Quantum Mechanics: Concepts and Applications · Problem 5.25

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · wave-function-normalization-lz-probabilities

Problem

Consider a system whose wave function is given by ψ(θ, φ)=12Y00(θ, φ)+13Y11(θ, φ)+12Y1,−1(θ, φ)+16Y22(θ, φ).\psi(\theta,\ \varphi) = \frac{1}{2}Y_{00}(\theta,\ \varphi) + \frac{1}{\sqrt{3}}Y_{11}(\theta,\ \varphi) + \frac{1}{2}Y_{1,-1}(\theta,\ \varphi) + \frac{1}{\sqrt{6}}Y_{22}(\theta,\ \varphi). (a) Is ψ(θ, φ)\psi(\theta,\ \varphi) normalized? (b) Is ψ(θ, φ)\psi(\theta,\ \varphi) an eigenstate of L^2\hat{L}^2 or L^z\hat{L}_z? (c) Calculate L^±ψ(θ, φ)\hat{L}_\pm\psi(\theta,\ \varphi) and ⟨ψ∣L^±∣ψ⟩\langle \psi | \hat{L}_\pm | \psi \rangle. (d) If a measurement of the zz-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 00, ℏ\hbar, −ℏ-\hbar, and 2ℏ2\hbar.

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