Energy eigenvalues and eigenvectors for bound quantum systems using parametric equations of motion

Abstract

This paper considers the solution of a family of Schrodinger equations, characterized by one or more continuous parameters in the Hamiltonian. From a solution of the Schrodinger equation at initial parameter values all other solutions may be obtained by integrating a set of ordinary differential equations in the parameter space of the quantum system. Specifically, the parametric equations for energy eigenvalues and eigenstates are explored. Existing parameteric equations are generalized to include nonlinear parameters in the Hamiltonian and systems with degenerate eigenstates. The connections between this method and more traditional methods like perturbation theory and the variational principle are examined. The method is illustrated with the study of the vibrational energies of hydrogen fluoride calculated by deforming continuously the solutions of a harmonic oscillator to those of a Morse oscillator. In another example several coupled diatomic electronic states are considered where the deformation parameter is the bond length. It is demonstrated that no modification of the method is required to treat degeneracies or avoided level crossings.