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## 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

[itex]E_2 = \frac{16 \pi^2 \hbar^2}{m}[/itex], 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.