STUDIES IN PERTURBATION THEORY. PART I. AN ELEMENTARY ITERATION-VARIATION PROCEDURE FOR SOLVING THE SCHRODINGER EQUATION BY PARTITIONING TECHNIQUE

Journal Article · Tue Jan 01 04:00:00 EST 1963 · Journal of Molecular Spectroscopy

DOI:https://doi.org/10.1016/0022-2852(63)90151-6· OSTI ID:4705991

Loewdin, P

The fundamental Schrodinger equation in quantum mechanics may be transformed to a discrete representation by introducing a complete set of basic functions in which the eigenfunctions may be developed. The eigenvalues are then determined by solving a secular equation, and this problem is attacked by a partitioning that leads to an implicit relation for the energy E of the form E = f(E), which corresponds to the Schrodinger- Brillouin perturbation formula but has a more condensed form. The first-order iteration process based on the relation E/sup (k+1)/ = f{E/sup (k)/} is studied, and it is shown that, independent of whether this process is convergent or not, one can go over to a secondorder process that appears to be closely connected with the variational expression. This second-order iterative procedure turns out to be very convenient for numerical work, particularly since it treats degenerate eigenvalues just as easily as the single eigenvalues. The general behavior of the curve y = E-f(E) is discussed, and the method is illustrated by a few numerical examples. In an appendix, a brief survey of the classification of iteration processes is also given. (auth)

• Research Organization: Uppsala Univ.; Univ. of Florida, Gainesville
• NSA Number: NSA-17-024132
• OSTI ID: 4705991
• Journal Information: Journal of Molecular Spectroscopy, Journal Name: Journal of Molecular Spectroscopy Journal Issue: 1-6 Vol. Vol: 10; ISSN 0022-2852
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

EIGENFUNCTIONS
EIGENVALUES
ENERGY
EQUATIONS
NUMERICALS
PERTURBATION THEORY
PHYSICS
QUANTUM MECHANICS
SCHROEDINGER EQUATION