Quantum Mechanics: Concepts and Applications · Problem 7.7

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-expectation-values

Problem

Consider a particle whose wave function is given by ψ(r⃗)=(15Y11(θ,φ)−15Y1−1(θ,φ)+12Y10(θ,φ))f(r),\psi(\vec{r}) = \left(\frac{1}{\sqrt{5}}Y_{11}(\theta,\varphi) - \frac{1}{5}Y_{1-1}(\theta,\varphi) + \frac{1}{\sqrt{2}}Y_{10}(\theta,\varphi)\right)f(r), where f(r)f(r) is a normalized radial function, i.e., ∫0∞r2f2(r) dr=1\int_0^\infty r^2 f^2(r)\,dr = 1. (a) Calculate the expectation values of L^2\hat{L}^2, L^z\hat{L}_z, and L^x\hat{L}_x in this state. (b) Calculate the expectation value of V(θ)=2cos⁡2θV(\theta) = 2\cos^2\theta in this state. (c) Find the probability that the particle will be found in the state ml=0m_l = 0.
Intuitive explanation

This problem tests your understanding of how angular momentum eigenstates combine in a superposition. The key insight is that spherical harmonics YlmY_{l m} are simultaneous eigenstates of L^2\hat{L}^2 and L^z\hat{L}_z, so expectation values decompose cleanly across the superposition—you don't need to compute integrals, just identify which terms contribute. For operators like L^x\hat{L}_x or potentials depending on θ\theta, you'll need the ladder operator relations or angular dependence of the harmonics, but orthogonality still eliminates most cross terms.

Intuitive explanation

This problem tests your understanding of how angular momentum eigenstates combine in a superposition. The key insight is that spherical harmonics YlmY_{l m} are simultaneous eigenstates of L^2\hat{L}^2 and L^z\hat{L}_z, so expectation values decompose cleanly across the superposition—you don't need to compute integrals, just identify which terms contribute. For operators like L^x\hat{L}_x or potentials depending on θ\theta, you'll need the ladder operator relations or angular dependence of the harmonics, but orthogonality still eliminates most cross terms.

Related theory

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