Quantum Mechanics: Concepts and Applications · Problem 5.19

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-state-lz-measurement

Problem

Consider a system which is in the state ψ(θ, φ)=213Y3,−3+313Y3,−2+313Y30+313Y32+213Y33.\psi(\theta,\ \varphi) = \sqrt{\frac{2}{13}}Y_{3,-3} + \sqrt{\frac{3}{13}}Y_{3,-2} + \sqrt{\frac{3}{13}}Y_{30} + \sqrt{\frac{3}{13}}Y_{32} + \sqrt{\frac{2}{13}}Y_{33}. (a) If L^z\hat{L}_z were measured, what values will one obtain and with what probabilities? (b) If after a measurement of L^z\hat{L}_z we find lz=2ℏl_z = 2\hbar, calculate the uncertainties ΔLx\Delta L_x and ΔLy\Delta L_y and their product ΔLxΔLy\Delta L_x \Delta L_y. (c) Find ⟨ψ∣L^x∣ψ⟩\langle \psi | \hat{L}_x | \psi \rangle and ⟨ψ∣L^y∣ψ⟩\langle \psi | \hat{L}_y | \psi \rangle.

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