Clebsch–Gordan coefficients

Clebsch–Gordan coefficients are expansion amplitudes that convert product states of two angular momenta into states of definite total angular momentum. They encode how the angular momenta combine.

Angular momentum addition: The quantum-mechanical rules for combining angular momentum operators and their possible total values. The coefficients implement this combination for two angular momenta.
Tensor product: A construction that combines vector spaces into a space of ordered pairs of their vectors. The uncoupled basis lives in the tensor product of the two angular-momentum spaces.
Atomic term symbol: A label for an atomic state's total spin, orbital, and total angular momentum. Atomic states are classified by coupling constituent angular momenta.
Alfred Clebsch: A German mathematician whose work on invariant theory and algebraic forms influenced the study of group representations. His name is attached to the coefficients used to combine angular-momentum representations.
Racah W coefficient: A coefficient describing the recoupling of three angular momenta between different coupling orders. It changes how three angular momenta are grouped, rather than transforming a pair's product basis.
Wigner 3-j symbol: A rotationally invariant symbol related to Clebsch–Gordan coefficients by a phase and normalization factor. It packages the same coupling amplitudes in a form with more transparent permutation symmetries.
Spin quantum number: A quantum number specifying the intrinsic angular momentum of a particle or system. It labels each constituent representation whose states are coupled.
LS coupling: An atomic coupling scheme that combines electron orbital angular momenta and spins separately before combining their totals. Its basis transformations use Clebsch–Gordan coefficients at each coupling step.
Paul Gordan: A German mathematician known for work in invariant theory and the theory of algebraic forms. The classical Clebsch–Gordan theorem bears his name alongside Clebsch's.
Wigner 6-j symbol: A symbol expressing transformations between alternative coupling schemes for three angular momenta. It captures recoupling relations beyond the two-angular-momentum coefficients.

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