Operator (physics) - Citizendia

In physics, an operator is a function acting on the space of physical states. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function A state of matter (or physical state, or form of matter) has physical properties which are qualitatively different from other states of matter As a result of its application on a physical state, another physical state is obtained, very often along with some extra relevant information.

The simplest example of the utility of operators is the study of symmetry. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Because of this, they are a very useful tool in classical mechanics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In quantum mechanics, on the other hand, they are an intrinsic part of the formulation of the theory. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons

1 Operators in classical mechanics
2 Concept of generator
3 The exponential map
4 Operators in quantum mechanics
5 General mathematical properties of quantum operators
6 See also

Operators in classical mechanics

Let us consider a classical mechanics system led by a certain hamiltonian $H (q, p)$ , function of the generalized coordinates $q$ and its conjugate momenta. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent Let us consider this function to be invariant under the action of a certain group of transformations $G$ , i. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element e. , if $S\in G$ , $H (S (q, p)) = H (q, p)$ . The elements of $G$ are physical operators, which map physical states among themselves.

An easy example is given by space translations. The hamiltonian of a translationally invariant problem does not change under the transformation $q\to T_a q=q+a$ . Other straightforward symmetry operators are the ones implementing rotations.

If the physical system is described by a function, as in classical field theories, the translation operator is generalized in a straightforward way:

$f(x) \to T_a f(x)=f(x-a).$

Notice that the transformation inside the parenthesis should be the inverse of the transformation done on the coordinates.

Concept of generator

If the transformation is infinitesimal, the operator action should be of the form

I + ε A

where $I$ is the identity operator, $ε$ is a small parameter, and $A$ 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, $T a f (x) = f (x - a)$ . If $a = ε$ is infinitesimal, then we may write

$T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).$

This formula may be rewritten as

T ε f (x) = (I - ε D) f (x)

where $D$ is the generator of the translation group, which happens to be just 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 differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine In the case of the translations the idea works like this.

The translation for a finite value of $a$ may be obtained by repeated application of the infinitesimal translation:

$T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)$

with the $\cdots$ standing for the application $N$ times. If $N$ is large, each of the factors may be considered to be infinitesimal:

$T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x).$

But this limit may be rewritten as an exponential:

T a f (x) = exp( - a D) f (x).

To be convinced of the validity of this formal expression, we may expand the exponential in a power series:

$T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x).$

The right-hand side may be rewritten as

$f(x) - a f'(x) + {a^2\over 2!} f''(x) - {a^3\over 3!} f'''(x) + \cdots$

which is just the Taylor expansion of $f (x - a)$ , which was our original value for $T a f (x)$ .

Operators in quantum mechanics

Once the interest of the operators in classical mechanics has been exposed, it has to be said that it is in quantum mechanics where they reach their full interest. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The mathematical description of quantum mechanics is built upon the concept of operator.

Physical pure states in quantum mechanics are unit-norm vectors in a certain vector space (a Hilbert space). In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added This article assumes some familiarity with Analytic geometry and the concept of a limit. Time evolution in this vector space is given by the application of a certain operator, called the evolution operator. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) Since the norm of the physical state should stay fixed, the evolution operator should be unitary. Informally a unitary transformation is a transformation that respects the Dot product: the dot product of two vectors before the transformation is equal to their Any other symmetry, mapping a physical state into another, should keep this restriction.

Any observable, i. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical e. , any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. In Mathematics, an element x of a Star-algebra is self-adjoint if x^*=x In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The values which may come up as the result of the experiment are the eigenvalues of the operator. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue.

General mathematical properties of quantum operators

The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand-Naimark theorem. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of Bounded operators

Contents

Operators in classical mechanics

Concept of generator

The exponential map

Operators in quantum mechanics

General mathematical properties of quantum operators

See also