Properties of 3jm- and 3nj-Symbols

Abstract

Basic elements in the theory of angular momentum are the Clebsch—Gordan coefficients for coupling two states characterized by j₁,m₁ and j₂,m₂ into a new state with quantum numbers J,M. The numbers j and their projections, or magnetic quantum numbers¹, m have integer or half-odd integer values. Hence, in an exponent to (−1) the quantum numbers must combine to integers as in j ± m or j₁+j₂+J, for example.The notation used for the corresponding Clebsch—Gordan coefficient is (j₁m₁j₂ m ₂| JM) and it can be non-zero only if j ₁, j ₂, J fulfil the triangle condition Δ{j ₁,j ₂,J} equivalent to the condition j ₁ + j ₂ ≥ J ≥ |j ₁ − j ₂| and, furthermore, provided that m ₁ + m ₂ = M. The Clebsch—Gordan coefficients are chosen to be purely real and constitute a unitary transformation.