Posted by Steve Simon on January 27, 2013, 12:33 pm, in reply to "Momentum"
Message modified by board administrator January 27, 2013, 3:13 pm
Hi Mo,
First of all, congratulations on your Olympic medals. I had no idea you were studying physics!
The question you ask is an extremely good one, and a very important one too -- and we will return to it many times throughout the course. It is a bit early in the course to try to give a decent explanation of this issue, but (conceding that full explanation may have to wait a few weeks) let me try to clarify a bit (and hopefully things will get clearer soon).
In quantum mechanics, you can think of particles as waves or waves as particles. So in our solids, let us think about both electrons and vibrations (phonons) as particles for simplicity.
Now also note that momentum of any particle is not generally conserved --- only total momentum of the universe. (For example, one particle may transfer its momentum to another). Given this, at first examination you might think that there should be no momentum conservation of particles in solids at all -- since an electron, or a phonon, can crash into a nucleus and transfer its momentum to the nucleus. Of course the total momentum of the entire system (the entire solid with all the electrons and all the phonons) is indeed conserved, but one would not expect the momentum of every electron in the entire system as being conserved.
The big surprise of this course is that, if one considers only crystal momentum (momentum modulo the reciprocal lattice) then you CAN think about this quantity as being conserved. The presence of the nuclei to crash into does not disrupt crystal momentum. This was already hinted at when we discovered that electrons have an unreasonably long mean free path -- they don't crash into nuclei at all (as long as we are willing to talk in terms of crystal momentum)!
We saw our first example of crystal momentum in the last lecture -- we found that the momentum of phonons is only conserved modulo 2pi/a. But given that a phonon might transfer its momentum to any of the other atoms, you might be surprised that there is anything like momentum conservation for these particles at all! But indeed, even if we write down much more complicated models, we will always find that crystal momentum (momentum modulo the reciprocal lattice) remains conserved. As I mentioned, this is a result of the fact that the original system is periodic in real space.
To give an example that we will study later in the course, you might imagine firing an electron into a solid. Once in the solid, the electron has momentum conserved only modulo 2pi/a. So it can come out of the solid later with different momentum. But one should not worry about this, because the missing momentum is taken up by the rest of the solid (the nuclei etc), so the total momentum of the universe remains conserved.
I hope this helps!
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