ENERGY EIGENVALUES AND EIGENVECTORS FOR BOUND QUANTUM-SYSTEMS USING PARAMETRIC EQUATIONS OF MOTION

This paper considers the solution of a family of Schrodinger equations , characterized by one or more continuous parameters in the Hamiltonia n. From a solution of the Schrodinger equation at initial parameter va lues all other solutions may be obtained by integrating a set of ordin ary differential equations in the parameter space of the quantum syste m. Specifically, the parametric equations for energy eigenvalues and e igenstates are explored. Existing parametric equations are generalized to include nonlinear parameters in the Hamiltonian and systems with d egenerate eigenstates. The connections between this method and more tr aditional methods like perturbation theory and the variational princip le are examined. The method is illustrated with the study of the vibra tional energies of hydrogen fluoride calculated by deforming continuou sly the solutions of a harmonic oscillator to those of a Morse oscilla tor. 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 d egeneracies or avoided level crossings.