Question

The hydrogen atom wave functions are written as

Answer 1

${\displaystyle \psi _{n\ell m}(r,\theta ,\phi )={\sqrt {{\left({\frac {2}{na_{0}^{*}}}\right)}^{3}{\frac {(n-\ell -1)!}{2n(n+\ell )!}}}}e^{-\rho /2}\rho ^{\ell }L_{n-\ell -1}^{2\ell +1}(\rho )Y_{\ell }^{m}(\theta ,\phi )}\\ \text{where:}\\ {\rho ={2r \over {na_{0}^{*}}}},\\ {\displaystyle a_{0}^{*}}\text{ is the reduced Bohr radius},{ a_{0}^{*}={{4\pi \epsilon _{0}\hbar ^{2}} \over {\mu e^{2}}}},\\ {\displaystyle L_{n-\ell -1}^{2\ell +1}(\rho )} \text{is a generalized Laguerre polynomial of degree }{\displaystyle n-\ell -1}, \text{and}\\ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )}\text{ is a spherical harmonic function of degree }{\displaystyle \ell }\text{ and order} \:m. \\\text{Note that the generalized Laguerre polynomials are defined} \\\text{differently by different authors.} \\\text{The usage here is consistent with the definitions used by Messiah, and Mathematica}.\\\text{In other places, the Laguerre polynomial includes a factor of }{\displaystyle (n+\ell )!},\\\text{ or the generalized Laguerre polynomial appearing in the hydrogen wave function is} \\{\displaystyle L_{n+\ell }^{2\ell +1}(\rho )} \text{instead.}$