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Oscillator and hydrogenic matrix elements by operator algebra
Author(s) -
Witschel W.,
Grosswendt B.
Publication year - 1974
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560080409
Subject(s) - operator (biology) , harmonic oscillator , mathematics , bessel function , matrix (chemical analysis) , ladder operator , mathematical physics , angular momentum , algebra over a field , simple (philosophy) , gauss , element (criminal law) , integer (computer science) , quantum mechanics , mathematical analysis , physics , pure mathematics , compact operator , materials science , repressor , law , chemistry , computer science , composite material , biochemistry , political science , transcription factor , programming language , extension (predicate logic) , gene , philosophy , epistemology
Occupation number representation of the two‐dimensional harmonic oscillator and some operator formulae are used in a simple algebraic derivation of complicated integrals. The calculation of full oscillator‐ and radial integrals of r̂ w and exp ( \documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document} ), where w is an arbitrary positive or negative integer, are performed by an integral transform, leading to a generalized Gauss matrix element. Thus it is possible, because of the back transformation, to derive from one generalized Gauss matrix element all matrix elements which are permitted by the selection rules. Some integrals of r̂ w and exp ( \documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document} ), Laguerre polynomials, and Bessel functions are completely new. For the already known integrals, the mathematical labour is considerably reduced. The relation between the two‐dimensional oscillator and the hydrogen atom and their angular momentum properties are discussed. A survey on the various methods applied to the oscillator problem, from complex integration to noncompact Lie groups, and a comprehensive bibliography on this important spectroscopic field are given.