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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Q-extension of the linear harmonic oscillator

Mesuma K. Atakishiyeva, Natig M. Atakishiyev, Carlos Villegas‐Blas

Published Jul 6, 1998• 0 citations
Publisher

Abstract

Overlap integrals over the full real line for a family of q- extensions of the linear harmonic oscillator wave functions in quantum mechanics are evaluated. In particular, an explicit form of the squared norms for these q-wave functions is obtained. The classical Fourier-Gauss transform connects the families with different values 0 < q < 1 and q < 1 of the deformation parameter q. An explicit expansion of the q-Hermite polynomials of Rogers in terms of the ordinary Hermite polynomials emerges as a by-product.

DOI
10.1117/12.312640

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Overlap integrals over the full real line for a family of q- extensions of the linear harmonic oscillator wave functions in quantum mechanics are evaluated. In particular, an explicit form of the squared norms for these q-wave functions is obtained. The classical Fourier-Gauss transform connects the families with different values 0 < q < 1 and q < 1 of the deformation parameter q. An explicit expansion of the q-Hermite polynomials of Rogers in terms of the ordinary Hermite polynomials emerges as a by-product.
We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bound, whereas the spectra of position and of momentum are a denumerable non-degenerate set of points in [− 1, 1] that depends on the deformation parameter q ∊ (0, 1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q -Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit when q → 1 − , one recovers the ordinary quantum harmonic oscillator.
In quantum mechanics the linear harmonic oscillator representatives of the canonical observables q and p are certain well-known infinite square matrices, qmn and pmn, with denumerable sets of rows and columns. It is pedagogically useful to investigate various problems which arise when one considers the finite matrices Qmn and Pmn which result when qmn and pmn are truncated after the Nth row and column, where N( 2≥2) is some preassigned positive integer. So that the enquiry may be more instructive, it begins with an abstract formulation of the problem of the “truncated harmonic oscillator.” Formally, the latter has the usual Hamiltonian H = 12( Q2+P2 ) , but Q and P are not canonical—their commutator [Q,P] now differing from i by a certain projection operator. The spectrum of H and the (finite) basis { |n〉} generated by H are found by standard methods, though there are now some complications which are absent from the theory of the linear harmonic oscillator. The diagonalization of Q and P presents no difficulties if full use is made of the classical theory of Hermite polynomials. The spectrum of Q is the set of N zeros hs of the Nth Hermite polynomial. Although the wavefunctions 〈 Q′|n〉 are therefore functions of the discrete variable hs, one can formally contemplate them as functions of a continuous real variable h; and some curious intimations of Schrödinger representation appear. The analog of a component of orbital angular momentum is considered, and its spectrum obtained. Finally, the asymptotic limit N → ∞ is briefly discussed, in order thus explicitly to dispel the impression that this limit might somehow represent the “usual” linear harmonic oscillator.
It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.
We discuss some results from q-series that can account for the foundations for the introduction of orthogonal polynomials on the circle and on the line, namely the Rogers-Szegö and Stieltjes-Wigert polynomials. These polynomials are explicitly written and their orthogonality is verified. Explicit realizations of the raising and lowering operators for these polynomials are introduced in analogy to those of the Hermite polynomials that are shown to obey the q-commutation relations associated with the q-deformed harmonic oscillator.