Talk:Phy5646

From PhyWiki
Revision as of 11:28, 29 April 2010 by ChorChan (talk | contribs)
Jump to navigation Jump to search

Question: A monoatomic atom undergo spontaneous emission. It changes from an excited state 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 |e\rangle} with 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 l=0} to an intermediate state 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 |i\rangle} with 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 l=1} emitting a photon with wave vector 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 \mathbf{k}} , and then to the ground state 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 |g\rangle} with 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 l=0} emitting a photon with wave vector 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 \mathbf{k'}} . Find the probability that the angle between the two wavevectors to be 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 \theta } .

Ans:

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 \mathcal{H}=\frac{1}{2m}\left(p-\frac{e}{c}A(r)\right)^2+V(r)+\sum_{k,\hat{\lambda_k}}\hbar\omega_{k}\left(\hat{a}_{k\hat{\lambda_k}}^{\dagger}\hat{a}_{k\hat{\lambda_k}}+\frac{1}{2}\right) }

where 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 \mathbf{\hat{A}(r)}=\frac{1}{\sqrt{V}}\sum_{k,\lambda_k}\left[\sqrt{\frac{2\pi\hbar}{\omega_{k}}}c\;\left(\hat{a}_{k,\hat{\lambda_k}}\hat{\lambda_k}e^{ik\cdot r}+\hat{a}^{\dagger}_{k,\hat{\lambda_k}}\hat{\lambda^*_k}e^{-ik\cdot r}\right)\right]}



Using second order time dependent perturbation theory , we can write the wavefunction in Dirac picture 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 |\chi(t)\rangle\approx|\chi_0\rangle+\frac{1}{i\hbar}\int_{-\infty}^{t}dt'\mathcal{H}'_I(t')|\chi_0\rangle+ \frac{1}{(i\hbar)^2}\int_{-\infty}^{t}dt'\int_{-\infty}^{t'}dt''\mathcal{H}'_I(t')\mathcal{H}'_I(t'')|\chi_0\rangle }

where

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 \mathcal{H}'_I(t)=e^{\frac{i}{\hbar}\mathcal{H}^{(at)}_0t}\left( -\frac{e}{mc}A(r,t)\cdot p+\frac{e^2}{2mc^2}A(r,t)\cdot A(r,t)\right)e^{-\frac{i}{\hbar}\mathcal{H}^{(at)}_0t}}

with

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 \mathbf{A(r,t)}=\frac{1}{\sqrt{V}}\sum_{k,\lambda_k}\left[\sqrt{\frac{2\pi\hbar}{\omega_{k}}}c\;\left(\hat{a}_{k,\hat{\lambda_k}}\hat{\lambda_k}e^{ik\cdot r-i\omega_{k} t}+\hat{a}^{\dagger}_{k,\hat{\lambda_k}}\hat{\lambda^*_k}e^{-ik\cdot r+i\omega_{k}t}\right)\right]}

Since the system has rotational symmetry, so the internal eigenstate 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 |n,l,m\rangle} , where 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 |n\rangle=|e\rangle,|i\rangle,|g\rangle} . The initial photon field is null 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 |\phi\rangle} . Initially 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 l=0} , so 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 m=0} , so intial state 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 |\chi_0\rangle=|e,0,0;\phi\rangle} and final state 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 |g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' }\rangle} . Therefore


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 \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' }|\chi(t)\rangle\approx \frac{1}{i\hbar}\int_{-\infty}^{t}dt' \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' } |\mathcal{H}'_I(t') |e,0,0;\phi\rangle+ \frac{1}{(i\hbar)^2}\int_{-\infty}^{t}dt'\int_{-\infty}^{t'}dt'' \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' } |\mathcal{H}'_I(t')\mathcal{H}'_I(t'')|e,0,0;\phi\rangle }

where to change from state 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 |e\rangle} to 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 |i\rangle} to 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 |g\rangle} , we need two momentum operator, so the first term must be zero. And so

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 \begin{align} & \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'} }|\chi(t)\rangle \\ &\approx \frac{1}{(i\hbar)^2}\int_{-\infty}^{t}dt'\int_{-\infty}^{t'}dt'' \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' } |\mathcal{H}'_I(t')\mathcal{H}'_I(t'')|e,0,0;\phi\rangle \\ &=\frac{1}{(i\hbar)^2}\int_{-\infty}^{t}dt'\int_{-\infty}^{t'}dt'' \sum_{m=-1,0,1} \langle g,0,0;1_{\lambda_k ,k },1_{\lambda '_{k'},k' } |\mathcal{H}'_I(t') |i,1,m;1_{\lambda ,k } \rangle \langle i,1,m;1_{\lambda ,k }| \mathcal{H}'_I(t'')|e,0,0;\phi\rangle \\ &=\sum_{m=-1,0,1} f(t,w_k,w_k') \left(\langle g,0,0|p|i,1,m\rangle \cdot \hat{\lambda^{'*}_{k'}} \right)\left( \langle i,1,m|p|e,0,0\rangle\cdot \hat{\lambda^{*}_{k}}\right ) \end{align} }

Retrieved from "http://wiki.physics.fsu.edu/index.php?title=Talk:Phy5646&oldid=8167"