The semiclassical treatment of some quantum problems

Abstract

The semiclassical treatment of quantummechanical problems is discussed making use of a generalization of the Hamilton-Jacobi partial differential equation to cases in which only some of the variables of the problem are treated classically. The quantum equations are put into a form in which the effects not covered by the semiclassical approximation are concentrated into one set of terms. This enables one to calculate corrections to the semiclassical theory and to demonstrate that compensations of the decrease of scattering in the coherent (“elastic”) channel by the increase caused by the presence of incoherent (“inelastic”) channels which is exact in the semiclassical approximation holds in a certain approximation quantum-mechanically. By means of these relationships it is shown that some and presumably the main contributions to the dynamical effects of molecular electrons interacting with protons in experiments on proton-proton and proton-neutron scattering are negligibly small.