Kh. Yeon et al., RELATIONS OF CANONICAL AND UNITARY TRANSFORMATIONS FOR A GENERAL TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM, Physical review. A, 55(6), 1997, pp. 4023-4029
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.