RELATIONS OF CANONICAL AND UNITARY TRANSFORMATIONS FOR A GENERAL TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM

We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classica l equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transfor mations in quantum mechanics. The wave functions and the propagators a re evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q an d P, but the values of the functions of q and the kinetic momentum p(k ), are equal to those of the other Q and P-k in classical mechanics. W e prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamilton ians.