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Europhysics Letters (EPL)

Geometric Phase and Exact Energy Spectrum for a Quantum System with Two Interacting Parts

Jing‐Bo Xu, Xiao-Chun Gao, Tiezheng Qian

Published Jan 20, 1994• 1 citation
Publisher

Abstract

A new method is introduced to solve the stationary-state Schrödinger equation for a spinning solenoid. The study of the exact energy spectrum of the systems is closely related to the study of the Aharonov-Anandan phase of a time-dependent system.
DOI
10.1209/0295-5075/25/3/002

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Quasi-energy spectral series ( epsilon nu (h(cross)), Psi ( epsilon nu )) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross) -1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma epsilon ( nu ) to the quasi-energy spectrum is calculated. It is shown that the gamma epsilon ( nu ) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.
We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-1∕21∕21∕2 problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.