I'm stuck on a 2D square potential well:$$ \frac{-\hbar^2}{2m}(\frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2})\Psi(x, y) + U(x, y)\Psi(x, y) = E\Psi(x, y)$$ $$U(x, y) =\begin{cases}0 & 0 < x < a, 0 < y < a\\ \infty & otherwise\end{cases} $$
I want to find all independent solutions of that problem. All that i have is a pretty nice solution based on the assumption that wave function can be factored as $\Psi(x, y) = \Psi_x(x)\Psi_y(y)$. But i haven't found anywhere the proof that that there are no other independent solutions of that problem. I'll be glad if somebody help me with the proof or another independent solution.