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.](../../../../math/c/c/1/cc17fac086de2cececed643f6a54cbbf.png)
Contents |
[edit] Recoupling of four angular momentum vectors
Coupling of two angular momenta 
and 
is the construction of simultaneous eigenfunctions of 
and Jz, where 
, 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 , , 
, and 
may be written as
Alternatively, one may first couple and to 
and and to 
, before coupling and to 
:
Both sets of functions provide a complete, orthonormal basis for the space with dimension (2j1 + 1)(2j2 + 1)(2j4 + 1)(2j5 + 1) spanned by
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 (m9):
[edit] Symmetry relations
A 9 − j symbol is invariant under reflection in either diagonal:
The permutation of any two rows or any two columns yields a phase factor ( − 1)S, where
For example:
[edit] Special case
When j9 = 0 the 9-j symbol is proportional to a 6-j symbol:
[edit] Orthogonality relation
The 9-j symbols satisfy this orthogonality relation:
The symbol {j1j2j3} is equal to one if the triad (j1j2j3) satisfies the triangular conditions and zero otherwise.
[edit] See also
[edit] External links
- Anthony Stone’s Wigner coefficient calculator (Gives exact answer)
- 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical answer)
[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.
- Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. (1988). Quantum Theory of Angular Momentum. Singapore: World Scientific. ISBN 9971-50-107-4.
