I'd really appreciate any insight on any of this since I've hit a wall. It is about the Fermi gas. --- 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?