Quantum Mechanics - Angular Momentum

At a given instant of time, a rigid rotator is in the state ψ(θ, ϕ) = √(¾π) sin θ sin ϕ, where θ is the polar angle relative to the z-axis and ϕ is the azimuthal angle. Measurement will find which of the following possible values of the z-component of the angular momentum L_z?

A. 0
B. ħ/2, −ħ/2
C. ħ, −ħ
D. 2ħ, −2ħ
E. ħ, 0, −ħ

(GR9677 #52)

Solution:

The eigenstates of L_z  is L_zY_l^m = mħY_l^m 

Possible values = eigen values of mħ.

L_z = −iħ ∂/∂ϕ → the information about m is contained in the sin ϕ term and it's proportional to e^imϕ:

sin ϕ  = (e^iϕ − e^−iϕ) / 2i

 → m = ±1

Thus, the eigenvalues are ±ħ

Answer: C