Phy5646/hydrogen atom lifetime lifetime

Excited Hydrogen Atom Lifetime.

We start with the wavefunctions of the ground and first excited state of the hydrogen atom.

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

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

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

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

We must evaluate equations of the form $<\psi _{100}|r|\psi _{2ab}>$

Exploiting the symmetry of the wavefunctions we find that the only non-zero element for the z compoent is,

<math> <\psi_{100} |z| \psi_{210}>= -\dfrac{1}{4 \sqrt{2} \pi a_o^4}\int r^4 e^{-3r/2a} \cos(\theta)^2 \sin(\theta) dr d\theta d\phi