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 Υₙₗₘ(P, Θ, Φ), in which P, Θ, and Φ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates r, θ, and φ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${{Υ}}ₙₗₘ(P, {Θ}, {Φ})=\{{1}{{(2{π})}1/2}{e}^{±{}im{Φ}}\} \{{({(2l+1)(l{-}m)!}{2(l+m)!})}1/2{{P}ₗ}ᵐ(cos{Θ})\}$ $\{{{π}{2}²ˡ⁺⁴l!}{{({γ}h)}3/2}{({n(n{-}l{-}1)!}{(n+l)!})}1/2{{{ζ}}ˡ}{{({{ζ}}²+1)}ˡ⁺²}{C}_{n{-}l{-}1}ˡ⁺¹({{{ζ}}²{-}1}{{{ζ}}²+1})\}$ in which ${ζ}=({2{π}}{{γ}h})P$, with ${γ}=({4{{π}}²{μ}{e}²Z}{n{h}²})=({Z}{n{a}₀})$. The probability ${{Ξ}}ₙₗ(P)dP$ 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.