Two Simple Solutions of the Schrödinger Equation

Jochen Pade

Mario Schönberg

Paul Caylor McKinney

Abstract Inseparable Schrödinger equations usually cannot be solved analytically. Most often their solutions are obtained numerically. It would be helpful to have an inseparable Schrödinger equation whose solution can be completely calculated. Perhaps more can be learned about appropriate wave function solutions of complex problems if the solutions of an inseparable Schrödinger equation, solved in terms of simple functions, are available. Such a study is possible using the Dirac delta function as the potential energy for two particles moving on a line. The Dirac delta function problem leads naturally to a consideration of the wave functions' symmetry properties. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

R. McWeeny

Anirban Bose

A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptomatic solution of the Schrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained. This equation has a recursive property which has been exploited to obtain the solutions of the equation. It is all about reconstruction of the equation with different solutions of that unknown function by repetitive differentiation and that ultimately leads to the wave functions of the Schrodinger equation for harmonic oscillator.

Bruce W. Shore, Michael Austin Johnson

We demonstrate the manner in which solutions of the (coherent) Schrödinger equation for multistep excitation approach those of an (incoherent) rate equation. For a two-level system, coherence produces oscillations about the mean rate-equation values. In more general N-level sequences the two types of solutions agree in the limit described by Ackerhalt and Eberly. We point out instances in which a simple rate equation cannot duplicate results of the Schrödinger equation.