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Journal of Physics A Mathematical and General

Evaluation of the Bloch density matrix for a charged oscillator in a magnetic field by canonical transformations

Mohammed A. Z. Habeeb

Published Nov 11, 1987• 6 citations
Publisher

Abstract

The problem of a charged particle moving in an anisotropic harmonic oscillator potential in the presence of a constant external magnetic field is reduced to that of a particle in an anisotropic oscillator by using canonical transformations. The Bloch density matrix for the original problem is then exactly evaluated by transforming the well known result for the anisotropic oscillator with the aid of the integral kernel corresponding to the canonical transformation under consideration.

DOI
10.1088/0305-4470/20/16/029

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The problem of a charged particle moving in an anisotropic harmonic oscillator potential in the presence of a constant external magnetic field is reduced to that of a particle in an anisotropic oscillator by using canonical transformations. The Bloch density matrix for the original problem is then exactly evaluated by transforming the well known result for the anisotropic oscillator with the aid of the integral kernel corresponding to the canonical transformation under consideration.
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.
We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.