Mathematical Formulation of Quantum Mechanics - Mathematical Structure of Quantum Mechanics - Postulates of Quantum Mechanics

Postulates of Quantum Mechanics

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 . Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.
• The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J.M. Jauch, Foundations of quantum mechanics, section 11-7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
• Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem (supersymmetry is another matter entirely).
• Physical observables are represented by Hermitian matrices 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 H is

By spectral theory, we can associate a probability measure to the values of A in any state ψ. 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 normalized to be of trace 1. The expected value of A in the state is

If is the orthogonal projector onto the one-dimensional subspace of H spanned by, then

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.

One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.

Superselection sectors. The correspondence between states and rays needs to be refined somewhat to take into account so-called superselection sectors. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.