Do we need Fermi function at all ?
In MSEE237 (NRL) we have been having enthusiastic discussions over this so I thought it could be a good idea to share it with the rest of the community (hopefully some experts) through nanoHUB.
The question that arose in our discussion is this: Quantum mechanical theory by itself, seems comprehensive in the sense that it can explain the time evolution of any system given the boundary/initial conditions if we adopt the simple Schrodinger picture (as opposed to Heisenberg picture)
So, for instance, in a simple particle-in-a-box problem, we can solve the Schrodinger equation analytically and obtain the discrete energy states rather easily.
However, there seems to be a problem when it comes to calculating electron densities even for this simple problem. The solutions to the Schrodinger equation DO NOT yield information about the probable populations of the discrete levels, it only defines the electron probabilities ( psi_n psi_n^*) for a given energy level.
So, if we were to put 50 electrons in our little 1D box (assuming our box is wide enough they don’t interact with each other), we would not be able to calculate their distribution over the allowed energies, without referring to the Fermi-Dirac statistics.
But this is confusing. Because QM theory should be self-consistent and we shouldn’t need to bring in the concept of Fermi-statistics which totally emerges from an apparently different field of physics. AND therefore, it is NOT and should NOT be fundamentally related to Quantum Mechanical theory.
In my opinion, there must be a way to use Schrodinger equation to calculate the so-called ‘exact’ statistics and Fermi-Dirac must be a very good approximation to this ‘exactly’ correct statistics. But I have yet to understand how this ‘exact’ statistics can come out naturally only from Schrodinger equation. Because even in 1-D what we obtain is a set of energies, there’s apparently no information about the population of these states in these results.
My colleagues do not share my opinion when I challenge the fundamental existence of fermi function
We would be very happy if you contributed to our debate with your valuable comments.