Multipole expansions and Fock symmetry of the Hydrogen atom

TL;DR

This paper develops a method using multipole expansions of four-dimensional hyperspherical harmonics to compute hydrogen atom matrix elements, simplifying calculations related to Fock symmetry and enabling recursive relations.

Contribution

It introduces a novel application of multipole expansions for hyperspherical harmonics to evaluate hydrogen atom matrix elements explicitly and efficiently.

Findings

Derived explicit formulas for hydrogen matrix elements.

Presented a new operator representation as derivatives of an elementary function.

Established recurrency relations for matrix elements across quantum states.

Abstract

The main difficulty in utilizing the O(4) symmetry of the Hydrogen atom in practical calculations is the dependence of the Fock stereographic projection on energy. This is due to the fact that the wave functions of the states with different energies are proportional to the hyperspherical harmonics (HSH) corresponding to different points on the hypersphere. Thus, the calculation of the matrix elements reduces to the problem of re-expanding HSH in terms of HSH depending on different points on the hypersphere. We solve this problem by applying the technique of multipole expansions for four-dimensional HSH. As a result, we obtain the multipole expansions whose coefficients are the matrix elements of the boost operator taken between hydrogen wave functions (i.e. hydrogen form-factors). The explicit expressions for those coefficients are derived. It is shown that the hydrogen matrix elements can be presented as derivatives of an elementary function. Such an operator representation is convenient for the derivation of recurrency relations connecting matrix elements between states corresponding to different values of the quantum numbers $n$ and $l$.