I've been meaning to ask this question for a while but I thought I'd google it first. I found out that this question was put to this forum a few years ago by peter038, without a definite resolution. So I thought maybe we could try again. Part of the problem is to decide exactly what is the best question to ask. Where do we start? We want to ask the simplest possible question and yet we don't want to make it TOO simple. We know that electrons have spin 1/2 (so two of them can occupy the same state) and we know that they repel each other. Both of these are unwanted complications; so the temptation is to ask the question for spinless, non-interacting electrons. Is this a good question to ask? My guess is that any number of such particles would then be able to fit into the ground state. This partially defeats the purpose of my question, since what I am really interested in is how the energy levels fill up one after another. But even if that's the case, I still have to start by asking the question: what does the wave function look like for two spinless, non-repelling electrons in a potential well, if one of them is in the ground state and the other is in the first excited state?