Related to the essay, but I figured I'd go through what I considered to be the three most important equations of the course. Reckon it would be interesting to see if they differ from others opinions. I apologise for the notation in advance, it'll probably be set out how I would write it in matlab
Fermi wave vector: n = (kf^3)/(3*pi^2)
Might seem like a bit of an odd one, but the construction of the Fermi surface and the subsequent Fermi wave vector has so many implications in all of the solid state that we dealt with. This definition first appeared in chapter 2 using the Sommerfield model, but was still used once the crystal lattice was introduced.
Bloch's Theorem: psi_nk(r+N*a) = exp(i*N*k.*a)psi_nk(r)
Bloch rocks, need I say more?
Semiclassical equations of motion: dr/dt = 1/hbar * de_n(k)/dk
hbar* dk/dt = -e(E + 1/c * dr/dt * B)
Grouped these two together, but with them we can determine the dynamics of electrons in magnetic and electric fields with certain energy levels defined by e_n(k). Requires just a few assumptions about position and momentum.
Anyway they're my three equations, and with them i feel you can describe most of the dynamics of electrons and ions in a crystal lattice. Any major disagreements with my choices?
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I'd question how "fundamental" the wavevector equation (first equation) is.
ReplyDeleteThat comes from the volume of the Fermi surface.
by definition, n is the number of state, taken by taking a volume in k-space.
We could argue that expression of Fermi energy in terms of wavevector (in terms of hbar*k) is more simple...
NIce post.
ReplyDeleteFirst equation is a specific example of a very profound result from quantum many-body theory known as Luttinger's theorem
http://en.wikipedia.org/wiki/Luttinger%27s_theorem
the volume enclosed by a material's Fermi surface is directly proportional to the particle density.
My lectures did not emphasize how fundamental this result is, but should have.