lecture_3.md

Atoms and bonds

Looking back

So far we have:

• Introduced the k-space (reciprocal space)
• Postulated electron and phonon dispersion relations

As a result we:

• Understood how phonons store heat (Debye model)
• Understood how electrons conduct (Drude model) and store heat/energy (Sommerfeld model)

We used our best guess as a starting point, and there are several mysteries:

• Why is there a cutoff frequency? Why are there no more phonon modes?
• Why do electrons not scatter off from every single atom in the Drude model? Atoms are charged and should provide a lot of scattering.
• Why are some materials not metals? (Think if you know a crystal that isn't a metal)

A quick review of atoms

Why chemistry is not physics

Chemists are merely solving the Schrödinger equation, they are a domain in physics
—An arrogant physicist

Everything is described by the Schrödinger equation:

H\psi = E\psi,

with H the sum of kinetic energy and Coulomb interaction, so for hydrogen we have:

H=-\hbar^2\frac{\partial^2}{2m\partial {\mathbf r^2}} - \frac{e^2}{4\pi\varepsilon_0|r|}

for helium it becomes more complex: \psi({\mathbf r})\rightarrow \psi({\mathbf {r_1, r_2}}), so

H=-\hbar^2\frac{\partial^2}{2m\partial {\mathbf r_1^2}} -\hbar^2\frac{\partial^2}{2m\partial {\mathbf r_2^2}}- \frac{2e^2}{4\pi\varepsilon_0|r_1|} - \frac{2e^2}{4\pi\varepsilon_0|r_2|} + \frac{e^2}{4\pi\varepsilon_0|r_1 - r_2|},

which means we need to find eigenvalues and eigenvectors of a 6-dimensional differential equation!

"Mundane" copper has 29 electrons, so to find the electronic spectrum of Copper we would need to solve an 87-dimensional Schrödinger equation, and there is no way in the world we can do so.

This exponential growth in complexity with the number of interacting quantum particles is why many-body quantum physics is very much an open area in solid state physics.