Phys622: Homework Assignment 5
04/01
==> Due on Thursday 04/10 in class <==
(1) Sakurai Problem 6.1a (part (a) only).
(2) An electron on a lattice, revisited.
We return to the problem of
``an electron on a lattice'' that we did last semester.
Instead of a closed system of N sites, we now include periodic
boundary conditions in our model to represent the extended nature of
the crystal. That is, the system repeats itself after N sites:
..... o----o----o-- .... --o----o - - x -- x .....
1 2 3 N-1 N N+1 N+2
Site N+1 is equivalent to site 1, N+2 to 2,
and so on. Put another way, the N sites can be thought of as forming
a ring with site 1 being near-neighbors with sites 2 and
N.
The state of the electron is still 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.
- To be specific, let's choose N=35 and study the system
numerically.
Here are some Maple commands
in case you find them useful
.
- Obtain all the energy eigenvalues. What is the degeneracy for
each energy level?
- Examine the eigenvector corresponding to the
ground-state wave function and explain its structure.
- Plot the wave function for the first excited state. If there is
a degeneracy, show all the degenerate wave functions.
Explain the structure of the wave function(s).
- In (1), if we had chosen an even number for N, how would
the level of degeneracy be for each energy level?
- Now let us solve the same problem analytically for any N.
You will notice that this model is essentially the same as the
tight-binding model discussed in Sakurai, except that we impose
periodic boundary conditions. In other words, we are considering a
ring while Sakurai considered an infinite lattice.
- Determine the energy levels and their wave functions. There should be
N of these.
- Now set N=35 and show that the numerical results in (1)
are completely consistent with the analytical results here.
(3) Sakurai Problem 6.4.