Rotational Invariance and Angular Momentum — R. Shankar
Physics — Quantum Mechanics · finite-rotations
Problem
Consider the family of states ∣jj⟩,…,∣jm⟩,…,∣j,−j⟩. One regards them as states of the same magnitude but different orientation of angular momentum. If one takes this remark literally, i.e., in the classical sense, one is led to believe that one may rotate these into each other, as is the case for classical states with these properties. Consider the family ∣1,1⟩,∣1,0⟩,∣1,−1⟩. It may seem, for example, that the state with zero angular momentum along the z axis, ∣1,0⟩, may be obtained by rotating ∣1,1⟩ by some suitable (21π?) angle about the x axis. Using D(1)[R(θxi)] from part (2) in the last exercise show that
∣1,0⟩=D(1)[R(θxi)]∣1,1⟩for any θx
The error stems from the fact that classical reasoning should be applied to ⟨J⟩, which responds to rotations like an ordinary vector, and not directly to ∣jm⟩, which is a vector in Hilbert space. Verify that ⟨J⟩ responds to rotations like its classical counterpart, by showing that ⟨J⟩ in the state D(1)[R(θxi)]∣1,1⟩ is ℏ[−sinθxi+cosθxk].