Wave-Mechanical Perturbation Calculation by Means of the Laplace Transform (with an Application to the Stark Effect)

A. G. Makhanek, V. S. Korol'kov, Pavel A Apanasevich

George Birtwistle

Philip L. Bowers

Kenneth Watson

A Laplace transform relation between the distribution function $\mathrm{exp}{\ensuremath{-}\ensuremath{\beta}H}$ in quantum statistical mechanics and the Green's function of the wave equation has been noted by Koppe. This relationship may be extended to the stationary-state scattering equation. General properties of the transformation are discussed. Applications to the evaluation of the virial coefficients are considered. A further application of the perturbation methods of Brueckner and his collaborators is described.

Paul Gombás, D. Kisdi

M. J. Englefield

Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.