Calculating Average Energy of a quantum state

[Z90E532]

Homework Statement

Given a wave function that is the super position of the two lowest energies of a particle in an infinite square well $\Psi = \frac{\sqrt{2}}{\sqrt{3}}\psi _1 + \frac{1}{\sqrt{3}}\psi _2$, find $\langle E \rangle$.

Homework Equations

The Attempt at a Solution

I'm not sure how to proceed with this problem. I understand that we basically need to find the coefficients $c_n$ from $\langle H \rangle = \sum |c_n|^2 E_n$, but I'm not sure how to find $E_n$. The energy of each state is known to be $E_n = \frac{n^2 \pi ^2 \hbar ^2}{2mL^2}$, but without the problem giving the length of the box, I can't see how we can use this.

 

[BvU]

Two possibilities:

1. Continue with $L$ as a parameter that stays in the answer. Same for $m, \hbar, \pi$, (There is no need for a numerical value in this exercise).
2. Continue and perhaps some of these divide out (for example because of normalization constants)

In both cases: continue

PS my money definitely isn't on case 2