Models describing electrons on a crystal lattice are very
important to understanding various phenomena in solids. Here
we consider a model in which an electron lives on a one-dimensional
lattice of N sites. The sites are labeled by
i=1,2, ....,N.
The system looks like
o----o----o-- .... --o----o
1 2 3 N-1 N
The state of the electron is then a vector of dimension N.
The Hamiltonian is given by an N by N matrix whose elements
are:
/ - 1, if i and j are near-neighbors;
H_{ij} = |
\ 0 , otherwise.
Physically, the electron can be thought of as hopping from site
to site through a near-neighbor hopping. As you see, the Hamiltonian
resembles the one we obtained in class when we discretized
the problem of a particle in a box.
- Justify the Hamiltonian above by relating it to the
time-independent Schroedinger wave equation.
To be specific, we now let N=7.
Suppose we prepare the electron in a state |a> with equal
amplitude for all N sites, i.e., a(1)=a(2)=....=a(N).
- What is the lowest value we can find if we measure the energy of the
electron? With what probability?
- List all the possible (i.e.,
with non-zero probability) energy values that we could
find in such a measurement.
You will need to do some computing to answer the last two
questions. This can be done with any programming environment you
choose. If you have not done any programming before, I recommend
Maple. (You may find the
Maple quick
reference page at Indiana helpful.)
Here are some Maple commands
for your reference.
Here is a simpler version, which I
used for the particle in a box, for those
who do not yet feel comfortable playing around in Maple.
On the Physics
Camelot cluster, you can start a Maple session by typing
xmaple and run these commands by, e.g., copying each line
with the mouse to the Maple command prompt.