id sid tid token lemma pos american_scientific_journal-1069 1 1 265 265 NUM american_scientific_journal-1069 1 2 american american ADJ american_scientific_journal-1069 1 3 scientific scientific ADJ american_scientific_journal-1069 1 4 research research NOUN american_scientific_journal-1069 1 5 journal journal NOUN american_scientific_journal-1069 1 6 for for ADP american_scientific_journal-1069 1 7 engineering engineering NOUN american_scientific_journal-1069 1 8 , , PUNCT american_scientific_journal-1069 1 9 technology technology NOUN american_scientific_journal-1069 1 10 , , PUNCT american_scientific_journal-1069 1 11 and and CCONJ american_scientific_journal-1069 1 12 sciences science NOUN american_scientific_journal-1069 1 13 ( ( PUNCT american_scientific_journal-1069 1 14 asrjets asrjet NOUN american_scientific_journal-1069 1 15 ) ) PUNCT american_scientific_journal-1069 1 16 issn issn PROPN american_scientific_journal-1069 1 17 ( ( PUNCT american_scientific_journal-1069 1 18 print print NOUN american_scientific_journal-1069 1 19 ) ) PUNCT american_scientific_journal-1069 1 20 2313 2313 NUM american_scientific_journal-1069 1 21 - - SYM american_scientific_journal-1069 1 22 4410 4410 NUM american_scientific_journal-1069 1 23 , , PUNCT american_scientific_journal-1069 1 24 issn issn PROPN american_scientific_journal-1069 1 25 ( ( PUNCT american_scientific_journal-1069 1 26 online online ADJ american_scientific_journal-1069 1 27 ) ) PUNCT american_scientific_journal-1069 1 28 2313 2313 NUM american_scientific_journal-1069 1 29 - - SYM american_scientific_journal-1069 1 30 4402 4402 NUM american_scientific_journal-1069 1 31 Β© Β© PROPN american_scientific_journal-1069 1 32 global global ADJ american_scientific_journal-1069 1 33 society society NOUN american_scientific_journal-1069 1 34 of of ADP american_scientific_journal-1069 1 35 scientific scientific ADJ american_scientific_journal-1069 1 36 research research NOUN american_scientific_journal-1069 1 37 and and CCONJ american_scientific_journal-1069 1 38 researchers researcher NOUN american_scientific_journal-1069 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-1069 1 40 how how SCONJ american_scientific_journal-1069 1 41 fractional fractional ADJ american_scientific_journal-1069 1 42 charge charge NOUN american_scientific_journal-1069 1 43 on on ADP american_scientific_journal-1069 1 44 an an DET american_scientific_journal-1069 1 45 electron electron NOUN american_scientific_journal-1069 1 46 in in ADP american_scientific_journal-1069 1 47 the the DET american_scientific_journal-1069 1 48 momentum momentum NOUN american_scientific_journal-1069 1 49 space space NOUN american_scientific_journal-1069 1 50 is be AUX american_scientific_journal-1069 1 51 quantized quantize VERB american_scientific_journal-1069 1 52 ? ? PUNCT american_scientific_journal-1069 2 1 saleem saleem PROPN american_scientific_journal-1069 2 2 iqbal iqbal PROPN american_scientific_journal-1069 3 1 a a PRON american_scientific_journal-1069 3 2 * * X american_scientific_journal-1069 3 3 , , PUNCT american_scientific_journal-1069 3 4 farhana farhana PROPN american_scientific_journal-1069 3 5 sarwar sarwar PROPN american_scientific_journal-1069 3 6 b b PROPN american_scientific_journal-1069 3 7 , , PUNCT american_scientific_journal-1069 3 8 syed syed ADJ american_scientific_journal-1069 3 9 mohsin mohsin PROPN american_scientific_journal-1069 3 10 raza raza PROPN american_scientific_journal-1069 3 11 c c PROPN american_scientific_journal-1069 3 12 , , PUNCT american_scientific_journal-1069 3 13 abdul abdul PROPN american_scientific_journal-1069 3 14 rehman rehman PROPN american_scientific_journal-1069 3 15 d d PROPN american_scientific_journal-1069 3 16 a a PRON american_scientific_journal-1069 3 17 , , PUNCT american_scientific_journal-1069 3 18 d d PROPN american_scientific_journal-1069 3 19 department department NOUN american_scientific_journal-1069 3 20 of of ADP american_scientific_journal-1069 3 21 mathematics mathematics PROPN american_scientific_journal-1069 3 22 , , PUNCT american_scientific_journal-1069 3 23 university university NOUN american_scientific_journal-1069 3 24 of of ADP american_scientific_journal-1069 3 25 balochistan balochistan PROPN american_scientific_journal-1069 3 26 , , PUNCT american_scientific_journal-1069 3 27 quetta quetta PROPN american_scientific_journal-1069 3 28 87300 87300 NUM american_scientific_journal-1069 3 29 , , PUNCT american_scientific_journal-1069 3 30 pakistan pakistan PROPN american_scientific_journal-1069 3 31 b b PROPN american_scientific_journal-1069 3 32 department department PROPN american_scientific_journal-1069 3 33 of of ADP american_scientific_journal-1069 3 34 mathematics mathematics PROPN american_scientific_journal-1069 3 35 f.g.girls f.g.girl NOUN american_scientific_journal-1069 3 36 degree degree NOUN american_scientific_journal-1069 3 37 college college NOUN american_scientific_journal-1069 3 38 , , PUNCT american_scientific_journal-1069 3 39 madrissa madrissa NOUN american_scientific_journal-1069 3 40 road road NOUN american_scientific_journal-1069 3 41 , , PUNCT american_scientific_journal-1069 3 42 quetta quetta PROPN american_scientific_journal-1069 3 43 , , PUNCT american_scientific_journal-1069 3 44 cantt cantt PROPN american_scientific_journal-1069 3 45 , , PUNCT american_scientific_journal-1069 3 46 87300 87300 NUM american_scientific_journal-1069 3 47 , , PUNCT american_scientific_journal-1069 3 48 pakistan pakistan PROPN american_scientific_journal-1069 3 49 . . PUNCT american_scientific_journal-1069 4 1 c c PROPN american_scientific_journal-1069 4 2 department department PROPN american_scientific_journal-1069 4 3 of of ADP american_scientific_journal-1069 4 4 physics physics PROPN american_scientific_journal-1069 4 5 , , PUNCT american_scientific_journal-1069 4 6 university university NOUN american_scientific_journal-1069 4 7 of of ADP american_scientific_journal-1069 4 8 balochistan balochistan PROPN american_scientific_journal-1069 4 9 , , PUNCT american_scientific_journal-1069 4 10 quetta quetta PROPN american_scientific_journal-1069 4 11 87300 87300 NUM american_scientific_journal-1069 4 12 , , PUNCT american_scientific_journal-1069 4 13 pakistan pakistan PROPN american_scientific_journal-1069 4 14 a a DET american_scientific_journal-1069 4 15 email email NOUN american_scientific_journal-1069 4 16 : : PUNCT american_scientific_journal-1069 5 1 saleemiqbal81@yahoo.com saleemiqbal81@yahoo.com X american_scientific_journal-1069 5 2 b b X american_scientific_journal-1069 5 3 email email NOUN american_scientific_journal-1069 5 4 : : PUNCT american_scientific_journal-1069 5 5 f_saleem10@yahoo.com f_saleem10@yahoo.com PROPN american_scientific_journal-1069 5 6 c c PROPN american_scientific_journal-1069 5 7 email email NOUN american_scientific_journal-1069 5 8 : : PUNCT american_scientific_journal-1069 5 9 smraza7@yahoo.com smraza7@yahoo.com X american_scientific_journal-1069 6 1 d d NOUN american_scientific_journal-1069 6 2 email email NOUN american_scientific_journal-1069 6 3 : : PUNCT american_scientific_journal-1069 6 4 abdul_maths@yahoo.com abdul_maths@yahoo.com X american_scientific_journal-1069 6 5 abstract abstract ADJ american_scientific_journal-1069 6 6 with with ADP american_scientific_journal-1069 6 7 our our PRON american_scientific_journal-1069 6 8 conjecture conjecture NOUN american_scientific_journal-1069 6 9 on on ADP american_scientific_journal-1069 6 10 charge charge NOUN american_scientific_journal-1069 6 11 quantization quantization NOUN american_scientific_journal-1069 6 12 ( ( PUNCT american_scientific_journal-1069 6 13 quantum quantum ADJ american_scientific_journal-1069 6 14 dipole dipole NOUN american_scientific_journal-1069 6 15 moment moment NOUN american_scientific_journal-1069 6 16 in in ADP american_scientific_journal-1069 6 17 a a DET american_scientific_journal-1069 6 18 momentum momentum NOUN american_scientific_journal-1069 6 19 space space NOUN american_scientific_journal-1069 6 20 ) ) PUNCT american_scientific_journal-1069 6 21 and and CCONJ american_scientific_journal-1069 6 22 using use VERB american_scientific_journal-1069 6 23 fractional fractional ADJ american_scientific_journal-1069 6 24 fourier fourier NOUN american_scientific_journal-1069 6 25 transform transform NOUN american_scientific_journal-1069 6 26 ( ( PUNCT american_scientific_journal-1069 6 27 frft frft ADJ american_scientific_journal-1069 6 28 ) ) PUNCT american_scientific_journal-1069 6 29 analysis analysis NOUN american_scientific_journal-1069 6 30 on on ADP american_scientific_journal-1069 6 31 hermite hermite ADJ american_scientific_journal-1069 6 32 polynomials polynomial NOUN american_scientific_journal-1069 6 33 ( ( PUNCT american_scientific_journal-1069 6 34 usually usually ADV american_scientific_journal-1069 6 35 used use VERB american_scientific_journal-1069 6 36 for for ADP american_scientific_journal-1069 6 37 quantum quantum ADJ american_scientific_journal-1069 6 38 oscillators oscillator NOUN american_scientific_journal-1069 6 39 ) ) PUNCT american_scientific_journal-1069 6 40 , , PUNCT american_scientific_journal-1069 6 41 we we PRON american_scientific_journal-1069 6 42 obtained obtain VERB american_scientific_journal-1069 6 43 energy energy NOUN american_scientific_journal-1069 6 44 profiles profile NOUN american_scientific_journal-1069 6 45 ( ( PUNCT american_scientific_journal-1069 6 46 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 6 47 ) ) PUNCT american_scientific_journal-1069 6 48 for for ADP american_scientific_journal-1069 6 49 fractional fractional ADJ american_scientific_journal-1069 6 50 quantum quantum ADJ american_scientific_journal-1069 6 51 states state NOUN american_scientific_journal-1069 6 52 on on ADP american_scientific_journal-1069 6 53 the the DET american_scientific_journal-1069 6 54 continuously continuously ADV american_scientific_journal-1069 6 55 changing change VERB american_scientific_journal-1069 6 56 surface surface NOUN american_scientific_journal-1069 6 57 of of ADP american_scientific_journal-1069 6 58 the the DET american_scientific_journal-1069 6 59 electron electron NOUN american_scientific_journal-1069 6 60 . . PUNCT american_scientific_journal-1069 7 1 the the DET american_scientific_journal-1069 7 2 charge charge NOUN american_scientific_journal-1069 7 3 on on ADP american_scientific_journal-1069 7 4 an an DET american_scientific_journal-1069 7 5 electron electron NOUN american_scientific_journal-1069 7 6 as as ADP american_scientific_journal-1069 7 7 a a DET american_scientific_journal-1069 7 8 physical physical ADJ american_scientific_journal-1069 7 9 constant constant NOUN american_scientific_journal-1069 7 10 and and CCONJ american_scientific_journal-1069 7 11 a a DET american_scientific_journal-1069 7 12 single single ADJ american_scientific_journal-1069 7 13 entity entity NOUN american_scientific_journal-1069 7 14 is be AUX american_scientific_journal-1069 7 15 degenerate degenerate ADJ american_scientific_journal-1069 7 16 because because SCONJ american_scientific_journal-1069 7 17 it it PRON american_scientific_journal-1069 7 18 always always ADV american_scientific_journal-1069 7 19 resides reside VERB american_scientific_journal-1069 7 20 on on ADP american_scientific_journal-1069 7 21 the the DET american_scientific_journal-1069 7 22 surface surface NOUN american_scientific_journal-1069 7 23 . . PUNCT american_scientific_journal-1069 8 1 the the DET american_scientific_journal-1069 8 2 charge charge NOUN american_scientific_journal-1069 8 3 is be AUX american_scientific_journal-1069 8 4 fractionally fractionally ADV american_scientific_journal-1069 8 5 quantized quantize VERB american_scientific_journal-1069 8 6 in in ADP american_scientific_journal-1069 8 7 momentum momentum NOUN american_scientific_journal-1069 8 8 space space NOUN american_scientific_journal-1069 8 9 . . PUNCT american_scientific_journal-1069 9 1 the the DET american_scientific_journal-1069 9 2 continuous continuous ADJ american_scientific_journal-1069 9 3 charging charging NOUN american_scientific_journal-1069 9 4 surface surface NOUN american_scientific_journal-1069 9 5 of of ADP american_scientific_journal-1069 9 6 the the DET american_scientific_journal-1069 9 7 electron electron NOUN american_scientific_journal-1069 9 8 is be AUX american_scientific_journal-1069 9 9 due due ADJ american_scientific_journal-1069 9 10 to to ADP american_scientific_journal-1069 9 11 competition competition NOUN american_scientific_journal-1069 9 12 between between ADP american_scientific_journal-1069 9 13 the the DET american_scientific_journal-1069 9 14 centrifugal centrifugal ADJ american_scientific_journal-1069 9 15 and and CCONJ american_scientific_journal-1069 9 16 electodynamic electodynamic ADJ american_scientific_journal-1069 9 17 potentials potential NOUN american_scientific_journal-1069 9 18 . . PUNCT american_scientific_journal-1069 10 1 the the DET american_scientific_journal-1069 10 2 fractional fractional ADJ american_scientific_journal-1069 10 3 quantized quantize VERB american_scientific_journal-1069 10 4 states state NOUN american_scientific_journal-1069 10 5 of of ADP american_scientific_journal-1069 10 6 charges charge NOUN american_scientific_journal-1069 10 7 in in ADP american_scientific_journal-1069 10 8 the the DET american_scientific_journal-1069 10 9 momentum momentum NOUN american_scientific_journal-1069 10 10 space space NOUN american_scientific_journal-1069 10 11 are be AUX american_scientific_journal-1069 10 12 the the DET american_scientific_journal-1069 10 13 manifestations manifestation NOUN american_scientific_journal-1069 10 14 of of ADP american_scientific_journal-1069 10 15 gyroscopic gyroscopic NOUN american_scientific_journal-1069 10 16 constants constant NOUN american_scientific_journal-1069 10 17 , , PUNCT american_scientific_journal-1069 10 18 𝑔2 𝑔2 VERB american_scientific_journal-1069 10 19 ℏ𝑐 ℏ𝑐 PRON american_scientific_journal-1069 10 20 ( ( PUNCT american_scientific_journal-1069 10 21 0.2 0.2 NUM american_scientific_journal-1069 10 22 βˆ’ βˆ’ NUM american_scientific_journal-1069 10 23 0.8 0.8 NUM american_scientific_journal-1069 10 24 ) ) PUNCT american_scientific_journal-1069 10 25 ; ; PUNCT american_scientific_journal-1069 10 26 twisting twisting NOUN american_scientific_journal-1069 10 27 and and CCONJ american_scientific_journal-1069 10 28 twigging twigging NOUN american_scientific_journal-1069 10 29 of of ADP american_scientific_journal-1069 10 30 energy energy NOUN american_scientific_journal-1069 10 31 profiles profile NOUN american_scientific_journal-1069 10 32 ( ( PUNCT american_scientific_journal-1069 10 33 quantum quantum ADJ american_scientific_journal-1069 10 34 electrodynamic electrodynamic ADJ american_scientific_journal-1069 10 35 behavior behavior NOUN american_scientific_journal-1069 10 36 ) ) PUNCT american_scientific_journal-1069 10 37 , , PUNCT american_scientific_journal-1069 10 38 oscillatory oscillatory ADJ american_scientific_journal-1069 10 39 behavior behavior NOUN american_scientific_journal-1069 10 40 of of ADP american_scientific_journal-1069 10 41 energy energy NOUN american_scientific_journal-1069 10 42 associated associate VERB american_scientific_journal-1069 10 43 with with ADP american_scientific_journal-1069 10 44 degeneracy degeneracy NOUN american_scientific_journal-1069 10 45 and and CCONJ american_scientific_journal-1069 10 46 indeed indeed ADV american_scientific_journal-1069 10 47 the the DET american_scientific_journal-1069 10 48 position position NOUN american_scientific_journal-1069 10 49 of of ADP american_scientific_journal-1069 10 50 fractional fractional PROPN american_scientific_journal-1069 10 51 quanta quanta PROPN american_scientific_journal-1069 10 52 in in ADP american_scientific_journal-1069 10 53 terms term NOUN american_scientific_journal-1069 10 54 of of ADP american_scientific_journal-1069 10 55 rotational rotational ADJ american_scientific_journal-1069 10 56 vector vector NOUN american_scientific_journal-1069 10 57 , , PUNCT american_scientific_journal-1069 10 58 𝛼(𝑑 𝛼(𝑑 PROPN american_scientific_journal-1069 10 59 , , PUNCT american_scientific_journal-1069 10 60 πœ” πœ” PRON american_scientific_journal-1069 10 61 ) ) PUNCT american_scientific_journal-1069 10 62 in in ADP american_scientific_journal-1069 10 63 complex complex ADJ american_scientific_journal-1069 10 64 plane plane NOUN american_scientific_journal-1069 10 65 . . PUNCT american_scientific_journal-1069 11 1 keywords keyword NOUN american_scientific_journal-1069 11 2 : : PUNCT american_scientific_journal-1069 11 3 fractional fractional ADJ american_scientific_journal-1069 11 4 fourier fourier NOUN american_scientific_journal-1069 11 5 transform transform NOUN american_scientific_journal-1069 11 6 , , PUNCT american_scientific_journal-1069 11 7 fractional fractional ADJ american_scientific_journal-1069 11 8 charge charge NOUN american_scientific_journal-1069 11 9 quantization quantization NOUN american_scientific_journal-1069 11 10 , , PUNCT american_scientific_journal-1069 11 11 hermite hermite ADJ american_scientific_journal-1069 11 12 polynomials polynomial VERB american_scientific_journal-1069 11 13 1 1 NUM american_scientific_journal-1069 11 14 . . PUNCT american_scientific_journal-1069 12 1 introduction introduction NOUN american_scientific_journal-1069 12 2 we we PRON american_scientific_journal-1069 12 3 attempted attempt VERB american_scientific_journal-1069 12 4 to to PART american_scientific_journal-1069 12 5 decipher decipher VERB american_scientific_journal-1069 12 6 a a DET american_scientific_journal-1069 12 7 new new ADJ american_scientific_journal-1069 12 8 idea idea NOUN american_scientific_journal-1069 12 9 based base VERB american_scientific_journal-1069 12 10 on on ADP american_scientific_journal-1069 12 11 fractional fractional ADJ american_scientific_journal-1069 12 12 charge charge NOUN american_scientific_journal-1069 12 13 quantization quantization NOUN american_scientific_journal-1069 12 14 on on ADP american_scientific_journal-1069 12 15 an an DET american_scientific_journal-1069 12 16 electron electron NOUN american_scientific_journal-1069 12 17 by by ADP american_scientific_journal-1069 12 18 using use VERB american_scientific_journal-1069 12 19 fractional fractional ADJ american_scientific_journal-1069 12 20 fourier fourier NOUN american_scientific_journal-1069 12 21 transform transform NOUN american_scientific_journal-1069 12 22 . . PUNCT american_scientific_journal-1069 13 1 the the DET american_scientific_journal-1069 13 2 charge charge NOUN american_scientific_journal-1069 13 3 on on ADP american_scientific_journal-1069 13 4 an an DET american_scientific_journal-1069 13 5 electron electron NOUN american_scientific_journal-1069 13 6 , , PUNCT american_scientific_journal-1069 13 7 being be AUX american_scientific_journal-1069 13 8 a a DET american_scientific_journal-1069 13 9 physical physical ADJ american_scientific_journal-1069 13 10 constant constant ADJ american_scientific_journal-1069 13 11 and and CCONJ american_scientific_journal-1069 13 12 single single ADJ american_scientific_journal-1069 13 13 entity entity NOUN american_scientific_journal-1069 13 14 , , PUNCT american_scientific_journal-1069 13 15 is be AUX american_scientific_journal-1069 13 16 fractionally fractionally ADV american_scientific_journal-1069 13 17 quantized quantize VERB american_scientific_journal-1069 13 18 due due ADP american_scientific_journal-1069 13 19 to to ADP american_scientific_journal-1069 13 20 momentum momentum NOUN american_scientific_journal-1069 13 21 impact impact NOUN american_scientific_journal-1069 13 22 by by ADP american_scientific_journal-1069 13 23 photons photon NOUN american_scientific_journal-1069 13 24 . . PUNCT american_scientific_journal-1069 14 1 ------------------------------------------------------ ------------------------------------------------------ PUNCT american_scientific_journal-1069 15 1 * * PUNCT american_scientific_journal-1069 15 2 corresponding correspond VERB american_scientific_journal-1069 15 3 author author NOUN american_scientific_journal-1069 15 4 . . PUNCT american_scientific_journal-1069 16 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-1069 16 2 mailto:saleemiqbal81@yahoo.com mailto:saleemiqbal81@yahoo.com PROPN american_scientific_journal-1069 16 3 mailto:f_saleem10@yahoo.com mailto:f_saleem10@yahoo.com X american_scientific_journal-1069 17 1 mailto:smraza7@yahoo.com mailto:smraza7@yahoo.com PROPN american_scientific_journal-1069 17 2 american american PROPN american_scientific_journal-1069 17 3 scientific scientific PROPN american_scientific_journal-1069 17 4 research research PROPN american_scientific_journal-1069 17 5 journal journal NOUN american_scientific_journal-1069 17 6 for for ADP american_scientific_journal-1069 17 7 engineering engineering NOUN american_scientific_journal-1069 17 8 , , PUNCT american_scientific_journal-1069 17 9 technology technology NOUN american_scientific_journal-1069 17 10 , , PUNCT american_scientific_journal-1069 17 11 and and CCONJ american_scientific_journal-1069 17 12 sciences science NOUN american_scientific_journal-1069 17 13 ( ( PUNCT american_scientific_journal-1069 17 14 asrjets asrjet NOUN american_scientific_journal-1069 17 15 ) ) PUNCT american_scientific_journal-1069 17 16 ( ( PUNCT american_scientific_journal-1069 17 17 2015 2015 NUM american_scientific_journal-1069 17 18 ) ) PUNCT american_scientific_journal-1069 17 19 volume volume NOUN american_scientific_journal-1069 17 20 14 14 NUM american_scientific_journal-1069 17 21 , , PUNCT american_scientific_journal-1069 17 22 no no DET american_scientific_journal-1069 17 23 2 2 NUM american_scientific_journal-1069 17 24 , , PUNCT american_scientific_journal-1069 17 25 pp pp ADV american_scientific_journal-1069 17 26 265 265 NUM american_scientific_journal-1069 17 27 - - SYM american_scientific_journal-1069 17 28 272 272 NUM american_scientific_journal-1069 17 29 266 266 NUM american_scientific_journal-1069 17 30 when when SCONJ american_scientific_journal-1069 17 31 a a DET american_scientific_journal-1069 17 32 photon photon NOUN american_scientific_journal-1069 17 33 collides collide VERB american_scientific_journal-1069 17 34 with with ADP american_scientific_journal-1069 17 35 an an DET american_scientific_journal-1069 17 36 electron electron NOUN american_scientific_journal-1069 17 37 , , PUNCT american_scientific_journal-1069 17 38 the the DET american_scientific_journal-1069 17 39 morphology morphology NOUN american_scientific_journal-1069 17 40 of of ADP american_scientific_journal-1069 17 41 a a DET american_scientific_journal-1069 17 42 bounded bounded ADJ american_scientific_journal-1069 17 43 electron electron NOUN american_scientific_journal-1069 17 44 is be AUX american_scientific_journal-1069 17 45 changed change VERB american_scientific_journal-1069 17 46 due due ADJ american_scientific_journal-1069 17 47 to to ADP american_scientific_journal-1069 17 48 inelastic inelastic ADJ american_scientific_journal-1069 17 49 collision collision NOUN american_scientific_journal-1069 17 50 . . PUNCT american_scientific_journal-1069 18 1 the the DET american_scientific_journal-1069 18 2 electron electron PROPN american_scientific_journal-1069 18 3 quanta quanta PROPN american_scientific_journal-1069 18 4 is be AUX american_scientific_journal-1069 18 5 stretched stretch VERB american_scientific_journal-1069 18 6 , , PUNCT american_scientific_journal-1069 18 7 as as ADP american_scientific_journal-1069 18 8 a a DET american_scientific_journal-1069 18 9 consequence consequence NOUN american_scientific_journal-1069 18 10 of of ADP american_scientific_journal-1069 18 11 which which PRON american_scientific_journal-1069 18 12 , , PUNCT american_scientific_journal-1069 18 13 the the DET american_scientific_journal-1069 18 14 wave wave NOUN american_scientific_journal-1069 18 15 length length NOUN american_scientific_journal-1069 18 16 , , PUNCT american_scientific_journal-1069 18 17 πœ† πœ† ADP