Hydrogen atom in the momentum representation

Abstract

The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables ${p}_{r}$, ${p}_{\ensuremath{\theta}}$, ${p}_{\ensuremath{\varphi}}$ which are associated with the spherical polar variables $r$, $t(={e}^{i\ensuremath{\pi}cos\ensuremath{\theta}})$ and $v({e}^{i\ensuremath{\varphi}})$. The functions may be written as ${{\ensuremath{\varphi}}_{n,l,m}({p}_{r},{p}_{\ensuremath{\theta}},{p}_{\ensuremath{\varphi}})={\ensuremath{\alpha}}_{n,l}({p}_{r}){\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})\ensuremath{\delta}({p}_{\ensuremath{\varphi}}+m),}{{\ensuremath{\alpha}}_{n,l}({p}_{r})={\left(\frac{n}{i{p}_{0}}\right)}^{3l+z}\ensuremath{\Sigma}\stackrel{n\ensuremath{-}l\ensuremath{-}1}{k=0}\left({n\ensuremath{-}l\ensuremath{-}1}{k}\right)\frac{{2}^{k\ensuremath{-}l+1}}{(2l+k+1)(2l+k)\ensuremath{\cdots}(l+k+2)}{\left(\frac{\frac{i{p}_{0}}{n}}{{p}_{r}\ensuremath{-}\frac{i{p}_{0}}{n}}\right)}^{l+k+2},}{{\ensuremath{\beta}}_{l}^{m}({p}_{\ensuremath{\theta}})=\ensuremath{\Sigma}\stackrel{E(\frac{l}{2})}{j=E(\frac{m}{2})}{{a}_{j}{p}_{\ensuremath{\theta}}}^{\ensuremath{-}\frac{(m+1)}{2}}{J}_{\frac{j\ensuremath{-}(m\ensuremath{-}1)}{2}}({p}_{\ensuremath{\theta}}),}$ where $\ensuremath{\delta}$ is the Dirac delta function and ${p}_{0}=Z{a}_{0}^{\ensuremath{-}1}$ (${a}_{0}$ is the Bohr radius). ${J}_{\ensuremath{\nu}}({p}_{0})$ are Bessel functions of integer or half-integer order. The radial functions are shown to have poles up to order $n+1$ at ${p}_{r}=i\frac{{p}_{0}}{n}$.