Wigner–Eckart theorem

The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart, who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the principles of conservation of energy, momentum, and angular momentum. Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator T ( k ) {\displaystyle T^{(k)}} and two states of angular momenta j {\displaystyle j} and j ′ {\displaystyle j'} , there exists a constant ⟨ j ‖ T ( k ) ‖ j ′ ⟩ {\displaystyle \langle j\|T^{(k)}\|j'\rangle } such that for all m {\displaystyle m} , m ′ {\displaystyle m'} , and q {\displaystyle q} , the following equation is satisfied: ⟨ j m | T q ( k ) | j ′ m ′ ⟩ = ⟨ j ′ m ′ k q | j m ⟩ ⟨ j ‖ T ( k ) ‖ j ′ ⟩ , {\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle =\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle ,} where T q ( k ) {\displaystyle T_{q}^{(k)}} is the q-th component of the spherical tensor operator T ( k ) {\displaystyle T^{(k)}} of rank k, | j m ⟩ {\displaystyle |jm\rangle } denotes an eigenstate of total angular momentum J2 and its z component Jz, ⟨ j ′ m ′ k q | j m ⟩ {\displaystyle \langle j'm'kq|jm\rangle } is the Clebsch–Gordan coefficient for coupling j′ with k to get j, ⟨ j ‖ T ( k ) ‖ j ′...