the Clebsch-Gordan coefficients
/ KLEBSH-GOR-dahn /
The Clebsch-Gordan coefficients are the exchange rate between the two ways of describing combined angular momentum. Once you decide to add two angular momenta, you have two competing vocabularies -- one that specifies each piece's projection separately, one that specifies the total -- and you constantly need to translate. The Clebsch-Gordan coefficients are precisely the numbers in that dictionary: the amplitude for finding a particular combined (j, m) state inside a particular pair of individual (m_1, m_2) states.
Concretely, they expand a coupled state in the uncoupled basis: |j, m> = sum over m_1, m_2 of <j_1 m_1; j_2 m_2 | j m> times |j_1 m_1> |j_2 m_2>, where the numbers <j_1 m_1; j_2 m_2 | j m> are the Clebsch-Gordan coefficients. They are nonzero only when m = m_1 + m_2 (projections simply add) and when j lies in the allowed range from |j_1 - j_2| to j_1 + j_2 (the triangle condition). By convention they are chosen real, and they form an orthogonal transformation between the two complete bases, so the same table also runs the translation backwards. You look them up in tables or generate them with the lowering operator J_- = J_1- + J_2-, starting from the stretched top state |j_1+j_2, j_1+j_2> = |j_1 j_1>|j_2 j_2>.
You meet these coefficients wherever quantized angular momenta are combined and then probed. They set the relative intensities of atomic spectral lines within a multiplet and encode the selection rules for transitions; they appear in the Wigner-Eckart theorem, which cleanly separates the geometry (a Clebsch-Gordan coefficient) from the physics (a single reduced matrix element) of any tensor operator; and they are ubiquitous in nuclear and particle physics wherever isospin or spin must be coupled. They are, in short, the concrete arithmetic behind the abstract statement that angular momenta add.
For two spin-1/2 particles the m=0 states illustrate the coefficients directly: the triplet |1,0> = (1/sqrt(2))(|up,down> + |down,up>) and the singlet |0,0> = (1/sqrt(2))(|up,down> - |down,up>). The 1/sqrt(2) factors are Clebsch-Gordan coefficients, and the relative minus sign is what makes the singlet antisymmetric under particle exchange.
The 1/sqrt(2) weights mixing up-down and down-up are Clebsch-Gordan coefficients; the sign distinguishes triplet from singlet.
Conventions and sign choices (the Condon-Shortley phase convention) vary between textbooks, so always check which convention a table uses before combining results. The coefficients vanish unless both m = m_1 + m_2 and the triangle inequality on j hold -- these two selection rules do most of the practical work.