Acta Polytechnica doi:10.14311/AP.2014.54.0149 Acta Polytechnica 54(2):149–155, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap LAPLACE-RUNGE-LENZ VECTOR IN QUANTUM MECHANICS IN NONCOMMUTATIVE SPACE Peter Prešnajder∗, Veronika Gáliková, Samuel Kováčik Commenius University Bratislava, Faculty of Mathematics, Physics and Informatics ∗ corresponding author: presnajder.peter@gmail.com Abstract. The object under scrutiny is the dynamical symmetry connected with conservation of the Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of noncommutative quantum mechanics (NCQM). The considered noncommutative configuration space has such a “fuzzy” structure that the rotational invariance is not spoilt. An analogy with the LRL vector in the NCQM is brought to provide our results and also a comparison with the standard QM predictions. Keywords: noncommutative space, Coulomb-Kepler problem, symmetry. 1. Introduction Our main goal we are after is to investigate the exis- tence of dynamical symmetry of the Coulomb-Kepler problem in the quantum mechanics in noncommu- tative space, and possibly to find the generalization of the so-called Laplace-Runge-Lenz (LRL) vector for this case. Before actually starting, we briefly look at the history of the LRL vector. We call this Laplace-Runge-Lenz vector, as it is commonly named nowadays, but as far as we know, the first ones to make a mention of it were Jakob Hermann and Johann Bernoulli in the letters they exchanged in 1710, see [1], [2]. So the name “Hermann-Bernoulli vector” would be more proper. Much later, in 1799 the vector was rediscovered by Laplace in his Celestial Mechanics [3]. Then it appeared in a popular German textbook on vectors by C. Runge [4], which was referenced by W. Lenz in his paper on the (old) quantum mechanical treatment of the Kepler problem or hydrogen atom [5]. Now back to physics. The Coulomb-Kepler problem is all about the motion of a particle in a field of a central force proportional to (r−2). The corresponding Newton equation for a body of mass m reads m~̇v = −q ~r r3 . (1) Here q denotes a constant which specifies the magni- tude of the force applied. The system with a central force definitely is rotationally symmetric and the or- bital momentum, ~L = m~r ×~v, (2) is conserved in any central field. However, as to the symmetries, more has to be said in this case, due to the fact that not only is the force central, but in addition it has the inverse square dependence of the distance. Besides the components of ~L, the Coulomb- Kepler problem has three additional integrals of mo- tion, namely those represented by the conserved LRL vector, ~A = ~L×~v + q ~r r . (3) When the motion of a planet around the Sun is con- sidered, the conservation of the given quantity has to do with the constant eccentricity of the orbit and the position of the perihelion. Another well-known system characterized by Cou- lomb potential is the hydrogen atom. Obviously a need for the use of quantum mechanics arises here. There are, however, several ways to address the issue. In 1926 Wolfgang Pauli published his paper on the sub- ject [6]. He used the LRL vector to find the spectrum of a hydrogen atom using modern quantum mechan- ics and the hidden dynamical symmetry of the prob- lem, without knowledge of the explicit solution of the Schrödinger equation. It turned out that the LRL vector can be found among the hermitian operators acting in the Hilbert space considered in an almost complete analogy with the classical case, the only sub- tlety to deal with being the fact that the cross product needs to be properly symmetrized, resulting in A QM k = 1 2 εijk(Livj + vjLi) + q xk r , (4) where vj = −i~m∂j stands for velocity operator. Im- portantly the operators Li and Ai commute with the Hamiltonian, i.e. are conserved with respect to time evolution, and as to their mutual commutation re- lations, there would