9-j symbol

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Wigner's $9 - j$ symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients involving four angular momenta

$[(2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)]^\frac{1}{2} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$

[edit] Recoupling of four angular momentum vectors

Coupling of two angular momenta $\mathbf{j}_1$ and $\mathbf{j}_2$ is the construction of simultaneous eigenfunctions of $\mathbf{J}^2$ and $J z$ , where $\mathbf{J}=\mathbf{j}_1+\mathbf{j}_2$ , as explained in the article on Clebsch-Gordan coefficients.

Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors $\mathbf{j}_1$ , $\mathbf{j}_2$ , $\mathbf{j}_4$ , and $\mathbf{j}_5$ may be written as

$| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle.$

Alternatively, one may first couple $\mathbf{j}_1$ and $\mathbf{j}_4$ to $\mathbf{j}_7$ and $\mathbf{j}_2$ and $\mathbf{j}_5$ to $\mathbf{j}_8$ , before coupling $\mathbf{j}_7$ and $\mathbf{j}_8$ to $\mathbf{j}_9$ :

$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle.$

Both sets of functions provide a complete, orthonormal basis for the space with dimension $(2 j 1 + 1)(2 j 2 + 1)(2 j 4 + 1)(2 j 5 + 1)$ spanned by

$|j_1 m_1\rangle |j_2 m_2\rangle |j_4 m_4\rangle |j_5 m_5\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5.$

Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions. As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number ( $m 9$ ):

$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_{j_3}\sum_{j6} | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$

[edit] See also

[edit] References

Biedenharn, L. C.; van Dam, H. (1965). Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers. New York: Academic Press. ISBN 0120960567.

Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9.

Condon, Edward U.; Shortley, G. H. (1970). "Chapter 3", The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4.

Messiah, Albert (1981). Quantum Mechanics (Volume II), 12th edition, New York: North Holland Publishing. ISBN 0-7204-0045-7.

Brink, D. M.; Satchler, G. R. (1993). "Chapter 2", Angular Momentum, 3rd edition, Oxford: Clarendon Press. ISBN 0-19-851759-9.

Zare, Richard N. (1988). "Chapter 2", Angular Momentum. New York: John Wiley. ISBN 0-471-85892-7.

Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0201135078.

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

9-j symbol

From Wikipedia, the free encyclopedia

[edit] Recoupling of four angular momentum vectors

[edit] See also

[edit] References

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