角運動量の公式集
しょっちゅう忘れてしまう角運動量の公式。僕がよく使う順に掲載します。
Wigner-Eckert theorem
\begin{equation}
\langle j' m' |O_{JM}|jm \rangle
=\frac{(-1)^{j-m}}{\sqrt{2J+1}}\langle j' m' j -m | JM \rangle
\langle j' || O_{J} || j \rangle
\label{eq:WE}
\end{equation}
Transformation of Clebsch-Gordan coefficient to 3j symbol
\begin{equation}
\langle j_{1} m_{1} j_{2} m_{2} | J M \rangle
=(-1)^{j_{1}-j_{2}-M}\sqrt{2J+1}
\left(
\begin{array}{ccc}
j_{1} & j_{2} & J \\
m_{1} & m_{2} & -M \\
\end{array}
\right)
\label{eq:2}
\end{equation}
Special case of 3j symbol
\begin{equation}
\left(
\begin{array}{ccc}
j & j & 0 \\
m & -m & 0 \\
\end{array}
\right)
=(-1)^{j-m}(2j+1)^{-1/2}
\label{eq:3}
\end{equation}
Orthogonlity of 3j symbol
\begin{eqnarray}
\sum_{m_{1}m_{2}}
\left(
\begin{array}{ccc}
j_{1} & j_{2} & j_{3} \\
m_{1} & m_{2} & m_{3} \\
\end{array}
\right)
\left(
\begin{array}{ccc}
j_{1} & j_{2} & j_{3}' \\
m_{1} & m_{2} & m_{3}' \\
\end{array}
\right)
=(2j_{3}+1)^{-1/2}\delta_{j_{3}j_{3}'}\delta_{m_{3}m_{3}'}
\end{eqnarray}
Symmetry of Crebsch-Gordan coefficient
\begin{eqnarray}
\langle j_{1} m_{1} j_{2} m_{2} | j_{3} m_{3} \rangle
=(-1)^{j_{1}-j_{3}+m_{2}}\sqrt{\frac{2j_{3}+1}{2j_{1}+1}}\langle j_{3}m_{3}j_{2}-m_{2} | j_{1} m_{1} \rangle
\end{eqnarray}
The addition theorem of spherical harmonics
\begin{equation}
P_{l}(\theta)=\frac{4\pi}{2l+1}\sum_{m}Y_{lm}(\hat{r})Y_{lm}^{*}(\hat{r}'),
{\rm where}\; \theta \;{\rm is \; angle \; between}\; \vec{r} \;{\rm and}\; \vec{r}'.
\end{equation}
Contraction rule
\begin{equation}
Y_{l_{1}m_{1}}(\hat{r})Y_{l_{2}m_{2}}(\hat{r})
=\sum_{lm}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}}
\left(
\begin{array}{ccc}
l_{1} & l_{2} & l \\
m_{1} & m_{2} & m \\
\end{array}
\right)
Y_{lm}^{*}(\hat{r})
\left(
\begin{array}{ccc}
l_{1} & l_{2} & l \\
0 & 0 & 0 \\
\end{array}
\right)
\end{equation}
Useful ansatz
\begin{equation}
\langle l'm' | Y_{LM} | lm \rangle
=\sqrt{\frac{(2l'+1)(2l+1)(2L+1)}{4\pi}}
\left(
\begin{array}{ccc}
l' & l & L \\
m' & m & M \\
\end{array}
\right)
\left(
\begin{array}{ccc}
l' & l & L \\
0 & 0 & 0 \\
\end{array}
\right)
\end{equation}
\begin{equation}
\langle l' || Y_{LM} || l \rangle
=(-1)^{l'}\sqrt{\frac{(2l'+1)(2l+1)(2L+1)}{4\pi}}
\left(
\begin{array}{ccc}
l' & L & l \\
0 & 0 & 0 \\
\end{array}
\right)
\end{equation}
