{"id":485,"date":"2013-07-08T13:59:00","date_gmt":"2013-07-08T16:59:00","guid":{"rendered":"http:\/\/263@sampa.if.usp.br\/~suaide\/blog\/"},"modified":"2025-06-14T09:39:00","modified_gmt":"2025-06-14T12:39:00","slug":"simetrias-e-leis-de-conservacao","status":"publish","type":"post","link":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/2013\/07\/08\/simetrias-e-leis-de-conservacao\/","title":{"rendered":"Simetrias e leis de conserva\u00e7\u00e3o"},"content":{"rendered":"<p><![CDATA[ <center>nn<a href=\"http:\/\/sampa.if.usp.br&#038;te=print.html\" target=\"_top\">Vers\u00e3o para impress\u00e3o<\/a>n\n\nnn <strong><em>Notas de aula s\u00e3o propriedade intelectual. Sendo assim, qualquer uso, no todo ou em parte, deve ter a origem referenciada apropriadamente, ap\u00f3s autoriza\u00e7\u00e3o de seu autor.n\n\nnn <span style=\"color: red;\">O texto a seguir corresponde \u00e0 anota\u00e7\u00f5es parciais de aula. N\u00e3o \u00e9 um texto em forma final, completo e totalmente revisado. Nesse caso, esse texto n\u00e3o tem como objetivo substituir livros sobre o assunto.  Assim, esse texto deve ser entendido apenas como um guia de estudo para o aluno acompanhar a disciplina. <\/span> <\/em><\/strong>n<\/center>\n\n<style>n             .start-eq             { counter-reset: figures; }n             .numbered-eq q        { counter-increment: figures; }n             .numbered-eq q:before { content: '(' counter(figures)')'; }n             .numbered-eq q:after  { content: ''; }n             <\/style>\n\n\n\n<div class =\"start-eq\"><\/div>\n\nn<br \/>n<br \/>nNa aula passada introduzimos alguns n\u00fameros qu\u00e2nticos que s\u00e3o utilizados para caracterizar rea\u00e7\u00f5es envolvendo part\u00edculas. Na maioria dos casos estes n\u00fameros qu\u00e2nticos s\u00e3o conservados por\u00e9m, principalmente em intera\u00e7\u00f5es fracas e eletromagn\u00e9ticas, alguns desses n\u00fameros qu\u00e2nticos n\u00e3o se conservam. A depend\u00eancia do processo f\u00edsico para decidir se h\u00e1 ou n\u00e3o conserva\u00e7\u00e3o de n\u00fameros qu\u00e2nticos deve estar presente em alguma caracter\u00edstica fundamental das intera\u00e7\u00f5es que regem estes processos. Por conta disso, entender os princ\u00edpios que estabelecem a conserva\u00e7\u00e3o ou n\u00e3o de propriedades f\u00edsicas de um sistema \u00e9 um passo importante no aprendizado das intera\u00e7\u00f5es fundamentais. Vamos ver que a conserva\u00e7\u00e3o de uma grandeza f\u00edsica est\u00e1 intimamente ligada a uma invari\u00e2ncia por uma transforma\u00e7\u00e3o qualquer do sistema, isto \u00e9, uma simetria apresentada pelo sistema. No in\u00edcio do s\u00e9c. XX, Emmy Noether publicou um teorema, que leva seu nome, que relacionava a presen\u00e7a de uma simetria em um sistema f\u00edsico a uma lei de conserva\u00e7\u00e3o. Para cada simetria presente no sistema h\u00e1 uma lei de conserva\u00e7\u00e3o. Do ponto de vista experimental, o inverso \u00e9 mais apropriado: para cada lei de conserva\u00e7\u00e3o deve haver uma simetria no sistema f\u00edsico. Isto \u00e9 interessante porque a observa\u00e7\u00e3o de fen\u00f4menos resultam na observa\u00e7\u00e3o dessas leis de conserva\u00e7\u00e3o, que guiam o caminho dos pesquisadores para tentar explicar fundamentalmente as intera\u00e7\u00f5es respeitando as devidas simetrias. Por exemplo, a conserva\u00e7\u00e3o de diversos n\u00fameros qu\u00e2nticos em intera\u00e7\u00f5es fortes estabelecem condi\u00e7\u00f5es de contorno nas teorias formuladas para estas intera\u00e7\u00f5es. Vamos ver que h\u00e1 uma representa\u00e7\u00e3o bastante \u00fatil para representar estas simetrias, atrav\u00e9s de teoria de grupos, um ramo da matem\u00e1tica fundamental para o estudo de simetrias.n<br \/>n<br \/>nAlgumas das simetrias mais fundamentais est\u00e3o listadas na tabela 1, abaixo. n<br \/>n<br \/>n<center>nTabela 1 &#8211; Algumas simetrias fundamentais.<br  \/>n\n\n<table border=\"0\" width=\"400\">n\n\n<tr>\n\n<td colspan=\"2\">\n\n<hr  \/>\n\n<\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td><center><b>Simetria<\/b><\/center><\/td>\n\n\n\n<td><center><b>Lei de conserva\u00e7\u00e3o<\/b><\/center><\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td colspan=\"2\">\n\n<hr  \/>\n\n<\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td><center>Transla\u00e7\u00e3o em tempo<\/center><\/td>\n\n\n\n<td><center>Conserva\u00e7\u00e3o de energia<\/center><\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td><center>Transla\u00e7\u00e3o no espa\u00e7o<\/center><\/td>\n\n\n\n<td><center>Conserva\u00e7\u00e3o do momento<\/center><\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td><center>Rota\u00e7\u00e3o<\/center><\/td>\n\n\n\n<td><center>Conserva\u00e7\u00e3o de momento angular<\/center><\/td>\n\n<\/tr>\n\nn\n\n<tr>\n\n<td colspan=\"2\">\n\n<hr  \/>\n\n<\/td>\n\n<\/tr>\n\nn<\/table>\n\n<\/center>n<br \/>n<br \/>nTomemos como exemplo a transla\u00e7\u00e3o espacial. Imagine um sistema de 2 part\u00edculas sujeitas a uma for\u00e7a conservativa que dependa apenas da dist\u00e2ncia entre elas. Neste caso, poder\u00edamos escrever que o potencial de intera\u00e7\u00e3o \u00e9, em um sistema de coordenadas qualquer:n<br \/>n<br \/>n<center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/e3b6c0b353120a62f7ad19059de91d1e.png' title='V=V(\\vec{r}_1-\\vec{r}_2)' alt='V=V(\\vec{r}_1-\\vec{r}_2)' align=absmiddle><\/center>n<br \/>n<br \/>nFa\u00e7amos agora uma transla\u00e7\u00e3o no sistema de coordenadas, ou seja:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/2298a9174f0a705ad04899e26a7c2c17.png' title='\\vec{r}_1 \\rightarrow \\vec{r}_1\\text{'}= \\vec{r}_1 + \\vec{a}' alt='\\vec{r}_1 \\rightarrow \\vec{r}_1\\text{'}= \\vec{r}_1 + \\vec{a}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/2bdf4ea17718750a0c7bed0f3d781bd0.png' title='\\vec{r}_2 \\rightarrow \\vec{r}_2\\text{'}= \\vec{r}_2 + \\vec{a}' alt='\\vec{r}_2 \\rightarrow \\vec{r}_2\\text{'}= \\vec{r}_2 + \\vec{a}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnNeste caso, o potencial de intera\u00e7\u00e3o torna-se:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/74a32d85184d8cf0a06694e92f6f3e7d.png' title='V(\\vec{r}_1-\\vec{r}_2) \\rightarrow