Exact solution of non-adiabatic model Hamiltonians in solid state physics and optics

Abstract

Simple non-adiabatic model Hamiltonians are treated using the realisation of the vibrational modes on Bargmann's Hilbert space of analytic functions. In this formulation the Schrodinger equation is a system of linear first-order differential equations. The energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions. The solutions are given in terms of Neumann series; the recurrence relations for the expansion coefficients have a simple structure. Under particular conditions for the interaction constant they allow for terminating Neumann series (isolated exact solutions). In the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction. The continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on Worpitzky's theorem and its relation to conformal mapping. The eigenvalues are calculated; a physical interpretation of the solutions which makes use of intuitive arguments is also given.