Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quan...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2016
1. Verfasser:	Rastelli, G.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2016
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spelling	Rastelli, G. 2019-02-16T09:15:55Z 2019-02-16T09:15:55Z 2016 Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator / G. Rastelli // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81S05; 81R12; 70H06 DOI:10.3842/SIGMA.2016.081 https://nasplib.isofts.kiev.ua/handle/123456789/147848 We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not. I am grateful to the referees of this article for their comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator
spellingShingle	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator Rastelli, G.
title_short	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator
title_full	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator
title_fullStr	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator
title_full_unstemmed	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator
title_sort	born-jordan and weyl quantizations of the 2d anisotropic harmonic oscillator
author	Rastelli, G.
author_facet	Rastelli, G.
publishDate	2016
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/147848
citation_txt	Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator / G. Rastelli // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 15 назв. — англ.
work_keys_str_mv	AT rastellig bornjordanandweylquantizationsofthe2danisotropicharmonicoscillator
first_indexed	2025-11-26T15:27:56Z
last_indexed	2025-11-26T15:27:56Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 081, 7 pages Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator Giovanni RASTELLI Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italy E-mail: giovanni.rastelli@unito.it Received July 15, 2016, in final form August 15, 2016; Published online August 17, 2016 http://dx.doi.org/10.3842/SIGMA.2016.081 Abstract. We apply the Born–Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born–Jordan formula, when producing different operators than the Weyl’s one, does not. Key words: Born–Jordan quantization; Weyl quantization; superintegrable systems; ex- tended systems 2010 Mathematics Subject Classification: 81S05; 81R12; 70H06 1 Introduction Max Born and Pascual Jordan proposed a first model of quantization in [3], restricted to one-dimensional systems, a model generalized to n-dimensional systems, together with Werner Heisenberg, in [4]. Hermann Weyl introduced in [14, 15] a general quantization scheme based on the Fourier transform formula. The Born–Jordan and Weyl quantizations produce in general different operators from the same classical function of momenta and coordinates. The two quan- tization methods, however, coincide on natural Hamiltonians in n-dimensional Euclidean spaces. This property and many others of the two quantizations are studied in the recent book [9], that is our main source for the following discussion. We consider here the Born–Jordan and Weyl quantizations applied to the algebra of constants of motion of the 2D anisotropic harmonic oscillator. This system admits the maximal number of three functionally independent constants of motion (and it is said to be superintegrable) whenever the ratio of the parameters is a rational number. Otherwise, the independent constants of motion are just two. Our aim here is to check the behaviour of the Born–Jordan and Weyl quantizations of the anisotropic harmonic oscillator with respect to the integrability and superintegrability of the resulting quantum system for some particular choice of the ratio of the parameters. This aspect of the two quantization procedures is not considered in [9]. We use here the same expression for the classical 2D anisotropic harmonic oscillator and its independent constants of motion that is employed in [7]. The nD anisotropic harmonic oscillator is there considered as an “extended system”, a particular structure of some natural Hamiltonians that allows the existence of polynomial constants of motion of higher degree, described in [7, 8]. The interest in using such a construction here comes from other our studies in progress on the quantization of extended systems (see for example [6]). We consider briefly also the factorization in annihilation-creation operators for the 2D aniso- tropic harmonic oscillator as given in [11], to check how the corresponding classical integrals are mailto:giovanni.rastelli@unito.it http://dx.doi.org/10.3842/SIGMA.2016.081 2 G. Rastelli