Initial wave function for a particle in an infinite square well A particle of mass m in the infinite square well (of width a) started out in the left half of the well and is (at t=0) equally likely to be found at any point in that region.
What is its initial wave function $$\Psi\left ( x,0 \right )$$ 
(Assume the wave function is real.)
Getting started,
The solution to the time-independent Schrodinger's equation(or just the n stationary state) is 
$$\psi_{n}\left ( x,0 \right )=A_{n}Sin\left ( k_{n}x \right )$$
where 
$$k_{n}=\frac{n\pi}{a}$$
To obtain $$A_{n}$$, subject $$\psi\left ( x,0 \right ) $$to the normalisation condition.
Recall: $$\int_{0}^{a}\left | \Psi\left ( x,0 \right ) \right |^{2}dx=\int_{0}^{a}\Psi^{\dagger }\left ( x,0 \right )\Psi\left ( x,0 \right )dx=\int_{0}^{a}\psi^{\dagger}\left ( x,0 \right )\psi\left ( x,0 \right )dx=1$$
working out, the constant we get $$|A_{n}|=\sqrt{\frac{2}{a}}$$
Therefore, 
$$\Psi_{n}\left(x,0\right)=\psi_{n}\left ( x,0 \right )=\sqrt{\frac{2}{a}}Sin\left ( k_{n}x \right )$$
I'd figured this was the solution on the domain of interest but looking at the solution sheet, the solution is given to be
$$\Psi\left( x,0 \right)=\left\{\begin{matrix}
\sqrt{\frac{2}{a}} & x \in\left [ 0,\frac{a}{2} \right ]\\ 
0 &everywhere 
\end{matrix}\right.$$
I think I might be close but missing a crucial piece of concept or an assumption. 
Would appreciate a point in the right direction.
 A: You're making it much harder than it is. It's actually an easy problem, and where you seem to be going wrong is assuming it's a hard problem and pulling out a bunch of extra tools that aren't needed.
Forget about the eigenvectors of the Hamiltonian (the $\psi_n$); they don't play a role in the thing you're trying to solve. Yes, the wave function can be expressed as a superposition of all of the $\psi_n$ functions (where the sum converges everywhere except where the wave function is discontinuous, as expected for a sine transform). But this is irrelevant to the problem you're trying to solve. Expressing the wave function as a superposition of the eigenvectors is a separate problem.
If we just rewrite the problem you're actually being asked to solve without the extraneous details, it's:
A particle is equally likely to be anywhere in the region $[0,a/2]$ and has zero probability of being found outside of that region. Its wave function is everywhere real and non-negative (yes, you do need this non-negativity assumption to answer this question, even though it's not in the problem statement). What's the wave function?
That's it. Forget about the Hamiltonian and its eigenvectors. Forget about the fact that it's in a square well at all. At this point you simply write down the answer, up to a normalization constant:
$\Psi\left( x,0 \right)=\left\{\begin{matrix}
C & x \in\left [ 0,\frac{a}{2} \right ]\\ 
0 &elsewhere 
\end{matrix}\right.$
for some constant $C$ (where I've corrected the typo "everywhere" to "elsewhere"). Then you just normalize it to find $C$ (which I see you already know how to do), taking $C$ to be real and positive.
There really isn't any more to it than that. That's the whole solution.
