated associate VERB ap-7718 335 36 with with ADP ap-7718 335 37 the the DET ap-7718 335 38 regularized regularize VERB ap-7718 335 39 caldirola caldirola PROPN ap-7718 335 40 - - PUNCT ap-7718 335 41 kanai kanai PROPN ap-7718 335 42 mass mass PROPN ap-7718 335 43 term term PROPN ap-7718 335 44 µrck(t µrck(t PROPN ap-7718 335 45 ) ) PUNCT ap-7718 335 46 . . PUNCT ap-7718 336 1 the the DET ap-7718 336 2 rest rest NOUN ap-7718 336 3 of of ADP ap-7718 336 4 parameters parameter NOUN ap-7718 336 5 have have AUX ap-7718 336 6 been be AUX ap-7718 336 7 fixed fix VERB ap-7718 336 8 as as ADP ap-7718 336 9 w0 w0 PROPN ap-7718 336 10 = = PROPN ap-7718 336 11 a a PRON ap-7718 336 12 = = SYM ap-7718 336 13 b b NOUN ap-7718 336 14 = = SYM ap-7718 336 15 1 1 NUM ap-7718 336 16 , , PUNCT ap-7718 336 17 w1 w1 NOUN ap-7718 336 18 = = SYM ap-7718 336 19 2 2 NUM ap-7718 336 20 , , PUNCT ap-7718 336 21 µ0 µ0 NOUN ap-7718 336 22 = = SYM ap-7718 336 23 0.3 0.3 NUM ap-7718 336 24 , , PUNCT ap-7718 336 25 and and CCONJ ap-7718 336 26 κ κ X ap-7718 336 27 = = NOUN ap-7718 336 28 0.5 0.5 NUM ap-7718 336 29 . . PUNCT ap-7718 336 30 motion motion NOUN ap-7718 336 31 for for ADP ap-7718 336 32 the the DET ap-7718 336 33 family family NOUN ap-7718 336 34 of of ADP ap-7718 336 35 time time NOUN ap-7718 336 36 - - PUNCT ap-7718 336 37 dependent dependent ADJ ap-7718 336 38 mass mass NOUN ap-7718 336 39 oscillators oscillator NOUN ap-7718 336 40 . . PUNCT ap-7718 337 1 this this PRON ap-7718 337 2 was be AUX ap-7718 337 3 achieved achieve VERB ap-7718 337 4 by by ADP ap-7718 337 5 exploiting exploit VERB ap-7718 337 6 the the DET ap-7718 337 7 preservation preservation NOUN ap-7718 337 8 of of ADP ap-7718 337 9 first first ADJ ap-7718 337 10 - - PUNCT ap-7718 337 11 integrals integral NOUN ap-7718 337 12 on on ADP ap-7718 337 13 the the DET ap-7718 337 14 initial initial ADJ ap-7718 337 15 stationary stationary ADJ ap-7718 337 16 oscillator oscillator NOUN ap-7718 337 17 model model NOUN ap-7718 337 18 . . PUNCT ap-7718 338 1 since since SCONJ ap-7718 338 2 several several ADJ ap-7718 338 3 constants constant NOUN ap-7718 338 4 of of ADP ap-7718 338 5 motion motion NOUN ap-7718 338 6 are be AUX ap-7718 338 7 already already ADV ap-7718 338 8 known know VERB ap-7718 338 9 for for ADP ap-7718 338 10 the the DET ap-7718 338 11 initial initial ADJ ap-7718 338 12 system system NOUN ap-7718 338 13 , , PUNCT ap-7718 338 14 the the DET ap-7718 338 15 corresponding correspond VERB ap-7718 338 16 counterparts counterpart NOUN ap-7718 338 17 for for ADP ap-7718 338 18 the the DET ap-7718 338 19 time time NOUN ap-7718 338 20 - - PUNCT ap-7718 338 21 dependent dependent ADJ ap-7718 338 22 model model NOUN ap-7718 338 23 are be AUX ap-7718 338 24 straightforwardly straightforwardly ADV ap-7718 338 25 constructed construct VERB ap-7718 338 26 by by ADP ap-7718 338 27 implementing implement VERB ap-7718 338 28 the the DET ap-7718 338 29 appropriate appropriate ADJ ap-7718 338 30 mappings mapping NOUN ap-7718 338 31 . . PUNCT ap-7718 339 1 notably notably ADV ap-7718 339 2 , , PUNCT ap-7718 339 3 three three NUM ap-7718 339 4 different different ADJ ap-7718 339 5 constants constant NOUN ap-7718 339 6 of of ADP ap-7718 339 7 motion motion NOUN ap-7718 339 8 were be AUX ap-7718 339 9 identified identify VERB ap-7718 339 10 , , PUNCT ap-7718 339 11 one one NUM ap-7718 339 12 that that PRON ap-7718 339 13 admits admit VERB ap-7718 339 14 an an DET ap-7718 339 15 orthogonal orthogonal ADJ ap-7718 339 16 set set NOUN ap-7718 339 17 of of ADP ap-7718 339 18 eigensolutions eigensolution NOUN ap-7718 339 19 , , PUNCT ap-7718 339 20 another another PRON ap-7718 339 21 that that PRON ap-7718 339 22 permits permit VERB ap-7718 339 23 non non ADJ ap-7718 339 24 - - ADJ ap-7718 339 25 orthogonal orthogonal ADJ ap-7718 339 26 eigensolutions eigensolution NOUN ap-7718 339 27 , , PUNCT ap-7718 339 28 and and CCONJ ap-7718 339 29 the the DET ap-7718 339 30 third third ADJ ap-7718 339 31 one one NOUN ap-7718 339 32 that that PRON ap-7718 339 33 does do AUX ap-7718 339 34 not not PART ap-7718 339 35 admit admit VERB ap-7718 339 36 finite finite ADJ ap-7718 339 37 - - ADJ ap-7718 339 38 norm norm ADJ ap-7718 339 39 solutions solution NOUN ap-7718 339 40 . . PUNCT ap-7718 340 1 interestingly interestingly ADV ap-7718 340 2 , , PUNCT ap-7718 340 3 the the DET ap-7718 340 4 non non ADJ ap-7718 340 5 - - ADJ ap-7718 340 6 orthogonal orthogonal ADJ ap-7718 340 7 eigensolutions eigensolution NOUN ap-7718 340 8 are be AUX ap-7718 340 9 actually actually ADV ap-7718 340 10 coherent coherent ADJ ap-7718 340 11 states state NOUN ap-7718 340 12 , , PUNCT ap-7718 340 13 for for SCONJ ap-7718 340 14 they they PRON ap-7718 340 15 are be AUX ap-7718 340 16 constructed construct VERB ap-7718 340 17 from from ADP ap-7718 340 18 the the DET ap-7718 340 19 annihilation annihilation NOUN ap-7718 340 20 operator operator NOUN ap-7718 340 21 of of ADP ap-7718 340 22 the the DET ap-7718 340 23 time time NOUN ap-7718 340 24 - - PUNCT ap-7718 340 25 dependent dependent ADJ ap-7718 340 26 oscillator oscillator NOUN ap-7718 340 27 . . PUNCT ap-7718 341 1 furthermore furthermore ADV ap-7718 341 2 , , PUNCT ap-7718 341 3 by by ADP ap-7718 341 4 exploiting exploit VERB ap-7718 341 5 the the DET ap-7718 341 6 underlying underlie VERB ap-7718 341 7 weyl weyl PROPN ap-7718 341 8 - - PUNCT ap-7718 341 9 heisenberg heisenberg PROPN ap-7718 341 10 algebra algebra PROPN ap-7718 341 11 fulfilled fulfil VERB ap-7718 341 12 by by ADP ap-7718 341 13 the the DET ap-7718 341 14 quantum quantum ADJ ap-7718 341 15 invariants invariant NOUN ap-7718 341 16 , , PUNCT ap-7718 341 17 it it PRON ap-7718 341 18 was be AUX ap-7718 341 19 possible possible ADJ ap-7718 341 20 to to PART ap-7718 341 21 find find VERB ap-7718 341 22 exact exact ADJ ap-7718 341 23 expressions expression NOUN ap-7718 341 24 for for ADP ap-7718 341 25 the the DET ap-7718 341 26 expectation expectation NOUN ap-7718 341 27 values value NOUN ap-7718 341 28 of of ADP ap-7718 341 29 the the DET ap-7718 341 30 position position NOUN ap-7718 341 31 and and CCONJ ap-7718 341 32 momentum momentum NOUN ap-7718 341 33 observables observable NOUN ap-7718 341 34 . . PUNCT ap-7718 342 1 the the DET ap-7718 342 2 latter latter ADJ ap-7718 342 3 revealed reveal VERB ap-7718 342 4 the the DET ap-7718 342 5 coherent coherent ADJ ap-7718 342 6 states state NOUN ap-7718 342 7 are be AUX ap-7718 342 8 represented represent VERB ap-7718 342 9 by by ADP ap-7718 342 10 gaussian gaussian ADJ ap-7718 342 11 wavepacket wavepacket NOUN ap-7718 342 12 whose whose DET ap-7718 342 13 maximum maximum NOUN ap-7718 342 14 follows follow VERB ap-7718 342 15 the the DET ap-7718 342 16 corresponding corresponding ADJ ap-7718 342 17 classical classical ADJ ap-7718 342 18 trajectory trajectory NOUN ap-7718 342 19 . . PUNCT ap-7718 343 1 besides besides SCONJ ap-7718 343 2 the the DET ap-7718 343 3 latter latter ADJ ap-7718 343 4 properties property NOUN ap-7718 343 5 , , PUNCT ap-7718 343 6 it it PRON ap-7718 343 7 was be AUX ap-7718 343 8 also also ADV ap-7718 343 9 found find VERB ap-7718 343 10 that that SCONJ ap-7718 343 11 , , PUNCT ap-7718 343 12 in in ADP ap-7718 343 13 general general ADJ ap-7718 343 14 , , PUNCT ap-7718 343 15 the the DET ap-7718 343 16 schrödinger schrödinger ADJ ap-7718 343 17 - - PUNCT ap-7718 343 18 robertson robertson PROPN ap-7718 343 19 uncertainty uncertainty NOUN ap-7718 