Plane wave expansion
\begin{equation}
e^{i\vec{k}\cdot\vec{r}}
=4\pi\sum_{lm}i^{l}j_{l}(kr)Y_{lm}(\hat{k})Y_{lm}^{*}(\hat{r})
\end{equation}
3j to 6j
\begin{equation}
\begin{split}
\sum_{\mu_{1}\mu_{2}\mu_{3}}(-1)^{\mu_{1}+\mu_{2}+\mu_{3}}
&
\left(
\begin{array}{ccc}
j_{1} & l_{2} & l_{3} \\
m_{1} & \mu_{2} & -\mu_{3} \\
\end{array}
\right)
\left(
\begin{array}{ccc}
l_{1} & j_{2} & l_{3} \\
-\mu_{1} & m_{2} & \mu_{3} \\
\end{array}
\right)
\left(
\begin{array}{ccc}
l_{1} & l_{2} & j_{3} \\
\mu_{1} & -\mu_{2} & m_{3} \\
\end{array}
\right)\\
&
=(-1)^{l_{1}+l_{2}+l_{3}}
\left(
\begin{array}{ccc}
j_{1} & j_{2} & j_{3} \\
m_{1} & m_{2} & m_{3} \\
\end{array}
\right)
\left\{
\begin{array}{ccc}
j_{1} & j_{2} & j_{3} \\
l_{1} & l_{2} & l_{3} \\
\end{array}
\right\}
\end{split}
\end{equation}
Useful ansatz
\begin{equation}
\left\{
\begin{array}{ccc}
j_{1} & j_{2} & j_{3} \\
0 & j_{3} & j_{2} \\
\end{array}
\right\}
=(-1)^{j_{1}+j_{2}+j_{3}}\sqrt{(2j_{2}+1)(2j_{3}+1)}
\end{equation}
When \( X_{L}=\left[O_{L_{1}}O_{L_{2}} \right]^{L} \)
\begin{equation}
\langle j'||X_{L}||j\rangle
=(2L+1)(-1)^{j'+j+L}\sum_{j''}
\left\{
\begin{array}{ccc}
L_{1} & L_{2} & L\\
j & j' & j''
\end{array}
\right\}
\langle j'||O_{L_{1}}||j''\rangle\langle j'' || O_{L_{2}} || j \rangle
\end{equation}
When \( X_{J}=\left[T_{L}U_{S} \right]^{J} \)
\begin{equation}
\langle j'(l' s') ||X_{J}|| j(l s) \rangle
=\sqrt{(2j_{1}'+1)(2j+1)(2J+1)}
\left\{
\begin{array}{ccc}
l' & l & L\\
s' & s & S\\
j' & j & J
\end{array}
\right\}
\langle l'||T_{L}||l\rangle \langle s' || U_{S} || s \rangle
\end{equation}
\begin{equation}
\langle j'm'(l' s') | T_{k} \cdot U_{k}| jm(l s) \rangle
=(-1)^{l+s'+j}
\left\{
\begin{array}{ccc}
j & s' & l'\\
k & l & s\\
\end{array}
\right\}
\langle l' || T_{k} || l \rangle \langle s' || U_{k} || s \rangle
\delta_{jj'}\delta_{mm'}
\end{equation}
Other cases
\begin{equation}
\langle j'(l' s) || T_{k} || j(l s) \rangle
=(-1)^{l'+s+j+k}\sqrt{(2j'+1)(2j+1)}
\left\{
\begin{array}{ccc}
l' & j' & s\\
j & l & k\\
\end{array}
\right\}
\langle l' || T_{k} || l \rangle
\end{equation}
\begin{equation}
\langle j'(l s') || U_{k} || j(l s) \rangle
=(-1)^{l+s+j'+k}\sqrt{(2j'+1)(2j+1)}
\left\{
\begin{array}{ccc}
s' & j' & l\\
j & s & k\\
\end{array}
\right\}
\langle s' || U_{k} || s \rangle
\end{equation}