Talk:Phy5646

sol by chor hoi chan

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 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 \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} }

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 f(t,w_k,w_k') } is independent of the direction of the emitted photons.

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} }

Taking away the radial dependent part, ie, take away 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,|i\rangle,|g\rangle} parts, and 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} }

we know

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 l',m'|z|l,m\rangle \neq 0 } only if 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'=m } ,

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 l',m'|r_+|l,m\rangle \neq 0 } only if 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'=m+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 \langle l',m'|r_-|l,m\rangle \neq 0 } only if 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'=m-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 \begin{align} \langle 1,m|\mathbf{r}|0,0\rangle &= \langle 1,m|\hat{x}x+\hat{y}y+\hat{z}z|0,0\rangle \\ &= \frac{\hat{x}}{2}\langle 1,m|r_+ + r_-|0,0\rangle + \frac{\hat{y}}{2i}\langle 1,m|r_+ - r_-|0,0\rangle + \hat{z}\langle 1,m|z|0,0\rangle \end{align} }

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 1,-1|\mathbf{r}|0,0\rangle &= \frac{1}{2}(\hat{x}+i\hat{y})\langle 1,-1|r_-|0,0\rangle \\ &= \frac{1}{2}(\hat{x}+i\hat{y}) r \int d\Omega sin\theta e^{-i\phi} {Y_{1}^{-1}}^{*} Y_{0}^{0} \\ &= \frac{1}{2}(\hat{x}+i\hat{y}) r \sqrt{\frac{2}{3}} \end{align} }

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 1,1|\mathbf{r}|0,0\rangle &= \frac{1}{2}(\hat{x}-i\hat{y})\langle 1,1|r_+|0,0\rangle \\ &= \frac{1}{2}(\hat{x}-i\hat{y}) r \int d\Omega sin\theta e^{i\phi} {Y_{1}^{1}}^{*} Y_{0}^{0} \\ &= -\frac{1}{2}(\hat{x}-i\hat{y}) r \sqrt{\frac{2}{3}} \end{align} }

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 1,0|\mathbf{r}|0,0\rangle &= \hat{z}\langle 1,0|z|0,0\rangle \\ &= \hat{z} r \int d\Omega cos\theta {Y_{1}^{0}}^{*} Y_{0}^{0} \\ &= \hat{z} r \sqrt{\frac{1}{3}} \end{align} }

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 \sum_{m=-1,0,1} g(t,w_k,w_k') \left(\langle 0,0|\mathbf{p}|1,m\rangle \cdot \hat{\lambda^{'*}_{k'}} \right)\left( \langle 1,m|\mathbf{p}|0,0\rangle\cdot \hat{\lambda^{*}_{k}}\right )\\ &= g(t,w_k,w_k')\frac{1}{3}\left [ (\hat{z} \cdot\hat{\lambda^{'*}_{k'}})( \hat{z} \cdot\hat{\lambda^{*}_{k}} ) + \frac{1}{2} (\hat{x}+i\hat{y} )\cdot\hat{\lambda^{'*}_{k'}}( \hat{x}-i\hat{y}) \cdot\hat{\lambda^{*}_{k}} + \frac{1}{2} (\hat{x}-i\hat{y} )\cdot\hat{\lambda^{'*}_{k'}}( \hat{x}+i\hat{y}) \cdot\hat{\lambda^{*}_{k}} \right ] \\ &= g(t,w_k,w_k')\frac{1}{3}\left [ (\hat{z} \cdot\hat{\lambda^{'*}_{k'}})( \hat{z} \cdot\hat{\lambda^{*}_{k}} ) + (\hat{x} \cdot\hat{\lambda^{'*}_{k'}})( \hat{x} \cdot\hat{\lambda^{*}_{k}}) + (\hat{y} \cdot\hat{\lambda^{'*}_{k'}})( \hat{y} \cdot\hat{\lambda^{*}_{k}} ) \right ] \\ &= g(t,w_k,w_k')\frac{1}{3}\left [ \hat{\lambda^{'*}_{k'}}\cdot\hat{\lambda^{*}_{k}} \right ] \\ \end{align} }

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 g(t,w_k,w_k') } is independent of the direction of the emitted photons.

Probability that the 2 outgoing photons with one pointing along 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 \hat{k'} } and another one pointing along 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 \hat{k} } 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 \begin{align} P(\hat{k'},\hat{k})&= \frac{1}{9} |g(t,w_k,w_k')|^{2} \sum_{\lambda '_{k'},\lambda_k} \left | \hat{\lambda^{'*}_{k'}}\cdot\hat{\lambda^{*}_{k}} \right |^{2} \\ \end{align} }

WOLG, choose 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 \hat{k}, \hat{k'} } both lie on the x-z plane. And choose 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 \hat{\lambda '_{1k'}} ,\hat{\lambda_{1k}} } both pointing along 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 \hat{y} } . Then 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 \hat{\lambda '_{2k'}},\hat{\lambda_{2k}} } both lie on the x-plane and the angle between 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 \hat{\lambda '_{2k'}},\hat{\lambda_{2k}} } is same as the angle between ${\hat {\lambda '_{1k'}}},{\hat {\lambda _{1k}}}$, and the question requires this angle to be $\gamma$.

${\begin{aligned}{\hat {\lambda _{1k'}^{'*}}}\cdot {\hat {\lambda _{1k}^{*}}}&=1\\{\hat {\lambda _{2k'}^{'*}}}\cdot {\hat {\lambda _{2k}^{*}}}&=cos\gamma \\{\hat {\lambda _{1k'}^{'*}}}\cdot {\hat {\lambda _{2k}^{*}}}&=0\\{\hat {\lambda _{2k'}^{'*}}}\cdot {\hat {\lambda _{1k}^{*}}}&=0\\\end{aligned}}$

$P({\hat {k'}},{\hat {k}})={\frac {1}{9}}|g(t,w_{k},w_{k}')|^{2}(1+cos^{2}\gamma )\equiv P(\gamma )$

Upon normalization, required probability $P(\gamma )={\frac {2}{3\pi }}(1+cos^{2}\gamma )$

Note that we can also take the first emitted photon to go along z-axis right in the beginning.