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10/15/08
ECE 495N, Fall’08
ME118, MWF 1130A – 1220P
HW#5: Due Wednesday Oct.22 in class.
Problem 1: Use the principles of bandstructure
to write down the eigenvalues and
eigenvectors of the matrix (a, b are real numbers)
a
ib
−
ib
−
ib
a
ib
ib
−
ib
a
Problem 2:
Benzene molecule consists of six carbon atoms arranged at the corners of a
regular hexagon of side ‘a’. Assume (1) one orbital per carbon atom as basis function ; (2)
the overlap matrix [S] is a (6x6) identity matrix; and (3) the Hamiltonian matrix is given by
H
n
,
n
=
ε
(site energy)
H
n
,
m
=
t
if n, m are neighboring atoms
H
n
,
m
=
0
if n, m are NOT nearest neighbors
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Unformatted text preview: ’ and ‘t’ ? What are the corresponding eigenvectors? Problem 3: In class we have seen that an infinitely long linear 1-D lattice (lattice constant: a) with a Hamiltonian = has a dispersion relation E ( k ) = + 2 cos ka where − π < < + (1) Do the same problem using a unit cell of two atoms (instead of one) and show that ( ) = ± 2 cos where − /2 < < + /2 (2) Are (1) and (2) equivalent? Explain. <-- a -->...
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- Spring '08
- S.Datta
- Linear Algebra, Matrices, Atom, Fundamental physics concepts, ib − ib −, a ib
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