quantized by applying again Born–Jordan and Weyl procedures. The examples are confronted with those arising from the extension procedure. We can apply here the simpler formulas for Born–Jordan and Weyl quantizations for mono- mials in coordinates and momenta discussed in [9]. The quantization procedures of classical quantities are not exhausted by Born–Jordan’s and Weyl’s approaches. Several techniques have been developed since the beginning of the quantum era, many of them specific for the particular system considered. Indeed, there is no unique way to assign quantum operators to classical quantities in a mea- ningful way. Hermiticity of the operator is, usually, a necessary requirement and it is obtained by some symmetrization procedure, however not uniquely determined, see for example [13]. The problem of preserving the algebra of the constants of motion of a Hamiltonian system after quantization is object of many recent studies, see for example [1, 5, 10, 12] and references therein, for solutions in flat and non-flat manifolds. For superintegrability and quantization in classical and quantum systems see also [12], in particular for the definition of quantum superintegrable systems, and [5]. 2 Born–Jordan and Weyl quantizations of monomials In [3, 4, 9] the Born–Jordan quantization of monomials in coordinates (xi, pi) is determined by the following general rules, [x̂i, p̂j ] = i~δij , [x̂i, x̂j ] = 0, [p̂i, p̂j ] = 0, where δij = 1 for i = j and zero otherwise, for any quantization of the coordinates xi → x̂i and of the momenta pi → p̂i, and by xri p s i → 1 s+ 1 s∑ k=0 p̂s−k i x̂ri p̂ k i , (1) for the monomials with same indices. When the indices are different, the operators commute by the general quantization rules of above and their quantization is therefore straightforward. For the Weyl quantization [9, 14], the general rules are the same of above and, for the monomials, we have instead xri p s i → 1 2s s∑ k=0 ( s k ) p̂s−k i x̂ri p̂ k i . (2) The standard realization of the operators x̂i and p̂i, at least for Cartesian coordinates, are x̂iφ = xiφ, p̂iφ = −i~ ∂ ∂xi φ, for any function φ(xj). We employ in the following this standard quantization of the canonical coordinates. It is easy to check that for r = s = 1 the Born–Jordan and Weyl quantizations of monomials coincide, but differ for r, s ≥ 2. Many properties of the Born–Jordan quantization, generalized to any function of coordinates and momenta, and of the Weyl quantization are considered into details in [9] and the different characteristics are discussed. We focus here our analysis on the effect of the two quantizations on the first integrals of a particular superintegrable system. We check in some examples if the quantized first integrals commute with the Hamiltonian operator, that is, if the algebraic structure of the constants of motion is preserved by the different formulas of quantization, an issue not considered in [9]. Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator 3 3 The superintegrable 2D anisotropic harmonic oscillator In order to express the first integrals of the system, we can write the Hamiltonian of the su- perintegrable 2D anisotropic harmonic oscillator in the form of an extended Hamiltonian [7] as follows Hm,n = 1 2 ( p2u + (m n )2 p2x ) + ω2 (m n )2 ( x2 + u2 ) , (3) where (x, u) are coordinates in the Euclidean plane, ω ∈ R and m,n ∈ N \ {0}. Two independent first integrals of Hm,n are Hm,n itself and L = 1 2 p2x + ω2x2, that is associated with the separability of the Hamilton–Jacobi equation of Hm,n in coordina- tes (u, x). If we put Gn = [n−1 2 ]∑ k=0 ( n 2k + 1 )( −2ω2 )k x2k+1pn−2k−1 x , (4) where [a] denotes the integer part of a, and XL is the Hamiltonian vector field of the function L, then, adapting to our case the more general theorem proved in [7], we have Proposition 1. For any couple of positive integers (m,n), the function Km,n is a first integral of Hm,n, where Km,n = Pm,nGn +Dm,nXL(Gn), with Pm,n = [m/2]∑ k=0 ( m 2k )( −m n u )2k pm−2k u ( −2ω2 )k , Dm,n = 1 n [(m−1)/2]∑ k=0 ( m 2k + 1 ) ( −m n u )2k+1 pm−2k−1 u ( −2ω2 )k , m > 1, and D1,n = −m n2u. We can introduce the usual Cartesian coordinates (x, y) by leaving x, px unchanged and putting u = n m y, pu = m n py, (5) so that Hm,n = (m n )2(1 2 ( p2x + p2y ) + ω2 ( x2 + ( n m )2 y2 )) . In the following, we consider the Hm,n of above by dropping the negligible overall factor ( m n )2 . The classical and quantum superintegrability of the nD anisotropic harmonic oscillator has been studied in [11]. The quantization of the classical system is there obtained