Kronig-Penny Model: Exploring Electron Behavior in Solids

[mamunalsiraj]

Hi everybody,

I have participated in a course about solid state physics and have been introduced with Kronig-Penny model. Now in this model, so far
I understood we solve the nature of wave function of an electron inside an array of potential barrier. It very nicely explains the concept of conduction band and forbidden band inside solid along with the widths of the bands. Now my question is: in my solid I have lot of electrons (some tightly bonded and some not) and so far I understood, if I had one electron in my system, Kronig penny model describes the energy for that situation like hydrogen atom. So what happens when have many electons inside the system.
1)Should not I have as many as wave equations as electrons and should not they be superimposed on each other , then how the solution should look like?
or 2) Do they collectively behave the same way as the single electron so it is not necessary to calculate ?

I am not physics major. Hence might be naive in understanding. I would very much appreciate if somebody share their thoughts about it.

 

[OhYoungLions]

Your post would have been better places in the Solid State section of the forum.
There are a couple of layers to answer your questions.

If you ignore the fact that electrons repel each other by Coulomb repulsion, then the solution is simple. In this case, you write your solutions to the multi-electron Hamiltonian as one big wavefunction that keeps all the information about all the electrons in one function. This total multi-electron wavefunction must be antisymmetric with respect to interchange of any two electrons (which is related to the Pauli exclusion principle). In practice, this means you have to write the state as a Slater Determinant (look this up if you don't know what it is).

When you solve the Kronig-Penny model, you find possible one-electron states, and in order to build a state for multiple electrons, you just have to decide which of those states are occupied by your multiple electrons, and write a Slater determinant of such states. This is analogous to writing electron configurations for atoms - in that case, you decide which "orbitals" are occupied.

So, if you neglect the interactions between the electrons, the multi-electron solutions are easy to write. The behaviour of each electron is essentially independent of the others, apart from obeying the Pauli principle. One only has to solve the one-electron problem, and this gives you all the information you need to write the solutions of the multi-electron problem. It turns out (for some deep reasons) that this is a good approximation for thinking about the properties of usual metals like Copper.

The situation is considerably more difficult if you include the repulsion between the electrons! You can get all sorts of weird solutions, like superconductors and magnets, etc. (Most of the time, you cannot even solve the equations without approximations.) This is an active topic of research, and probably beyond the scope of your knowledge. :-)

 

[mamunalsiraj]

Finally somebody answered my question. @OhYoungLions:Thank you very much for the reply. I was getting nervous about asking the wrong question.