Multipole matrix elements for hydrogen atom

Abstract

The orthogonality relations of the eigenfunctions in the discrete and continuum states for the hydrogen-like atom are proved. The multipole matrix elements ⟨n, l|ra|m, j⟩ are given as the analytic expressions which are represented by Appell's hypergeometric function of two variables. For the particular case, the electric dipole and electric quadrupole matrix elements ⟨n, l|r|m, l − 1⟩, ⟨n, l|r2|m, l⟩ and ⟨n, l|r2|m, l − 2⟩ are represented as the sum of the hypergeometric functions and the electric dipole matrix elements ⟨n, l|r|m, l − 1⟩ are consistent with the results obtained by Gordon.