Today's problem: energies of states in a crystal

So for today's problem, try to finish that by calculating E vs k for that 1-D chain of attractive delta functions and assuming phi_0 is just the g.s. for one delta function. Does that make sense?

Don't forget to graph it; that is important! And think about the energy scale. Is it positive or negative? Are the energies above or below those for the isolated atom? Is Ek real?

Also, you might want to think about the range of k. What if k is 2 pi/a ? or 3 pi / a ? Is there some limit on k or something?

Reflections on QM

Perhaps it would be good for people to share their reflections about QM as comments here. These could focus on what seems most compelling and interested to you, what you would like to remember in 10 years or so or ...?

PS. Please look at the previous post (from today) also.

For Tuesday: more about the periodic table, states in solids, U and the Schro eqn.

So far we have been mostly considering an idealized periodic table, but actually, as some people pointed out, the d-levels don't even start in the correct row, etc. We can discuss this a little more on Tuesday. Perhaps the key things that survive from the idealized picture are the s-p degeneracies (which are v. important!) and the fact that there are always 6 p states, 10 d states, etc. in a group. Anybody have an idea as to why the d-states are shifted out of their expected row? (comments are welcome)

Here is a problem that might be good preparation for understanding electron states in crystals:
Consider a potential that consists of 4 equal-strength and equally-spaced attractive delta functions all in a line (in 1 dimension). Suppose they are relatively far apart*. How many different (orthogonal) low-energy (one-electron) states are there and what do they look like?
(You could warm up with the 2 delta function problem, and then maybe try 3 or 6 delta functions. 3 is a little tricky. *can you define "relatively far apart"?

Is there anything else we should be working on or thinking about for this last week? [After this week it will be just you and the Schrodinger equation* traveling through the world together for the rest of your life. I hope you will be ready for that. (*and a normalization condition)]