Energy eigenvalue for particle in a box

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 teh mean energy.

Thanks in advance.


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:

[tex]\psi = a_1 \psi_1 + a_2 \psi_2 + a_3 \psi_3 + ...[/tex]

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

The energy eigenfunctions are orthogonal, which means that

[tex]\int {\psi^*_j \psi_k dx} = 0[/tex]

whenever [itex]j \ne k[/itex]. Suppose that the eigenfunctions are also normalized so that

[tex]\int {\psi^*_k \psi_k dx} = 1[/tex]

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

[tex]\psi^*_2 \psi = a_1 \psi^*_2 \psi_1 + a_2 \psi^*_2 \psi_2 + a_3 \psi^*_2 \psi_3 + ...[/tex]

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

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