On matrix elements of the vector physical quantities

TL;DR

This paper revisits classical angular momentum techniques in quantum mechanics to derive formulas for matrix elements of vector quantities, applying them to symmetric systems and exploring algebraic relations in complex Coulomb problems.

Contribution

It provides a modern re-derivation of angular momentum matrix element formulas and explores their application to quantum systems with additional constraints and symmetries.

Findings

Derived general formulas for matrix elements of vector operators.

Applied techniques to quantum systems with spherical symmetry.

Analyzed algebraic structures in Coulomb problems with two centers.

Abstract

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 $\hat{\bf J}^{2}$ and vector-operator $\hat{\bf A}$ 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 E. Fermi and A. 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).