Excited Hydrogen Atom Lifetime.
We start with the wavefunctions of the ground and first excited state of the hydrogen atom.
ψ 100 = e − r / a o π a o 3 {\displaystyle \psi _{100}={\dfrac {e^{-r/a_{o}}}{\sqrt {\pi a_{o}^{3}}}}}
ψ 200 = e − r / 2 a o 32 π a o 3 ( 2 − r a o ) {\displaystyle \psi _{200}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left(2-{\dfrac {r}{a_{o}}}\right)}
ψ 210 = e − r / 2 a o 32 π a o 3 ( r a o ) c o s ( θ ) {\displaystyle \psi _{210}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left({\dfrac {r}{a_{o}}}\right)cos(\theta )}
ψ 21 ± 1 = e − r / 2 a o 64 π a o 3 ( r a o ) s i n ( θ ) e ± i ϕ {\displaystyle \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 }}
