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 ′ ⟩ {\displaystyle \langle j\|T^{(k)}\|j'\rangle } denotes some value that does not depend on m, m′, nor q and is referred to as the reduced matrix element. The Wigner–Eckart theorem states indeed that operating with a spherical tensor operator of rank k on an angular momentum eigenstate is like adding a state with angular momentum k to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch–Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book Quantum Mechanics by Cohen–Tannoudji, Diu and Laloe.
Background and overview
Motivating example: position operator matrix elements for 4d → 2p transition Let's say we want to calculate transition dipole moments for an electron transition from a 4d to a 2p orbital of a hydrogen atom, i.e. the matrix elements of the form ⟨ 2 p , m 1 | r i | 4 d , m 2 ⟩ {\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle } , where ri is either the x, y, or z component of the position operator, and m1, m2 are the magnetic quantum numbers that distinguish different orbitals within the 2p or 4d subshell. If we do this directly, it involves calculating 45 different integrals: there are 3 possibilities for m1 (−1, 0, 1), 5 possibilities for m2 (−2, −1, 0, 1, 2), and 3 possibilities for i, so the total is 3 × 5 × 3 = 45. The Wigner–Eckart theorem allows one to obtain the same information after evaluating just one of those 45 integrals (any of them can be used, as long as it is nonzero). Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of Clebsch–Gordan coefficients, which can be easily looked up in a table or computed by hand or computer.
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