A particle of mass is constrained to move on a rigid circular ring of radius

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

A particle of mass mm is constrained to move on a rigid circular ring of radius RR, with position measured by arc length ss around the ring, s∈[0,L)s \in [0,L) where L=2πRL = 2\pi R is the circumference. Translation by an arc length aa along the ring is generated by the momentum exactly as on an infinite line, T^(a)=e−iap^/ℏ\hat T(a) = e^{-ia\hat p/\hbar}, and the free-particle Hamiltonian (no potential acts on the ring) is H^=p^2/2m\hat H = \hat p^2/2m.Because the ring closes on itself, translating all the way around it must return every state to itself up to at most an overall phase: T^(L)∣ψ⟩=eiα∣ψ⟩\hat T(L)|\psi\rangle = e^{i\alpha}|\psi\rangle for every state ∣ψ⟩|\psi\rangle, where α\alpha is a fixed real number, the same for every state (a property of the ring, not of the particular state). ((a)) Working in the momentum representation, use T^(L)∣p⟩=e−iLp/ℏ∣p⟩\hat T(L)|p\rangle = e^{-iLp/\hbar}|p\rangle together with the boundary condition above to show that the allowed momentum eigenvalues form the discrete set pn=ℏR(n−α2π),n=0,±1,±2,…p_n = \frac{\hbar}{R}\left(n - \frac{\alpha}{2\pi}\right), \qquad n = 0,\pm1,\pm2,\dots ((b)) Take the periodic case α=0\alpha = 0. Write down the energy eigenvalues EnE_n and identify the two lowest distinct energies. ((c)) State, with a one-line reason, whether the position or the momentum representation makes finding this spectrum easier, given that [H^,p^]=0[\hat H,\hat p] = 0 for a free particle on the line or on the ring. ((d)) An electron of mass m=9.109×10−31 kgm = 9.109\times10^{-31}\,\mathrm{kg} is confined to such a ring with R=2.0 nmR = 2.0\,\mathrm{nm} and α=0\alpha = 0. Compute, in electronvolts, the energy of the first excited level above the ground state.

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