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ECE495N-F08-HW_5 - ’ and ‘t’ What are the...

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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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