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 ϒ_nlm (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 / (2π)^{1 / 2}e^±imΦ} {((2l+1)(l-m)! / 2(l+m)!)^{1 / 2}P_l^m(cosΘ)} {π2^2l+4l! / (γh)^{3 / 2}(n(n-l-1)! / (n+l)!)^{1 / 2}ζ^l / (ζ²+1)^l+2C_n-l-1^l+1(ζ²-1 / ζ²+1)} in which ζ=(2π / γh)P, with γ=(4π²μe²Z / nh²)=(Z / na₀). 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.