- 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 $A$ to be measured, and the state $sigma$ of the system. The expectation value of $A$ in the state $sigma$ is denoted as $langle\; A\; angle\_sigma$.Mathematically, $A$ is a

self-adjoint operator on aHilbert space . In the most commonly used case in quantum mechanics, $sigma$ is apure state , described by a normalized [*This article always takes $psi$ to be of norm 1. For non-normalized vectors, $psi$ has to be replaced with $psi\; /\; |psi|$ in all formulas.*] vector $psi$ in the Hilbert space. The expectation value of $A$ in the state $psi$ is defined as(1) $langle\; A\; angle\_psi\; =\; langle\; psi\; |\; A\; |\; psi\; angle$.

If dynamics is considered, either the vector $psi$ or the operator $A$ 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 $A$ has a complete set of

eigenvector s $phi\_j$, witheigenvalue s $a\_j$, then (1) can be expressed as(2) $langle\; A\; angle\_psi\; =\; sum\_j\; a\_j\; |langle\; psi\; |\; phi\_j\; angle|^2$.

This expression is similar to the

arithmetic mean , and illustrates the physical meaning of the mathematical formalism: The eigenvalues $a\_j$ are the possible outcomes of the experiment, [*It is assumed here that the eigenvalues are non-degenerate.*] and their corresponding coefficient $|langle\; psi\; |\; phi\_j\; angle|^2$ is the probability that this outcome will occur; it is often called the "transition probability".A particularly simple case arises when $A$ 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) $langle\; A\; angle\_psi\; =\; |\; A\; psi\; |^2$.

In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator $Q$ in quantum mechanics. This operator does not have

eigenvalue s, but has a completelycontinuous spectrum . In this case, the vector $psi$ can be written as a complex-valued function $psi(x)$ on the spectrum of $Q$ (usually the real line). For the expectation value of the position operator, one then has the formula(4) $langle\; Q\; angle\_psi\; =\; int\; ,\; x\; ,\; |psi(x)|^2\; ,\; dx$.

A similar formula holds for the

momentum operator $P$, in systems where it has continuous spectrum.All the above formulae are valid for pure states $sigma$ only. Prominently in

thermodynamics , also "mixed states" are of importance; theseare described by a positivetrace-class operator $ho\; =\; sum\_i\; ho\_i\; |\; psi\_i\; angle\; langle\; psi\_i\; |$, the "statistical operator" or "density matrix ". The expectation value then can be obtained as(5) $langle\; A\; angle\_\; ho\; =\; mathrm\{Trace\}\; (\; ho\; A)\; =\; sum\_i\; ho\_i\; langle\; psi\_i\; |\; A\; |\; psi\_i\; angle=\; sum\_i\; ho\_i\; langle\; A\; angle\_\{psi\_i\}$.

**General formulation**In general, quantum states $sigma$ 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 $A$ is then simply given by(6) $langle\; A\; angle\_sigma\; =\; sigma(A)$.

If the algebra of observables acts irreducibly on a

Hilbert space , and if $sigma$ is a "normal functional", that is, it is continuous in theultraweak topology , then it can be written as:$sigma\; (cdot)\; =\; mathrm\{Trace\}\; (\; ho\; ;\; cdot)$

with a positive

trace-class operator $ho$ of trace 1. This gives formula (5) above. In the case of apure state , $ho=\; |psi\; anglelanglepsi|$ is a projection onto a unit vector $psi$. Then $sigma\; =\; langle\; psi\; |cdot\; ;\; psi\; angle$, which gives formula (1) above.$A$ 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 $A$ in a

spectral decomposition ,:$A\; =\; int\; a\; ,\; mathrm\{d\}P(a)$

with a projector-valued measure $P$. For the expectation value of $A$ in a pure state $sigma=langlepsi\; |\; cdot\; ,\; psi\; angle$, this means

:$langle\; A\; angle\_sigma\; =\; int\; a\; ;\; mathrm\{d\}\; langle\; psi\; |\; P(a)\; psi\; angle$,

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*] and as charged states in

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 = 978-3540170938quantum field theory . [*cite book*] In these cases, the expectation value is determined only by the more general formula (6).

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**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 $mathcal\{H\}\; =\; L^2(mathbb\{R\})$, the space of square-integrable functions on the real line. Vectors $psiinmathcal\{H\}$ are represented by functions $psi(x)$, calledwave functions . The scalar product is given by $langle\; psi\_1|\; psi\_2\; angle\; =\; int\; psi\_1(x)^ast\; psi\_2(x)\; ,\; mathrm\{d\}x$. The wave functions have a direct interpretation as a probability distribution::$p(x)\; dx\; =\; psi^*(x)psi(x)\; dx$

gives the probability of finding the particle in an infinitesimal interval of length $dx$ about some point $x$.

As an observable, consider the position operator $Q$, which acts on wavefunctions $psi$ by

:$(Q\; psi)\; (x)\; =\; x\; psi(x)$.

The expectation value, or mean value of measurements, of $Q$ performed on a very large number of "identical" independent systems will be given by

: $langle\; Q\; angle\_psi\; =\; langle\; psi\; |\; Q\; psi\; angle\; =int\_\{-infty\}^\{infty\}\; psi^ast(x)\; ,\; x\; ,\; psi(x)\; ,\; mathrm\{d\}x=\; int\_\{-infty\}^\{infty\}\; x\; ,\; p(x)\; ,\; mathrm\{d\}x$.

It should be noted that the expectation only exists if the integral converges, which is not the case for all vectors $psi$. This is because the position operator is unbounded, and $psi$ has to be chosen from its

domain of definition .In general, the expectation of any observable can be calculated by replacing $Q$ with the appropriate operator. For example, to calculate the average momentum, one uses the momentum operator "in

configuration space ", $P\; =\; -ihbar,d/dx$. Explicitly, its expectation value is: $langle\; P\; angle\_psi\; =\; -ihbar\; int\_\{-infty\}^\{infty\}\; psi^ast(x)\; ,\; frac\{dpsi\}\{dx\}(x)\; ,\; mathrm\{d\}x$.

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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