The particle in a box problem is a common question that people are taught in order to get some practice using Schrödinger’s equation. For this kind of problem one usually solves the equation for energy eigenvalues
$$ \hat{H} \psi_n (x) = E_n \psi_n (x)\;, $$ where you get some $\psi_n (x)$ with their respective quantized values of energy $E_n$.
My question is, what is the real state of the particle? I guess the $\psi_n (x)$ are the space dependent part of some stationary state. So, intuition gives me the answer: $$ \Psi (x,t) = \sum_{n=1}^{\infty}{\alpha_n \psi_n (x) e^{-i E_n t / \hbar}}. $$
However, if this is true, what are the values for each $\alpha_n$? The solutions for $\psi_n (x)$ let $n$ to run in the set $\{n \in \mathbb{N} \; \vert \; n > 0\}$, so it looks something annoying to think of infinite states with same probability for all of them, and those probabilities restricted to sum one.