Approaching the problem o 1D well that changes size

[Emspak]

Homework Statement

You have a potential well, it's 1-dimensional and has a width of 0 to a. All of a sudden the wall of the well is pushed inward so that it's half as wide. Now the well is only extending from 0 to a/2.

in the well is a particle (mass m) that is in the first excited state n=2.

We want to know the following:

- What are the allowed energies and eigenfunctions in our new (smaller) infinite well?
- what is the probability of finding our particle in the ground state? The first excited state?

Homework Equations

We know the ground state of a particle (the wave function) is
$\psi_0 = \sqrt{ \frac{2}{L}} \sin \left( \frac{n \pi}{L} x \right)$ and that $E_n = \frac{\pi^2 \hbar^2}{2mL^2}n^2$

The Attempt at a Solution

What I did here was look first at the equation for E. I can just plug in n=2 and a/2 = L and get

$E_2 = \frac{16 \pi^2 \hbar^2}{m}$, which tells me the n=2 energy.

So I want to know th probability of the ground state, though. For that I need the series expansion

$\sum c_n \psi_n$

where

$c_n = \sqrt{ \frac{2}{a}} \int_0^{\frac{a}{2}} \sin \left(\frac{n\pi x}{a} \right) \sqrt{ \frac{2}{L}} \sin \left(\frac{n\pi x}{L}\right) dx$

and making sure that I plug in a/2 for L:
$c_n = \sqrt{ \frac{2}{a}} \int_0^{\frac{a}{2}} \sin \left(\frac{n\pi x}{a} \right) \sqrt{ \frac{4}{a}} \sin \left(\frac{2 n\pi x}{a}\right) dx$
pull out the constants:
$c_n = \frac{\sqrt{8}}{a} \int_0^{\frac{a}{2}} \sin \left(\frac{n\pi x}{a} \right) \sin \left(\frac{2 n\pi x}{a}\right) dx$

and we have here a pretty well-behaved function. A trig substitution / identity gives me:
$c_n = \frac{\sqrt{8}}{a} \int_0^{\frac{a}{2}} \frac{1}{2} \left[\cos \left(\frac{n\pi}{a}-\frac{2 n\pi}{a} x \right) -\cos \left(\frac{n\pi}{a}+\frac{2 n\pi}{a} x \right) \right] dx$
$= \frac{\sqrt{8}}{2a} \int_0^{\frac{a}{2}} \cos \left(\frac{ n\pi}{a} x \right) -\cos \left(\frac{3 n\pi}{a} x \right) dx$
$= \frac{\sqrt{8}}{2a} \left[ -\frac{a \sin \left(\frac{ n\pi}{a} x \right)}{n\pi} -\frac{ a \sin \left(\frac{3 n\pi}{a} x \right)}{3n\pi} \right]^{\frac{a}{2}}_0$
$= \frac{\sqrt{8}}{2a} \left[ -\frac{a \sin \left(\frac{ n\pi}{2} \right)}{n\pi} -\frac{ a \sin \left(\frac{3 n\pi}{2} \right)}{3n\pi}\right] = -\frac{\sqrt{8}}{2n\pi} \left[ \sin \left(\frac{ n\pi}{2} \right) +\frac{ \sin \left(\frac{3n\pi}{2} \right)}{3}\right]$

So I see that the sine terms are
n=1 --> 2/3
n=3 --> -2/3
n=5 --> 2/3

and so on. So I want to know the probability that the particle is in the first excited state, n=2. It turns out that it can't be there, because $c_n$ is zero wherever n is even. But at n=1 $(c_1)^2 = \frac{8}{4\pi^2}\frac{4}{9} = \frac{32}{36\pi^2}$. That's my probability for that particular state. At $(c_3)^2 = \frac{8}{36\pi^2}\frac{4}{9} = \frac{32}{324\pi^2}$, et cetera.

Ayhow I am checking to see if I approached this thing right, and didn't make a dumb mistake.

 

[BvU]

in the well is a particle (mass m) that is in the first excited state n=2.

This tells you the wave function $\psi$. There is a 2 in there, but no longer an n.
To get the $c_{n'}$ you need to evaluate $\int \psi^*_{n'}\; \psi$.

I have a problem with this exercise, because pushing in the wall can't leave the wave function as it was: it has to remain normalized. And I think there's no telling what that does to the energy of the particle.

(The reverse problem allows one to extend the wave function with 0 in the 'new' range, thus avoiding this renormalization issue)

 

[Emspak]

As I recall the problem was stated as an infinite potential is inserted at a/2, if that helps, and the idea was w could just ignore the part of the well on the right. But if I hear you right I should set up [itex]\int \psi^*_n' \psi [\itex] as $\int \psi^*_n' \psi_2$ ?

 

[Emspak]

This tells you the wave function $\psi$. There is a 2 in there, but no longer an n.
To get the $c_{n'}$ you need to evaluate $\int \psi^*_{n'}\; \psi$.

I have a problem with this exercise, because pushing in the wall can't leave the wave function as it was: it has to remain normalized. And I think there's no telling what that does to the energy of the particle.

(The reverse problem allows one to extend the wave function with 0 in the 'new' range, thus avoiding this renormalization issue)

As I recall the problem was stated as an infinite potential is inserted at a/2, if that helps, and the idea was w could just ignore the part of the well on the right. But if I hear you right I should set up [itex]\int \psi^*_n' \psi [\itex] as $\int \psi^*_{n'} \psi_2$ ? And then the issue is figuring out what n' is? (Or can I go with 1 since the well is half-size, and the wave is going to be cut in half -- foe purposes of this problem I don't think we had to renormalize).

 

[Emspak]

i just tried getting the dangd Latex to work and it's buggy today. but i think you ought to be able to see what i was asking.

 

[BvU]

Yeah, you need curly brackets around {n'} :)

Cutting the wave function in half makes the probablility the particle is somewhere equal to one half, which I find hard to swallow.
Otherwise, yes, I agree: that's what I tried to bring across. n (for the full width well) is fixed to 2 by the exercise statement.