The enclosed 5 pages of jpeg notes contain a calculation on the origin of bands from quantum atoms using Bloch eigenstates. The key thing is how a band of states emerges from a single quantized state, and how the width of that band depends on the overlap integral t.
For homework (see below*) you may rederive this result (optional) and also, most importantly, graph Eq as a function of q (from -pi/a to pi/a) (not optional). Before you graph you should be sure of the sign of every term in your equation, and also have some sense of what their relative magnitudes might be. (Also units.)
PS: Please note the page numbers. I am not sure how to control the order of multiple page uploads here, but the pages are numbered...
PPS. Please feel free to ask and answer questions here!
* Homework summary as of 3-31-10 (HW#1):
1. a) Sketch the two lowest energy states of a potential which consists of two equal finite square wells.
b) Sketch the four lowest energy states of a potential which consists of four equally spaced, equal strength finite square wells.
c) Sketch the six lowest energy states of a potential which consists of six equally spaced, equal strength finite square wells.
d) (extra credit/optional) Sketch the three lowest energy states of a potential which consists of three equally spaced, equal strength finite square wells.
e) (optional) Sketch the fifth lowest energy state of a potential which consists of four equally spaced, equal strength finite square wells.
2. What are the units of t and I (see notes). Graph Eq vs q from -pi/a to pi/a. Which is larger, 2t or E0 ?
Hi - If I were using a finite potential well with a zero-potential at the top of the well (-V_0 at the bottom of the well - ie -13.6 eV) would the wavefunction decay upwards to zero when it is in the "forbidden" region?
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Ethan
not sure if this is still relevant, but for finite square wells the wave function decays exponentially to zero for positions outside the well and is a Sin inside, so the ground state looks kind of like a gaussian.
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