Energy State Probability Particle in a Box

[GrantB]

Homework Statement

Show that the probability of obtaining E_n 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 |c_n|² = 240/(n⁶\pi⁶)[1-(-1)ⁿ]²

Homework Equations

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

The probability is c_n squared.

Shouldn't have to use eigenvalues and eigenfunctions.

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.

 

[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.