Properties of finite Gaussians and the discrete-continuous transition

TL;DR

This paper explores finite Gaussian functions in quantum systems, analyzing their properties, uncertainty relations, phase-space distributions, and dynamics, while examining the transition to continuous systems as dimension increases.

Contribution

It develops a comprehensive framework for finite Gaussians using Jacobi theta functions, extending continuous case results to finite-dimensional quantum systems.

Findings

Finite Gaussians satisfy uncertainty relations similar to continuous case.

Finite Gaussians' Wigner distributions resemble their continuous counterparts.

As system dimension increases, finite Gaussians recover continuous-limit behaviors.

Abstract

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterised by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension $d$ of the system increases.