by introdu- cing creation and annihilation operators. This technique is widely in use today (see for exam- ple [5] and [2], where the anisotropic harmonic oscillator is generalized to 2D constant-curvature 4 G. Rastelli manifolds obtaining new classical and quantum superintegrable systems) and can have some application towards the quantization of extended systems. The Jauch–Hill Hamiltonian of the anisotropic harmonic oscillator is HJH = 1 2 ( p21 M1 + p22 M2 +M1ω 2 1q 2 1 +M2ω 2 2q 2 2 ) , and coincide with Hm,n if we put x = √ M1q1, p1 = √ M1px, y = √ M2q2, p2 = √ M2py, with mω2 = nω1, ω1 = √ 2ω. The classical first integrals given in [11] become F1(m,n) = 1 2 ( bn1b ∗m 2 + b∗n1 b m 2 ) , F2(m,n) = − i 2 ( bn1b ∗m 2 − b∗n1 bm2 ) , (6) where b1 = 1√ 2ω1 (px − iω1x) , b∗1 = 1√ 2ω1 (px + iω1x) , and similarly b2, b ∗ 2 in function of y, py, ω2. The corresponding quantum operators are obtained simply by substituting px, py with p̂x, p̂y in the expressions of above. The first integrals obtained with the two methods of above are different, having for example, up to constant factors, K1,1 = xpy − ypx, F1(1, 1) = pxpy + ω2xy. 4 The two quantizations of the constants of motion of the oscillator The Born–Jordan and Weyl quantizations of both Hm,n and L clearly coincide, being in both cases Hm,n → Ĥm,n = −~2 2 ( ∂2 ∂x2 + ∂2 ∂y2 ) + ω2 ( x2 + ( n m )2 y2 ) , L→ L̂ = −~2 2 ∂2 ∂x2 + ω2x2. The operators are clearly independent and commuting, so that the integrability of the system is preserved by both the quantizations. The Born–Jordan and Weyl quantizations of K1,1, K2,1 and K3,1 coincide and commute with the corresponding Hamiltonian operators Ĥm,n. Things become different for (m,n) = (4, 1). Indeed Proposition 2. The first integral K4,1 of H4,1 is K4,1 = 256 ( xp4y − ypxp3y − 3 4 ω2xy2p2y + ω2 8 y3pxpy + ω4 64 xy4 ) . (7) Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator 5 By applying to K4,1 the Weyl quantization formula (2), we have the Weyl operator K̂W 4,1 = 256 ( ~4 ( x ∂4 ∂y4 − y ∂4 ∂x∂y3 ) − 3~4 2 ∂3 ∂x∂y2 + ~2ω2 8 ( 6xy2 ∂2 ∂y2 − y3 ∂2 ∂x∂y ) + ~2ω2y ( 3 2 x ∂ ∂y − 3 16 y ∂ ∂x ) + ω4 64 xy4 + 3~2ω2 8 x ) , (8) and, by applying to K4,1 the formula (1), we get the Born–Jordan operator K̂BJ 4,1 = K̂W 4,1 + 32~2ω2x. (9) The computation of the commutators gives [ Ĥ4,1, K̂ W 4,1 ] = 0, [ Ĥ4,1, K̂ BJ 4,1 ] = −32~4ω2 ∂ ∂x . (10) Proof. We have from (4) G1 = x, therefore XLG1 = px. The application of Proposition 1 to (3) with (m,n) = (4, 1) followed by the transformation of coordinates (5) gives P4,1 = 256p4y − 192ω2y2p2y + 4ω2y, D4,1 = −256yp3y + 32ω2y3py, and we get (7). We observe now that, both for Born–Jordan and Weyl quantizations, we have K̂4,1 = P̂4,1x̂+ D̂4,1p̂x, where the order of the operators is immaterial, because P4,1 and D4,1 depend uniquely on (py, y). Therefore, we need to apply the two quantizations to P4,1 and D4,1 only, since the quantizations coincide on x and px. By applying (1) and (2) to the monomials y2p2y, yp 3 y and y3py, we have P̂4,1 = 256p̂4y − 192ω2Q̂1 + 4ω2ŷ, D̂4,1 = −256Q̂2 + 32ω2Q̂3, where Q̂2 = i ~3 2 ( 2y ∂3 ∂y3 + 3 ∂2 ∂y2 ) , Q̂3 = −i ~ 2 y2 ( 2y ∂ ∂y + 3 ) , coincide for both quantizations, and, for the Weyl case, Q̂1 = − ~2 2 ( 2y2 ∂2 ∂y2 + 4y ∂ ∂y + 1 ) , while, for the Born–Jordan case, Q̂1 = − ~2 3 ( 3y2 ∂2 ∂y2 + 6y ∂ ∂y + 2 ) . We obtain in this way (8) and (9). We can finally compute the commutators of Ĥ4,1 with K̂W 4,1 and K̂BJ 4,1 with the help of the formula[ Ĥ4,1, K̂4,1 ] = p̂x ( −i~P̂4,1 + [ Ĥ4,1, D̂4,1 ]) + x̂ ( 2ω2i~D̂4,1 + [ Ĥ4,1, P̂4,1 ]) , after making the suitable substitutions, obtaining the (10). � 6 G. Rastelli We computed the quantizations for several other values of (m,n), for example (5, 1), (6, 1), (1, 4), (3, 4), such that K̂BJ m,n 6= K̂W m,n, obtaining always commutation with Ĥm,n for K̂W m,n and no commutation with Ĥm,n for K̂BJ m,n. In these last cases, it can be observed that both ω and ~ always appear as factors in the commutator, meaning that for ω = 0 and for ~→ 0 the operators commute. Analogous results are obtained from the quantizations of the first integrals F1(m,n) or F2(m,n) given in (6) for several values of (m,n): the Weyl formula produces symmetry operators of the Hamiltonian, the Born–Jordan one, when giving different operators, does not. Actually, one can conjecture that the Weyl quantizations of Km,n and Fi(m,n) always com- mute with the Hamiltonian operator Ĥm,n. It can be observed, as one of the referees of this article pointed out, that “for the Jauch– Hill approach the quantization using creation and annihilation operators shows very