Effective generating functions for quantum canonical transformations

An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using $e^{\mathrm{iF}}$ as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (ħ→0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of ħ. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schrödinger wave-functional formalism.