V(\\vec{r}_1\\text{'}-\\vec{r}_2\\text{'})=V(\\vec{r}_1 + \\vec{a} -\\vec{r}_2 -  \\vec{a})= V(\\vec{r}_1-\\vec{r}_2)' alt='V(\\vec{r}_1-\\vec{r}_2) \\rightarrow V(\\vec{r}_1\\text{'}-\\vec{r}_2\\text{'})=V(\\vec{r}_1 + \\vec{a} -\\vec{r}_2 -  \\vec{a})= V(\\vec{r}_1-\\vec{r}_2)' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnOu seja, o potencial de intera\u00e7\u00e3o n\u00e3o \u00e9 alterado por uma transla\u00e7\u00e3o de referencial. Ou seja, podemos interpretar que o sistema f\u00edsico \u00e9 o mesmo independente da escolha da or\u00edgem do sistema de coordenadas (j\u00e1 que as energias cin\u00e9ticas tamb\u00e9m independem desta escolha &#8211; mostre isso). Esta simetria d\u00e1 or\u00edgem a uma lei de conserva\u00e7\u00e3o bastante interessante. Podemos escrever que a for\u00e7a total agindo sobre o sistema \u00e9:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0005ff5af127787ee55996aed63402d0.png' title='\\vec{F}=\\vec{F}_1+\\vec{F}_2 = -\\vec \\nabla_1 V(\\vec{r}_1-\\vec{r}_2) - \\vec \\nabla_2 V(\\vec{r}_1-\\vec{r}_2)' alt='\\vec{F}=\\vec{F}_1+\\vec{F}_2 = -\\vec \\nabla_1 V(\\vec{r}_1-\\vec{r}_2) - \\vec \\nabla_2 V(\\vec{r}_1-\\vec{r}_2)' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nne, como <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/45d577248619e80947e2e9b12b1fa068.png' title='\\vec \\nabla_1 V = - \\vec \\nabla_2 V' alt='\\vec \\nabla_1 V = - \\vec \\nabla_2 V' align=absmiddle>, por conta da diferen\u00e7a <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f5afc02ad6808a77f8bb49b672ab4ad5.png' title='\\vec{r}_1-\\vec{r}_2' alt='\\vec{r}_1-\\vec{r}_2' align=absmiddle>, resulta que a for\u00e7a total \u00e9 nula e, neste caso:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/9753c1f189a3e2122d25424c7720ab4b.png' title='\\vec{F}=0=\\frac{d}{dt}\\vec{P}_{total} \\rightarrow \\vec{P}_{total} = cte' alt='\\vec{F}=0=\\frac{d}{dt}\\vec{P}_{total} \\rightarrow \\vec{P}_{total} = cte' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnOu seja, o momento total do sistema se conserva. Podemos expandir um pouco mais esta argumenta\u00e7\u00e3o olhando o sistema do ponto de vista do formalismo lagrangeano. Definindo a lagrangeana do sistema como sendo <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f798ab44bef1cadade43191a28b2fa6b.png' title='L=T-V' alt='L=T-V' align=absmiddle>, sendo <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/b023692c67cc6d509edcaee15087b0a6.png' title='T=T(\\dot{q}_i,q_i)' alt='T=T(\\dot{q}_i,q_i)' align=absmiddle> a energia cin\u00e9tica e <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/8d67f71a331a3a2cb7aaf3646bc0b2f6.png' title='V=V(\\dot{q}_i,q_i)' alt='V=V(\\dot{q}_i,q_i)' align=absmiddle>, a energia potencial. <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/5e8f5516e90d2fff1ff85a93b4c7492b.png' title='\\dot{q}_i,q_i' alt='\\dot{q}_i,q_i' align=absmiddle> s\u00e3o, respectivamente, as velocidades e posi\u00e7\u00f5es generalizadas. As equa\u00e7\u00f5es do movimento podem ser escritas resolvendo as equa\u00e7\u00f5es:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/46151ecc37da60ec4df47547cb91c1e6.png' title='\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}_i}-\\frac{\\partial L}{\\partial q_i} = 0' alt='\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}_i}-\\frac{\\partial L}{\\partial q_i} = 0' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnonde definimos o momento associado \u00e0 coordenada <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/da326f7200e158a864695985b2e2f095.png' title='q_i' alt='q_i' align=absmiddle> como sendo:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/6c138f349c8f90584741e07ab2c39a54.png' title='p_i = \\frac{\\partial L}{\\partial \\dot{q}_i}' alt='p_i = \\frac{\\partial L}{\\partial \\dot{q}_i}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnAssim:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c4b9436f496752e17fb54a17cd1c9276.png' title='\\frac{d}{dt}p_i-\\frac{\\partial L}{\\partial q_i} = 0' alt='\\frac{d}{dt}p_i-\\frac{\\partial L}{\\partial q_i} = 0' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnNa equa\u00e7\u00e3o acima, se a lagrangeana do sistema n\u00e3o depender explicitamente da coordenada generalizada <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/da326f7200e158a864695985b2e2f095.png' title='q_i' alt='q_i' align=absmiddle>, o momento associado a esta coordenada ser\u00e1 uma constante do sistema, ou seja, \u00e9 uma grandeza conservada. Isto \u00e9 equivalente a dizer que uma transforma\u00e7\u00e3o do tipo:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/97ae5c368d13e352faef5edcb259c7d5.png' title='q_i \\rightarrow q_i +\\delta q_i' alt='q_i \\rightarrow q_i +\\delta q_i' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnN\u00e3o altera a lagrangeana do sistema, ou seja ela \u00e9 sim\u00e9trica por uma transforma\u00e7\u00e3o deste tipo. No nosso exemplo, escrevendo a lagrangeana em termos das coordenadas <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/8d55321f485d68c666d704a8df57087b.png' title='\\vec{R}' alt='\\vec{R}' align=absmiddle>, a posi\u00e7\u00e3o do centro de momentos e <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/30d15731506533f20bd1225da1c58aac.png' title='\\vec{r}' alt='\\vec{r}' align=absmiddle>, a posi\u00e7\u00e3o relativa entre as part\u00edculas 1 e 2, pode-se mostrar que a lagrangeana resultante n\u00e3o depende explicitamente de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/8d55321f485d68c666d704a8df57087b.png' title='\\vec{R}' alt='\\vec{R}' align=absmiddle> e, consequentemente, seu momento associado, neste caso, o momento total, \u00e9 conservado. n<br \/>n<br \/>nPrecisamos transportar esses conhecimentos acerca de simetrias para o mundo qu\u00e2ntico. Apesar de o formalismo lagrangeano ser mais apropriado para tratar sistemas qu\u00e2nticos relativ\u00edsticos, vamos olhar a quest\u00e3o de constantes de movimento e simetrias usando o formalismo hamiltoniano, que \u00e9 mais familiar aqueles que t\u00eam apenas cursos introdut\u00f3rios de mec\u00e2nica qu\u00e2ntica. <br \/>n<br \/>nUma grandeza f\u00edsica qualquer pode ser obtida atrav\u00e9s do valor esperado do seu operador em rela\u00e7\u00e3o ao estado qu\u00e2ntico estudado, ou seja:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/82ddc7749867b4039a75d67ff19f8650.png' title='\\langle A \\rangle_{\\psi} =   \\langle \\psi | \\hat A | \\psi \\rangle = \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)}' alt='\\langle A \\rangle_{\\psi} =   \\langle \\psi | \\hat A | \\psi \\rangle = \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>  nnSe a grandeza \u00e9 uma constante do movimento, a sua varia\u00e7\u00e3o no tempo \u00e9 nula, ou seja:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/bcdd33415d556c95dc40234257a70618.