Phy5646/hydrogen atom lifetime

The First Excited State of Hydrogen

Let us use time-dependent perturbation theory to determine the lifetime of the first excited state of the hydrogen atom. We will began by using the wave funtions

$\psi _{100}={\dfrac {e^{-r/a_{o}}}{\sqrt {\pi a_{o}^{3}}}}$

$\psi _{200}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left(2-{\dfrac {r}{a_{o}}}\right)$

$\psi _{210}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left({\dfrac {r}{a_{o}}}\right)cos(\theta )$

$\psi _{21\pm 1}={\dfrac {e^{-r/2a_{o}}}{\sqrt {64\pi a_{o}^{3}}}}\left({\dfrac {r}{a_{o}}}\right)sin(\theta )e^{\pm i\phi }$

In computing the lifetime we must evaluate matrix elements of the form $<\psi _{100}|r|\psi _{2ab}>$

All the wavefunctions except for $\psi _{210}$ are even in z, therefore the only nonzero matrix element for z is,

$<\psi _{100}|z|\psi _{210}>=-{\dfrac {1}{4\pi a_{o}^{4}{\sqrt {2}}}}\int r^{4}e^{-3r/2a_{o}}\cos(\theta )^{2}\sin(\theta )drd\theta d\phi$

Integrating over all space we obtain;

<math><\psi_{100} |z| \psi_{210}>=-\dfrac{2^8}{3^5 \sqrt{2}}a_o