Quasi-Classical Transformation Theory

Abstract

The quasi-classical theory is the theory of the WKB approximation as defined by Van Vleck. This approximate wave theory is invariant under canonical transformations. We derive the transformation laws for wave functions and operators in this theory. The connection problem of the ordinary configuration space is discussed from the point of view of Van Vleck. The usual method of WKB quantization is then contrasted with another method of quantization in a classical configuration space of creation and destruction operators. In this new configuration space we find that, for some problems, the quasi-classical solutions are exact solutions of the Schr\"odinger equation. Finally, we show that the canonical transformations of the quasi-classical theory may be put into one-to-one correspondence with a group of approximate unitary transformations.