Quantum Mechanics - Infinite Potential Well

[TheBaker]

Homework Statement

A particle is trapped in an infinite potential well, with the infinite walls at ±a. At time t=0, the wavefunction of the particle is

\psi = \frac{1}{\sqrt{2a}}

between -a and a, and 0 otherwise.

Find the probability that the Energy of the particle is \frac{9 \bar{h}^2 \pi^2}{8ma^2}

Homework Equations

E_n = \frac{n^2\bar{h}^2\pi^2}{8ma}

\psi = A \cos{\frac{(2r+1) \pi x}{2a}} for |x| < a
\psi = 0 otherwise

The Attempt at a Solution

I've calculated the above equations, but I'm unsure how to get from them to the probability of the particle having a certain energy. This could be really simple and it's me just having a brain dead moment, but any help would be very much appreciated.

 

[CPL.Luke]

btw there is also a sin solution with an argument (in your notation) 2r(pi)x/a

personally I prefer the notation n(pi)x/a n even

however the cos solution you wrote is the one you want with r=1, that 2r+1 thing is just a way of writing n so that n is always odd.

so just take the projection of psi at t=0 on your cos function and square the answer

 

[TheBaker]

The sin solution isn't valid because this well has even parity (i.e. it's symmetric).

How do I find A? Presumably I need to use the initial condition of Psi, but I found when doing that that A is x dependent, when it should be a constant.