Introduction to Quantum Mechanics · Problem 7.41

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · particle_on_ring_perturbation

Problem

A free particle of mass

m

is confined to a ring of circumference

L

such that

\psi(x + L) = \psi(x)

. The unperturbed Hamiltonian is

H^0 = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2},

to which we add a perturbation

H' = V_0 \cos\left(2\pi \frac{x}{L}\right).

(a) Show that the unperturbed states may be written

\psi^0_n(x) = \frac{1}{\sqrt{L}} e^{i2\pi n x/L}

for

n = 0, \pm 1, \pm 2

and that, apart from

n = 0

, all of these states are two-fold degenerate. (b) Find a general expression for the matrix elements of the perturbation:

H'_{mn} = \langle \psi^0_m | H' | \psi^0_n \rangle.

n = \pm 1

. Construct the matrix

\mathbf{W}

and calculate the first-order energy corrections,

E^1

. Note that the degeneracy does not lift at first order. Therefore, diagonalizing

\mathbf{W}

does not tell us what the "good" states are. (d) Construct the matrix

\mathsf{W}^2

(Problem 7.40) for the states

n = \pm 1

, and show that the degeneracy lifts at second order. What are the good linear combinations of the states with

n = \pm 1

? (e) What are the energies, accurate to second order, for these states?