operator(redirected from Mathematical operator)
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a mathematical concept denoting, in its most general sense, a correspondence between the elements of two sets X and Y such that with each element x in X there is associated an element y in Y. An equivalent meaning is conveyed by the terms “operation,” “mapping,” “transformation,” and “function.” The element y is called the image of x, and x is called the preimage of y. When X and Y are sets of numbers, the term “function” is usually used. An operator that maps an infinite-dimensional space to a set of real or complex numbers is called a functional. The most important class of operators is that of linear operators in normed linear spaces. Differential and integral operators play an important role in physics and mathematical analysis. Various properties of operators, operations on operators, and the use of operators in the solution of various mathematical problems are studied in operator theory.
(in quantum theory), a mathematical concept widely used in the mathematical apparatus of quantum mechanics and quantum field theory to compare a given state vector or wave function ψ with other given vectors or functions ψʹ. The relation between ψ and ψʹ is written in the form ψʹ = L̂ψ where L̂ is an operator. In quantum mechanics, operators L̂ such as the coordinate and momentum operators, that act on the state vector or wave function ψ—that is, the quantity that describes the physical state of a system—correspond to such physical quantities L as coordinates, momentum, angular momentum, and energy.
The simplest types of operators acting on a wave function ψ(x), where x is the coordinate of a particle, are (1) the multiplication operator—for example, the coordinate operator x̂, x̂ψ = xψ and (2) the differential operator—for example, the momentum operator p̂, p̂ψ = – iℏ(∂ψ/∂x), where i is the imaginary unit and ℏ is Planck’s constant. If ψ is a vector whose components can be represented as a column of numbers, then the operator is a matrix.
Linear operators are mainly used in quantum mechanics. They have the followingʹ property: if L̂ψ1 = ψʹ1 and L̂ψ2 =ψʹ2 then L̂(c1ψ1 + c2ψ2) = c1ψʹ, c2ψʹ2, where c1 and c2 are complex numbers. This property reflects the superposition principle, one of the fundamental principles of quantum mechanics.
The essential properties of the operator L̂ are defined by the equation L̂ψn = λnψn where λn is a number. The solutions of this equation ψn are called the eigenfunctions (eigenvectors) of the operator L̂. In quantum mechanics, the wave eigenfunctions (state eigenvectors) describe states in which a given physical quantity L has a certain value λn. The numbers λn are called the eigenvalues of the operator [com], and their set is called the spectrum of the operator. The spectrum may be continuous or discrete. In the former case, the equation defining ψn has a solution for any value of λn (in a specified region), but in the latter case solutions exist only for certain discrete values of λn. The operator spectrum may also be mixed: partly continuous, partly discrete. For example, the coordinate and momentum operators have continuous spectra, but the energy operator may have a continuous, discrete, or mixed spectrum, depending on the nature of the forces acting in the system. The discrete eigenvalues of the energy operator are called energy levels.
The eigenfunctions and eigenvalues of the operators of physical quantities must satisfy certain requirements. Since directly measurable physical quantities always assume real values, the corresponding quantum-mechanical operators must have real eigenvalues. Further, since one possible eigenvalue of a physical quantity should be obtained as a result of measurement of the quantity in any state ψ, it is necessary that an arbitrary wave function or state vector can be represented as a linear combination of eigenfunctions or vectors ψn of the operator of the physical quantity; in other words, the set of eigenfunctions or vectors must make up a complete system. The eigenfunctions and eigenvalues of self-adjoint operators, or Hermitian operators, have these properties.
Algebraic operations may be performed with operators. In particular, the product of the operators L̂1 and L̂2 is understood to be the operator L̂ =L̂1L̂2 whose action on a vector or function ψ gives L̂ψ = ψ”, if L̂2ψ = ψʹ and L̂1 ψʹ = ψ”. The product of operators usually depends on the order of the factors —that is, L̂1 L̂2.≠ L̂2 L̂1 The algebra of operators differs in this regard from the ordinary algebra of numbers. The possibility of changing the order of the factors in the product of two operators is closely connected with the possibility of the simultaneous measurement of the physical quantities to which the operators correspond. The equation L̂1L̂2 ≠ L̂2L̂1 is a necessary and sufficient condition for the simultaneous measurability of physical quantities.
The equations of quantum mechanics can be formally written in precisely the same form as the equations of classical mechanics (the Heisenberg representation in quantum mechanics) if the physical quantities entering to the equations of classical mechanics are replaced by the corresponding operators. The whole difference between quantum and classical mechanics then reduces to the difference between the algebras used. In quantum mechanics operators are therefore sometimes called q-numbers, in contrast to c-numbers—that is, the conventional numbers with which classical mechanics deals.
Operators can be not only multiplied but also raised to a power. Series may be formed from them, and functions of operators may be considered. The product of Hermitian operators is usually non-Hermitian. Non-Hermitian operators, of which unitary operators are an important class, are also used in quantum mechanics. Unitary operators do not change the norms (“lengths”) of vectors or the “angles” between them. The invariance of the norm of a state vector means that the vector’s components can be interpreted as probability amplitudes in both the initial and transformed functions. Therefore, the operation of a unitary operator describes the development of a quantum-mechanical system over time, and also the system’s displacement as a whole in space and its rotation, mirror reflection, and so on. The transformations performed by unitary operators (unitary transformations) play the same role in quantum mechanics as do canonical transformations in classical mechanics.
The complex conjugation operator, which is nonlinear, is also used in quantum mechanics. The product of such an operator and a unitary operator is called an antiunitary operator. Antiuni-tary operators describe the transformation of time reversal and some other transformations.
The method of second quantization is used extensively in the theory of quantum systems consisting of identical particles. In this method, states with an indefinite or variable number of particles are considered, and operators are introduced whose action on a state vector with a given number of particles leads to a state vector with the number of particles altered by unity (particle production and absorption operators). The particle production or annihilation operator at a given point x , [com]ψ(x), is formally similar to the wave function ψ(x), like the q- and c-numbers corresponding to the same physical quantity in quantum and classical mechanics, respectively. Such operators form quantized fields, which play a fundamental role in relativistic quantum theories (quantum electrodynamics, the theory of elementary particles).
REFERENCESSee References under QUANTUM FIELD THEORY and QUANTUM MECHANICS.
V. B. BERESTETSKII
operator(1) An organization that provides communications services. See network operator.
(2) In programming, a symbol used to perform an arithmetic or logical operation. See arithmetic operator and Boolean operator.
(3) A person who operates a computer in a datacenter and performs such activities as commanding the operating system, mounting disks and tapes and placing paper in the printer. Operators may also write the job control language (JCL), which schedules the daily work for the computer. See system development life cycle, job descriptions and salary survey.