Maybe we should mention that we are using the time-independent Schrödinger equation here, which is different from the one on Schrodinger wave equation. Or maybe that page should be changed.
Also, is it a coincidence that all ψn are real-valued functions? Shouldn't complex numbers show up? --AxelBoldt
As for the absense of complex numbers, my understanding is that complex numbers, indicate the "phase" of the system and arise in the time-dependent equation. Since the particle in a box is a time-independent problem and corresponds to a standing wave there is no time-dependent portion and hence no complex component to the solution. If anyone more knowledgeable than myself on this subject can confirm or deny this explanation, I would greatly appreciate it.--Matt Stoker