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:''This article is about operators in [[mathematics]], for other kinds of operators see [[operator (disambiguation)]].''

L' '''operatore densità''', o '''matrice densità''', è utilizzato in [[Meccanica quantistica]] per descrivere lo stato statistico di un [[sistema quantistico]]. Il formalismo venne introdotto da [[John von Neumann]] (altre fonti sostengono che venne introdotto indipendentemente anche da [[Lev Landau]] e [[Felix Bloch]] ) nel 1927.

E' l'analogo quantistico della [[distribuzione di probabilità]] nello [[spazio delle fasi]] in [[meccanica classica]].

La necessità di una descrizione statistica emerge perchè non è possibile descrivere un sistema quantistico che sia sottoposto ad una generica [[operazione quantistica]], come ad esempio una [[misura]], usando esclusivamente stati rappresentati da [[notazione bra-ket|vettori ket]].

In [[mathematics]], an '''operator''' is some kind of [[function (mathematics)|function]]; if it comes with a specified type of [[operand]] as [[function ___domain]], it is no more than another way of talking of functions of a given type. The most frequently met usage is a mapping between [[vector space]]s; this kind of operator is distinguished by taking one vector and returning another. For example, consider an [[enlargement]], say by a factor of √2; such as is required to take one size of paper to another. It can also be applied geometrically to vectors as operands.

Un sistema in generale è detto essere in uno [[stato misto]], eccetto nel caso lo stato non sia riducibile ad una [[combinazione convessa]] di altri stati. In questo caso lo stato è detto [[stato puro]].

In many important cases, operators transform functions into other functions. We also say an operator ''maps'' a function to another. The operator itself is a function, but has an attached ''type'' indicating the correct operand, and the kind of function returned. This extra data can be defined formally, using [[type theory]]; but in everyday usage saying ''operator'' flags its significance. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the ___domain and codomain.

Situazioni tipiche in cui un operatore densità è richiesto includono: uno stato quantistico in equilibrio termico ( a temperature finite) e nel caso di [[entanglement quantistico| entanglement]] tra due sistemi, in tal caso ogni sistema è in uno stato misto anche se lo stato del sistema complessivo può essere puro. Si veda [[meccanica statistica quantistica]].

==Operators and levels of abstraction==

== Formalismo ==

To begin with, the usage of '''''operator''''' in mathematics is subsumed in the usage of ''[[function (mathematics)|function]]'': an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these:

L'operatore densità, comunemente chiamato &rho;, è un operatore sullo [[Spazio di Hilbert]] del sistema in questione. Nel caso speciale di uno stato puro è dato dall'[[operatore di proiezione]] dello stato. Per uno stato misto , dove il sistema è nello stato <math> |\psi_j \rang </math> con probabilità p_j, l'operatore densità è la somma dei proiettori, pesata con le appropriate probabilità:

*To draw attention to the [[function ___domain]], which may itself consist of [[vector (spatial)|vector]]s or [[function (mathematics)|function]]s, rather than just numbers. The [[expectation]] operator in [[probability theory]], for example, has [[random variable]]s as ___domain (and is also a [[functional (mathematics)|functional]]).

:<math> \rho = \sum_j p_j |\psi_j \rang \lang \psi_j| </math>

*To draw attention to the fact that the ___domain consists of pairs or [[tuple]]s of some sort, in which case ''operator'' is synonymous with the usual mathematical sense of [[operation]].

*To draw attention to the function codomain; for example a ''vector-valued function'' might be called an operator.

A single operator might conceivably qualify under all three of these. Other important ideas are:

L'operatore densità è utilizzato per calcolare i [[valore di aspettazione|valori di aspettazione]] di ogni osservabile A del sistema, mediato su tutti i differenti stati <math> |\psi_j \rang </math>. Come si può vedere l'operazione è equivalente a prendere la traccia del prodotto tra &rho; e A:

*[[Overloading]], in which for example [[addition]], say, is thought of as a single ''operator'' able to act on numbers, vectors, matrices ... .

