Use of the coherent states in evaluating the canonical density matrix for a harmonic oscillator

Abstract

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.