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Concept
Wigner–Eckart theorem
Summary
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 laws of conservation of energy, momentum, and angular momentum.
Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator and two states of angular momenta and , there exists a constant such that for all , , and , the following equation is satisfied:
where
is the q-th component of the spherical tensor operator of rank k,
denotes an eigenstate of total angular momentum J2 and its z component Jz,
is the Clebsch–Gordan coefficient for coupling j′ with k to get j,
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.
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.
Official source
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