be pretty good views of them forming a closed algebra so(4), if it were not for the commutator [Ai,Aj] ∝ LkH. However, restricting ourselves to the subspace HE spanned by eigenvectors of H corresponding to the eigenvalue E, H can be re- placed by its eigenvalue, which is a c-number. Besides enabling the algebra to close, we have also dragged the energy into the very definition of the algebra gen- erators. This, together with the theory related to the relevant Casimir operators, has a direct impact on the H-atom energy spectrum. In this way Pauli found the correct formulas for the hydrogen atom spectrum even before Schrödinger. Now the search for the analogy in noncommutative quantum mechanics (NCQM) begins. The rest of this paper is organized in the following way: In Section 2 149 http://dx.doi.org/10.14311/AP.2014.54.0149 http://ojs.cvut.cz/ojs/index.php/ap P. Presnajder, V. Galikova, S. Kovacik Acta Polytechnica we give a brief introduction to the quantum mechan- ics in a spherically symmetric noncommutative space. The Hilbert space of wave functions in NC space and the NC generalizations of important operators (Hamil- tonian, angular momentum, coordinate and velocity) are introduced. When we use these, a generalization of the dynamical symmetry of the Coulomb-Kepler problem in NCQM is presented in Section 3. Section 4 provides conclusions. We skip all the detailed and lengthy calculations that can be found in our recently published paper [7]. 2. Basics of noncommutative quantum mechanics To begin, it should be made clear how to introduce some uncertainty principle into the configuration space for the Coulomb problem without spoiling the key feature which allows us to find the exact solution - the rotational invariance. The uncertainty is expressed as nontrivial commu- tation relations for the NC analogs of the former Cartesian coordinates (obviously we have to abandon c-numbers), defined in a spherically symmetrical way. The coordinates in the NC configuration space R3λ are realized in terms of 2 pairs of boson annihila- tion and creation operators aα, a+α , α = 1, 2, satisfying [aα,a+β ] = δαβ, [aα,aβ] = [a + α ,a + β ] = 0. (5) They act in an auxiliary Fock space F spanned by the normalized vectors |n1,n2〉 = (a+1 ) n1 (a+2 ) n2 √ n1! n2! |0〉. (6) The normalized vacuum state is denoted as |0〉 ≡ |0, 0〉. The noncommutative coordinates xj, j = 1, 2, 3, and the NC analog of the Euclidean distance from the origin are given as xj = λa+σja ≡ λσ j αβa + αaβ, r = λ(N + 1), (7) where σj are the Pauli matrices, N = a+αaα is the number operator in the Fock space, and λ is a length parameter. Its magnitude is not fixed within our model. Naturally it has a connection with the small- est distance relevant in the given noncommutative configuration space denoted as R3λ. The key rotation- ally invariant relations in the theory are [xi,xj] = 2iλεijkxk, [xi,r] = 0, r2−x2j = λ 2. (8) The first equation defines a noncommutative or fuzzy sphere that appeared a long time ago in various con- texts [8], e.g., quantization on a sphere = nonflat phase space, a simple model of NC manifolds. All these models considered a single fuzzy sphere. Here we deal with an infinite sequence of fuzzy spheres dynam- ically via an NC analog of the (radial) Schrödinger equation that is introduced below. We remark that while the first equation in (8) is postulated, the other two follow from the construction of R3λ. While constructing NC quantum mechanics, firstly we have to decide on a Hilbert space Hλ of states (see also [9]). The suitable choice is a linear space of normally-ordered analytic functions containing the same number of creation and annihilation operators: Ψ = ∑ Cm1m2n1n2 (a + 1 ) m1 (a+2 ) m2an11 a n2 2 , (9) which possess finite weighted Hilbert-Schmidt norm ‖Ψ‖2 = 4πλ2Tr[rΨ†Ψ]. (10) (The summation in (9) is over nonnegative integers satisfying m1 + m2 = n1 + n2.) Our NC