The Momentum Distribution in Hydrogen-Like Atoms

Abstract

The probability density that an electron have certain momenta is given by the square of the absolute magnitude of a momentum eigenfunction ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})$, in which $P$, $\ensuremath{\Theta}$, and $\ensuremath{\Phi}$ are spatial polar coordinates of the total momentum vector referred to the same axes as the coordinates $r$, $\ensuremath{\theta}$, and $\ensuremath{\varphi}$ of the electron. The following general expression for these functions for a hydrogen-like atom is obtained: ${\ensuremath{\Upsilon}}_{\mathrm{nlm}}(P, \ensuremath{\Theta}, \ensuremath{\Phi})=\left\{\frac{1}{{(2\ensuremath{\pi})}^{\frac{1}{2}}}{e}^{\ifmmode\pm\else\textpm\fi{}\mathrm{im}\ensuremath{\Phi}}\right\} \left\{{\left(\frac{(2l+1)(l\ensuremath{-}m)!}{2(l+m)!}\right)}^{\frac{1}{2}}{{P}_{l}}^{m}(cos\ensuremath{\Theta})\right\}$ $\left\{\frac{\ensuremath{\pi}{2}^{2l+4}l!}{{(\ensuremath{\gamma}h)}^{\frac{3}{2}}}{\left(\frac{n(n\ensuremath{-}l\ensuremath{-}1)!}{(n+l)!}\right)}^{\frac{1}{2}}\frac{{\ensuremath{\zeta}}^{l}}{{({\ensuremath{\zeta}}^{2}+1)}^{l+2}}{C}_{n\ensuremath{-}l\ensuremath{-}1}^{l+1}\left(\frac{{\ensuremath{\zeta}}^{2}\ensuremath{-}1}{{\ensuremath{\zeta}}^{2}+1}\right)\right\}$ in which $\ensuremath{\zeta}=(\frac{2\ensuremath{\pi}}{\ensuremath{\gamma}h})P$, with $\ensuremath{\gamma}=(\frac{4{\ensuremath{\pi}}^{2}\ensuremath{\mu}{e}^{2}Z}{n{h}^{2}})=(\frac{Z}{n{a}_{0}})$. The probability ${\ensuremath{\Xi}}_{\mathrm{nl}}(P)\mathrm{dP}$ 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.