343 20 relation relation NOUN ap-7718 343 21 saturates saturate VERB ap-7718 343 22 for for ADP ap-7718 343 23 all all DET ap-7718 343 24 times time NOUN ap-7718 343 25 , , PUNCT ap-7718 343 26 whereas whereas SCONJ ap-7718 343 27 the the DET ap-7718 343 28 heisenberg heisenberg PROPN ap-7718 343 29 one one NOUN ap-7718 343 30 gets get AUX ap-7718 343 31 minimized minimize VERB ap-7718 343 32 only only ADV ap-7718 343 33 for for ADP ap-7718 343 34 some some DET ap-7718 343 35 times time NOUN ap-7718 343 36 . . PUNCT ap-7718 344 1 still still ADV ap-7718 344 2 , , PUNCT ap-7718 344 3 two two NUM ap-7718 344 4 special special ADJ ap-7718 344 5 time time NOUN ap-7718 344 6 - - PUNCT ap-7718 344 7 dependent dependent ADJ ap-7718 344 8 hamiltonians hamiltonian NOUN ap-7718 344 9 exist exist VERB ap-7718 344 10 so so SCONJ ap-7718 344 11 that that SCONJ ap-7718 344 12 the the DET ap-7718 344 13 heisenberg heisenberg PROPN ap-7718 344 14 inequality inequality PROPN ap-7718 344 15 saturates saturate VERB ap-7718 344 16 at at ADP ap-7718 344 17 all all DET ap-7718 344 18 times time NOUN ap-7718 344 19 , , PUNCT ap-7718 344 20 one one NUM ap-7718 344 21 of of ADP ap-7718 344 22 which which PRON ap-7718 344 23 is be AUX ap-7718 344 24 the the DET ap-7718 344 25 stationary stationary ADJ ap-7718 344 26 limit limit NOUN ap-7718 344 27 case case NOUN ap-7718 344 28 , , PUNCT ap-7718 344 29 as as SCONJ ap-7718 344 30 expected expect VERB ap-7718 344 31 . . PUNCT ap-7718 345 1 remarkably remarkably ADV ap-7718 345 2 , , PUNCT ap-7718 345 3 the the DET ap-7718 345 4 newly newly ADV ap-7718 345 5 introduced introduce VERB ap-7718 345 6 regularized regularize VERB ap-7718 345 7 caldirola caldirola PROPN ap-7718 345 8 - - PUNCT ap-7718 345 9 kanai kanai PROPN ap-7718 345 10 mass mass PROPN ap-7718 345 11 term term NOUN ap-7718 345 12 admits admit VERB ap-7718 345 13 exact exact ADJ ap-7718 345 14 solutions solution NOUN ap-7718 345 15 that that PRON ap-7718 345 16 regularize regularize VERB ap-7718 345 17 the the DET ap-7718 345 18 unusual unusual ADJ ap-7718 345 19 behavior behavior NOUN ap-7718 345 20 observed observe VERB ap-7718 345 21 in in ADP ap-7718 345 22 the the DET ap-7718 345 23 conventional conventional ADJ ap-7718 345 24 caldirola caldirola PROPN ap-7718 345 25 - - PUNCT ap-7718 345 26 kanai kanai PROPN ap-7718 345 27 case case NOUN ap-7718 345 28 . . PUNCT ap-7718 346 1 more more ADV ap-7718 346 2 precisely precisely ADV ap-7718 346 3 , , PUNCT ap-7718 346 4 the the DET ap-7718 346 5 variances variance NOUN ap-7718 346 6 become become VERB ap-7718 346 7 bounded bounded ADJ ap-7718 346 8 as as ADV ap-7718 346 9 well well ADV ap-7718 346 10 as as ADP ap-7718 346 11 the the DET ap-7718 346 12 expectation expectation NOUN ap-7718 346 13 values value NOUN ap-7718 346 14 . . PUNCT ap-7718 347 1 this this PRON ap-7718 347 2 allows allow VERB ap-7718 347 3 obtaining obtain VERB ap-7718 347 4 localization localization NOUN ap-7718 347 5 of of ADP ap-7718 347 6 particles particle NOUN ap-7718 347 7 , , PUNCT ap-7718 347 8 which which PRON ap-7718 347 9 is be AUX ap-7718 347 10 desired desire VERB ap-7718 347 11 in in ADP ap-7718 347 12 physical physical ADJ ap-7718 347 13 implementations implementation NOUN ap-7718 347 14 such such ADJ ap-7718 347 15 as as ADP ap-7718 347 16 traps trap NOUN ap-7718 347 17 of of ADP ap-7718 347 18 charged charge VERB ap-7718 347 19 particles particle NOUN ap-7718 347 20 . . PUNCT ap-7718 348 1 acknowledgements acknowledgement NOUN ap-7718 348 2 the the DET ap-7718 348 3 author author NOUN ap-7718 348 4 acknowledges acknowledge VERB ap-7718 348 5 the the DET ap-7718 348 6 support support NOUN ap-7718 348 7 from from ADP ap-7718 348 8 the the DET ap-7718 348 9 project project NOUN ap-7718 348 10 “ " PUNCT ap-7718 348 11 physicist physicist NOUN ap-7718 348 12 on on ADP ap-7718 348 13 the the DET ap-7718 348 14 move move NOUN ap-7718 348 15 ii ii PROPN ap-7718 348 16 ” " PUNCT ap-7718 348 17 ( ( PUNCT ap-7718 348 18 kineó kineó PROPN ap-7718 348 19 ii ii PROPN ap-7718 348 20 ) ) PUNCT ap-7718 348 21 , , PUNCT ap-7718 348 22 czech czech PROPN ap-7718 348 23 republic republic NOUN ap-7718 348 24 , , PUNCT ap-7718 348 25 grant grant VERB ap-7718 348 26 no no INTJ ap-7718 348 27 . . PUNCT ap-7718 348 28 cz.02.2.69/0.0/0.0/18_053/0017163 cz.02.2.69/0.0/0.0/18_053/0017163 PROPN ap-7718 348 29 . . PUNCT ap-7718 349 1 this this DET ap-7718 349 2 research research NOUN ap-7718 349 3 has have AUX ap-7718 349 4 been be AUX ap-7718 349 5 also also ADV ap-7718 349 6 funded fund VERB ap-7718 349 7 by by ADP ap-7718 349 8 consejo consejo PROPN ap-7718 349 9 nacional nacional PROPN ap-7718 349 10 de de X ap-7718 349 11 ciencia ciencia PROPN ap-7718 349 12 y y PROPN ap-7718 349 13 tecnología tecnología PROPN ap-7718 349 14 ( ( PUNCT ap-7718 349 15 conacyt conacyt NOUN ap-7718 349 16 ) ) PUNCT ap-7718 349 17 , , PUNCT ap-7718 349 18 mexico mexico PROPN ap-7718 349 19 , , PUNCT ap-7718 349 20 grant grant VERB ap-7718 349 21 no no PRON ap-7718 349 22 . . PUNCT ap-7718 350 1 a1 a1 NOUN ap-7718 350 2 - - PUNCT ap-7718 350 3 s-24569 s-24569 NOUN ap-7718 350 4 . . PUNCT ap-7718 351 1 references reference NOUN ap-7718 351 2 [ [ X ap-7718 351 3 1 1 NUM ap-7718 351 4 ] ] X ap-7718 351 5 f. f. PROPN ap-7718 351 6 cooper cooper PROPN ap-7718 351 7 , , PUNCT ap-7718 351 8 a. a. PROPN ap-7718 351 9 khare khare PROPN ap-7718 351 10 , , PUNCT ap-7718 351 11 u. u. PROPN ap-7718 351 12 sukhatme sukhatme PROPN ap-7718 351 13 . . PUNCT ap-7718 352 1 supersymmetry supersymmetry NOUN ap-7718 352 2 in in ADP ap-7718 352 3 quantum quantum ADJ ap-7718 352 4 mechanics mechanic NOUN ap-7718 352 5 . . PUNCT ap-7718 353 1 world world PROPN ap-7718 353 2 scientific scientific PROPN ap-7718 353 3 , , PUNCT ap-7718 353 4 singapore singapore PROPN ap-7718 353 5 , , PUNCT ap-7718 353 6 2001 2001 NUM ap-7718 353 7 . . PUNCT ap-7718 354 1 [ [ X ap-7718 354 2 2 2 NUM ap-7718 354 3 ] ] PUNCT ap-7718 354 4 f. f. PROPN ap-7718 354 5 schwabl schwabl PROPN ap-7718 354 6 . . PUNCT ap-7718 355 1 quantum quantum ADJ ap-7718 355 2 mechanics mechanic NOUN ap-7718 355 3 . . PUNCT ap-7718 356 1 springer springer NOUN ap-7718 356 2 - - PUNCT ap-7718 356 3 verlag verlag PROPN ap-7718 356 4 , , PUNCT ap-7718 356 5 berlin berlin PROPN ap-7718 356 6 , , PUNCT ap-7718 356 7 3rd 3rd ADJ ap-7718 356 8 edn edn NOUN ap-7718 356 9 . . PUNCT ap-7718 356 10 , , PUNCT ap-7718 356 11 2002 2002 NUM ap-7718 356 12 . . PUNCT ap-7718 357 1 [ [ X ap-7718 357 2 3 3 NUM ap-7718 357 3 ] ] PUNCT ap-7718 357 4 a. a. PROPN ap-7718 357 5 bohm bohm PROPN ap-7718 357 6 , , PUNCT ap-7718 357 7 a. a. NOUN ap-7718 357 8 mostafazadeh mostafazadeh PROPN ap-7718 357 9 , , PUNCT ap-7718 357 10 h. h. PROPN ap-7718 357 11 koizumi koizumi PROPN ap-7718 357 12 , , PUNCT ap-7718 357 13 et et PROPN ap-7718 357 14 al al PROPN ap-7718 357 15 . . PUNCT ap-7718 358 1 the the DET ap-7718 358 2 geometric geometric ADJ ap-7718 358 3 phase phase NOUN ap-7718 358 4 in in ADP ap-7718 358 5 quantum quantum ADJ ap-7718 358 6 systems system NOUN ap-7718 358 7 : : PUNCT ap-7718 358 8 foundations foundation NOUN ap-7718 358 9 , , PUNCT ap-7718 358 10 mathematical mathematical ADJ ap-7718 358 11 concepts concept NOUN ap-7718 358 12 , , PUNCT ap-7718 358 13 and and CCONJ ap-7718 358 14 applications application NOUN ap-7718 358 15 in in ADP ap-7718 358 16 molecular molecular ADJ ap-7718 358 17 and and CCONJ ap-7718 358 18 condensed