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Physical Review A

Exact evaluation of the propagator beyond and at caustics for a charged particle in a constant magnetic field

Bin Kang Cheng

Published Sep 1, 1984• 5 citations
Publisher

Abstract

It is first shown that the propagator of a charged particle in a constant external magnetic field is related to the propagator of a one-dimensional driven harmonic oscillator with memory. The memory term can be removed by introducing the auxiliary force term and then by integrating it out. By using this new formula for a driven harmonic oscillator, a generalization of Horv\'athy-Feynman formula, we finally evaluate exactly the propagator beyond and at caustics of our dynamical system.

DOI
10.1103/physreva.30.1491

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•Aug 27, 2010•0 citations
It is first shown that the propagator of a charged particle in a constant external magnetic field is related to the propagator of a one-dimensional driven harmonic oscillator with memory. The memory term can be removed by introducing the auxiliary force term and then by integrating it out. By using this new formula for a driven harmonic oscillator, a generalization of Horv'athy-Feynman formula, we finally evaluate exactly the propagator beyond and at caustics of our dynamical system.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.