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =  0' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =  0' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnExpandindo (11) a partir de (10), temos:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/44b34ce1479befc9322bfa82d2017f51.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =  \\frac{d}{dt} \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)}' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =  \\frac{d}{dt} \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/7b08a2c30a09eb7dfbb6b1bb5a8ecbd9.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\int{d\\vec{r} \\frac{\\partial}{\\partial t}\\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)} + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\frac{\\partial}{\\partial t}\\hat A \\psi(\\vec{r},t)}  + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\frac{\\partial}{\\partial t}\\psi(\\vec{r},t)}' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\int{d\\vec{r} \\frac{\\partial}{\\partial t}\\psi^*(\\vec{r},t) \\hat A \\psi(\\vec{r},t)} + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\frac{\\partial}{\\partial t}\\hat A \\psi(\\vec{r},t)}  + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\frac{\\partial}{\\partial t}\\psi(\\vec{r},t)}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnA partir da equa\u00e7\u00e3o de Schr\u00f6dinger dependente do tempo, lembrando que estamos trabalhando em unidades naturais:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/a28682e8a4fc9888c173c785ad8d1558.png' title='i\\frac{\\partial}{\\partial t} \\psi(\\vec{r},t) = \\hat H\\psi(\\vec{r},t)' alt='i\\frac{\\partial}{\\partial t} \\psi(\\vec{r},t) = \\hat H\\psi(\\vec{r},t)' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnPodemos reescrever (12) como: nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/b8da0cc3235f8e519245f7e72dbae00a.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\int{d\\vec{r} \\left(\\frac{1}{i}\\hat H\\psi(\\vec{r},t)\\right)^* \\hat A \\psi(\\vec{r},t)} + \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi} + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\frac{1}{i}\\hat H\\psi(\\vec{r},t)}' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\int{d\\vec{r} \\left(\\frac{1}{i}\\hat H\\psi(\\vec{r},t)\\right)^* \\hat A \\psi(\\vec{r},t)} + \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi} + \\int{d\\vec{r} \\psi^*(\\vec{r},t) \\hat A \\frac{1}{i}\\hat H\\psi(\\vec{r},t)}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnSabendo que <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/3cac99275fc7e6f4921a129f9ce9ae6a.png' title='\\hat H' alt='\\hat H' align=absmiddle> \u00e9 hermitiano, podemos rescrever a primeira integral e, reorganizando as integrais acima, temos:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/9ff65ac08ec0bdd07525331e940bc17b.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi}   +  i\\int{d\\vec{r} \\psi^*(\\vec{r},t) (\\hat H\\hat A -\\hat A\\hat H) \\psi^*(\\vec{r},t) }' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi}   +  i\\int{d\\vec{r} \\psi^*(\\vec{r},t) (\\hat H\\hat A -\\hat A\\hat H) \\psi^*(\\vec{r},t) }' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/b46996c517256ce5a0a88a11fc1dca4e.png' title='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi}   + i  \\langle [\\hat H,\\hat A] \\rangle_{\\psi}' alt='\\frac{d}{dt} \\langle A \\rangle_{\\psi} =   \\langle \\frac{\\partial}{\\partial t}\\hat A \\rangle_{\\psi}   + i  \\langle [\\hat H,\\hat A] \\rangle_{\\psi}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnA equa\u00e7\u00e3o (17), mostra que, caso o operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d5ac2ffc2b907daf433a2343b2a8f4c3.png' title='\\hat A' alt='\\hat A' align=absmiddle> n\u00e3o dependa explicitamente do tempo e caso <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/b870cfc751ba215141e286fae074de54.png' title='[\\hat H,\\hat A]=0' alt='[\\hat H,\\hat A]=0' align=absmiddle>, ent\u00e3o o valor m\u00e9dio de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/7fc56270e7a70fa81a5935b72eacbe29.png' title='A' alt='A' align=absmiddle> \u00e9 constante. Nesse caso, podemos dizer que essa grandeza \u00e9 uma constante do movimento. Essa igualdade \u00e9 particularmente importante pois permite obter claramente quais s\u00e3o as constantes de movimento de um sistema qu\u00e2ntico. n\n\nnUm aspecto interessante da equa\u00e7\u00e3o (17) \u00e9 a consequ\u00eancia que, se houver estados qu\u00e2nticos que sejam, simultaneamente, auto-estados de um operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/7fc56270e7a70fa81a5935b72eacbe29.png' title='A' alt='A' align=absmiddle> e da Hamiltoniana do sistema, a grandeza <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/7fc56270e7a70fa81a5935b72eacbe29.png' title='A' alt='A' align=absmiddle> ser\u00e1 uma constante do movimento. Para mostrar isso, basta aplicar o comutador desse operador e da Hamiltoniana nesses auto-estados, supondo <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/1f28756b58bd4b077330f8042b9d3168.png' title='\\lambda_A' alt='\\lambda_A' align=absmiddle> e <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/2fc545d1702367bbc973e7bde389d5de.png' title='\\lambda_H' alt='\\lambda_H' align=absmiddle> auto-valores de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d5ac2ffc2b907daf433a2343b2a8f4c3.png' title='\\hat A' alt='\\hat A' align=absmiddle> e <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/3cac99275fc7e6f4921a129f9ce9ae6a.png' title='\\hat