# Probability of finding a particle in the ground state

1. Jun 1, 2010

### Sam Harrison

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

A particle is prepared in the state $$\psi (x) = \frac{1}{\sqrt{L}}$$ in a region $$0 < x < L$$ 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.

2. Relevant 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.

3. The attempt at a solution

$$\left| \psi (x) \right|^{2} = \frac{1}{L}$$

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

2. Jun 1, 2010

### 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?