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Question: A monoatomic atom undergo spontaneous emission. It changes from an excited state <math>|e\rangle</math> with <math>l=0</math> to an intermediate state <math>|i\rangle</math> with <math>l=1</math> emitting a photon with wave vector <math>\mathbf{k}</math>, and then to the ground state <math>|g\rangle</math> with <math>l=0</math> emitting a photon with wave vector <math>\mathbf{k'}</math>. Find the probability that the angle between the two wavevectors to be <math> \ | Question: A monoatomic atom undergo spontaneous emission. It changes from an excited state <math>|e\rangle</math> with <math>l=0</math> to an intermediate state <math>|i\rangle</math> with <math>l=1</math> emitting a photon with wave vector <math>\mathbf{k}</math>, and then to the ground state <math>|g\rangle</math> with <math>l=0</math> emitting a photon with wave vector <math>\mathbf{k'}</math>. Find the probability that the angle between the two wavevectors to be <math> \gamma </math>. | ||
Ans: | Ans: | ||
<math>\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) </math> | <math>\mathcal{H}=\frac{1}{2m}\left(\mathbf{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) </math> | ||
where | where | ||
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<math>\mathcal{H}'_I(t)=e^{\frac{i}{\hbar}\mathcal{H}^{(at)}_0t}\left( | <math>\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}</math> | -\frac{e}{mc}A(r,t)\cdot \mathbf{p}+\frac{e^2}{2mc^2}A(r,t)\cdot A(r,t)\right)e^{-\frac{i}{\hbar}\mathcal{H}^{(at)}_0t}</math> | ||
with | with | ||
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&=\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 \\ | &=\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 ) | &=\sum_{m=-1,0,1} f(t,w_k,w_k') \left(\langle g,0,0|\mathbf{p}|i,1,m\rangle \cdot \hat{\lambda^{'*}_{k'}} \right)\left( \langle i,1,m|\mathbf{p}|e,0,0\rangle\cdot \hat{\lambda^{*}_{k}}\right ) | ||
\end{align} </math> | \end{align} </math> | ||
Here | |||
<math>\begin{align} \frac{\mathbf{p}}{m}&=\frac{i}{\hbar} [\mathcal{H}^{(at)}_0,\mathbf{r}] \\ | |||
\langle i,1,m| \frac{\mathbf{p}}{m} |e,0,0\rangle &=\frac{i}{\hbar} \langle i,1,m| [\mathcal{H}^{(at)}_0,\mathbf{r}] |e,0,0\rangle \\ | |||
&=\frac{i}{\hbar} (E_{i}-E_{e}) \langle i,1,m|\mathbf{r} |e,0,0\rangle \\ | |||
\langle g,0,0| \frac{\mathbf{p}}{m} |i,1,m\rangle &=\frac{i}{\hbar} (E_{g}-E_{i}) \langle g,0,0| \mathbf{r} |i,1,m\rangle | |||
\end{align} </math> | |||
Define | |||
<math>\begin{align} | |||
r_+&=x+iy=rsin\theta e^{i\phi} \\ | |||
r_-&=x-iy=rsin\theta e^{-i\phi} \\ | |||
x&=(r_+ + r_-)/2 \\ | |||
y&=(r_+ - r_-)/2i \end{align} </math> | |||
Revision as of 11:55, 29 April 2010
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 \gamma } .
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(\mathbf{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 \mathbf{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|\mathbf{p}|i,1,m\rangle \cdot \hat{\lambda^{'*}_{k'}} \right)\left( \langle i,1,m|\mathbf{p}|e,0,0\rangle\cdot \hat{\lambda^{*}_{k}}\right ) \end{align} }
Here
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} \frac{\mathbf{p}}{m}&=\frac{i}{\hbar} [\mathcal{H}^{(at)}_0,\mathbf{r}] \\ \langle i,1,m| \frac{\mathbf{p}}{m} |e,0,0\rangle &=\frac{i}{\hbar} \langle i,1,m| [\mathcal{H}^{(at)}_0,\mathbf{r}] |e,0,0\rangle \\ &=\frac{i}{\hbar} (E_{i}-E_{e}) \langle i,1,m|\mathbf{r} |e,0,0\rangle \\ \langle g,0,0| \frac{\mathbf{p}}{m} |i,1,m\rangle &=\frac{i}{\hbar} (E_{g}-E_{i}) \langle g,0,0| \mathbf{r} |i,1,m\rangle \end{align} }
Define
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} r_+&=x+iy=rsin\theta e^{i\phi} \\ r_-&=x-iy=rsin\theta e^{-i\phi} \\ x&=(r_+ + r_-)/2 \\ y&=(r_+ - r_-)/2i \end{align} }