Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved (2S−2S) systems

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 a0, where a0 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 (8 states for each atom), the Hamiltonian matrix 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 leadingorder 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 χ (a0/R) 6 , where E h and L are the Hartree and Lamb shift energies, respectively, and χ = E h /L ≈ 6.22 × 10 6 is their ratio.

Long-Range Interactions for Hydrogen: 6P–1S and 6P–2S Systems

“…The atomic states can be classified according to the quantum number F z ; the z component of the total angular momentum commutes [11] with the total Hamiltonian given in Equation (3). Within the 6P 1/2 -6P 3/2 -1S 1/2 system, the states in the manifold F z = 3 are given as follows,…”

“…An adjacency graph [11,24] shows that the matrix is irreducible, which in particular implies that there are no hidden symmetries in the Hamiltonian matrix H F z =2 , which would otherwise lead to a further decomposition into irreducible submatrices.…”

“…A completely different situation is encountered when both atoms are in excited states or when the excited state is accessible from the ground state via an allowed electric-dipole transition [11]. In this case, one encounters nonvanishing first-order van der Waals interaction matrix elements instead of second-order effects.…”

“…For the 2S-2P system, this problem has been analyzed in [12], on the basis of nonrelativistic Schrödinger theory. However, in order to evaluate the distance-dependent frequency shift of hyperfine-resolved transitions, one has to invoke a more sophisticated analysis, which has recently been performed in [11]. A complete rediagonalization of the total Hamiltonian, comprising Lamb shift, fine-structure and hyperfine effects, becomes necessary.…”

Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-degenerate States: Applied Graph Theory

“…For example, in a (24 × 24) matrix, as has been recently encountered in our analysis of the 2S-2S hyperfine-resolved interactions in hydrogen [3], entries can follow a rather irregular pattern, and the analysis then becomes far less trivial. The possibility to break up a matrix into irreducible subcomponents is equivalent to a search for "hidden" symmetries of the interaction which imply that only sublevels of specific symmetry are coupled.…”