Gaussian density operator properties for the harmonic oscillator

An operator whose position representation is a Gaussian function is referred to as a Gaussian operator. In common with the harmonic oscillator Hamiltonian operator, the eigenvectors of the Gaussian operator are a product of Hermite polynomial functions and a Gaussian weighting factor. It is demonstrated that the thermal equilibrium density operator of a one-dimensional harmonic oscillator can be expressed in Gaussian operator form, including its eigenvalues. These can be obtained without first solving for the eigenfunctions of the harmonic oscillator Hamiltonian.