Rotational Invariance and Angular Momentum — R. Shankar
Physics — Quantum Mechanics · euler-angles
Problem
*Euler Angles.* Rather than parametrize an arbitrary rotation by the angle θ, which describes a single rotation by θ about an axis parallel to θ, we may parametrize it by three angles, γ, β, and α called Euler angles, which define three successive rotations:
U[R(α,β,γ)]=e−iαJz/ℏe−iβJy/ℏe−iγJz/ℏ
(1) Construct D(1)[R(α,β,γ)] explicitly as a product of three 3×3 matrices. (Use the result from Exercise 12.5.5 with Jx→Jy.)
(2) Let it act on ∣1,1⟩ and show that ⟨J⟩ in the resulting state is
⟨J⟩=ℏ(sinβcosαi+sinβsinαj+cosβk)
(3) Show that for no value of α, β, and γ can one rotate ∣1,1⟩ into just ∣1,0⟩.
(4) Show that one can always rotate any ∣1,m⟩ into a linear combination that involves ∣1,m′⟩, i.e.,
⟨1,m′∣D(1)[R(α,β,γ)]∣1,m⟩=0
for some α, β, γ and any m, m′.
(5) To see that one can occasionally rotate ∣jm⟩ into ∣jm′⟩, verify that a 180∘ rotation about the y axis applied to ∣1,1⟩ turns it into ∣1,−1⟩.