condense VERB ap-7718 358 19 matter matter NOUN ap-7718 358 20 physics physics NOUN ap-7718 358 21 . . PUNCT ap-7718 359 1 springer springer NOUN ap-7718 359 2 - - PUNCT ap-7718 359 3 verlag verlag PROPN ap-7718 359 4 , , PUNCT ap-7718 359 5 berlin berlin PROPN ap-7718 359 6 , , PUNCT ap-7718 359 7 2003 2003 NUM ap-7718 359 8 . . PUNCT ap-7718 360 1 [ [ X ap-7718 360 2 4 4 X ap-7718 360 3 ] ] PUNCT ap-7718 360 4 w. w. PROPN ap-7718 360 5 paul paul PROPN ap-7718 360 6 . . PUNCT ap-7718 361 1 electromagnetic electromagnetic ADJ ap-7718 361 2 traps trap NOUN ap-7718 361 3 for for ADP ap-7718 361 4 charged charge VERB ap-7718 361 5 and and CCONJ ap-7718 361 6 neutral neutral ADJ ap-7718 361 7 particles particle NOUN ap-7718 361 8 . . PUNCT ap-7718 362 1 reviews review NOUN ap-7718 362 2 of of ADP ap-7718 362 3 modern modern ADJ ap-7718 362 4 physics physics NOUN ap-7718 362 5 62(3):531–540 62(3):531–540 PROPN ap-7718 362 6 , , PUNCT ap-7718 362 7 1990 1990 NUM ap-7718 362 8 . . PUNCT ap-7718 363 1 https://doi.org/10.1103/revmodphys.62.531 https://doi.org/10.1103/revmodphys.62.531 PROPN ap-7718 363 2 . . PUNCT ap-7718 364 1 [ [ X ap-7718 364 2 5 5 NUM ap-7718 364 3 ] ] PUNCT ap-7718 364 4 m. m. NOUN ap-7718 364 5 combescure combescure NOUN ap-7718 364 6 . . PUNCT ap-7718 365 1 a a DET ap-7718 365 2 quantum quantum NOUN ap-7718 365 3 particle particle NOUN ap-7718 365 4 in in ADP ap-7718 365 5 a a DET ap-7718 365 6 quadrupole quadrupole NOUN ap-7718 365 7 radio radio NOUN ap-7718 365 8 - - PUNCT ap-7718 365 9 frequency frequency NOUN ap-7718 365 10 trap trap NOUN ap-7718 365 11 . . PUNCT ap-7718 366 1 annales annales PROPN ap-7718 366 2 de de ADP ap-7718 366 3 l’ihp l’ihp NOUN ap-7718 366 4 physique physique NOUN ap-7718 366 5 théorique théorique PROPN ap-7718 366 6 44(3):293–314 44(3):293–314 PROPN ap-7718 366 7 , , PUNCT ap-7718 366 8 1986 1986 NUM ap-7718 366 9 . . PUNCT ap-7718 367 1 http http ADJ ap-7718 367 2 : : PUNCT ap-7718 367 3 //www.numdam.org //www.numdam.org ADJ ap-7718 367 4 / / SYM ap-7718 367 5 item item NOUN ap-7718 367 6 / / SYM ap-7718 367 7 aihpa_1986__44_3_293_0/. aihpa_1986__44_3_293_0/. NOUN ap-7718 368 1 [ [ X ap-7718 368 2 6 6 NUM ap-7718 368 3 ] ] PUNCT ap-7718 368 4 d. d. PROPN ap-7718 368 5 e. e. PROPN ap-7718 368 6 pritchard pritchard PROPN ap-7718 368 7 . . PUNCT ap-7718 369 1 cooling cool VERB ap-7718 369 2 neutral neutral ADJ ap-7718 369 3 atoms atom NOUN ap-7718 369 4 in in ADP ap-7718 369 5 a a DET ap-7718 369 6 magnetic magnetic ADJ ap-7718 369 7 trap trap NOUN ap-7718 369 8 for for ADP ap-7718 369 9 precision precision NOUN ap-7718 369 10 spectroscopy spectroscopy NOUN ap-7718 369 11 . . PUNCT ap-7718 370 1 physical physical ADJ ap-7718 370 2 review review NOUN ap-7718 370 3 letters letter NOUN ap-7718 370 4 51:1336–1339 51:1336–1339 NUM ap-7718 370 5 , , PUNCT ap-7718 370 6 1983 1983 NUM ap-7718 370 7 . . PUNCT ap-7718 371 1 https://doi.org/10.1103/physrevlett.51.1336 https://doi.org/10.1103/physrevlett.51.1336 PROPN ap-7718 371 2 . . PUNCT ap-7718 372 1 [ [ X ap-7718 372 2 7 7 X ap-7718 372 3 ] ] X ap-7718 372 4 r. r. PROPN ap-7718 372 5 j. j. PROPN ap-7718 372 6 glauber glauber PROPN ap-7718 372 7 . . PUNCT ap-7718 372 8 quantum quantum PROPN ap-7718 372 9 theory theory NOUN ap-7718 372 10 of of ADP ap-7718 372 11 optical optical ADJ ap-7718 372 12 coherence coherence NOUN ap-7718 372 13 , , PUNCT ap-7718 372 14 chap chap NOUN ap-7718 372 15 . . PUNCT ap-7718 373 1 the the DET ap-7718 373 2 quantum quantum ADJ ap-7718 373 3 mechanics mechanic NOUN ap-7718 373 4 of of ADP ap-7718 373 5 trapped trap VERB ap-7718 373 6 wavepackets wavepacket NOUN ap-7718 373 7 , , PUNCT ap-7718 373 8 pp pp ADV ap-7718 373 9 . . PUNCT ap-7718 374 1 577–594 577–594 NUM ap-7718 374 2 . . PUNCT ap-7718 375 1 john john PROPN ap-7718 375 2 wiley wiley PROPN ap-7718 375 3 & & CCONJ ap-7718 375 4 sons sons PROPN ap-7718 375 5 , , PUNCT ap-7718 375 6 ltd ltd PROPN ap-7718 375 7 , , PUNCT ap-7718 375 8 2006 2006 NUM ap-7718 375 9 . . PUNCT ap-7718 376 1 https://doi.org/10.1002/9783527610075.ch15 https://doi.org/10.1002/9783527610075.ch15 NOUN ap-7718 376 2 . . PUNCT ap-7718 377 1 [ [ X ap-7718 377 2 8 8 NUM ap-7718 377 3 ] ] X ap-7718 377 4 b. b. PROPN ap-7718 377 5 m. m. PROPN ap-7718 377 6 mihalcea mihalcea PROPN ap-7718 377 7 , , PUNCT ap-7718 377 8 s. s. PROPN ap-7718 377 9 lynch lynch PROPN ap-7718 377 10 . . PUNCT ap-7718 378 1 investigations investigation NOUN ap-7718 378 2 on on ADP ap-7718 378 3 dynamical dynamical ADJ ap-7718 378 4 stability stability NOUN ap-7718 378 5 in in ADP ap-7718 378 6 3d 3d NUM ap-7718 378 7 quadrupole quadrupole NOUN ap-7718 378 8 ion ion NOUN ap-7718 378 9 traps trap NOUN ap-7718 378 10 . . PUNCT ap-7718 379 1 applied apply VERB ap-7718 379 2 sciences science NOUN ap-7718 379 3 11(7):2938 11(7):2938 NUM ap-7718 379 4 , , PUNCT ap-7718 379 5 2021 2021 NUM ap-7718 379 6 . . PUNCT ap-7718 380 1 https://doi.org/10.3390/app11072938 https://doi.org/10.3390/app11072938 ADJ ap-7718 380 2 . . PUNCT ap-7718 381 1 [ [ X ap-7718 381 2 9 9 NUM ap-7718 381 3 ] ] X ap-7718 381 4 h. h. PROPN ap-7718 381 5 r. r. PROPN ap-7718 381 6 lewis lewis PROPN ap-7718 381 7 , , PUNCT ap-7718 381 8 w. w. PROPN ap-7718 381 9 b. b. PROPN ap-7718 381 10 riesenfled riesenfle VERB ap-7718 381 11 . . PUNCT ap-7718 382 1 an an DET ap-7718 382 2 exact exact ADJ ap-7718 382 3 quantum quantum NOUN ap-7718 382 4 theory theory NOUN ap-7718 382 5 of of ADP ap-7718 382 6 the the DET ap-7718 382 7 time time NOUN ap-7718 382 8 - - PUNCT ap-7718 382 9 dependent dependent ADJ ap-7718 382 10 harmonic harmonic ADJ ap-7718 382 11 oscillator oscillator NOUN ap-7718 382 12 and and CCONJ ap-7718 382 13 of of ADP ap-7718 382 14 a a DET ap-7718 382 15 charged charge VERB ap-7718 382 16 particle particle NOUN ap-7718 382 17 in in ADP ap-7718 382 18 a a DET ap-7718 382 19 time time NOUN ap-7718 382 20 - - PUNCT ap-7718 382 21 dependent dependent ADJ ap-7718 382 22 electromagnetic electromagnetic ADJ ap-7718 382 23 219 219 NUM ap-7718 382 24 https://doi.org/10.1103/revmodphys.62.531 https://doi.org/10.1103/revmodphys.62.531 PROPN ap-7718 382 25 http://www.numdam.org/item/aihpa_1986__44_3_293_0/ http://www.numdam.org/item/aihpa_1986__44_3_293_0/ PROPN ap-7718 382 26 http://www.numdam.org/item/aihpa_1986__44_3_293_0/ http://www.numdam.org/item/aihpa_1986__44_3_293_0/ PROPN ap-7718 382 27 https://doi.org/10.1103/physrevlett.51.1336 https://doi.org/10.1103/physrevlett.51.1336 VERB ap-7718 382 28 https://doi.org/10.1002/9783527610075.ch15 https://doi.org/10.1002/9783527610075.ch15 PROPN ap-7718 383 1 https://doi.org/10.3390/app11072938 https://doi.org/10.3390/app11072938 PROPN ap-7718 383 2 kevin kevin PROPN ap-7718 383 3 zelaya zelaya PROPN ap-7718 383 4 acta acta PROPN ap-7718 383 5 polytechnica polytechnica PROPN ap-7718 383 6 field field NOUN ap-7718 383 7 . . PUNCT ap-7718 384 1 journal journal PROPN ap-7718 384 2 of of ADP ap-7718 384 3 mathematical mathematical ADJ ap-7718 384 4 physics physics NOUN ap-7718 384 5 10(8):1458–1473 10(8):1458–1473 NUM ap-7718 384 6 , , PUNCT ap-7718 384 7 1969 1969 NUM ap-7718 384 8 . . PUNCT ap-7718 385 1 https://doi.org/10.1063/1.1664991 https://doi.org/10.1063/1.1664991 X ap-7718 385 2 . . PUNCT ap-7718 386 1 [ [ X ap-7718 386 2 10 10 NUM ap-7718 386 3 ] ] X ap-7718 386 4 v. v. PROPN ap-7718 386 5 v. v. CCONJ ap-7718 386 6 dodonov dodonov PROPN ap-7718 386 7 , , PUNCT ap-7718 386 8 v. v. PROPN ap-7718 386 9 