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 Selection rules
4 Scalar invariant
5 Orthogonality Relations
6 Relation to integrals of spin-weighted spherical harmonics
7 See also
8 External links
9 References

[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

Anthony Stone’s Wigner coefficient calculator (Gives exact answer)
Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator (Numerical)
369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)

[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.

Categories: Rotational symmetry | Representation theory of Lie groups | Quantum mechanics

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3-jm symbol

From Wikipedia, the free encyclopedia

Contents

[edit] Inverse relation

[edit] Symmetry properties

[edit] Selection rules

[edit] Scalar invariant

[edit] Orthogonality Relations

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

[edit] See also

[edit] External links

[edit] References

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