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.