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arXiv:0711.3089 · Journal of Physics A Mathematical and Theoretical

A discrete quantum model of the harmonic oscillator

Natig M Atakishiyev, Anatoliy U Klimyk, Kurt Bernardo Wolf

Published Feb 12, 2008• 21 citations
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Abstract

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.

DOI
10.1088/1751-8113/41/8/085201

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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.
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An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
We constructed a new exactly solvable model of the non-relativistic quantum linear harmonic oscillator with the position-dependent mass Mx=2m0/eαx+1. We solved the Schrödinger equation for this model in the absence and presence of an external homogeneous field. We showed that in both cases the wave functions of the bound states are expressed through Jacobi polynomials, and the corresponding energy spectra are non-equidistant. We also showed that in the limit α→0 the wave functions and energy spectra coincide with the corresponding expressions for a linear harmonic oscillator with the constant mass m0 in the absence and presence of an external homogeneous field. We present also two new limit relations that reduce the Jacobi polynomials directly to the Hermite polynomials with shifted and non-shifted arguments. The proofs of these limit relations are based on the method of mathematical induction.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.