Probability of finding a particle in the ground state

[Sam Harrison]

Homework Statement

A particle is prepared in the state [tex]\psi (x) = \frac{1}{\sqrt{L}}[/tex] in a region [tex]0 < x < L[/tex] between two hard walls (particle in a box). Calculate the probability that the particle is found in the ground state when its energy is measured.

Homework Equations

This question is worth 10 marks, so I presume it's not as simple as squaring the wave function to find the probability. However I'm just not sure what else to do, a hint in the right direction would be greatly appreciated.

The Attempt at a Solution

[tex]\left| \psi (x) \right|^{2} = \frac{1}{L}[/tex]

That's all I can think of doing. I've checked the given wave function is normalised, which it is.

 

[sebb1e]

You need to use Fourier Series. You know the normalised solution to Schrodingers equation is root(2/L)sin(nPix/L)

Let the initial wave function be f(x)=1/root(L)

So f(x)=root(2/L)[sum of (c_n)sin(nPix/L)

So c_n=root(2/L)Integral 0 to L of f(x)sin(nPix/L) by Fourier methods

The probability of a particular state is the square of c_n if the wave function is normalised. I make your answer 8/(Pi)^2

Does that make sense?