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)