Phy5646/hydrogen atom lifetime lifetime: Difference between revisions
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<math><\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 </math> | <math><\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 </math> | ||
Our | Our equation for <math>\omega</math> is as follows; | ||
<math>\omega= \dfrac{E_{2}-E_{1}}{\hbar}= -\dfrac{3 E_{1}}{4 \hbar}</math> | <math>\omega= \dfrac{E_{2}-E_{1}}{\hbar}= -\dfrac{3 E_{1}}{4 \hbar}</math> | ||
This | This yields | ||
<math> R= -\dfrac{2^(10)}{3^8}(\dfrac{E_{1}}{m c^2})^2 \dfrac{c}{a_o}= 6.27x10^8 1/s </math> | <math> R= -\dfrac{2^(10)}{3^8}(\dfrac{E_{1}}{m c^2})^2 \dfrac{c}{a_o}= 6.27x10^8 1/s </math> | ||
This gives a value fore the lifetime of the <math> \psi_{210} </math>and <math>\psi_{21 \pm1}</math> states as <math> \tau= \dfrac{1}{r}= 1. | This gives a value fore the lifetime of the <math> \psi_{210} </math>and <math>\psi_{21 \pm1}</math> states as <math> \tau= \dfrac{1}{r}= 1.60\times10^{-9}s </math> | ||
The <math> \psi_{200}</math> state had matrix elements of 0, this implies that the lifetime is; | The <math> \psi_{200}</math> state had matrix elements of 0, this implies that the lifetime is; | ||
Latest revision as of 16:27, 27 April 2010
Excited Hydrogen Atom Lifetime.
(Submitted by group 3)
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;
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{\pi}{\epsilon \hbar^2} |p|^2 \rho(\omega) }
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}}
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}|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 }
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 equation 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 \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 yields
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}= 1.60\times10^{-9}s }
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.