Operator (physics)
In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory. Operators in classical mechanics. In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian formula_1 or  equivalently the Hamiltonian formula_2, a function of the generalized coordinates "q", generalized velocities formula_3 and its conjugate momenta: If either "L" or "H" is independent of a generalized coordinate "q", meaning the "L" and "H" do not change when "q" is changed, which in turn means the dynamics of the particle are still the same even when "q" changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate "q" is a symmetry). Operators in classical mechanics are related to these symmetries. More  technically, when "H" is invariant under the action of a certain group of transformations "G": the elements of "G" are physical operators, which map physical states among themselves. Table of classical mechanics operators. where formula_6 is the rotation matrix about an axis defined by the unit vector formula_7 and angle "θ". Generators. If the transformation is infinitesimal, the operator action should be of the form where formula_9 is the identity operator, formula_10 is a parameter with a small value, and formula_11 will depend on the transformation at hand, and is called a generator of the group. Again, as a simple  example, we will derive the generator of the space translations on 1D functions. As it was stated, formula_12. If formula_13 is infinitesimal, then we may write This formula may be rewritten as where formula_16 is the generator of the translation group, which in this case happens to be the "derivative" operator. Thus, it is said that the generator of translations is the derivative. The exponential map. The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this. The translation for a finite value of  formula_17 may be obtained by repeated application of the infinitesimal translation: with the formula_19 standing for the application formula_20 times. If formula_20 is large, each of the factors may be considered to be infinitesimal: But this limit may be rewritten as an exponential: To be convinced of the validity of this formal expression, we may expand the exponential in a power series: The right-hand side may be rewritten as which is just the Taylor expansion of formula_26, which was our original value for formula_27. The mathematical properties of physical operators are a topic of great importance in itself. For  further information, see C*-algebra and Gelfand-Naimark theorem. Operators in quantum mechanics. The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator. Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator. Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come  up as the result of the experiment. Mathematically this means the operators must be Hermitian. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details about Hermitian operators. In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators. In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can  be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction. Wavefunction. The wavefunction must be square-integrable (see "L" spaces), meaning: and normalizable, so that: Two cases of eigenstates (and eigenvalues) are: Linear operators in wave mechanics. Let be the wavefunction for a quantum system, and formula_40 be any linear operator for some observable (such as position, momentum, energy, angular momentum etc.). If is an eigenfunction of the operator formula_40, then where is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable has a measured value . If  is an eigenfunction of a given operator formula_40, then a definite quantity (the eigenvalue ) will be observed if a measurement of the observable is made on the state . Conversely, if is not an eigenfunction of formula_40, then it has no eigenvalue for formula_40, and the observable does not have a single definite value in that case. Instead, measurements of the observable will yield each eigenvalue with a certain probability (related to the decomposition of relative to the orthonormal eigenbasis of formula_40). In bra–ket notation the above can be written; that are equal if formula_48 is an eigenvector, or eigenket  of the observable . Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below). An operator in "n"-dimensional space can be written: where e are basis vectors corresponding to each component operator "A". Each component will yield a corresponding eigenvalue formula_50. Acting this on the wave function : in which we have used formula_52 In bra–ket notation: Commutation of operators on "Ψ". If two observables "A  and B" have linear operators formula_54 and formula_55, the commutator is defined by, The commutator is itself a (composite) operator. Acting the commutator on "ψ" gives: If "ψ" is an eigenfunction with eigenvalues "a" and "b" for observables "A" and "B" respectively, and if the operators commute: then the observables "A" and "B" can be measured simultaneously with infinite precision i.e. uncertainties formula_59, formula_60 simultaneously. "ψ" is then said to be the simultaneous eigenfunction of A and B. To illustrate this: It shows that measurement of A and B does not cause any shift of state i.e. initial and final  states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state ("ψ") of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision. If the operators do not commute: they can't be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables, even if "ψ" is an eigenfunction the above relation holds.. Notable pairs are position-and-momentum  and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as "L" and "L", or "s" and "s" etc.). Expectation values of operators on "Ψ". The expectation value (equivalently the average or mean value) is the average measurement of an observable, for particle in region "R". The expectation value formula_64 of the operator formula_54 is calculated from: This can be generalized to any function "F" of an operator: An example of "F" is the 2-fold action of "A" on "ψ", i.e. squaring an operator or doing it twice: Hermitian  operators. The definition of a Hermitian operator is: Following from this, in bra–ket notation: Important properties of Hermitian operators include: Operators in matrix mechanics. An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a linear transformation (aka transition matrix) between bases. Each basis element formula_71 can be connected to another, by the expression: which is a matrix element: A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal. In matrix form, operators allow real eigenvalues to be found, corresponding  to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the characteristic polynomial: where "I" is the "n" × "n" identity matrix, as an operator it corresponds to the identity operator. For a discrete basis: while for a continuous basis: Inverse of an operator. A non-singular operator formula_40 has an inverse formula_78 defined by: If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator  is non-singular if and only if its determinant is nonzero: and hence the determinant is zero for a singular operator. Table of QM operators. The operators used in quantum mechanics are collected in the table below (see for example,). The bold-face vectors with circumflexes are not unit vectors, they are 3-vector operators; all three spatial components taken together. Examples of applying quantum operators. The procedure for extracting information from a wave function is as follows. Consider the momentum "p" of a particle as an example. The momentum operator in position basis in one dimension is: Letting this  act on "ψ" we obtain: if "ψ" is an eigenfunction of formula_83, then the momentum eigenvalue "p" is the value of the particle's momentum, found by: For three dimensions the momentum operator uses the nabla operator to become: In Cartesian coordinates (using the standard Cartesian basis vectors e, e, e) this can be written; that is: The process of finding eigenvalues is the same. Since this is a vector and operator equation, if "ψ" is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting formula_88 on "ψ" obtains