Time-dependent invariants and quantum mechanics in two dimensions

Abstract

A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.