Phys. 6124 Assignment 9
Problem 1
Find the energy levels and wave functions for a quantum particle in a weak periodic potential, such that U(x+π) =
U(x). With the proper choice of physical parameters this means that we have to find the eigenvalues and eigenfunctions
of the ODE
−d2
dx2y(x) + ²U(x)y(x) = Ey(x),
subject to the condition y(x+π) = y(x). In the absence of the potential the eigenfunctions are yn(x) = Ceinx , with
n-integer, and the respective eigenvalues are En=n2.Cis some normalization constant.
a) Find the first order corrections to the eigenvalues. Write the secular equation for the general case (remember, any
periodic function can be expanded in the Fourier series) and solve it for the special case U(x) = sin2x. Also determine
the 0th order approximation to the eigenfunctions of the perturbed problem.
b) By analogy with the non-degenerate case derive the perturbation expansion for the first order correction to the
eigenvectors (eigenfunctions).
c) Find the first order corrections to yn(x) with the special choice of U(x) above.
Problem 2
Given the differential equation ˙x+x+²x2= 0 subject to the initial condition x= 1 at t= 0
a) Compute the approximate solution x(t, ²) using perturbation theory (assuming |²| ¿ 1) up to terms of O(²3).
b) Compute the exact solution x(t, ²) using separation of variables.
c) Perform a series expansion of the exact solution for small ²and compare with the perturbation solution. Do your
expansions agree?