My understanding of QM is very naive, but Schrodinger equation seems really simple to me. I could probably even explain it to a 5 year old. I'd show a few pictures of standing waves on a drum surface: the first two energy levels create simple shapes, higher levels create more sophisticated shapes. Then I'd say that an electron is such a wave. Then I'd introduce the idea of electric potential: I'd put a heavy thing in the center and say that this is how the atom's nucleus warps the electric field; and this changes the wave patterns somewhat, but the idea remains the same. Then I'd say that often multiple standing waves mix together and we get a quasipattern that oscillates between a few standing waves. That's really it.