GRE Question (QM, Perturbation theory?)

[HungryChemist]

Homework Statement

Initially, you have a one dimensional square well potential with infinitely high potential fixed at x = 0 and x = a. In the lowest energy state, the wave function is proportional to sin (kx). If the potential is altered slightly by introducing a small bulge(symmetric about x = a/2) in the middle, which of the following is true of the ground state?

a. The energy of the ground state remains unchanged.
b. The energy of the ground state is increased.
c. The energy of the ground state is decreased.
d. The original ground state splits into two states of lower energy
e. the original ground state splits into two states of higher energy

Homework Equations

This has to do with perturbation theory but I am not sure. Since this is GRE question, I am guessing there must be a cleaver way of solving this problem rather than solving the Schrödinger's equation.

The Attempt at a Solution

I eliminated a, d, and e so far and left with b and c as candidates. I thought that to introduce a new potential to a system( however small) requires certain amount of work which then will increase the energy of system. So, I would go for the choice b as a final answer. But then, I wouldn't be surprise if I am entirely wrong. Can someone help?

 

[Avodyne]

Do you know the formula (from what is called "Rayleigh-Schrodinger" or "stationary state" perturbation theory) for the first-order change in an energy eigenvalue due to a small change in the hamiltonian?

 

[HungryChemist]

I looked them up just now, but it seems quiet handful of mathematics involved and I wish to postpone studying them formally when my course get there. However now, I am looking for some quick guidance to solve this problem (since this is GRE problem, you've got a minute and a half to solve). Am I being too optimistic?

 

[Avodyne]

No. You don't need to do a computation. The relevant formula is
$$\Delta E_n = \langle \psi_n | \Delta H | \psi_n \rangle$$
where $$|\psi_n\rangle$$ is the unperturbed eigenstate. So, what does this formula reduce to (in terms of an integral over x) in the case that the change in the hamiltonian is a positive change in the potential energy V(x)? What does this tell you about the sign of $$\Delta E_n$$? (Note: I am assuming that the "small bulge" represents a local increase in the potential energy.)