It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean).
It is a fundamental concept in all areas of quantum physics.
In quantum theory, an experimental setup is described by the observable
is a self-adjoint operator on a separable complex Hilbert space.
The evolution of the expectation value does not depend on this choice, however.
then (1) can be expressed as[1] This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues
This physically corresponds to a "yes-no" type of experiment.
In this case, the expectation value is the probability that the experiment results in "1", and it can be computed as In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the position operator
This is formally achieved by projecting the state vector
It happens that the eigenvectors of the position operator form a complete basis for the vector space of states, and therefore obey a completeness relation in quantum mechanics:
The above may be used to derive the common, integral expression for the expected value (4), by inserting identities into the vector expression of expected value, then expanding in the position basis: Where the orthonormality relation of the position basis vectors
The last line uses the modulus of a complex valued function to replace
The expectation value may then be stated, where x is unbounded, as the formula A similar formula holds for the momentum operator, in systems where it has continuous spectrum.
Prominently in thermodynamics and quantum optics, also mixed states are of importance; these are described by a positive trace-class operator
The expectation value then can be obtained as In general, quantum states
are described by positive normalized linear functionals on the set of observables, mathematically often taken to be a C*-algebra.
is then given by 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
In the general case, its spectrum will neither be entirely discrete nor entirely continuous.
In non-relativistic theories of finitely many particles (quantum mechanics, in the strict sense), the states considered are generally normal[clarification needed].
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,[3] and as charged states in quantum field theory.
[4] In these cases, the expectation value is determined only by the more general formula (6).
As an example, consider a quantum mechanical particle in one spatial dimension, in the configuration space representation.
, the space of square-integrable functions on the real line.
The wave functions have a direct interpretation as a probability distribution: gives the probability of finding the particle in an infinitesimal interval of length
performed on a very large number of identical independent systems will be given by
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
The expectation value, in particular as presented in the section "Formalism in quantum mechanics", is covered in most elementary textbooks on quantum mechanics.