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Journal of Physics A Mathematical and General

Path integral evaluation of the Bloch density matrix for an oscillator in an electric and magnetic field

M. L. Glasser

Published Feb 21, 1987• 5 citations
Publisher

Abstract

The Bloch density matrix for an electron in a three-dimensional oscillator potential in the presence of parallel electric and magnetic fields is presented in cylindrical coordinates.
DOI
10.1088/0305-4470/20/3/002

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A new method is proposed in evaluating the Bloch density matrix for a three-dimensional charged harmonic oscillator placed in a constant magnetic field. The method first reduces the operator exp (- beta H) to a product of several factors of a simple nature by using the commutation relations among the Hamiltonian components and by performing some transformations on the Hamiltonian, and then calculates the matrix element. The method is especially efficient when the system has a rotational symmetry around the axis parallel to the magnetic field, but it is also useful in other cases. A generalised problem in which a uniform electric field coexists is also discussed. To make the whole discussion consistent, the density matrix for a one-dimensional harmonic oscillator, which plays an essential role in the whole problem, is recalculated in the spirit of this method, and it is shown that the present method derives the well known form of the Bloch density matrix for this system in a quite elementary way, without reference to any advanced knowledge of eigenfunctions.