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Journal of Physics A Mathematical and General

Maslov's complex germ method and Berry's phase

Andrey Yu. Trifonov, A A Yevseyevich

Published Sep 21, 1994• 4 citations
Publisher

Abstract

In the adiabatic approximation the connection of the Berry phase with the quasi-classical trajectory-coherent states of the Schrodinger-type equation (with the arbitrary scalar h(cross)-(pseudo) differential operator) and the Dirac equation in the external periodic electromagnetic field is studied.
DOI
10.1088/0305-4470/27/18/034

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In this paper we show that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the Schrödinger equation, such that the quantum-mechanical averages of the coordinates and the momenta with respect to these states are general exact solutions of the classical Hamiltonian equations. Such states are called trajectory-coherent states. The wave functions of the trajectory-coherent states are obtained by the complex germ method by V. P. Maslov. The simplest properties of these states are studied.
It is shown that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the SchrSdinger equation such that the mean quantum-mechanic al coordinates and momenta of these states are enact general solutions of the classical Hamilton equations. Such states are called trajectory-coher ent states. The wave functions of trajectory-coher ent states are obtained by Maslov's complex germ method. The simplest properties of these states are studied.
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.
The quantum dynamics of time-dependent Hamiltonians of the kind H(t)=U(t)H(0)U(t)° is studied in the framework of generalized coherent states. It is shown that the quantum and the classical dynamics are isomorphic and that the phase of ‖ψ(t)〉 is simply the classical action. In adiabatic approximation it is easy to extract Berry’s phase from this. A discussion of generalized coherent states (CS’s) necessary to deal with excited states is given; in particular for SU(2), SU(1,1) and E(2) their structure is completely characterized, showing that the deep connection with geometric quantization extends to these generalized CS’s. This technique may be employed to estimate higher-order correction to the adiabatic approximation.
The investigation of the Dirac equation by the WKB-Maslov method (complex sprout method) in Minkowski-Cartan space with an arbitrary external electromagnetic field is carried out. The Ehrenfest-type complete set of quasi-classical trajectory-coherent states (TCS) for the Dirac equation up to the order of O(h(cross) 3 2/) at to 0 is constructed. These states have the form of wavepackets moving along the worldline of the charged particle. From the Dirac equation a classical relativistic spin equation is derived by the use of the TCS which coincides with the generalization of the well known Bargmann-Michel-Telegdi equation in the case of an arbitrary external torsion field.