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Journal of Physics A Mathematical and General

The coherent-state method of evaluating the density matrix for an oscillator in a constant magnetic field

Gong Ren-shan

Published Jun 7, 1990• 1 citation
Publisher

Abstract

The coherent-state method is proposed for evaluating the Bloch density matrix for a charged isotropic oscillator placed in a constant magnetic field. The method first reduces the Bloch equation, satisfied by the density matrix e - beta H , to a partial differential equation by means of the coherent-state representation, and then calculates the density matrix. A generalised problem in which a uniform electric field coexists is also discussed.

DOI
10.1088/0305-4470/23/11/041

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The coherent-state method is proposed for evaluating the Bloch density matrix for a charged isotropic oscillator placed in a constant magnetic field. The method first reduces the Bloch equation, satisfied by the density matrix e - beta H , to a partial differential equation by means of the coherent-state representation, and then calculates the density matrix. A generalised problem in which a uniform electric field coexists is also discussed.
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.
An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.
We introduce a novel method to find exact density operators for a spin-1/2 particle in time-dependent magnetic fields by using the one-mode bosonic representation of su(2)su(2)su(2) and the connection with a time-dependent oscillator. As illustrative examples, we apply the method to find the density operators for constant and/or oscillating magnetic fields, which turn out to be time-dependent in general.