wave functions Ψ are themselves operators on the Fock space mentioned above, so the relations which occur are to be looked at as operator equalities (we could put |n1,n2〉 on both sides of every such equation). We can move to the definition of the operators acting on NC wave functions Ψ ∈ Hλ. To avoid potential confusion, we have decided to leave the NC coordinates and the NC wave functions Ψ (operators on the Fock space) as they are, and to denote the operators acting on Ψ with a hat from now on. The generators of rotations in Hλ, orbital mo- mentum operators, are defined as L̂jΨ = 1 2λ (xjΨ − Ψxj), j = 1, 2, 3. (11) They are hermitian and obey the usual commutation relations [L̂i, L̂j]Ψ ≡ (L̂iL̂j − L̂jL̂i)Ψ = iεijkL̂kΨ. (12) The standard eigenfunctions Ψjm, j = 0, 1, 2, . . . , m = −j, . . . , +j, satisfying L̂2i Ψjm = j(j + 1)Ψjm, L̂3Ψjm = mΨjm, (13) are given by Ψjm = ∑ (jm) (a+1 ) m1 (a+2 ) m2 m1! m2! Rj(r) an11 (−a2) n2 n1! n2! . (14) The summation goes over all nonnegative integers that satisfy m1 + m2 = n1 + n2 = j,m1 −m2 −n1 + n2 = 2m. For any fixed Rj(r) equation (14) defines a represen- tation space for a unitary irreducible representation with spin j. The NC analog of the usual Laplace operator is ∆̂λΨ = − 1 λr [â+α , [âα, Ψ]] = − 1 λ2(N + 1) [â+α , [âα, Ψ]]. (15) 150 vol. 54 no. 2/2014 LRL Vector in Quantum Mechanics in Noncommutative Space As to the operator Û, the NC analog of the central potential, it is defined simply as the multiplication of the NC wave function by U(r): (ÛΨ)(r) = U(r)Ψ = ΨU(r). (16) Since any term of Ψ ∈Hλ consists of the same number of creation and annihilation operators (any commuta- tor of such a term with r is zero), there is no difference between left and right multiplication by U(r). So finally here is the definition of our NC Hamil- tonian: ĤΨ = ~2 2mλr [â+α , [âα, Ψ]] − q r Ψ. (17) The coordinate operator x̂j acts on Ψ symmet- rically as x̂jΨ = 1 2 (xjΨ + Ψxj). (18) As to the velocity operator, clearly it should be in some relation with the evolution of the coordinate operator. The NC analog of the time derivative is proportional to the commutator of the quantity con- sidered with H; so the components of the velocity operator are given by V̂jΨ = −i[x̂j,Ĥ]Ψ. (19) Both sets of NC observables, V̂j and x̂j, have been introduced in [10]. As we see below, they are well adapted to the construction of the NC analog of the LRL vector. Based on what has been briefly summarized above, the NC analog of the Schrödinger equation with the Coulomb potential in R3λ can be postulated: ~2 2mλr [â+α , [âα, Ψ]] − q r Ψ = EΨ. (20) To avoid overloading the formulas, we usually set m = 1, ~ = 1 below. 3. Dynamical symmetry in NCQM It is time to address the NCQM version of the Coulomb-Kepler problem. Our task is to find sen- sible analogs of the three components Ai of the LRL vector in such a way that all the requirements regard- ing commutation relations are met (the commutator with the Hamiltonian has to be zero because of the conservation law and relations among all components of ~A and ~L are supposed to correspond to the rele- vant symmetry). Recall the subtlety which had to be taken into account when the standard QM version of the LRL vector had been built based on the classical model. The cross product of velocity and angular momentum needed symmetrization due to their non- vanishing commutator. The NC operators that we are going to use when constructing the analog of the cross product part, i.e. V̂i, L̂i, do not commute either, so some adjustment of this sort will also have to be made. However, there is also another, “potential” part of the LRL vector, which is proportional to ~r/r. The cor- responding NC analogs of xi and Ψ do not commute either, so we resolve the ordering in a similar way as in the cross product case — we take x̂k instead of xk: Âk = 1 2 εijk(L̂iV̂j + V̂jL̂i) + q x̂k r . (21) It has turned out that