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.

Contents

[edit] Inverse relation

The inverse relation can be found by noting that j1 - j2 - m3 is an integer 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] Selection rules

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

m_1+m_2+m_3=0\,
j_1+j_2 + j_3\, is integer
|m_i| \le j_i
|j_1-j_2|\le j_3 \le j_1+j_2.

[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] Orthogonality Relations

(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'}.

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

\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}})=(-1)^{m_1+s_1} \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}

This should be checked for phase conventions of the harmonics.

[edit] See also

[edit] External links

[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, Princeton, 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.