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Adiabatic Theorem and Anomalous Commutators

Shinji Iida, Hiroshi Kuratsuji

Published Jan 1, 1987• 0 citations
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Abstract

Abstract We show that the geometrical phase factor recently found in the quantum adiabatic theorem modifies the canonical structure of the phase space. This gives rise to the anomalous terms in the quantum mechanical commutation relations.

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Adiabatic theorem and anomalous commutators

Shinji Iida, Hiroshi Kuratsuji

Physics Letters B•Jan 1, 1987•18 citations
Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of a two-dimensional simple harmonic oscillator in the Moyal plane, where momentum noncommutativity are also considered along with spatial noncommutativity. For that we introduce a modified form of Bopp's shift, that bridges the noncommutative phase-space operators with their effective commutative counterparts, having their dependence on the noncommutative parameters, and study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to Hannay's angle of a classically analogous system, by studying the evolution of the coherent state of this system.