# Homework Help: Prob of part being in Lowest Energy Level after Potential Change

1. Apr 13, 2015

### 12x4

First off sorry for the badly worded title.

1. The problem statement, all variables and given/known data
Beginning of Question:

Consider a single quantum particle of mass M trapped in the infinite square well potential, V(x), given by

V(x)= 0 if 0 < x < L
infinity otherwise

The wave function for a particle in the n-th energy level is: Ψn(x) = √(2/L) sin(nπx/L)

a.) I found the expected position and momentum of a particle in the n-th energy level.

b.) I calculated the expectation value for the energy of a particle in the n-th energy level using the hamiltonian.

Bit of Question I'm stuck on:

c.)
Suppose that the particle initially starts in the lowest energy level and the potential is instantaneously changed to:

V(x) = 0 if 0 < x < L/2
infinity otherwise

Find the probability that the particle ends up in the lowest energy level of the new potential.

2. Relevant equations

3. The attempt at a solution.

I'm not exactly sure what to do here. I assume it must be something along the lines of:
1st - Finding the lowest allowed energy level
2nd - Finding the probability that the particle would be in this state.

I was thinking that I might be able to use a method along these lines:
Renormalise the wave equation first to account for the change in potential?
Then repeat what I did in part b.) to find the expectation value of the energy of the particle in the n-th energy level?
I would surely then be able to find the lowest expected value for the energy?
And then I would be able to find the probability that a particle is in that state?

Or have I got totally the wrong idea here? It seems as though I'm ignoring the fact that the potential changed instantaneously.

2. Apr 13, 2015

### BvU

2. relevant equations. Nothing ?

a) What did you find ?
b) What did you find ? Isn't that what you put in in the first place ?
c) This is very strange (not your fault): your renormalization idea set me thinking about how to bring about this potential change -- without affecting the particle wave function. After all, the particle has to be somewhere - so what happens to the wave function in the [L/2, L] section ? Usually this kind of exercise expands the size of the box, so you can extend $\Psi$ with zero.

3. Your attempt isn't really an attempt: you are musing, considering, ...

The basic idea is to assume some wave function (in this case the lowest energy steady state wavefunction associated with the [0,L] box; the wavefunction you might mention under 2. relevant equations) and claim that that wavefunction stays the same during the instantaneous change in potential. Then expand the old $\Psi$ in terms of the eigenfunctions (steady state wave functions) associated with the [0,L/2] box -- see under 2: relevant equations )

I see all kinds of problems on the way, so we might need a real expert. I know one...