3-jm symbol

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Wigner 3-jm symbols, also called 3j symbols, are related to Clebsch-Gordan coefficients through

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

1 Inverse relation
2 Symmetry properties
3 Scalar invariant
4 References

[edit] Inverse relation

The inverse relation can be found by noting that j₁ - j₂ - m₃ is an integral number and making the substitution $m_3 \rightarrow -m_3$

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

[edit] Symmetry properties

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:

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

[edit] Scalar invariant

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.

[edit] References

E. P. Wigner, On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups, unpublished (1940). Reprinted in: L. C. Biedenharn and H. van Dam, Quantum Theory of Angular Momentum, Academic Press, New York (1965).
A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd edition, Princeton University Press, Pinceton, 1960.
D. M. Brink and G. R. Satchler, Angular Momentum, 3rd edition, Clarendon, Oxford, 1993.
L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981.
D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific Publishing Co., Singapore, 1988.