Propagators and Canonical Transformations

Alfredo Luis, Jan Peřina

We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.

G. F. Torres del Castillo, H. Martı́nez, Ramiro Julián Sánchez +1 more

The transformation of the wave functions induced by a given canonical transformation in the classical phase space, (q i,p i) - (Q i, Pi), is considered. In the examples presented here, the kernel of the integral transform turns out to be essentially exp (q i, Qi) is defined by Pi dQi In the case of the time evolution, which is a canonical transformation, the kernel of the transform is the propagator, and is obtained directly by making use of the solution to the classical equations of motion.

Walter Greiner, J. Reinhardt

Chung‐In Um

In a general time-dependent quadratic Hamiltonian, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The values of some functions of canonical variables q and p are generally not equal to the values of same functions of the other canonical variables Q and P. The wave functions, propagators and uncertainty relations are evaluated explicitly using the dynamical invariant method.

Walter Dittrich, Martin Reuter