I haven't had time to really put together the worksheet I wanted for this topic, but I thought I'd post what I had for those who may be interested in Markov processes, matrix manipulations and quantum mechanics. I'm attaching two files. The first illustrates how to use eigenvectors and eigenvalues to decompose a stochastic matrix. The decomposition allows one to understand the steady state or equilibrium state of a system as well as time evolve the system in a numerically efficient and robust manner. During the course of putting together this worksheet, I found that there are situations where the nth eigenvalue computed by the eigenvals function may not correspond to the nth eigenvector computed by the eigenvecs function (2001i, sp2, Mathsoft is now aware of the situation).
The second worksheet is the middle of a conversation (which may make things a bit hard to follow) I had with one of our summer students that I throw in for a couple of reasons. First, there is a connection between probability and quantum mechanics that can be tackled several ways. Second, it shows another example where one may want to raise a matrix to a high power. This worksheet looks at band structures using quantum scattering theory rather than the more traditional band structure math. As such, it could use time evolution and space evolution propagators that are very similar to the Markov stochastic matrices. I'll illustrate this someday when I get time, but it will be awhile. In the meantime, those puzzled by the mysteries of quantum mechanics may find it interesting to think about the problem of probability from the perspective of optics.
Robert
The second worksheet is the middle of a conversation (which may make things a bit hard to follow) I had with one of our summer students that I throw in for a couple of reasons. First, there is a connection between probability and quantum mechanics that can be tackled several ways. Second, it shows another example where one may want to raise a matrix to a high power. This worksheet looks at band structures using quantum scattering theory rather than the more traditional band structure math. As such, it could use time evolution and space evolution propagators that are very similar to the Markov stochastic matrices. I'll illustrate this someday when I get time, but it will be awhile. In the meantime, those puzzled by the mysteries of quantum mechanics may find it interesting to think about the problem of probability from the perspective of optics.
Robert