i. i. NOUN ap-7718 386 10 man’ko man’ko PROPN ap-7718 386 11 . . PUNCT ap-7718 387 1 coherent coherent ADJ ap-7718 387 2 states state NOUN ap-7718 387 3 and and CCONJ ap-7718 387 4 the the DET ap-7718 387 5 resonance resonance NOUN ap-7718 387 6 of of ADP ap-7718 387 7 a a DET ap-7718 387 8 quantum quantum NOUN ap-7718 387 9 damped damped NOUN ap-7718 387 10 oscillator oscillator NOUN ap-7718 387 11 . . PUNCT ap-7718 388 1 physical physical ADJ ap-7718 388 2 review review NOUN ap-7718 388 3 a a DET ap-7718 388 4 20:550–560 20:550–560 PROPN ap-7718 388 5 , , PUNCT ap-7718 388 6 1979 1979 NUM ap-7718 388 7 . . PUNCT ap-7718 389 1 https://doi.org/10.1103/physreva.20.550 https://doi.org/10.1103/physreva.20.550 PROPN ap-7718 389 2 . . PUNCT ap-7718 390 1 [ [ X ap-7718 390 2 11 11 NUM ap-7718 390 3 ] ] X ap-7718 390 4 v. v. PROPN ap-7718 390 5 v. v. CCONJ ap-7718 390 6 dodonov dodonov PROPN ap-7718 390 7 , , PUNCT ap-7718 390 8 o. o. PROPN ap-7718 390 9 v. v. PROPN ap-7718 390 10 man’ko man’ko PROPN ap-7718 390 11 , , PUNCT ap-7718 390 12 v. v. PROPN ap-7718 390 13 i. i. NOUN ap-7718 390 14 man’ko man’ko PROPN ap-7718 390 15 . . PUNCT ap-7718 391 1 quantum quantum PROPN ap-7718 391 2 nonstationary nonstationary ADJ ap-7718 391 3 oscillator oscillator NOUN ap-7718 391 4 : : PUNCT ap-7718 391 5 models model NOUN ap-7718 391 6 and and CCONJ ap-7718 391 7 applications application NOUN ap-7718 391 8 . . PUNCT ap-7718 392 1 journal journal NOUN ap-7718 392 2 of of ADP ap-7718 392 3 russian russian ADJ ap-7718 392 4 laser laser NOUN ap-7718 392 5 research research NOUN ap-7718 392 6 16(1):1 16(1):1 NUM ap-7718 392 7 – – PUNCT ap-7718 392 8 56 56 NUM ap-7718 392 9 , , PUNCT ap-7718 392 10 1995 1995 NUM ap-7718 392 11 . . PUNCT ap-7718 393 1 https://doi.org/10.1007/bf02581075 https://doi.org/10.1007/bf02581075 PROPN ap-7718 393 2 . . PUNCT ap-7718 394 1 [ [ X ap-7718 394 2 12 12 NUM ap-7718 394 3 ] ] X ap-7718 394 4 i. i. PROPN ap-7718 394 5 ramos ramos PROPN ap-7718 394 6 - - PROPN ap-7718 394 7 prieto prieto PROPN ap-7718 394 8 , , PUNCT ap-7718 394 9 a. a. PROPN ap-7718 394 10 r. r. PROPN ap-7718 394 11 urzúa urzúa PROPN ap-7718 394 12 , , PUNCT ap-7718 394 13 m. m. NOUN ap-7718 394 14 fernández fernández PROPN ap-7718 394 15 - - PUNCT ap-7718 394 16 guasti guasti PROPN ap-7718 394 17 , , PUNCT ap-7718 394 18 h. h. PROPN ap-7718 394 19 m. m. PROPN ap-7718 394 20 moya moya PROPN ap-7718 394 21 - - PUNCT ap-7718 394 22 cessa cessa NOUN ap-7718 394 23 . . PUNCT ap-7718 395 1 ermakov ermakov PROPN ap-7718 395 2 - - PUNCT ap-7718 395 3 lewis lewis PROPN ap-7718 395 4 invariant invariant PROPN ap-7718 395 5 for for ADP ap-7718 395 6 two two NUM ap-7718 395 7 coupled couple VERB ap-7718 395 8 oscillators oscillator NOUN ap-7718 395 9 . . PUNCT ap-7718 396 1 journal journal PROPN ap-7718 396 2 of of ADP ap-7718 396 3 physics physics PROPN ap-7718 396 4 : : PUNCT ap-7718 396 5 conference conference NOUN ap-7718 396 6 series series PROPN ap-7718 396 7 1540(1):012009 1540(1):012009 NUM ap-7718 396 8 , , PUNCT ap-7718 396 9 2020 2020 NUM ap-7718 396 10 . . PUNCT ap-7718 397 1 https://doi.org/10.1088/1742-6596/1540/1/012009 https://doi.org/10.1088/1742-6596/1540/1/012009 X ap-7718 397 2 . . PUNCT ap-7718 398 1 [ [ X ap-7718 398 2 13 13 NUM ap-7718 398 3 ] ] X ap-7718 398 4 v. v. PROPN ap-7718 398 5 v. v. ADP ap-7718 398 6 dodonov dodonov PROPN ap-7718 398 7 . . PROPN ap-7718 398 8 invariant invariant ADJ ap-7718 398 9 quantum quantum ADJ ap-7718 398 10 states state NOUN ap-7718 398 11 of of ADP ap-7718 398 12 quadratic quadratic ADJ ap-7718 398 13 hamiltonians hamiltonian NOUN ap-7718 398 14 . . PUNCT ap-7718 399 1 entropy entropy PROPN ap-7718 399 2 23(5):634 23(5):634 NUM ap-7718 399 3 , , PUNCT ap-7718 399 4 2021 2021 NUM ap-7718 399 5 . . PUNCT ap-7718 400 1 https://doi.org/10.3390/e23050634 https://doi.org/10.3390/e23050634 X ap-7718 400 2 . . PUNCT ap-7718 401 1 [ [ X ap-7718 401 2 14 14 NUM ap-7718 401 3 ] ] X ap-7718 401 4 v. v. PROPN ap-7718 401 5 v. v. ADP ap-7718 401 6 dodonov dodonov PROPN ap-7718 401 7 , , PUNCT ap-7718 401 8 m. m. PROPN ap-7718 401 9 b. b. PROPN ap-7718 401 10 horovits horovits PROPN ap-7718 401 11 . . PUNCT ap-7718 402 1 energy energy NOUN ap-7718 402 2 and and CCONJ ap-7718 402 3 magnetic magnetic ADJ ap-7718 402 4 moment moment NOUN ap-7718 402 5 of of ADP ap-7718 402 6 a a DET ap-7718 402 7 quantum quantum NOUN ap-7718 402 8 charged charge VERB ap-7718 402 9 particle particle NOUN ap-7718 402 10 in in ADP ap-7718 402 11 time time NOUN ap-7718 402 12 - - PUNCT ap-7718 402 13 dependent dependent ADJ ap-7718 402 14 magnetic magnetic ADJ ap-7718 402 15 and and CCONJ ap-7718 402 16 electric electric ADJ ap-7718 402 17 fields field NOUN ap-7718 402 18 of of ADP ap-7718 402 19 circular circular ADJ ap-7718 402 20 and and CCONJ ap-7718 402 21 plane plane NOUN ap-7718 402 22 solenoids solenoid NOUN ap-7718 402 23 . . PUNCT ap-7718 403 1 entropy entropy PROPN ap-7718 403 2 23(12):1579 23(12):1579 NUM ap-7718 403 3 , , PUNCT ap-7718 403 4 2021 2021 NUM ap-7718 403 5 . . PUNCT ap-7718 404 1 https://doi.org/10.3390/e23121579 https://doi.org/10.3390/e23121579 NUM ap-7718 404 2 . . PUNCT ap-7718 405 1 [ [ X ap-7718 405 2 15 15 NUM ap-7718 405 3 ] ] X ap-7718 405 4 k. k. PROPN ap-7718 405 5 zelaya zelaya PROPN ap-7718 405 6 , , PUNCT ap-7718 405 7 v. v. PROPN ap-7718 405 8 hussin hussin PROPN ap-7718 405 9 . . PUNCT ap-7718 406 1 time time NOUN ap-7718 406 2 - - PUNCT ap-7718 406 3 dependent dependent ADJ ap-7718 406 4 rational rational ADJ ap-7718 406 5 extensions extension NOUN ap-7718 406 6 of of ADP ap-7718 406 7 the the DET ap-7718 406 8 parametric parametric ADJ ap-7718 406 9 oscillator oscillator NOUN ap-7718 406 10 : : PUNCT ap-7718 406 11 quantum quantum NOUN ap-7718 406 12 invariants invariant NOUN ap-7718 406 13 and and CCONJ ap-7718 406 14 the the DET ap-7718 406 15 factorization factorization NOUN ap-7718 406 16 method method NOUN ap-7718 406 17 . . PUNCT ap-7718 407 1 journal journal PROPN ap-7718 407 2 of of ADP ap-7718 407 3 physics physics PROPN ap-7718 407 4 a a PRON ap-7718 407 5 : : PUNCT ap-7718 407 6 mathematical mathematical ADJ ap-7718 407 7 and and CCONJ ap-7718 407 8 theoretical theoretical ADJ ap-7718 407 9 53(16):165301 53(16):165301 NUM ap-7718 407 10 , , PUNCT ap-7718 407 11 2020 2020 NUM ap-7718 407 12 . . PUNCT ap-7718 408 1 https://doi.org/10.1088/1751-8121/ab78d1 https://doi.org/10.1088/1751-8121/ab78d1 PROPN ap-7718 408 2 . . PUNCT ap-7718 409 1 [ [ X ap-7718 409 2 16 16 NUM ap-7718 409 3 ] ] PUNCT ap-7718 409 4 k. k. PROPN ap-7718 409 5 zelaya zelaya PROPN ap-7718 409 6 . . PUNCT ap-7718 410 1 nonstationary nonstationary ADJ ap-7718 410 2 deformed deform VERB ap-7718 410 3 singular singular ADJ ap-7718 410 4 oscillator oscillator NOUN ap-7718 410 5 : : PUNCT ap-7718 410 6 quantum quantum NOUN ap-7718 410 7 invariants invariant NOUN ap-7718 410 8 and and CCONJ ap-7718 410 9 the the DET ap-7718 410 10 factorization factorization NOUN ap-7718 410 11 method method NOUN ap-7718 410 12 . . PUNCT ap-7718 411 1 journal journal PROPN ap-7718 411 2 of of ADP ap-7718 411 3 physics physics PROPN ap-7718 411 4 : : PUNCT ap-7718 411 5 conference conference NOUN ap-7718 411 6 series series NOUN ap-7718 411 7 1540:012017 1540:012017 NUM ap-7718 411 8 , , PUNCT ap-7718 411 9 2020 2020 NUM ap-7718 411 10 . . PUNCT ap-7718 412 1 https://doi.org/10.1088/1742-6596/1540/1/012017 https://doi.org/10.1088/1742-6596/1540/1/012017 