The quantum-mechanical properties of the strongly non-linear quantum oscillator in the Pöeschl-Teller model are considered. In the first place, the energy spectrum and its dependence upon the confinement parameter (i.e., the width of the “box”) are studied. Moreover, on the grounds of the Hellman-Feynman theorem the pressure operator in this model is obtained and (along with the energy spectrum) is studied in two main approximations: the “particle in the box” and “linear (harmonic) oscillator” for large and low values of the main quantum number; the critical value is also evaluated. Semi-classical approximation as well as perturbation theory for the Pöeschl-Teller are also considered. The results obtained here are intended for future thermodynamic calculations: first of all, for the generalization of the well-known Bloch result for the linear harmonic oscillator in the thermostat. To this end, the density matrix for the Pöeschl-Teller oscillator will be calculated and the full Carnot cycle conducted.

We present the motivation, formulation, and modified proof of the Bogoliubov-Zubarev theorem connecting the pressure of a dynamical object with its energy within the framework of a classical...

It is shown that many important and still relevant results in the field of nonequilibrium statistical mechanics were derived by Bogoliubov (jointly with N.M. Krylov) early in the prewar years and were completely published in a still insufficiently known monograph of 1945. Among these findings were the concept of time hierarchy for stochastic evolution, the concept of quasi-averages that remove symmetry relative to time inversion, and the mechanism of dynamic relaxation of a physical object in a thermostat.