american_scientific_journal-1069 18 18 increases increase NOUN american_scientific_journal-1069 18 19 and and CCONJ american_scientific_journal-1069 18 20 the the DET american_scientific_journal-1069 18 21 frequency frequency NOUN american_scientific_journal-1069 18 22 , , PUNCT american_scientific_journal-1069 18 23 𝜈 𝜈 PRON american_scientific_journal-1069 18 24 decreases decrease VERB american_scientific_journal-1069 18 25 , , PUNCT american_scientific_journal-1069 18 26 by by ADP american_scientific_journal-1069 18 27 maintaining maintain VERB american_scientific_journal-1069 18 28 , , PUNCT american_scientific_journal-1069 18 29 𝑐 𝑐 PROPN american_scientific_journal-1069 18 30 = = SYM american_scientific_journal-1069 18 31 πœ†πœˆ. πœ†πœˆ. VERB american_scientific_journal-1069 18 32 the the DET american_scientific_journal-1069 18 33 oscillatory oscillatory ADJ american_scientific_journal-1069 18 34 frequency frequency NOUN american_scientific_journal-1069 18 35 of of ADP american_scientific_journal-1069 18 36 the the DET american_scientific_journal-1069 18 37 bounded bounded ADJ american_scientific_journal-1069 18 38 electron electron NOUN american_scientific_journal-1069 18 39 decreases decrease VERB american_scientific_journal-1069 18 40 on on ADP american_scientific_journal-1069 18 41 the the DET american_scientific_journal-1069 18 42 hypothetical hypothetical ADJ american_scientific_journal-1069 18 43 wall wall NOUN american_scientific_journal-1069 18 44 of of ADP american_scientific_journal-1069 18 45 the the DET american_scientific_journal-1069 18 46 electron electron NOUN american_scientific_journal-1069 18 47 string string NOUN american_scientific_journal-1069 18 48 . . PUNCT american_scientific_journal-1069 19 1 this this DET american_scientific_journal-1069 19 2 hypothetical hypothetical ADJ american_scientific_journal-1069 19 3 wall wall NOUN american_scientific_journal-1069 19 4 behaves behave VERB american_scientific_journal-1069 19 5 like like ADP american_scientific_journal-1069 19 6 an an DET american_scientific_journal-1069 19 7 adiabatic adiabatic ADJ american_scientific_journal-1069 19 8 wall wall NOUN american_scientific_journal-1069 19 9 . . PUNCT american_scientific_journal-1069 20 1 due due ADP american_scientific_journal-1069 20 2 to to ADP american_scientific_journal-1069 20 3 overwhelming overwhelming ADJ american_scientific_journal-1069 20 4 centrifugal centrifugal ADJ american_scientific_journal-1069 20 5 potential potential NOUN american_scientific_journal-1069 20 6 as as SCONJ american_scientific_journal-1069 20 7 compare compare ADJ american_scientific_journal-1069 20 8 to to ADP american_scientific_journal-1069 20 9 electrodynamic electrodynamic ADJ american_scientific_journal-1069 20 10 potential potential NOUN american_scientific_journal-1069 20 11 , , PUNCT american_scientific_journal-1069 20 12 the the DET american_scientific_journal-1069 20 13 electron electron PROPN american_scientific_journal-1069 20 14 quanta quanta PROPN american_scientific_journal-1069 20 15 string string NOUN american_scientific_journal-1069 20 16 is be AUX american_scientific_journal-1069 20 17 self self NOUN american_scientific_journal-1069 20 18 ----twisted ----twiste VERB american_scientific_journal-1069 20 19 and and CCONJ american_scientific_journal-1069 20 20 twigged twig VERB american_scientific_journal-1069 20 21 ( ( PUNCT american_scientific_journal-1069 20 22 swirling swirling ADJ american_scientific_journal-1069 20 23 effect effect NOUN american_scientific_journal-1069 20 24 ) ) PUNCT american_scientific_journal-1069 20 25 . . PUNCT american_scientific_journal-1069 21 1 the the DET american_scientific_journal-1069 21 2 charge charge NOUN american_scientific_journal-1069 21 3 on on ADP american_scientific_journal-1069 21 4 electron electron PROPN american_scientific_journal-1069 21 5 quanta quanta PROPN american_scientific_journal-1069 21 6 is be AUX american_scientific_journal-1069 21 7 distributed distribute VERB american_scientific_journal-1069 21 8 on on ADP american_scientific_journal-1069 21 9 twigs twig NOUN american_scientific_journal-1069 21 10 ( ( PUNCT american_scientific_journal-1069 21 11 sub sub NOUN american_scientific_journal-1069 21 12 quanta quanta PROPN american_scientific_journal-1069 21 13 ) ) PUNCT american_scientific_journal-1069 21 14 . . PUNCT american_scientific_journal-1069 22 1 these these DET american_scientific_journal-1069 22 2 twigs twig NOUN american_scientific_journal-1069 22 3 are be AUX american_scientific_journal-1069 22 4 beaded bead VERB american_scientific_journal-1069 22 5 sub sub ADJ american_scientific_journal-1069 22 6 - - ADJ american_scientific_journal-1069 22 7 quanta quanta ADJ american_scientific_journal-1069 22 8 on on ADP american_scientific_journal-1069 22 9 an an DET american_scientific_journal-1069 22 10 electron electron NOUN american_scientific_journal-1069 22 11 string string NOUN american_scientific_journal-1069 22 12 . . PUNCT american_scientific_journal-1069 23 1 the the DET american_scientific_journal-1069 23 2 charge charge NOUN american_scientific_journal-1069 23 3 on on ADP american_scientific_journal-1069 23 4 electron electron NOUN american_scientific_journal-1069 23 5 , , PUNCT american_scientific_journal-1069 23 6 which which PRON american_scientific_journal-1069 23 7 is be AUX american_scientific_journal-1069 23 8 a a DET american_scientific_journal-1069 23 9 single single ADJ american_scientific_journal-1069 23 10 entity entity NOUN american_scientific_journal-1069 23 11 and and CCONJ american_scientific_journal-1069 23 12 constant constant ADJ american_scientific_journal-1069 23 13 , , PUNCT american_scientific_journal-1069 23 14 is be AUX american_scientific_journal-1069 23 15 distributed distribute VERB american_scientific_journal-1069 23 16 on on ADP american_scientific_journal-1069 23 17 these these DET american_scientific_journal-1069 23 18 sub sub NOUN american_scientific_journal-1069 23 19 quanta quanta PROPN american_scientific_journal-1069 23 20 ( ( PUNCT american_scientific_journal-1069 23 21 twigs twig NOUN american_scientific_journal-1069 23 22 ) ) PUNCT american_scientific_journal-1069 23 23 and and CCONJ american_scientific_journal-1069 23 24 hence hence ADV american_scientific_journal-1069 23 25 the the DET american_scientific_journal-1069 23 26 fractional fractional ADJ american_scientific_journal-1069 23 27 charge charge NOUN american_scientific_journal-1069 23 28 quantization quantization NOUN american_scientific_journal-1069 23 29 . . PUNCT american_scientific_journal-1069 24 1 remember remember VERB american_scientific_journal-1069 24 2 that that SCONJ american_scientific_journal-1069 24 3 each each PRON american_scientific_journal-1069 24 4 of of ADP american_scientific_journal-1069 24 5 these these DET american_scientific_journal-1069 24 6 sub sub NOUN american_scientific_journal-1069 24 7 - - NOUN american_scientific_journal-1069 24 8 quanta quanta ADJ american_scientific_journal-1069 24 9 on on ADP american_scientific_journal-1069 24 10 an an DET american_scientific_journal-1069 24 11 electron electron NOUN american_scientific_journal-1069 24 12 has have VERB american_scientific_journal-1069 24 13 an an DET american_scientific_journal-1069 24 14 integrated integrate VERB american_scientific_journal-1069 24 15 oscillatory oscillatory ADJ american_scientific_journal-1069 24 16 effect effect NOUN american_scientific_journal-1069 24 17 ( ( PUNCT american_scientific_journal-1069 24 18 discovered discover VERB american_scientific_journal-1069 24 19 in in ADP american_scientific_journal-1069 24 20 this this DET american_scientific_journal-1069 24 21 paper paper NOUN american_scientific_journal-1069 24 22 ) ) PUNCT american_scientific_journal-1069 24 23 , , PUNCT american_scientific_journal-1069 24 24 .i.e .i.e X american_scientific_journal-1069 24 25 . . PUNCT american_scientific_journal-1069 25 1 ,2𝑛𝑓 ,2𝑛𝑓 PUNCT american_scientific_journal-1069 25 2 , , PUNCT american_scientific_journal-1069 25 3 where where SCONJ american_scientific_journal-1069 25 4 0.1 0.1 NUM american_scientific_journal-1069 25 5 ≀ ≀ NOUN american_scientific_journal-1069 25 6 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 25 7 ≀ ≀ NUM american_scientific_journal-1069 25 8 0.9 0.9 NUM american_scientific_journal-1069 25 9 , , PUNCT american_scientific_journal-1069 25 10 and and CCONJ american_scientific_journal-1069 25 11 are be AUX american_scientific_journal-1069 25 12 beaded bead VERB american_scientific_journal-1069 25 13 on on ADP american_scientific_journal-1069 25 14 an an DET american_scientific_journal-1069 25 15 electron electron NOUN american_scientific_journal-1069 25 16 quanta quanta PROPN american_scientific_journal-1069 25 17 string string PROPN american_scientific_journal-1069 25 18 . . PUNCT american_scientific_journal-1069 26 1 the the DET american_scientific_journal-1069 26 2 momentum momentum NOUN american_scientific_journal-1069 26 3 impact impact NOUN american_scientific_journal-1069 26 4 of of ADP american_scientific_journal-1069 26 5 a a DET american_scientific_journal-1069 26 6 photon photon NOUN american_scientific_journal-1069 26 7 on on ADP american_scientific_journal-1069 26 8 a a DET american_scientific_journal-1069 26 9 bounded bound VERB american_scientific_journal-1069 26 10 electron electron NOUN american_scientific_journal-1069 26 11 causes cause VERB american_scientific_journal-1069 26 12 stretching stretch VERB american_scientific_journal-1069 26 13 . . PUNCT american_scientific_journal-1069 27 1 this this DET american_scientific_journal-1069 27 2 stretching stretching NOUN american_scientific_journal-1069 27 3 is be AUX american_scientific_journal-1069 27 4 a a DET american_scientific_journal-1069 27 5 manifestation manifestation NOUN american_scientific_journal-1069 27 6 of of ADP american_scientific_journal-1069 27 7 quantum quantum ADJ american_scientific_journal-1069 27 8 mechanical mechanical ADJ american_scientific_journal-1069 27 9 scattering scattering NOUN american_scientific_journal-1069 27 10 ( ( PUNCT american_scientific_journal-1069 27 11 inelastic inelastic ADJ american_scientific_journal-1069 27 12 scattering scattering NOUN american_scientific_journal-1069 27 13 ) ) PUNCT american_scientific_journal-1069 27 14 which which PRON american_scientific_journal-1069 27 15 hold hold VERB american_scientific_journal-1069 27 16 true true ADJ american_scientific_journal-1069 27 17 for for ADP american_scientific_journal-1069 27 18 compton compton PROPN american_scientific_journal-1069 27 19 and and CCONJ american_scientific_journal-1069 27 20 photoelectric photoelectric ADJ american_scientific_journal-1069 27 21 effects effect NOUN american_scientific_journal-1069 27 22 , , PUNCT american_scientific_journal-1069 27 23 too too ADV american_scientific_journal-1069 27 24 . . PUNCT american_scientific_journal-1069 28 1 the the DET american_scientific_journal-1069 28 2 stretching stretching NOUN american_scientific_journal-1069 28 3 , , PUNCT american_scientific_journal-1069 28 4 twisting twisting NOUN american_scientific_journal-1069 28 5 and and CCONJ american_scientific_journal-1069 28 6 twigging twigging NOUN american_scientific_journal-1069 28 7 holds hold VERB american_scientific_journal-1069 28 8 true true ADJ american_scientific_journal-1069 28 9 for for ADP american_scientific_journal-1069 28 10 quantized quantize VERB american_scientific_journal-1069 28 11 particles particle NOUN american_scientific_journal-1069 28 12 , , PUNCT american_scientific_journal-1069 28 13 but but CCONJ american_scientific_journal-1069 28 14 not not PART american_scientific_journal-1069 28 15 for for ADP american_scientific_journal-1069 28 16 free free ADJ american_scientific_journal-1069 28 17 particles particle NOUN american_scientific_journal-1069 28 18 as as SCONJ american_scientific_journal-1069 28 19 the the DET american_scientific_journal-1069 28 20 case case NOUN american_scientific_journal-1069 28 21 is be AUX american_scientific_journal-1069 28 22 for for ADP american_scientific_journal-1069 28 23 compton compton PROPN american_scientific_journal-1069 28 24 and and CCONJ american_scientific_journal-1069 28 25 photoelectric photoelectric ADJ american_scientific_journal-1069 28 26 effects effect NOUN american_scientific_journal-1069 28 27 . . PUNCT american_scientific_journal-1069 29 1 that that PRON american_scientific_journal-1069 29 2 is be AUX american_scientific_journal-1069 29 3 why why SCONJ american_scientific_journal-1069 29 4 eisenstein eisenstein PROPN american_scientific_journal-1069 29 5 of of ADP american_scientific_journal-1069 29 6 caltech caltech PROPN american_scientific_journal-1069 29 7 ( ( PUNCT american_scientific_journal-1069 29 8 usa usa PROPN american_scientific_journal-1069 29 9 ) ) PUNCT american_scientific_journal-1069 29 10 , , PUNCT american_scientific_journal-1069 29 11 on on ADP american_scientific_journal-1069 29 12 the the DET american_scientific_journal-1069 29 13 basis basis NOUN american_scientific_journal-1069 29 14 of of ADP american_scientific_journal-1069 29 15 experimental experimental ADJ american_scientific_journal-1069 29 16 results result NOUN american_scientific_journal-1069 29 17 , , PUNCT american_scientific_journal-1069 29 18 considered consider VERB american_scientific_journal-1069 29 19 quasi quasi ADJ american_scientific_journal-1069 29 20 particle particle NOUN american_scientific_journal-1069 29 21 nature nature NOUN american_scientific_journal-1069 29 22 of of ADP american_scientific_journal-1069 29 23 bounded bounded ADJ american_scientific_journal-1069 29 24 electron electron NOUN american_scientific_journal-1069 29 25 due due ADP american_scientific_journal-1069 29 26 to to ADP american_scientific_journal-1069 29 27 its its PRON american_scientific_journal-1069 29 28 morphological morphological ADJ american_scientific_journal-1069 29 29 changes change NOUN american_scientific_journal-1069 29 30 . . PUNCT american_scientific_journal-1069 30 1 2 2 X american_scientific_journal-1069 30 2 . . X american_scientific_journal-1069 30 3 theory theory NOUN american_scientific_journal-1069 30 4 the the DET american_scientific_journal-1069 30 5 quantum quantum ADJ american_scientific_journal-1069 30 6 dipole dipole NOUN american_scientific_journal-1069 30 7 moments moment NOUN american_scientific_journal-1069 30 8 lead lead VERB american_scientific_journal-1069 30 9 to to PART american_scientific_journal-1069 30 10 charge charge VERB american_scientific_journal-1069 30 11 quantization quantization NOUN american_scientific_journal-1069 30 12 [ [ X american_scientific_journal-1069 30 13 1 1 NUM american_scientific_journal-1069 30 14 , , PUNCT american_scientific_journal-1069 30 15 2 2 NUM american_scientific_journal-1069 30 16 , , PUNCT american_scientific_journal-1069 30 17 3 3 NUM american_scientific_journal-1069 30 18 , , PUNCT american_scientific_journal-1069 30 19 4 4 NUM american_scientific_journal-1069 30 20 ] ] SYM american_scientific_journal-1069 30 21 π‘₯ π‘₯ NOUN american_scientific_journal-1069 30 22 = = PUNCT american_scientific_journal-1069 30 23 β„Žπ‘ž β„Žπ‘ž NOUN american_scientific_journal-1069 30 24 ( ( PUNCT american_scientific_journal-1069 30 25 1 1 NUM american_scientific_journal-1069 30 26 ) ) PUNCT american_scientific_journal-1069 30 27 where where SCONJ american_scientific_journal-1069 30 28 π‘₯ π‘₯ PROPN american_scientific_journal-1069 30 29 is be AUX american_scientific_journal-1069 30 30 the the DET american_scientific_journal-1069 30 31 quantum quantum ADJ american_scientific_journal-1069 30 32 dipole dipole NOUN american_scientific_journal-1069 30 33 moment moment NOUN american_scientific_journal-1069 30 34 , , PUNCT american_scientific_journal-1069 30 35 π‘ž π‘ž X american_scientific_journal-1069 30 36 the the DET american_scientific_journal-1069 30 37 charge charge NOUN american_scientific_journal-1069 30 38 and and CCONJ american_scientific_journal-1069 30 39 β„Ž β„Ž PROPN american_scientific_journal-1069 30 40 is be AUX american_scientific_journal-1069 30 41 the the DET american_scientific_journal-1069 30 42 planck planck NOUN american_scientific_journal-1069 30 43 ’s ’s PART american_scientific_journal-1069 30 44 constant constant ADJ american_scientific_journal-1069 30 45 ( ( PUNCT american_scientific_journal-1069 30 46 quantum quantum ADJ american_scientific_journal-1069 30 47 action action NOUN american_scientific_journal-1069 30 48 ) ) PUNCT american_scientific_journal-1069 30 49 the the DET american_scientific_journal-1069 30 50 matter matter NOUN american_scientific_journal-1069 30 51 energy energy NOUN american_scientific_journal-1069 30 52 such such ADJ american_scientific_journal-1069 30 53 as as ADP american_scientific_journal-1069 30 54 of of ADP american_scientific_journal-1069 30 55 an an DET american_scientific_journal-1069 30 56 electron electron NOUN american_scientific_journal-1069 30 57 exists exist VERB american_scientific_journal-1069 30 58 in in ADP american_scientific_journal-1069 30 59 the the DET american_scientific_journal-1069 30 60 form form NOUN american_scientific_journal-1069 30 61 of of ADP american_scientific_journal-1069 30 62 transverse transverse NOUN american_scientific_journal-1069 30 63 wave wave NOUN american_scientific_journal-1069 30 64 . . PUNCT american_scientific_journal-1069 31 1 this this DET american_scientific_journal-1069 31 2 energy energy NOUN american_scientific_journal-1069 31 3 is be AUX american_scientific_journal-1069 31 4 oscillatory oscillatory ADJ american_scientific_journal-1069 31 5 ( ( PUNCT american_scientific_journal-1069 31 6 quantum quantum ADJ american_scientific_journal-1069 31 7 action action NOUN american_scientific_journal-1069 31 8 ) ) PUNCT american_scientific_journal-1069 31 9 . . PUNCT american_scientific_journal-1069 32 1 and and CCONJ american_scientific_journal-1069 32 2 configures configure VERB american_scientific_journal-1069 32 3 a a DET american_scientific_journal-1069 32 4 space space NOUN american_scientific_journal-1069 32 5 called call VERB american_scientific_journal-1069 32 6 a a DET american_scientific_journal-1069 32 7 wave wave NOUN american_scientific_journal-1069 32 8 packet packet NOUN american_scientific_journal-1069 32 9 or or CCONJ american_scientific_journal-1069 32 10 β€œ " PUNCT american_scientific_journal-1069 32 11 quanta quanta PROPN american_scientific_journal-1069 32 12 ” " PUNCT american_scientific_journal-1069 32 13 . . PUNCT american_scientific_journal-1069 33 1 we we PRON american_scientific_journal-1069 33 2 consider consider VERB american_scientific_journal-1069 33 3 that that SCONJ american_scientific_journal-1069 33 4 the the DET american_scientific_journal-1069 33 5 charge charge NOUN american_scientific_journal-1069 33 6 of of ADP american_scientific_journal-1069 33 7 an an DET american_scientific_journal-1069 33 8 electron electron NOUN american_scientific_journal-1069 33 9 is be AUX american_scientific_journal-1069 33 10 treated treat VERB american_scientific_journal-1069 33 11 as as ADP american_scientific_journal-1069 33 12 its its PRON american_scientific_journal-1069 33 13 density density NOUN american_scientific_journal-1069 33 14 which which PRON american_scientific_journal-1069 33 15 is be AUX american_scientific_journal-1069 33 16 not not PART american_scientific_journal-1069 33 17 only only ADV american_scientific_journal-1069 33 18 smeared smear VERB american_scientific_journal-1069 33 19 on on ADP american_scientific_journal-1069 33 20 the the DET american_scientific_journal-1069 33 21 surface surface NOUN american_scientific_journal-1069 33 22 but but CCONJ american_scientific_journal-1069 33 23 also also ADV american_scientific_journal-1069 33 24 inside inside ADP american_scientific_journal-1069 33 25 the the DET american_scientific_journal-1069 33 26 volume volume NOUN american_scientific_journal-1069 33 27 despite despite SCONJ american_scientific_journal-1069 33 28 the the DET american_scientific_journal-1069 33 29 fact fact NOUN american_scientific_journal-1069 33 30 that that SCONJ american_scientific_journal-1069 33 31 charges charge NOUN american_scientific_journal-1069 33 32 always always ADV american_scientific_journal-1069 33 33 reside reside VERB american_scientific_journal-1069 33 34 on on ADP american_scientific_journal-1069 33 35 the the DET american_scientific_journal-1069 33 36 surface surface NOUN american_scientific_journal-1069 33 37 . . PUNCT american_scientific_journal-1069 34 1 with with ADP american_scientific_journal-1069 34 2 momentum momentum NOUN american_scientific_journal-1069 34 3 impact impact NOUN american_scientific_journal-1069 34 4 , , PUNCT american_scientific_journal-1069 34 5 the the DET american_scientific_journal-1069 34 6 electron electron NOUN american_scientific_journal-1069 34 7 quanta quanta PROPN american_scientific_journal-1069 34 8 is be AUX american_scientific_journal-1069 34 9 first first ADV american_scientific_journal-1069 34 10 stretched stretch VERB american_scientific_journal-1069 34 11 , , PUNCT american_scientific_journal-1069 34 12 twisted twisted ADJ american_scientific_journal-1069 34 13 and and CCONJ american_scientific_journal-1069 34 14 then then ADV american_scientific_journal-1069 34 15 twigged twig VERB american_scientific_journal-1069 34 16 . . PUNCT american_scientific_journal-1069 35 1 we we PRON american_scientific_journal-1069 35 2 envisage envisage VERB american_scientific_journal-1069 35 3 the the DET american_scientific_journal-1069 35 4 electron electron NOUN american_scientific_journal-1069 35 5 like like ADP american_scientific_journal-1069 35 6 a a DET american_scientific_journal-1069 35 7 flexible flexible ADJ american_scientific_journal-1069 35 8 ball ball NOUN american_scientific_journal-1069 35 9 , , PUNCT american_scientific_journal-1069 35 10 the the DET american_scientific_journal-1069 35 11 surface surface NOUN american_scientific_journal-1069 35 12 of of ADP american_scientific_journal-1069 35 13 which which PRON american_scientific_journal-1069 35 14 would would AUX american_scientific_journal-1069 35 15 vary vary VERB american_scientific_journal-1069 35 16 continuously continuously ADV american_scientific_journal-1069 35 17 due due ADJ american_scientific_journal-1069 35 18 to to ADP american_scientific_journal-1069 35 19 competing compete VERB american_scientific_journal-1069 35 20 centrifugal centrifugal ADJ american_scientific_journal-1069 35 21 and and CCONJ american_scientific_journal-1069 35 22 electrodynamic electrodynamic ADJ american_scientific_journal-1069 35 23 potential potential NOUN american_scientific_journal-1069 35 24 . . PUNCT american_scientific_journal-1069 36 1 the the DET american_scientific_journal-1069 36 2 coupling couple VERB american_scientific_journal-1069 36 3 constant constant ADJ american_scientific_journal-1069 36 4 𝑒2 𝑒2 PROPN american_scientific_journal-1069 36 5 ℏ𝑐 ℏ𝑐 X american_scientific_journal-1069 36 6 ~ ~ PUNCT american_scientific_journal-1069 36 7 1 1 NUM american_scientific_journal-1069 36 8 137 137 NUM american_scientific_journal-1069 36 9 on on ADP american_scientific_journal-1069 36 10 an an DET american_scientific_journal-1069 36 11 electron electron NOUN american_scientific_journal-1069 36 12 is be AUX american_scientific_journal-1069 36 13 overwhelmed overwhelm VERB american_scientific_journal-1069 36 14 by by ADP american_scientific_journal-1069 36 15 the the DET american_scientific_journal-1069 36 16 gyroscopic gyroscopic NOUN american_scientific_journal-1069 36 17 constant constant ADJ american_scientific_journal-1069 36 18 , , PUNCT american_scientific_journal-1069 36 19 𝑔2 𝑔2 VERB american_scientific_journal-1069 36 20 ℏ𝑐 ℏ𝑐 PRON american_scientific_journal-1069 36 21 ( ( PUNCT american_scientific_journal-1069 36 22 0.2 0.2 NUM american_scientific_journal-1069 36 23 βˆ’ βˆ’ NOUN american_scientific_journal-1069 36 24 0.8 0.8 NUM american_scientific_journal-1069 36 25 ) ) PUNCT american_scientific_journal-1069 36 26 due due ADP american_scientific_journal-1069 36 27 to to ADP