X ap-7718 412 2 . . PUNCT ap-7718 413 1 [ [ X ap-7718 413 2 17 17 NUM ap-7718 413 3 ] ] PUNCT ap-7718 413 4 k. k. PROPN ap-7718 413 5 zelaya zelaya PROPN ap-7718 413 6 , , PUNCT ap-7718 413 7 i. i. PROPN ap-7718 413 8 marquette marquette PROPN ap-7718 413 9 , , PUNCT ap-7718 413 10 v. v. PROPN ap-7718 413 11 hussin hussin PROPN ap-7718 413 12 . . PUNCT ap-7718 414 1 fourth fourth ADJ ap-7718 414 2 painlevé painlevé NOUN ap-7718 414 3 and and CCONJ ap-7718 414 4 ermakov ermakov ADJ ap-7718 414 5 equations equation NOUN ap-7718 414 6 : : PUNCT ap-7718 414 7 quantum quantum NOUN ap-7718 414 8 invariants invariant NOUN ap-7718 414 9 and and CCONJ ap-7718 414 10 new new ADJ ap-7718 414 11 exactly exactly ADV ap-7718 414 12 - - PUNCT ap-7718 414 13 solvable solvable ADJ ap-7718 414 14 time time NOUN ap-7718 414 15 - - PUNCT ap-7718 414 16 dependent dependent ADJ ap-7718 414 17 hamiltonians hamiltonian NOUN ap-7718 414 18 . . PUNCT ap-7718 415 1 journal journal PROPN ap-7718 415 2 of of ADP ap-7718 415 3 physics physics PROPN ap-7718 415 4 a a PRON ap-7718 415 5 : : PUNCT ap-7718 415 6 mathematical mathematical ADJ ap-7718 415 7 and and CCONJ ap-7718 415 8 theoretical theoretical ADJ ap-7718 415 9 54(1):015206 54(1):015206 NUM ap-7718 415 10 , , PUNCT ap-7718 415 11 2020 2020 NUM ap-7718 415 12 . . PUNCT ap-7718 416 1 https://doi.org/10.1088/1751-8121/abcab8 https://doi.org/10.1088/1751-8121/abcab8 X ap-7718 416 2 . . PUNCT ap-7718 417 1 [ [ X ap-7718 417 2 18 18 NUM ap-7718 417 3 ] ] X ap-7718 417 4 v. v. PROPN ap-7718 417 5 g. g. PROPN ap-7718 417 6 bagrov bagrov PROPN ap-7718 417 7 , , PUNCT ap-7718 417 8 b. b. PROPN ap-7718 417 9 f. f. PROPN ap-7718 417 10 samsonov samsonov PROPN ap-7718 417 11 , , PUNCT ap-7718 417 12 l. l. PROPN ap-7718 417 13 a. a. PROPN ap-7718 417 14 shekoyan shekoyan PROPN ap-7718 417 15 . . PUNCT ap-7718 418 1 darboux darboux VERB ap-7718 418 2 transformation transformation NOUN ap-7718 418 3 for for ADP ap-7718 418 4 the the DET ap-7718 418 5 nonsteady nonsteady ADV ap-7718 418 6 schrödinger schrödinger ADJ ap-7718 418 7 equation equation NOUN ap-7718 418 8 . . PUNCT ap-7718 419 1 russian russian ADJ ap-7718 419 2 physics physics PROPN ap-7718 419 3 journal journal PROPN ap-7718 419 4 38(7):706–712 38(7):706–712 PROPN ap-7718 419 5 , , PUNCT ap-7718 419 6 1995 1995 NUM ap-7718 419 7 . . PUNCT ap-7718 420 1 https://doi.org/10.1007/bf00560273 https://doi.org/10.1007/bf00560273 PROPN ap-7718 420 2 . . PUNCT ap-7718 421 1 [ [ X ap-7718 421 2 19 19 NUM ap-7718 421 3 ] ] PUNCT ap-7718 421 4 k. k. PROPN ap-7718 421 5 zelaya zelaya PROPN ap-7718 421 6 , , PUNCT ap-7718 421 7 o. o. PROPN ap-7718 421 8 rosas rosas PROPN ap-7718 421 9 - - PUNCT ap-7718 421 10 ortiz ortiz PROPN ap-7718 421 11 . . 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PUNCT ap-7718 425 1 [ [ X ap-7718 425 2 20 20 NUM ap-7718 425 3 ] ] PUNCT ap-7718 425 4 a. a. PROPN ap-7718 425 5 contreras contreras PROPN ap-7718 425 6 - - PUNCT ap-7718 425 7 astorga astorga PROPN ap-7718 425 8 . . PUNCT ap-7718 426 1 a a DET ap-7718 426 2 time time NOUN ap-7718 426 3 - - PUNCT ap-7718 426 4 dependent dependent ADJ ap-7718 426 5 anharmonic anharmonic ADJ ap-7718 426 6 oscillator oscillator NOUN ap-7718 426 7 . . PUNCT ap-7718 427 1 journal journal PROPN ap-7718 427 2 of of ADP ap-7718 427 3 physics physics PROPN ap-7718 427 4 : : PUNCT ap-7718 427 5 conference conference NOUN ap-7718 427 6 series series NOUN ap-7718 427 7 839:012019 839:012019 NUM ap-7718 427 8 , , PUNCT ap-7718 427 9 2017 2017 NUM ap-7718 427 10 . . PUNCT ap-7718 428 1 https://doi.org/10.1088/1742-6596/839/1/012019 https://doi.org/10.1088/1742-6596/839/1/012019 X ap-7718 428 2 . . PUNCT ap-7718 429 1 [ [ X ap-7718 429 2 21 21 NUM ap-7718 429 3 ] ] X ap-7718 429 4 r. r. NOUN ap-7718 429 5 razo razo NOUN ap-7718 429 6 , , PUNCT ap-7718 429 7 s. s. PROPN ap-7718 429 8 cruz cruz PROPN ap-7718 429 9 y y PROPN ap-7718 429 10 cruz cruz PROPN ap-7718 429 11 . . PUNCT ap-7718 430 1 new new ADJ ap-7718 430 2 confining confining ADJ ap-7718 430 3 optical optical ADJ ap-7718 430 4 media medium NOUN ap-7718 430 5 generated generate VERB ap-7718 430 6 by by ADP ap-7718 430 7 darboux darboux ADJ ap-7718 430 8 transformations transformation NOUN ap-7718 430 9 . . PUNCT ap-7718 431 1 journal journal PROPN ap-7718 431 2 of of ADP ap-7718 431 3 physics physics PROPN ap-7718 431 4 : : PUNCT ap-7718 431 5 conference conference NOUN ap-7718 431 6 series series PROPN ap-7718 431 7 1194:012091 1194:012091 NUM ap-7718 431 8 , , PUNCT ap-7718 431 9 2019 2019 NUM ap-7718 431 10 . . PUNCT ap-7718 432 1 https://doi.org/10.1088/1742-6596/1194/1/012091 https://doi.org/10.1088/1742-6596/1194/1/012091 PROPN ap-7718 432 2 . . PUNCT ap-7718 433 1 [ [ X ap-7718 433 2 22 22 NUM ap-7718 433 3 ] ] PUNCT ap-7718 433 4 j. j. PROPN ap-7718 433 5 cen cen PROPN ap-7718 433 6 , , PUNCT ap-7718 433 7 a. a. NOUN ap-7718 433 8 fring fring NOUN ap-7718 433 9 , , PUNCT ap-7718 433 10 t. t. PROPN ap-7718 433 11 frith frith PROPN ap-7718 433 12 . . PUNCT ap-7718 434 1 time time NOUN ap-7718 434 2 - - PUNCT ap-7718 434 3 dependent dependent ADJ ap-7718 434 4 darboux darboux NOUN ap-7718 434 5 ( ( PUNCT ap-7718 434 6 supersymmetric supersymmetric ADJ ap-7718 434 7 ) ) PUNCT ap-7718 434 8 transformations transformation NOUN ap-7718 434 9 for for ADP ap-7718 434 10 non non ADJ ap-7718 434 11 - - ADJ ap-7718 434 12 hermitian hermitian ADJ ap-7718 434 13 quantum quantum NOUN ap-7718 434 14 systems system NOUN ap-7718 434 15 . . PUNCT ap-7718 435 1 journal journal PROPN ap-7718 435 2 of of ADP ap-7718 435 3 physics physics PROPN ap-7718 435 4 a a PRON ap-7718 435 5 : : PUNCT ap-7718 435 6 mathematical mathematical ADJ ap-7718 435 7 and and CCONJ ap-7718 435 8 theoretical theoretical ADJ ap-7718 435 9 52(11):115302 52(11):115302 NUM ap-7718 435 10 , , PUNCT ap-7718 435 11 2019 2019 NUM ap-7718 435 12 . . PUNCT ap-7718 436 1 https://doi.org/10.1088/1751-8121/ab0335 https://doi.org/10.1088/1751-8121/ab0335 X ap-7718 436 2 . . PUNCT ap-7718 437 1 [ [ X ap-7718 437 2 23 23 NUM ap-7718 437 3 ] ] PUNCT ap-7718 437 4 a. a. PROPN ap-7718 437 5 contreras contreras PROPN ap-7718 437 6 - - PUNCT ap-7718 437 7 astorga astorga PROPN ap-7718 437 8 , , PUNCT ap-7718 437 9 v. v. ADP ap-7718 437 10 jakubský jakubský PROPN ap-7718 437 11 . . PUNCT ap-7718 438 1 photonic photonic NOUN ap-7718 438 2 systems system NOUN ap-7718 438 3 with with ADP ap-7718 438 4 two two NUM ap-7718 438 5 - - PUNCT ap-7718 438 6 dimensional dimensional ADJ ap-7718 438 7 landscapes landscape NOUN ap-7718 438 8 of of ADP ap-7718 438 9 complex complex ADJ ap-7718 438 10 refractive refractive ADJ ap-7718 438 11 index index NOUN ap-7718 438 12 via via ADP ap-7718 438 13 time time NOUN ap-7718 438 14 - - PUNCT ap-7718 438 15 dependent dependent ADJ ap-7718 438 16 supersymmetry supersymmetry NOUN ap-7718 438 17 . . PUNCT ap-7718 439 1 physical physical ADJ ap-7718 439 2 review review NOUN ap-7718 439 3 a a DET ap-7718 439 4 99:053812 99:053812 NUM ap-7718 439 5 , , PUNCT ap-7718 439 6 2019 2019 NUM ap-7718 439 7 . . PUNCT ap-7718 440 1 https://doi.org/10.1103/physreva.99.053812 https://doi.org/10.1103/physreva.99.053812 NOUN ap-7718 440 2 . . PUNCT ap-7718 441 1 [ [ X ap-7718 441 2 24 24 NUM ap-7718 441 3 ] ] PUNCT ap-7718 441 4 a. a. PROPN ap-7718 441 5 contreras contreras PROPN ap-7718 441 6 - - PUNCT ap-7718 441 7 astorga astorga PROPN ap-7718 441 8 , , PUNCT ap-7718 441 9 v. v. PROPN ap-7718 441 10 jakubský jakubský PROPN ap-7718 441 11 . . PUNCT ap-7718 442 1 multimode multimode VERB ap-7718 442 2 two two NUM ap-7718 442 3 - - PUNCT ap-7718 442 4 dimensional dimensional ADJ ap-7718 442 5 pt pt ADJ ap-7718 442 6 - - PUNCT ap-7718 442 7 symmetric symmetric ADJ ap-7718 442 8 waveguides waveguide NOUN ap-7718 442 9 . . PUNCT ap-7718 443 1 journal journal NOUN ap-7718 443 2 of of ADP ap-7718 443 3 physics physics PROPN ap-7718 443 4 : : PUNCT ap-7718 443 5 conference conference NOUN ap-7718 443 6 series series NOUN ap-7718 443 7 1540:012018 1540:012018 NUM ap-7718 443 8 , , PUNCT ap-7718 443 9 2020 2020 NUM ap-7718 443 10 . . PUNCT ap-7718 444 1 https://doi.org/10.1088/1742-6596/1540/1/012018 https://doi.org/10.1088/1742-6596/1540/1/012018 X ap-7718 444 2 . . PUNCT ap-7718 445 1 [ [ X ap-7718 445 2 25 25 NUM ap-7718 445 3 ] ] PUNCT ap-7718 445 4 w.