Homework 2. due Friday, Feb 10 (or Monday Feb 13 in my mailbox)

Edited Saturday,  Feb 11.
Note added: I think maybe the expectation value of x,  \(\langle x \rangle\) can be calculated as  \(\langle n \rangle \times a\). Is that true? Similarly, the expectation value \(\langle x^2 \rangle\) is probably \(\langle n^2 \rangle \times a^2\). This could make the calculation of the "size" of the wave-function quicker.  Please let me know if this seems correct or not. This is just off the top of my head.  Also, please take the poll.



2nd note added: I think the key thing to look at is the large time behavior of the size of the wave function.  I think you can calculated the size by taking the square root of \(\langle x^2 \rangle - \langle x \rangle^2\). Does that make sense?  I think that that is also proportional to: \(\langle n^2 \rangle - \langle n \rangle^2\).

This homework focuses on using a matrix approach to study electron wave-functions in crystals.  Please start working on this soon. It has a open ended nature. Questions and comments here would be really helpful: Does the problem makes sense to you? Is it clear how to proceed? Are the symbols well-defined and clear? And perhaps most importantly: does the formalism make sense? Do you see how it is related to and different from the Schrodinger wave equation approach in which super-positions of energy eigenstates, each with their own individual time-dependent factor \(e^{-iE_k t/\hbar}\), are used to create general (wave-packet) wave-functions (wave-functions with time-dependent expectation values)?

These problems may be a little vague, and may require consideration and clarification as you work through them. I would be very happy to see comments here from many people within the next few days.  I feel like if you give yourself some time to think things over, by starting soon, then it will go better.  Also, please feel free to email me with any results and questions you have.

Consider a large matrix of the form:
$$ \begin{matrix}
E_1 &-\gamma &0 &0
\\ -\gamma &E_1 &-\gamma &0 &0
\\ 0 & -\gamma &E_1 &-\gamma &0 &0
\\ 0 &0 & -\gamma &E_1 &-\gamma &0 &0
\\ 0 & 0 & 0 & -\gamma &E_1 &-\gamma &0
\\ \end{matrix} $$
As we discussed in class today, the eigenfunctions and eigenvalues of this matrix are in one-to-one correspondence with Bloch states and their energies provided \(-\gamma\) has the value of the overlap integral from tight binding theory. (\(\gamma\) has units of energy as does \(E_1\).)

1. (warm-up problem)  a) Write the equivalent of a Bloch state, with wave-vector k, in vector language, i.e., as a sequence of complex coefficients. (Don't over think this one. It can be easy and short.)
b) Show that this vector has an eigenvalue \(E_k = E_1 -2 \gamma cos(ak)\).
(For this problem you can assume the matrix is infinite so that boundary issues disappear. Multiply the infinite matrix times the infinite vector.)

2. Numerical problem. Starting with a localized state (wave-function non-zero on just one lattice site), preferably not near a boundary:  (For this is is probably best to use a numerical value for the unit-less quantity  \( \gamma t / \hbar\).  [Recall that \(\hbar c = \) ...  eV-nm  (what is the value that goes there?).  Although I suppose you don't really need that for the numerics, just a value for the unit-less quantity of about 0.01 or... what do you find works well for numerical iteration here?]
a) Look at the evolution of the wave-function vs time. (Could you make a giff showing the wave-function as a function of time. That would be really cool and much appreciated. Maybe use different colors for the real and imaginary parts...)
b) Calculate and plot the expectation value of \(\langle x^2 \rangle - \langle x \rangle^2\)
(perhaps look at the "width" of the wave-function as a function of time?  How do you define the width of a wave-function (in terms of the graph or of expectation values)? Please do that in the comments here.
c) Try different initial states. For example, try to make a traveling wave, (or just go on to the next problem if you prefer).

3. Consider the case where E_1 varies from site to site. You can assume a Gaussian distribution of values for E_1 centered on zero and assume random variation from site to site.  (And thus random variation along the diagonal of the matrix, which is where E_1 appears. (If anyone can expand and clarify on this part in the comments, that would be much appreciated.)
a) Look at the evolution of the wave-function vs time.(make graphs)
b) Try to determine: at what level of disorder the fundamental nature of the crystal energy eigenstates changes in a profound way? I think you can discover this by looking at the behavior of the size of the wave-function as a function of time, particularly the asymptotic behavior.
--Added note: For this problem, perhaps the disorder does not necessarily have to be of a Gaussian form. Maybe someone could try instead (or in addition) a binary disorder where there are just two possible energies for the diagonal matrix element. Apply those in a pattern that is random (decided by a "coin flip") and see what level of disorder leads to localization in that model?  Can you guess which causes localization more readily: a gaussian of width E (for the distribution of diagonal matrix elements), or a difference of E between two binary choices?  You can vote on the poll the right.