A Wigner distribution function for finite oscillator systems

Abstract

We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the su(2) oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.

The Wigner distribution function for the $\mathfrak {su}(2)$ finite oscillator and Dyck paths

“…We shall recall some general notation and concepts, introduced in [1]. Consider a representation space V of dimension N + 1, and denote the (normalized) eigenvectors of Ĥ by |n , with Ĥ|n = E n |n (n = 0, 1, .…”

“…In the continuous case, the Wigner function W n (p, q) is a distribution function in (p, q)phase-space such that the expectation value for a classical phase-space function G(p, q) coincides with the quantum mechanical expectation value of the suitably ordered operator expression Ĝ( p, q) for the nth stationary state. In the discrete case, the Wigner function W (n; p k , q l ) is (for every n) a function of the discrete values (p k , q l ) such that [1] n| Ĝ( p, q)|n =…”

“…In a previous paper [1], a new definition of a Wigner distribution function for a finite quantum system [2][3][4] was developed, in particular for a quantum system in which the position operator q has a finite multiplicity-free spectrum {q 0 , q 1 , . .…”

“…, q N }, so it can be considered as an (N + 1) × (N + 1) matrix (the Wigner matrix). The assumption in [1] is that-as in the continuous case [5][6][7][8]-the distribution averages for all 'physical observables' of the form p a q b (a, b = 0, . .…”

“…In [1], the discrete Wigner function for the su(2) oscillator was computed for some particular values of j, and the matrix plots of these functions were considered. This led to quite remarkable observations, in particular when compared to plots of continuous Wigner functions for the canonical oscillator.…”