Finding the eigen function for an infinite square well (quantum mechanics)

[ElijahRockers]

Homework Statement

Quantum mechanics is absolutely confusing me.

A proton is confined in an infinite square well of length 10^-5nm.

Calculate the wavelength and energy associated with the photon that is emitted when the proton undergoes a transition from the first excited state (n=2) to the ground state (n=1).

In what region of the electromagnetic spectrum does this wavelength belong?

The Attempt at a Solution

I'm not really sure what I'm doing at all.

But I started with the time independent Schrödinger equation. For region I and III, (where the potential is infinite) then the eigenfunction must be 0.

So I put in a potential of zero for the second region and got

\frac{d^2\psi}{dx^2} + k^2\psi = 0 where k^2 = \frac{2mE}{\hbar}

Looks like a second order diff eq, so I tried to solve it. Solutions to the characteristic equation were ±ik...

From here I am a little stumped, I didn't take notes, and I can't remember what the solution was that he used in class.

But anyway if I'm just going by my normal diff eq understanding,

\psi = Acos(kx)+Bsin(kx)

What's throwing me off is that I recall he had exponential solutions with imaginary components in them. These would oscillate, and so does my solution, but even if I had my solution in his form, what next?

Where does the length of the well come into play? What does n=2 -> n=1 mean?
Am I making this more complicated than it should be?

 

[ehild]

The wavefunction has to vanish at the boundaries of the well. http://en.wikipedia.org/wiki/Particle_in_a_box That means some restriction for the possible values of k.

ehild