# Energy State Probability Particle in a Box

1. Sep 21, 2011

### GrantB

1. The problem statement, all variables and given/known data

Show that the probability of obtaining En for a particle in a box with wave function

$\Psi$(x) = $\sqrt{\frac{30}{L^{5}}}$(x)(L-x) for 0 < x < L

and $\Psi$(x) = 0 for everywhere else

is given by

|cn|2 = 240/(n6$\pi$6)[1-(-1)n]2

2. Relevant equations

cn = $\int$$\psi$$^{*}_{n}$$\Psi$(x)dx

The probability is cn squared.

Shouldn't have to use eigenvalues and eigenfunctions.

3. The attempt at a solution

I used the integral from (2) and used the given uppercase Psi and used the sqrt(2/L)sin(n*pi*x/L) lowercase psi (conjugate), from 0 to L.

The integral quickly turned messy with integration by parts and such.

I would like to know if I am on the right track here... If I am, I will just work through the integral until I get the right answer.

I'm hoping there is a much easier way to do this.

Thanks!

P.S. Sorry this is a repost from another section. I wasn't getting any responses from the Intro Physics section so thought I would try here.

2. Sep 21, 2011

### diazona

Yep, you're on the right track. Yes, the integral takes a little bit of work, but it is of the form
$$\int(\text{polynomial in }u)\sin u\;\mathrm{d}u$$
and those integrals are doable.