Wigner–Eckart theorem

The Wigner–Eckart theorem factors matrix elements of spherical tensor operators into Clebsch–Gordan coefficients fixed by angular-momentum symmetry and reduced matrix elements independent of magnetic quantum numbers.

Spherical tensor operator: A set of operators whose components transform under rotations like the states of a fixed angular momentum. The theorem applies to operators whose components transform this way.
Rotation group: The group of spatial rotations, represented in quantum mechanics by operators acting on states. Rotational symmetry is the foundation of the theorem.
Electric dipole transition: A transition between quantum states mediated by interaction with the electric dipole operator. Its transition amplitudes use rank-one tensor matrix elements and angular selection rules.
Eugene Wigner: A Hungarian-American theoretical physicist whose work established major applications of symmetry in quantum mechanics. Wigner helped formulate the theorem’s role in quantum angular momentum.
Clebsch–Gordan coefficients: Numbers that combine two angular momenta into states of definite total angular momentum. They supply the universal angular-momentum factors in the theorem.
Angular momentum operator: A quantum operator generating rotations and carrying angular momentum quantum numbers. Its eigenstates and algebra underpin the theorem’s matrix elements.
Atomic spectroscopy: The study of atoms through their absorption, emission, and scattering of electromagnetic radiation. The theorem organizes angular factors in spectral transition amplitudes.
Carl Eckart: An American theoretical physicist known for contributions to quantum mechanics and molecular structure. Eckart independently developed the result associated with the theorem.
Reduced matrix element: A matrix element with magnetic-quantum-number dependence removed using rotational symmetry. It carries the operator-specific information left after the theorem’s factorization.
Irreducible representation: A representation with no proper nonzero invariant subspace. The theorem follows from how rotation operators act on irreducible spaces.

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Multipole matrix elementsRelated: It separates multipole transition strengths into symmetry-fixed factors and state-dependent quantities.