The Momentum Distribution in Hydrogen-Like Atoms

The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ₍₋₌ (P, , ), in which P, , and are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates r, , and of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ₍₋₌ (P, , ) =\1{{ (2) ^1{2}}e^\} \{ ( (2l+1) (l{-m) !2 (l+m) !) }^1{2}{P₋}^m (cos) \} \{{2^2l+4l!}{ () ^3{2}} (n (n{-l-1) ! (n+l) !) }^1{2}{^l}{ ({^2+1) }^l+2}C₍-₋-₁^l+1 ({^2-1}{^2+1}) \} in which = (2) P, with = (4{^2e^2Z}n{h^2}) = (Zn{a₀}). The probability ₍₋ (P) dP that the electron have a total momentum lying within the limits P and P+dP is also evaluated, and it is shown that the root mean square of the total momentum is equal to the momentum of the electron in a circular Bohr orbit with the same total quantum number.