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QUASICLASSICAL TRAJECTORY-COHERENT STATES OF A NONRELATIVISTIC PARTICL)~ IN AN ARBITRARY ELECTROMAGNETIC FIELD

L.D. LANDAU

Published Jan 1, 1982• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
We solve the quantum mechanical problem of the motion of charged particle in a quadrupole field that varies arbitrarily with time. The time-dependent wave functions are shown to be in a simple one-to-one correspondence with the wave functions of an ordinary static-field harmonic oscillator. We define and generate a set of coherent states for the charged particle that provide Gaussian localization about the classical particle trajectories. When the field variation is periodic in time, these minimum uncertainty states are shown to pulsate periodically in size and in phase. Under the action of external fields, coherent states retain their coherence. We discuss the way in which the Kapitza pseudopotential approach approximates the present exact results.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.