easily that the quantum extensions of F1 and F2 are still integrals. It is less obvious to prove that this quantization is nothing but Weyl’s one”. Remark 3. The failure of the Born–Jordan quantization formula in reproducing the algebra of constants of motion at the quantum level is, actually, restricted to the particular set of generators of the algebra that we choose. We do not know in general if another choice of independent first integrals can lead to different results. Remark 4. The failure in reproducing the algebra of the constants of motion after quantization appears also in the case of natural Hamiltonian systems on curved manifolds. In these cases, quantum corrections of the Hamiltonian operator are necessary in order to preserve the integrable or superintegrable algebraic structure [1, 10]. 5 Conclusions From the examples computed, it appears that the Born–Jordan quantization formula fails in preserving the high-degree constants of motion of the 2D anisotropic harmonic oscillator, and therefore its superintegrability, to the quantum level, differently from the Weyl formula. A study of this problem in full generality is then desirable. Acknowledgements I am grateful to the referees of this article for their comments and suggestions. References [1] Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Quantum mechanics on spaces of non- constant curvature: the oscillator problem and superintegrability, Ann. Physics 326 (2011), 2053–2073, arXiv:1102.5494. [2] Ballesteros Á., Herranz F.J., Kuru Ş., Negro J., The anisotropic oscillator on curved spaces: a new exactly solvable model, Ann. Physics 373 (2016), 399–423, arXiv:1605.02384. [3] Born M., Jordan P., Zur Quantenmechanik, Z. Phys. 34 (1925), 858–888. [4] Born M., Heisenberg W., Jordan P., Zur Quantenmechanik. II, Z. Phys. 35 (1925), 557–615. [5] Celeghini E., Kuru Ş., Negro J., del Olmo M.A., A unified approach to quantum and classical TTW systems based on factorizations, Ann. Physics 332 (2013), 27–37. [6] Chanu C., Degiovanni L., Rastelli G., Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization, J. Phys. Conf. Ser. 343 (2012), 012101, 15 pages, arXiv:1111.0030. http://dx.doi.org/10.1016/j.aop.2011.03.002 http://arxiv.org/abs/1102.5494 http://dx.doi.org/10.1016/j.aop.2016.07.006 http://arxiv.org/abs/1605.02384 http://dx.doi.org/10.1007/BF01328531 http://dx.doi.org/10.1007/BF01379806 http://dx.doi.org/10.1016/j.aop.2013.01.008 http://dx.doi.org/10.1088/1742-6596/343/1/012101 http://arxiv.org/abs/1111.0030 Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator 7 [7] Chanu C.M., Degiovanni L., Rastelli G., Extensions of Hamiltonian systems dependent on a rational pa- rameter, J. Math. Phys. 55 (2014), 122703, 11 pages, arXiv:1310.5690. [8] Chanu C.M., Degiovanni L., Rastelli G., The Tremblay–Turbiner–Winternitz system as extended Hamilto- nian, J. Math. Phys. 55 (2014), 122701, 8 pages, arXiv:1404.4825. [9] de Gosson M.A., Born–Jordan quantization. Theory and applications, Fundamental Theories of Physics, Vol. 182, Springer, Cham, 2016. [10] Duval C., Valent G., Quantum integrability of quadratic Killing tensors, J. Math. Phys. 46 (2005), 053516, 22 pages, math-ph/0412059. [11] Jauch J.M., Hill E.L., On the problem of degeneracy in quantum mechanics, Phys. Rev. 57 (1940), 641–645. [12] Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694. [13] Post S., Winternitz P., General Nth order integrals of motion in the Euclidean plane, J. Phys. A: Math. Theor. 48 (2015), 405201, 24 pages, arXiv:1501.00471. [14] Weyl H., Quantenmechanik und Gruppentheorie, Z. Phys. 46 (1927), 1–46. [15] Weyl H., The theory of groups and quantum mechanics, Dover, New York, 1950. http://dx.doi.org/10.1063/1.4904452 http://arxiv.org/abs/1310.5690 http://dx.doi.org/10.1063/1.4903508 http://arxiv.org/abs/1404.4825 http://dx.doi.org/10.1007/978-3-319-27902-2 http://dx.doi.org/10.1063/1.1899986 http://arxiv.org/abs/math-ph/0412059 http://dx.doi.org/10.1103/PhysRev.57.641 http://dx.doi.org/10.1088/1751-8113/46/42/423001 http://dx.doi.org/10.1088/1751-8113/46/42/423001 http://arxiv.org/abs/1309.2694 http://dx.doi.org/10.1088/1751-8113/48/40/405201 http://dx.doi.org/10.1088/1751-8113/48/40/405201 http://arxiv.org/abs/1501.00471 http://dx.doi.org/10.1007/BF02055756 1 Introduction 2 Born–Jordan and Weyl quantizations of monomials 3 The superintegrable 2D anisotropic harmonic oscillator 4 The two quantizations of the constants of motion of the oscillator 5 Conclusions References

Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

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