H' alt='\\hat H' align=absmiddle>, respectivamente:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/45f9be8305d638ef8f3eddf029770029.png' title='[\\hat H,\\hat A]|\\psi\\rangle = (\\hat H\\hat A - \\hat A\\hat H)|\\psi\\rangle = \\hat H\\hat A|\\psi\\rangle - \\hat A\\hat H|\\psi\\rangle = \\hat H\\lambda_A|\\psi\\rangle - \\hat A\\lambda_H|\\psi\\rangle' alt='[\\hat H,\\hat A]|\\psi\\rangle = (\\hat H\\hat A - \\hat A\\hat H)|\\psi\\rangle = \\hat H\\hat A|\\psi\\rangle - \\hat A\\hat H|\\psi\\rangle = \\hat H\\lambda_A|\\psi\\rangle - \\hat A\\lambda_H|\\psi\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/10f47135ddedb234232d9dcd3483c8ed.png' title='[\\hat H,\\hat A]|\\psi\\rangle =  \\lambda_A\\hat H|\\psi\\rangle - \\lambda_H\\hat A|\\psi\\rangle =  \\lambda_A\\lambda_H|\\psi\\rangle - \\lambda_H\\lambda_A|\\psi\\rangle = 0' alt='[\\hat H,\\hat A]|\\psi\\rangle =  \\lambda_A\\hat H|\\psi\\rangle - \\lambda_H\\hat A|\\psi\\rangle =  \\lambda_A\\lambda_H|\\psi\\rangle - \\lambda_H\\lambda_A|\\psi\\rangle = 0' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/949df32287e072b88deb2f9b0eda0b30.png' title='[\\hat H,\\hat A]|\\psi\\rangle = 0 \\rightarrow [\\hat H,\\hat A] = 0' alt='[\\hat H,\\hat A]|\\psi\\rangle = 0 \\rightarrow [\\hat H,\\hat A] = 0' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnBom, neste ponto estabelecemos uma condi\u00e7\u00e3o para o valor esperado de uma grandeza ser uma constante de movimento. Basta que seu operador correspondente comute com a hamiltoniana do sistema e ele n\u00e3o dependa explicitamente do tempo. Precisamos agora estabelecer que a hamiltoniana do sistema possua uma simetria relacionada a este operador, de modo a estabelecer v\u00ednculos entre uma grandeza conservada e uma simetria em sistemas qu\u00e2nticos. Como discutimos, uma simetria torna-se evidente quando o sistema n\u00e3o se modifica perante uma transforma\u00e7\u00e3o.n<br \/>n<br \/>nVamos tentar construir operadores que realizem uma transforma\u00e7\u00e3o no sistema, seja ela de rota\u00e7\u00e3o, deslocamento espacial ou temporal, ou outra transforma\u00e7\u00e3o qualquer. Vamos fazer isso olhando exemplos simples e dai, tentar retirar uma forma para estas opera\u00e7\u00f5es de transforma\u00e7\u00e3o. Vamos come\u00e7ar por uma transforma\u00e7\u00e3o temporal. A evolu\u00e7\u00e3o temporal de um estado qu\u00e2ntico pode ser escrita atrav\u00e9s de:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/030614d886e542f861e6907e4268c7c1.png' title='i\\frac{\\partial}{\\partial t}| \\psi(t) \\rangle = \\hat H | \\psi(t) \\rangle' alt='i\\frac{\\partial}{\\partial t}| \\psi(t) \\rangle = \\hat H | \\psi(t) \\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnSe a hamiltoniana n\u00e3o depende explicitamente do tempo, ent\u00e3o podemos dizer que a solu\u00e7\u00e3o da equa\u00e7\u00e3o (21) pode ser escrita como:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0fa38bc272692217aba3b67b75536350.png' title='| \\psi(t) \\rangle = \\exp(-i\\hat H(t-t_0)) | \\psi(t_0) \\rangle' alt='| \\psi(t) \\rangle = \\exp(-i\\hat H(t-t_0)) | \\psi(t_0) \\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnPodemos reescrever (22) como:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/71ad2a3dd3e29d43a0a2d358292a5473.png' title='| \\psi(t_0+\\Delta t) \\rangle = \\exp(-i\\hat H\\Delta t) | \\psi(t_0) \\rangle' alt='| \\psi(t_0+\\Delta t) \\rangle = \\exp(-i\\hat H\\Delta t) | \\psi(t_0) \\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnou seja, o operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/766702e7b9c69159b1cfac03ac9412ae.png' title='\\exp(-i\\hat H\\Delta t)' alt='\\exp(-i\\hat H\\Delta t)' align=absmiddle> realiza uma transforma\u00e7\u00e3o temporal do tipo <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/778430986cd16c55eb43b958393a8f9a.png' title='| \\psi(t_0) \\rangle \\rightarrow | \\psi(t_0+\\Delta t) \\rangle' alt='| \\psi(t_0) \\rangle \\rightarrow | \\psi(t_0+\\Delta t) \\rangle' align=absmiddle> no estado qu\u00e2ntico do sistema.<br \/>n<br \/>nVamos usar esta mesma estrutura para tentar construir um operador similar para uma transforma\u00e7\u00e3o espacial. Considere agora o estado <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d3fb397fea373c58eaf8ddcde93ae549.png' title='|x\\rangle' alt='|x\\rangle' align=absmiddle>, autoestado do operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c243886a288804343eee2af0ad8dcebc.png' title='\\hat x' alt='\\hat x' align=absmiddle> com autovalor <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/9dd4e461268c8034f5c8564e155c67a6.png' title='x' alt='x' align=absmiddle>. Deste modo, podemos escrever que:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d68425dc6e59987c937be531ad479efc.png' title='\\hat x|x\\rangle = x|x\\rangle' alt='\\hat x|x\\rangle = x|x\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/8d95506bc75dad57be3392720eea4076.png' title='\\langle x'|x\\rangle=\\delta(x-x')' alt='\\langle x'|x\\rangle=\\delta(x-x')' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnVamos considerar o estado qu\u00e2ntico <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/a43357bc44c55cc5c9cb453d6167b86b.png' title='|?\\rangle = \\exp(-i\\hat p_x \\Delta x)|x\\rangle' alt='|?\\rangle = \\exp(-i\\hat p_x \\Delta x)|x\\rangle' align=absmiddle>. Qual o valor esperado do operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c243886a288804343eee2af0ad8dcebc.png' title='\\hat x' alt='\\hat x' align=absmiddle> para este estado? Precisamos calcular, ent\u00e3o, a grandeza:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/4b9394cbd5f40755bba47a2ccd9a191a.png' title='\\langle ?|\\hat x|?\\rangle = \\langle x| \\exp(i\\hat p_x \\Delta x) \\hat x \\exp(-i\\hat p_x \\Delta x)|x\\rangle' alt='\\langle ?|\\hat x|?