## Band Structure p.8

We see here the wavefunctions for various values of *V*_{0}
and **k**. The probability densities for the four lowest eigenvectors
are presented in "tour" order:
**k**=(*k*_{x},*k*_{y})=

(max,0), (2,0), (0,0), (0,2), (0,max)

As you recall for the free lattice, the eigenfunctions are:

exp(*i***G·r**)

so the probability density is uniform. However, our tour
of **k**-space is along high symmetry lines: there is much degeneracy.
Any slight perturbation will mix the degenerate wavefunctions
to produce linear combinations of the above waves. Linear
combinations of the above do *not* produce uniform
probability density. SO here is what is expected for
*V*_{0} near zero:

*V*_{0}=0

Note that the non-degenerate wavefunctions have not been mixed and so have
uniform probability densities.
The upper rhs eigenfunctions were seen to have a degeneracy involving the nearest neighbors
in the two and ten o'clock positions: what you see displayed is just the plus and
minus linear combination of those two exp(*i***G·r**). It will be a nice
problem to figure out some of the other linear combinations!

*V*_{0}=16

*V*_{0}=64

Notice that the ground state is fairly independent of **k**. (The gray-scale is constant only within
a column of four. Thus even though **k**=0 ground state looks dimmer than the rest, they are really
all similar.
*V*_{0}=256

Here is the summary: we display wavefunctions taken at a fairly
random **k**-space point (not a tour-point; in fact the median of the tour triangle):
about **k**=(0.7,2.4). We display below in succession *V*_{0}=16,64,256.

Notice that we go from a fairly uniform electron density, to something approximating bound states.
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