Quantum Mechanics (1) · Problem 5.7

Rotations and angular momentum

Physics — Quantum Mechanics · clebsch-gordan-series-proof

Problem

Prove the Clebsch-Gordan series theorem by carrying out the following computation. Let

j_1

and

j_2

index two irreps of

SU(2)

. Show that the coefficients in the Clebsch-Gordan series

D^{j_1}(u) \otimes D^{j_2}(u) = \sum_{j=|j_1-j_2|}^{j_1+j_2} \oplus D^j(u) \tag{5.206}

are given by

C_{j_1 j_2 j} = \frac{1}{\pi} \int_0^{2\pi} d\theta \sin\left[\left(j_1 + \frac{1}{2}\right)\theta\right] \sin\left[\left(j_2 + \frac{1}{2}\right)\theta\right] \sum_{m=-j}^{j} e^{-im\theta}, \tag{5.210}

etc.