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 ${\Upsilon }_{\mathrm{nlm}}(P, \Theta , \Phi )$, in which $P$, $\Theta$, and $\Phi$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates $r$, $\theta$, and $\phi$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\Upsilon }_{\mathrm{nlm}}(P, \Theta , \Phi )=\left\{\frac{1}{{\left(2\mathrm{}\pi \right)}^{\frac{1}{2}}}{e}^{\pm \mathrm{im}\Phi }\right\} \left\{{\left(\frac{\mathrm{}(2l+1)\mathrm{}(l-m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{P_l}^m\left(cos\Theta \right)\right\}$ $\left\{\frac{\pi 2^{2l+4}l!}{{\left(\gamma h\right)}^{\frac{3}{2}}}{\left(\frac{n(n-l-1\mathrm{})!}{\mathrm{}(n+l)!}\right)}^{\frac{1}{2}}\frac{{\zeta }^l}{{({\zeta }^2+1)}^{l+2\mathrm{}}}{C}_{n-l-1\mathrm{}}^{l+1\mathrm{}}\left(\frac{{\zeta }^2-1\mathrm{}}{{\zeta }^2+1\mathrm{}}\right)\right\}$ in which $\zeta =\left(\frac{2\pi }{\gamma h}\right)P$, with $\gamma =\left(\frac{4{\pi }^2\mu e^{2}Z}{n{h}^2}\right)=\left(\frac{Z}{n{a}_0}\right)$. The probability ${\Xi }_{\mathrm{nl}}\mathrm{}\left(P\right)\mathrm{dP}$ 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.