Method of generating functions for a quantum oscillator

A method of generating functions is developed for studying a quantum oscillator with a variable frequency $\omega(t)$ subject to the influence of an external force $f(t)$. The method is used to obtain an explicit expression for the transition probabilities $w_{mn}$ between states $|n,\omega_{-}\rangle$ and $|m,\omega_{+}\rangle$, containing a definite number of quanta at the start $(n)$ and end $(m)$ of the process. The Heisenberg representation is discussed and the associated geometrical interpretation of the dynamical variables on the phase plane. By means of the phase plane, formulas are obtained for $w_{mn}$ in the quasiclassical limit (strongly degenerate oscillator for which $m,n\gg 1$). The application of the method of generating functions to the problem of the relaxation of a quantum oscillator interacting with a thermostat is discussed.