Quantum Mechanics: Concepts and Applications · Problem 7.17

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · wigner-eckart-theorem-spherical-tensors

Problem

Consider the tensor A(θ,φ)=cos⁡θsin⁡θcos⁡φA(\theta,\varphi) = \cos\theta\sin\theta\cos\varphi. (a) Calculate all the matrix elements Am′m=⟨l, m′∣A∣l, m⟩A_{m'm} = \langle l,\,m'\mid A\mid l,\,m\rangle for l=1l = 1. (b) Express A(θ,φ)A(\theta,\varphi) in terms of the components of a spherical tensor of rank 2 (i.e., in terms of Y2m(θ,φ)Y_{2m}(\theta,\varphi)). (c) Calculate again all the matrix elements Am′mA_{m'm}, but this time using the Wigner–Eckart theorem. Compare these results with those obtained in (a). (The Clebsch–Gordan coefficients may be obtained from tables.)

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