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Journal of Physics A Mathematical and General

On the eigenvalues of S. pi for arbitrary spin in a constant magnetic field

J. Jayaraman, Miguel A. Oliveira

Published Sep 7, 1989• 1 citation
Publisher

Abstract

Utilising the intimate connection of a charged particle in a homogeneous magnetic field to that of a harmonic oscillator, the authors demonstrate explicitly that the eigenvalue spectrum for the matrix operator S. pi for arbitrary spin in a uniform magnetic field in the z direction is governed, for its discrete part, by that of a Hermitian matrix defined on the space of the particle number and spin operators and is thus constrained to be real for any intensity of the external magnetic field. The present analysis generalises to arbitrary spin their previously reported results applied to the case of a spin-1 particle.

DOI
10.1088/0305-4470/22/17/018

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Utilising the intimate connection of a charged particle in a homogeneous magnetic field to that of a harmonic oscillator, the authors demonstrate explicitly that the eigenvalue spectrum for the matrix operator S. pi for arbitrary spin in a uniform magnetic field in the z direction is governed, for its discrete part, by that of a Hermitian matrix defined on the space of the particle number and spin operators and is thus constrained to be real for any intensity of the external magnetic field. The present analysis generalises to arbitrary spin their previously reported results applied to the case of a spin-1 particle.
Exploiting the intimate connection of the problem of a charged spin-1 particle in a homogeneous magnetic field to that of a harmonic oscillator, it is demonstrated that the eigenvalues of the matrix operator S. pi for spin-1 and for a constant magnetic field are governed by a Hermitian matrix defined on the space of the particle number n and spin and are thus constrained to be real for any intensity of the external magnetic field H, thereby contradicting a recent affirmation of Weaver (1978) that for n=0, complex eigenvalues are present for sufficiently intense magnetic fields. The errors in Weaver's analysis which have led to complex eigenvalues as well as to an excessive number of eigenvalues compared with that demanded by the spin degrees of freedom are pointed out.
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.