Energy eigenvalue for particle in a box

[rias]

Hello all, I'm stuck on this question, and I would appericate if someone can tell me how to start cracking the problem.

I have a infinite square well, and is given a wavefunction that exist inside the well. The problem is to find the probability that a measurement of the energy will result in a certain given energy eigenvalue. Also I have to find the mean energy.


Thanks in advance.

 

[jtbell]

The wavefunction that you're given is not one of the eigenfunctions of the inifinite square well, right?

In general, any wave function that satisfies the boundary conditions for a physical situation (such as the infinite square well) can be written as a linear combination of the energy eigenfunctions:

$$\psi = a_1 \psi_1 + a_2 \psi_2 + a_3 \psi_3 + ...$$

where $\psi_1$ has energy eigenvalue $E_1$, and $a^*_1 a_1$ gives the probability that the particle has energy $E_1$. So your problem is to find out what the $a_k$ are, or at least some of them.

The energy eigenfunctions are orthogonal, which means that

$$\int {\psi^*_j \psi_k dx} = 0$$

whenever $j \ne k$. Suppose that the eigenfunctions are also normalized so that

$$\int {\psi^*_k \psi_k dx} = 1$$

for any k. Take the first equation above and multiply through by (say) $\psi^*_2$:

$$\psi^*_2 \psi = a_1 \psi^*_2 \psi_1 + a_2 \psi^*_2 \psi_2 + a_3 \psi^*_2 \psi_3 + ...$$

Then integrate both sides. Look at what happens to the integrals on the right side, and you should see how to calculate $a_2$, and how to generalize this method to any value of k.