Phy5646/Using Wigner-Eckart Theorem to get selection rule of spontaneous emission

Posted by Student Team #2 (Yuhui Zhang, Hang Chen, Xu Jiang)

The formula for the spontaneously emitted power of a given polarization $\lambda$ is:

${\frac {dP_{\lambda }}{d\Omega }}={\frac {\omega ^{4}e^{2}}{2\pi c^{3}}}|d_{0n}\cdot {\hat {\lambda ^{*}}}|^{2},$,

where $d_{0n}=<\alpha 'l'm'\left|{\overset {\rightharpoonup }{r}}\right|\alpha lm>$ is the matrix element of electric dipole moment operator.

Use the Wigner-Eckart Theorem to get the selection rule of spontaneous emission.

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Solution:

Transform $\lambda _{1}^{1}=-{\frac {1}{\sqrt {2}}}\left(\lambda _{x}+i\lambda _{y}\right);\lambda _{-1}^{1}={\frac {1}{\sqrt {2}}}\left(\lambda _{x}-i\lambda _{y}\right);\lambda _{0}^{1}=\lambda _{z},$

and $T_{1}^{1}=-{\frac {1}{\sqrt {2}}}(x+iy);T_{-1}^{1}={\frac {1}{\sqrt {2}}}(x-iy);T_{0}^{1}=z.$

$\therefore {\hat {\lambda }}\cdot {\overset {\rightharpoonup }{r}}={\hat {\lambda _{x}}}x+{\hat {\lambda _{y}}}y+{\hat {\lambda _{z}}}z=\sum _{q=-1}^{1}\lambda _{q}^{*(1)}T_{q}^{(1)}$

$\therefore d_{0n}\cdot {\hat {\lambda ^{*}}}=<\alpha 'l'm'\left|{\overset {\rightharpoonup }{r}}\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}=\sum _{q=-1}^{1}\lambda _{q}^{*(1)}<\alpha 'l'm'\left|T_{q}^{(1)}\right|\alpha lm>={\frac {<\alpha 'l'\left\|T^{(1)}\right\|\alpha l>}{\sqrt {2l'+1}}}\sum _{q=-1}^{1}\lambda _{q}^{*(1)}<1qlm|1ll'm'>$

In this equation, we have Clebsch-Gordan coefficient : $<1qlm\left|1ll'm'>\right.$

By the selection rule of Clebsch-Gordan coefficient, we know that :

$m'=m+q,$

$l'=l+1,l,|l-1|.$

We know that q = -1, 0, 1, and

$<\alpha 'l'm'\left|{\overset {\rightharpoonup }{r}}\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}=<\alpha 'l'm'\left|P^{2}{\overset {\rightharpoonup }{r}}P^{2}\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}=<\alpha 'l'm'\left|P\left(P{\overset {\rightharpoonup }{r}}P\right)P\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}=-(-1)^{l'}(-1)^{l}<\alpha 'l'm'\left|{\overset {\rightharpoonup }{r}}\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}.$

So if $l=l',<\alpha 'l'm'\left|{\overset {\rightharpoonup }{r}}\right|\alpha lm>\cdot {\hat {\lambda ^{*}}}=0.$

Above all we get the selection rule of spontaneous emission :

$l'=l\pm 1;m'=-1,0,1.$