The Constraints and Spectra of a Deformed Quantum Mechanics

Abstract

We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR's) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave potentials in one dimension and isotropic power-law potentials in higher dimensions. Among other conclusions, we find that in a subset of the studied MCR's, the leading order energy shifts of bound states are of opposite sign compared to those obtained using string-theory motivated MCR's, and thus these two cases are more easily distinguishable in potential experiments.

30 pages inclusive of 7 figures

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Cited by in corpus (9)

• The Path Integral Quantization corresponding to the Deformed Heisenberg Algebra
• Emergence of Cosmic Space and Minimal Length in Quantum Gravity
• Deformed Gazeau-Klauder Schrödinger-Cat States with Modified Commutation Relations
• Quantum phase transitions of the Dirac oscillator in the Anti-Snyder model
• Non-Relativistic Anti-Snyder Model and Some Applications
• Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics
• Quantum probes for universal gravity corrections
• Generalized Relativistic Wave Equations with Intrinsic Maximum Momentum
• Scattering and Bound States of a Deformed Quantum Mechanics