Representation of canonical transformations in quantum mechanics

Abstract

The transformation of the wave functions induced by a given canonical transformation in the classical phase space, $(q^{i}, p_{i}) \rightarrow (Q^{i}, P_{i})$, is considered. In the examples presented here, the kernel of the integral transform turns out to be essentially $\exp ({\rm i} \Lambda/\hbar)$, where $\Lambda(q^{i}, Q^{i})$ is defined by $P_{i} {\rm d}Q^{i} = p_{i} {\rm d}q^{i} + {\rm d} \Lambda$. 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.