- Expectation value (quantum mechanics)
quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. It is a fundamental concept in all areas of quantum physics.
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 classical measure theory.
Formalism in quantum mechanics
In quantum theory, an experimental setup is described by the
observableto be measured, and the state of the system. The expectation value of in the state is denoted as .
Mathematically, is a
self-adjointoperator on a Hilbert space. In the most commonly used case in quantum mechanics, is a pure 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
If dynamics is considered, either the vector or the operator is taken to be time-dependent, depending on whether the
Schrödinger pictureor Heisenberg pictureis used. The time-dependence of the expectation value does not depend on this choice, however.
If has a complete set of
eigenvectors , with eigenvalues , then (1) can be expressed as
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
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
eigenvalues, but has a completely continuous 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
A similar formula holds for the
momentumoperator , 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 positive trace-classoperator , the "statistical operator" or " density matrix". 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. The expectation value of an observable is then simply 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
with a positive
trace-classoperator 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
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 mechanicsof infinitely extended media, [cite book
last = Bratteli
first = Ola
coauthors = Robinson, Derek W
title = Operator Algebras and Quantum Statistical Mechanics 1
publisher = Springer
date = 1987
id = 2nd edition
isbn = 978-3540170938] and as charged states in
quantum field theory. [cite book
last = Haag
first = Rudolf
authorlink = Rudolf Haag
title = Local Quantum Physics
publisher = Springer
date = 1996
pages = Chapter IV
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 spacerepresentation. Here the Hilbert space is , the space of square-integrable functions on the real line. Vectors are represented by functions , called wave 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
observableand its value can be directly measured in experiment.
Heisenberg's uncertainty principle
Notes and references
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
title = Lectures on Quantum Theory: Mathematical and Structural Foundations
publisher = Imperial College Press
date = 1995
isbn = 978-1860940019
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