Phy5646/hydrogen atom lifetime lifetime: Difference between revisions
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For the integrations over x and y we note that all the wavefunctions are even in these variables except for <math>\psi_{21 \pm 1}</math> | For the integrations over x and y we note that all the wavefunctions are even in these variables except for <math>\psi_{21 \pm 1}</math> | ||
<math><\psi_{100}|x|\psi_{21 \pm1}>=\mp \dfrac{1}{8 pi a_o^4}\int r^4 e^{-3r/2a} \sin(\theta)^3 (\cos(\phi)\pm i \sin(\phi))\cos(\phi)dr d\theta d\phi | <math><\psi_{100}|x|\psi_{21 \pm1}>=\mp \dfrac{1}{8 pi a_o^4}\int r^4 e^{-3r/2a} \sin(\theta)^3 (\cos(\phi)\pm i \sin(\phi))\cos(\phi)dr d\theta d\phi = \mp \dfrac{2^7}{3^5}a_o | ||
Revision as of 16:56, 18 April 2010
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}}
We must evaluate equations of the form 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}|r| \psi_{2ab}> }
Exploiting the symmetry of the wavefunctions we find that the only non-zero element for the z compoent 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 <\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 }
Integrating over all space we find;
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} |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 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}}
<math><\psi_{100}|x|\psi_{21 \pm1}>=\mp \dfrac{1}{8 pi a_o^4}\int r^4 e^{-3r/2a} \sin(\theta)^3 (\cos(\phi)\pm i \sin(\phi))\cos(\phi)dr d\theta d\phi = \mp \dfrac{2^7}{3^5}a_o