-h w.-h NOUN ap-7718 445 5 . . PUNCT ap-7718 446 1 steeb steeb PROPN ap-7718 446 2 . . PUNCT ap-7718 447 1 invertible invertible ADJ ap-7718 447 2 point point NOUN ap-7718 447 3 transformations transformation NOUN ap-7718 447 4 and and CCONJ ap-7718 447 5 nonlinear nonlinear ADJ ap-7718 447 6 differential differential ADJ ap-7718 447 7 equations equation NOUN ap-7718 447 8 . . PUNCT ap-7718 448 1 world world NOUN ap-7718 448 2 scientific scientific ADJ ap-7718 448 3 publishing publishing NOUN ap-7718 448 4 , , PUNCT ap-7718 448 5 singapore singapore PROPN ap-7718 448 6 , , PUNCT ap-7718 448 7 1993 1993 NUM ap-7718 448 8 . . PUNCT ap-7718 449 1 https://doi.org/10.1142/1987 https://doi.org/10.1142/1987 X ap-7718 449 2 . . PUNCT ap-7718 450 1 [ [ X ap-7718 450 2 26 26 NUM ap-7718 450 3 ] ] PUNCT ap-7718 450 4 k. k. PROPN ap-7718 450 5 zelaya zelaya PROPN ap-7718 450 6 , , PUNCT ap-7718 450 7 o. o. PROPN ap-7718 450 8 rosas rosas PROPN ap-7718 450 9 - - PUNCT ap-7718 450 10 ortiz ortiz PROPN ap-7718 450 11 . . PUNCT ap-7718 451 1 quantum quantum ADJ ap-7718 451 2 nonstationary nonstationary ADJ ap-7718 451 3 oscillators oscillator NOUN ap-7718 451 4 : : PUNCT ap-7718 451 5 invariants invariant NOUN ap-7718 451 6 , , PUNCT ap-7718 451 7 dynamical dynamical ADJ ap-7718 451 8 algebras algebra NOUN ap-7718 451 9 and and CCONJ ap-7718 451 10 coherent coherent ADJ ap-7718 451 11 states state NOUN ap-7718 451 12 via via ADP ap-7718 451 13 point point NOUN ap-7718 451 14 transformations transformation NOUN ap-7718 451 15 . . PUNCT ap-7718 452 1 physica physica NOUN ap-7718 452 2 scripta scripta PROPN ap-7718 452 3 95(6):064004 95(6):064004 NUM ap-7718 452 4 , , PUNCT ap-7718 452 5 2020 2020 NUM ap-7718 452 6 . . PUNCT ap-7718 453 1 https://doi.org/10.1088/1402-4896/ab5cbf https://doi.org/10.1088/1402-4896/ab5cbf NOUN ap-7718 453 2 . . PUNCT ap-7718 454 1 [ [ X ap-7718 454 2 27 27 NUM ap-7718 454 3 ] ] PUNCT ap-7718 454 4 k. k. PROPN ap-7718 454 5 zelaya zelaya PROPN ap-7718 454 6 , , PUNCT ap-7718 454 7 v. v. PROPN ap-7718 454 8 hussin hussin PROPN ap-7718 454 9 . . PUNCT ap-7718 455 1 point point NOUN ap-7718 455 2 transformations transformation NOUN ap-7718 455 3 : : PUNCT ap-7718 455 4 exact exact ADJ ap-7718 455 5 solutions solution NOUN ap-7718 455 6 of of ADP ap-7718 455 7 the the DET ap-7718 455 8 quantum quantum ADJ ap-7718 455 9 time time NOUN ap-7718 455 10 - - PUNCT ap-7718 455 11 dependent dependent ADJ ap-7718 455 12 mass mass ADJ ap-7718 455 13 nonstationary nonstationary ADJ ap-7718 455 14 oscillator oscillator NOUN ap-7718 455 15 . . PUNCT ap-7718 456 1 in in ADP ap-7718 456 2 m. m. PROPN ap-7718 456 3 b. b. PROPN ap-7718 456 4 paranjape paranjape PROPN ap-7718 456 5 , , PUNCT ap-7718 456 6 r. r. PROPN ap-7718 456 7 mackenzie mackenzie PROPN ap-7718 456 8 , , PUNCT ap-7718 456 9 z. z. PROPN ap-7718 456 10 thomova thomova PROPN ap-7718 456 11 , , PUNCT ap-7718 456 12 et et PROPN ap-7718 456 13 al al PROPN ap-7718 456 14 . . PUNCT ap-7718 457 1 ( ( PUNCT ap-7718 457 2 eds ed NOUN ap-7718 457 3 . . PUNCT ap-7718 457 4 ) ) PUNCT ap-7718 457 5 , , PUNCT ap-7718 457 6 quantum quantum NOUN ap-7718 457 7 theory theory NOUN ap-7718 457 8 and and CCONJ ap-7718 457 9 symmetries symmetry NOUN ap-7718 457 10 , , PUNCT ap-7718 457 11 pp pp ADV ap-7718 457 12 . . PUNCT ap-7718 458 1 295–303 295–303 NUM ap-7718 458 2 . . PUNCT ap-7718 459 1 springer springer NOUN ap-7718 459 2 international international ADJ ap-7718 459 3 publishing publishing NOUN ap-7718 459 4 , , PUNCT ap-7718 459 5 cham cham NOUN ap-7718 459 6 , , PUNCT ap-7718 459 7 2021 2021 NUM ap-7718 459 8 . . PUNCT ap-7718 460 1 https://doi.org/10.1007/978-3-030-55777-5_28 https://doi.org/10.1007/978-3-030-55777-5_28 PROPN ap-7718 460 2 . . PUNCT ap-7718 461 1 [ [ X ap-7718 461 2 28 28 NUM ap-7718 461 3 ] ] X ap-7718 461 4 a. a. NOUN ap-7718 461 5 fring fring PROPN ap-7718 461 6 , , PUNCT ap-7718 461 7 r. r. PROPN ap-7718 461 8 tenney tenney PROPN ap-7718 461 9 . . PUNCT ap-7718 462 1 exactly exactly ADV ap-7718 462 2 solvable solvable ADJ ap-7718 462 3 time time NOUN ap-7718 462 4 - - PUNCT ap-7718 462 5 dependent dependent ADJ ap-7718 462 6 non non ADJ ap-7718 462 7 - - ADJ ap-7718 462 8 hermitian hermitian ADJ ap-7718 462 9 quantum quantum NOUN ap-7718 462 10 systems system NOUN ap-7718 462 11 from from ADP ap-7718 462 12 point point NOUN ap-7718 462 13 transformations transformation NOUN ap-7718 462 14 . . PUNCT ap-7718 463 1 physics physics NOUN ap-7718 463 2 letters letter NOUN ap-7718 463 3 a a DET ap-7718 463 4 410:127548 410:127548 NOUN ap-7718 463 5 , , PUNCT ap-7718 463 6 2021 2021 NUM ap-7718 463 7 . . PUNCT ap-7718 464 1 https://doi.org/10.1016/j.physleta.2021.127548 https://doi.org/10.1016/j.physleta.2021.127548 NOUN ap-7718 464 2 . . PUNCT ap-7718 465 1 [ [ X ap-7718 465 2 29 29 NUM ap-7718 465 3 ] ] PUNCT ap-7718 465 4 k. k. PROPN ap-7718 465 5 zelaya zelaya PROPN ap-7718 465 6 , , PUNCT ap-7718 465 7 o. o. PROPN ap-7718 465 8 rosas rosas PROPN ap-7718 465 9 - - PUNCT ap-7718 465 10 ortiz ortiz PROPN ap-7718 465 11 . . PUNCT ap-7718 466 1 exact exact ADJ ap-7718 466 2 solutions solution NOUN ap-7718 466 3 for for ADP ap-7718 466 4 time time NOUN ap-7718 466 5 - - PUNCT ap-7718 466 6 dependent dependent ADJ ap-7718 466 7 non non ADJ ap-7718 466 8 - - ADJ ap-7718 466 9 hermitian hermitian ADJ ap-7718 466 10 oscillators oscillator NOUN ap-7718 466 11 : : PUNCT ap-7718 466 12 classical classical ADJ ap-7718 466 13 and and CCONJ ap-7718 466 14 quantum quantum ADJ ap-7718 466 15 pictures picture NOUN ap-7718 466 16 . . PUNCT ap-7718 467 1 quantum quantum ADJ ap-7718 467 2 reports report NOUN ap-7718 467 3 3(3):458–472 3(3):458–472 NUM ap-7718 467 4 , , PUNCT ap-7718 467 5 2021 2021 NUM ap-7718 467 6 . . PUNCT ap-7718 468 1 https://doi.org/10.3390/quantum3030030 https://doi.org/10.3390/quantum3030030 PROPN ap-7718 468 2 . . PUNCT ap-7718 469 1 [ [ X ap-7718 469 2 30 30 NUM ap-7718 469 3 ] ] X ap-7718 469 4 v. v. CCONJ ap-7718 469 5 aldaya aldaya PROPN ap-7718 469 6 , , PUNCT ap-7718 469 7 f. f. PROPN ap-7718 469 8 cossío cossío PROPN ap-7718 469 9 , , PUNCT ap-7718 469 10 j. j. PROPN ap-7718 469 11 guerrero guerrero PROPN ap-7718 469 12 , , PUNCT ap-7718 469 13 f. f. PROPN ap-7718 469 14 f. f. PROPN ap-7718 469 15 lópez lópez PROPN ap-7718 469 16 - - PUNCT ap-7718 469 17 ruiz ruiz NOUN ap-7718 469 18 . . PUNCT ap-7718 470 1 the the DET ap-7718 470 2 quantum quantum PROPN ap-7718 470 3 arnold arnold PROPN ap-7718 470 4 transformation transformation PROPN ap-7718 470 5 . . PUNCT ap-7718 471 1 journal