Noncommutative quantum mechanics in a time-dependent background

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.

21 pages, 6 figures

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• The non-Hermitian Swanson model with a time-dependent metric
• Energy-dependent noncommutative quantum mechanics
• Nonclassicality versus entanglement in a noncommutative space
• Lewis-Riesenfeld quantization and SU(1,1) coherent states for 2D damped harmonic oscillator
• A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length
• Metric versus observable operator representation, higher spin models
• Time-independent approximations for time-dependent optical potentials
• Analytical Solution of (2+1) Dimensional Dirac Equation in Time-Dependent Noncommutative Phase-Space
• Entropy and Information of a harmonic oscillator in a time-varying electric field in 2D and 3D noncommutative spaces
• Klein-Gordon theory in noncommutative phase space
• Exact solutions of a damped harmonic oscillator in a time dependent noncommutative space
• Exact solution of damped harmonic oscillator with a magnetic field in a time dependent noncommutative space
• An introductory review on resource theories of generalized nonclassical light
• Relativistic Ermakov-Milne-Pinney Systems and First Integrals
• Lewis and Berry phases for a gravitational wave interacting with a quantum harmonic oscillator
• Time-dependent massless Dirac fermions in graphene
• Quantum harmonic oscillator in a time dependent noncommutative background
• Dynamic noncommutative BTZ black holes
• Evolution operator for time-dependent non-Hermitian Hamiltonians
• Thermal Properties of Gauge-Invariant Graphene in Noncommutative Phase-Space