Finite-Dimensional Quantum Mechanics
Or, What I Learned in San Diego
When we learn about quantum mechanics at the graduate level, it's easy to
become lost in all the technicalities which arise (necessarily!) in the
infinite-dimensional case: unbounded operators, domain restrictions, singular
measures, abstract spectral theory, etc.
Fortunately, at the AMS/MAA joint meetings in San Diego this winter I saw a
splendid little talk about a two-state quantum mechanical system (involving
oscillation of solar neutrinos, as it happens) which is completely manageable
at the elementary level. As well, I suddenly realized a few weeks ago that the
Schroedinger equation can be solved numerically on a discrete mesh using the
discrete Laplacian and a simple Runge-Kutta method, using just a couple dozen
lines of Matlab code. I'll show you the results of both experiments. This
means actual numbers -- decimal places and all -- and some nice Matlab plots.
As you will see, things become much clear in finite-dimensional cases -- the
math reduces to straightforward notions in probability, finite-dimensional
linear algebra, and differential equations. We will see tangible
demonstrations of the fundamental notions of quantum mechanics: state spaces,
time evolution, and Hermitian and unitary operators.
John Kerl
Graduate Colloquium
March 5, 2008