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Kerem Yunus Camsari

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.

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    Hi, first of all, I am a naive graduate student so, please understand my language is not that elegant.

    To my understanding, quantum mechanics is explaining the behavior of quantum particle-wave, for example, fermion, or boson. Here, the schrodinger equation explains it through wavefunction. Though, the problem is when we have more than two fermions or bosons.

    For fermion case, Fermi-Dirac statics should be applied which satisfies the Pauli’s exclusion principle and for Boson’s case, Bose-Einstein statistics is to be used. These are what schrodinger equation itself cannot explain.

    Well, let me think in a different way. You could think that electrons can be anywhere and with any momentum and with arbitrary energy according to schrodinger equation. As you know wavefunction can just give you the probability that electrons can have in a real space or k space which follows the uncertainty principle, which I believe just comes out from the characteristics of Fourier transform.

    Well, what is energy? Energy is different matter in a sense that it is related to time or lifetime. If some electron has broad range of energy then the lifetime is short and vice versa.

    Schrodinger equation can give us the possible energy states that one electron or many electrons can occupy(if you solve many-body schrodinger equation). So in this sense, you can argue that schrodinger equation can explain everything.

    Well, but there is one more thing to understand that is reservoir. If you have only electron system you can say that schrodinger equation can explain everything. But, if you have reservoir and reservoir interacts with electron system. The picture is whole lot different.

    You should explain how these two interacts with each other. There, the statistics kicks in. I think prof. Datta has explained this in his class. But, I can’t remember exactly. His class(ECE612) is also provided in nanohub(Atom to transistor) so you can go and listen to him but, I think this is mentioned in recent class(last year) so you may not find this in online class(2006Aug).

    Anyways, it is very fun to think about this kind of stuff. Thank you.

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    SungGeun Kim

    Ok. I am the one who wrote the above answer which can mislead a little bit. I first said that statistics comes in because there are many particles, which is not true and I also said that statistics is needed when you think about big reservoir whose energy states are almost continuous and interact with electron system so that electrons can occupy certain level where the statistics comes in, which I believe is true. There maybe someone else who can explain more clearly.

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    Marco Antonio Garcia

    This is not an answer but i think that this is a great question that should get a lot of people thinking. I know that i will research this to try to answer it for you but to be quite honest, I’m a sophomore in high school so I do not honestly believe I will be able to. However, I do want to thank you for getting me a little more interested in this subject.

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