Phys621: Homework Assignment 1

9/14

==> Due on Tuesday 9/26 in class <==


(1) An electron on a lattice.

  1. Models describing electrons on a crystal lattice are very important to understanding various phenomena in solids. Here we consider a model in which an electron lives on a one-dimensional lattice of N sites. The sites are labeled by i=1,2, ....,N. The system looks like
         o----o----o-- .... --o----o
         1    2    3         N-1   N
    
    The state of the electron is then 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.
    
    Physically, the electron can be thought of as hopping from site to site through a near-neighbor hopping. As you see, the Hamiltonian resembles the one we obtained in class when we discretized the problem of a particle in a box.

    To be specific, we now let N=7. Suppose we prepare the electron in a state |a> with equal amplitude for all N sites, i.e., a(1)=a(2)=....=a(N).

    You will need to do some computing to answer the last two questions. This can be done with any programming environment you choose. If you have not done any programming before, I recommend Maple. (You may find the Maple quick reference page at Indiana helpful.) Here are some Maple commands for your reference. Here is a simpler version, which I used for the particle in a box, for those who do not yet feel comfortable playing around in Maple. On the Physics Camelot cluster, you can start a Maple session by typing xmaple and run these commands by, e.g., copying each line with the mouse to the Maple command prompt.

  2. Now let us look at the question of degeneracy in Sakurai Problem 1.14, part a. Without computing the eigenvectors or eigenvalues, answer the question (Is there any degeneracy?) and justify your answer.

(2) Sakurai Problem 1.6 (Problem 6 of Chapter 1).

(3) Sakurai Problem 1.5.