Principles of Quantum Mechanics · Problem 12.3.3

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-lz

Problem

A particle is described by a wave function ψ(ρ,ϕ)=A e−ρ2/2Δ2cos⁡2ϕ\psi(\rho, \phi) = A\, e^{-\rho^2/2\Delta^2} \cos^2\phi Show (by expressing cos⁡2ϕ\cos^2\phi in terms of Φm\Phi_m) that P(lz=0)=2/3P(l_z = 0) = 2/3 P(lz=2ℏ)=1/6P(l_z = 2\hbar) = 1/6 P(lz=−2ℏ)=1/6P(l_z = -2\hbar) = 1/6 (Hint: Argue that the radial part e−ρ2/2Δ2e^{-\rho^2/2\Delta^2} is irrelevant here.)

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