:<math> \operatorname{tr}[\rho A]=\sum_j p_j \lang \psi_j|A|\psi_j \rang </math>

*Operators are often in practice just [[partial function]]s, a common phenomenon in the theory of [[differential equation]]s since there is no guarantee that the [[derivative]] of a function exists.

*Use of higher operations on operators, meaning that operators are themselves combined.

These are abstract ideas from mathematics, and [[computer science]]. They may however also be encountered in [[quantum mechanics]]. There [[Dirac]] drew a clear distinction between [[q-number]] or operator quantities, and [[c-number]]s which are conventional [[complex number]]s. The manipulation of q-numbers from that point on became basic to theoretical physics.

Le probabilità p_j sono non-negative e normalizzate. Per l'operatore densità questo significa che &rho; è un [[operatore positivo]] e che la traccia di &rho; è uguale a uno.

== Describing operators ==

== Formulazione in termini di algebre C* ==

Operators are described usually by the number of operands:

E' ora in generale accettato che la descrizione della meccanica quantistica in cui tutti gli [[operatore autoaggiunto | operatori autoaggiunti]] rappresentino osservabili non è mantenibile.

Gli osservabili sono oggi identificati con elementi di un'astratta [[algebra C*]] ''A'' e gli stati vengono rappresentati da [[funzionale| funzionali]] positivi su ''A''. In questo formalismo gli stati puri sono gli [[estremale| estremali]] dell'[[insieme convesso]] degli stati. Si noti che attraverso la [[costruzione GNS]] è possibile recuperare lo spazio di Hilbert che realizza ''A'' come un'algebra di operatori.

* [[monadic]] or [[unary operator|unary]] operators take one argument.

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* [[dyadic]] or [[binary operator|binary]] operators take two arguments.

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* [[triadic]] or [[ternary operator|ternary/trinary/tertiary]] operators take three arguments.

The number of operands is also called the '''''arity''''' of the operator. If an operator has an arity given as ''n''-ary (or ''n''-adic), then it takes ''n'' arguments. In programming, outside than [[functional programming]], the -ary terms are more often used than the other variants. See [[arity]] for an extensive list of the -ary endings.

=== Notations ===

There are three major ''systematic'' ways of writing operators and their arguments. These are

* '''prefix''': where the operator name comes ''first'' and the arguments follow, for example:

::Q(''x''₁, ''x''₂,...,''x''_n).

: In [[prefix notation]], the brackets are sometimes omitted if it is known that Q is an ''n''-ary operator.

* '''postfix''': where the operator name comes ''last'' and the arguments precede, for example:

::(''x''₁, ''x''₂,...,''x''_n) Q

: In [[postfix notation]], the brackets are sometimes omitted if it is known that Q is an ''n''-ary operator.

* '''[[infix]]''': where the operator name comes ''between'' the arguments. This is awkward and not commonly used for operators other than binary operators. Infix style is written, for example, as

:: ''x''₁ Q ''x''₂.

For operators on a single argument, prefix notation such as &minus;7 is most common, but postfix such as 5! ([[factorial]]) or ''x''* is also usual.

There are other notations commonly met. Writing [[exponent]]s such as 2⁸ is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a circumflex is written over the operator name. In certain circumstances, they are written ''unlike'' functions, when an operator has a single [[parameter|argument]] or ''operand''. For example, if the operator name is ''Q ''and the operand a function ''f'', we write ''Qf'' and not usually ''Q''(''f''); this latter notation may however be used for clarity if there is a product &mdash; for instance, ''Q''(''fg''). Later on we will use ''Q'' to denote a general operator, and ''x''_i to denote the ''i''-th argument.

Notations for operators include the following. If ''f''(''x'') is a function of ''x'' and ''Q'' is the general operator we can write ''Q'' acting on ''f'' as ''(Qf)(''x'')'' also.

Operators are often written in [[calligraphy]] to differentiate them from standard functions. For instance, the [[Fourier transform]] (an operator on functions) of f(t) (a function of t), which produces another function F(&omega;) (a function of &omega;), would be represented as <math>\mathcal{F}(f(t)) = F(\omega).</math>

== Examples of mathematical operators==

This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.