journal PROPN ap-7718 471 2 of of ADP ap-7718 471 3 physics physics PROPN ap-7718 471 4 a a PRON ap-7718 471 5 : : PUNCT ap-7718 471 6 mathematical mathematical ADJ ap-7718 471 7 and and CCONJ ap-7718 471 8 theoretical theoretical ADJ ap-7718 471 9 44(6):065302 44(6):065302 NUM ap-7718 471 10 , , PUNCT ap-7718 471 11 2011 2011 NUM ap-7718 471 12 . . PUNCT ap-7718 472 1 https://doi.org/10.1088/1751-8113/44/6/065302 https://doi.org/10.1088/1751-8113/44/6/065302 NOUN ap-7718 472 2 . . PUNCT ap-7718 473 1 [ [ X ap-7718 473 2 31 31 NUM ap-7718 473 3 ] ] X ap-7718 473 4 n. n. PROPN ap-7718 473 5 ünal ünal PROPN ap-7718 473 6 . . PUNCT ap-7718 474 1 quasi quasi ADJ ap-7718 474 2 - - ADJ ap-7718 474 3 coherent coherent ADJ ap-7718 474 4 states state NOUN ap-7718 474 5 for for ADP ap-7718 474 6 the the DET ap-7718 474 7 hermite hermite ADJ ap-7718 474 8 oscillator oscillator NOUN ap-7718 474 9 . . PUNCT ap-7718 475 1 journal journal PROPN ap-7718 475 2 of of ADP ap-7718 475 3 mathematical mathematical ADJ ap-7718 475 4 physics physics NOUN ap-7718 475 5 59(6):062104 59(6):062104 PROPN ap-7718 475 6 , , PUNCT ap-7718 475 7 2018 2018 NUM ap-7718 475 8 . . PUNCT ap-7718 476 1 https://doi.org/10.1063/1.5016897 https://doi.org/10.1063/1.5016897 INTJ ap-7718 476 2 . . PUNCT ap-7718 477 1 [ [ X ap-7718 477 2 32 32 NUM ap-7718 477 3 ] ] PUNCT ap-7718 477 4 i. i. PROPN ap-7718 477 5 ramos ramos PROPN ap-7718 477 6 - - PROPN ap-7718 477 7 prieto prieto PROPN ap-7718 477 8 , , PUNCT ap-7718 477 9 m. m. NOUN ap-7718 477 10 fernández fernández PROPN ap-7718 477 11 - - PUNCT ap-7718 477 12 guasti guasti PROPN ap-7718 477 13 , , PUNCT ap-7718 477 14 h. h. PROPN ap-7718 477 15 m. m. PROPN ap-7718 477 16 moya moya PROPN ap-7718 477 17 - - PUNCT ap-7718 477 18 cessa cessa NOUN ap-7718 477 19 . . PUNCT ap-7718 478 1 quantum quantum PROPN ap-7718 478 2 harmonic harmonic NOUN ap-7718 478 3 oscillator oscillator NOUN ap-7718 478 4 with with ADP ap-7718 478 5 time time NOUN ap-7718 478 6 - - PUNCT ap-7718 478 7 dependent dependent ADJ ap-7718 478 8 mass mass NOUN ap-7718 478 9 . . PUNCT ap-7718 479 1 modern modern ADJ ap-7718 479 2 physics physics PROPN ap-7718 479 3 letters letters PROPN ap-7718 479 4 b b PROPN ap-7718 479 5 32(20):1850235 32(20):1850235 NUM ap-7718 479 6 , , PUNCT ap-7718 479 7 2018 2018 NUM ap-7718 479 8 . . PUNCT ap-7718 480 1 https://doi.org/10.1142/s0217984918502354 https://doi.org/10.1142/s0217984918502354 NUM ap-7718 480 2 . . PUNCT ap-7718 481 1 [ [ X ap-7718 481 2 33 33 NUM ap-7718 481 3 ] ] PUNCT ap-7718 481 4 f. f. PROPN ap-7718 481 5 olver olver PROPN ap-7718 481 6 , , PUNCT ap-7718 481 7 d. d. PROPN ap-7718 481 8 lozier lozier PROPN ap-7718 481 9 , , PUNCT ap-7718 481 10 r. r. PROPN ap-7718 481 11 boisvert boisvert PROPN ap-7718 481 12 , , PUNCT ap-7718 481 13 c. c. PROPN ap-7718 481 14 clark clark PROPN ap-7718 481 15 . . PUNCT ap-7718 482 1 nist nist PROPN ap-7718 482 2 handbook handbook PROPN ap-7718 482 3 of of ADP ap-7718 482 4 mathematical mathematical ADJ ap-7718 482 5 functions function NOUN ap-7718 482 6 . . PUNCT ap-7718 483 1 cambridge cambridge PROPN ap-7718 483 2 university university PROPN ap-7718 483 3 press press PROPN ap-7718 483 4 , , PUNCT ap-7718 483 5 new new PROPN ap-7718 483 6 york york PROPN ap-7718 483 7 , , PUNCT ap-7718 483 8 2010 2010 NUM ap-7718 483 9 . . PUNCT ap-7718 484 1 isbn isbn PROPN ap-7718 484 2 0521140633 0521140633 NUM ap-7718 484 3 . . PUNCT ap-7718 485 1 [ [ X ap-7718 485 2 34 34 NUM ap-7718 485 3 ] ] PUNCT ap-7718 485 4 v. v. PROPN ap-7718 485 5 p. p. PROPN ap-7718 485 6 ermakov ermakov PROPN ap-7718 485 7 . . PUNCT ap-7718 486 1 second second ADJ ap-7718 486 2 - - PUNCT ap-7718 486 3 order order NOUN ap-7718 486 4 differential differential ADJ ap-7718 486 5 equations equation NOUN ap-7718 486 6 : : PUNCT ap-7718 486 7 conditions condition NOUN ap-7718 486 8 of of ADP ap-7718 486 9 complete complete ADJ ap-7718 486 10 integrability integrability NOUN ap-7718 486 11 . . PUNCT ap-7718 487 1 universita universita PROPN ap-7718 487 2 izvestia izvestia PROPN ap-7718 487 3 kiev kiev PROPN ap-7718 487 4 series series PROPN ap-7718 487 5 iii iii PROPN ap-7718 487 6 20(9):1–25 20(9):1–25 NUM ap-7718 487 7 , , PUNCT ap-7718 487 8 1880 1880 NUM ap-7718 487 9 . . PUNCT ap-7718 488 1 [ [ X ap-7718 488 2 35 35 NUM ap-7718 488 3 ] ] PUNCT ap-7718 488 4 a. a. NOUN ap-7718 488 5 o. o. PROPN ap-7718 488 6 harin harin PROPN ap-7718 488 7 . . PUNCT ap-7718 489 1 “ " PUNCT ap-7718 489 2 second second ADJ ap-7718 489 3 - - PUNCT ap-7718 489 4 order order NOUN ap-7718 489 5 differential differential ADJ ap-7718 489 6 equations equation NOUN ap-7718 489 7 : : PUNCT ap-7718 489 8 conditions condition NOUN ap-7718 489 9 of of ADP ap-7718 489 10 complete complete ADJ ap-7718 489 11 integrability integrability NOUN ap-7718 489 12 ” " PUNCT ap-7718 489 13 ( ( PUNCT ap-7718 489 14 english english ADJ ap-7718 489 15 translation translation NOUN ap-7718 489 16 ) ) PUNCT ap-7718 489 17 . . PUNCT ap-7718 490 1 applicable applicable ADJ ap-7718 490 2 analysis analysis NOUN ap-7718 490 3 and and CCONJ ap-7718 490 4 discrete discrete ADJ ap-7718 490 5 mathematics mathematic NOUN ap-7718 490 6 2:123–145 2:123–145 NUM ap-7718 490 7 , , PUNCT ap-7718 490 8 2008 2008 NUM ap-7718 490 9 . . PUNCT ap-7718 491 1 [ [ X ap-7718 491 2 36 36 NUM ap-7718 491 3 ] ] X ap-7718 491 4 d. d. PROPN ap-7718 491 5 schuch schuch PROPN ap-7718 491 6 . . PUNCT ap-7718 491 7 quantum quantum PROPN ap-7718 491 8 theory theory NOUN ap-7718 491 9 from from ADP ap-7718 491 10 a a DET ap-7718 491 11 nonlinear nonlinear ADJ ap-7718 491 12 perspective perspective NOUN ap-7718 491 13 : : PUNCT ap-7718 491 14 riccati riccati PROPN ap-7718 491 15 equations equation NOUN ap-7718 491 16 in in ADP ap-7718 491 17 fundamental fundamental ADJ ap-7718 491 18 physics physic NOUN ap-7718 491 19 . . PUNCT ap-7718 492 1 spinger spinger PROPN ap-7718 492 2 , , PUNCT ap-7718 492 3 cham cham NOUN ap-7718 492 4 , , PUNCT ap-7718 492 5 2018 2018 NUM ap-7718 492 6 . . 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PUNCT ap-7718 498 1 62 62 NUM ap-7718 498 2 no no INTJ ap-7718 498 3 . . PUNCT ap-7718 499 1 1/2022 1/2022 NUM ap-7718 499 2 td td NOUN ap-7718 499 3 mass mass NOUN ap-7718 499 4 oscillators oscillator NOUN ap-7718 499 5 : : PUNCT ap-7718 499 6 constants constant NOUN ap-7718 499 7 of of ADP ap-7718 499 8 motion motion NOUN ap-7718 499 9 and and CCONJ ap-7718 499 10 semiclasical semiclasical ADJ ap-7718 499 11 states state NOUN ap-7718 499 12 [ [ X ap-7718 499 13 37 37 NUM ap-7718 499 14 ] ] PUNCT ap-7718 499 15 a. a. NOUN ap-7718 499 16 o. o. PROPN ap-7718 499 17 barut barut PROPN ap-7718 499 18 , , PUNCT ap-7718 499 19 l. l. PROPN ap-7718 499 20 girardello girardello PROPN ap-7718 499 21 . . PUNCT ap-7718 500 1 new new ADJ ap-7718 500 2 “ " PUNCT ap-7718 500 3 coherent coherent ADJ ap-7718 500 4 ” " PUNCT ap-7718 500 5 states state NOUN ap-7718 500 6 associated associate VERB ap-7718 500 7 with with ADP ap-7718 500 8 non non ADJ ap-7718 500 9 - - ADJ ap-7718 500 10 compact compact ADJ ap-7718 500 11 groups group NOUN ap-7718 500 12 . . PUNCT ap-7718 501 1 communications communication NOUN ap-7718 501 2 in in ADP ap-7718 501 3 mathematical mathematical ADJ ap-7718 501 4 physics physics NOUN ap-7718 501 5 21(1):41–55 21(1):41–55 NUM ap-7718 501 6 , , PUNCT ap-7718 501 7 1971 1971 NUM ap-7718 501 8 . . PUNCT ap-7718 502 1 https://doi.org/10.1007/bf01646483 