besides coping with the or- dering dilemma, nothing more needs to be done - that is, except for doing the calculations to show that our definition of Âk has been a good choice. So now we are going to take the NC analogs of the Hamiltonian, velocity, angular momentum and position operators, and build the NC LRL vector according to (21). Then we have to move to the next task - evaluate the commutator [Âi,Ĥ], examine the commutation relations between Âk and L̂k, searching for the signs of a higher dynamical symmetry. As soon as the symmetry group is recognized, we can construct the corresponding Casimir operators. All these crucial operators: the Hamiltonian, veloc- ity, angular momentum and position operators, have been defined already in terms of creation and anni- hilation operators a+α , aα; knowing the commutation relations for these, one can calculate all that is re- quired. However, after writing it all down and trying to make heads and tails of it, we soon realize that the problem is not assigned in the most friendly way. This definitely seems to be a case in which introduc- ing some auxiliary quantities may help. There are certain combinations of a+α , aα that occur often in our expressions, and separating them the right way makes the calculations more manageable. 3.1. Auxiliary operators We have to examine how the operators considered here act on the wave function Ψ. They are expressed in terms of â+α , âα. In general it makes a difference whether the creation and annihilation operators act from the right or the left and the following notation seems to be useful to keep track of it: âαΨ = aαΨ, b̂αΨ = Ψaα, (22) â+α Ψ = a + α Ψ, b̂ + α Ψ = Ψa + α . (23) An advantage of this notation is the fact that now we do not have to drag Ψ into the formulas just to make clear which side the operators act from. The relevant commutation relations are (see (5)) [âα, â+β ] = δαβ, [b̂α, b̂ + β ] = −δαβ. (24) The other commutators are zero. This, when kept in mind, spares a lot of paper during the calculations. 151 P. Presnajder, V. Galikova, S. Kovacik Acta Polytechnica As it was already mentioned, we use the position operator in the form x̂iΨ = 1 2 (xiΨ + Ψxi) = λ 2 σiαβ(â + α âβ + b̂βb̂ + α )Ψ, r̂Ψ = 1 2 (rΨ + Ψr) = λ 2 ((â+α âα + 1) + (b̂αb̂ + α + 1))Ψ. (25) The following sequences of operators appear often and their role is important enough to admit that they deserve some notation on their own: Ŵk = σkαβ(â + α âβ − â + α b̂β − âβb̂ + α + b̂ + α b̂β), Ŵ = â+α âα − â + α b̂α − b̂ + α âα + b̂ + α b̂α, Ŵ ′k = Ŵk − 2λEx̂k, Ŵ ′ = Ŵ − 2λEr̂, (26) where E is energy and λ is the NC parameter already mentioned. Note that the only difference between Ŵ ′k and Ŵk is the constant multiplying one of their terms. Ŵ ′ and Ŵ are related similarly. 3.2. NC operators revisited Now we rewrite the Hamiltonian, the velocity operator and the NC LRL vector in terms of the new auxiliary operators which have been introduced. Ĥ = 1 2λr̂ (â+α âα + b̂ + α b̂α − â + α b̂α − âαb̂ + α ) − q r̂ = 1 2λr̂ Ŵ − q r̂ , (27) V̂i = −i[x̂i,Ĥ] = i 2r̂ σiαβ(â + α b̂β − âβb̂ + α ) (28) Âk = 1 2 εijk(L̂iV̂j + V̂jL̂i) + q x̂k r̂ = 1 2r̂λ (r̂Ŵ ′k − x̂k(Ŵ ′ − 2λq)). (29) Deriving equations (28) and (29) involves somewhat laborious calculations. The details can be found in [7]. This gives us an opportunity to write the NC Schrödinger equation in the following way: ( 1 2λr̂ Ŵ− q r̂ −E ) ΨE = 1 2λr̂ (Ŵ ′−2λq)ΨE = 0. (30) ΨE belongs to HEλ , i.e. to the subspace spanned by the eigenvectors of the Hamiltonian. The important quantity for us is Âk|HE λ , the LRL vector as it acts on the solutions of the Schrödinger equation: Âk|HE λ = 1 2r̂λ (r̂Ŵ ′k−x̂k (Ŵ ′ − 2λq)︸ ︷︷ ︸ see Eq. (30) ) = 1 2λ Ŵ ′k. (31) When dealing with calculations related to the con- servation of Âk, we just need to ascertain whether the following commutator with the Hamiltonian vanishes: ˙̂ W ′k = i [ Ĥ0 − q r̂ ,Ŵ ′k ] = i [ 1 2r̂λ Ŵ ′ − q r̂ ,Ŵ ′k ] = i 2r̂λ [ Ŵ ′,Ŵ ′k ] + i [1 r̂ ,Ŵ ′k ](Ŵ ′ 2λ −q ) = 0. (32) The second term in the second-to-last line vanishes when acting on vectors from HEλ (and we are not so interested in the rest of Hλ). The first term propor- tional to [Ŵ ′,Ŵ ′k] does not contribute either. The calculations proving this involve more steps and can be found in [7]. The equation above encourages one to search for the underlying SO(4) symmetry, since the LRL vector conservation makes its components suitable candi- dates for half of its generators, the remaining three consisting of the components of the angular momen- tum. Once again we have to ask the reader either to check [7] for details or simply to believe that the following holds: [Âi, Âj] = iεijk(−2E + λ2E2)L̂k (33) There is nothing but a constant in the way, as long as we let the operator [Âi, Âj] act upon the vectors from HEλ with the energy fixed. Eq. (33) and [L̂i, L̂j] = iεijkL̂k, [L̂i, Âj] = iεijkÂk (34) define Lie algebra relations corresponding to a particu- lar symmetry group, the actual form of which depends on the sign of the E-dependent factor in (33). The relevant relations for L̂i have already been mentioned, the formula for the mixed commutator [L̂i, Âj] follows from the fact that Âj, j = 1, 2, 3, are components of a vector. There are three independent cases: • SO(4) symmetry: −2E + λ2E2 > 0 ⇐⇒ E < 0 or E > 2/λ2; • SO(3, 1) symmetry: −2E + λ2E2 < 0 ⇐⇒ 0 < E < 2/λ2; • E(3) Euclidean group: −2E + λ2E2 = 0 ⇐⇒ E = 0 or E = 2/λ2. The admissible values of E should correspond to the unitary representations of the symmetry in ques- tion. This requirement guarantees that the generators L̂j and Âj are realized as hermitian operators, and consequently correspond to physical observables. The Casimir operators in the mentioned cases are Ĉ′1 = L̂jÂj, Ĉ′2 = ÂiÂi + (−2E + λ 2E2)(L̂iL̂i + 1). (35) 152 vol. 54 no. 2/2014 LRL Vector in Quantum Mechanics in Noncommutative Space The prime indicates that we are not using the standard normalization of Casimir operators. Now, we need to calculate their values in HEλ . The first Casimir vanishes in all cases due to the fact that Ĉ′1ΨE ∼ rΨE − ΨEr = 0. The second Casimir operator is somewhat more demanding. According to (31) we have Ĉ′2ΨE = (Ŵ ′iŴ ′i 4λ2 + (−2E + λ2E2)(L̂iL̂i + 1) ) ΨE = 1 4λ2 Ŵ ′Ŵ ′ΨE, (36) where we used the quadratic identity Ŵ ′iŴ ′ i + 4λ 2(−2E + λ2E2)(L̂iL̂i + 1) = Ŵ ′2. (37) According to the Schrödinger equation, (Ŵ ′)2ΨE = 4λ2q2ΨE, and we are left with Ĉ′2ΨE = q 2ΨE. (38) Since both Casimir operators take constant val- ues Ĉ′1 = 0 and Ĉ′2 = q2 in HEλ , we are deal- ing with irreducible representations of the dynam- ical symmetry group G that are unitary for partic- ular values of energy. In all the cases considered, G = SO(4), SO(3, 1),E(3), the unitary irreducible representations are well known. The corresponding systems of eigenfunctions that span the representation space have been found in [9]. Here we do not repeat their construction, but we restrict ourselves to brief comments pointing out some interesting aspects. 3.3. Bound states – the case of SO(4) symmetry −2E + λ2E2 > 0 In this case we rescale the LRL vector as K̂j = Âj√ −2E + λ2E2 = Ŵ ′j 2λ √ −2E + λ2E2 . (39) After this step Eqs. (33), (34) turn into the following relations: [L̂i, L̂j] = iεijkL̂k, [L̂i,K̂j] = iεijkK̂k, [K̂i,K̂j] = iεijkL̂k. (40) Thus we have got the representation of the so(4) algebra. The relevant normalized Casimir operators are Ĉ1 = L̂jK̂j, Ĉ2 = K̂iK̂i + L̂iL̂i + 1. (41) As we have stated already, the Ĉ1 acting on an eigen- function of the Hamiltonian returns zero. As to Ĉ2, we know that for so(4), under the condition that the first Casimir is zero, the second Casimir has to be equal to n2 for some integer n = j + 1,j + 2, . . . (with j(j + 1) corresponding to the square of the angular momentum). At the same time, according to (38) it is related to the