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quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the expectation value is the probabilistic
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
.


Operational definition

Consider an operator A. The expectation value is then \langle A \rangle = \langle \psi , A , \psi \rangle in
Dirac notation Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
with , \psi \rangle a normalized state vector.


Formalism in quantum mechanics

In quantum theory, an experimental setup is described by the
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
A to be measured, and the
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
\sigma of the system. The expectation value of A in the state \sigma is denoted as \langle A \rangle_\sigma. Mathematically, A is a
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...
operator on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. In the most commonly used case in quantum mechanics, \sigma is a
pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
, described by a normalized vector \psi in the Hilbert space. The expectation value of A in the state \psi is defined as If dynamics is considered, either the vector \psi or the operator A is taken to be time-dependent, depending on whether the
Schrödinger picture In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
or
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
is used. The evolution of the expectation value does not depend on this choice, however. If A has a complete set of
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s \phi_j, with
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s a_j, then () can be expressed as This expression is similar to the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
, and illustrates the physical meaning of the mathematical formalism: The eigenvalues a_j are the possible outcomes of the experiment, and their corresponding coefficient , \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 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 In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the
position operator In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues ...
X in quantum mechanics. This operator has a completely
continuous spectrum In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable ...
, with eigenvalues and eigenvectors depending on a continuous parameter, x. Specifically, the operator X acts on a spatial vector , x \rangle as X , x \rangle = x , x\rangle. In this case, the vector \psi can be written as a
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
function \psi(x) on the spectrum of X (usually the real line). This is formally achieved by projecting the state vector , \psi \rangle onto the eigenvalues of the operator, as in the discrete case \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 closure relation: \int , x \rangle \langle x, \, dx \equiv \mathbb The above may be used to derive the common, integral expression for the expected value (), by inserting identities into the vector expression of expected value, then expanding in the position basis: \begin \langle X \rangle_ &= \langle \psi , X , \psi \rangle = \langle \psi , \mathbb X \mathbb, \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 Where the orthonormality relation of the position basis vectors \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 \psi^*\psi with , \psi, ^2, which is a common substitution in quantum-mechanical integrals. The expectation value may then be stated, where is unbounded, as the formula A similar formula holds for the
momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensi ...
P, in systems where it has continuous spectrum. All the above formulas are valid for pure states \sigma only. Prominently in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
and
quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
, also ''mixed states'' are of importance; these are described by a positive trace-class operator \rho = \sum_i p_i , \psi_i \rangle \langle \psi_i , , the ''statistical operator'' or ''
density matrix In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
''. The expectation value then can be obtained as


General formulation

In general, quantum states \sigma are described by positive normalized
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
s on the set of observables, mathematically often taken to be a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
. The expectation value of an observable A is then given by If the algebra of observables acts irreducibly on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, and if \sigma is a ''normal functional'', that is, it is continuous in the
ultraweak topology In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set ''B''(''H'') of bounded operators on a Hilbert space is the weak-* topology obta ...
, then it can be written as \sigma (\cdot) = \operatorname (\rho \; \cdot) with a positive trace-class operator \rho of trace 1. This gives formula () above. In the case of a
pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
, \rho= , \psi\rangle\langle\psi, is a projection onto a unit vector \psi. Then \sigma = \langle \psi , \cdot \; \psi\rangle, which gives formula () 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 \, dP(a) with a projector-valued measure P. For the expectation value of A in a pure state \sigma = \langle \psi , \cdot \, \psi \rangle, this means \langle A \rangle_\sigma = \int a \; d \langle \psi , P(a) \psi\rangle , which may be seen as a common generalization of formulas () and () 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 In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin– Schwinger state or, more commonly, a KMS s ...
s in
quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
of infinitely extended media, and as charged states in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. In these cases, the expectation value is determined only by the more general formula ().


Example in configuration space

As an example, consider a quantum mechanical particle in one spatial dimension, in the configuration space representation. Here the Hilbert space is \mathcal = L^2(\mathbb), the space of square-integrable functions on the real line. Vectors \psi\in\mathcal are represented by functions \psi(x), called
wave functions A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
. The scalar product is given by \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: 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 \rangle_\psi = \langle \psi , Q , \psi \rangle = \int_^ \psi^\ast(x) \, x \, \psi(x) \, dx = \int_^ x \, p(x) \, dx . The expectation value 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 = -i \hbar \, \frac. Explicitly, its expectation value is \langle P \rangle_\psi = -i\hbar \int_^ \psi^\ast(x) \, \frac \, dx. Not all operators in general provide a measurable value. An operator that has a pure real expectation value is called an
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
and its value can be directly measured in experiment.


See also

*
Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the ...
*
Uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
*
Virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...


Notes


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 = , title = Lectures on Quantum Theory: Mathematical and Structural Foundations , publisher = Imperial College Press , date = 1995 , location = , pages = , url = https://archive.org/details/lecturesonquantu0000isha , doi = , id = , isbn = 978-1-86094-001-9 , url-access = registration Quantum mechanics de:Erwartungswert#Quantenmechanischer Erwartungswert