Noncommutative quantum mechanics in a time-dependent background

TL;DR

This paper explores a noncommutative quantum system with a time-dependent structure constant, solving the Schrödinger equation analytically, and analyzing uncertainty relations and coherent states in various background fields.

Contribution

It introduces a method to solve the time-dependent Schrödinger equation on a noncommutative space with dynamic structure constants, using invariants and nonlinear equations.

Findings

Explicit solutions for time-dependent Schrödinger equation in noncommutative space.

Generalized Heisenberg uncertainty relations with time-dependent bounds.

Coherent states' uncertainty properties depend on background fields.

Abstract

We investigate a quantum mechanical system on a noncommutative space for which the structure constant is explicitly time-dependent. Any autonomous Hamiltonian on such a space acquires a time-dependent form in terms of the conventional canonical variables. We employ the Lewis-Riesenfeld method of invariants to construct explicit analytical solutions for the corresponding time-dependent Schroedinger equation. The eigenfunctions are expressed in terms of the solutions of variants of the nonlinear Ermakov-Pinney equation and discussed in detail for various types of background fields. We utilize the solutions to verify a generalized version of Heisenberg's uncertainty relations for which the lower bound becomes a time-dependent function of the background fields. We study the variance for various states including standard Glauber coherent states with their squeezed versions and Gaussian Klauder coherent states resembling a quasi-classical behaviour. No type of coherent states appears to be optimal in general with regard to achieving minimal uncertainties, as this feature turns out to be background field dependent.