\\rangle = \\langle x| \\exp(i\\hat p_x \\Delta x) \\hat x \\exp(-i\\hat p_x \\Delta x)|x\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnPara calcular (26) precisamos resolver as exponenciais do elemento de matriz acima. No fundo, pretendemos resolver algo do tipo (a troca de sinal \u00e9 para poder batizar o resultado mais tarde):nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0c82a9587641c84c58ddcccd65e6b3bf.png' title='\\hat Y(s) = \\exp(-s\\hat B)\\hat A \\exp(s\\hat B)' alt='\\hat Y(s) = \\exp(-s\\hat B)\\hat A \\exp(s\\hat B)' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnExpandindo o operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c57c0b23a130737606eb6cc633be6753.png' title='\\hat Y(s)' alt='\\hat Y(s)' align=absmiddle> em uma s\u00e9rie de Taylor em torno de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/787d0b6e5d9e7525a7054c6f96c377ea.png' title='s=0' alt='s=0' align=absmiddle>, temos que:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/5305ac0f0473ea2142a8e64dfbd5086f.png' title='\\hat Y(s) = \\sum_{n=0}^{\\infty}{\\frac{1}{n!}\\frac{d^n\\hat Y(s=0)}{ds^n} s^n' alt='\\hat Y(s) = \\sum_{n=0}^{\\infty}{\\frac{1}{n!}\\frac{d^n\\hat Y(s=0)}{ds^n} s^n' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnO primeiro termo vale <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f5a93fe2df8b1c132d60572a20f66e83.png' title='\\hat Y^0 = \\hat Y(s=0) = \\hat A' alt='\\hat Y^0 = \\hat Y(s=0) = \\hat A' align=absmiddle>. O segundo termo (<img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/6d24e2bc97c5e4283dd8e34674afe7ea.png' title='n=1' alt='n=1' align=absmiddle>) vale:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/4327e5f6a17037be996143ef2c3f63a0.png' title='\\hat Y^1 &#038;= \\frac{d\\hat Y(s=0)}{ds} s' alt='\\hat Y^1 &#038;= \\frac{d\\hat Y(s=0)}{ds} s' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/38bb0f4161aa7af2f00e02dfb25d6ab2.png' title='\\hat Y^1 &#038;= \\left(-\\exp(-s\\hat B)\\hat B\\hat A\\exp(s\\hat B)+\\exp(-s\\hat B)\\hat A\\hat B\\exp(s\\hat B) \\right)_{s=0} s' alt='\\hat Y^1 &#038;= \\left(-\\exp(-s\\hat B)\\hat B\\hat A\\exp(s\\hat B)+\\exp(-s\\hat B)\\hat A\\hat B\\exp(s\\hat B) \\right)_{s=0} s' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/a131fd5bfaa6cbd18c96b377226ec6b1.png' title='\\hat Y^1 &#038;= (-\\hat B\\hat A +\\hat A \\hat B)s = [\\hat A,\\hat B]s' alt='\\hat Y^1 &#038;= (-\\hat B\\hat A +\\hat A \\hat B)s = [\\hat A,\\hat B]s' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnO terceiro termo, <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c303081f7a16f603112b0375bdc84883.png' title='n=2' alt='n=2' align=absmiddle> pode ser calculado da mesma maneira e \u00e9 f\u00e1cil mostrar que:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/bf98dd9e9cca7a6a29bcddc944e39e34.png' title='\\hat Y^2 = \\frac{1}{2!}[ [\\hat A,\\hat B],\\hat B] s^2' alt='\\hat Y^2 = \\frac{1}{2!}[ [\\hat A,\\hat B],\\hat B] s^2' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nne assim por diante. A expres\u00e3o gen\u00e9rica para resolver este conjunto de exponenciais \u00e9 dada (no caso de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/73bbe012edfb61eca43444d61fefe937.png' title='s=1' alt='s=1' align=absmiddle>)  por:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d1aa6088a73301920be658c3653be2c1.png' title='\\exp(-\\hat B)\\hat A\\exp(\\hat B) = \\hat A + [\\hat A,\\hat B] +\\frac{1}{2!}[ [\\hat A,\\hat B],\\hat B] + ...' alt='\\exp(-\\hat B)\\hat A\\exp(\\hat B) = \\hat A + [\\hat A,\\hat B] +\\frac{1}{2!}[ [\\hat A,\\hat B],\\hat B] + ...' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnque \u00e9 conhecido como teorema de Baker-Hausdorff. No nosso caso, precisamos apenas dos dois primeiros termos de (33), j\u00e1 que o comutador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/e397701505dfacade3a406574d580084.png' title='[\\hat x,\\hat p_x]' alt='[\\hat x,\\hat p_x]' align=absmiddle> \u00e9 um n\u00famero, o que anula todos os termos subsequentes. Assim, sabendo que <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d80c982f1fa96cce4c97fe5e047f4a9b.png' title='[\\hat x, \\hat p_x]=i' alt='[\\hat x, \\hat p_x]=i' align=absmiddle>, resolvendo (26) obtemos:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0379af85b06fd3829763a21b87d0311b.png' title='\\langle ?|\\hat x|?\\rangle = \\langle x| \\hat x + (-i\\Delta x)[\\hat x,\\hat p_x]|x\\rangle' alt='\\langle ?|\\hat x|?\\rangle = \\langle x| \\hat x + (-i\\Delta x)[\\hat x,\\hat p_x]|x\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/31d05192392a04dced03bdf8cbb7f0de.png' title='\\langle ?|\\hat x|?\\rangle =\\langle x|\\hat x|x\\rangle +\\Delta x \\langle x|x\\rangle' alt='\\langle ?|\\hat x|?\\rangle =\\langle x|\\hat x|x\\rangle +\\Delta x \\langle x|x\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/eedda964f8c8c877247fd00550eca63b.png' title='\\langle ?|\\hat x|?\\rangle = x + \\Delta x' alt='\\langle ?|\\hat x|?\\rangle = x + \\Delta x' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnDe (36) podemos ver claramente que o estado <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/cc393f3ca27936b5ea74d007526293b0.png' title='|?\\rangle' alt='|?\\rangle' align=absmiddle> tamb\u00e9m \u00e9 um autoestado do operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c243886a288804343eee2af0ad8dcebc.png' title='\\hat x' alt='\\hat x' align=absmiddle>, com autovalor <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/36b7d3ed3d16c8bf780c9bca5e221f5e.png' title='x+\\Delta x' alt='x+\\Delta x' align=absmiddle>, o que nos permite concluir que o operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/8e1ca1319405f443c38a03ed2379a511.png' title='\\exp(-i\\hat p_x \\Delta x)' alt='\\exp(-i\\hat p_x \\Delta x)' align=absmiddle> \u00e9 respons\u00e1vel por uma transla\u00e7\u00e3o espacial do sistema em <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/b56546a86ab832a9b2a5b15f96519319.png' title='\\Delta x' alt='\\Delta x' align=absmiddle>, ou seja, de forma similar \u00e0 transla\u00e7\u00e3o temporal:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0d496cc790c1549d94ee19a94db5716b.png' title='|x+\\Delta x\\rangle = \\exp(-i \\hat p_x \\Delta x)|x\\rangle' alt='|x+\\Delta x\\rangle = \\exp(-i \\hat p_x \\Delta x)|x\\rangle' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>  nnOs operadores de deslocamento temporal e espacial t\u00eam estrutura muito similares, na forma:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/83f0beae261a1711c1477639804a46aa.png' title='\\hat o = \\exp(-i\\hat q \\Delta a)' alt='\\hat o = \\exp(-i\\hat q \\Delta a)' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnonde <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/dddadfc73bb8f7b216dc844db56d9eb9.png' title='\\hat q' alt='\\hat q' align=absmiddle> \u00e9 um operador hermitiano e <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f8a2102664419c44aa434cf216441a03.png' title='\\Delta a' alt='\\Delta a' align=absmiddle> \u00e9 a amplitude da transforma\u00e7\u00e3o, que \u00e9 uma quantidade real. O operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/ce553a8638b99b1a5463cf9e9470c2d3.png' title='\\hat o' alt='\\hat o' align=absmiddle> \u00e9 unit\u00e1rio, ou seja, <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f2ac6e45a303e2e2af0bc4010474f9af.png' title='\\hat o^* \\hat o = \\hat 1' alt='\\hat o^* \\hat o = \\hat 1' align=absmiddle> e, por conta disso n\u00e3o alterna as normaliza\u00e7\u00f5es e produtos internos de estados qu\u00e2nticos. De fato, transforma\u00e7\u00f5es em mec\u00e2nica qu\u00e2ntica s\u00e3o, na grande maioria, descritas por operadores unit\u00e1rios nesta forma. Uma exce\u00e7\u00e3o \u00e9 a opera\u00e7\u00e3o de revers\u00e3o temporal, onde o operador \u00e9 anti-unit\u00e1rio.<br \/>n<br \/>nTemos dois ingredientes na m\u00e3o. A condi\u00e7\u00e3o para uma grandeza ser conservada em mec\u00e2nica qu\u00e2ntica \u00e9 que o seu operador comute com a hamiltoniana do sistema e n\u00e3o dependa explicitamente do tempo. E que uma transforma\u00e7\u00e3o em um sistema pode ser conduzida por um operador unit\u00e1rio como o descrito em (38). Precisamos agora conectar estes dois peda\u00e7os de informa\u00e7\u00e3o e mostrar que uma transforma\u00e7\u00e3o em um sistema evidencia uma simetria se o gerador desta transforma\u00e7\u00e3o, o operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/dddadfc73bb8f7b216dc844db56d9eb9.png' title='\\hat q' alt='\\hat q' align=absmiddle> em (38), representar uma grandeza que \u00e9 conservada. Para isto, vamos imaginar que uma transforma\u00e7\u00e3o <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f8a2102664419c44aa434cf216441a03.png' title='\\Delta a' alt='\\Delta a' align=absmiddle> pode ser tratada como uma sucess\u00e3o de infinitas transforma\u00e7\u00f5es infinitesimais, ou seja, podemos utilizar a rela\u00e7\u00e3o:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/170dda7a95cbbce9925cf16d79f8c130.png' title='\\exp(x) = \\lim_{n\\rightarrow\\infty}{\\left(1 + \\frac{x}{n}\\right)^n}' alt='\\exp(x) = \\lim_{n\\rightarrow\\infty}{\\left(1 + \\frac{x}{n}\\right)^n}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnNeste caso, uma transforma\u00e7\u00e3o como a mostrada em (38) torna-se:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/31321e3f1ae740cf4820c63525b92cfb.png' title='\\exp(-i\\hat q\\Delta a) = \\lim_{n\\rightarrow\\infty}{\\left(1 -i\\hat q \\frac{\\Delta a}{n}\\right)^n}=1-i\\hat q \\frac{\\Delta a}{n} + ...' alt='\\exp(-i\\hat q\\Delta a) = \\lim_{n\\rightarrow\\infty}{\\left(1 -i\\hat q \\frac{\\Delta a}{n}\\right)^n}=1-i\\hat q \\frac{\\Delta a}{n} + ...' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnNo caso de uma transforma\u00e7\u00e3o infinitesimal, onde <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/df0092f0fab128d4e927083569209150.png' title='\\Delta a \\rightarrow \\delta a' alt='\\Delta a \\rightarrow \\delta a' align=absmiddle>, podemos tomar apenas os dois primeiros termos na expans\u00e3o em Taylor de (40) e escrever o operador como sendo:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/3f6d2b4756cef9b24b2de01980ee5e4d.png' title='\\hat o = 1-i\\hat q \\delta a' alt='\\hat o = 1-i\\hat q \\delta a' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnUma transforma\u00e7\u00e3o como a de (41) altera o valor esperado da hamiltoniana do sistema na forma:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c2c16511c7706d25e0365133a3dafe9a.png' title='< H > = \\langle a|\\hat H|a\\rangle \\rightarrow <h'>=\\langle a+\\delta a|\\hat H|a+\\delta a\\rangle&#8217; alt=&#8217;< H > = \\langle a|\\hat H|a\\rangle \\rightarrow <h'>=\\langle a+\\delta a|\\hat H|a+\\delta a\\rangle&#8217; align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnque podemos resolver da seguinte forma, lembrando que <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/dddadfc73bb8f7b216dc844db56d9eb9.png' title='\\hat q' alt='\\hat q' align=absmiddle> \u00e9 hermitiano:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/0a719f35792c78f58842d6b16dfc405c.png' title='<h'>=\\langle a+\\delta a|\\hat H|a+\\delta a\\rangle = \\langle a|(1-i\\hat q\\delta a)^* \\hat H (1-i\\hat q\\delta a)|a\\rangle&#8217; alt=&#8217;<h'>=\\langle a+\\delta a|\\hat H|a+\\delta a\\rangle = \\langle a|(1-i\\hat q\\delta a)^* \\hat H (1-i\\hat q\\delta a)|a\\rangle&#8217; align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/25c5a3c94d90398c803786792b523821.png' title='<h'>= \\langle a|(1+i\\hat q^*\\delta a) \\hat H (1-i\\hat q\\delta a)|a\\rangle&#8217; alt=&#8217;<h'>= \\langle a|(1+i\\hat q^*\\delta a) \\hat H (1-i\\hat q\\delta a)|a\\rangle&#8217; align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/20f29caa9e29f6b0c65c639c7cbc6d8c.png' title='<h'>= \\langle a| \\hat H |a\\rangle + i\\delta a( \\langle a|\\hat q\\hat H|a\\rangle &#8211; \\langle a|\\hat H\\hat q|a\\rangle) + \\mathcal{O}(\\delta a^2)&#8217; alt=&#8217;<h'>= \\langle a| \\hat H |a\\rangle + i\\delta a( \\langle a|\\hat q\\hat H|a\\rangle &#8211; \\langle a|\\hat H\\hat q|a\\rangle) + \\mathcal{O}(\\delta a^2)&#8217; align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/cd73a867ddfd4ca7bd780b8708d3cdba.png' title='<h'>= < H > + i\\delta a \\langle a|[\\hat q,\\hat H]|a\\rangle  + \\mathcal{O}(\\delta a^2)&#8217; alt=&#8217;<h'>= < H > + i\\delta a \\langle a|[\\hat q,\\hat H]|a\\rangle  + \\mathcal{O}(\\delta a^2)&#8217; align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnisto \u00e9, se <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c30af7a1f3f91398b619c83c8505372c.png' title='[\\hat q,\\hat H]=0' alt='[\\hat q,\\hat H]=0' align=absmiddle>, em uma transforma\u00e7\u00e3o infinitesimal, <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/5b20fbecf8cd21cc10b3d9daecdf3025.png' title='< H > \\rightarrow <h'> = < H > + \\mathcal{O}(\\delta a^2)&#8217; alt=&#8217;< H > \\rightarrow <h'> = < H > + \\mathcal{O}(\\delta a^2)&#8217; align=absmiddle>, o sistema \u00e9 invariante pela transforma\u00e7\u00e3o gerada pelo operador <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/dddadfc73bb8f7b216dc844db56d9eb9.png' title='\\hat q' alt='\\hat q' align=absmiddle>, evidenciando uma simetria no sistema por esta transforma\u00e7\u00e3o. Isso implica, como vimos, que <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/11c568f3a51589f202bd522da1685972.png' title='< q >&#8216; alt=&#8217;< q >&#8216; align=absmiddle> \u00e9 uma constante do movimento, uma grandeza conservada. Apesar de n\u00f3s termos mostrado esta rela\u00e7\u00e3o at\u00e9 primeira ordem na expans\u00e3o do operador de transforma\u00e7\u00e3o, esta condi\u00e7\u00e3o \u00e9 exata. Isto \u00e9 f\u00e1cil de ser demonstrado, bastando calcular o valor esperado da hamiltoniana para o operador na sua forma exponencial, aplicando o teorema de Baker-Hausdorff. Fa\u00e7am isso como exerc\u00edcio.n<br \/>n<br \/>nQuando n\u00f3s fizemos a expans\u00e3o do operador transforma\u00e7\u00e3o em (40) n\u00f3s deixamos impl\u00edcito que uma transforma\u00e7\u00e3o qualquer pode ser decomposta em uma s\u00e9rie infinita de pequenas transforma\u00e7\u00f5es. Claro, estas transforma\u00e7\u00f5es dependem de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/f8a2102664419c44aa434cf216441a03.png' title='\\Delta a' alt='\\Delta a' align=absmiddle>, que pode ser t\u00e3o pequeno quanto queiramos. Situa\u00e7\u00f5es como estas s\u00e3o classificadas como transforma\u00e7\u00f5es cont\u00ednuas. Por\u00e9m, nem todas as transforma\u00e7\u00f5es s\u00e3o cont\u00ednuas, algumas delas (por exemplo, a paridade) n\u00e3o podem ser descritas como uma s\u00e9rie de transforma\u00e7\u00f5es infinitesimais e s\u00e3o normalmente chamadas de transforma\u00e7\u00f5es discretas. Vamos tratar desse tipo de transforma\u00e7\u00e3o em breve.n<br \/>n<br \/>nnVamos agora considerar a seguinte situa\u00e7\u00e3o: imagine um sistema f\u00edsico qualquer que possa ser representado pela Eq. de Schr\u00f6dinger, abaixo:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/2636deb9e999ae8140f296eba6d030d6.png' title='\\hat H \\psi(\\vec{r}) = -\\frac{1}{2m}\\nabla ^2 \\psi(\\vec{r}) + V(\\vec{r})\\psi(\\vec{r}) = E\\psi(\\vec{r})' alt='\\hat H \\psi(\\vec{r}) = -\\frac{1}{2m}\\nabla ^2 \\psi(\\vec{r}) + V(\\vec{r})\\psi(\\vec{r}) = E\\psi(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnImagine agora uma transforma\u00e7\u00e3o do tipo: nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/85de99f12d7a0c1587736a89cb30ae2c.png' title='\\psi(\\vec{r}) \\rightarrow \\psi'(\\vec{r}) = \\exp(i\\alpha)\\psi(\\vec{r})' alt='\\psi(\\vec{r}) \\rightarrow \\psi'(\\vec{r}) = \\exp(i\\alpha)\\psi(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnonde <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/7b7f9dbfea05c83784f8b85149852f08.png' title='\\alpha' alt='\\alpha' align=absmiddle> \u00e9 uma constante qualquer. Claramente esta transforma\u00e7\u00e3o, uma simples mudan\u00e7a de fase, mant\u00e9m (47) invariante. Esta transforma\u00e7\u00e3o se aplica igualmente em todos os pontos espaciais-temporais e, como vimos, uma transforma\u00e7\u00e3o deste tipo, quando mant\u00e9m o sistema invariante, est\u00e1 relacionada a conserva\u00e7\u00e3o de um n\u00famero qu\u00e2ntico do sistema. Transforma\u00e7\u00f5es deste tipo, que s\u00e3o aplicadas igualmente em todo o espa\u00e7o-tempo, s\u00e3o chamadas de transforma\u00e7\u00f5es globais.<br \/>n<br \/>nPor outro lado, se a amplitude da transforma\u00e7\u00e3o depender das coordenadas espa\u00e7o-temporais, isto \u00e9, sua magnitude depender de cada ponto do espa\u00e7o, a transforma\u00e7\u00e3o \u00e9 denominada de local. Seja, por exemplo, a transforma\u00e7\u00e3o:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/d839cb47a748cddd9f91f5a28802fd2a.png' title='\\psi(\\vec{r}) \\rightarrow \\psi'(\\vec{r}) = \\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r})' alt='\\psi(\\vec{r}) \\rightarrow \\psi'(\\vec{r}) = \\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnPara verificar se esta transforma\u00e7\u00e3o mant\u00e9m (47) sim\u00e9trica, basta eu verificar se eu posso isolar a transforma\u00e7\u00e3o de fase no gradiente deste estado, ou seja :nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/6e77495048899ae14e906ac9c420c6a6.png' title='\\vec{\\nabla}\\left( \\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r}) \\right) = \\exp(i\\alpha(\\vec{r}))\\left( i\\psi(\\vec{r})\\vec{\\nabla}\\alpha(\\vec{r}) +  \\vec{\\nabla}\\psi(\\vec{r}) \\right) \\ne \\exp(i\\alpha(\\vec{r}))\\vec{\\nabla}\\psi(\\vec{r})' alt='\\vec{\\nabla}\\left( \\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r}) \\right) = \\exp(i\\alpha(\\vec{r}))\\left( i\\psi(\\vec{r})\\vec{\\nabla}\\alpha(\\vec{r}) +  \\vec{\\nabla}\\psi(\\vec{r}) \\right) \\ne \\exp(i\\alpha(\\vec{r}))\\vec{\\nabla}\\psi(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnPor conta da depend\u00eancia de <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/c1d7aa10c5d94ad9d197a49d9901fc0c.png' title='\\alpha(\\vec{r})' alt='\\alpha(\\vec{r})' align=absmiddle> com as coordenadas espaciais, (47) n\u00e3o mant\u00e9m a simetria do sistema. Neste caso dizemos que a Eq. de Schr\u00f6dinger n\u00e3o mant\u00e9m a simetria por uma transforma\u00e7\u00e3o de fase local. Para tornar o sistema sim\u00e9trico por transforma\u00e7\u00f5es locais \u00e9 necess\u00e1rio introduzir, deliberadamente, for\u00e7as reais no problema. Neste problema especial, isto pode ser feito modificando, \u00e0 m\u00e3o, o operador gradiente de tal forma que:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/63d1a16f0bbd5c516c40da2d16e82d9f.png' title='\\vec{\\nabla}\\rightarrow \\vec{\\nabla} - i\\vec{A}(\\vec{r})' alt='\\vec{\\nabla}\\rightarrow \\vec{\\nabla} - i\\vec{A}(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center> nnEstabelecendo a condi\u00e7\u00e3o de que o termo <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/5918939fd2abb6b651362d7659864d0c.png' title='\\vec{A}(\\vec{r})' alt='\\vec{A}(\\vec{r})' align=absmiddle> se transforme na forma:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/2ce053486c8c8bc45c6eace377ccb721.png' title='\\vec{A}(\\vec{r}) \\rightarrow \\vec{A}(\\vec{r})+\\vec{\\nabla}\\alpha(\\vec{r})' alt='\\vec{A}(\\vec{r}) \\rightarrow \\vec{A}(\\vec{r})+\\vec{\\nabla}\\alpha(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnVamos verificar se estas modifica\u00e7\u00f5es mant\u00e9m a simetria no sistema. Vamos ent\u00e3o calcular a transforma\u00e7\u00e3o no nosso novo operador gradiente, ou seja:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/e7d85d4bbe3d62ab1456e69b4a30db9d.png' title='\\begin{matrix}rn(\\vec{\\nabla} - i\\vec{A}(\\vec{r}))\\psi(\\vec{r}) &#038; \\rightarrow \\left(\\vec{\\nabla} - i(\\vec{A}(\\vec{r})+\\vec{\\nabla}\\alpha(\\vec{r}) )\\right)\\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r}) \\\\ rn &#038; = \\exp(i\\alpha(\\vec{r})) (\\vec{\\nabla} - i\\vec{A}(\\vec{r}))\\psi(\\vec{r})rn\\end{matrix}' alt='\\begin{matrix}rn(\\vec{\\nabla} - i\\vec{A}(\\vec{r}))\\psi(\\vec{r}) &#038; \\rightarrow \\left(\\vec{\\nabla} - i(\\vec{A}(\\vec{r})+\\vec{\\nabla}\\alpha(\\vec{r}) )\\right)\\exp(i\\alpha(\\vec{r}))\\psi(\\vec{r}) \\\\ rn &#038; = \\exp(i\\alpha(\\vec{r})) (\\vec{\\nabla} - i\\vec{A}(\\vec{r}))\\psi(\\vec{r})rn\\end{matrix}' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnOu seja, se escrevermos a Eq. de Schr\u00f6dinger modificada, de forma que:nn<center> <br style=\"line-height: .75em;\" >n            \n\n<table class=\"numbered-eq\" border=\"0\" width = \"99%\">n            \n\n<tr>n              \n\n<td><center><img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/e5548af4a4e37334349d41f08abfdc4d.png' title='\\hat H \\psi(\\vec{r}) = -\\frac{1}{2m}\\left(\\nabla ^2 -i\\vec{A}(\\vec{r})\\right) \\psi(\\vec{r}) + V(\\vec{r})\\psi(\\vec{r}) = E\\psi(\\vec{r})' alt='\\hat H \\psi(\\vec{r}) = -\\frac{1}{2m}\\left(\\nabla ^2 -i\\vec{A}(\\vec{r})\\right) \\psi(\\vec{r}) + V(\\vec{r})\\psi(\\vec{r}) = E\\psi(\\vec{r})' align=absmiddle><\/center><\/td>\n\nn              \n\n<td width = \"30\" ><center> <q class=\"numbered-eq\"><\/q><\/center><\/td>\n\nn            <\/tr>\n\nn            <\/table>\n\n <br style=\"line-height: .75em;\" >n            <\/center>nnnela se mant\u00e9m sim\u00e9trica por transforma\u00e7\u00f5es locais. Para recuperar esta simetria foi necess\u00e1rio introduzir um termo de potencial vetorial, <img src='http:\/\/picard.if.usp.br\/sampa\/blog\/pivotx\/extensions\/renderlatex\/pictures\/5918939fd2abb6b651362d7659864d0c.png' title='\\vec{A}(\\vec{r})' alt='\\vec{A}(\\vec{r})' align=absmiddle>. Este termo \u00e9 bastante similar ao potencial vetor encontrado nas Eq. Maxwell para o eletromagnetismo. Esta introdu\u00e7\u00e3o \u00e9 interpretada como a necessidade de introduzir novos campos na hamiltoniana do sistema de modo a manter o sistema sim\u00e9trico por transforma\u00e7\u00f5es locais. Estes campos s\u00e3o denominados campos de calibre e levam \u00e0 introdu\u00e7\u00e3o de for\u00e7as f\u00edsicas reais no problema. <br \/>n<br \/>nEm um problema f\u00edsico mais complexo, sujeito a transforma\u00e7\u00f5es locais mais elaboradas, pode ser necess\u00e1rio introduzir v\u00e1rios campos de calibre para preservar a simetria do sistema por estas transforma\u00e7\u00f5es locais. Isso implica na adi\u00e7\u00e3o de for\u00e7as f\u00edsicas adicionais no sistema para manter estas simetrias. Podemos pensar da maneira invertida: for\u00e7as f\u00edsicas s\u00e3o originadas em um sistema de modo a preservar a simetria desses sistemas em transforma\u00e7\u00f5es locais. Este princ\u00edpio, conhecido como princ\u00edpio de calibre (ou, do Ingl\u00eas, <i>Gauge Principle<\/i>) \u00e9 fundamental na constru\u00e7\u00e3o de teorias f\u00edsicas para as intera\u00e7\u00f5es fundamentais.nn\n\n<h2>Leitura recomendada<\/h2>\n\nn\n\n<ol>n\n\n<li> Cap\u00edtulo X do livro &#8220;Introducion to Nuclear and Particle Physics&#8221;, Ashok Das e Thomas Ferbel.n\n\n<li> Cap\u00edtulo 4 do livro &#8220;Introduction to Elementary Particles&#8221;, David Griffths.n<\/ol>\n\nn<br \/>n\n\n<h2>Exerc\u00edcios<\/h2>\n\nn\n\n<ol>n\n\n<li> Mostre (32)n\n\n<li> Mostre (53)n<\/ol>\n\n ]]><\/p>\n","protected":false},"excerpt":{"rendered":"<p>nnVers\u00e3o para impress\u00e3on nn Notas de aula s\u00e3o propriedade intelectual. Sendo assim, qualquer uso, no todo ou em parte, deve ter a origem referenciada apropriadamente, ap\u00f3s autoriza\u00e7\u00e3o de seu autor.n nn O texto a seguir corresponde \u00e0 anota\u00e7\u00f5es parciais de aula. N\u00e3o \u00e9 um texto em forma final, completo e totalmente revisado. Nesse caso, esse [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-485","post","type-post","status-publish","format-standard","hentry","category-notas_particulas"],"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"suaide","author_link":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/author\/suaide\/"},"uagb_comment_info":0,"uagb_excerpt":"nnVers\u00e3o para impress\u00e3on nn Notas de aula s\u00e3o propriedade intelectual. Sendo assim, qualquer uso, no todo ou em parte, deve ter a origem referenciada apropriadamente, ap\u00f3s autoriza\u00e7\u00e3o de seu autor.n nn O texto a seguir corresponde \u00e0 anota\u00e7\u00f5es parciais de aula. N\u00e3o \u00e9 um texto em forma final, completo e totalmente revisado. Nesse caso, esse&hellip;","_links":{"self":[{"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/485","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/comments?post=485"}],"version-history":[{"count":1,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/485\/revisions"}],"predecessor-version":[{"id":498,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/485\/revisions\/498"}],"wp:attachment":[{"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/media?parent=485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/categories?post=485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/picard.if.usp.br\/wordpress\/index.php\/wp-json\/wp\/v2\/tags?post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}