A discrete quantum model of the harmonic oscillator

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.

Models of damped oscillators in quantum mechanics

“…It is worth applying the well-known factorization technique (see, for example, [2], [3], [4], [10] and [26]) to the Hamiltonian (3.7). The corresponding ladder operators can be found in the forms…”

Group Theory in Finite Hamiltonian Systems

“…We can postulate instead that the key commutator close into a different Lie algebra. In one space dimension we can have the spin SU(2) group (β = 1) [1,2], the Lorentz group (β = −1) SU(1, 1) [3], the group of Euclidean motions ISO(2) (β = 0) [4], or any three-parameter algebra, or q-algebra [5,6,7,8], which contracts to HW. This postulate thus entails the correspondence principle: when the number and density of points increase without limit in a discrete Hamiltonian system, the limit should be some well-known continuous Hamiltonian system [9,10].…”

Thermo Field Dynamics on a Quantum Computer

“…Discrete quantum mechanics is also developed by Singh in [5], by Atakishiyev in [6], by Lorente in [7], and by Barker in [8].…”