Wigner 3-j symbols

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In quantum mechanics, the Wigner 3-j symbols, also called 3j or 3-jm symbols, are related to Clebsch–Gordan coefficients through


\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
\equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle.

Inverse relation[edit]

The inverse relation can be found by noting that j1j2m3 is an integer and making the substitution  m_3 \rightarrow -m_3 :


\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{-j_1+j_2-m_3}\sqrt{2j_3+1}
\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & -m_3
\end{pmatrix}.

Symmetry properties[edit]

The symmetry properties of 3j symbols are more convenient than those of Clebsch–Gordan coefficients. A 3j symbol is invariant under an even permutation of its columns:


\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
=
\begin{pmatrix}
  j_2 & j_3 & j_1\\
  m_2 & m_3 & m_1
\end{pmatrix}
=
\begin{pmatrix}
  j_3 & j_1 & j_2\\
  m_3 & m_1 & m_2
\end{pmatrix}.

An odd permutation of the columns gives a phase factor:


\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
  j_2 & j_1 & j_3\\
  m_2 & m_1 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
  j_1 & j_3 & j_2\\
  m_1 & m_3 & m_2
\end{pmatrix}.

Changing the sign of the m quantum numbers also gives a phase:


\begin{pmatrix}
  j_1 & j_2 & j_3\\
  -m_1 & -m_2 & -m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}.

Regge symmetries also give


\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
=
\begin{pmatrix}
  j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\
  j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3
\end{pmatrix}.

\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
  \frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\
  j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2}
\end{pmatrix}.

Regge symmetries account for a total of 72 symmetries.[1] These are best displayed by the definition of a Regge symbol which is a one to one correspondence between it and a 3j symbol and assumes the properties of a semi-magic square[2]


R=
\begin{array}{|ccc|}
  \hline
    -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\
    j_1-m_1 & j_2-m_2 & j_3-m_3\\
    j_1+m_1 & j_2+m_2 & j_3+m_3\\
  \hline
\end{array}

whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. This can be used to devise an effective storage scheme.[3]

Selection rules[edit]

The Wigner 3j is zero unless all these conditions are satisfied:

m_1+m_2+m_3=0\,
j_1+j_2 + j_3\text{ is an integer} \,  \text{(or an even integer if} \,m_1=m_2=m_3=0)\,
|m_i| \le j_i \,
|j_1-j_2|\le j_3 \le j_1+j_2. \,

Scalar invariant[edit]

The contraction of the product of three rotational states with a 3j symbol,


  \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3}
  |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle
\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix},

is invariant under rotations.

Orthogonality relations[edit]


(2j+1)\sum_{m_1 m_2}
\begin{pmatrix}
  j_1 & j_2 & j\\
  m_1 & m_2 & m
\end{pmatrix}
\begin{pmatrix}
  j_1 & j_2 & j'\\
  m_1 & m_2 & m'
\end{pmatrix}
=\delta_{j j'}\delta_{m m'}.

\sum_{j m} (2j+1)
\begin{pmatrix}
  j_1 & j_2 & j\\
  m_1 & m_2 & m
\end{pmatrix}
\begin{pmatrix}
  j_1 & j_2 & j\\
  m_1' & m_2' & m
\end{pmatrix}
=\delta_{m_1 m_1'}\delta_{m_2 m_2'}.

Relation to spherical harmonics[edit]

The 3jm symbols give the integral of the products of three spherical harmonics


\begin{align}
& {} \quad \int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\
&  =
\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}
\begin{pmatrix}
  l_1 & l_2 & l_3 \\[8pt]
  0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
  l_1 & l_2 & l_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
\end{align}

with l_1, l_2 and l_3 integers.

Relation to integrals of spin-weighted spherical harmonics[edit]

Similar relations exist for the spin-weighted spherical harmonics:


\begin{align}
& {} \quad \int d{\mathbf{\hat n}}\,{}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}})
\,{}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}})\, {}_{s_3} Y_{j_3m_3}({\mathbf{\hat
n}}) \\[8pt]
& = \sqrt{\frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4\pi}}
\begin{pmatrix}
  j_1 & j_2 & j_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
\begin{pmatrix}
  j_1 & j_2 & j_3\\
  -s_1 & -s_2 & -s_3
\end{pmatrix}
\end{align}

Recursion relations[edit]


\begin{align}
& {} \quad -\sqrt{(l_3\mp s_3)(l_3\pm s_3+1)} 
\begin{pmatrix}
  l_1 & l_2 & l_3\\
  s_1 & s_2 & s_3\pm 1
\end{pmatrix}
 \\
& = \sqrt{(l_1\mp s_1)(l_1\pm s_1+1)} 
\begin{pmatrix}
  l_1 & l_2 & l_3\\
  s_1 \pm 1 & s_2 & s_3
\end{pmatrix}
+\sqrt{(l_2\mp s_2)(l_2\pm s_2+1)} 
\begin{pmatrix}
  l_1 & l_2 & l_3\\
  s_1 & s_2 \pm 1 & s_3
\end{pmatrix}
\end{align}

Asymptotic expressions[edit]

For l_1\ll l_2,l_3 a non-zero 3-j symbol has


\begin{pmatrix}
  l_1 & l_2 & l_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
 \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{2l_3+1}}

where \cos(\theta) = -2m_3/(2l_3+1) and d^l_{mn} is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by


\begin{pmatrix}
  l_1 & l_2 & l_3\\
  m_1 & m_2 & m_3
\end{pmatrix}
 \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}}

where \cos(\theta) = (m_2-m_3)/(l_2+l_3+1).

Other properties[edit]

\sum_m (-1)^{j-m}
\begin{pmatrix}
  j & j & J\\
  m & -m & 0
\end{pmatrix} = \sqrt{2j+1}~ \delta_{J0}

\frac{1}{2} \int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \, dx = 
\begin{pmatrix}
  l & l_1 & l_2 \\
  0 & 0 & 0
\end{pmatrix} ^2

See also[edit]

References[edit]

  1. ^ Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento 10: 544. doi:10.1007/BF02859841. 
  2. ^ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932. 
  3. ^ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932. 

External links[edit]