Circle with Field

A particle with charge e and mass m is confined to move on the circumference of a circle of radius r. The only term in the Hamiltonian is the kinetic energy, so the eigenfunctions and eigenvalues are

(i)

(ii)

where ϕ is the angle around the circle. An electric field E is imposed in the plane of the circle. Find the perturbed energy levels up to O(|E|²).

SOLUTION

The perturbation is V(ϕ) = -e|E|r cos ϕ if we assume the field is in the x-direction. The same result is obtained if we assume the perturbation is in the y-direction (V(ϕ) = -e|E|r sin ϕ). In order to do perturbation theory, we need to find the matrix element of the perturbation between different eigenstates. For first-order perturbation theory we need

(1)

The eigenvalues are unchanged to first-order in the field E.

To do second-order perturbation theory, we need off-diagonal matrix elements:

(2)

If we recall that cos ϕ = (e^iϕ + -e^iϕ)/2, then we see that n – m can only

equal ±1 for the integral to be nonzero. In doing second-order perturbation

theory for the state |n⟩, the only permissible intermediate states are m = n ± 1:

(3)

This solution is valid for states n > 0. For the ground state, with n = 0, the n – 1, state does not exist, so the answer is

(4)