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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{100}= \dfrac{e^{-r/a_o}}{\sqrt{\pi a_o^3}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{200}= \dfrac{e^{-r/2a_o}}{\sqrt{32\pi a_o^3}} \left( 2-\dfrac{r}{a_o} \right)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{210}= \dfrac{e^{-r/2a_o}}{\sqrt{32\pi a_o^3}} \left( \dfrac{r}{a_o} \right) cos(\theta)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}

The transistion rate is given by the Fermi Golden rule;

$R={\dfrac {\pi }{\epsilon \hbar ^{2}}}|p|^{2}\rho (\omega )$

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,

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

Integrating over all space we find;

$<\psi _{100}|z|\psi _{210}>=-({\dfrac {2^{8}}{3^{5}{\sqrt {2}}}})a_{o}$

For the integrations over x and y we note that all the wavefunctions are even in these variables except for $\psi _{21\pm 1}$

$<\psi _{100}|x|\psi _{21\pm 1}>=\mp {\dfrac {1}{8pia_{o}^{4}}}\int r^{4}e^{-3r/2a}\sin(\theta )^{3}(\cos(\phi )\pm i\sin(\phi ))\cos(\phi )drd\theta d\phi =\mp {\dfrac {2^{7}}{3^{5}}}a_{o}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\psi_{100}|y|\psi_{21 \pm1}>= \mp \dfrac{1}{8 \pi a_o^4} 4! (\dfrac{2a_o}{3})^5 \dfrac{4}{3} \int(\cos(\phi)\pm i \sin(\phi))\sin(\phi) d\phi=-i\dfrac{2^7}{3^5}a_o }

Our ewuation for \omega is as follows;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega= \dfrac{E_{2}-E_{1}}{\hbar}= -\dfrac{3 E_{1}}{4 \hbar}}

This yeilds'

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R= -\dfrac{2^(10)}{3^8}(\dfrac{E_{1}}{m c^2})^2 \dfrac{c}{a_o}= 6.27x10^8 1/s }

This gives a value fore the lifetime of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{210} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{21 \pm1}} states as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau= \dfrac{1}{r}= 1.60x10^-9s }

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{200}} state had matrix elements of 0, this implies that the lifetime is;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau= \dfrac{1}{0}= \infty }

This implies that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{200}} state is stable.