Consider a system which is described by the state
ψ(θ,φ)=83Y11(θ,φ)+81Y10(θ,φ)+AY1,−1(θ,φ),
where A is a real constant.
(a) Calculate A so that ∣ψ⟩ is normalized.
(b) Find L^+ψ(θ,φ).
(c) Calculate the expectation values of L^x and L^2 in the state ∣ψ⟩.
(d) Find the probability associated with a measurement that gives zero for the z-component of the angular momentum.
(e) Calculate ⟨Φ∣L^z∣ψ⟩ and ⟨Φ∣L^−∣ψ⟩ where
Φ(θ,φ)=158Y11(θ,φ)+154Y10(θ,φ)+153Y2,−1(θ,φ).