american_scientific_journal-1069 36 28 pronounced pronounce VERB american_scientific_journal-1069 36 29 centrifugal centrifugal ADJ american_scientific_journal-1069 36 30 potential potential NOUN american_scientific_journal-1069 36 31 . . PUNCT american_scientific_journal-1069 37 1 this this PRON american_scientific_journal-1069 37 2 causes cause VERB american_scientific_journal-1069 37 3 the the DET american_scientific_journal-1069 37 4 charge charge NOUN american_scientific_journal-1069 37 5 on on ADP american_scientific_journal-1069 37 6 an an DET american_scientific_journal-1069 37 7 electron electron NOUN american_scientific_journal-1069 37 8 to to PART american_scientific_journal-1069 37 9 become become VERB american_scientific_journal-1069 37 10 degenerate degenerate ADJ american_scientific_journal-1069 37 11 and and CCONJ american_scientific_journal-1069 37 12 fractionally fractionally ADV american_scientific_journal-1069 37 13 quantized quantize VERB american_scientific_journal-1069 37 14 on on ADP american_scientific_journal-1069 37 15 its its PRON american_scientific_journal-1069 37 16 surface surface NOUN american_scientific_journal-1069 37 17 and and CCONJ american_scientific_journal-1069 37 18 hence hence ADV american_scientific_journal-1069 37 19 the the DET american_scientific_journal-1069 37 20 charge charge NOUN american_scientific_journal-1069 37 21 quantization quantization NOUN american_scientific_journal-1069 37 22 . . PUNCT american_scientific_journal-1069 38 1 the the DET american_scientific_journal-1069 38 2 depth depth NOUN american_scientific_journal-1069 38 3 of of ADP american_scientific_journal-1069 38 4 the the DET american_scientific_journal-1069 38 5 quantum quantum ADJ american_scientific_journal-1069 38 6 well well NOUN american_scientific_journal-1069 38 7 of of ADP american_scientific_journal-1069 38 8 an an DET american_scientific_journal-1069 38 9 electron electron NOUN american_scientific_journal-1069 38 10 is be AUX american_scientific_journal-1069 38 11 equivalent equivalent ADJ american_scientific_journal-1069 38 12 to to ADP american_scientific_journal-1069 38 13 its its PRON american_scientific_journal-1069 38 14 radius radius NOUN american_scientific_journal-1069 38 15 . . PUNCT american_scientific_journal-1069 39 1 with with ADP american_scientific_journal-1069 39 2 fractional fractional ADJ american_scientific_journal-1069 39 3 charge charge NOUN american_scientific_journal-1069 39 4 quantization quantization NOUN american_scientific_journal-1069 39 5 , , PUNCT american_scientific_journal-1069 39 6 the the DET american_scientific_journal-1069 39 7 envelope envelope NOUN american_scientific_journal-1069 39 8 of of ADP american_scientific_journal-1069 39 9 energy energy NOUN american_scientific_journal-1069 39 10 associated associate VERB american_scientific_journal-1069 39 11 with with ADP american_scientific_journal-1069 39 12 an an DET american_scientific_journal-1069 39 13 electron electron NOUN american_scientific_journal-1069 39 14 ( ( PUNCT american_scientific_journal-1069 39 15 electron electron PROPN american_scientific_journal-1069 39 16 quanta quanta PROPN american_scientific_journal-1069 39 17 ) ) PUNCT american_scientific_journal-1069 39 18 is be AUX american_scientific_journal-1069 39 19 twisted twist VERB american_scientific_journal-1069 39 20 and and CCONJ american_scientific_journal-1069 39 21 twigged twig VERB american_scientific_journal-1069 39 22 to to PART american_scientific_journal-1069 39 23 smear smear VERB american_scientific_journal-1069 39 24 the the DET american_scientific_journal-1069 39 25 density density NOUN american_scientific_journal-1069 39 26 ( ( PUNCT american_scientific_journal-1069 39 27 charge charge NOUN american_scientific_journal-1069 39 28 ) ) PUNCT american_scientific_journal-1069 39 29 of of ADP american_scientific_journal-1069 39 30 energy energy NOUN american_scientific_journal-1069 39 31 in in ADP american_scientific_journal-1069 39 32 its its PRON american_scientific_journal-1069 39 33 fractional fractional ADJ american_scientific_journal-1069 39 34 components component NOUN american_scientific_journal-1069 39 35 , , PUNCT american_scientific_journal-1069 39 36 as as ADP american_scientific_journal-1069 39 37 a a DET american_scientific_journal-1069 39 38 consequences consequence NOUN american_scientific_journal-1069 39 39 of of ADP american_scientific_journal-1069 39 40 which which PRON american_scientific_journal-1069 39 41 , , PUNCT american_scientific_journal-1069 39 42 the the DET american_scientific_journal-1069 39 43 fractional fractional ADJ american_scientific_journal-1069 39 44 charges charge NOUN american_scientific_journal-1069 39 45 float float VERB american_scientific_journal-1069 39 46 on on ADP american_scientific_journal-1069 39 47 their their PRON american_scientific_journal-1069 39 48 respective respective ADJ american_scientific_journal-1069 39 49 broken broken ADJ american_scientific_journal-1069 39 50 β€œ " PUNCT american_scientific_journal-1069 39 51 quanta quanta PROPN american_scientific_journal-1069 39 52 ” " PUNCT american_scientific_journal-1069 39 53 only only ADV american_scientific_journal-1069 39 54 on on ADP american_scientific_journal-1069 39 55 the the DET american_scientific_journal-1069 39 56 surface surface NOUN american_scientific_journal-1069 39 57 . . PUNCT american_scientific_journal-1069 40 1 each each PRON american_scientific_journal-1069 40 2 of of ADP american_scientific_journal-1069 40 3 the the DET american_scientific_journal-1069 40 4 broken broken ADJ american_scientific_journal-1069 40 5 sub sub NOUN american_scientific_journal-1069 40 6 - - ADJ american_scientific_journal-1069 40 7 quanta quanta ADJ american_scientific_journal-1069 40 8 is be AUX american_scientific_journal-1069 40 9 woven weave VERB american_scientific_journal-1069 40 10 in in ADP american_scientific_journal-1069 40 11 a a DET american_scientific_journal-1069 40 12 string string NOUN american_scientific_journal-1069 40 13 due due ADP american_scientific_journal-1069 40 14 to to ADP american_scientific_journal-1069 40 15 whirling whirl VERB american_scientific_journal-1069 40 16 and and CCONJ american_scientific_journal-1069 40 17 swirling swirling ADJ american_scientific_journal-1069 40 18 effects effect NOUN american_scientific_journal-1069 40 19 ( ( PUNCT american_scientific_journal-1069 40 20 electro electro VERB american_scientific_journal-1069 40 21 weak weak ADJ american_scientific_journal-1069 40 22 interaction interaction NOUN american_scientific_journal-1069 40 23 ) ) PUNCT american_scientific_journal-1069 40 24 on on ADP american_scientific_journal-1069 40 25 an an DET american_scientific_journal-1069 40 26 electron electron NOUN american_scientific_journal-1069 40 27 . . PUNCT american_scientific_journal-1069 41 1 these these DET american_scientific_journal-1069 41 2 broken break VERB american_scientific_journal-1069 41 3 sub sub ADJ american_scientific_journal-1069 41 4 - - ADJ american_scientific_journal-1069 41 5 quanta quanta ADJ american_scientific_journal-1069 41 6 are be AUX american_scientific_journal-1069 41 7 degenerate degenerate ADJ american_scientific_journal-1069 41 8 fractional fractional ADJ american_scientific_journal-1069 41 9 charged charge VERB american_scientific_journal-1069 41 10 quantized quantize VERB american_scientific_journal-1069 41 11 states state NOUN american_scientific_journal-1069 41 12 in in ADP american_scientific_journal-1069 41 13 the the DET american_scientific_journal-1069 41 14 momentum momentum NOUN american_scientific_journal-1069 41 15 space space NOUN american_scientific_journal-1069 41 16 . . PUNCT american_scientific_journal-1069 42 1 each each PRON american_scientific_journal-1069 42 2 of of ADP american_scientific_journal-1069 42 3 sub sub NOUN american_scientific_journal-1069 42 4 - - ADJ american_scientific_journal-1069 42 5 quanta quanta ADJ american_scientific_journal-1069 42 6 would would AUX american_scientific_journal-1069 42 7 have have VERB american_scientific_journal-1069 42 8 the the DET american_scientific_journal-1069 42 9 oscillatory oscillatory ADJ american_scientific_journal-1069 42 10 behavior behavior NOUN american_scientific_journal-1069 42 11 . . PUNCT american_scientific_journal-1069 43 1 american american ADJ american_scientific_journal-1069 43 2 scientific scientific ADJ american_scientific_journal-1069 43 3 research research PROPN american_scientific_journal-1069 43 4 journal journal NOUN american_scientific_journal-1069 43 5 for for ADP american_scientific_journal-1069 43 6 engineering engineering NOUN american_scientific_journal-1069 43 7 , , PUNCT american_scientific_journal-1069 43 8 technology technology NOUN american_scientific_journal-1069 43 9 , , PUNCT american_scientific_journal-1069 43 10 and and CCONJ american_scientific_journal-1069 43 11 sciences science NOUN american_scientific_journal-1069 43 12 ( ( PUNCT american_scientific_journal-1069 43 13 asrjets asrjet NOUN american_scientific_journal-1069 43 14 ) ) PUNCT american_scientific_journal-1069 43 15 ( ( PUNCT american_scientific_journal-1069 43 16 2015 2015 NUM american_scientific_journal-1069 43 17 ) ) PUNCT american_scientific_journal-1069 43 18 volume volume NOUN american_scientific_journal-1069 43 19 14 14 NUM american_scientific_journal-1069 43 20 , , PUNCT american_scientific_journal-1069 43 21 no no DET american_scientific_journal-1069 43 22 2 2 NUM american_scientific_journal-1069 43 23 , , PUNCT american_scientific_journal-1069 43 24 pp pp ADV american_scientific_journal-1069 43 25 265 265 NUM american_scientific_journal-1069 43 26 - - SYM american_scientific_journal-1069 43 27 272 272 NUM american_scientific_journal-1069 43 28 267 267 NUM american_scientific_journal-1069 43 29 almeida almeida PROPN american_scientific_journal-1069 44 1 [ [ X american_scientific_journal-1069 44 2 5 5 NUM american_scientific_journal-1069 44 3 ] ] PUNCT american_scientific_journal-1069 44 4 defines define VERB american_scientific_journal-1069 44 5 the the DET american_scientific_journal-1069 44 6 fractional fractional ADJ american_scientific_journal-1069 44 7 fourier fourier NOUN american_scientific_journal-1069 44 8 transform transform NOUN american_scientific_journal-1069 44 9 ( ( PUNCT american_scientific_journal-1069 44 10 frft frft ADJ american_scientific_journal-1069 44 11 ) ) PUNCT american_scientific_journal-1069 44 12 of of ADP american_scientific_journal-1069 44 13 a a DET american_scientific_journal-1069 44 14 function function NOUN american_scientific_journal-1069 44 15 π‘₯(𝑑 π‘₯(𝑑 NOUN american_scientific_journal-1069 44 16 ) ) PUNCT american_scientific_journal-1069 44 17 , , PUNCT american_scientific_journal-1069 44 18 with with ADP american_scientific_journal-1069 44 19 angle angle NOUN american_scientific_journal-1069 44 20 𝛼 𝛼 X american_scientific_journal-1069 44 21 ( ( PUNCT american_scientific_journal-1069 44 22 𝑑 𝑑 PROPN american_scientific_journal-1069 44 23 is be AUX american_scientific_journal-1069 44 24 the the DET american_scientific_journal-1069 44 25 time time NOUN american_scientific_journal-1069 44 26 and and CCONJ american_scientific_journal-1069 44 27 𝑒 𝑒 PROPN american_scientific_journal-1069 44 28 is be AUX american_scientific_journal-1069 44 29 the the DET american_scientific_journal-1069 44 30 frequency frequency NOUN american_scientific_journal-1069 44 31 ) ) PUNCT american_scientific_journal-1069 44 32 as as ADP american_scientific_journal-1069 44 33 ℱ𝛼[π‘₯(𝑑 ℱ𝛼[π‘₯(𝑑 NOUN american_scientific_journal-1069 44 34 ) ) PUNCT american_scientific_journal-1069 44 35 ] ] PUNCT american_scientific_journal-1069 45 1 = = SYM american_scientific_journal-1069 45 2 𝑋𝛼(𝑒 𝑋𝛼(𝑒 NOUN american_scientific_journal-1069 45 3 ) ) PUNCT american_scientific_journal-1069 45 4 = = SYM american_scientific_journal-1069 45 5 ∫ ∫ NOUN american_scientific_journal-1069 45 6 π‘₯(𝑑 π‘₯(𝑑 PROPN american_scientific_journal-1069 45 7 ) ) PUNCT american_scientific_journal-1069 45 8 ∞ ∞ NUM american_scientific_journal-1069 45 9 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-1069 45 10 𝐾𝛼(𝑑 𝐾𝛼(𝑑 NOUN american_scientific_journal-1069 45 11 , , PUNCT american_scientific_journal-1069 45 12 𝑒)𝑑𝑑 𝑒)𝑑𝑑 PROPN american_scientific_journal-1069 45 13 = = PRON american_scientific_journal-1069 45 14 { { PUNCT american_scientific_journal-1069 45 15 √ √ NUM american_scientific_journal-1069 45 16 1βˆ’π‘— 1βˆ’π‘— NUM american_scientific_journal-1069 45 17 cot𝛼 cot𝛼 PROPN american_scientific_journal-1069 45 18 2πœ‹ 2πœ‹ NUM american_scientific_journal-1069 45 19 𝑒𝑗 𝑒𝑗 PROPN american_scientific_journal-1069 45 20 𝑒2 𝑒2 PROPN american_scientific_journal-1069 45 21 2 2 NUM american_scientific_journal-1069 45 22 cot𝛼 cot𝛼 NOUN american_scientific_journal-1069 45 23 ∫ ∫ PROPN american_scientific_journal-1069 46 1 π‘₯(𝑑)𝑒𝑗 π‘₯(𝑑)𝑒𝑗 NUM american_scientific_journal-1069 46 2 𝑑2 𝑑2 ADJ american_scientific_journal-1069 46 3 2 2 NUM american_scientific_journal-1069 46 4 cotπ›Όβˆ’π‘—π‘’π‘‘ cotπ›Όβˆ’π‘—π‘’π‘‘ PROPN american_scientific_journal-1069 46 5 csc𝛼𝑑𝑑 csc𝛼𝑑𝑑 NOUN american_scientific_journal-1069 46 6 𝑖𝑓 𝑖𝑓 ADP american_scientific_journal-1069 46 7 𝛼 𝛼 PROPN american_scientific_journal-1069 46 8 𝑖𝑠 𝑖𝑠 NOUN american_scientific_journal-1069 46 9 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ NOUN american_scientific_journal-1069 46 10 π‘Ž π‘Ž NOUN american_scientific_journal-1069 46 11 π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ NOUN american_scientific_journal-1069 46 12 π‘œπ‘“ π‘œπ‘“ ADP american_scientific_journal-1069 46 13 πœ‹ πœ‹ NOUN american_scientific_journal-1069 46 14 ∞ ∞ PROPN american_scientific_journal-1069 46 15 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-1069 46 16 π‘₯(𝑑 π‘₯(𝑑 NOUN american_scientific_journal-1069 46 17 ) ) PUNCT american_scientific_journal-1069 46 18 𝑖𝑓 𝑖𝑓 VERB american_scientific_journal-1069 46 19 𝛼 𝛼 NOUN american_scientific_journal-1069 46 20 𝑖𝑠 𝑖𝑠 NOUN american_scientific_journal-1069 46 21 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ NOUN american_scientific_journal-1069 46 22 π‘Ž π‘Ž NOUN american_scientific_journal-1069 46 23 π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ NOUN american_scientific_journal-1069 46 24 π‘œπ‘“ π‘œπ‘“ ADP american_scientific_journal-1069 46 25 2πœ‹ 2πœ‹ NOUN american_scientific_journal-1069 46 26 π‘₯(βˆ’π‘‘ π‘₯(βˆ’π‘‘ ADV american_scientific_journal-1069 46 27 ) ) PUNCT american_scientific_journal-1069 46 28 𝑖𝑓 𝑖𝑓 VERB american_scientific_journal-1069 47 1 𝛼 𝛼 NOUN american_scientific_journal-1069 47 2 + + CCONJ american_scientific_journal-1069 47 3 πœ‹ πœ‹ NOUN american_scientific_journal-1069 47 4 𝑖𝑠 𝑖𝑠 NOUN american_scientific_journal-1069 47 5 π‘›π‘œπ‘‘ π‘›π‘œπ‘‘ NOUN american_scientific_journal-1069 48 1 π‘Ž π‘Ž NOUN american_scientific_journal-1069 48 2 π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ π‘šπ‘’π‘™π‘‘π‘–π‘π‘™π‘’ NOUN american_scientific_journal-1069 48 3 π‘œπ‘“ π‘œπ‘“ ADP american_scientific_journal-1069 48 4 2πœ‹ 2πœ‹ NOUN american_scientific_journal-1069 48 5 ( ( PUNCT american_scientific_journal-1069 48 6 2 2 NUM american_scientific_journal-1069 48 7 ) ) PUNCT american_scientific_journal-1069 48 8 in in ADP american_scientific_journal-1069 48 9 our our PRON american_scientific_journal-1069 48 10 case case NOUN american_scientific_journal-1069 48 11 we we PRON american_scientific_journal-1069 48 12 consider consider VERB american_scientific_journal-1069 48 13 a a DET american_scientific_journal-1069 48 14 function function NOUN american_scientific_journal-1069 48 15 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-1069 48 16 ) ) PUNCT american_scientific_journal-1069 48 17 and and CCONJ american_scientific_journal-1069 48 18 frequency frequency NOUN american_scientific_journal-1069 48 19 𝑒 𝑒 X american_scientific_journal-1069 48 20 = = NOUN american_scientific_journal-1069 48 21 πœ” πœ” PRON american_scientific_journal-1069 48 22 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-1069 48 23 ) ) PUNCT american_scientific_journal-1069 48 24 = = PUNCT american_scientific_journal-1069 49 1 𝐻𝑛𝑓(π‘₯)𝑒π‘₯𝑝 𝐻𝑛𝑓(π‘₯)𝑒π‘₯𝑝 INTJ american_scientific_journal-1069 49 2 ( ( PUNCT american_scientific_journal-1069 49 3 βˆ’ βˆ’ NUM american_scientific_journal-1069 49 4 π‘₯2 π‘₯2 NOUN american_scientific_journal-1069 49 5 2 2 NUM american_scientific_journal-1069 49 6 ) ) PUNCT american_scientific_journal-1069 49 7 ( ( PUNCT american_scientific_journal-1069 49 8 3 3 X american_scientific_journal-1069 49 9 ) ) PUNCT american_scientific_journal-1069 49 10 with with ADP american_scientific_journal-1069 49 11 equation equation NOUN american_scientific_journal-1069 49 12 ( ( PUNCT american_scientific_journal-1069 49 13 1 1 NUM american_scientific_journal-1069 49 14 ) ) PUNCT american_scientific_journal-1069 49 15 π‘₯ π‘₯ PRON american_scientific_journal-1069 49 16 ≑ ≑ PROPN american_scientific_journal-1069 49 17 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 49 18 , , PUNCT american_scientific_journal-1069 49 19 0.1 0.1 NUM american_scientific_journal-1069 49 20 ≀ ≀ NOUN american_scientific_journal-1069 49 21 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 49 22 ≀ ≀ NUM american_scientific_journal-1069 49 23 0.9 0.9 NUM american_scientific_journal-1069 49 24 ( ( PUNCT american_scientific_journal-1069 49 25 4 4 X american_scientific_journal-1069 49 26 ) ) PUNCT american_scientific_journal-1069 49 27 we we PRON american_scientific_journal-1069 49 28 know know VERB american_scientific_journal-1069 49 29 that that SCONJ american_scientific_journal-1069 49 30 the the DET american_scientific_journal-1069 49 31 frft frft NOUN american_scientific_journal-1069 49 32 of of ADP american_scientific_journal-1069 49 33 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-1069 49 34 ) ) PUNCT american_scientific_journal-1069 49 35 is be AUX american_scientific_journal-1069 49 36 given give VERB american_scientific_journal-1069 49 37 as as SCONJ american_scientific_journal-1069 49 38 follows follow VERB american_scientific_journal-1069 49 39 [ [ X american_scientific_journal-1069 49 40 5 5 NUM american_scientific_journal-1069 49 41 ] ] PUNCT american_scientific_journal-1069 49 42 ℱ𝛼[𝑓(π‘₯ ℱ𝛼[𝑓(π‘₯ NUM american_scientific_journal-1069 49 43 ) ) PUNCT american_scientific_journal-1069 49 44 ] ] PUNCT american_scientific_journal-1069 50 1 = = PUNCT american_scientific_journal-1069 50 2 𝑒 𝑒 PROPN american_scientific_journal-1069 50 3 𝑖𝑛𝑓𝛼𝐻𝑛𝑓(π‘₯)exp 𝑖𝑛𝑓𝛼𝐻𝑛𝑓(π‘₯)exp PROPN american_scientific_journal-1069 50 4 ( ( PUNCT american_scientific_journal-1069 50 5 βˆ’ βˆ’ PROPN american_scientific_journal-1069 50 6 π‘₯2 π‘₯2 NOUN american_scientific_journal-1069 50 7 2 2 NUM american_scientific_journal-1069 50 8 ) ) PUNCT american_scientific_journal-1069 50 9 ( ( PUNCT american_scientific_journal-1069 50 10 5 5 X american_scientific_journal-1069 50 11 ) ) PUNCT american_scientific_journal-1069 50 12 using use VERB american_scientific_journal-1069 50 13 eq eq NOUN american_scientific_journal-1069 50 14 ( ( PUNCT american_scientific_journal-1069 50 15 4 4 NUM american_scientific_journal-1069 50 16 ) ) PUNCT american_scientific_journal-1069 50 17 on on ADP american_scientific_journal-1069 50 18 eq eq X american_scientific_journal-1069 50 19 ( ( PUNCT american_scientific_journal-1069 50 20 3 3 NUM american_scientific_journal-1069 50 21 ) ) PUNCT american_scientific_journal-1069 50 22 , , PUNCT american_scientific_journal-1069 50 23 we we PRON american_scientific_journal-1069 50 24 have have VERB american_scientific_journal-1069 50 25 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-1069 50 26 ) ) PUNCT american_scientific_journal-1069 50 27 = = PUNCT american_scientific_journal-1069 51 1 𝐻𝑛𝑓(β„Žπ‘ž)𝑒π‘₯𝑝 𝐻𝑛𝑓(β„Žπ‘ž)𝑒π‘₯𝑝 NOUN american_scientific_journal-1069 52 1 [ [ X american_scientific_journal-1069 52 2 ( ( PUNCT american_scientific_journal-1069 52 3 βˆ’ βˆ’ PROPN american_scientific_journal-1069 52 4 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 52 5 √2 √2 PROPN american_scientific_journal-1069 52 6 ) ) PUNCT american_scientific_journal-1069 52 7 2 2 NUM american_scientific_journal-1069 52 8 ] ] PUNCT american_scientific_journal-1069 52 9 ( ( PUNCT american_scientific_journal-1069 52 10 6 6 X american_scientific_journal-1069 52 11 ) ) PUNCT american_scientific_journal-1069 52 12 using use VERB american_scientific_journal-1069 52 13 eq eq NOUN american_scientific_journal-1069 52 14 ( ( PUNCT american_scientific_journal-1069 52 15 5 5 NUM american_scientific_journal-1069 52 16 ) ) PUNCT american_scientific_journal-1069 52 17 ℱ𝛼 ℱ𝛼 PROPN american_scientific_journal-1069 52 18 [ [ X american_scientific_journal-1069 52 19 𝐻𝑛𝑓(β„Žπ‘ž)𝑒π‘₯𝑝 𝐻𝑛𝑓(β„Žπ‘ž)𝑒π‘₯𝑝 NOUN american_scientific_journal-1069 52 20 [ [ X american_scientific_journal-1069 52 21 ( ( PUNCT american_scientific_journal-1069 52 22 βˆ’ βˆ’ PROPN american_scientific_journal-1069 52 23 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 52 24 √2 √2 PROPN american_scientific_journal-1069 52 25 ) ) PUNCT american_scientific_journal-1069 52 26 2 2 NUM american_scientific_journal-1069 52 27 ] ] PUNCT american_scientific_journal-1069 52 28 ] ] PUNCT american_scientific_journal-1069 52 29 = = PUNCT american_scientific_journal-1069 52 30 𝑒𝑖𝑛𝑓𝛼𝐻𝑛𝑓(β„Žπ‘ž)exp 𝑒𝑖𝑛𝑓𝛼𝐻𝑛𝑓(β„Žπ‘ž)exp X american_scientific_journal-1069 53 1 [ [ X american_scientific_journal-1069 53 2 ( ( PUNCT american_scientific_journal-1069 53 3 βˆ’ βˆ’ PROPN american_scientific_journal-1069 53 4 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 53 5 √2 √2 PROPN american_scientific_journal-1069 53 6 ) ) PUNCT american_scientific_journal-1069 53 7 2 2 NUM american_scientific_journal-1069 53 8 ] ] PUNCT american_scientific_journal-1069 53 9 ( ( PUNCT american_scientific_journal-1069 53 10 7 7 X american_scientific_journal-1069 53 11 ) ) PUNCT american_scientific_journal-1069 53 12 where where SCONJ american_scientific_journal-1069 53 13 𝛼 𝛼 NOUN american_scientific_journal-1069 53 14 is be AUX american_scientific_journal-1069 53 15 the the DET american_scientific_journal-1069 53 16 angle angle NOUN american_scientific_journal-1069 53 17 of of ADP american_scientific_journal-1069 53 18 rotation rotation NOUN american_scientific_journal-1069 53 19 in in ADP american_scientific_journal-1069 53 20 the the DET american_scientific_journal-1069 53 21 complex complex ADJ american_scientific_journal-1069 53 22 plane plane NOUN american_scientific_journal-1069 53 23 ( ( PUNCT american_scientific_journal-1069 53 24 𝑑 𝑑 NOUN american_scientific_journal-1069 53 25 , , PUNCT american_scientific_journal-1069 53 26 πœ” πœ” NOUN american_scientific_journal-1069 53 27 ) ) PUNCT american_scientific_journal-1069 53 28 with with ADP american_scientific_journal-1069 53 29 changing change VERB american_scientific_journal-1069 53 30 surface surface NOUN american_scientific_journal-1069 53 31 and and CCONJ american_scientific_journal-1069 53 32 indeed indeed ADV american_scientific_journal-1069 53 33 the the DET american_scientific_journal-1069 53 34 shape shape NOUN american_scientific_journal-1069 53 35 of of ADP american_scientific_journal-1069 53 36 an an DET american_scientific_journal-1069 53 37 electron electron NOUN american_scientific_journal-1069 53 38 , , PUNCT american_scientific_journal-1069 53 39 𝛼 𝛼 PROPN american_scientific_journal-1069 53 40 is be AUX american_scientific_journal-1069 53 41 also also ADV american_scientific_journal-1069 53 42 changed change VERB american_scientific_journal-1069 53 43 in in ADP american_scientific_journal-1069 53 44 ( ( PUNCT american_scientific_journal-1069 53 45 𝑑 𝑑 NOUN american_scientific_journal-1069 53 46 , , PUNCT american_scientific_journal-1069 53 47 πœ” πœ” NOUN american_scientific_journal-1069 53 48 ) ) PUNCT american_scientific_journal-1069 53 49 coordinates coordinate NOUN american_scientific_journal-1069 53 50 , , PUNCT american_scientific_journal-1069 53 51 πœ” πœ” X american_scientific_journal-1069 53 52 = = VERB american_scientific_journal-1069 53 53 2πœ‹πœˆ. 2πœ‹πœˆ. NUM american_scientific_journal-1069 53 54 we we PRON american_scientific_journal-1069 53 55 are