Gaussian type functions defined on a finite set and finite oscillators

TL;DR

This paper explores finite Gaussian functions on discrete sets, analyzing their Fourier transforms, Wigner functions, and how they define finite quantum oscillators, bridging continuous and discrete quantum models.

Contribution

It introduces a framework for finite Gaussian functions and their role in defining finite-dimensional quantum oscillators, extending the classical Gaussian-based quantum harmonic oscillator model.

Findings

Finite Gaussians have well-defined Fourier transforms and Wigner functions.

Finite Gaussians can be used to construct finite quantum oscillators.

The study bridges continuous and discrete quantum harmonic oscillator models.

Abstract

The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators.