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Progress of Theoretical Physics

On the Extra Phase Accompanying Berry's Phase in Terms of Coherent State Path Integral

Takahiro Fukui

Published Apr 1, 1992• 7 citations
Publisher

Abstract

In the WKB approximation for coupled channel equations, it has been pointed out by Yabana and Horiuchi that the semi-classical quantization condition should be modified by Berry's phase and another extra phase. Recently Littlejohn and Flynn have rederived them in a quite elegant way. We concentrate on this extra phase in this paper, and derive it for a simple model by means of the path integral method in the generalized coherent state representation. It is shown that it appears as a quantum fluctuation around classical orbits.

DOI
10.1143/ptp.87.927

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In the WKB approximation for coupled channel equations, it has been pointed out by Yabana and Horiuchi that the semi-classical quantization condition should be modified by Berry's phase and another extra phase. Recently Littlejohn and Flynn have rederived them in a quite elegant way. We concentrate on this extra phase in this paper, and derive it for a simple model by means of the path integral method in the generalized coherent state representation. It is shown that it appears as a quantum fluctuation around classical orbits.
Toward the classical description of quantum fluctuations in quantal systems, the Maslov phase occurring in a semi-classical quantization rule is investigated in a framework of the time-dependent variational principle with squeezed coherent states. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to Berry's phases or canonical phases including dynamical effects. The Maslov phase appears as a winding number in this framework. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.
Multi-dimensional coupled-channel equation is treated by the WKB approximation for the main purpose of deriving the semi-classical quantization rule for it. A somewhat general type of the multi-dimensional coupled-channel Hamiltonian is presented which enables us to derive the semiclassical quantization rule for the energy spectra. As in the one-dimensional case, the WKB wave function has an additional phase besides the usual phase ∫Σpsdxs, and the quantization rule should include the contribution from this additional phase. The structure of the WKB wave function in the adiabatic representation is also discussed.
A general framework for time-dependent variational approach in terms of squeezed coherent states is constructed with the aim of describing quantal systems by means of classical mechanics including higher order quantal effects with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. The Maslov phase occurring in a semi-classical quantization rule is investigated in this framework. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to the Berry phase. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.