In quantum mechanics , the expectation value is the probabilistic expected value of the result (measurement) of an experiment. 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 .

Operational definition [ edit ]

Consider an operator ${\displaystyle A}$. The expectation value is then ${\displaystyle \langle A\rangle =\langle \psi |A|\psi \rangle }$ in Dirac notation with ${\displaystyle |\psi \rangle }$ a normalized state vector.

Formalism in quantum mechanics [ edit ]

In quantum theory, an experimental setup is described by the observable ${\displaystyle A}$ to be measured, and the state ${\displaystyle \sigma }$ of the system. The expectation value of ${\displaystyle A}$ in the state ${\displaystyle \sigma }$ is denoted as ${\displaystyle \langle A\rangle _{\sigma }}$.

Mathematically, ${\displaystyle A}$ is a self-adjoint operator on a separable complex Hilbert space . In the most commonly used case in quantum mechanics, ${\displaystyle \sigma }$ is a pure state , described by a normalized ^[a] vector ${\displaystyle \psi }$ in the Hilbert space. The expectation value of ${\displaystyle A}$ in the state ${\displaystyle \psi }$ is defined as

$${\displaystyle \langle A\rangle _{\psi }=\langle \psi |A|\psi \rangle .}$$

( 1 )

If dynamics is considered, either the vector ${\displaystyle \psi }$ or the operator ${\displaystyle A}$ is taken to be time-dependent, depending on whether the Schrodinger picture or Heisenberg picture is used. The evolution of the expectation value does not depend on this choice, however.

If ${\displaystyle A}$ has a complete set of eigenvectors ${\displaystyle \phi _{j}}$, with eigenvalues ${\displaystyle a_{j}}$, then ( 1 ) can be expressed as ^[1]

$${\displaystyle \langle A\rangle _{\psi }=\sum _{j}a_{j}|\langle \psi |\phi _{j}\rangle |^{2}.}$$

( 2 )

This expression is similar to the arithmetic mean , and illustrates the physical meaning of the mathematical formalism: The eigenvalues ${\displaystyle a_{j}}$ are the possible outcomes of the experiment, ^[b] and their corresponding coefficient ${\displaystyle |\langle \psi |\phi _{j}\rangle |^{2}}$ is the probability that this outcome will occur; it is often called the transition probability .

A particularly simple case arises when ${\displaystyle A}$ 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

$${\displaystyle \langle A\rangle _{\psi }=\|A|\psi \rangle \|^{2}.}$$

( 3 )

In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the position operator ${\displaystyle X}$ in quantum mechanics. This operator has a completely continuous spectrum , with eigenvalues and eigenvectors depending on a continuous parameter, ${\displaystyle x}$. Specifically, the operator ${\displaystyle X}$ acts on a spatial vector ${\displaystyle |x\rangle }$ as ${\displaystyle X|x\rangle =x|x\rangle }$. ^[2] In this case, the vector ${\displaystyle \psi }$ can be written as a complex-valued function ${\displaystyle \psi (x)}$ on the spectrum of ${\displaystyle X}$ (usually the real line). This is formally achieved by projecting the state vector ${\displaystyle |\psi \rangle }$ onto the eigenvalues of the operator, as in the discrete case ${\textstyle \psi (x)\equiv \langle x|\psi \rangle }$. 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:

${\displaystyle {\begin{aligned}\langle X\rangle _{\psi }&=\langle \psi |X|\psi \rangle =\langle \psi |\mathbb {I} X\mathbb {I} |\psi \rangle \\&=\iint \langle \psi |x\rangle \langle x|X|x'\rangle \langle x'|\psi \rangle dx\ dx'\\&=\iint \langle x|\psi \rangle ^{*}x'\langle x|x'\rangle \langle x'|\psi \rangle dx\ dx'\\&=\iint \langle x|\psi \rangle ^{*}x'\delta (x-x')\langle x'|\psi \rangle dx\ dx'\\&=\int \psi (x)^{*}x\psi (x)dx=\int x\psi (x)^{*}\psi (x)dx=\int x|\psi (x)|^{2}dx\end{aligned}}}$

Where the orthonormality relation of the position basis vectors ${\displaystyle \langle x|x'\rangle =\delta (x-x')}$, reduces the double integral to a single integral. The last line uses the modulus of a complex valued function to replace ${\displaystyle \psi ^{*}\psi }$ with ${\displaystyle |\psi |^{2}}$, which is a common substitution in quantum-mechanical integrals.

