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arXiv · 1407.4843

Noncommutative quantum mechanics in a time-dependent background

Sanjib Dey·Andreas Fring
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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.

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Sanjib Dey, Andreas Fring. 2014-07-17. Noncommutative quantum mechanics in a time-dependent background. https://doi.org/10.1103/physrevd.90.084005

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