# Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved 2S-2S system

## TL;DR

This paper analyzes the hyperfine-resolved long-range interactions between two excited hydrogen atoms in metastable states, focusing on van der Waals shifts and symmetry reductions to determine interaction energies.

## Contribution

It provides a detailed Hamiltonian analysis and calculates leading-order van der Waals shifts for hyperfine states in the 2S-2S hydrogen system, considering quasi-degenerate levels.

## Key findings

- Interaction energy scales as (a_0/R)^6 with hyperfine effects
- Hamiltonian symmetry analysis reduces complexity from 64 to 12 dimensions
- Leading-order shifts are proportional to 1/R^3 and 1/R^6

## Abstract

The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasi-degenerate 2P_{1/2} levels which are removed from the 2S states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift and the hyperfine effects. The van der Waals shift becomes commensurate with the 2S-2P_{3/2} fine-structure splitting only for close approach (R < 100 a_0, where a_0 is the Bohr radius) and one may thus restrict the discussion to the levels with n=2 and J=1/2 to good approximation. Because each S or P state splits into an F=1 triplet and an F=0 hyperfine singlet (eight states for each atom), the Hamiltonian matrix {\em a priori} is of dimension 64. A careful analysis of symmetries the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states which are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to 1/R^3 and 1/R^6 and are evaluated within the hyperfine manifolds. When both atoms are metastable 2S states, we find an interaction energy of order E_h chi (a_0/R)^6, where E_h and L are the Hartree and Lamb shift energies, respectively, and chi = E_h/L ~ 6.22 \times 10^6 is their ratio.