**Homework 9**

Matrix Diagonalization:

Use a matrix diagonalization program to obtain all eigenvalues for a 50-site
chain (N=50) of non-interacting bosons with hopping parameter "t" (=
1). The Hamiltonian for this problem can be represented by a matrix, H_(i,j),
with H_(i,i+1) = H_(i+1,i) = t, H(1,N) = H(N,1) = t (due to periodic boundary
conditions), and all other matrix elements equal to zero.

(1) Show how this matrix representation is derived from the tight-binding
Hamiltonian with periodic boundary conditions. **(20 points) **

(2) Use a diagonalization routine of your choice (you may use Numerical
Recipes, netlib, ...) to obtain all N energy eigenvalues of this Hamiltonian
matrix. Compare them to the exact solutions: E_n = - 2*t*cos(k_n), with k_n=
2*Pi*n/N (n=-N/2+1, ..., N/2). **(40 points) **

(3) Approximate the density of states, N(E), for this Hamiltonian in the
following way: for each energy eigenvalue, E_n, draw a Lorentzian peak, P(E_n),
centered at (E - E_n): P(E_n) = 1/[(E - E_n)^2 + delta]. Let E go from -6 to 6
in increments of 0.01. This gives you an array with 1200 entries for P(E_n).
"delta" is taken to be a small number (larger than the step size in
E), e.g. delta = 0.05. Then add up all the peaks to obtain the total density of
states. It should look like this
. **(40 points) **