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Em [[física]], um '''operador''' é uma [[função]] atuando sobre o espaço de [[Estados físicos da matéria|estados físicos]]. Como resultado desta aplicação sobre um estado físico, outro estado físico é obtido, muito frequentemente conjuntamente com alguma informação extra relevante. <br />
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O mais simples exemplo da utilidade de operadores é o estudo da [[simetria]]. Por causa disto, eles são ferramentas muito úteis em [[mecânica clássica]]. Em [[mecânica quântica]], por outro lado, eles são uma parte intrínseca da formulação da teoria.<br />
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==Operators in classical mechanics==<br />
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Let us consider a classical mechanics system led by a certain [[hamiltonian]] <math>H(q,p)</math>, <br />
function of the generalized coordinates <math>q</math> and its [[conjugate momenta]]. Let us consider<br />
this function to be invariant under the action of a certain [[group (mathematics)|group]] of transformations <math>G</math>, i.e., if <math>S\in G</math>, <math>H(S(q,p))=H(q,p)</math>. The elements of <math>G</math> are physical operators, which map physical states among themselves.<br />
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An easy example is given by space translations. The hamiltonian of a translationally invariant problem does not change under the transformation <math>q\to T_a q=q+a</math>. Other straightforward symmetry operators are the ones implementing rotations.<br />
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If the physical system is described by a function, as in classical field theories, the translation operator is generalized in a straightforward way:<br />
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: <math>f(x) \to T_a f(x)=f(x-a).</math><br />
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Notice that the transformation inside the parenthesis should be the [[inverse]] of the transformation done on the coordinates.<br />
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==Concept of generator==<br />
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If the transformation is infinitesimal, the operator action should be of the form<br />
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: <math> I + \epsilon A </math><br />
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where <math>I</math> is the identity operator, <math>\epsilon</math> is a small parameter, and <math>A</math> 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.<br />
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As it was stated, <math>T_a f(x)=f(x-a)</math>. If <math>a=\epsilon</math> is infinitesimal, then we may write<br />
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: <math>T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).</math><br />
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This formula may be rewritten as<br />
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: <math>T_\epsilon f(x) = (I-\epsilon D) f(x)</math><br />
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where <math>D</math> 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.<br />
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==The exponential map==<br />
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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.<br />
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The translation for a finite value of <math>a</math> may be obtained by repeated application of the infinitesimal translation:<br />
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: <math>T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)</math><br />
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with the <math>\cdots</math> standing for the application <math>N</math> times. If <math>N</math> is large, each of the factors may be considered to be infinitesimal:<br />
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: <math>T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x).</math><br />
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But this limit may be rewritten as an exponential:<br />
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: <math>T_a f(x)= \exp(-aD) f(x).</math><br />
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To be convinced of the validity of this formal expression, we may expand the exponential in a power series:<br />
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: <math>T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x).</math><br />
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The right-hand side may be rewritten as<br />
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: <math>f(x) - a f'(x) + {a^2\over 2!} f''(x) - {a^3\over 3!} f'''(x) + \cdots</math><br />
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which is just the Taylor expansion of <math>f(x-a)</math>, which was our original value for <math>T_a f(x)</math>.<br />
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==Operators in quantum mechanics==<br />
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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. The mathematical description of quantum mechanics is built upon the concept of operator.<br />
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Physical [[pure state]]s in quantum mechanics are unit-norm vectors in a certain [[vector space]] (a [[Hilbert space]]). Time evolution in this vector space is given by the application of a certain operator, called the [[evolution operator]]. Since the norm of the physical state should stay fixed, the evolution operator should be [[unitary transformation|unitary]]. Any other symmetry, mapping a physical state into another, should keep this restriction.<br />
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Any [[observable]], i.e., any quantity which can be measured in a physical experiment, should be associated with a [[self-adjoint]] [[linear operator]]. The values which may come up as the result of the experiment are the [[eigenvalue]]s of the operator. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue.<br />
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==General mathematical properties of quantum operators==<br />
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The mathematical properties of physical operators are a topic of great importance in itself. For further information, see [[C*-algebra]] and [[Gelfand-Naimark theorem]].<br />
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==See also==<br />
<div class="references-small" style="-moz-column-count:3; column-count:3;"><br />
*[[Position operator]]<br />
*[[Momentum operator#Momentum in quantum mechanics|Momentum Operator]]<br />
*[[Annihilation operator]]<br />
*[[Creation operator]]<br />
*[[Evolution operator]]<br />
*[[Hamiltonian (quantum mechanics)|Hamiltonian operator]]<br />
*[[Ladder operator]]<br />
*[[Bounded linear operator]]<br />
*[[Representation theory]]<br />
</div><br />
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[[Categoria:Física]]<br />
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[[de:Operator (Mathematik)#Operatoren der Physik]]</div>Calimero0000