Quantum mechanics on non commutative spaces and squeezed states: a functional approach

TL;DR

This paper explores quantum mechanics on noncommutative spaces, proposing a variational approach to identify quasi-classical states, and demonstrates its effectiveness through models with minimal length and noncommutative planes.

Contribution

It introduces a variational principle-based method to find quasi-classical states in noncommutative quantum mechanics, extending traditional Gaussian solutions.

Findings

Recovered Gaussian states in standard quantum mechanics

Identified new states as products of Gaussians and hypergeometrics

Derived and solved second order PDEs for specific models

Abstract

We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics. We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.