Quantum Mechanical Formulas with Applications to Bound States

Martin Fraas, Bruno Nachtergaele, Simone Warzel +1 more

Orest Hryhorchak

A method for determining the quantum wave impedance function for a quantum-mechanical system with an arbitrary piecewise constant potential has been developed.Using this method, both the well-known iterative formula [A.N. Khondker, M. R. Khan, A. F. M. Anwar, J. Appl.Phys.63, 5191 (1988)] and an alternative approach for calculating quantum wave impedance were derived.This approach was applied to both scattering and bound state cases.The general form of the wave functions for bound states, as well as the transmission and reection coecients in the scattering case, were obtained for systems with a potential consisting of an arbitrary number of rectangular barriers and wells.The application of the developed method was demonstrated by describing symmetric rectangular double-well and double-barrier quantum-mechanical systems.

Il-W Kim, U. Sukhatme

The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.

Weder, Ricardo, Pavel Exner

Thomas Hoffmann-Ostenhof, Maria Hoffmann-Ostenhof, Gottfried Olbrich

A variational lower bound to the overlap of the ground state psi 0 of a quantum mechanical system with an approximate wavefunction phi is given. A simplified non-variational formula is deduced from this lower bound to mod ( phi mod psi 0 ) mod . Both versions are illustrated with an application to the hydrogen atom and are compared with the lower bounds of Eckart and Wang. The non-variational version turns out to be a useful alternative to Eckart's bound.