be AUX american_scientific_journal-1069 53 56 dealing deal VERB american_scientific_journal-1069 53 57 with with ADP american_scientific_journal-1069 53 58 fractional fractional ADJ american_scientific_journal-1069 53 59 quantum quantum ADJ american_scientific_journal-1069 53 60 oscillators oscillator NOUN american_scientific_journal-1069 53 61 and and CCONJ american_scientific_journal-1069 53 62 hence hence ADV american_scientific_journal-1069 53 63 with with ADP american_scientific_journal-1069 53 64 fractional fractional ADJ american_scientific_journal-1069 53 65 charge charge NOUN american_scientific_journal-1069 53 66 distributions distribution NOUN american_scientific_journal-1069 53 67 so so SCONJ american_scientific_journal-1069 53 68 that that SCONJ american_scientific_journal-1069 53 69 fourier fourier NOUN american_scientific_journal-1069 53 70 transform transform NOUN american_scientific_journal-1069 53 71 ( ( PUNCT american_scientific_journal-1069 53 72 ft ft NOUN american_scientific_journal-1069 53 73 ) ) PUNCT american_scientific_journal-1069 53 74 should should AUX american_scientific_journal-1069 53 75 not not PART american_scientific_journal-1069 53 76 enter enter VERB american_scientific_journal-1069 53 77 in in ADP american_scientific_journal-1069 53 78 our our PRON american_scientific_journal-1069 53 79 analysis analysis NOUN american_scientific_journal-1069 53 80 . . PUNCT american_scientific_journal-1069 54 1 for for ADP american_scientific_journal-1069 54 2 this this DET american_scientific_journal-1069 54 3 purpose purpose NOUN american_scientific_journal-1069 54 4 we we PRON american_scientific_journal-1069 54 5 set set VERB american_scientific_journal-1069 54 6 𝛼 𝛼 PRON american_scientific_journal-1069 54 7 β‰  β‰  NOUN american_scientific_journal-1069 54 8 πœ‹ πœ‹ ADP american_scientific_journal-1069 54 9 2 2 NUM american_scientific_journal-1069 54 10 . . PUNCT american_scientific_journal-1069 55 1 with with ADP american_scientific_journal-1069 55 2 𝛼 𝛼 NOUN american_scientific_journal-1069 55 3 = = SYM american_scientific_journal-1069 55 4 1 1 NUM american_scientific_journal-1069 55 5 , , PUNCT american_scientific_journal-1069 55 6 ( ( PUNCT american_scientific_journal-1069 55 7 𝛼 𝛼 X american_scientific_journal-1069 55 8 = = X american_scientific_journal-1069 55 9 π‘Žπœ‹ π‘Žπœ‹ PROPN american_scientific_journal-1069 55 10 2 2 NUM american_scientific_journal-1069 55 11 ) ) PUNCT american_scientific_journal-1069 55 12 we we PRON american_scientific_journal-1069 55 13 get get VERB american_scientific_journal-1069 55 14 the the DET american_scientific_journal-1069 55 15 frft frft NOUN american_scientific_journal-1069 55 16 to to PART american_scientific_journal-1069 55 17 change change VERB american_scientific_journal-1069 55 18 into into ADP american_scientific_journal-1069 55 19 ft ft PROPN american_scientific_journal-1069 55 20 . . PUNCT american_scientific_journal-1069 56 1 the the DET american_scientific_journal-1069 56 2 frft frft ADJ american_scientific_journal-1069 56 3 analysis analysis NOUN american_scientific_journal-1069 56 4 is be AUX american_scientific_journal-1069 56 5 a a DET american_scientific_journal-1069 56 6 time time NOUN american_scientific_journal-1069 56 7 frequency frequency NOUN american_scientific_journal-1069 56 8 distribution distribution NOUN american_scientific_journal-1069 56 9 and and CCONJ american_scientific_journal-1069 56 10 an an DET american_scientific_journal-1069 56 11 extension extension NOUN american_scientific_journal-1069 56 12 of of ADP american_scientific_journal-1069 56 13 the the DET american_scientific_journal-1069 56 14 classical classical ADJ american_scientific_journal-1069 56 15 ft ft NOUN american_scientific_journal-1069 56 16 . . PUNCT american_scientific_journal-1069 56 17 considering consider VERB american_scientific_journal-1069 56 18 the the DET american_scientific_journal-1069 56 19 schrodinger schrodinger NOUN american_scientific_journal-1069 56 20 ’s ’s PART american_scientific_journal-1069 56 21 equation equation NOUN american_scientific_journal-1069 56 22 for for ADP american_scientific_journal-1069 56 23 oscillatory oscillatory ADJ american_scientific_journal-1069 56 24 quanta quanta NOUN american_scientific_journal-1069 56 25 of of ADP american_scientific_journal-1069 56 26 fractional fractional ADJ american_scientific_journal-1069 56 27 charges charge NOUN american_scientific_journal-1069 56 28 on on ADP american_scientific_journal-1069 56 29 the the DET american_scientific_journal-1069 56 30 continuous continuous ADJ american_scientific_journal-1069 56 31 changing change VERB american_scientific_journal-1069 56 32 surface surface NOUN american_scientific_journal-1069 56 33 of of ADP american_scientific_journal-1069 56 34 the the DET american_scientific_journal-1069 56 35 electron electron NOUN american_scientific_journal-1069 57 1 [ [ X american_scientific_journal-1069 57 2 6 6 NUM american_scientific_journal-1069 57 3 , , PUNCT american_scientific_journal-1069 57 4 7 7 NUM american_scientific_journal-1069 57 5 , , PUNCT american_scientific_journal-1069 57 6 8 8 NUM american_scientific_journal-1069 57 7 ] ] PUNCT american_scientific_journal-1069 57 8 . . PUNCT american_scientific_journal-1069 58 1 βˆ’ βˆ’ PROPN american_scientific_journal-1069 58 2 β„Ž2 β„Ž2 ADJ american_scientific_journal-1069 58 3 2πœ‡ 2πœ‡ NUM american_scientific_journal-1069 58 4 𝑑2πœ“(π‘₯ 𝑑2πœ“(π‘₯ PROPN american_scientific_journal-1069 58 5 ) ) PUNCT american_scientific_journal-1069 58 6 𝑑π‘₯2 𝑑π‘₯2 NOUN american_scientific_journal-1069 58 7 + + CCONJ american_scientific_journal-1069 58 8 1 1 NUM american_scientific_journal-1069 58 9 2 2 NUM american_scientific_journal-1069 58 10 π‘˜π‘₯2πœ“(π‘₯ π‘˜π‘₯2πœ“(π‘₯ ADJ american_scientific_journal-1069 58 11 ) ) PUNCT american_scientific_journal-1069 58 12 = = SYM american_scientific_journal-1069 58 13 πΈπœ“(π‘₯ πΈπœ“(π‘₯ NOUN american_scientific_journal-1069 58 14 ) ) PUNCT american_scientific_journal-1069 58 15 ( ( PUNCT american_scientific_journal-1069 58 16 8) 8) NUM american_scientific_journal-1069 58 17 american american ADJ american_scientific_journal-1069 58 18 scientific scientific ADJ american_scientific_journal-1069 58 19 research research NOUN american_scientific_journal-1069 58 20 journal journal NOUN american_scientific_journal-1069 58 21 for for ADP american_scientific_journal-1069 58 22 engineering engineering NOUN american_scientific_journal-1069 58 23 , , PUNCT american_scientific_journal-1069 58 24 technology technology NOUN american_scientific_journal-1069 58 25 , , PUNCT american_scientific_journal-1069 58 26 and and CCONJ american_scientific_journal-1069 58 27 sciences science NOUN american_scientific_journal-1069 58 28 ( ( PUNCT american_scientific_journal-1069 58 29 asrjets asrjet NOUN american_scientific_journal-1069 58 30 ) ) PUNCT american_scientific_journal-1069 58 31 ( ( PUNCT american_scientific_journal-1069 58 32 2015 2015 NUM american_scientific_journal-1069 58 33 ) ) PUNCT american_scientific_journal-1069 58 34 volume volume NOUN american_scientific_journal-1069 58 35 14 14 NUM american_scientific_journal-1069 58 36 , , PUNCT american_scientific_journal-1069 58 37 no no DET american_scientific_journal-1069 58 38 2 2 NUM american_scientific_journal-1069 58 39 , , PUNCT american_scientific_journal-1069 58 40 pp pp ADV american_scientific_journal-1069 58 41 265 265 NUM american_scientific_journal-1069 58 42 - - SYM american_scientific_journal-1069 58 43 272 272 NUM american_scientific_journal-1069 58 44 268 268 NUM american_scientific_journal-1069 58 45 where where SCONJ american_scientific_journal-1069 58 46 πœ” πœ” ADP american_scientific_journal-1069 58 47 = = SYM american_scientific_journal-1069 58 48 ( ( PUNCT american_scientific_journal-1069 58 49 π‘˜ π‘˜ X american_scientific_journal-1069 58 50 πœ‡ πœ‡ X american_scientific_journal-1069 58 51 ) ) PUNCT american_scientific_journal-1069 58 52 1 1 NUM american_scientific_journal-1069 58 53 2 2 NUM american_scientific_journal-1069 58 54 , , PUNCT american_scientific_journal-1069 58 55 π‘˜ π‘˜ X american_scientific_journal-1069 58 56 the the DET american_scientific_journal-1069 58 57 restoring restore VERB american_scientific_journal-1069 58 58 constant constant NOUN american_scientific_journal-1069 58 59 πœ‡ πœ‡ ADP american_scientific_journal-1069 58 60 the the DET american_scientific_journal-1069 58 61 reduced reduce VERB american_scientific_journal-1069 58 62 mass mass NOUN american_scientific_journal-1069 58 63 of of ADP american_scientific_journal-1069 58 64 an an DET american_scientific_journal-1069 58 65 electron electron NOUN american_scientific_journal-1069 58 66 . . PUNCT american_scientific_journal-1069 59 1 using use VERB american_scientific_journal-1069 59 2 the the DET american_scientific_journal-1069 59 3 dimensionless dimensionless NOUN american_scientific_journal-1069 59 4 variable variable NOUN american_scientific_journal-1069 59 5 πœ‰ πœ‰ X american_scientific_journal-1069 59 6 = = SYM american_scientific_journal-1069 59 7 𝛼π‘₯ 𝛼π‘₯ PROPN american_scientific_journal-1069 59 8 , , PUNCT american_scientific_journal-1069 59 9 𝛼 𝛼 X american_scientific_journal-1069 59 10 = = PUNCT american_scientific_journal-1069 59 11 ( ( PUNCT american_scientific_journal-1069 59 12 πœ‡π‘˜ πœ‡π‘˜ NOUN american_scientific_journal-1069 59 13 ℏ2 ℏ2 NOUN american_scientific_journal-1069 59 14 ) ) PUNCT american_scientific_journal-1069 59 15 1 1 NUM american_scientific_journal-1069 59 16 4 4 NUM american_scientific_journal-1069 59 17 = = SYM american_scientific_journal-1069 59 18 ( ( PUNCT american_scientific_journal-1069 59 19 πœ‡πœ” πœ‡πœ” ADP american_scientific_journal-1069 59 20 ℏ ℏ PROPN american_scientific_journal-1069 59 21 ) ) PUNCT american_scientific_journal-1069 59 22 1 1 NUM american_scientific_journal-1069 59 23 2 2 NUM american_scientific_journal-1069 59 24 ( ( PUNCT american_scientific_journal-1069 59 25 9 9 NUM american_scientific_journal-1069 59 26 ) ) PUNCT american_scientific_journal-1069 59 27 we we PRON american_scientific_journal-1069 59 28 shall shall AUX american_scientific_journal-1069 59 29 replace replace VERB american_scientific_journal-1069 59 30 dimensionless dimensionless NOUN american_scientific_journal-1069 59 31 , , PUNCT american_scientific_journal-1069 59 32 Ξ± Ξ± NOUN american_scientific_journal-1069 59 33 with with ADP american_scientific_journal-1069 59 34 rotation rotation NOUN american_scientific_journal-1069 59 35 vector vector NOUN american_scientific_journal-1069 59 36 in in ADP american_scientific_journal-1069 59 37 coordinates coordinate NOUN american_scientific_journal-1069 59 38 , , PUNCT american_scientific_journal-1069 59 39 ( ( PUNCT american_scientific_journal-1069 59 40 𝑑 𝑑 NOUN american_scientific_journal-1069 59 41 , , PUNCT american_scientific_journal-1069 59 42 πœ” πœ” NOUN american_scientific_journal-1069 59 43 ) ) PUNCT american_scientific_journal-1069 59 44 . . PUNCT american_scientific_journal-1069 60 1 substitution substitution NOUN american_scientific_journal-1069 60 2 of of ADP american_scientific_journal-1069 60 3 eq eq NOUN american_scientific_journal-1069 60 4 ( ( PUNCT american_scientific_journal-1069 60 5 9 9 NUM american_scientific_journal-1069 60 6 ) ) PUNCT american_scientific_journal-1069 60 7 in in ADP american_scientific_journal-1069 60 8 eq eq NOUN american_scientific_journal-1069 60 9 ( ( PUNCT american_scientific_journal-1069 60 10 8) 8) NUM american_scientific_journal-1069 60 11 , , PUNCT american_scientific_journal-1069 60 12 we we PRON american_scientific_journal-1069 60 13 have have VERB american_scientific_journal-1069 60 14 𝑑2πœ“(πœ‰ 𝑑2πœ“(πœ‰ NOUN american_scientific_journal-1069 60 15 ) ) PUNCT american_scientific_journal-1069 60 16 π‘‘πœ‰2 π‘‘πœ‰2 NOUN american_scientific_journal-1069 60 17 + + CCONJ american_scientific_journal-1069 61 1 ( ( PUNCT american_scientific_journal-1069 61 2 πœ† πœ† PRON american_scientific_journal-1069 61 3 βˆ’ βˆ’ PROPN american_scientific_journal-1069 61 4 πœ‰2)πœ“(πœ‰ πœ‰2)πœ“(πœ‰ NOUN american_scientific_journal-1069 61 5 ) ) PUNCT american_scientific_journal-1069 61 6 = = SYM american_scientific_journal-1069 61 7 0 0 NUM american_scientific_journal-1069 61 8 ( ( PUNCT american_scientific_journal-1069 61 9 10 10 NUM american_scientific_journal-1069 61 10 ) ) PUNCT american_scientific_journal-1069 61 11 where where SCONJ american_scientific_journal-1069 61 12 πœ† πœ† NOUN american_scientific_journal-1069 61 13 is be AUX american_scientific_journal-1069 61 14 also also ADV american_scientific_journal-1069 61 15 dimensionless dimensionless NOUN american_scientific_journal-1069 61 16 , , PUNCT american_scientific_journal-1069 61 17 but but CCONJ american_scientific_journal-1069 61 18 considered consider VERB american_scientific_journal-1069 61 19 as as ADP american_scientific_journal-1069 61 20 binding bind VERB american_scientific_journal-1069 61 21 energy energy NOUN american_scientific_journal-1069 61 22 of of ADP american_scientific_journal-1069 61 23 the the DET american_scientific_journal-1069 61 24 quantum quantum ADJ american_scientific_journal-1069 61 25 system system NOUN american_scientific_journal-1069 61 26 . . PUNCT american_scientific_journal-1069 62 1 for for ADP american_scientific_journal-1069 62 2 large large ADJ american_scientific_journal-1069 62 3 |πœ‰| |πœ‰| NOUN american_scientific_journal-1069 62 4 , , PUNCT american_scientific_journal-1069 62 5 it it PRON american_scientific_journal-1069 62 6 is be AUX american_scientific_journal-1069 62 7 readily readily ADV american_scientific_journal-1069 62 8 verified verify VERB american_scientific_journal-1069 62 9 that that SCONJ american_scientific_journal-1069 62 10 the the DET american_scientific_journal-1069 62 11 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 62 12 πœ“(πœ‰ πœ“(πœ‰ PART american_scientific_journal-1069 62 13 ) ) PUNCT american_scientific_journal-1069 62 14 = = PUNCT american_scientific_journal-1069 62 15 πœ‰π‘ƒπ‘’Β±πœ‰ πœ‰π‘ƒπ‘’Β±πœ‰ NOUN american_scientific_journal-1069 62 16 2 2 NUM american_scientific_journal-1069 62 17 exists exist VERB american_scientific_journal-1069 62 18 . . PUNCT american_scientific_journal-1069 63 1 the the DET american_scientific_journal-1069 63 2 asymptotic asymptotic ADJ american_scientific_journal-1069 63 3 analysis analysis NOUN american_scientific_journal-1069 63 4 provides provide VERB american_scientific_journal-1069 63 5 us we PRON american_scientific_journal-1069 63 6 an an DET american_scientific_journal-1069 63 7 indication indication NOUN american_scientific_journal-1069 63 8 for for ADP american_scientific_journal-1069 63 9 valid valid ADJ american_scientific_journal-1069 63 10 solutions solution NOUN american_scientific_journal-1069 63 11 to to ADP american_scientific_journal-1069 63 12 eq eq NOUN american_scientific_journal-1069 63 13 ( ( PUNCT american_scientific_journal-1069 63 14 10 10 NUM american_scientific_journal-1069 63 15 ) ) PUNCT american_scientific_journal-1069 63 16 and and CCONJ american_scientific_journal-1069 63 17 for for ADP american_scientific_journal-1069 63 18 all all PRON american_scientific_journal-1069 63 19 πœ‰ πœ‰ AUX american_scientific_journal-1069 63 20 having have VERB american_scientific_journal-1069 63 21 the the DET american_scientific_journal-1069 63 22 form form NOUN american_scientific_journal-1069 63 23 πœ‰π‘ƒ πœ‰π‘ƒ ADV american_scientific_journal-1069 63 24 where where SCONJ american_scientific_journal-1069 63 25 p p NOUN american_scientific_journal-1069 63 26 is be AUX american_scientific_journal-1069 63 27 the the DET american_scientific_journal-1069 63 28 polynomial polynomial ADJ american_scientific_journal-1069 63 29 . . PUNCT american_scientific_journal-1069 64 1 thus thus ADV american_scientific_journal-1069 64 2 πœ“(πœ‰ πœ“(πœ‰ X american_scientific_journal-1069 64 3 ) ) PUNCT american_scientific_journal-1069 64 4 = = PUNCT american_scientific_journal-1069 64 5 π‘’βˆ’ π‘’βˆ’ NOUN american_scientific_journal-1069 64 6 πœ‰2 πœ‰2 ADJ american_scientific_journal-1069 64 7 2 2 NUM american_scientific_journal-1069 64 8 𝐻(πœ‰ 𝐻(πœ‰ NOUN american_scientific_journal-1069 64 9 ) ) PUNCT american_scientific_journal-1069 64 10 ( ( PUNCT american_scientific_journal-1069 64 11 11 11 NUM american_scientific_journal-1069 64 12 ) ) PUNCT american_scientific_journal-1069 64 13 using use VERB american_scientific_journal-1069 64 14 eq eq NOUN american_scientific_journal-1069 64 15 ( ( PUNCT american_scientific_journal-1069 64 16 11 11 NUM american_scientific_journal-1069 64 17 ) ) PUNCT american_scientific_journal-1069 64 18 in in ADP american_scientific_journal-1069 64 19 eq eq NOUN american_scientific_journal-1069 64 20 ( ( PUNCT american_scientific_journal-1069 64 21 10 10 NUM american_scientific_journal-1069 64 22 ) ) PUNCT american_scientific_journal-1069 64 23 , , PUNCT american_scientific_journal-1069 64 24 we we PRON american_scientific_journal-1069 64 25 have have VERB american_scientific_journal-1069 64 26 hermite hermite ADJ american_scientific_journal-1069 64 27 equation equation NOUN american_scientific_journal-1069 64 28 𝑑2𝐻 𝑑2𝐻 NOUN american_scientific_journal-1069 64 29 π‘‘πœ‰2 π‘‘πœ‰2 NOUN american_scientific_journal-1069 64 30 βˆ’ βˆ’ PROPN american_scientific_journal-1069 64 31 2πœ‰ 2πœ‰ NOUN american_scientific_journal-1069 64 32 𝑑𝐻 𝑑𝐻 ADJ american_scientific_journal-1069 64 33 π‘‘πœ‰ π‘‘πœ‰ ADP american_scientific_journal-1069 64 34 + + X american_scientific_journal-1069 64 35 ( ( PUNCT american_scientific_journal-1069 64 36 πœ† πœ† DET american_scientific_journal-1069 64 37 βˆ’ βˆ’ PROPN american_scientific_journal-1069 64 38 1)𝐻 1)𝐻 NOUN american_scientific_journal-1069 65 1 = = SYM american_scientific_journal-1069 65 2 0 0 PUNCT american_scientific_journal-1069 65 3 ( ( PUNCT american_scientific_journal-1069 65 4 12 12 NUM american_scientific_journal-1069 65 5 ) ) PUNCT american_scientific_journal-1069 65 6 we we PRON american_scientific_journal-1069 65 7 assume assume VERB american_scientific_journal-1069 65 8 a a DET american_scientific_journal-1069 65 9 solution solution NOUN american_scientific_journal-1069 65 10 to to ADP american_scientific_journal-1069 65 11 eq eq NOUN american_scientific_journal-1069 65 12 ( ( PUNCT american_scientific_journal-1069 65 13 12 12 NUM american_scientific_journal-1069 65 14 ) ) PUNCT american_scientific_journal-1069 65 15 in in ADP american_scientific_journal-1069 65 16 the the DET american_scientific_journal-1069 65 17 form form NOUN american_scientific_journal-1069 65 18 of of ADP american_scientific_journal-1069 65 19 a a DET american_scientific_journal-1069 65 20 finite finite ADJ american_scientific_journal-1069 65 21 polynomial polynomial NOUN american_scientific_journal-1069 65 22 𝐻(πœ‰ 𝐻(πœ‰ NOUN american_scientific_journal-1069 65 23 ) ) PUNCT american_scientific_journal-1069 65 24 = = PUNCT american_scientific_journal-1069 66 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-1069 66 2 π‘Žπ‘  π‘Žπ‘  ADP american_scientific_journal-1069 66 3 𝑁 𝑁 PROPN american_scientific_journal-1069 66 4 𝑠=0 𝑠=0 PUNCT american_scientific_journal-1069 66 5 πœ‰2𝑠 πœ‰2𝑠 NUM american_scientific_journal-1069 66 6 ; ; PUNCT american_scientific_journal-1069 66 7 𝑠 𝑠 NUM american_scientific_journal-1069 66 8 β‰₯ β‰₯ X american_scientific_journal-1069 66 9 0 0 NUM american_scientific_journal-1069 66 10 ( ( PUNCT american_scientific_journal-1069 66 11 13 13 NUM american_scientific_journal-1069 66 12 ) ) PUNCT american_scientific_journal-1069 66 13 using use VERB american_scientific_journal-1069 66 14 eq eq NOUN american_scientific_journal-1069 66 15 ( ( PUNCT american_scientific_journal-1069 66 16 13 13 NUM american_scientific_journal-1069 66 17 ) ) PUNCT american_scientific_journal-1069 66 18 in in ADP american_scientific_journal-1069 66 19 eq eq NOUN american_scientific_journal-1069 66 20 ( ( PUNCT american_scientific_journal-1069 66 21 12 12 NUM american_scientific_journal-1069 66 22 ) ) PUNCT american_scientific_journal-1069 66 23 , , PUNCT american_scientific_journal-1069 66 24 we we PRON american_scientific_journal-1069 66 25 obtain obtain VERB american_scientific_journal-1069 66 26 a a DET american_scientific_journal-1069 66 27 recursion recursion NOUN american_scientific_journal-1069 66 28 formula formula NOUN american_scientific_journal-1069 66 29 to to ADP american_scientific_journal-1069 66 30 reproducing reproduce VERB american_scientific_journal-1069 66 31 the the DET american_scientific_journal-1069 66 32 shape shape NOUN american_scientific_journal-1069 66 33 of of ADP american_scientific_journal-1069 66 34 the the DET american_scientific_journal-1069 66 35 energy energy NOUN american_scientific_journal-1069 66 36 profiles profile NOUN american_scientific_journal-1069 66 37 oscillating oscillate VERB american_scientific_journal-1069 66 38 within within ADP american_scientific_journal-1069 66 39 the the DET american_scientific_journal-1069 66 40 momentum momentum NOUN american_scientific_journal-1069 66 41 quantized quantize VERB american_scientific_journal-1069 66 42 space space NOUN american_scientific_journal-1069 66 43 𝜎 𝜎 NOUN american_scientific_journal-1069 67 1 𝑠+2= 𝑠+2= ADJ american_scientific_journal-1069 67 2 2𝑠+1βˆ’πœ† 2𝑠+1βˆ’πœ† NUM american_scientific_journal-1069 67 3 ( ( PUNCT american_scientific_journal-1069 67 4 𝑠+2)(𝑠+1 𝑠+2)(𝑠+1 NUM american_scientific_journal-1069 67 5 ) ) PUNCT american_scientific_journal-1069 67 6 π‘Žπ‘  π‘Žπ‘  NOUN american_scientific_journal-1069 67 7 , , PUNCT american_scientific_journal-1069 67 8 𝑠β‰₯0 𝑠β‰₯0 NOUN american_scientific_journal-1069 67 9 ( ( PUNCT american_scientific_journal-1069 67 10 14 14 NUM american_scientific_journal-1069 67 11 ) ) PUNCT american_scientific_journal-1069 67 12 for for ADP american_scientific_journal-1069 67 13 an an DET american_scientific_journal-1069 67 14 upper upper ADJ american_scientific_journal-1069 67 15 cut cut NOUN american_scientific_journal-1069 67 16 off off ADP american_scientific_journal-1069 67 17 and and CCONJ american_scientific_journal-1069 67 18 the the DET american_scientific_journal-1069 67 19 coefficients coefficient NOUN american_scientific_journal-1069 67 20 so so SCONJ american_scientific_journal-1069 67 21 that that SCONJ american_scientific_journal-1069 67 22 the the DET american_scientific_journal-1069 67 23 polynomial polynomial ADJ american_scientific_journal-1069 67 24 equation equation NOUN american_scientific_journal-1069 67 25 ( ( PUNCT american_scientific_journal-1069 67 26 14 14 NUM american_scientific_journal-1069 67 27 ) ) PUNCT american_scientific_journal-1069 67 28 is be AUX american_scientific_journal-1069 67 29 not not PART american_scientific_journal-1069 67 30 an an DET american_scientific_journal-1069 67 31 infinite infinite ADJ american_scientific_journal-1069 67 32 series series NOUN american_scientific_journal-1069 67 33 , , PUNCT american_scientific_journal-1069 67 34 we we PRON american_scientific_journal-1069 67 35 have have VERB american_scientific_journal-1069 67 36 to to PART american_scientific_journal-1069 67 37 insert insert VERB american_scientific_journal-1069 67 38 a a DET american_scientific_journal-1069 67 39 condition condition NOUN american_scientific_journal-1069 67 40 πœ† πœ† ADP american_scientific_journal-1069 67 41 = = PUNCT american_scientific_journal-1069 67 42 2𝑛𝑓 2𝑛𝑓 NOUN american_scientific_journal-1069 68 1 + + CCONJ american_scientific_journal-1069 68 2 1 1 NUM american_scientific_journal-1069 68 3 , , PUNCT american_scientific_journal-1069 68 4 0.1 0.1 NUM american_scientific_journal-1069 68 5 ≀ ≀ NUM american_scientific_journal-1069 68 6 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 68 7 ≀ ≀ NUM american_scientific_journal-1069 68 8 0.9 0.9 NUM american_scientific_journal-1069 68 9 ⟹ ⟹ NUM american_scientific_journal-1069 68 10 1.2 1.2 NUM american_scientific_journal-1069 68 11 ≀ ≀ NOUN american_scientific_journal-1069 68 12 πœ† πœ† DET american_scientific_journal-1069 68 13 ≀ ≀ NUM american_scientific_journal-1069 68 14 2.8 2.8 NUM american_scientific_journal-1069 68 