Clebsch-Gordan coefficient



Representation theory


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Given a tensor product ρ j 1ρ j 2\rho_{j_1} \otimes \rho_{j_2} of two representations of some Lie group GG (by default often the special orthogonal group SO(3)SO(3)) and given its decomposition into a direct sum of irreducible representations {ρ t tot}\{\rho_{t_{tot}}\} by an isomorphism

ρ j 1ρ j 2j totC j 1j 2 j totρ j tot \rho_{j_1} \otimes \rho_{j_2} \stackrel{\simeq}{\longrightarrow} \underset{j_{tot}}{\oplus} C_{j_1 j_2}{}^{j_{tot}} \rho_{j_{tot}}

then the matrix elements of this linear map in some standard basis are called – in the physics literature – the Clebsch-Gordan coefficients or equivalently (up to a constant) the Wigner 3j symbols .

Specifically for G=SO(3)G = SO(3) the rotation group in 3-dimensional Cartesian space, then the standard basis elements of the representation ρ j\rho_{j} of total angular momentum jj \in \mathbb{N} are traditionally denoted

|j,mρ j,jmj |j,m\rangle \in \rho_{j} \;\;\,,\;\; -j \leq m \leq j

and their inner product is traditionally denoted by

j 1,m 1|j 2,m 2. \langle j_1, m_1 | j_2, m_2\rangle \in \mathbb{C} \,.

In these basis elements that above matrix then has components given by

(j 1,m 1)(j 2,m 2)|j tot,m tot. \langle (j_1, m_1)\otimes(j_2,m_2) | j_{tot}, m_{tot}\rangle \in \mathbb{C} \,.

These expressions are specifically the Clebsch-Gordan coefficients as they appear in the physics literature.


Lecture notes include for instance

  • M. Tuckerman, Quantum mechanics and dynamics – Addition of angular momenta – The general problem)

Last revised on January 6, 2017 at 06:24:05. See the history of this page for a list of all contributions to it.