energy: K̂iK̂i + L̂iL̂i + 1 = q2 λ2E2 − 2E = n2. (42) Now solving the quadratic equation for energy we obtain two discrete sets of solutions depending on n: E = 1 λ2 ∓ 1 λ2 √ 1 + κ2n, κn = qλ n . (43) The first set of eigenfunctions of the Hamiltonian in (17) for energies E < 0 (i.e. negative sign in front of the square root in (43)) has been found for Coulomb attractive potential, i.e. q > 0 in (17): EIλn = 1 λ2 − 1 λ2 √ 1 + κ2n. (44) These eigenvalues possess a smooth standard limit for λ → 0 EIλn = 1 λ2 − 1 λ2 ( 1 + 1 2 κ2n − 1 24 κ4n + · · · ) →− q2 2n2 = − q2m 2n2~2 . (45) This spectrum coincides (in the commutative limit λ → 0) with the spectrum for Coulomb attractive po- tential, q > 0, that was found by Pauli using algebraic methods prior to solving Schrödinger equation for the hydrogen atom. The full set of eigenfunctions of (17) for energies E < 0 was constructed in [9] by explicitly solving the NC Schrödinger equation. The radial NC wave functions defined in (14) are given in terms of the hypergeometric function RIλn = (Ωn) NF(−n,−N, 2j + 2,−2κnΩ−1n ), Ωn = κn − √ 1 + κ2n + 1 κn + √ 1 + κ2n − 1 , (46) where N = a+αaα controls the radial NC variable. The second set of very unexpected solutions corre- sponds to energies (43) with positive sign EIIλn = 1 λ2 + 1 λ2 √ 1 + κ2n > 2 λ2 . (47) The corresponding radial NC wave functions have been found in [9] solving NC Schrödinger equation for a Coulomb repulsive potential, q < 0 in (17). These radial NC wave functions are closely related to those given above RIIλn = (−Ωn) NF(−n,−N, 2j + 2, 2κnΩ−1n ). (48) Both SO(4) representations, the representation for Coulomb attractive potential with EIλn < 0 and that for ultra-high energies EIIλn > 2/λ 2 for Coulomb re- pulsive potential, are unitary equivalent as in both representations the Casimir operators take the same values, Ĉ1 = 0 and Ĉ1 = n2. However, physically they are quite distinct: In the commutative limit λ → 0 the first bound states persist and reduce to the stan- dard ones, while the extraordinary bound states at ultra-high energies disappear from the Hilbert space. 153 P. Presnajder, V. Galikova, S. Kovacik Acta Polytechnica 3.4. Coulomb scattering – the case 2E −λ2E2 > 0 In this case we rescale the LRL vector as K̂j = Âj√ 2E −λ2E2 = Ŵ ′j 2λ √ 2E −λ2E2 . (49) After this step we obtain equations [L̂i, L̂j] = iεijkL̂k, [L̂i,K̂j] = iεijkK̂k, [K̂i,K̂j] = −iεijkL̂k. (50) So this time we have obtained the representation of the so(3, 1) algebra. The relevant normalized Casimir operators are Ĉ1 = L̂jK̂j, Ĉ2 = K̂iK̂i − L̂iL̂i. (51) In our case Ĉ1 = 0, so we are dealing with SO(3, 1) unitary representations that are labeled by the value of second Casimir operator Ĉ2. Rewriting (38) in terms of K̂j we obtain relation between energy E and the eigenvalue τ of Ĉ2: K̂iK̂i − L̂iL̂i = 1 + q2 2E −λ2E2 = τ > 1. (52) Thus we are dealing with the spherical principal series of SO(3, 1) unitary representations, see e.g [11]. The scattering NC wave functions have been con- structed in [9] for any admissible energy E ∈ (0, 2/λ2), and from their asymptotic behavior the partial wave S-matrix has been derived Sλj (E) = Γ(j + 1 − iq p ) Γ(j + 1 + iq p ) , p = √ 2E −λ2E2. (53) It can be easily seen that such S-matrix possesses poles at energies E = EIλn for Coulomb attractive potential and poles at energies E = EIIλn for Coulomb repulsive potential, where both EIλn and E II λn coincide with (44) and (47) given above. As for energies Eλ∓ = 1 λ2 ( 1 ∓ √ 1 − λ2q2 τ − 1 ) (54) the Casimir operator values coincide, the correspond- ing representations are unitarily equivalent. This re- lates the scattering for low energies 0 < E < 1/λ2 to that at high energies 1/λ2 < E < 2/λ2. We skip the limiting cases of the scattering at the edges E = 0 and E = 2/λ2 of the admissible interval of energies, where the SO(3, 1) group contracts to the Euclidean group E(3) = SO(3) . T(3) of isometries of 3D space. The corresponding NC Hamiltonian eigenstates are given in [9]. 