In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.
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Operational definition
Consider an operator A. So the expectation value is
Formalism in quantum mechanics
In quantum theory, an experimental setup is described by the observable
Mathematically,
(1)
If dynamics is considered, either the vector
If
(2)
This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues
A particularly simple case arises when
(3)
In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator
(4)
A similar formula holds for the momentum operator
All the above formulas are valid for pure states
(5)
General formulation
In general, quantum states
(6)
If the algebra of observables acts irreducibly on a Hilbert space, and if
with a positive trace-class operator
with a projector-valued measure
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).
Example in configuration space
As an example, let us consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is
gives the probability of finding the particle in an infinitesimal interval of length
As an observable, consider the position operator
The expectation value, or mean value of measurements, of
The expectation value only exists if the integral converges, which is not the case for all vectors
In general, the expectation of any observable can be calculated by replacing
Not all operators in general provide a measureable value. An operator that has a pure real expectation value is called an observable and its value can be directly measured in experiment.