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Archive for July, 2011

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 there is a conceptual explanation of this connection somewhere, probably coming from statistical mechanics, but I don’t know it. I suppose the approximate scale invariance of the zeta distribution is relevant to its appearance in many real-life statistics, as described in Terence Tao’s blog post on the subject here.

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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 \psi in some Hilbert space L^2(X) (up to multiplication by a constant), and time evolution is given by

\displaystyle \psi \mapsto e^{ \frac{H}{i \hbar} t} \psi

where H is a self-adjoint operator on L^2(X) called the Hamiltonian. Observables are given by other self-adjoint operators F, and at least in the case when F has discrete spectrum measurement can be described as follows: if \psi_k is a unit eigenvector of F with eigenvalue F_k, then F takes the value F_k upon measurement with probability \left| \langle \psi, \psi_k \rangle \right|^2; moreover, the state vector \psi is projected onto \psi_k.

The Heisenberg picture is an alternate way of understanding time evolution which de-emphasizes the role of the state vector. Instead of transforming the state vector, we transform observables, and this point of view allows us to talk about time evolution (independent of measurement) without mentioning state vectors at all: we can work entirely with the algebra of bounded operators. This point of view is attractive because, among other things, once we isolate what properties we need this algebra to have we can abstract them to a more general setting such as that of von Neumann algebras.

In order to get a feel for the kind of observables people actually care about, we won’t study a finite toy model in this post: instead we’ll work through some classical (!) one-dimensional examples.

(more…)

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