Expectation Value (quantum Mechanics) - General Formulation

General Formulation

In general, quantum states are described by positive normalized linear functionals on the set of observables, mathematically often taken to be a C* algebra. The expectation value of an observable is then given by

(6) .

If the algebra of observables acts irreducibly on a Hilbert space, and if is a normal functional, that is, it is continuous in the ultraweak topology, then it can be written as

with a positive trace-class operator of trace 1. This gives formula (5) above. In the case of a pure state, is a projection onto a unit vector . Then, which gives formula (1) above.

is assumed to be a self-adjoint operator. In the general case, its spectrum will neither be entirely discrete nor entirely continuous. Still, one can write in a spectral decomposition,

with a projector-valued measure . For the expectation value of in a pure state, this means

,

which may be seen as a common generalization of formulas (2) and (4) above.

In non-relativistic theories of finitely many particles (quantum mechanics, in the strict sense), the states considered are generally normal. However, in other areas of quantum theory, also non-normal states are in use: They appear, for example. in the form of KMS states in quantum statistical mechanics of infinitely extended media, and as charged states in quantum field theory. In these cases, the expectation value is determined only by the more general formula (6).