On a Class of Non-separable Quantum-Mechanical Eigenvalue Problems: Analytical and Technical Considerations within the Frame of a Born Expansion Method
✍ Scribed by S. Barcza
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 571 KB
- Volume
- 110
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
An idea of Born is reviewed and elaborated to non-separable quantum-mechanical eigenvalue problems in which the Schrödinger equation can be solved exactly for a subconfiguration. (By subconfiguration we mean a subsystem in which one dynamic variable of the whole system is considered as parameter; derivations with respect to this variable are omitted.) The eigenfunctions in the subconfiguration (e.g., the eigenfunctions of a Born-Oppenheimer approximation) are used as a basis to expand the eigenfunction of the complete problem. By analytical methods it is shown how to construct the complete ensemble of solutions which can be systematically mapped and classified by their analytical betiaviour in one of the singularities (in a regular singularity). A modification of the Numerov procedure is given to the numerical solution of the coupled second-order ordinary differential equations which arise from our treatment. The analytical asymptotic solutions are used to bridge over the asymptotic regions in which the error of the Numerov procedure is large. As a concrete example the comprehensive asymitotic analysis of the Schrödinger equation of a hydrogen-like ion in strong homogeneous magnetic field is presented, practical methods and computational aspects are discussed, and finally a few actual numerical results are reported: some energy levels are given as a function of field strength. 1994 Academic Piess. Inc.