=== Linear operators ===

''Main article'': [[Linear transformation]]

The most common kind of operator encountered are ''linear operators''. In talking about linear operators, the operator is signified generally by the letters ''T'' or ''L''. Linear operators are those which satisfy the following conditions; take the general operator ''T'', the function acted on under the operator ''T'', written as f(x), and the constant a:

:<math>T\ (f(x)+g(x)) = T\ (f(x))+T\ (g(x))</math>

:<math>T\ (af(x)) = a\,T\ (f(x))</math>

Many operators are linear. For example, the differential operator and Laplacian operator, which we will see later.

Linear operators are also known as [[linear transformation]]s or linear mappings. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with [[nonlinearity]]).

Such an example of a linear transformation between vectors in '''R'''² is reflection, given a vector '''x'''=(x₁, x₂)

:Q(x₁, x₂)=(-x₁, x₂)

We can also make sense of linear operators between generalisations of finite-[[dimension|dimensional]] vector spaces. For example, there is a large body of work dealing with linear operators on [[Hilbert space#Bounded Operators|Hilbert spaces]] and on [[Banach space#Linear operators|Banach spaces]]. See also [[operator algebra]].

=== Operators in probability theory ===

''Main article'': [[Probability theory]]

Operators are also involved in probability theory.

Such operators as expectation, variance, covariance, [[factorial]]s, et al.

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=== Operators in calculus ===

[[Calculus]] is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator very clearly. The key operator studied is the [[differential operator]]. It is ''linear'', as are many of the operators constructed from it.

==== The differential operator ====

''Main article'': [[Differential operator]]

The [[differential operator]] is an operator which is fundamentally used in Calculus to denote the action of taking a derivative. Common notations are such ''d/dx'', y'(x) to denote the derivative of y(x). However here we will use the notation that is closest to the operator notation we have been using, that is, using D ''f'' to represent the action of taking the derivative of f.

==== Integral operators ====

Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration.

===== Convolution =====

''Main article'': [[Convolution]]

The ''convolution'' of two functions is a mapping from two functions to one other, defined by an integral as follows:

If ''x''₁=f(t) and ''x''₂=g(t), define the operator Q such that;

: <math>Q\ x_1\ x_2\ = \int f(t) g(\tau - t) dt</math>

which we write as <math>(f*g)(\tau)</math>.

===== Fourier transform =====

''Main article'': [[Fourier transform]]

The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. It is another integral operator; it is useful mainly because it converts a function on one (spatial) ___domain to a function on another (frequency) ___domain, in a way that is effectively [[invertible]]. Nothing significant is lost, because there is an inverse transform operator. In the simple case of [[periodic function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:

:<math> f(t) = {a_0 \over 2} + \sum_{n=1}^{\infty}{ a_n \cos ( \omega n t ) + b_n \sin ( \omega n t ) } </math>

When dealing with general function '''R->C''', the transform takes up an [[integral]] form:

:<math> f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega ) \cdot \exp {( i \omega t )} \cdot d \omega } </math>

===== Laplacian transform =====

''Main article:'' [[Laplace transform]]

The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations.

Given f=f(s), it is defined by:

: <math>F(s) = (\mathcal{L}f)(s) =\int_0^\infty e^{-st} f(t)\,dt.</math>

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=== Fundamental operators on scalar and vector fields===

''Main articles:'' [[vector calculus]], [[scalar field]], [[gradient]], [[divergence]], and [[curl]]

Three main operators are key to [[vector calculus]], the operator &nabla;, known as [[gradient]], where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the [[divergence]] is an operator that measures a vector field's tendency to originate from or converge upon a given point. [[Curl]], in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point.

==Operators in physics==

''Main article:'' [[Operator (physics)]]

In [[physics]], an operator often takes on a more specialized meaning than in mathematics. Operators as [[observable]]s are a key part of the theory of [[quantum mechanics]]. In that context ''operator'' often means a [[linear transformation]] from a [[Hilbert space]] to another, or (more abstractly) an element of a [[C*-algebra]].

==See also==

* [[Logical operator]]

* [[function (mathematics)]]

* [[unary operation]]

* [[binary operation]]

* [[ternary operation]]

* [[relational operator]]

* [[Common operator notation|common operator notation]].

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