On matrix elements of the vector physical quantities

Methods of angular momenta are modified and used to solve some actual problems in quantum mechanics. In particular, we re-derive some known formulas for analytical and numerical calculations of matrix elements of the vector physical quantities. These formulas are applied to a large number of quantum...

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Veröffentlicht in:The European physical journal. D, Atomic, molecular, and optical physics Jg. 79; H. 2; S. 13
1. Verfasser: Frolov, Alexei M.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2025
Springer Nature B.V
Schlagworte:
Algebra
Angular momentum
Applications of Nonlinear Dynamics and Chaos Theory
Atomic
Commutators
Euclidean geometry
Euclidean space
Exact solutions
Mathematical analysis
Matrices (mathematics)
Molecular
Operators (mathematics)
Optical and Plasma Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Information Technology
Quantum mechanics
Quantum Physics
Quantum theory
Regular Article - Atomic Physics
Spectroscopy/Spectrometry
Spintronics
Vectors (mathematics)
Wave functions
ISSN:1434-6060, 1434-6079
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Zusammenfassung:Methods of angular momenta are modified and used to solve some actual problems in quantum mechanics. In particular, we re-derive some known formulas for analytical and numerical calculations of matrix elements of the vector physical quantities. These formulas are applied to a large number of quantum systems which have an explicit spherical symmetry. Multiple commutators of different powers of the angular momenta J ^ 2 and vector-operator A ^ are determined in the general form. Calculations of the expectation values averaged over orbital angular momenta are also described in detail. This effective and elegant old technique, which was successfully used by Enrico Fermi and Aage Bohr, is almost forgotten in modern times. We also discuss quantum systems with additional relations (or constraints) between some vector-operators and orbital angular momentum. For similar systems such relations allow one to obtain some valuable additional information about their properties, including the bound state spectra, correct asymptotics of actual wave functions, etc. As an example of unsolved problems we consider applications of the algebras of angular momenta to investigation of the one-electron, two-center (Coulomb) problem ( Q 1 , Q 2 ) . For this problem it is possible to obtain the closed analytical solutions which are written as the ‘correct’ linear combinations of products of the two one-electron wave functions of the hydrogen-like ions with the nuclear charges Q 1 + Q 2 and Q 1 - Q 2 , respectively. However, in contrast with the usual hydrogen-like ions such hydrogenic wave functions must be constructed in three-dimensional pseudo-Euclidean space with the metric ( - 1 , - 1 , 1 ). Graphical abstract
Bibliographie:ObjectType-Article-1
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ISSN:1434-6060
1434-6079
DOI:10.1140/epjd/s10053-025-00966-3