homework help:discrete schodinger eq. Energy Eigenstates?

tiger_striped_cat

Hello all,

Before I start. You should note:
-I'm not just looking for a solution
-I've been working on this for hours
-I've checked probably about 10 quantum books
-I've scoured the internet
-My professor can't (or won't) help me (and two other students) so I can't ask around. We three are doing harder problems in the class, so there isn't anyone I can turn to.
-We all get full credit regardless on how well we do on these sets.

Here's the problem: Baym's Lectures on Quanum Mechanics Chapter 3, problem 1 (verbatim):

1. In the "discrete" Schrodinger equation (3-12) [which is given by:

[tex]i\frac{\partial\psi_{i}(t)}{\partial t}=w_{i,i-1}[\psi_{i,i-1}(t)-\psi_{i}(t)]+w_{i,i+1}[\psi_{i,i+1}-\psi_{i}(t)]+\frac{v_{i}}{\hbar}\psi_{i}(t)[/tex]

]

for a free particle, [tex] w_{i,i-1}=w_{i,i+1}\equiv w [/tex], and [tex]v_{i} = 0[/tex]. Assume periodic boundar conditions, that is, [tex]\psi_{N+1}(t)=\psi_{1}(t)[/tex], and generally [tex]\psi_{N+j}(t)=\psi_{j}(t)[/tex], where N is a large integer.
(a) Show that there are N linearly independent solutions of this Schrodinger equation.
(b) Find the N normalized energy eigenstates, that is, those solutions that vary in time with a fixed frequency. What are teh possible energy values for the particle? Show that these go over into the free particle energies, [tex]p^2/2m[/tex], in the continuum limit, i.e., as [tex]\lambda[/tex], the size of the intervals [tex]\rightarrow[/tex]0, but [tex]N\lambda[/tex] remains fixed.
(c) We can define the propagator matrix, K, by [tex]\psi_{j}(t)=\sum_{k} K_{jk}(t,t')\psi_{k}(t')[/tex] in analogy with (3-91). [Which is given by:

[tex]<r|\Psi(t)> = \int d^3r' K(rt,r't')<r'|\Psi(t')>[/tex]

Write down an explicit expression for the matrix elements [tex]K_{jk}(t,t')[/tex].


__________
Now, equation 3-12 simplifies to:

[tex]i\frac{\partial\psi_{i}(t)}{\partial t}=w[\psi_{i,i-1}(t)+\psi_{i,i+1}-2\psi_{i}(t)][/tex].

So I have N coupled differential equations. I think I can argue for part (a) by showing that I have these N equations that form a vector equation. And that the tridiagonal matrix formed by the coefficients of the psi's can be diagonalized. Or agrgue in this vein somehow.

But part (b) is killing me. I've been thinking about it. And if any of you know solid state you might have ran into problems like this before. The problem is that there are N equations. I might be able to deal with diagonalizing a matrix in Mathematica with some known size, but with some general N equations, I'm a little lost. I found a resource that says you can solve the discrete SE using Bloch's Theorem. The explanation was very shotty and I still don't see how to get the energy eigenvalues. And the resource isn't readily available anyways.

If you want to help me out you could try and wikipedia/mathworld/google Bloch's Theorem or "discrete schrodinger equation," but it didn't seem to help much.

Thank you for reading!

tiger_striped_cat

anyone??? (this space is for filler because I can't post a one word reply on this forum)