Normalizing a wave function and finding probability density

1. The problem statement, all variables and given/known data

A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function $\psi = 1\phi_1 + 2\phi_2$ where $\phi_i$ are the stationary states.
a) Normalize the wave function.
b) What is the probability to find the particle between x=L/4 and x=3L/4?
c) Calculate the expectation value of the Hamiltonian operator $\langle \hat{H} \rangle$

2. Relevant equations

I wasn’t sure Schrödinger’s equation was necessary here: $i \hbar \frac{\partial \psi}{\partial t} = – \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V(x)\psi(x,t)$
but I did remember that the solution to any wavefunction — at least the stationary state — is going to be $\psi(x) = A sin(\frac{n\pi x}{a})$ where a is the length of your "box" for the particle. So in this case a=L.

3. The attempt at a solution

We have a situation where V(x) = 0 0 < x < L and V(x) = infinity outside of that. So the V(x) term for inside the well disappears (it’s zero).

The probability of the particle being at any point from 0 to L is 1. So I need to integrate the wave functions squared over that interval. By the superposition principle it is OK to just add them.

$\psi = 1\phi_1 + 2\phi_2$
$\psi = (1\phi_1 + 2\phi_2)(1\phi_1^* + 2\phi_2^*)$

multiply this out
$\psi = (1\phi_1^* \phi_1 + 2\phi_1 \phi_2^* + 2\phi_1^* \phi_2 + 4\phi_2^*\phi_2)$

SInce the phi functions are eigenvalues, the ones on the diagonal of the matrix are the only ones not zero. So we get
$\psi = (1\phi_1^* \phi_1 + 4\phi_2^*\phi_2) = (1 + 4)$

because the complex conjugate of a function multiplied by a function is 1.

That makes the whole thing add up to five. and since the probability of finding the particle on the interval 0 to x is

$$\int^L_0 |\psi|^2 dx = 1 \rightarrow \int^L_0 |5|^2 dx = 1 \rightarrow 25x = 1$$

so x = 1/5 for the whole interval, (since that is the square root of 1/25) so normalizing the wave function I should get

$\psi = \frac{1}{5}\phi_1 + \frac{2}{5}\phi_2$

and for the probability that the particle is at L/4 and 3/4 L

(25L/4) and (75L/4)

Now, if someone could tell me where I am getting lot and doing this completely wrong 🙂

Actually I know this is wrong, because the probabilities should add up to one, at least with the stationary states.

After that I get even more confused. I think — and I stress think — I have some vague idea of how to get expectation values, but whenever someone mentions stuff like "Hamiltonian operator" I want to run away and hide. 🙂 More seriously, I am trying to grasp what exactly is meant by the notation $\langle \hat{H} \rangle$ – some of it is a notational question, but i get confused because I am never sure if they want the Hamiltonian like what you do in mechanics or something else. I feel like if someone could explain that I’d be a lot further along.