https://doi.org/10.1007/bf01646483 X ap-7718 502 2 . . PUNCT ap-7718 503 1 [ [ X ap-7718 503 2 38 38 NUM ap-7718 503 3 ] ] PUNCT ap-7718 503 4 r. r. PROPN ap-7718 503 5 j. j. PROPN ap-7718 503 6 glauber glauber PROPN ap-7718 503 7 . . PUNCT ap-7718 504 1 coherent coherent ADJ ap-7718 504 2 and and CCONJ ap-7718 504 3 incoherent incoherent ADJ ap-7718 504 4 states state NOUN ap-7718 504 5 of of ADP ap-7718 504 6 the the DET ap-7718 504 7 radiation radiation NOUN ap-7718 504 8 field field NOUN ap-7718 504 9 . . PUNCT ap-7718 505 1 physical physical ADJ ap-7718 505 2 review review PROPN ap-7718 505 3 131:2766–2788 131:2766–2788 NUM ap-7718 505 4 , , PUNCT ap-7718 505 5 1963 1963 NUM ap-7718 505 6 . . PUNCT ap-7718 506 1 https://doi.org/10.1103/physrev.131.2766 https://doi.org/10.1103/physrev.131.2766 X ap-7718 506 2 . . PUNCT ap-7718 507 1 [ [ X ap-7718 507 2 39 39 NUM ap-7718 507 3 ] ] PUNCT ap-7718 507 4 a. a. NOUN ap-7718 507 5 perelomov perelomov NOUN ap-7718 507 6 . . PUNCT ap-7718 508 1 generalized generalize VERB ap-7718 508 2 coherent coherent ADJ ap-7718 508 3 states state NOUN ap-7718 508 4 and and CCONJ ap-7718 508 5 their their PRON ap-7718 508 6 applications application NOUN ap-7718 508 7 . . PUNCT ap-7718 509 1 springer springer NOUN ap-7718 509 2 - - PUNCT ap-7718 509 3 verlag verlag PROPN ap-7718 509 4 , , PUNCT ap-7718 509 5 berlin berlin PROPN ap-7718 509 6 , , PUNCT ap-7718 509 7 1986 1986 NUM ap-7718 509 8 . . PUNCT ap-7718 510 1 [ [ X ap-7718 510 2 40 40 NUM ap-7718 510 3 ] ] PUNCT ap-7718 510 4 h. h. PROPN ap-7718 510 5 p. p. PROPN ap-7718 510 6 robertson robertson PROPN ap-7718 510 7 . . PUNCT ap-7718 511 1 the the DET ap-7718 511 2 uncertainty uncertainty NOUN ap-7718 511 3 principle principle NOUN ap-7718 511 4 . . PUNCT ap-7718 512 1 physical physical ADJ ap-7718 512 2 review review PROPN ap-7718 512 3 34:163–164 34:163–164 PROPN ap-7718 512 4 , , PUNCT ap-7718 512 5 1929 1929 NUM ap-7718 512 6 . . PUNCT ap-7718 513 1 https://doi.org/10.1103/physrev.34.163 https://doi.org/10.1103/physrev.34.163 X ap-7718 513 2 . . PUNCT ap-7718 514 1 [ [ X ap-7718 514 2 41 41 NUM ap-7718 514 3 ] ] X ap-7718 514 4 e. e. PROPN ap-7718 514 5 schrödinger schrödinger PROPN ap-7718 514 6 . . PUNCT ap-7718 515 1 zum zum PROPN ap-7718 515 2 heisenbergschen heisenbergschen PROPN ap-7718 515 3 unscharfeprinzip unscharfeprinzip PROPN ap-7718 515 4 . . PUNCT ap-7718 516 1 proceedings proceeding NOUN ap-7718 516 2 of of ADP ap-7718 516 3 the the DET ap-7718 516 4 prussian prussian PROPN ap-7718 516 5 academy academy PROPN ap-7718 516 6 of of ADP ap-7718 516 7 sciences sciences PROPN ap-7718 516 8 19:296–303 19:296–303 NUM ap-7718 516 9 , , PUNCT ap-7718 516 10 1929 1929 NUM ap-7718 516 11 . . PUNCT ap-7718 517 1 [ [ X ap-7718 517 2 42 42 NUM ap-7718 517 3 ] ] PUNCT ap-7718 517 4 p. p. NOUN ap-7718 517 5 caldirola caldirola PROPN ap-7718 517 6 . . PUNCT ap-7718 518 1 forze forze PROPN ap-7718 518 2 non non PROPN ap-7718 518 3 conservative conservative ADJ ap-7718 518 4 nella nella PROPN ap-7718 518 5 meccanica meccanica PROPN ap-7718 518 6 quantistica quantistica PROPN ap-7718 518 7 . . PUNCT ap-7718 519 1 il il PROPN ap-7718 519 2 nuovo nuovo PROPN ap-7718 519 3 cimento cimento PROPN ap-7718 519 4 18:393–400 18:393–400 NUM ap-7718 519 5 , , PUNCT ap-7718 519 6 1941 1941 NUM ap-7718 519 7 . . PUNCT ap-7718 520 1 https://doi.org/10.1007/bf02960144 https://doi.org/10.1007/bf02960144 X ap-7718 520 2 . . PUNCT ap-7718 521 1 [ [ X ap-7718 521 2 43 43 NUM ap-7718 521 3 ] ] X ap-7718 521 4 e. e. PROPN ap-7718 521 5 kanai kanai PROPN ap-7718 521 6 . . PROPN ap-7718 522 1 on on ADP ap-7718 522 2 the the DET ap-7718 522 3 quantization quantization NOUN ap-7718 522 4 of of ADP ap-7718 522 5 the the DET ap-7718 522 6 dissipative dissipative NOUN ap-7718 522 7 systems system NOUN ap-7718 522 8 . . PUNCT ap-7718 523 1 progress progress NOUN ap-7718 523 2 of of ADP ap-7718 523 3 theoretical theoretical ADJ ap-7718 523 4 physics physics NOUN ap-7718 523 5 3(4):440–442 3(4):440–442 NUM ap-7718 523 6 , , PUNCT ap-7718 523 7 1948 1948 NUM ap-7718 523 8 . . PUNCT ap-7718 524 1 https://doi.org/10.1143/ptp/3.4.440 https://doi.org/10.1143/ptp/3.4.440 NOUN ap-7718 524 2 . . PUNCT ap-7718 525 1 [ [ X ap-7718 525 2 44 44 NUM ap-7718 525 3 ] ] PUNCT ap-7718 525 4 m. m. NOUN ap-7718 525 5 dernek dernek PROPN ap-7718 525 6 , , PUNCT ap-7718 525 7 n. n. PROPN ap-7718 525 8 ünal ünal NOUN ap-7718 525 9 . . PUNCT ap-7718 526 1 quasi quasi ADJ ap-7718 526 2 - - ADJ ap-7718 526 3 coherent coherent ADJ ap-7718 526 4 states state NOUN ap-7718 526 5 for for ADP ap-7718 526 6 damped damped NOUN ap-7718 526 7 and and CCONJ ap-7718 526 8 forced force VERB ap-7718 526 9 harmonic harmonic ADJ ap-7718 526 10 oscillator oscillator NOUN ap-7718 526 11 . . PUNCT ap-7718 527 1 journal journal PROPN ap-7718 527 2 of of ADP ap-7718 527 3 mathematical mathematical ADJ ap-7718 527 4 physics physics NOUN ap-7718 527 5 54(9):092102 54(9):092102 NUM ap-7718 527 6 , , PUNCT ap-7718 527 7 2013 2013 NUM ap-7718 527 8 . . PUNCT ap-7718 528 1 https://doi.org/10.1063/1.4819261 https://doi.org/10.1063/1.4819261 PROPN ap-7718 528 2 . . PUNCT ap-7718 529 1 221 221 NUM ap-7718 529 2 https://doi.org/10.1007/bf01646483 https://doi.org/10.1007/bf01646483 VERB ap-7718 529 3 https://doi.org/10.1103/physrev.131.2766 https://doi.org/10.1103/physrev.131.2766 ADP ap-7718 529 4 https://doi.org/10.1103/physrev.34.163 https://doi.org/10.1103/physrev.34.163 X ap-7718 529 5 https://doi.org/10.1007/bf02960144 https://doi.org/10.1007/bf02960144 X ap-7718 529 6 https://doi.org/10.1143/ptp/3.4.440 https://doi.org/10.1143/ptp/3.4.440 PROPN ap-7718 529 7 https://doi.org/10.1063/1.4819261 https://doi.org/10.1063/1.4819261 PROPN ap-7718 529 8 acta acta PROPN ap-7718 529 9 polytechnica polytechnica PROPN ap-7718 529 10 62(1):211–221 62(1):211–221 PROPN ap-7718 529 11 , , PUNCT ap-7718 529 12 2022 2022 NUM ap-7718 529 13 1 1 NUM ap-7718 529 14 introduction introduction NOUN ap-7718 529 15 2 2 NUM ap-7718 529 16 materials material NOUN ap-7718 529 17 and and CCONJ ap-7718 529 18 methods method NOUN ap-7718 529 19 2.1 2.1 NUM ap-7718 529 20 point point NOUN ap-7718 529 21 transformations transformation NOUN ap-7718 529 22 3 3 NUM ap-7718 529 23 results result NOUN ap-7718 529 24 : : PUNCT ap-7718 529 25 constants constant NOUN ap-7718 529 26 of of ADP ap-7718 529 27 motion motion NOUN ap-7718 529 28 and and CCONJ ap-7718 529 29 semiclassical semiclassical ADJ ap-7718 529 30 states state NOUN ap-7718 529 31 3.1 3.1 NUM ap-7718 529 32 semiclassical semiclassical ADJ ap-7718 529 33 dynamics dynamic NOUN ap-7718 529 34 4 4 NUM ap-7718 529 35 discussion discussion NOUN ap-7718 529 36 : : PUNCT ap-7718 529 37 conventional conventional ADJ ap-7718 529 38 and and CCONJ ap-7718 529 39 regularized regularize VERB ap-7718 529 40 caldirola caldirola PROPN ap-7718 529 41 - - PUNCT ap-7718 529 42 kanai kanai PROPN ap-7718 529 43 oscillators oscillator VERB ap-7718 529 44 4.1 4.1 NUM ap-7718 529 45 caldirola caldirola PROPN ap-7718 529 46 - - PUNCT ap-7718 529 47 kanai kanai PROPN ap-7718 529 48 case case NOUN ap-7718 529 49 4.2 4.2 NUM ap-7718 529 50 regularized regularize VERB ap-7718 529 51 caldirola caldirola PROPN ap-7718 529 52 - - PUNCT ap-7718 529 53 kanai kanai PROPN ap-7718 529 54 5 5 PROPN ap-7718 529 55 conclusions conclusion NOUN ap-7718 529 56 acknowledgements acknowledgement NOUN ap-7718 529 57 references reference NOUN