4. Conclusions This paper deals with the Coulomb-Kepler problem in noncommutative space. We have found the NC analog of the LRL vector; its components, together with those of the NC angular momentum operator, supply the algebra of generators of a symmetry group. It is interesting that the formula for the NC version of the LRL vector looks very much like the one from standard QM, that is, when written in terms of the proper NC observables: NC angular momentum, NC velocity, symmetrized NC coordinate and NC radial distance. It is quite remarkable that the SO(4) symmetry has appeared twice: Firstly, not so surprisingly, when addressing the problem of the bound states for neg- ative energies in the case of the attractive Coulomb potential. These have an analog in standard quantum mechanics, and our result for negative energy bound states indeed coincides with the well known QM pre- diction if the commutative limit λ → 0 is applied. The second appearance of the SO(4) symmetry is probably not so expected, for we have found a set of bound states for positive energies above a certain ultra-high value in the case that the potential is repulsive. How- ever, there is again no discrepancy between QM and NCQM, since the unexpected ultra-high energy bound states disappear from the Hilbert space in the above mentioned commutative limit. When examining the scattering (relevant for the interval of energies between zero and the mentioned critical ultra-high value), SO(3, 1) is the symmetry to be considered. The scattering is usually characterized by the S-matrix. In the NC version of the problem this object has exactly those poles in complex energy plane which correspond to the bound states (of both kinds) mentioned above. This goes, however, beyond the scope of this paper. To summarize, there are basically two ways of ex- amining the hydrogen energy spectrum: by solving some differential equation in Schrödinger fashion or by looking for an underlying symmetry and using an algebraic approach à la Pauli. Both possibilities were tried in NCQM (the aim of this paper has been mainly to provide some outline of the latter option), for details see [7], [9]. We are glad to find out that both approaches (agreeing in standard QM) lead to the same outcomes also in NCQM. Acknowledgements The author V. G. is indebted to Comenius University for support received from grant No. UK/545/2013. References [1] Jakob Hermann, Giornale de Letterati D’Italia 2 (1710) 447; Jakob Hermann, Extrait d’une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710, Histoire de l’academie royale des sciences (Paris) 1732: 519; 154 vol. 54 no. 2/2014 LRL Vector in Quantum Mechanics in Noncommutative Space [2] Johann Bernoulli, Extrait de la Réponse de M. Bernoulli à M. 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Prešnajder, Int. J. Theor. Phys. 35 (1996) 231 doi: 10.1007/BF02083810. [9] Veronika Gáliková and Peter Prešnajder, Coulomb problem in non-commutative quantum mechanics J. Math. Phys. 54, 052102 (2013) doi: 10.1063/4803457; [10] S. Kováčik and P. Prešnajder, The velocity operator in quantum mechanics in noncommutative space, J. Math. Phys. 54, 102103 (2013) doi: 1063/1.4826355; [11] A. O. Barut and R. Raçzka, Theory of Group Representations and Applications, Polish Scientific Publishers, 1977. 155 http://dx.doi.org/10.1063/1.4835615 http://dx.doi.org/1007/BF01609397 http://dx.doi.org/10.1063/1.529418 http://dx.doi.org/10.1007/BF00745155 http://dx.doi.org/10.1007/BF02083810 http://dx.doi.org/10.1063/4803457 http://dx.doi.org/1063/1.4826355 Acta Polytechnica 54(2):149–155, 2014 1 Introduction 2 Basics of noncommutative quantum mechanics 3 Dynamical symmetry in NCQM 3.1 Auxiliary operators 3.2 NC operators revisited 3.3 Bound states 3.4 Coulomb scattering 4 Conclusions Acknowledgements References