Suppose I can solve time-dependent Schrödinger equation for several 1D particles (currently 3). I'd like to see, what an electronic hole is and how it behaves — in a series of numerical experiments.
I know that a hole is lack of electron in a sea of occupied electron states, but I think I don't quite understand, what it really is: do I need to have electron repulsion to see any useful effects? Do I need much more electrons for such a simulation than 3? What initial conditions should I use to see the hole? I'm thinking of using a model "crystal" potential with 3 cells — would this be enough?
As an example of what I'd like to actually see is exciton, namely how its wave packet wanders across the crystal. For an electron in conductivity band, taken as a wavepacket, I'd expect it to look like increase in probability density in the place where the wavepacket is located, and for hole it might be a similar decrease in charge density, so I guess the exciton should look like an increase of charge density in a wave packet and decrease around this wave packet. Is it right? Or is it actually invisible as a charge density?
Another example would be seeing excitonic states in band gap, but for this I don't really understand, what to compare it with — should I use some single-particle approximation to compute band structure without excitonic states, and then after computation of true energy states search for extra states in band gap, or how should I go about finding them?