15 ( ( PUNCT american_scientific_journal-1069 68 16 15 15 NUM american_scientific_journal-1069 68 17 ) ) PUNCT american_scientific_journal-1069 68 18 with with ADP american_scientific_journal-1069 68 19 dimensionless dimensionless NOUN american_scientific_journal-1069 68 20 eigen eigen PROPN american_scientific_journal-1069 68 21 value value NOUN american_scientific_journal-1069 68 22 , , PUNCT american_scientific_journal-1069 68 23 πœ† πœ† NOUN american_scientific_journal-1069 68 24 = = PUNCT american_scientific_journal-1069 68 25 2𝐸 2𝐸 NOUN american_scientific_journal-1069 68 26 β„πœ” β„πœ” ADJ american_scientific_journal-1069 68 27 and and CCONJ american_scientific_journal-1069 68 28 measuring measure VERB american_scientific_journal-1069 68 29 the the DET american_scientific_journal-1069 68 30 variation variation NOUN american_scientific_journal-1069 68 31 of of ADP american_scientific_journal-1069 68 32 electron electron NOUN american_scientific_journal-1069 68 33 radius radius NOUN american_scientific_journal-1069 68 34 , , PUNCT american_scientific_journal-1069 68 35 .i.e .i.e PUNCT american_scientific_journal-1069 68 36 . . PUNCT american_scientific_journal-1069 68 37 , , PUNCT american_scientific_journal-1069 68 38 depth depth NOUN american_scientific_journal-1069 68 39 of of ADP american_scientific_journal-1069 68 40 the the DET american_scientific_journal-1069 68 41 quantum quantum ADJ american_scientific_journal-1069 68 42 well well NOUN american_scientific_journal-1069 68 43 in in ADP american_scientific_journal-1069 68 44 units unit NOUN american_scientific_journal-1069 68 45 of of ADP american_scientific_journal-1069 68 46 ( ( PUNCT american_scientific_journal-1069 68 47 ℏ ℏ PROPN american_scientific_journal-1069 68 48 πœ‡πœ” πœ‡πœ” PROPN american_scientific_journal-1069 68 49 ) ) PUNCT american_scientific_journal-1069 68 50 1 1 NUM american_scientific_journal-1069 68 51 2 2 NUM american_scientific_journal-1069 68 52 where where SCONJ american_scientific_journal-1069 68 53 πœ”2 πœ”2 ADV american_scientific_journal-1069 68 54 = = SYM american_scientific_journal-1069 68 55 π‘˜ π‘˜ PRON american_scientific_journal-1069 68 56 πœ” πœ” NOUN american_scientific_journal-1069 68 57 , , PUNCT american_scientific_journal-1069 68 58 we we PRON american_scientific_journal-1069 68 59 can can AUX american_scientific_journal-1069 68 60 ascertain ascertain VERB american_scientific_journal-1069 68 61 that that SCONJ american_scientific_journal-1069 68 62 𝐸𝑛𝑓 𝐸𝑛𝑓 PROPN american_scientific_journal-1069 68 63 = = PUNCT american_scientific_journal-1069 68 64 ( ( PUNCT american_scientific_journal-1069 68 65 𝑛𝑓 𝑛𝑓 X american_scientific_journal-1069 68 66 + + CCONJ american_scientific_journal-1069 68 67 1 1 NUM american_scientific_journal-1069 68 68 2 2 NUM american_scientific_journal-1069 68 69 ) ) PUNCT american_scientific_journal-1069 68 70 β„πœ” β„πœ” NOUN american_scientific_journal-1069 68 71 = = X american_scientific_journal-1069 68 72 ( ( PUNCT american_scientific_journal-1069 68 73 𝑛𝑓 𝑛𝑓 X american_scientific_journal-1069 68 74 + + CCONJ american_scientific_journal-1069 68 75 1 1 NUM american_scientific_journal-1069 68 76 2 2 NUM american_scientific_journal-1069 68 77 ) ) PUNCT american_scientific_journal-1069 68 78 β„πœˆ β„πœˆ NOUN american_scientific_journal-1069 68 79 ; ; PUNCT american_scientific_journal-1069 68 80 ℏ ℏ NOUN american_scientific_journal-1069 68 81 = = SYM american_scientific_journal-1069 68 82 β„Ž β„Ž PART american_scientific_journal-1069 68 83 2πœ‹ 2πœ‹ NOUN american_scientific_journal-1069 68 84 , , PUNCT american_scientific_journal-1069 68 85 𝑛 𝑛 PROPN american_scientific_journal-1069 68 86 = = SYM american_scientific_journal-1069 68 87 0 0 NUM american_scientific_journal-1069 68 88 ( ( PUNCT american_scientific_journal-1069 68 89 16 16 NUM american_scientific_journal-1069 68 90 ) ) PUNCT american_scientific_journal-1069 68 91 american american ADJ american_scientific_journal-1069 68 92 scientific scientific ADJ american_scientific_journal-1069 68 93 research research NOUN american_scientific_journal-1069 68 94 journal journal NOUN american_scientific_journal-1069 68 95 for for ADP american_scientific_journal-1069 68 96 engineering engineering NOUN american_scientific_journal-1069 68 97 , , PUNCT american_scientific_journal-1069 68 98 technology technology NOUN american_scientific_journal-1069 68 99 , , PUNCT american_scientific_journal-1069 68 100 and and CCONJ american_scientific_journal-1069 68 101 sciences science NOUN american_scientific_journal-1069 68 102 ( ( PUNCT american_scientific_journal-1069 68 103 asrjets asrjet NOUN american_scientific_journal-1069 68 104 ) ) PUNCT american_scientific_journal-1069 68 105 ( ( PUNCT american_scientific_journal-1069 68 106 2015 2015 NUM american_scientific_journal-1069 68 107 ) ) PUNCT american_scientific_journal-1069 68 108 volume volume NOUN american_scientific_journal-1069 68 109 14 14 NUM american_scientific_journal-1069 68 110 , , PUNCT american_scientific_journal-1069 68 111 no no DET american_scientific_journal-1069 68 112 2 2 NUM american_scientific_journal-1069 68 113 , , PUNCT american_scientific_journal-1069 68 114 pp pp ADV american_scientific_journal-1069 68 115 265 265 NUM american_scientific_journal-1069 68 116 - - SYM american_scientific_journal-1069 68 117 272 272 NUM american_scientific_journal-1069 68 118 269 269 NUM american_scientific_journal-1069 68 119 using use VERB american_scientific_journal-1069 68 120 eq eq NOUN american_scientific_journal-1069 68 121 ( ( PUNCT american_scientific_journal-1069 68 122 16 16 NUM american_scientific_journal-1069 68 123 ) ) PUNCT american_scientific_journal-1069 68 124 with with ADP american_scientific_journal-1069 68 125 the the DET american_scientific_journal-1069 68 126 collection collection NOUN american_scientific_journal-1069 68 127 of of ADP american_scientific_journal-1069 68 128 even even ADV american_scientific_journal-1069 68 129 and and CCONJ american_scientific_journal-1069 68 130 odd odd ADJ american_scientific_journal-1069 68 131 cases case NOUN american_scientific_journal-1069 68 132 , , PUNCT american_scientific_journal-1069 68 133 the the DET american_scientific_journal-1069 68 134 physically physically ADV american_scientific_journal-1069 68 135 acceptable acceptable ADJ american_scientific_journal-1069 68 136 solution solution NOUN american_scientific_journal-1069 68 137 of of ADP american_scientific_journal-1069 68 138 eq eq NOUN american_scientific_journal-1069 68 139 ( ( PUNCT american_scientific_journal-1069 68 140 10 10 NUM american_scientific_journal-1069 68 141 ) ) PUNCT american_scientific_journal-1069 68 142 corresponding correspond VERB american_scientific_journal-1069 68 143 to to ADP american_scientific_journal-1069 68 144 eigenvalues eigenvalues PROPN american_scientific_journal-1069 68 145 ( ( PUNCT american_scientific_journal-1069 68 146 eq(16 eq(16 PROPN american_scientific_journal-1069 68 147 ) ) PUNCT american_scientific_journal-1069 68 148 ) ) PUNCT american_scientific_journal-1069 68 149 are be AUX american_scientific_journal-1069 68 150 given give VERB american_scientific_journal-1069 68 151 by by ADP american_scientific_journal-1069 68 152 πœ“π‘›π‘“(πœ‰ πœ“π‘›π‘“(πœ‰ PROPN american_scientific_journal-1069 68 153 ) ) PUNCT american_scientific_journal-1069 68 154 = = PUNCT american_scientific_journal-1069 68 155 𝑒 𝑒 PROPN american_scientific_journal-1069 68 156 βˆ’ βˆ’ PROPN american_scientific_journal-1069 68 157 πœ‰2 πœ‰2 PROPN american_scientific_journal-1069 68 158 2 2 NUM american_scientific_journal-1069 68 159 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 68 160 ) ) PUNCT american_scientific_journal-1069 68 161 ; ; PUNCT american_scientific_journal-1069 68 162 0.1 0.1 NUM american_scientific_journal-1069 68 163 ≀ ≀ NOUN american_scientific_journal-1069 68 164 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 68 165 ≀ ≀ NUM american_scientific_journal-1069 68 166 0.9 0.9 NUM american_scientific_journal-1069 68 167 ( ( PUNCT american_scientific_journal-1069 68 168 17 17 NUM american_scientific_journal-1069 68 169 ) ) PUNCT american_scientific_journal-1069 68 170 where where SCONJ american_scientific_journal-1069 68 171 the the DET american_scientific_journal-1069 68 172 function function NOUN american_scientific_journal-1069 68 173 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 68 174 ) ) PUNCT american_scientific_journal-1069 68 175 are be AUX american_scientific_journal-1069 68 176 polynomials polynomial NOUN american_scientific_journal-1069 68 177 of of ADP american_scientific_journal-1069 68 178 order order NOUN american_scientific_journal-1069 68 179 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 68 180 . . PUNCT american_scientific_journal-1069 69 1 moreover moreover ADV american_scientific_journal-1069 69 2 , , PUNCT american_scientific_journal-1069 69 3 the the DET american_scientific_journal-1069 69 4 polynomials polynomial NOUN american_scientific_journal-1069 69 5 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ NOUN american_scientific_journal-1069 69 6 ) ) PUNCT american_scientific_journal-1069 69 7 satisfy satisfy VERB american_scientific_journal-1069 69 8 the the DET american_scientific_journal-1069 69 9 hermite hermite ADJ american_scientific_journal-1069 69 10 equation equation NOUN american_scientific_journal-1069 69 11 ( ( PUNCT american_scientific_journal-1069 69 12 eq(12 eq(12 PROPN american_scientific_journal-1069 69 13 ) ) PUNCT american_scientific_journal-1069 69 14 ) ) PUNCT american_scientific_journal-1069 69 15 with with ADP american_scientific_journal-1069 69 16 πœ† πœ† PRON american_scientific_journal-1069 69 17 = = PUNCT american_scientific_journal-1069 69 18 2𝑛𝑓 2𝑛𝑓 NOUN american_scientific_journal-1069 69 19 + + CCONJ american_scientific_journal-1069 69 20 1 1 NUM american_scientific_journal-1069 69 21 , , PUNCT american_scientific_journal-1069 69 22 [ [ X american_scientific_journal-1069 69 23 eq(15 eq(15 NOUN american_scientific_journal-1069 69 24 ) ) PUNCT american_scientific_journal-1069 69 25 ] ] PUNCT american_scientific_journal-1069 69 26 . . PUNCT american_scientific_journal-1069 70 1 𝑑2𝐻𝑛𝑓 𝑑2𝐻𝑛𝑓 PROPN american_scientific_journal-1069 70 2 π‘‘πœ‰2 π‘‘πœ‰2 VERB american_scientific_journal-1069 70 3 βˆ’ βˆ’ NOUN american_scientific_journal-1069 70 4 2πœ‰ 2πœ‰ NUM american_scientific_journal-1069 70 5 𝑑𝐻𝑛𝑓 𝑑𝐻𝑛𝑓 PROPN american_scientific_journal-1069 70 6 π‘‘πœ‰ π‘‘πœ‰ ADP american_scientific_journal-1069 70 7 + + ADJ american_scientific_journal-1069 70 8 2𝑛𝐻𝑛𝑓 2𝑛𝐻𝑛𝑓 NUM american_scientific_journal-1069 70 9 = = SYM american_scientific_journal-1069 70 10 0 0 NUM american_scientific_journal-1069 70 11 , , PUNCT american_scientific_journal-1069 70 12 0.1 0.1 NUM american_scientific_journal-1069 70 13 ≀ ≀ NOUN american_scientific_journal-1069 70 14 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 70 15 ≀ ≀ NUM american_scientific_journal-1069 70 16 0.9 0.9 NUM american_scientific_journal-1069 70 17 ( ( PUNCT american_scientific_journal-1069 70 18 18 18 NUM american_scientific_journal-1069 70 19 ) ) PUNCT american_scientific_journal-1069 70 20 where where SCONJ american_scientific_journal-1069 70 21 the the DET american_scientific_journal-1069 70 22 function function NOUN american_scientific_journal-1069 70 23 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 70 24 ) ) PUNCT american_scientific_journal-1069 70 25 are be AUX american_scientific_journal-1069 70 26 hermite hermite ADJ american_scientific_journal-1069 70 27 polynomial polynomial ADJ american_scientific_journal-1069 70 28 . . PUNCT american_scientific_journal-1069 71 1 their their PRON american_scientific_journal-1069 71 2 constant constant ADJ american_scientific_journal-1069 71 3 is be AUX american_scientific_journal-1069 71 4 traditionally traditionally ADV american_scientific_journal-1069 71 5 chosen choose VERB american_scientific_journal-1069 71 6 so so SCONJ american_scientific_journal-1069 71 7 that that SCONJ american_scientific_journal-1069 71 8 the the DET american_scientific_journal-1069 71 9 highest high ADJ american_scientific_journal-1069 71 10 power power NOUN american_scientific_journal-1069 71 11 of of ADP american_scientific_journal-1069 71 12 πœ‰ πœ‰ PRON american_scientific_journal-1069 71 13 appear appear VERB american_scientific_journal-1069 71 14 with with ADP american_scientific_journal-1069 71 15 the the DET american_scientific_journal-1069 71 16 coefficients coefficient NOUN american_scientific_journal-1069 71 17 of of ADP american_scientific_journal-1069 71 18 2𝑛𝑓 2𝑛𝑓 NOUN american_scientific_journal-1069 71 19 in in ADP american_scientific_journal-1069 71 20 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 71 21 ) ) PUNCT american_scientific_journal-1069 71 22 . . PUNCT american_scientific_journal-1069 72 1 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 72 2 ) ) PUNCT american_scientific_journal-1069 72 3 = = SYM american_scientific_journal-1069 72 4 2 2 NUM american_scientific_journal-1069 72 5 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 72 6 ; ; PUNCT american_scientific_journal-1069 72 7 0.1 0.1 NUM american_scientific_journal-1069 72 8 ≀ ≀ NOUN american_scientific_journal-1069 72 9 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 72 10 ≀ ≀ NUM american_scientific_journal-1069 72 11 0.9 0.9 NUM american_scientific_journal-1069 72 12 ( ( PUNCT american_scientific_journal-1069 72 13 19 19 NUM american_scientific_journal-1069 72 14 ) ) PUNCT american_scientific_journal-1069 72 15 eq eq NOUN american_scientific_journal-1069 72 16 ( ( PUNCT american_scientific_journal-1069 72 17 19 19 NUM american_scientific_journal-1069 72 18 ) ) PUNCT american_scientific_journal-1069 72 19 is be AUX american_scientific_journal-1069 72 20 consist consist VERB american_scientific_journal-1069 72 21 with with ADP american_scientific_journal-1069 72 22 the the DET american_scientific_journal-1069 72 23 following follow VERB american_scientific_journal-1069 72 24 definition definition NOUN american_scientific_journal-1069 72 25 of of ADP american_scientific_journal-1069 72 26 hermite hermite PROPN american_scientific_journal-1069 72 27 polynomials polynomial NOUN american_scientific_journal-1069 72 28 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 72 29 ) ) PUNCT american_scientific_journal-1069 73 1 = = PRON american_scientific_journal-1069 73 2 ( ( PUNCT american_scientific_journal-1069 73 3 βˆ’1 βˆ’1 NOUN american_scientific_journal-1069 73 4 ) ) PUNCT american_scientific_journal-1069 73 5 π‘›π‘“π‘’πœ‰ π‘›π‘“π‘’πœ‰ VERB american_scientific_journal-1069 73 6 2 2 NUM american_scientific_journal-1069 73 7 𝑑 𝑑 NOUN american_scientific_journal-1069 73 8 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 73 9 π‘‘πœ‰ π‘‘πœ‰ ADP american_scientific_journal-1069 73 10 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 73 11 π‘’βˆ’πœ‰ π‘’βˆ’πœ‰ PROPN american_scientific_journal-1069 73 12 2 2 NUM american_scientific_journal-1069 73 13 ( ( PUNCT american_scientific_journal-1069 73 14 20 20 NUM american_scientific_journal-1069 73 15 ) ) PUNCT american_scientific_journal-1069 73 16 𝐻𝑛𝑓(πœ‰ 𝐻𝑛𝑓(πœ‰ PROPN american_scientific_journal-1069 73 17 ) ) PUNCT american_scientific_journal-1069 73 18 = = PUNCT american_scientific_journal-1069 73 19 𝑒 𝑒 PROPN american_scientific_journal-1069 73 20 πœ‰2 πœ‰2 PROPN american_scientific_journal-1069 73 21 2 2 NUM american_scientific_journal-1069 73 22 ( ( PUNCT american_scientific_journal-1069 73 23 πœ‰ πœ‰ X american_scientific_journal-1069 73 24 βˆ’ βˆ’ PROPN american_scientific_journal-1069 73 25 𝑑 𝑑 NOUN american_scientific_journal-1069 73 26 π‘‘πœ‰ π‘‘πœ‰ PROPN american_scientific_journal-1069 73 27 ) ) PUNCT american_scientific_journal-1069 73 28 π‘›π‘“π‘’βˆ’ π‘›π‘“π‘’βˆ’ PROPN american_scientific_journal-1069 73 29 πœ‰2 πœ‰2 PROPN american_scientific_journal-1069 73 30 2 2 NUM american_scientific_journal-1069 73 31 ( ( PUNCT american_scientific_journal-1069 73 32 21 21 NUM american_scientific_journal-1069 73 33 ) ) PUNCT american_scientific_journal-1069 73 34 eq eq NOUN american_scientific_journal-1069 74 1 ( ( PUNCT american_scientific_journal-1069 74 2 20 20 NUM american_scientific_journal-1069 74 3 ) ) PUNCT american_scientific_journal-1069 74 4 and and CCONJ american_scientific_journal-1069 74 5 ( ( PUNCT american_scientific_journal-1069 74 6 21 21 NUM american_scientific_journal-1069 74 7 ) ) PUNCT american_scientific_journal-1069 74 8 show show VERB american_scientific_journal-1069 74 9 fractional fractional ADJ american_scientific_journal-1069 74 10 exponents exponent NOUN american_scientific_journal-1069 74 11 of of ADP american_scientific_journal-1069 74 12 𝑑 𝑑 PRON american_scientific_journal-1069 74 13 π‘‘πœ‰ π‘‘πœ‰ PROPN american_scientific_journal-1069 74 14 and and CCONJ american_scientific_journal-1069 74 15 ( ( PUNCT american_scientific_journal-1069 74 16 πœ‰ πœ‰ AUX american_scientific_journal-1069 74 17 βˆ’ βˆ’ PROPN american_scientific_journal-1069 74 18 𝑑 𝑑 NOUN american_scientific_journal-1069 74 19 π‘‘πœ‰ π‘‘πœ‰ PROPN american_scientific_journal-1069 74 20 ) ) PUNCT american_scientific_journal-1069 74 21 and and CCONJ american_scientific_journal-1069 74 22 can can AUX american_scientific_journal-1069 74 23 be be AUX american_scientific_journal-1069 74 24 dealt deal VERB american_scientific_journal-1069 74 25 either either CCONJ american_scientific_journal-1069 74 26 with with ADP american_scientific_journal-1069 74 27 heaviside heaviside ADJ american_scientific_journal-1069 74 28 approximation approximation NOUN american_scientific_journal-1069 74 29 or or CCONJ american_scientific_journal-1069 74 30 lypanov lypanov VERB american_scientific_journal-1069 74 31 exponents exponent NOUN american_scientific_journal-1069 74 32 for for ADP american_scientific_journal-1069 74 33 attractors attractor NOUN american_scientific_journal-1069 74 34 . . PUNCT american_scientific_journal-1069 75 1 the the DET american_scientific_journal-1069 75 2 lypanov lypanov NOUN american_scientific_journal-1069 75 3 exponents exponent NOUN american_scientific_journal-1069 75 4 for for ADP american_scientific_journal-1069 75 5 equations equation NOUN american_scientific_journal-1069 75 6 ( ( PUNCT american_scientific_journal-1069 75 7 20 20 NUM american_scientific_journal-1069 75 8 ) ) PUNCT american_scientific_journal-1069 75 9 and and CCONJ american_scientific_journal-1069 75 10 ( ( PUNCT american_scientific_journal-1069 75 11 21 21 NUM american_scientific_journal-1069 75 12 ) ) PUNCT american_scientific_journal-1069 75 13 will will AUX american_scientific_journal-1069 75 14 preferably preferably ADV american_scientific_journal-1069 75 15 show show VERB american_scientific_journal-1069 75 16 the the DET american_scientific_journal-1069 75 17 behavior behavior NOUN american_scientific_journal-1069 75 18 of of ADP american_scientific_journal-1069 75 19 attracting attract VERB american_scientific_journal-1069 75 20 the the DET american_scientific_journal-1069 75 21 fractional fractional ADJ american_scientific_journal-1069 75 22 quantum quantum NOUN american_scientific_journal-1069 75 23 states state NOUN american_scientific_journal-1069 75 24 in in ADP american_scientific_journal-1069 75 25 the the DET american_scientific_journal-1069 75 26 momentum momentum NOUN american_scientific_journal-1069 75 27 space space NOUN american_scientific_journal-1069 75 28 with with ADP american_scientific_journal-1069 75 29 a a DET american_scientific_journal-1069 75 30 string string NOUN american_scientific_journal-1069 75 31 , , PUNCT american_scientific_journal-1069 75 32 .i.e .i.e PUNCT american_scientific_journal-1069 75 33 . . PROPN american_scientific_journal-1069 75 34 , , PUNCT american_scientific_journal-1069 75 35 a a DET american_scientific_journal-1069 75 36 quantum quantum ADJ american_scientific_journal-1069 75 37 wire wire NOUN american_scientific_journal-1069 75 38 . . PUNCT american_scientific_journal-1069 76 1 we we PRON american_scientific_journal-1069 76 2 calculate calculate VERB american_scientific_journal-1069 76 3 the the DET american_scientific_journal-1069 76 4 values value NOUN american_scientific_journal-1069 76 5 of of ADP american_scientific_journal-1069 76 6 the the DET american_scientific_journal-1069 76 7 hermite hermite ADJ american_scientific_journal-1069 76 8 polynomials polynomial NOUN american_scientific_journal-1069 76 9 from from ADP american_scientific_journal-1069 76 10 eqs eqs X american_scientific_journal-1069 76 11 ( ( PUNCT american_scientific_journal-1069 76 12 19 19 NUM american_scientific_journal-1069 76 13 ) ) PUNCT american_scientific_journal-1069 76 14 , , PUNCT american_scientific_journal-1069 76 15 ( ( PUNCT american_scientific_journal-1069 76 16 20 20 NUM american_scientific_journal-1069 76 17 ) ) PUNCT american_scientific_journal-1069 76 18 , , PUNCT american_scientific_journal-1069 76 19 and and CCONJ american_scientific_journal-1069 76 20 ( ( PUNCT american_scientific_journal-1069 76 21 21 21 NUM american_scientific_journal-1069 76 22 ) ) PUNCT american_scientific_journal-1069 76 23 as as SCONJ american_scientific_journal-1069 76 24 shown show VERB american_scientific_journal-1069 76 25 below below ADP american_scientific_journal-1069 76 26 𝐻𝜊(β„Žπ‘ž 𝐻𝜊(β„Žπ‘ž NOUN american_scientific_journal-1069 76 27 ) ) PUNCT american_scientific_journal-1069 76 28 = = SYM american_scientific_journal-1069 76 29 1 1 NUM american_scientific_journal-1069 76 30 for for ADP american_scientific_journal-1069 76 31 𝑛 𝑛 NOUN american_scientific_journal-1069 76 32 = = SYM american_scientific_journal-1069 76 33 0 0 NUM american_scientific_journal-1069 76 34 and and CCONJ american_scientific_journal-1069 76 35 𝐻1(β„Žπ‘ž 𝐻1(β„Žπ‘ž PROPN american_scientific_journal-1069 76 36 ) ) PUNCT american_scientific_journal-1069 76 37 = = SYM american_scientific_journal-1069 76 38 2 2 NUM american_scientific_journal-1069 76 39 for for ADP american_scientific_journal-1069 76 40 𝑛 𝑛 NOUN american_scientific_journal-1069 76 41 = = SYM american_scientific_journal-1069 76 42 1 1 NUM american_scientific_journal-1069 76 43 , , PUNCT american_scientific_journal-1069 76 44 πœ‰ πœ‰ X american_scientific_journal-1069 76 45 ≑ ≑ PROPN american_scientific_journal-1069 76 46 β„Žπ‘ž β„Žπ‘ž ADP american_scientific_journal-1069 76 47 now now ADV american_scientific_journal-1069 76 48 for for ADP american_scientific_journal-1069 76 49 0.1 0.1 NUM american_scientific_journal-1069 76 50 ≀ ≀ NOUN american_scientific_journal-1069 76 51 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 76 52 ≀ ≀ NUM american_scientific_journal-1069 76 53 0.9 0.9 NUM american_scientific_journal-1069 76 54 𝐻0.1(β„Žπ‘ž 𝐻0.1(β„Žπ‘ž NOUN american_scientific_journal-1069 76 55 ) ) PUNCT american_scientific_journal-1069 77 1 = = SYM american_scientific_journal-1069 77 2 1.072 1.072 NUM american_scientific_journal-1069 77 3 , , PUNCT american_scientific_journal-1069 77 4 𝐻0.2(β„Žπ‘ž 𝐻0.2(β„Žπ‘ž NOUN american_scientific_journal-1069 77 5 ) ) PUNCT american_scientific_journal-1069 77 6 = = PUNCT american_scientific_journal-1069 78 1 1.149 1.149 NUM american_scientific_journal-1069 78 2 , , PUNCT american_scientific_journal-1069 78 3 𝐻0.3(β„Žπ‘ž 𝐻0.3(β„Žπ‘ž ADJ american_scientific_journal-1069 78 4 ) ) PUNCT american_scientific_journal-1069 78 5 = = SYM american_scientific_journal-1069 78 6 1.231 1.231 NUM american_scientific_journal-1069 78 7 , , PUNCT american_scientific_journal-1069 78 8 𝐻0.4(β„Žπ‘ž 𝐻0.4(β„Žπ‘ž NUM american_scientific_journal-1069 78 9 ) ) PUNCT american_scientific_journal-1069 78 10 = = SYM american_scientific_journal-1069 78 11 1.319 1.319 NUM american_scientific_journal-1069 78 12 , , PUNCT american_scientific_journal-1069 78 13 𝐻0.5(β„Žπ‘ž 𝐻0.5(β„Žπ‘ž NOUN american_scientific_journal-1069 78 14 ) ) PUNCT american_scientific_journal-1069 79 1 = = NOUN american_scientific_journal-1069 79 2 1.414 1.414 NUM american_scientific_journal-1069 79 3 , , PUNCT american_scientific_journal-1069 79 4 𝐻0.6(β„Žπ‘ž 𝐻0.6(β„Žπ‘ž NOUN american_scientific_journal-1069 79 5 ) ) PUNCT american_scientific_journal-1069 80 1 = = PUNCT american_scientific_journal-1069 80 2 1.516 1.516 NUM american_scientific_journal-1069 80 3 , , PUNCT american_scientific_journal-1069 80 4 𝐻0.7(β„Žπ‘ž 𝐻0.7(β„Žπ‘ž NOUN american_scientific_journal-1069 80 5 ) ) PUNCT american_scientific_journal-1069 81 1 = = SYM american_scientific_journal-1069 81 2 1.624 1.624 NUM american_scientific_journal-1069 81 3 , , PUNCT american_scientific_journal-1069 81 4 𝐻0.8(β„Žπ‘ž 𝐻0.8(β„Žπ‘ž NOUN american_scientific_journal-1069 81 5 ) ) PUNCT american_scientific_journal-1069 81 6 = = SYM american_scientific_journal-1069 81 7 1.741 1.741 NUM american_scientific_journal-1069 81 8 , , PUNCT american_scientific_journal-1069 81 9 𝐻0.9(β„Žπ‘ž 𝐻0.9(β„Žπ‘ž PROPN american_scientific_journal-1069 81 10 ) ) PUNCT american_scientific_journal-1069 82 1 = = SYM american_scientific_journal-1069 82 2 1.8666 1.8666 NUM american_scientific_journal-1069 82 3 . . PUNCT american_scientific_journal-1069 83 1 ( ( PUNCT american_scientific_journal-1069 83 2 22 22 NUM american_scientific_journal-1069 83 3 ) ) PUNCT american_scientific_journal-1069 83 4 the the DET american_scientific_journal-1069 83 5 generating generate VERB american_scientific_journal-1069 83 6 function function NOUN american_scientific_journal-1069 83 7 is be AUX american_scientific_journal-1069 83 8 given give VERB american_scientific_journal-1069 83 9 by by ADP american_scientific_journal-1069 83 10 the the DET american_scientific_journal-1069 83 11 following follow VERB american_scientific_journal-1069 83 12 relations relation NOUN american_scientific_journal-1069 83 13 𝐺(β„Žπ‘ž 𝐺(β„Žπ‘ž NOUN american_scientific_journal-1069 83 14 , , PUNCT american_scientific_journal-1069 83 15 𝑠 𝑠 NOUN american_scientific_journal-1069 83 16 ) ) PUNCT american_scientific_journal-1069 83 17 = = PUNCT american_scientific_journal-1069 83 18 π‘’βˆ’π‘  π‘’βˆ’π‘  ADV american_scientific_journal-1069 83 19 2 2 NUM american_scientific_journal-1069 83 20 + + NOUN american_scientific_journal-1069 83 21 2π‘ β„Žπ‘ž 2π‘ β„Žπ‘ž NUM american_scientific_journal-1069 83 22 βˆ‘ βˆ‘ ADV american_scientific_journal-1069 83 23 𝐻𝑛𝑓 𝐻𝑛𝑓 PROPN american_scientific_journal-1069 83 24 ( ( PUNCT american_scientific_journal-1069 83 25 β„Žπ‘ž)𝑠𝑛 β„Žπ‘ž)𝑠𝑛 X american_scientific_journal-1069 83 26 𝑛 𝑛 X american_scientific_journal-1069 83 27 ! ! NOUN american_scientific_journal-1069 83 28 ∞ ∞ NUM american_scientific_journal-1069 83 29 𝑛=0 𝑛=0 NOUN american_scientific_journal-1069 83 30 ( ( PUNCT american_scientific_journal-1069 83 31 23 23 NUM american_scientific_journal-1069 83 32 ) ) PUNCT american_scientific_journal-1069 83 33 american american ADJ american_scientific_journal-1069 83 34 scientific scientific ADJ american_scientific_journal-1069 83 35 research research NOUN american_scientific_journal-1069 83 36 journal journal NOUN american_scientific_journal-1069 83 37 for for ADP american_scientific_journal-1069 83 38 engineering engineering NOUN american_scientific_journal-1069 83 39 , , PUNCT american_scientific_journal-1069 83 40 technology technology NOUN american_scientific_journal-1069 83 41 , , PUNCT american_scientific_journal-1069 83 42 and and CCONJ american_scientific_journal-1069 83 43 sciences science NOUN american_scientific_journal-1069 83 44 ( ( PUNCT american_scientific_journal-1069 83 45 asrjets asrjet NOUN american_scientific_journal-1069 83 46 ) ) PUNCT american_scientific_journal-1069 83 47 ( ( PUNCT american_scientific_journal-1069 83 48 2015 2015 NUM american_scientific_journal-1069 83 49 ) ) PUNCT american_scientific_journal-1069 83 50 volume volume NOUN american_scientific_journal-1069 83 51 14 14 NUM american_scientific_journal-1069 83 52 , , PUNCT american_scientific_journal-1069 83 53 no no DET american_scientific_journal-1069 83 54 2 2 NUM american_scientific_journal-1069 83 55 , , PUNCT american_scientific_journal-1069 83 56 pp pp ADV american_scientific_journal-1069 83 57 265 265 NUM american_scientific_journal-1069 83 58 - - SYM american_scientific_journal-1069 83 59 272 272 NUM american_scientific_journal-1069 83 60 270 270 NUM american_scientific_journal-1069 83 61 the the DET american_scientific_journal-1069 83 62 above above ADJ american_scientific_journal-1069 83 63 relation relation NOUN american_scientific_journal-1069 83 64 says say VERB american_scientific_journal-1069 83 65 that that SCONJ american_scientific_journal-1069 83 66 if if SCONJ american_scientific_journal-1069 83 67 the the DET american_scientific_journal-1069 83 68 function function NOUN american_scientific_journal-1069 83 69 π‘’βˆ’π‘  π‘’βˆ’π‘  ADV american_scientific_journal-1069 83 70 2 2 NUM american_scientific_journal-1069 83 71 + + NOUN american_scientific_journal-1069 83 72 2π‘ β„Žπ‘ž 2π‘ β„Žπ‘ž PROPN american_scientific_journal-1069 83 73 is be AUX american_scientific_journal-1069 83 74 expanded expand VERB american_scientific_journal-1069 83 75 in in ADP american_scientific_journal-1069 83 76 a a DET american_scientific_journal-1069 83 77 power power NOUN american_scientific_journal-1069 83 78 series series NOUN american_scientific_journal-1069 83 79 in in ADP american_scientific_journal-1069 83 80 𝑠 𝑠 PROPN american_scientific_journal-1069 83 81 , , PUNCT american_scientific_journal-1069 83 82 the the DET american_scientific_journal-1069 83 83 coefficients coefficient NOUN american_scientific_journal-1069 83 84 of of ADP american_scientific_journal-1069 83 85 successive successive ADJ american_scientific_journal-1069 83 86 powers power NOUN american_scientific_journal-1069 83 87 of of ADP american_scientific_journal-1069 83 88 𝑠 𝑠 PROPN american_scientific_journal-1069 83 89 are be AUX american_scientific_journal-1069 83 90 just just ADV american_scientific_journal-1069 83 91 1 1 NUM american_scientific_journal-1069 83 92 𝑛 𝑛 NOUN american_scientific_journal-1069 83 93 ! ! PUNCT american_scientific_journal-1069 84 1 times time NOUN american_scientific_journal-1069 84 2 the the DET american_scientific_journal-1069 84 3 hermite hermite ADJ american_scientific_journal-1069 84 4 polynomial polynomial NOUN american_scientific_journal-1069 84 5 , , PUNCT american_scientific_journal-1069 84 6 𝐻𝑛𝑓 𝐻𝑛𝑓 PROPN american_scientific_journal-1069 84 7 . . PUNCT american_scientific_journal-1069 85 1 by by ADP american_scientific_journal-1069 85 2 using use VERB american_scientific_journal-1069 85 3 eq eq X american_scientific_journal-1069 85 4 ( ( PUNCT american_scientific_journal-1069 85 5 23 23 NUM american_scientific_journal-1069 85 6 ) ) PUNCT american_scientific_journal-1069 85 7 and and CCONJ american_scientific_journal-1069 85 8 𝑛 𝑛 ADJ american_scientific_journal-1069 85 9 = = SYM american_scientific_journal-1069 85 10 0 0 NUM american_scientific_journal-1069 85 11 , , PUNCT american_scientific_journal-1069 85 12 we we PRON american_scientific_journal-1069 85 13 can can AUX american_scientific_journal-1069 85 14 prove prove VERB american_scientific_journal-1069 85 15 that that SCONJ american_scientific_journal-1069 85 16 the the DET american_scientific_journal-1069 85 17 hermite hermite ADJ american_scientific_journal-1069 85 18 polynomials polynomial NOUN american_scientific_journal-1069 85 19 satisfy satisfy VERB american_scientific_journal-1069 85 20 the the DET american_scientific_journal-1069 85 21 recursion recursion NOUN american_scientific_journal-1069 85 22 relations relation NOUN american_scientific_journal-1069 85 23 by by ADP american_scientific_journal-1069 85 24 the the DET american_scientific_journal-1069 85 25 following follow VERB american_scientific_journal-1069 85 26 relations relation NOUN american_scientific_journal-1069 85 27 : : PUNCT american_scientific_journal-1069 85 28 𝐻𝑛𝑓+1(β„Žπ‘ž 𝐻𝑛𝑓+1(β„Žπ‘ž VERB american_scientific_journal-1069 85 29 ) ) PUNCT american_scientific_journal-1069 85 30 βˆ’ βˆ’ ADP american_scientific_journal-1069 86 1 2β„Žπ‘žπ»π‘›π‘“(β„Žπ‘ž 2β„Žπ‘žπ»π‘›π‘“(β„Žπ‘ž NUM american_scientific_journal-1069 86 2 ) ) PUNCT american_scientific_journal-1069 87 1 + + CCONJ american_scientific_journal-1069 87 2 2π‘›π‘“π»π‘›π‘“βˆ’1(β„Žπ‘ž 2π‘›π‘“π»π‘›π‘“βˆ’1(β„Žπ‘ž NUM american_scientific_journal-1069 87 3 ) ) PUNCT american_scientific_journal-1069 87 4 = = SYM american_scientific_journal-1069 87 5 0 0 X american_scientific_journal-1069 88 1 𝑑𝐻𝑛𝑓 𝑑𝐻𝑛𝑓 PROPN american_scientific_journal-1069 88 2 ( ( PUNCT american_scientific_journal-1069 88 3 β„Žπ‘ž β„Žπ‘ž NOUN american_scientific_journal-1069 88 4 ) ) PUNCT american_scientific_journal-1069 88 5 π‘‘β„Žπ‘ž π‘‘β„Žπ‘ž ADV american_scientific_journal-1069 88 6 = = SYM american_scientific_journal-1069 88 7 2π‘›π‘“π»π‘›π‘“βˆ’1(β„Žπ‘ž 2π‘›π‘“π»π‘›π‘“βˆ’1(β„Žπ‘ž X american_scientific_journal-1069 88 8 ) ) PUNCT american_scientific_journal-1069 88 9 = = SYM american_scientific_journal-1069 88 10 0 0 NUM american_scientific_journal-1069 88 11 ( ( PUNCT american_scientific_journal-1069 88 12 24 24 NUM american_scientific_journal-1069 88 13 ) ) PUNCT american_scientific_journal-1069 88 14 using use VERB american_scientific_journal-1069 88 15 eq eq NOUN american_scientific_journal-1069 88 16 ( ( PUNCT american_scientific_journal-1069 88 17 24 24 NUM american_scientific_journal-1069 88 18 ) ) PUNCT american_scientific_journal-1069 88 19 for for ADP american_scientific_journal-1069 88 20 each each PRON american_scientific_journal-1069 88 21 of of ADP american_scientific_journal-1069 88 22 the the DET american_scientific_journal-1069 88 23 fractional fractional ADJ american_scientific_journal-1069 88 24 discrete discrete ADJ american_scientific_journal-1069 88 25 and and CCONJ american_scientific_journal-1069 88 26 distinct distinct ADJ american_scientific_journal-1069 88 27 values value NOUN american_scientific_journal-1069 88 28 of of ADP american_scientific_journal-1069 88 29 𝐸𝑛𝑓 𝐸𝑛𝑓 PROPN american_scientific_journal-1069 88 30 given give VERB american_scientific_journal-1069 88 31 by by ADP american_scientific_journal-1069 88 32 πœ†π‘“ πœ†π‘“ NOUN american_scientific_journal-1069 88 33 = = NOUN american_scientific_journal-1069 88 34 2𝑛𝑓 2𝑛𝑓 NOUN american_scientific_journal-1069 88 35 + + CCONJ american_scientific_journal-1069 88 36 1 1 NUM american_scientific_journal-1069 88 37 , , PUNCT american_scientific_journal-1069 88 38 0.1 0.1 NUM american_scientific_journal-1069 88 39 ≀ ≀ NOUN american_scientific_journal-1069 88 40 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 88 41 ≀ ≀ NUM american_scientific_journal-1069 88 42 0.9 0.9 NUM american_scientific_journal-1069 88 43 , , PUNCT american_scientific_journal-1069 88 44 there there PRON american_scientific_journal-1069 88 45 is be VERB american_scientific_journal-1069 88 46 only only ADV american_scientific_journal-1069 88 47 one one NUM american_scientific_journal-1069 88 48 physically physically ADV american_scientific_journal-1069 88 49 acceptable acceptable ADJ american_scientific_journal-1069 88 50 solution solution NOUN american_scientific_journal-1069 88 51 for for ADP american_scientific_journal-1069 88 52 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 88 53 having have VERB american_scientific_journal-1069 88 54 oscillatory oscillatory ADJ american_scientific_journal-1069 88 55 behavior behavior NOUN american_scientific_journal-1069 88 56 in in ADP american_scientific_journal-1069 88 57 fractionally fractionally ADV american_scientific_journal-1069 88 58 quantized quantize VERB american_scientific_journal-1069 88 59 momentum momentum NOUN american_scientific_journal-1069 88 60 space space NOUN american_scientific_journal-1069 88 61 , , PUNCT american_scientific_journal-1069 88 62 .i.e .i.e PUNCT american_scientific_journal-1069 88 63 . . PUNCT american_scientific_journal-1069 88 64 , , PUNCT american_scientific_journal-1069 88 65 πœ“π‘›π‘“(β„Žπ‘ž πœ“π‘›π‘“(β„Žπ‘ž NOUN american_scientific_journal-1069 88 66 ) ) PUNCT american_scientific_journal-1069 88 67 = = SYM american_scientific_journal-1069 89 1 𝑁𝑛𝑓𝑒 𝑁𝑛𝑓𝑒 PROPN american_scientific_journal-1069 89 2 βˆ’π›Ό2 βˆ’π›Ό2 PROPN american_scientific_journal-1069 89 3 ( ( PUNCT american_scientific_journal-1069 89 4 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 89 5 √2 √2 PROPN american_scientific_journal-1069 89 6 ) ) PUNCT american_scientific_journal-1069 89 7 2 2 NUM american_scientific_journal-1069 89 8 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 89 9 ) ) PUNCT american_scientific_journal-1069 89 10 ( ( PUNCT american_scientific_journal-1069 89 11 25 25 NUM american_scientific_journal-1069 89 12 ) ) PUNCT american_scientific_journal-1069 89 13 considering consider VERB american_scientific_journal-1069 89 14 the the DET american_scientific_journal-1069 89 15 hermite hermite ADJ american_scientific_journal-1069 89 16 generating generating NOUN american_scientific_journal-1069 89 17 function function NOUN american_scientific_journal-1069 89 18 and and CCONJ american_scientific_journal-1069 89 19 equate equate VERB american_scientific_journal-1069 89 20 the the DET american_scientific_journal-1069 89 21 coefficient coefficient NOUN american_scientific_journal-1069 89 22 of of ADP american_scientific_journal-1069 89 23 equal equal ADJ american_scientific_journal-1069 89 24 parts part NOUN american_scientific_journal-1069 89 25 of of ADP american_scientific_journal-1069 89 26 𝑠 𝑠 PROPN american_scientific_journal-1069 89 27 and and CCONJ american_scientific_journal-1069 89 28 𝑑 𝑑 PROPN american_scientific_journal-1069 89 29 , , PUNCT american_scientific_journal-1069 89 30 the the DET american_scientific_journal-1069 89 31 normalized normalize VERB american_scientific_journal-1069 89 32 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 89 33 are be AUX american_scientific_journal-1069 89 34 given give VERB american_scientific_journal-1069 89 35 by by ADP american_scientific_journal-1069 89 36 πœ“(π‘₯ πœ“(π‘₯ NOUN american_scientific_journal-1069 89 37 ) ) PUNCT american_scientific_journal-1069 89 38 = = SYM american_scientific_journal-1069 89 39 ( ( PUNCT american_scientific_journal-1069 89 40 𝛼 𝛼 PROPN american_scientific_journal-1069 89 41 βˆšπœ‹2𝑛𝑛 βˆšπœ‹2𝑛𝑛 PROPN american_scientific_journal-1069 89 42 ! ! PUNCT american_scientific_journal-1069 89 43 ) ) PUNCT american_scientific_journal-1069 89 44 1 1 NUM american_scientific_journal-1069 89 45 2 2 NUM american_scientific_journal-1069 89 46 π‘’βˆ’π›Ό π‘’βˆ’π›Ό PROPN american_scientific_journal-1069 89 47 2π‘₯2 2π‘₯2 NUM american_scientific_journal-1069 89 48 2 2 NUM american_scientific_journal-1069 89 49 𝐻𝑛(𝛼π‘₯ 𝐻𝑛(𝛼π‘₯ NOUN american_scientific_journal-1069 89 50 ) ) PUNCT american_scientific_journal-1069 89 51 ( ( PUNCT american_scientific_journal-1069 89 52 26 26 NUM american_scientific_journal-1069 89 53 ) ) PUNCT american_scientific_journal-1069 89 54 for for ADP american_scientific_journal-1069 89 55 our our PRON american_scientific_journal-1069 89 56 case case NOUN american_scientific_journal-1069 89 57 π‘₯ π‘₯ X american_scientific_journal-1069 89 58 ≑ ≑ PROPN american_scientific_journal-1069 89 59 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 89 60 , , PUNCT american_scientific_journal-1069 89 61 0.17 0.17 NUM american_scientific_journal-1069 89 62 ≀ ≀ NUM american_scientific_journal-1069 89 63 𝛼 𝛼 PRON american_scientific_journal-1069 89 64 ≀ ≀ NOUN american_scientific_journal-1069 89 65 1.53 1.53 NUM american_scientific_journal-1069 89 66 where where SCONJ american_scientific_journal-1069 89 67 𝛼 𝛼 AUX american_scientific_journal-1069 89 68 = = X american_scientific_journal-1069 89 69 π‘Žπœ‹ π‘Žπœ‹ PROPN american_scientific_journal-1069 89 70 2 2 NUM american_scientific_journal-1069 89 71 ≑ ≑ PROPN american_scientific_journal-1069 89 72 πœ† πœ† ADP american_scientific_journal-1069 89 73 as as ADP american_scientific_journal-1069 89 74 π‘Ž π‘Ž DET american_scientific_journal-1069 89 75 β‰  β‰  PROPN american_scientific_journal-1069 89 76 1 1 NUM american_scientific_journal-1069 89 77 . . PUNCT american_scientific_journal-1069 90 1 the the DET american_scientific_journal-1069 90 2 rotation rotation NOUN american_scientific_journal-1069 90 3 vector vector NOUN american_scientific_journal-1069 90 4 will will AUX american_scientific_journal-1069 90 5 show show VERB american_scientific_journal-1069 90 6 that that SCONJ american_scientific_journal-1069 90 7 the the DET american_scientific_journal-1069 90 8 position position NOUN american_scientific_journal-1069 90 9 of of ADP american_scientific_journal-1069 90 10 the the DET american_scientific_journal-1069 90 11 fractional fractional ADJ american_scientific_journal-1069 90 12 quantized quantize VERB american_scientific_journal-1069 90 13 momentum momentum NOUN american_scientific_journal-1069 90 14 space space NOUN american_scientific_journal-1069 90 15 for for ADP american_scientific_journal-1069 90 16 charge charge NOUN american_scientific_journal-1069 90 17 quantization quantization NOUN american_scientific_journal-1069 90 18 on on ADP american_scientific_journal-1069 90 19 the the DET american_scientific_journal-1069 90 20 varying vary VERB american_scientific_journal-1069 90 21 surface surface NOUN american_scientific_journal-1069 90 22 of of ADP american_scientific_journal-1069 90 23 the the DET american_scientific_journal-1069 90 24 electron electron NOUN american_scientific_journal-1069 90 25 in in ADP american_scientific_journal-1069 90 26 terms term NOUN american_scientific_journal-1069 90 27 of of ADP american_scientific_journal-1069 90 28 radius radius NOUN american_scientific_journal-1069 90 29 . . PUNCT american_scientific_journal-1069 91 1 put put VERB american_scientific_journal-1069 91 2 𝑛 𝑛 PROPN american_scientific_journal-1069 91 3 ! ! PUNCT american_scientific_journal-1069 92 1 = = NOUN american_scientific_journal-1069 92 2 0 0 X american_scientific_journal-1069 92 3 ! ! PUNCT american_scientific_journal-1069 93 1 = = SYM american_scientific_journal-1069 93 2 1 1 NUM american_scientific_journal-1069 93 3 , , PUNCT american_scientific_journal-1069 93 4 the the DET american_scientific_journal-1069 93 5 exponent exponent NOUN american_scientific_journal-1069 93 6 𝑛 𝑛 PROPN american_scientific_journal-1069 93 7 in in ADP american_scientific_journal-1069 93 8 terms term NOUN american_scientific_journal-1069 93 9 of of ADP american_scientific_journal-1069 93 10 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 93 11 ( ( PUNCT american_scientific_journal-1069 93 12 fractional fractional ADJ american_scientific_journal-1069 93 13 exponents exponent NOUN american_scientific_journal-1069 93 14 ) ) PUNCT american_scientific_journal-1069 93 15 and and CCONJ american_scientific_journal-1069 93 16 the the DET american_scientific_journal-1069 93 17 subscript subscript NOUN american_scientific_journal-1069 93 18 𝑛 𝑛 PROPN american_scientific_journal-1069 93 19 in in ADP american_scientific_journal-1069 93 20 𝐻 𝐻 PROPN american_scientific_journal-1069 93 21 with with ADP american_scientific_journal-1069 93 22 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 93 23 , , PUNCT american_scientific_journal-1069 93 24 where where SCONJ american_scientific_journal-1069 93 25 0.1 0.1 NUM american_scientific_journal-1069 93 26 ≀ ≀ NOUN american_scientific_journal-1069 93 27 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 93 28 ≀ ≀ NUM american_scientific_journal-1069 93 29 0.9 0.9 NUM american_scientific_journal-1069 93 30 in in ADP american_scientific_journal-1069 93 31 eq eq NOUN american_scientific_journal-1069 93 32 ( ( PUNCT american_scientific_journal-1069 93 33 26 26 NUM american_scientific_journal-1069 93 34 ) ) PUNCT american_scientific_journal-1069 93 35 , , PUNCT american_scientific_journal-1069 93 36 we we PRON american_scientific_journal-1069 93 37 have have VERB american_scientific_journal-1069 93 38 πœ“π‘›π‘“(β„Žπ‘ž πœ“π‘›π‘“(β„Žπ‘ž NOUN american_scientific_journal-1069 93 39 ) ) PUNCT american_scientific_journal-1069 93 40 = = SYM american_scientific_journal-1069 94 1 ( ( PUNCT american_scientific_journal-1069 94 2 0.17≀𝛼≀1.53 0.17≀𝛼≀1.53 NOUN american_scientific_journal-1069 95 1 βˆšπœ‹2 βˆšπœ‹2 X american_scientific_journal-1069 95 2 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 95 3 .1 .1 NUM american_scientific_journal-1069 95 4 ) ) PUNCT american_scientific_journal-1069 95 5 1 1 NUM american_scientific_journal-1069 95 6 2 2 NUM american_scientific_journal-1069 95 7 𝑒 𝑒 ADP american_scientific_journal-1069 95 8 βˆ’π›Ό2 βˆ’π›Ό2 PROPN american_scientific_journal-1069 95 9 ( ( PUNCT american_scientific_journal-1069 95 10 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 95 11 √2 √2 PROPN american_scientific_journal-1069 95 12 ) ) PUNCT american_scientific_journal-1069 95 13 2 2 NUM american_scientific_journal-1069 95 14 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 95 15 ) ) PUNCT american_scientific_journal-1069 95 16 ( ( PUNCT american_scientific_journal-1069 95 17 27 27 NUM american_scientific_journal-1069 95 18 ) ) PUNCT american_scientific_journal-1069 95 19 using use VERB american_scientific_journal-1069 95 20 eq eq NOUN american_scientific_journal-1069 95 21 ( ( PUNCT american_scientific_journal-1069 95 22 22 22 NUM american_scientific_journal-1069 95 23 ) ) PUNCT american_scientific_journal-1069 95 24 , , PUNCT american_scientific_journal-1069 95 25 .i.e .i.e X american_scientific_journal-1069 95 26 . . PROPN american_scientific_journal-1069 95 27 , , PUNCT american_scientific_journal-1069 95 28 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž PROPN american_scientific_journal-1069 95 29 ) ) PUNCT american_scientific_journal-1069 95 30 = = SYM american_scientific_journal-1069 95 31 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 95 32 ) ) PUNCT american_scientific_journal-1069 95 33 in in ADP american_scientific_journal-1069 95 34 eq eq NOUN american_scientific_journal-1069 95 35 ( ( PUNCT american_scientific_journal-1069 95 36 27 27 NUM american_scientific_journal-1069 95 37 ) ) PUNCT american_scientific_journal-1069 95 38 and and CCONJ american_scientific_journal-1069 95 39 putting put VERB american_scientific_journal-1069 95 40 𝑛𝑓 𝑛𝑓 X american_scientific_journal-1069 95 41 = = PUNCT american_scientific_journal-1069 95 42 0.1 0.1 NUM american_scientific_journal-1069 95 43 ( ( PUNCT american_scientific_journal-1069 95 44 𝛼 𝛼 X american_scientific_journal-1069 95 45 = = SYM american_scientific_journal-1069 95 46 0.17 0.17 NUM american_scientific_journal-1069 95 47 ) ) PUNCT american_scientific_journal-1069 95 48 , , PUNCT american_scientific_journal-1069 95 49 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 95 50 = = NUM american_scientific_journal-1069 95 51 0.2 0.2 NUM american_scientific_journal-1069 95 52 ( ( PUNCT american_scientific_journal-1069 95 53 𝛼 𝛼 X american_scientific_journal-1069 95 54 = = NOUN american_scientific_journal-1069 95 55 0.34 0.34 NUM american_scientific_journal-1069 95 56 ) ) PUNCT american_scientific_journal-1069 95 57 , , PUNCT american_scientific_journal-1069 95 58 … … PUNCT american_scientific_journal-1069 95 59 , , PUNCT american_scientific_journal-1069 95 60 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 95 61 = = NUM american_scientific_journal-1069 95 62 0.9 0.9 NUM american_scientific_journal-1069 95 63 ( ( PUNCT american_scientific_journal-1069 95 64 𝛼 𝛼 X american_scientific_journal-1069 95 65 = = NOUN american_scientific_journal-1069 95 66 1.53 1.53 NUM american_scientific_journal-1069 95 67 ) ) PUNCT american_scientific_journal-1069 95 68 . . PUNCT american_scientific_journal-1069 96 1 we we PRON american_scientific_journal-1069 96 2 can can AUX american_scientific_journal-1069 96 3 reproduce reproduce VERB american_scientific_journal-1069 96 4 the the DET american_scientific_journal-1069 96 5 distribution distribution NOUN american_scientific_journal-1069 96 6 of of ADP american_scientific_journal-1069 96 7 the the DET american_scientific_journal-1069 96 8 fractional fractional ADJ american_scientific_journal-1069 96 9 quantized quantize VERB american_scientific_journal-1069 96 10 states state NOUN american_scientific_journal-1069 96 11 for for ADP american_scientific_journal-1069 96 12 charges charge NOUN american_scientific_journal-1069 96 13 on on ADP