Ramp in Square Well

A particle of mass m is bound in a square well where –a/2 < x < a/2.

a) What are the energy and eigenfunction of the ground state?

b) A small perturbation is added, V(x) = 2ε|x|/a. Use perturbation theory to calculate the change in the ground state energy to O(ε).

SOLUTION

a) For a particle bound in a square well that runs from –a/2 < x < a/2, the eigenfunction and eigenvalue for the lowest energy state are

(1)

(2)

The eigenfunction is symmetric and vanishes at the walls of the well.

b) We use first-order perturbation theory to calculate the change in energy from the perturbation:

(3)