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Irreducible representations of *SU*(2) are defined. Eigenstates of *J* _{3} to the eigenvalue *m* are introduced. Finite-dimensional representations have a so-called highest-weight state for which this eigenvalue is called *j* and the chapter shows that 2*j* must be a non-negative integer. It follows that these representations are (2*j*+1)-dimensional and that the quadratic Casimir operator has the eigenvalue *j*(*j*+1). Next addition of angular momentum is discussed, the Clebsch-Gordan series is described, and explicit values for Clebsch–Gordan coefficients are given when one of the angular momenta in the addition is 1/2 or 1. Spherical tensors are defined, the Wigner–Eckart theorem is demonstrated and reduced matrix elements are introduced. Inner and outer multiplicity is defined and *SU*(2) is shown to be multiplicity-free. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given.

*Keywords: *
irreducible representations;
highest-weight state;
quadratic Casimir;
angular momentum addition;
Clebsch–Gordan series;
Wigner–Eckart theorem;
multiplicity

*Chapter.*
*6811 words.*

*Subjects: *
Mathematical and Statistical Physics

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