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" Schrödinger equation (3-12) [which is given by:

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)

]

for a free particle, w_{i,i-1}=w_{i,i+1}\equiv w, and v_{i} = 0. Assume periodic boundar conditions, that is, \psi_{N+1}(t)=\psi_{1}(t), and generally \psi_{N+j}(t)=\psi_{j}(t), where N is a large integer.
(a) Show that there are N linearly independent solutions of this Schrödinger equation.
(b) Find the N normalized energy eigenstates, that is, those solutions that vary in time with a fixed frequency. What are the possible energy values for the particle? Show that these go over into the free particle energies, p^2/2m, in the continuum limit, i.e., as \lambda, the size of the intervals \rightarrow0, but N\lambda remains fixed.
(c) We can define the propagator matrix, K, by \psi_{j}(t)=\sum_{k} K_{jk}(t,t')\psi_{k}(t') in analogy with (3-91). [Which is given by:

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

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


__________
Now, equation 3-12 simplifies to:

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

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 Schrödinger 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)

 

[wais]

Hello there,

I understand your frustration and I will do my best to help you out with this problem. First off, let me say that it's great that you have already checked multiple sources and have put in a lot of effort to solve this problem on your own. That shows dedication and determination, which are great qualities to have in the field of quantum mechanics.

Now, let's tackle the problem. Part (a) is indeed about showing that there are N linearly independent solutions to the discrete Schrödinger equation. This can be done by considering the matrix formed by the coefficients of the psi's, as you mentioned. This matrix is a tridiagonal matrix, as you correctly pointed out, and can be diagonalized. This means that there are N eigenvectors of this matrix, which correspond to the N linearly independent solutions of the Schrödinger equation. This is a valid argument for part (a).

Moving on to part (b), the key here is to use Bloch's theorem. This theorem states that for a system with periodic boundary conditions, the solutions to the Schrödinger equation can be written as a product of a periodic function and a plane wave with a fixed wave vector. In this case, the periodic function is the psi's and the plane wave is e^{ikx}, where k is the wave vector.

Using this, we can write the solutions to the Schrödinger equation as:
\psi_{j}(x,t) = e^{ikx}\phi_{j}(t)
where \phi_{j}(t) is a periodic function with the same period as the potential.

Now, we can substitute this into the Schrödinger equation and we get:
i\frac{\partial\phi_{j}(t)}{\partial t} = w(e^{ik\lambda}+e^{-ik\lambda}-2)\phi_{j}(t)
where \lambda is the size of the intervals (as given in the problem).

Now, we can solve this equation for \phi_{j}(t) and we get:
\phi_{j}(t) = e^{-iE_{j}t}
where E_{j} = 2w(cos(k\lambda)-1). This is the energy eigenvalue corresponding to the jth solution.

Now, to find the normalized energy eigenstates, we need to find the values of k that satisfy the periodic boundary conditions. In other words, we need to find