The expectation value may then be stated, where x is unbounded, as the formula

$${\displaystyle \langle X\rangle _{\psi }=\int _{-\infty }^{\infty }\,x\,|\psi (x)|^{2}\,dx.}$$

( 4 )

A similar formula holds for the momentum operator , in systems where it has continuous spectrum.

All the above formulas are valid for pure states ${\displaystyle \sigma }$ only. Prominently in thermodynamics and quantum optics , also mixed states are of importance; these are described by a positive trace-class operator ${\textstyle \rho =\sum _{i}p_{i}|\psi _{i}\rangle \langle \psi _{i}|}$, the statistical operator or density matrix . The expectation value then can be obtained as

$${\displaystyle \langle A\rangle _{\rho }=\operatorname {Trace} (\rho A)=\sum _{i}p_{i}\langle \psi _{i}|A|\psi _{i}\rangle =\sum _{i}p_{i}\langle A\rangle _{\psi _{i}}.}$$

( 5 )

General formulation [ edit ]

In general, quantum states ${\displaystyle \sigma }$ 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 ${\displaystyle A}$ is then given by

$${\displaystyle \langle A\rangle _{\sigma }=\sigma (A).}$$

( 6 )

If the algebra of observables acts irreducibly on a Hilbert space , and if ${\displaystyle \sigma }$ is a normal functional , that is, it is continuous in the ultraweak topology , then it can be written as

with a positive trace-class operator ${\displaystyle \rho }$ of trace 1. This gives formula ( 5 ) above. In the case of a pure state , ${\displaystyle \rho =|\psi \rangle \langle \psi |}$ is a projection onto a unit vector ${\displaystyle \psi }$. Then ${\displaystyle \sigma =\langle \psi |\cdot \;\psi \rangle }$, which gives formula ( 1 ) above.

${\displaystyle A}$ 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 ${\displaystyle A}$ in a spectral decomposition ,

with a projection-valued measure ${\displaystyle P}$. For the expectation value of ${\displaystyle A}$ in a pure state ${\displaystyle \sigma =\langle \psi |\cdot \,\psi \rangle }$, 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 ^{[ 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 ).

Example in configuration space [ edit ]

As an example, consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is ${\displaystyle {\mathcal {H}}=L^{2}(\mathbb {R} )}$, the space of square-integrable functions on the real line. Vectors ${\displaystyle \psi \in {\mathcal {H}}}$ are represented by functions ${\displaystyle \psi (x)}$, called wave functions . The scalar product is given by ${\textstyle \langle \psi _{1}|\psi _{2}\rangle =\int \psi _{1}^{\ast }(x)\psi _{2}(x)\,dx}$. The wave functions have a direct interpretation as a probability distribution:

${\displaystyle \rho (x)dx=\psi ^{*}(x)\psi (x)dx}$

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

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

The expectation value, or mean value of measurements, of ${\displaystyle Q}$ 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 ${\displaystyle \psi }$. This is because the position operator is unbounded , and ${\displaystyle \psi }$ has to be chosen from its domain of definition .

In general, the expectation of any observable can be calculated by replacing ${\displaystyle Q}$ with the appropriate operator. For example, to calculate the average momentum, one uses the momentum operator in configuration space , ${\textstyle \mathbf {p} =-i\hbar \,{\frac {d}{dx}}}$. Explicitly, its expectation value is

${\displaystyle \langle \mathbf {p} \rangle _{\psi }=-i\hbar \int _{-\infty }^{\infty }\psi ^{\ast }(x)\,{\frac {d\psi (x)}{dx}}\,dx.}$

Not all operators in general provide a measurable 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 [ edit ]

• Rayleigh quotient
• Uncertainty principle
• Virial theorem

Notes [ edit ]

References [ edit ]

Further reading [ edit ]

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:

• Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations . Imperial College Press. ISBN   978-1-86094-001-9 .