Quantum Physics - Electron in a 1d Potential Well Question

[daleklama]

Homework Statement

This is a Quantum Physics problem.

An electron moves in a one-dimensional potential well such that the potential V = 0 for |x| ≤ a, and V = ∞ otherwise.

The system has energy eigenfunctions:

Un = a^(-1/2) cos (n∏x/2a), for n odd, and
Un = a^(-1/2) sin (n∏x/2a) for n even.

(Those are both for |x| ≤ a)

and Un = 0 for |x| > a.

The lowest energy level of the system is 37.6 eV.

At t=0, the wavefunction for the system is

ψ(x, t = 0) = a^(-1/2) (2 cos (∏x/2a) + sin (∏x/a)).

(1) Write the wavefunction as a linear combination of energy eigenfunctions, and hence normalise the wavefunction.

(2) What are the possible results of a measurement of electron energy?

Homework Equations

Not really sure!

The Attempt at a Solution

Okay, so I think I have 1) complete. The wavefunction as a linear combination of energy eigenfunctions is ψ = 2u1 + u2.

Also, I normalised it and got ψ (norm) = ψ/√5. I'm pretty sure I understand all this.

My problem is with (2). I know that the possible results of an energy measurement are either u1 or u2.

And u1 = E1 = 37.6 eV (given in question)
But I don't understand how to get E2. Apparently E2 = 4 (E1)?? Why would that be true.. I don't understand where that 4 came from? :/

Sorry, I'm an absolute beginner to Quantum Physics so I'm going very very slowly!

Any help would be appreciated :)

 

[Simon Bridge]

But I don't understand how to get E2. Apparently E2 = 4 (E1)?? Why would that be true.. I don't understand where that 4 came from? :/

... you should have this already in your notes when you covered the infinite square well in general. If not, then you'll have to put U2 through the Schrödinger equation to find the eigenvalue. (H-E₂)U₂=0.

I know that the possible results of an energy measurement are either u1 or u2.
And u1 = E1 = 37.6 eV

The possible outcomes of a measurement are the eigenvalues, not the wavefunctions. Important not to mix them up. It is never correct to write "u1=E1" for example.