identify the Clebsch–Gordan coefficient with the matrix element , then we may write Alternative conventions There are different conventions for the reduced matrix elements. One convention, used by Racah and Wigner, includes an additional phase and normalization factor, where the array denotes the 3-j symbol. (Since in practice is often an integer, the factor is sometimes omitted in literature.) With this choice of normalization, the reduced matrix element satisfies the relation: where the Hermitian adjoint is defined with the convention. Although this relation is not affected by the presence or absence of the phase factor in the definition of the reduced matrix element, it is affected by the phase convention for the Hermitian adjoint. Another convention for reduced matrix elements is that of Sakurai's Modern Quantum Mechanics: Example Consider the position expectation value . This matrix element is the expectation value of a Cartesian operator in a spherically symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using parity, although a slightly longer route will be taken.) We know that is one component of , which is a vector. Since vectors are rank-1 spherical tensor operators, it follows that must be some linear combination of a rank-1 spherical tensor with }. In fact, it can be shown that where we define the spherical tensors as and are spherical harmonics, which themselves are also spherical tensors of rank . Additionally, , and Therefore, The above expression gives us the matrix element for in the basis. To find the expectation value, we set , , and . The selection rule for and is for the spherical tensors. As we have , this makes the Clebsch–Gordan Coefficients zero, leading to the expectation value to be equal to zero. See also Tensor operator Landé g-factor References General External links J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, . Wigner–Eckart theorem Tensor Operators Representation theory of Lie groups Theorems in quantum mechanics Theorems in representation theory