A comparison of the hydrogenlike dipole radial matrix elements with overlap integrals and a step toward explicit expressions of the multipole matrix elements

摘要

For nonrelativistic radial hydrogenlike wavefunctions R_(nl)(r), the ratio　＜R_(vλ),|r|R_(nl)) (R_(vλ)|R_(nl)＞ of the dipole to the overlap integral is proportional to (l·ΔE)~(-1) for 1 =λ + 1.ΔE is the energy difference between the shells v and n. This property holds for discrete or continuous states, and can be derived as a particular case of the Feneuille recurrence formula (Feneuille 1970 C. R. Acad. Sci., Paris B 271 992). This relation and Feinberg's comparable one for ＜R_(vλ)|r~(-2)|R_(nl)＞ suggest the existence of explicit expressions for the matrix elements　＜R_(vλ)|r~t|R_(nl)＞ For v≠n we are not yet able to write down a general formula valid for any l and λ, but we show with examples (λ = 0, l = 1 and A = 1, l = 0) how explicit formulae can be obtained for fixed values of l and λ.