A theorem of fundamental importance in spectroscopy and angular momentum theory which provides both (1) an explicit form for the dependence of all matrix elements of irreducible tensors on the projection quantum numbers and (2) a formal expression of the conservation laws of angular momentum (Rose 1995).
The theorem states that the dependence of the matrix element Wigner 3j -symbol (or,
equivalently, the Clebsch-Gordan coefficient ),
given by
where Clebsch-Gordan coefficient and
See also Clebsch-Gordan Coefficient ,
Wigner 3j -Symbol
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References Cohen-Tannoudji, C.; Diu, B.; and Laloë, F. "Vector Operators: The WIgner-Eckart Theorem." Complement Quantum
Mechanics, Vol. 2. Eckart,
C. "The Application of Group Theory to the Quantum Dynamics of Monatomic Systems."
Rev. Mod. Phys. 2 , 305-380, 1930. Edmonds, A. R. Angular
Momentum in Quantum Mechanics, 2nd ed., rev. printing. Gordy, W. and Cook, R. L. Microwave
Molecular Spectra, 3rd ed. Messiah,
A. "Representation of Irreducible Tensor Operators: Wigner-Eckart Theorem."
§32 in Quantum
Mechanics, Vol. 2. Rose, M. E. "The Wigner-Eckart Theorem." §19
in Elementary
Theory of Angular Momentum. Shore,
B. W. and Menzel, D. H. "Tensor Operators and the Wigner-Eckart Theorem."
§6.4 in Principles
of Atomic Spectra. Wigner,
E. P. "Einige Folgerungen aus der Schrödingerschen Theorie für
die Termstrukturen." Z. Physik 43 , 624-652, 1927. Wigner,
E. P. Group
Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded and
improved ed. Wybourne, B. G.
Symmetry
Principles and Atomic Spectroscopy. Referenced on Wolfram|Alpha Wigner-Eckart Theorem
Cite this as:
Weisstein, Eric W. "Wigner-Eckart Theorem."
From MathWorld https://mathworld.wolfram.com/Wigner-EckartTheorem.html
Subject classifications
on the projection quantum numbers is entirely
contained in the Wigner 3j-symbol (or,
equivalently, the Clebsch-Gordan coefficient),
given by
is a Clebsch-Gordan coefficient and
is a set of tensor operators (Rose 1995, p. 85).
in