CLASSICAL PSEUDOPOTENTIAL FOR THE EQUILIBRIUM DENSITY-MATRIX

A variational procedure for the integral with respect to the conditional Wiener measure is considered which yields an approximation from below for the equilibrium density matrix. An harmonic oscillator is used as a trial system, the variables being not only the frequency, as usual but also the displacement in the state of equilibrium. The result is represented in the form of a majoring pseudopotential defined by a gaussian transformation of the initial potential. The procedure provides a correct Wigner correction to the diagonal element of the density matrix and also variational estimate obtained by the Ritz method for the ground state energy. For a system with one degree of freedom the pseudopotential is obtained in an explicit analytic form as an expansion in powers of anharmonicity, the coefficients being expressed in terms of Legendre functions of the second kind Q(l)(2n+1), where n is the mean phonon occupation number of the trial system.