Wigner-Eckart Theorem

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 $(j'm'\vert T_{LM}\vert jm)$ on the projection quantum numbers is entirely contained in the Wigner 3j-Symbol (or, equivalently, the Clebsch-Gordan Coefficient), given by

$$(j'm'\vert T_{LM}\vert jm)=C(jLj';mMm')(j'\vert\vert T_L\vert\vert j),$$

where $C(jLj';mMm')$ is a Clebsch-Gordan Coefficient and $T_{LM}$ is a set of tensor operators (Rose 1995, p. 85).

See also Clebsch-Gordan Coefficient, Wigner 3j-Symbol

References

Cohen-Tannoudji, C.; Diu, B.; and Laloë, F. ``Vector Operators: The WIgner-Eckart Theorem.'' Complement $D_X$ in Quantum Mechanics, Vol. 2. New York: Wiley, pp. 1048-1058, 1977.

Edmonds, A. R. Angular Momentum in Quantum Mechanics, 2nd ed., rev. printing. Princeton, NJ: Princeton University Press, 1968.

Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, p. 807, 1984.

Messiah, A. ``Representation of Irreducible Tensor Operators: Wigner-Eckart Theorem.'' §32 in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 573-575, 1962.

Rose, M. E. ``The Wigner-Eckart Theorem.'' §19 in Elementary Theory of Angular Momentum. New York: Dover, pp. 85-94, 1995.

Shore, B. W. and Menzel, D. H. ``Tensor Operators and the Wigner-Eckart Theorem.'' §6.4 in Principles of Atomic Spectra. New York: Wiley, pp. 285-294, 1968.

Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded and improved ed. New York: Academic Press, 1959.

Wybourne, B. G. Symmetry Principles and Atomic Spectroscopy. New York: Wiley, pp. 89 and 93-96, 1970.