nLab Clebsch-Gordan coefficient

Redirected from "Wigner 3j symbols".
Contents

Context

Representation theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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.

References

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 11:24:05. See the history of this page for a list of all contributions to it.