Principles of Quantum Mechanics · Problem 12.3.2

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-lz

Problem

Let us try to deduce the restriction on lzl_z from another angle. Consider a superposition of two allowed lzl_z eigenstates: ψ(ρ,ϕ)=A(ρ) eiϕlz/ℏ+B(ρ) eiϕlz′/ℏ\psi(\rho, \phi) = A(\rho)\, e^{i\phi l_z/\hbar} + B(\rho)\, e^{i\phi l_z'/\hbar} By demanding that upon a 2π2\pi rotation we get the same physical state (not necessarily the same state vector), show that lz−lz′=mℏl_z - l_z' = m\hbar, where mm is an integer. By arguing on the grounds of symmetry that the allowed values of lzl_z must be symmetric about zero, show that these values are either …,3ℏ/2,ℏ/2,−ℏ/2,−3ℏ/2,…\ldots, 3\hbar/2, \hbar/2, -\hbar/2, -3\hbar/2, \ldots or …,2ℏ,ℏ,0,−ℏ,−2ℏ,…\ldots, 2\hbar, \hbar, 0, -\hbar, -2\hbar, \ldots. It is not possible to restrict lzl_z any further this way.

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