Exact Quantum Dynamical Solutions for Oscillator-Like Systems

Abstract

The solution of the quantum dynamical equation $\frac{i\hslash \mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator $T$, is given, when the Hamiltonian $H$ is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schrödinger's equation in coordinate space is expressed by the Green's function corresponding to $T$. An example is given.