Understanding the Fermi sphere

[scigal89]

I'd really appreciate any insight on any of this since I've hit a wall. It is about the Fermi gas.
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My teacher did an example in class that didn't make much sense, and I'm trying to understand it. He had us take the real-part of the antiderivative of exp(ik(x-x'))dk, then evaluate it to obtain 2 functions, both like sinc functions. Tthe first is evaluated from 0 to k0 and I don't know if the other was supposed to be from 0 to 2k0 or k0 to 2k0 (that's part of my question). I assume we integrate the plane waves on these intervals because it's like periodic conditions in momentum-space since obviously integrating over all space is impractical since it would give a delta function... but why 0 to k0? Shouldn't it be -k0 to k0 for a sphere of radius k0? Otherwise, it seems like you are only getting half the sphere. What about the other integral, would it be 0 to 2k0 or k0 to 2k0? Are these probabilities? What are they physically?

 

[NateD]

The Fermi gas is a system of interacting particles in which the Pauli exclusion principle is at work. The particles have a specific energy, and due to the Pauli exclusion principle, no two particles can occupy the same energy state. This means that the particles must be distributed over different energy states. As such, one can use the real part of the antiderivative of exp(ik(x-x'))dk to obtain two functions which represent the probability of a particle being in a particular energy state. The first integral from 0 to k0 represents the probability of a particle having an energy up to k0, while the second integral from k0 to 2k0 represents the probability of a particle having an energy between k0 and 2k0. These probabilities are important for understanding the behaviour of the Fermi gas.