- Expectation value (quantum mechanics)
In
quantum mechanics , the expectation value is the predicted mean value of the result of an experiment. It is a fundamental concept in all areas ofquantum physics .Operational definition
Quantum physics shows an inherent statistical behaviour: The measured outcome of an experiment will generally not be the same if the experiment is repeated several times. Only the statistical mean of the measured values, averaged over a large number of runs of the experiment, is a repeatable quantity. Quantum theory does not, in fact, predict the result of individual measurements, but only their statistical mean. This predicted mean value is called the "expectation value".
While the computation of the mean value of experimental results is very much the same as in classical
statistics , its mathematical representation in the formalism of quantum theory differs significantly from classicalmeasure theory .Formalism in quantum mechanics
In quantum theory, an experimental setup is described by the
observable to be measured, and the state of the system. The expectation value of in the state is denoted as .Mathematically, is a
self-adjoint operator on aHilbert space . In the most commonly used case in quantum mechanics, is apure state , described by a normalized [This article always takes to be of norm 1. For non-normalized vectors, has to be replaced with in all formulas.] vector in the Hilbert space. The expectation value of in the state is defined as(1) .
If dynamics is considered, either the vector or the operator is taken to be time-dependent, depending on whether the
Schrödinger picture orHeisenberg picture is used. The time-dependence of the expectation value does not depend on this choice, however.If has a complete set of
eigenvector s , witheigenvalue s , then (1) can be expressed as(2) .
This expression is similar to the
arithmetic mean , and illustrates the physical meaning of the mathematical formalism: The eigenvalues are the possible outcomes of the experiment, [It is assumed here that the eigenvalues are non-degenerate.] and their corresponding coefficient is the probability that this outcome will occur; it is often called the "transition probability".A particularly simple case arises when is a projection, and thus has only the eigenvalues 0 and 1. 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
(3) .
In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator in quantum mechanics. This operator does not have
eigenvalue s, but has a completelycontinuous spectrum . In this case, the vector can be written as a complex-valued function on the spectrum of (usually the real line). For the expectation value of the position operator, one then has the formula(4) .
A similar formula holds for the
momentum operator , in systems where it has continuous spectrum.All the above formulae are valid for pure states only. Prominently in
thermodynamics , also "mixed states" are of importance; theseare described by a positivetrace-class operator , the "statistical operator" or "density matrix ". The expectation value then can be obtained as(5) .
General formulation
In general, quantum states are described by positive normalized
linear functional s on the set of observables, mathematically often taken to be aC* algebra . The expectation value of an observable is then simply 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 theultraweak 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 apure 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 state s inquantum statistical mechanics of infinitely extended media, [cite book
last = Bratteli
first = Ola
authorlink =
coauthors = Robinson, Derek W
title = Operator Algebras and Quantum Statistical Mechanics 1
publisher = Springer
date = 1987
location =
pages =
url =
doi =
id = 2nd edition
isbn = 978-3540170938] and as charged states inquantum field theory . [cite book
last = Haag
first = Rudolf
authorlink = Rudolf Haag
coauthors =
title = Local Quantum Physics
publisher = Springer
date = 1996
location =
pages = Chapter IV
url =
doi =
id =
isbn = 3-540-61451-6] 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 , the space of square-integrable functions on the real line. Vectors are represented by functions , calledwave functions . The scalar product is given by . The wave functions have a direct interpretation as a probability distribution::
gives the probability of finding the particle in an infinitesimal interval of length about some point .
As an observable, consider the position operator , which acts on wavefunctions by
:.
The expectation value, or mean value of measurements, of performed on a very large number of "identical" independent systems will be given by
: .
It should be noted that the expectation only exists if the integral converges, which is not the case for all vectors . This is because the position operator is unbounded, and has to be chosen from its
domain of definition .In general, the expectation of any observable can be calculated by replacing with the appropriate operator. For example, to calculate the average momentum, one uses the momentum operator "in
configuration space ", . Explicitly, its expectation value is: .
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.See also
*
Heisenberg's uncertainty principle
*Virial theorem Notes and references
Further reading
The expectation value, in particular as presented in the section "Formalism in quantum mechanics", is covered in most elementary textbooks on quantum mechanics.
For a discussion of conceptual aspects, see:
* cite book
last = Isham
first = Chris J
authorlink =
coauthors =
title = Lectures on Quantum Theory: Mathematical and Structural Foundations
publisher = Imperial College Press
date = 1995
location =
pages =
url =
doi =
id =
isbn = 978-1860940019
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