Poisson algebras and the classical limit

In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra (the algebra of observables of some quantum system) and a Hamiltonian , we obtain a derivation , which is (up to some scalar multiple) the infinitesimal generator of time evolution. This [...]

Read Full Post »

Update

I put up a post over at the StackOverflow blog describing a little of what I’ve been up to this summer. Curiously enough, the Zipf distribution which shows up in that post is the same as the zeta distribution that shows up when trying to motivate the definition of the Riemann zeta function. I’m sure [...]

Read Full Post »

The Heisenberg picture of quantum mechanics

In an earlier post we introduced the Schrödinger picture of quantum mechanics, which can be summarized as follows: the state of a quantum system is described by a unit vector in some Hilbert space (up to multiplication by a constant), and time evolution is given by where is a self-adjoint operator on called the Hamiltonian. [...]

Read Full Post »

The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and [...]

Read Full Post »

More about the Schrödinger equation on a finite graph

It looks like the finite graph model is not just a toy model! It’s called a continuous-time quantum random walk and is used in quantum computing in a way similar to how random walks on graphs are used in classical computing. The fact that quantum random walks mix sooner than classical random walks relates to [...]

Read Full Post »

The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation [...]

Read Full Post »

A little more about zeta functions and statistical mechanics

In the previous post we described the following result characterizing the zeta distribution. Theorem: Let be a probability distribution on . Suppose that the exponents in the prime factorization of are chosen independently and according to a geometric distribution, and further suppose that is monotonically decreasing. Then for some real . I have been thinking [...]

Read Full Post »

Zeta functions, statistical mechanics and Haar measure

An interesting result that demonstrates, among other things, the ubiquity of in mathematics is that the probability that two random positive integers are relatively prime is . A more revealing way to write this number is , where is the Riemann zeta function. A few weeks ago this result came up on math.SE in the [...]

Read Full Post »

Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile. The setup of the puzzle is as follows. Let be a weighted undirected graph, e.g. to each edge is associated a non-negative real number , and let be the corresponding weighted adjacency matrix. If is stochastic, one can interpret the [...]

Read Full Post »