american_scientific_journal-1069 96 14 the the DET american_scientific_journal-1069 96 15 surface surface NOUN american_scientific_journal-1069 96 16 of of ADP american_scientific_journal-1069 96 17 the the DET american_scientific_journal-1069 96 18 electron electron NOUN american_scientific_journal-1069 96 19 . . PUNCT american_scientific_journal-1069 97 1 in in ADP american_scientific_journal-1069 97 2 other other ADJ american_scientific_journal-1069 97 3 words word NOUN american_scientific_journal-1069 97 4 , , PUNCT american_scientific_journal-1069 97 5 we we PRON american_scientific_journal-1069 97 6 can can AUX american_scientific_journal-1069 97 7 either either CCONJ american_scientific_journal-1069 97 8 reproduce reproduce VERB american_scientific_journal-1069 97 9 the the DET american_scientific_journal-1069 97 10 shape shape NOUN american_scientific_journal-1069 97 11 of of ADP american_scientific_journal-1069 97 12 the the DET american_scientific_journal-1069 97 13 fractional fractional ADJ american_scientific_journal-1069 97 14 charge charge NOUN american_scientific_journal-1069 97 15 distribution distribution NOUN american_scientific_journal-1069 97 16 which which PRON american_scientific_journal-1069 97 17 are be AUX american_scientific_journal-1069 97 18 beaded bead VERB american_scientific_journal-1069 97 19 in in ADP american_scientific_journal-1069 97 20 a a DET american_scientific_journal-1069 97 21 string string NOUN american_scientific_journal-1069 97 22 on on ADP american_scientific_journal-1069 97 23 the the DET american_scientific_journal-1069 97 24 surface surface NOUN american_scientific_journal-1069 97 25 of of ADP american_scientific_journal-1069 97 26 the the DET american_scientific_journal-1069 97 27 electron electron NOUN american_scientific_journal-1069 97 28 or or CCONJ american_scientific_journal-1069 97 29 the the DET american_scientific_journal-1069 97 30 shape shape NOUN american_scientific_journal-1069 97 31 of of ADP american_scientific_journal-1069 97 32 the the DET american_scientific_journal-1069 97 33 garland garland NOUN american_scientific_journal-1069 97 34 with with ADP american_scientific_journal-1069 97 35 beads bead NOUN american_scientific_journal-1069 97 36 with with ADP american_scientific_journal-1069 97 37 fractional fractional ADJ american_scientific_journal-1069 97 38 charge charge NOUN american_scientific_journal-1069 97 39 quantization quantization NOUN american_scientific_journal-1069 97 40 . . PUNCT american_scientific_journal-1069 98 1 the the DET american_scientific_journal-1069 98 2 garland garland NOUN american_scientific_journal-1069 98 3 with with ADP american_scientific_journal-1069 98 4 beads bead NOUN american_scientific_journal-1069 98 5 ( ( PUNCT american_scientific_journal-1069 98 6 fractional fractional ADJ american_scientific_journal-1069 98 7 charge charge NOUN american_scientific_journal-1069 98 8 quantization quantization NOUN american_scientific_journal-1069 98 9 with with ADP american_scientific_journal-1069 98 10 sub sub PROPN american_scientific_journal-1069 98 11 quanta quanta PROPN american_scientific_journal-1069 98 12 ) ) PUNCT american_scientific_journal-1069 98 13 could could AUX american_scientific_journal-1069 98 14 be be AUX american_scientific_journal-1069 98 15 envisaged envisage VERB american_scientific_journal-1069 98 16 like like ADP american_scientific_journal-1069 98 17 a a DET american_scientific_journal-1069 98 18 quanta quanta PROPN american_scientific_journal-1069 98 19 wire wire NOUN american_scientific_journal-1069 98 20 . . PUNCT american_scientific_journal-1069 99 1 for for ADP american_scientific_journal-1069 99 2 ft ft PROPN american_scientific_journal-1069 99 3 representation representation NOUN american_scientific_journal-1069 99 4 of of ADP american_scientific_journal-1069 99 5 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 99 6 , , PUNCT american_scientific_journal-1069 99 7 the the DET american_scientific_journal-1069 99 8 eq eq NOUN american_scientific_journal-1069 99 9 ( ( PUNCT american_scientific_journal-1069 99 10 26 26 NUM american_scientific_journal-1069 99 11 ) ) PUNCT american_scientific_journal-1069 99 12 , , PUNCT american_scientific_journal-1069 99 13 we we PRON american_scientific_journal-1069 99 14 put put VERB american_scientific_journal-1069 99 15 𝛼 𝛼 NOUN american_scientific_journal-1069 99 16 = = NOUN american_scientific_journal-1069 99 17 πœ‹ πœ‹ X american_scientific_journal-1069 99 18 2 2 NUM american_scientific_journal-1069 99 19 , , PUNCT american_scientific_journal-1069 99 20 𝑛 𝑛 NOUN american_scientific_journal-1069 99 21 = = SYM american_scientific_journal-1069 99 22 0 0 NUM american_scientific_journal-1069 99 23 and and CCONJ american_scientific_journal-1069 99 24 1 1 NUM american_scientific_journal-1069 99 25 , , PUNCT american_scientific_journal-1069 99 26 we we PRON american_scientific_journal-1069 99 27 shall shall AUX american_scientific_journal-1069 99 28 then then ADV american_scientific_journal-1069 99 29 have have VERB american_scientific_journal-1069 99 30 the the DET american_scientific_journal-1069 99 31 asymptotic asymptotic ADJ american_scientific_journal-1069 99 32 variation variation NOUN american_scientific_journal-1069 99 33 of of ADP american_scientific_journal-1069 99 34 πœ“π‘›(β„Žπ‘ž πœ“π‘›(β„Žπ‘ž NOUN american_scientific_journal-1069 99 35 ) ) PUNCT american_scientific_journal-1069 99 36 with with ADP american_scientific_journal-1069 99 37 π‘₯ π‘₯ PROPN american_scientific_journal-1069 99 38 ≑ ≑ PROPN american_scientific_journal-1069 99 39 β„Žπ‘ž. β„Žπ‘ž. PROPN american_scientific_journal-1069 99 40 with with ADP american_scientific_journal-1069 99 41 arbitrary arbitrary ADJ american_scientific_journal-1069 99 42 values value NOUN american_scientific_journal-1069 99 43 of of ADP american_scientific_journal-1069 99 44 π‘₯ π‘₯ PRON american_scientific_journal-1069 99 45 starting start VERB american_scientific_journal-1069 99 46 from from ADP american_scientific_journal-1069 99 47 zero zero NUM american_scientific_journal-1069 99 48 ( ( PUNCT american_scientific_journal-1069 99 49 depth depth NOUN american_scientific_journal-1069 99 50 of of ADP american_scientific_journal-1069 99 51 the the DET american_scientific_journal-1069 99 52 quantum quantum ADJ american_scientific_journal-1069 99 53 well well NOUN american_scientific_journal-1069 99 54 of of ADP american_scientific_journal-1069 99 55 electron electron NOUN american_scientific_journal-1069 99 56 ) ) PUNCT american_scientific_journal-1069 99 57 to to PART american_scientific_journal-1069 99 58 radius radius VERB american_scientific_journal-1069 99 59 of of ADP american_scientific_journal-1069 99 60 an an DET american_scientific_journal-1069 99 61 electron electron NOUN american_scientific_journal-1069 99 62 , , PUNCT american_scientific_journal-1069 99 63 π‘₯ π‘₯ NOUN american_scientific_journal-1069 99 64 = = SYM american_scientific_journal-1069 99 65 π‘Ÿπ‘’(10 π‘Ÿπ‘’(10 ADJ american_scientific_journal-1069 99 66 βˆ’ βˆ’ PROPN american_scientific_journal-1069 99 67 15πœ‡π‘š 15πœ‡π‘š NOUN american_scientific_journal-1069 99 68 ) ) PUNCT american_scientific_journal-1069 99 69 and and CCONJ american_scientific_journal-1069 99 70 using use VERB american_scientific_journal-1069 99 71 eq eq X american_scientific_journal-1069 99 72 ( ( PUNCT american_scientific_journal-1069 99 73 26 26 NUM american_scientific_journal-1069 99 74 ) ) PUNCT american_scientific_journal-1069 99 75 with with ADP american_scientific_journal-1069 99 76 𝑛 𝑛 PROPN american_scientific_journal-1069 99 77 ! ! PUNCT american_scientific_journal-1069 100 1 = = NOUN american_scientific_journal-1069 100 2 0 0 X american_scientific_journal-1069 100 3 ! ! PUNCT american_scientific_journal-1069 101 1 = = SYM american_scientific_journal-1069 101 2 1 1 NUM american_scientific_journal-1069 101 3 , , PUNCT american_scientific_journal-1069 101 4 20.1≀𝑛𝑓≀0.9 20.1≀𝑛𝑓≀0.9 PRON american_scientific_journal-1069 101 5 , , PUNCT american_scientific_journal-1069 101 6 𝐻𝑛𝑓(𝛼π‘₯ 𝐻𝑛𝑓(𝛼π‘₯ NOUN american_scientific_journal-1069 101 7 ) ) PUNCT american_scientific_journal-1069 101 8 = = SYM american_scientific_journal-1069 101 9 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž PROPN american_scientific_journal-1069 101 10 ) ) PUNCT american_scientific_journal-1069 101 11 and and CCONJ american_scientific_journal-1069 101 12 0.17 0.17 NUM american_scientific_journal-1069 101 13 ≀ ≀ NUM american_scientific_journal-1069 102 1 𝛼 𝛼 PRON american_scientific_journal-1069 102 2 ≀ ≀ NUM american_scientific_journal-1069 102 3 1.53 1.53 NUM american_scientific_journal-1069 102 4 , , PUNCT american_scientific_journal-1069 102 5 we we PRON american_scientific_journal-1069 102 6 can can AUX american_scientific_journal-1069 102 7 reproduce reproduce VERB american_scientific_journal-1069 102 8 the the DET american_scientific_journal-1069 102 9 energy energy NOUN american_scientific_journal-1069 102 10 profile profile NOUN american_scientific_journal-1069 102 11 for for ADP american_scientific_journal-1069 102 12 each each PRON american_scientific_journal-1069 102 13 of of ADP american_scientific_journal-1069 102 14 the the DET american_scientific_journal-1069 102 15 fractional fractional ADJ american_scientific_journal-1069 102 16 states state NOUN american_scientific_journal-1069 102 17 inside inside ADP american_scientific_journal-1069 102 18 the the DET american_scientific_journal-1069 102 19 quantum quantum NOUN american_scientific_journal-1069 102 20 well well ADV american_scientific_journal-1069 102 21 . . PUNCT american_scientific_journal-1069 103 1 american american ADJ american_scientific_journal-1069 103 2 scientific scientific ADJ american_scientific_journal-1069 103 3 research research PROPN american_scientific_journal-1069 103 4 journal journal NOUN american_scientific_journal-1069 103 5 for for ADP american_scientific_journal-1069 103 6 engineering engineering NOUN american_scientific_journal-1069 103 7 , , PUNCT american_scientific_journal-1069 103 8 technology technology NOUN american_scientific_journal-1069 103 9 , , PUNCT american_scientific_journal-1069 103 10 and and CCONJ american_scientific_journal-1069 103 11 sciences science NOUN american_scientific_journal-1069 103 12 ( ( PUNCT american_scientific_journal-1069 103 13 asrjets asrjet NOUN american_scientific_journal-1069 103 14 ) ) PUNCT american_scientific_journal-1069 103 15 ( ( PUNCT american_scientific_journal-1069 103 16 2015 2015 NUM american_scientific_journal-1069 103 17 ) ) PUNCT american_scientific_journal-1069 103 18 volume volume NOUN american_scientific_journal-1069 103 19 14 14 NUM american_scientific_journal-1069 103 20 , , PUNCT american_scientific_journal-1069 103 21 no no DET american_scientific_journal-1069 103 22 2 2 NUM american_scientific_journal-1069 103 23 , , PUNCT american_scientific_journal-1069 103 24 pp pp ADV american_scientific_journal-1069 103 25 265 265 NUM american_scientific_journal-1069 103 26 - - SYM american_scientific_journal-1069 103 27 272 272 NUM american_scientific_journal-1069 103 28 271 271 NUM american_scientific_journal-1069 103 29 at at ADP american_scientific_journal-1069 103 30 π‘₯ π‘₯ NOUN american_scientific_journal-1069 103 31 = = NOUN american_scientific_journal-1069 103 32 π‘Ÿπ‘’ π‘Ÿπ‘’ NOUN american_scientific_journal-1069 103 33 we we PRON american_scientific_journal-1069 103 34 have have VERB american_scientific_journal-1069 103 35 the the DET american_scientific_journal-1069 103 36 brim brim NOUN american_scientific_journal-1069 103 37 of of ADP american_scientific_journal-1069 103 38 the the DET american_scientific_journal-1069 103 39 quantum quantum NOUN american_scientific_journal-1069 103 40 well well ADV american_scientific_journal-1069 103 41 . . PUNCT american_scientific_journal-1069 104 1 each each PRON american_scientific_journal-1069 104 2 of of ADP american_scientific_journal-1069 104 3 the the DET american_scientific_journal-1069 104 4 eigenfunctions eigenfunction NOUN american_scientific_journal-1069 104 5 for for ADP american_scientific_journal-1069 104 6 0.1 0.1 NUM american_scientific_journal-1069 104 7 ≀ ≀ NOUN american_scientific_journal-1069 104 8 𝑛𝑓 𝑛𝑓 PROPN american_scientific_journal-1069 104 9 ≀ ≀ NUM american_scientific_journal-1069 104 10 0.9 0.9 NUM american_scientific_journal-1069 104 11 will will AUX american_scientific_journal-1069 104 12 show show VERB american_scientific_journal-1069 104 13 the the DET american_scientific_journal-1069 104 14 whirling whirl VERB american_scientific_journal-1069 104 15 profile profile NOUN american_scientific_journal-1069 104 16 for for ADP american_scientific_journal-1069 104 17 energy energy NOUN american_scientific_journal-1069 104 18 whereas whereas SCONJ american_scientific_journal-1069 104 19 the the DET american_scientific_journal-1069 104 20 fractional fractional ADJ american_scientific_journal-1069 104 21 change change NOUN american_scientific_journal-1069 104 22 quantization quantization NOUN american_scientific_journal-1069 104 23 on on ADP american_scientific_journal-1069 104 24 the the DET american_scientific_journal-1069 104 25 surface surface NOUN american_scientific_journal-1069 104 26 of of ADP american_scientific_journal-1069 104 27 an an DET american_scientific_journal-1069 104 28 electron electron NOUN american_scientific_journal-1069 104 29 is be AUX american_scientific_journal-1069 104 30 a a DET american_scientific_journal-1069 104 31 swirling swirl VERB american_scientific_journal-1069 104 32 phenomenon phenomenon NOUN american_scientific_journal-1069 104 33 . . PUNCT american_scientific_journal-1069 105 1 on on ADP american_scientific_journal-1069 105 2 comparison comparison NOUN american_scientific_journal-1069 105 3 of of ADP american_scientific_journal-1069 105 4 eq eq NOUN american_scientific_journal-1069 105 5 ( ( PUNCT american_scientific_journal-1069 105 6 7 7 NUM american_scientific_journal-1069 105 7 ) ) PUNCT american_scientific_journal-1069 105 8 and and CCONJ american_scientific_journal-1069 105 9 ( ( PUNCT american_scientific_journal-1069 105 10 27 27 NUM american_scientific_journal-1069 105 11 ) ) PUNCT american_scientific_journal-1069 105 12 with with ADP american_scientific_journal-1069 105 13 condition condition NOUN american_scientific_journal-1069 105 14 that that SCONJ american_scientific_journal-1069 105 15 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž ADV american_scientific_journal-1069 105 16 ) ) PUNCT american_scientific_journal-1069 105 17 = = SYM american_scientific_journal-1069 105 18 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 105 19 ) ) PUNCT american_scientific_journal-1069 105 20 , , PUNCT american_scientific_journal-1069 105 21 we we PRON american_scientific_journal-1069 105 22 have have VERB american_scientific_journal-1069 105 23 𝑒 𝑒 PROPN american_scientific_journal-1069 105 24 βˆ’π›Ό2 βˆ’π›Ό2 PROPN american_scientific_journal-1069 105 25 ( ( PUNCT american_scientific_journal-1069 105 26 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 105 27 √2 √2 PROPN american_scientific_journal-1069 105 28 ) ) PUNCT american_scientific_journal-1069 105 29 2 2 NUM american_scientific_journal-1069 105 30 ≑ ≑ PROPN american_scientific_journal-1069 105 31 1 1 NUM american_scientific_journal-1069 105 32 we we PRON american_scientific_journal-1069 105 33 find find VERB american_scientific_journal-1069 105 34 𝑒𝑖𝑛𝑓𝛼 𝑒𝑖𝑛𝑓𝛼 NOUN american_scientific_journal-1069 105 35 = = PUNCT american_scientific_journal-1069 105 36 ( ( PUNCT american_scientific_journal-1069 105 37 𝛼 𝛼 X american_scientific_journal-1069 105 38 βˆšπœ‹2 βˆšπœ‹2 PROPN american_scientific_journal-1069 105 39 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 105 40 ) ) PUNCT american_scientific_journal-1069 105 41 1 1 NUM american_scientific_journal-1069 105 42 2 2 NUM american_scientific_journal-1069 105 43 π‘’βˆ’π›Ό π‘’βˆ’π›Ό PROPN american_scientific_journal-1069 105 44 2 2 NUM american_scientific_journal-1069 105 45 ( ( PUNCT american_scientific_journal-1069 105 46 28 28 NUM american_scientific_journal-1069 105 47 ) ) PUNCT american_scientific_journal-1069 105 48 considering consider VERB american_scientific_journal-1069 105 49 eq eq NOUN american_scientific_journal-1069 105 50 ( ( PUNCT american_scientific_journal-1069 105 51 27 27 NUM american_scientific_journal-1069 105 52 ) ) PUNCT american_scientific_journal-1069 105 53 with with ADP american_scientific_journal-1069 105 54 unitary unitary ADJ american_scientific_journal-1069 105 55 operator operator NOUN american_scientific_journal-1069 105 56 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž NOUN american_scientific_journal-1069 105 57 ) ) PUNCT american_scientific_journal-1069 105 58 = = SYM american_scientific_journal-1069 105 59 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 105 60 ) ) PUNCT american_scientific_journal-1069 105 61 ≑ ≑ PROPN american_scientific_journal-1069 105 62 π‘ˆπ‘œπ‘ π‘ˆπ‘œπ‘ PROPN american_scientific_journal-1069 105 63 , , PUNCT american_scientific_journal-1069 105 64 π‘ˆπ»π‘›π‘“(π›Όβ„Žπ‘ž)π‘ˆ π‘ˆπ»π‘›π‘“(π›Όβ„Žπ‘ž)π‘ˆ PROPN american_scientific_journal-1069 105 65 𝑇 𝑇 NOUN american_scientific_journal-1069 105 66 = = SYM american_scientific_journal-1069 105 67 πΌπ‘œπ‘ πΌπ‘œπ‘ PROPN american_scientific_journal-1069 105 68 with with ADP american_scientific_journal-1069 105 69 this this DET american_scientific_journal-1069 105 70 unitary unitary ADJ american_scientific_journal-1069 105 71 operator operator NOUN american_scientific_journal-1069 105 72 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž NOUN american_scientific_journal-1069 105 73 ) ) PUNCT american_scientific_journal-1069 105 74 = = SYM american_scientific_journal-1069 105 75 𝐻𝑛𝑓(π›Όβ„Žπ‘ž 𝐻𝑛𝑓(π›Όβ„Žπ‘ž NOUN american_scientific_journal-1069 105 76 ) ) PUNCT american_scientific_journal-1069 105 77 ≑ ≑ PROPN american_scientific_journal-1069 105 78 πΌπ‘œπ‘ πΌπ‘œπ‘ PROPN american_scientific_journal-1069 105 79 = = NOUN american_scientific_journal-1069 105 80 1 1 NUM american_scientific_journal-1069 105 81 𝐻𝑛𝑓(β„Žπ‘ž 𝐻𝑛𝑓(β„Žπ‘ž PROPN american_scientific_journal-1069 105 82 ) ) PUNCT american_scientific_journal-1069 105 83 converges converge NOUN american_scientific_journal-1069 105 84 to to ADP american_scientific_journal-1069 105 85 unity unity NOUN american_scientific_journal-1069 105 86 because because SCONJ american_scientific_journal-1069 105 87 a a DET american_scientific_journal-1069 105 88 new new ADJ american_scientific_journal-1069 105 89 space space NOUN american_scientific_journal-1069 105 90 is be AUX american_scientific_journal-1069 105 91 configured configure VERB american_scientific_journal-1069 105 92 for for ADP american_scientific_journal-1069 105 93 an an DET american_scientific_journal-1069 105 94 electron electron NOUN american_scientific_journal-1069 105 95 quanta quanta NOUN american_scientific_journal-1069 105 96 with with ADP american_scientific_journal-1069 105 97 twisting twisting NOUN american_scientific_journal-1069 105 98 and and CCONJ american_scientific_journal-1069 105 99 twigging twigging NOUN american_scientific_journal-1069 105 100 effects effect NOUN american_scientific_journal-1069 105 101 . . PUNCT american_scientific_journal-1069 106 1 gaussian gaussian ADJ american_scientific_journal-1069 106 2 like like ADP american_scientific_journal-1069 106 3 function function NOUN american_scientific_journal-1069 106 4 , , PUNCT american_scientific_journal-1069 106 5 .i.e .i.e PUNCT american_scientific_journal-1069 106 6 . . PUNCT american_scientific_journal-1069 106 7 , , PUNCT american_scientific_journal-1069 106 8 π‘’βˆ’ π‘’βˆ’ PRON american_scientific_journal-1069 106 9 𝛼2 𝛼2 VERB american_scientific_journal-1069 106 10 ( ( PUNCT american_scientific_journal-1069 106 11 β„Žπ‘ž β„Žπ‘ž PROPN american_scientific_journal-1069 106 12 √2 √2 NOUN american_scientific_journal-1069 106 13 ) ) PUNCT american_scientific_journal-1069 106 14 2 2 NUM american_scientific_journal-1069 106 15 ≑ ≑ PROPN american_scientific_journal-1069 106 16 π‘’βˆ’π›Ό π‘’βˆ’π›Ό PROPN american_scientific_journal-1069 106 17 2 2 NUM american_scientific_journal-1069 106 18 = = SYM american_scientific_journal-1069 106 19 1 1 NUM american_scientific_journal-1069 106 20 also also ADV american_scientific_journal-1069 106 21 converges converge VERB american_scientific_journal-1069 106 22 to to ADP american_scientific_journal-1069 106 23 unity unity NOUN american_scientific_journal-1069 106 24 for for ADP american_scientific_journal-1069 106 25 a a DET american_scientific_journal-1069 106 26 new new ADJ american_scientific_journal-1069 106 27 configured configure VERB american_scientific_journal-1069 106 28 space space NOUN american_scientific_journal-1069 106 29 . . PUNCT american_scientific_journal-1069 107 1 the the DET american_scientific_journal-1069 107 2 quanta quanta PROPN american_scientific_journal-1069 107 3 of of ADP american_scientific_journal-1069 107 4 electron electron NOUN american_scientific_journal-1069 107 5 initially initially ADV american_scientific_journal-1069 107 6 existed exist VERB american_scientific_journal-1069 107 7 in in ADP american_scientific_journal-1069 107 8 the the DET american_scientific_journal-1069 107 9 wiener wiener NOUN american_scientific_journal-1069 107 10 space space NOUN american_scientific_journal-1069 107 11 , , PUNCT american_scientific_journal-1069 107 12 but but CCONJ american_scientific_journal-1069 107 13 with with ADP american_scientific_journal-1069 107 14 twisting twisting NOUN american_scientific_journal-1069 107 15 and and CCONJ american_scientific_journal-1069 107 16 twigging twig VERB american_scientific_journal-1069 107 17 a a DET american_scientific_journal-1069 107 18 new new ADJ american_scientific_journal-1069 107 19 space space NOUN american_scientific_journal-1069 107 20 , , PUNCT american_scientific_journal-1069 107 21 .i.e .i.e PUNCT american_scientific_journal-1069 107 22 . . PROPN american_scientific_journal-1069 107 23 , , PUNCT american_scientific_journal-1069 107 24 a a DET american_scientific_journal-1069 107 25 wigner wigner ADJ american_scientific_journal-1069 107 26 space space NOUN american_scientific_journal-1069 107 27 is be AUX american_scientific_journal-1069 107 28 configured configure VERB american_scientific_journal-1069 107 29 . . PUNCT american_scientific_journal-1069 108 1 wiener wiener NOUN american_scientific_journal-1069 108 2 space space NOUN american_scientific_journal-1069 108 3 is be AUX american_scientific_journal-1069 108 4 transformed transform VERB american_scientific_journal-1069 108 5 in in ADP american_scientific_journal-1069 108 6 to to ADP american_scientific_journal-1069 108 7 a a DET american_scientific_journal-1069 108 8 wigner wigner ADJ american_scientific_journal-1069 108 9 space space NOUN american_scientific_journal-1069 108 10 which which PRON american_scientific_journal-1069 108 11 is be AUX american_scientific_journal-1069 108 12 a a DET american_scientific_journal-1069 108 13 reciprocal reciprocal ADJ american_scientific_journal-1069 108 14 space space NOUN american_scientific_journal-1069 108 15 . . PUNCT american_scientific_journal-1069 109 1 the the DET american_scientific_journal-1069 109 2 reciprocity reciprocity NOUN american_scientific_journal-1069 109 3 is be AUX american_scientific_journal-1069 109 4 a a DET american_scientific_journal-1069 109 5 manifestation manifestation NOUN american_scientific_journal-1069 109 6 of of ADP american_scientific_journal-1069 109 7 hyperbolic hyperbolic ADJ american_scientific_journal-1069 109 8 space space NOUN american_scientific_journal-1069 109 9 which which PRON american_scientific_journal-1069 109 10 depends depend VERB american_scientific_journal-1069 109 11 only only ADV american_scientific_journal-1069 109 12 on on ADP american_scientific_journal-1069 109 13 operators operator NOUN american_scientific_journal-1069 109 14 . . PUNCT american_scientific_journal-1069 110 1 thus thus ADV american_scientific_journal-1069 110 2 𝑒𝑖𝑛𝑓𝛼 𝑒𝑖𝑛𝑓𝛼 VERB american_scientific_journal-1069 110 3 = = PUNCT