Wave packet propagation / special project.

This is a post on wave packet propagation, in free space and in a crystal lattice, and a suggestion for a special project if anyone has time for that.
Can you guess what this first giff shows?
(ps. this post contained a number of errors originally. i edited it, hopefully fixing them, on Feb 26.)



A lot of our discussion of electron movement in semiconductors is based on a belief in electron states that are somewhat localized (maybe within 20 lattice sites or therabouts), which maintain their integrity for some bit of time, and can move in a crystal. This belief is based in wave-packet theory. Here is an outline of how one might make some single-electron wave-packet states in free space or in a crystal:
Free space:
In free space we would use the spatial energy eigenstates, $e^{ikx}$ to make the wave packet. (One always has to use the energy eigenstates. In a crystal they are more complex.) In free space they are all of the form $e^{ikx}$ and have energies of $E(k) = \hbar^2 k^2/(2m)$.

A typical wave-packet state is a mixed state made of lots of these energy eigenstates, each with its appropriate time dependence. For example.

$\Psi(x,t) = \int g(k) e^{-ikx} e^{-i \omega(k) t} dk$

where
$g(k) = \sqrt{30/\pi} e^{-30^2 (k-\pi/20)^2}$.

The E vs k relationship:
$E(k) = \hbar^2 k^2/(2m)$,

leads to something like:
$\omega (k)  = 1972 (3 \times 10^{18}) k^2/(1.022 \times 10^6)$
where k is in inverse angstroms (and c in there is in angstroms/second.)

For  an electron in a crystal the E vs k relationship is different:
$\omega (k) =  (E_o - (B/2) cos (kb))/\hbar$
              $= (-10/6.6-(2/6.6) cos(k)) \times 10^{16}$
for a crystal with b = 1 Angstrom,  B/2 = 2 eV and with $\hbar$ in eV-sec.
(With k in inverse angstroms.)
(Hmmm. It might be better to program with hbar in eV-picoseconds and then have t in picoseconds in the equation for Psi. (Smaller exponents))

Also, the energy eigenstates would be different for an electron in a crystal. (They would be the Bloch states created by a sum over all the lattice sites that we talk about a lot.) I think that at low k, near the bottom of the conduction band, these long wave-length states are pretty similar to free electron states. A bit surprising, but i think it is true.

Anyway, I think what either of those two possibilities will give you is a one-electron wave-packet state that starts out about 10 or 20 lattice spacings (Angstroms) wide and moves to the right as a function of time. The reason I think it moves to the right is that the k values that are used center around k=pi/20, which is positive.  I think the packet profile, g(k), is pretty narrow in k; that was my intention.

One could start by just integrating (over k) at t=0 to investigate the nature of $\Psi (x,0)$. The integral is from k = -infinity to infinity.

And then try a small finite t and integrate (k) again to see the difference.

Here is an intriguing result from using a pretty narrow (in k) gaussian "packet" of crystal states centered at k=pi/10.