Diffussion of a localized crystal state: electron dynamics in a lattice

This post is a continuation of the "thread" that started with the posts IP#3 and, more recently, "Spec.Prob.#1: subverting the dominant paradigm ".

Based on what we did in class last Thursday, we now have a sense that we can calculate time evolution of a state function without going through the tedious an unintuitive process of doing separation of variables, finding energy eigenstates and then writing the general state function as a superposition. We can instead use an alternate approach based on the formal solution of the time-dependent Schrodinger equation, H Psi(t) = i hbar d (Psi(t)/dt, which is:
Psi(t) = exp[i H t/hbar] Psi(o)
= [1 + i H t/hbar] Psi(o),
where the last line is approximate, but becomes arbitrarily accurate as t gets small.

A key question then, is when might this be useful?

An example of where it might be useful, as we discussed in class, is in the 1D crystal problem, especially the very interesting case in which an electron is initially localized at one site (at t=0) and we would like to understand the time-dependent behavior of the electron, that is, we would like to calculate how the state function of the electron changes as a function of time.

Calculating this time evolution, except for numerical errors and errors associated with the approximations one inevitably makes in dealing with infinite sums, should produce a result mathematically equivalent to a representation of the localized state as a superposition of energy eigenstates (that being the standard approach, the "dominant paradigm").

Looking at the time evolution in this new way, we can answer questions like:
What is the probability that the electron will "hop" from site i to the adjacent site i+1? or a time t?, or, to put it another way:
what is the characteristic time for an electron to hop from one sit to another?

For the two square-well problem, I think that one could get a good estimate of the time for an electron to hop (or "tunnel", as we have said in the past) from one well to another using essentially this method. (This is a problem that we discussed earlier in the quarter, but never actually solved quantitatively.) Things one would like to calculate include the characteristic tunneling time and its' dependence on the distance separating the wells. I think you might be able to do a reasonable "back of the envelope" estimate of that, i think, with the two key things being the wave-function exponential length scale outside the well, the distance between the wells, and perhaps the depth of the well (all of which will go into the overlap integral, right?).
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In addition to being able to do most things one can do with the original methodology based on superposition of eigenstates, this formalism makes it relatively easy (possible) to introduce disorder.
For example, here is a problem involving disorder that might interesting (though, i have never done this before or seen a solution or discussion of this problem. It may be related to "localization".):
Suppose the site at which the electron starts has a different energy than all the other sites. Let say it has a larger negative alpha, and thus a lower energy (a larger, more negative e_0). Then will the electron still diffuse away or will it remain localized?
This is part of the more general question of what effect will disorder have on the propagation or diffusion of an electron in a lattice?