The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.
- Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product \langle \phi | \psi \rangle. Rays (one-dimensional subspaces) in H are associated with states of the system.
- The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems.
- Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem.
- Physical observables are represented by densely-defined self-adjoint operators on H.
The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector | \psi \rangle \in H is
\langle\psi|A|\psi\rangle
By spectral theory, we can associate a probability measure to the values of A in any state \psi. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.
More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator \rho normalized to be of trace 1. The expected value of A in the state \rho is
\text{tr}\left(A\rho\right)
If \rho_\psi is the orthogonal projector onto the one-dimensional subspace of H spanned by \psi, then
\text{tr}\left(A\rho_\psi\right)=\langle\psi|A|\psi\rangle
Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.
