On the ‘‘series method’’ to solve Schrödinger’s equation for the linear oscillator

Gerald I. Kerley

This report discusses two new ideas for using perturbation methods to solve the time-independent Schrödinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix diagonalization methods. That allows a simple, compact derivation of the standard (Rayleigh-Schrödinger) equations. But it also leads to a new iterative solution method that is not based on the usual power series expansion. The iterative method can also be used when the unperturbed system is two-fold degenerate. The second concept is quite different from the first but compatible with it. It is based on the fact that one can replace the true Hamiltonian with a synthetic Hamiltonian having the same eigenvalues. This approach allows one to cancel part of the perturbation and to reduce the size of the off-diagonal matrix elements, giving better convergence of the series. These methods are illustrated by application to three perturbed harmonic oscillator problems--a 1-D oscillator with a linear perturbation, a 1-D oscillator with a quartic perturbation, and a perturbed 2-D oscillator, which involves degenerate states. The iterative method and the synthetic Hamiltonian method are shown to give significantly better results than the standard method.

Erich W. Schmid, Gerhard Spitz, Wolfgang Lösch

Erich W. Schmid, Gerhard Spitz, Wolfgang Lösch

Qi Mou

Symposium is a new calculation method on Hamiltonian mechanics. We use it to solve Schrodinger equation of linear harmonic oscillator on quantum mechanics and perform calculation with computer program made by ourselves,and compare with the ordinary calculation method. The result indicates that the symposium is a very advanced calculation method to solve partial differential equations such as Schrodinger equation.

Konstantin Eduardovich Plokhotnikov

Abstract The paper considers the method of the numerical solution of the Schrodinger equation, which can partly be attributed to the class of Monte Carlo methods. The method is presented and simultaneously illustrated by the examples of solving the one-dimensional and multidimensional Schrodinger equation in the problems of a linear one-dimensional oscillator, hydrogen atom, and benzene atom.

Mike Summers