Energy-dependent noncommutative quantum mechanics

Abstract

We propose a model of dynamical noncommutative quantum mechanics in which the noncommutative strengths, describing the properties of the commutation relations of the coordinate and momenta, respectively, are arbitrary energy dependent functions. The Schrödinger equation in the energy dependent noncommutative algebra is derived for a two dimensional system for an arbitrary potential. The resulting equation reduces in the small energy limit to the standard quantum mechanical one, while for large energies the effects of the noncommutativity become important. We investigate in detail three cases, in which the noncommutative strengths are determined by an independent energy scale, related to the vacuum quantum fluctuations, by the particle energy, and by a quantum operator representation, respectively. Specifically, in our study we assume an arbitrary power law energy-dependence of the noncommutative strength parameters, and of their algebra. In this case, in the quantum operator representation, the Schrödinger equation can be formulated mathematically as a fractional differential equation. For all our three models we analyze the quantum evolution of the free particle, and of the harmonic oscillator, respectively. The general solutions of the noncommutative Schrödinger equation as well as the expressions of the energy levels are explicitly obtained.

21 pages, no figures, accepted for publication in European Physical Journal C; references added

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• Upper bound on the momentum scale in noncommutative phase space of canonical type
• An Introduction to Noncommutative Physics
• Two-dimensional pauli equation in noncommutative phase-space
• Heat flow and noncommutative quantum mechanics in phase-space
• Klein-Gordon theory in noncommutative phase space
• Length-preserving biconnection gravity and its cosmological implications
• Covariant version of Pauli Hamiltonian, spin-induced non commutativity, Thomas precession and precession of spin
• Squeezed coherent states for gravitational well in noncommutative space
• Energy corrections due to the noncommutative phase-space of the charged isotropic harmonic oscillator in a uniform magnetic field in 3D
• Has the problem of the motion of a heavy symmetric top been solved in quadratures?
• Energy corrections due to the Non-commutative Phase-Space of the Charged Harmonic Oscillator in a constant magnetic field in 3D