Quantum Mechanics (1) · Problem 11

Rotations and angular momentum

Physics — Quantum Mechanics · su2-representation-reduction-multiplicities

Problem

We said that if we are given an arbitrary representation, D(u)D(u), of SU(2)SU(2), it may be reduced to a direct sum of irreps Dr(u)D^r(u): D(u)=∑r⊕Dr(u).(5.521)D(u) = \sum_r \oplus D^r(u). \tag{5.521} The multiplicities mjm_j (the number of irreducible representations DrD^r which belong to the equivalence class of irreducible representations characterized by index jj) are unique, and they are given by: mj=∫SU(2)d(u) χj ⁣(u−1)χ(u),(5.522)m_j = \int_{SU(2)} d(u)\, \chi^j\!\left(u^{-1}\right) \chi(u), \tag{5.522} where χ(u)=Tr[D(u)]\chi(u) = \mathrm{Tr}[D(u)]. (a) Suppose you are given a representation, with characters: χ[ue(θ)]=1+sin⁡32θ+2sin⁡74θcos⁡14θsin⁡12θ.(5.523)\chi[u_{\mathbf{e}}(\theta)] = 1 + \frac{\sin\frac{3}{2}\theta + 2\sin\frac{7}{4}\theta\cos\frac{1}{4}\theta}{\sin\frac{1}{2}\theta}. \tag{5.523} What irreducible representations appear in the reduction of this representation, with what multiplicities? (b) Does the representation we are given look like it could correspond to rotations of a physically realizable system? Discuss.

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