Density matrix of anharmonic oscillators by a thermodynamic variation–perturbation technique, a closed form approach

A general approximation technique for the density matrix of coupled anharmonic oscillators is developed. Starting with the Moyal–Weyl trace formalism for the Wigner phase space distribution function, a combination of the Gibbs–Bogoliubov inequality for the partition function and an operator perturbation technique is applied to the Moyal–Weyl trace. Introduction of generating functions and of operator techniques allows the closed form evaluation of the Moyal–Weyl formula for anharmonic oscillators. The Fourier transforms relating the Moyal–Weyl formula and the Wigner function as well as those relating the Wigner function and the density matrix can be given in terms of Hermite polynomials. The Moyal–Weyl formula as the crucial part of the technique is worked out in the first order thermodynamic variation perturbation approximation for the anharmonic quartic oscillator as well as for two coupled anharmonic oscillators as an example for arbitrary coupled oscillators.