In an earlier post we introduced the SchrÃ¶dinger 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. Observables are given by other self-adjoint operators , and at least in the case when has discrete spectrum measurement can be described as follows: if is a unit eigenvector of with eigenvalue , then takes the value upon measurement with probability ; moreover, the state vector is projected onto .

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.

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