american_scientific_journal-1069 110 4 ( ( PUNCT american_scientific_journal-1069 110 5 𝛼 𝛼 X american_scientific_journal-1069 110 6 βˆšπœ‹2 βˆšπœ‹2 PROPN american_scientific_journal-1069 110 7 𝑛𝑓 𝑛𝑓 NOUN american_scientific_journal-1069 110 8 ) ) PUNCT american_scientific_journal-1069 110 9 1 1 NUM american_scientific_journal-1069 110 10 2 2 NUM american_scientific_journal-1069 110 11 ( ( PUNCT american_scientific_journal-1069 110 12 29 29 NUM american_scientific_journal-1069 110 13 ) ) PUNCT american_scientific_journal-1069 110 14 this this DET american_scientific_journal-1069 110 15 reciprocal reciprocal ADJ american_scientific_journal-1069 110 16 space space NOUN american_scientific_journal-1069 110 17 is be AUX american_scientific_journal-1069 110 18 a a DET american_scientific_journal-1069 110 19 manifestation manifestation NOUN american_scientific_journal-1069 110 20 of of ADP american_scientific_journal-1069 110 21 reflection reflection NOUN american_scientific_journal-1069 110 22 under under ADP american_scientific_journal-1069 110 23 inversion inversion NOUN american_scientific_journal-1069 110 24 symmetry symmetry NOUN american_scientific_journal-1069 110 25 ( ( PUNCT american_scientific_journal-1069 110 26 orthogonality orthogonality NOUN american_scientific_journal-1069 110 27 is be AUX american_scientific_journal-1069 110 28 maintained maintain VERB american_scientific_journal-1069 110 29 ) ) PUNCT american_scientific_journal-1069 110 30 . . PUNCT american_scientific_journal-1069 111 1 3 3 X american_scientific_journal-1069 111 2 . . X american_scientific_journal-1069 111 3 conclusion conclusion NOUN american_scientific_journal-1069 111 4 we we PRON american_scientific_journal-1069 111 5 presented present VERB american_scientific_journal-1069 111 6 a a DET american_scientific_journal-1069 111 7 new new ADJ american_scientific_journal-1069 111 8 thesis thesis NOUN american_scientific_journal-1069 111 9 about about ADP american_scientific_journal-1069 111 10 the the DET american_scientific_journal-1069 111 11 morphology morphology NOUN american_scientific_journal-1069 111 12 of of ADP american_scientific_journal-1069 111 13 a a DET american_scientific_journal-1069 111 14 bounded bounded ADJ american_scientific_journal-1069 111 15 electron electron NOUN american_scientific_journal-1069 111 16 which which PRON american_scientific_journal-1069 111 17 suffers suffer VERB american_scientific_journal-1069 111 18 momentum momentum NOUN american_scientific_journal-1069 111 19 impact impact NOUN american_scientific_journal-1069 111 20 , , PUNCT american_scientific_journal-1069 111 21 as as ADP american_scientific_journal-1069 111 22 a a DET american_scientific_journal-1069 111 23 consequence consequence NOUN american_scientific_journal-1069 111 24 of of ADP american_scientific_journal-1069 111 25 which which PRON american_scientific_journal-1069 111 26 the the DET american_scientific_journal-1069 111 27 electron electron NOUN american_scientific_journal-1069 111 28 quanta quanta PROPN american_scientific_journal-1069 111 29 is be AUX american_scientific_journal-1069 111 30 first first ADV american_scientific_journal-1069 111 31 stretched stretch VERB american_scientific_journal-1069 111 32 twisted twisted ADJ american_scientific_journal-1069 111 33 and and CCONJ american_scientific_journal-1069 111 34 then then ADV american_scientific_journal-1069 111 35 twigged twig VERB american_scientific_journal-1069 111 36 . . PUNCT american_scientific_journal-1069 112 1 this this DET american_scientific_journal-1069 112 2 behavior behavior NOUN american_scientific_journal-1069 112 3 different different ADJ american_scientific_journal-1069 112 4 from from ADP american_scientific_journal-1069 112 5 , , PUNCT american_scientific_journal-1069 112 6 compton compton PROPN american_scientific_journal-1069 112 7 and and CCONJ american_scientific_journal-1069 112 8 photoelectric photoelectric ADJ american_scientific_journal-1069 112 9 effects effect NOUN american_scientific_journal-1069 112 10 , , PUNCT american_scientific_journal-1069 112 11 respectively respectively ADV american_scientific_journal-1069 112 12 . . PUNCT american_scientific_journal-1069 113 1 such such ADJ american_scientific_journal-1069 113 2 morphology morphology NOUN american_scientific_journal-1069 113 3 of of ADP american_scientific_journal-1069 113 4 a a DET american_scientific_journal-1069 113 5 bounded bounded PROPN american_scientific_journal-1069 113 6 electron electron NOUN american_scientific_journal-1069 113 7 quanta quanta PROPN american_scientific_journal-1069 113 8 is be AUX american_scientific_journal-1069 113 9 termed term VERB american_scientific_journal-1069 113 10 as as ADP american_scientific_journal-1069 113 11 quasi quasi ADJ american_scientific_journal-1069 113 12 particle particle NOUN american_scientific_journal-1069 113 13 . . PUNCT american_scientific_journal-1069 114 1 this this DET american_scientific_journal-1069 114 2 morphology morphology NOUN american_scientific_journal-1069 114 3 of of ADP american_scientific_journal-1069 114 4 an an DET american_scientific_journal-1069 114 5 electron electron NOUN american_scientific_journal-1069 114 6 can can AUX american_scientific_journal-1069 114 7 explained explain VERB american_scientific_journal-1069 114 8 fractional fractional ADJ american_scientific_journal-1069 114 9 charge charge NOUN american_scientific_journal-1069 114 10 quantization quantization NOUN american_scientific_journal-1069 114 11 , , PUNCT american_scientific_journal-1069 114 12 quantum quantum ADJ american_scientific_journal-1069 114 13 hall hall NOUN american_scientific_journal-1069 114 14 effect effect NOUN american_scientific_journal-1069 114 15 , , PUNCT american_scientific_journal-1069 114 16 giant giant ADJ american_scientific_journal-1069 114 17 magneto magneto NOUN american_scientific_journal-1069 114 18 resistance resistance NOUN american_scientific_journal-1069 114 19 and and CCONJ american_scientific_journal-1069 114 20 quantum quantum NOUN american_scientific_journal-1069 114 21 capacitance capacitance NOUN american_scientific_journal-1069 114 22 . . PUNCT american_scientific_journal-1069 115 1 reference reference NOUN american_scientific_journal-1069 115 2 [ [ X american_scientific_journal-1069 115 3 1 1 NUM american_scientific_journal-1069 115 4 ] ] PUNCT american_scientific_journal-1069 115 5 m. m. NOUN american_scientific_journal-1069 115 6 gormani gormani NOUN american_scientific_journal-1069 115 7 , , PUNCT american_scientific_journal-1069 115 8 fazlur fazlur PROPN american_scientific_journal-1069 115 9 rehman rehman PROPN american_scientific_journal-1069 115 10 , , PUNCT american_scientific_journal-1069 115 11 syed syed PROPN american_scientific_journal-1069 115 12 mohsin mohsin PROPN american_scientific_journal-1069 115 13 raza raza PROPN american_scientific_journal-1069 115 14 , , PUNCT american_scientific_journal-1069 115 15 and and CCONJ american_scientific_journal-1069 115 16 m.a m.a PROPN american_scientific_journal-1069 115 17 . . PROPN american_scientific_journal-1069 115 18 ahmed,β€œquantum ahmed,β€œquantum PROPN american_scientific_journal-1069 115 19 behaviour behaviour NOUN american_scientific_journal-1069 115 20 of of ADP american_scientific_journal-1069 115 21 dielectric dielectric NOUN american_scientific_journal-1069 115 22 in in ADP american_scientific_journal-1069 115 23 dolomite dolomite NOUN american_scientific_journal-1069 115 24 of of ADP american_scientific_journal-1069 115 25 balochistan balochistan PROPN american_scientific_journal-1069 115 26 , , PUNCT american_scientific_journal-1069 115 27 pakistan pakistan PROPN american_scientific_journal-1069 115 28 , , PUNCT american_scientific_journal-1069 115 29 ” " PUNCT american_scientific_journal-1069 115 30 jr.chem.soc.pak jr.chem.soc.pak NOUN american_scientific_journal-1069 115 31 , , PUNCT american_scientific_journal-1069 115 32 vol vol NOUN american_scientific_journal-1069 115 33 . . PROPN american_scientific_journal-1069 115 34 28 28 NUM american_scientific_journal-1069 115 35 , , PUNCT american_scientific_journal-1069 115 36 no no INTJ american_scientific_journal-1069 115 37 . . NOUN american_scientific_journal-1069 115 38 5 5 NUM american_scientific_journal-1069 115 39 , , PUNCT american_scientific_journal-1069 115 40 pp pp ADJ american_scientific_journal-1069 115 41 . . PUNCT american_scientific_journal-1069 116 1 414–416 414–416 NUM american_scientific_journal-1069 116 2 , , PUNCT american_scientific_journal-1069 116 3 2006 2006 NUM american_scientific_journal-1069 116 4 . . PUNCT american_scientific_journal-1069 117 1 american american ADJ american_scientific_journal-1069 117 2 scientific scientific ADJ american_scientific_journal-1069 117 3 research research PROPN american_scientific_journal-1069 117 4 journal journal NOUN american_scientific_journal-1069 117 5 for for ADP american_scientific_journal-1069 117 6 engineering engineering NOUN american_scientific_journal-1069 117 7 , , PUNCT american_scientific_journal-1069 117 8 technology technology NOUN american_scientific_journal-1069 117 9 , , PUNCT american_scientific_journal-1069 117 10 and and CCONJ american_scientific_journal-1069 117 11 sciences science NOUN american_scientific_journal-1069 117 12 ( ( PUNCT american_scientific_journal-1069 117 13 asrjets asrjet NOUN american_scientific_journal-1069 117 14 ) ) PUNCT american_scientific_journal-1069 117 15 ( ( PUNCT american_scientific_journal-1069 117 16 2015 2015 NUM american_scientific_journal-1069 117 17 ) ) PUNCT american_scientific_journal-1069 117 18 volume volume NOUN american_scientific_journal-1069 117 19 14 14 NUM american_scientific_journal-1069 117 20 , , PUNCT american_scientific_journal-1069 117 21 no no DET american_scientific_journal-1069 117 22 2 2 NUM american_scientific_journal-1069 117 23 , , PUNCT american_scientific_journal-1069 117 24 pp pp ADV american_scientific_journal-1069 117 25 265 265 NUM american_scientific_journal-1069 117 26 - - SYM american_scientific_journal-1069 117 27 272 272 NUM american_scientific_journal-1069 117 28 272 272 NUM american_scientific_journal-1069 118 1 [ [ X american_scientific_journal-1069 118 2 2 2 X american_scientific_journal-1069 118 3 ] ] PUNCT american_scientific_journal-1069 118 4 t. t. PROPN american_scientific_journal-1069 118 5 samina samina PROPN american_scientific_journal-1069 118 6 yousaf yousaf PROPN american_scientific_journal-1069 118 7 , , PUNCT american_scientific_journal-1069 118 8 syed syed PROPN american_scientific_journal-1069 118 9 mohsin mohsin PROPN american_scientific_journal-1069 118 10 raza raza PROPN american_scientific_journal-1069 118 11 , , PUNCT american_scientific_journal-1069 118 12 masoom masoom NOUN american_scientific_journal-1069 118 13 yasinzi yasinzi PROPN american_scientific_journal-1069 118 14 , , PUNCT american_scientific_journal-1069 118 15 and and CCONJ american_scientific_journal-1069 118 16 mujeeb mujeeb PROPN american_scientific_journal-1069 118 17 ur ur PROPN american_scientific_journal-1069 118 18 rehman rehman PROPN american_scientific_journal-1069 118 19 , , PUNCT american_scientific_journal-1069 118 20 β€œ " PUNCT american_scientific_journal-1069 118 21 absorption absorption NOUN american_scientific_journal-1069 118 22 of of ADP american_scientific_journal-1069 118 23 radiant radiant ADJ american_scientific_journal-1069 118 24 energy energy NOUN american_scientific_journal-1069 118 25 in in ADP american_scientific_journal-1069 118 26 water water NOUN american_scientific_journal-1069 118 27 : : PUNCT american_scientific_journal-1069 118 28 a a DET american_scientific_journal-1069 118 29 new new ADJ american_scientific_journal-1069 118 30 conjecture conjecture NOUN american_scientific_journal-1069 118 31 and and CCONJ american_scientific_journal-1069 118 32 theory theory NOUN american_scientific_journal-1069 118 33 of of ADP american_scientific_journal-1069 118 34 charge charge NOUN american_scientific_journal-1069 118 35 quantization quantization NOUN american_scientific_journal-1069 118 36 in in ADP american_scientific_journal-1069 118 37 chromotized chromotize VERB american_scientific_journal-1069 118 38 water water NOUN american_scientific_journal-1069 118 39 samples sample NOUN american_scientific_journal-1069 118 40 , , PUNCT american_scientific_journal-1069 118 41 ” " PUNCT american_scientific_journal-1069 118 42 science science NOUN american_scientific_journal-1069 118 43 international international PROPN american_scientific_journal-1069 118 44 lahore.pak lahore.pak PROPN american_scientific_journal-1069 118 45 , , PUNCT american_scientific_journal-1069 118 46 vol vol NOUN american_scientific_journal-1069 118 47 . . PROPN american_scientific_journal-1069 118 48 20 20 NUM american_scientific_journal-1069 118 49 , , PUNCT american_scientific_journal-1069 118 50 no no INTJ american_scientific_journal-1069 118 51 . . NOUN american_scientific_journal-1069 118 52 3 3 NUM american_scientific_journal-1069 118 53 , , PUNCT american_scientific_journal-1069 118 54 pp pp ADJ american_scientific_journal-1069 118 55 . . PUNCT american_scientific_journal-1069 119 1 189–195 189–195 NUM american_scientific_journal-1069 119 2 , , PUNCT american_scientific_journal-1069 119 3 2008 2008 NUM american_scientific_journal-1069 119 4 . . PUNCT american_scientific_journal-1069 120 1 [ [ X american_scientific_journal-1069 120 2 3 3 X american_scientific_journal-1069 120 3 ] ] PUNCT american_scientific_journal-1069 120 4 t. t. PROPN american_scientific_journal-1069 120 5 samina samina PROPN american_scientific_journal-1069 120 6 yousaf yousaf PROPN american_scientific_journal-1069 120 7 azeemi azeemi PROPN american_scientific_journal-1069 120 8 , , PUNCT american_scientific_journal-1069 120 9 syed syed PROPN american_scientific_journal-1069 120 10 mohsin mohsin PROPN american_scientific_journal-1069 120 11 raza raza PROPN american_scientific_journal-1069 120 12 , , PUNCT american_scientific_journal-1069 120 13 and and CCONJ american_scientific_journal-1069 120 14 m. m. NOUN american_scientific_journal-1069 120 15 ashfaq ashfaq PROPN american_scientific_journal-1069 120 16 ahmed ahmed PROPN american_scientific_journal-1069 120 17 , , PUNCT american_scientific_journal-1069 120 18 β€œ " PUNCT american_scientific_journal-1069 120 19 newly newly ADV american_scientific_journal-1069 120 20 developed develop VERB american_scientific_journal-1069 120 21 recursive recursive ADJ american_scientific_journal-1069 120 22 relationship relationship NOUN american_scientific_journal-1069 120 23 for for ADP american_scientific_journal-1069 120 24 fractional fractional ADJ american_scientific_journal-1069 120 25 quantum quantum ADJ american_scientific_journal-1069 120 26 states state NOUN american_scientific_journal-1069 120 27 and and CCONJ american_scientific_journal-1069 120 28 associated associate VERB american_scientific_journal-1069 120 29 energy energy NOUN american_scientific_journal-1069 120 30 eigen eigen PROPN american_scientific_journal-1069 120 31 values value NOUN american_scientific_journal-1069 120 32 , , PUNCT american_scientific_journal-1069 120 33 ” " PUNCT american_scientific_journal-1069 120 34 science science NOUN american_scientific_journal-1069 120 35 international international PROPN american_scientific_journal-1069 120 36 lahore.pak lahore.pak PROPN american_scientific_journal-1069 120 37 , , PUNCT american_scientific_journal-1069 120 38 vol vol NOUN american_scientific_journal-1069 120 39 . . PROPN american_scientific_journal-1069 120 40 20 20 NUM american_scientific_journal-1069 120 41 , , PUNCT american_scientific_journal-1069 120 42 no no INTJ american_scientific_journal-1069 120 43 . . NOUN american_scientific_journal-1069 120 44 4 4 NUM american_scientific_journal-1069 120 45 , , PUNCT american_scientific_journal-1069 120 46 pp pp ADJ american_scientific_journal-1069 120 47 . . PUNCT american_scientific_journal-1069 121 1 255–260 255–260 NUM american_scientific_journal-1069 121 2 , , PUNCT american_scientific_journal-1069 121 3 2008 2008 NUM american_scientific_journal-1069 121 4 . . PUNCT american_scientific_journal-1069 122 1 [ [ X american_scientific_journal-1069 122 2 4 4 X american_scientific_journal-1069 122 3 ] ] PUNCT american_scientific_journal-1069 122 4 fazlur fazlur PROPN american_scientific_journal-1069 122 5 rehman rehman PROPN american_scientific_journal-1069 122 6 , , PUNCT american_scientific_journal-1069 122 7 m. m. NOUN american_scientific_journal-1069 122 8 ashfaq ashfaq PROPN american_scientific_journal-1069 122 9 ahmed ahmed PROPN american_scientific_journal-1069 122 10 , , PUNCT american_scientific_journal-1069 122 11 and and CCONJ american_scientific_journal-1069 122 12 syed syed PROPN american_scientific_journal-1069 122 13 mohsin mohsin PROPN american_scientific_journal-1069 122 14 raza raza PROPN american_scientific_journal-1069 122 15 , , PUNCT american_scientific_journal-1069 122 16 β€œ " PUNCT american_scientific_journal-1069 122 17 quantum quantum ADJ american_scientific_journal-1069 122 18 theory theory NOUN american_scientific_journal-1069 122 19 of of ADP american_scientific_journal-1069 122 20 dielectric dielectric NOUN american_scientific_journal-1069 122 21 and and CCONJ american_scientific_journal-1069 122 22 its its PRON american_scientific_journal-1069 122 23 applications application NOUN american_scientific_journal-1069 122 24 to to ADP american_scientific_journal-1069 122 25 dolomite dolomite NOUN american_scientific_journal-1069 122 26 of of ADP american_scientific_journal-1069 122 27 balochistan balochistan PROPN american_scientific_journal-1069 122 28 , , PUNCT american_scientific_journal-1069 122 29 pakistan pakistan PROPN american_scientific_journal-1069 122 30 , , PUNCT american_scientific_journal-1069 122 31 ” " PUNCT american_scientific_journal-1069 122 32 science science NOUN american_scientific_journal-1069 122 33 international international ADJ american_scientific_journal-1069 122 34 lahore lahore PROPN american_scientific_journal-1069 122 35 . . PUNCT american_scientific_journal-1069 123 1 pak pak NOUN american_scientific_journal-1069 123 2 , , PUNCT american_scientific_journal-1069 123 3 vol vol NOUN american_scientific_journal-1069 123 4 . . PROPN american_scientific_journal-1069 123 5 21 21 NUM american_scientific_journal-1069 123 6 , , PUNCT american_scientific_journal-1069 123 7 no no INTJ american_scientific_journal-1069 123 8 . . NOUN american_scientific_journal-1069 123 9 1 1 NUM american_scientific_journal-1069 123 10 , , PUNCT american_scientific_journal-1069 123 11 pp pp ADJ american_scientific_journal-1069 123 12 . . PUNCT american_scientific_journal-1069 123 13 29 29 NUM american_scientific_journal-1069 123 14 – – PUNCT american_scientific_journal-1069 123 15 32 32 NUM american_scientific_journal-1069 123 16 , , PUNCT american_scientific_journal-1069 123 17 2009 2009 NUM american_scientific_journal-1069 123 18 . . PUNCT american_scientific_journal-1069 124 1 [ [ X american_scientific_journal-1069 124 2 5 5 NUM american_scientific_journal-1069 124 3 ] ] X american_scientific_journal-1069 124 4 l.b l.b PROPN american_scientific_journal-1069 124 5 . . PROPN american_scientific_journal-1069 124 6 almeida almeida PROPN american_scientific_journal-1069 124 7 , , PUNCT american_scientific_journal-1069 124 8 β€œ " PUNCT american_scientific_journal-1069 124 9 the the DET american_scientific_journal-1069 124 10 fractional fractional ADJ american_scientific_journal-1069 124 11 fourier fourier NOUN american_scientific_journal-1069 124 12 transform transform NOUN american_scientific_journal-1069 124 13 and and CCONJ american_scientific_journal-1069 124 14 time time NOUN american_scientific_journal-1069 124 15 - - PUNCT american_scientific_journal-1069 124 16 frequency frequency NOUN american_scientific_journal-1069 124 17 representations representation NOUN american_scientific_journal-1069 124 18 , , PUNCT american_scientific_journal-1069 124 19 ” " PUNCT american_scientific_journal-1069 124 20 ieee ieee NOUN american_scientific_journal-1069 124 21 transactions transaction NOUN american_scientific_journal-1069 124 22 on on ADP american_scientific_journal-1069 124 23 signal signal ADJ american_scientific_journal-1069 124 24 processing processing NOUN american_scientific_journal-1069 124 25 , , PUNCT american_scientific_journal-1069 124 26 vol vol NOUN american_scientific_journal-1069 124 27 . . PROPN american_scientific_journal-1069 124 28 42 42 NUM american_scientific_journal-1069 124 29 , , PUNCT american_scientific_journal-1069 124 30 no.11 no.11 NOUN american_scientific_journal-1069 124 31 , , PUNCT american_scientific_journal-1069 124 32 pp pp ADJ american_scientific_journal-1069 124 33 . . PUNCT american_scientific_journal-1069 124 34 3084–3091 3084–3091 NUM american_scientific_journal-1069 124 35 , , PUNCT american_scientific_journal-1069 124 36 1994 1994 NUM american_scientific_journal-1069 124 37 . . PUNCT american_scientific_journal-1069 125 1 [ [ X american_scientific_journal-1069 125 2 6 6 NUM american_scientific_journal-1069 125 3 ] ] X american_scientific_journal-1069 125 4 d.j d.j PROPN american_scientific_journal-1069 125 5 . . PROPN american_scientific_journal-1069 125 6 griffths griffths PROPN american_scientific_journal-1069 125 7 , , PUNCT american_scientific_journal-1069 125 8 introduction introduction NOUN american_scientific_journal-1069 125 9 to to ADP american_scientific_journal-1069 125 10 quantum quantum ADJ american_scientific_journal-1069 125 11 mechanics mechanic NOUN american_scientific_journal-1069 125 12 , , PUNCT american_scientific_journal-1069 125 13 2 2 NUM american_scientific_journal-1069 125 14 / / SYM american_scientific_journal-1069 125 15 e e NOUN american_scientific_journal-1069 125 16 , , PUNCT american_scientific_journal-1069 125 17 pearson pearson NOUN american_scientific_journal-1069 125 18 education,2005 education,2005 NOUN american_scientific_journal-1069 125 19 . . PUNCT american_scientific_journal-1069 126 1 [ [ X american_scientific_journal-1069 126 2 7 7 NUM american_scientific_journal-1069 126 3 ] ] X american_scientific_journal-1069 126 4 r.l r.l PROPN american_scientific_journal-1069 126 5 . . PROPN american_scientific_journal-1069 126 6 liboff liboff PROPN american_scientific_journal-1069 126 7 , , PUNCT american_scientific_journal-1069 126 8 introductory introductory ADJ american_scientific_journal-1069 126 9 quantum quantum NOUN american_scientific_journal-1069 126 10 mechanics mechanic NOUN american_scientific_journal-1069 126 11 , , PUNCT american_scientific_journal-1069 126 12 4 4 NUM american_scientific_journal-1069 126 13 / / SYM american_scientific_journal-1069 126 14 e e NOUN american_scientific_journal-1069 126 15 , , PUNCT american_scientific_journal-1069 126 16 pearson pearson PROPN american_scientific_journal-1069 126 17 education education PROPN american_scientific_journal-1069 126 18 , , PUNCT american_scientific_journal-1069 126 19 2003 2003 NUM american_scientific_journal-1069 126 20 . . PUNCT american_scientific_journal-1069 127 1 [ [ X american_scientific_journal-1069 127 2 8 8 NUM american_scientific_journal-1069 127 3 ] ] PUNCT american_scientific_journal-1069 127 4 s. s. PROPN american_scientific_journal-1069 127 5 gasiorowicz gasiorowicz PROPN american_scientific_journal-1069 127 6 , , PUNCT american_scientific_journal-1069 127 7 quantum quantum NOUN american_scientific_journal-1069 127 8 mechanics mechanic NOUN american_scientific_journal-1069 127 9 , , PUNCT american_scientific_journal-1069 127 10 3 3 NUM american_scientific_journal-1069 127 11 / / SYM american_scientific_journal-1069 127 12 e e NOUN american_scientific_journal-1069 127 13 , , PUNCT american_scientific_journal-1069 127 14 john john PROPN american_scientific_journal-1069 127 15 wiley wiley PROPN american_scientific_journal-1069 127 16 , , PUNCT american_scientific_journal-1069 127 17 new new PROPN american_scientific_journal-1069 127 18 york york PROPN american_scientific_journal-1069 127 19 , , PUNCT american_scientific_journal-1069 127 20 2003 2003 NUM