Gases quânticos sem interação

UFABC - Mecânica Estatística Curso 2018.1 Prof. Germán Lugones CAPÍTULO 6 Gases quânticos sem interação!1

Regime clássico e regime quântico Para um gás ideal clássico em equilíbrio térmico a uma temperatura T, a energia média de uma partícula é E=(3/2)kT e o comprimento de onda de de Broglie é B = h p = Por outro lado, o comprimento de onda térmico = h p 2mE = h p 2 mkt. h p 3mkT. é da mesma ordem de grandeza que! B. Por isso, tanto! quanto! B dão uma ideia da ``extensão espacial'' ou tamanho de uma partícula do gás. Para grandes temperaturas as partículas se tornam localizadas e tendem a ter um comportamento clássico. A baixas temperaturas as partículas ficam mais deslocalizadas" e se a densidade for grande o suficiente podemos ter um overlapping entre as suas funções de onda.!2

A distância média entre partículas pode ser estimada por d = v 1/3 = n 1/3 onde n=n/v é o número de partículas por unidade de volume, e v 1/n é o volume específico (volume ocupado por uma partícula do gás). A relação entre as grandezas d e! permite identificar os regimes clássico e quântico: Regime clássico. O tamanho das partículas é muito menor que a separação entre as mesmas, i.e.,! n -1/3, logo, 3 n 1!3

Regime quântico. O tamanho das partículas é da ordem ou maior que a separação entre as mesmas, i.e.,! n -1/3, logo, 3 n & 1!4

Bósons e Férmions Consideremos um gás quântico composto por N partículas livres sem spin (bósons ou férmions). Atenção! Obviamente, não existem férmions sem spin na natureza. O conceito de férmions sem spin indica apenas que temos partículas que verificam o principio de exclusão de Pauli, e que o número de partículas possível em cada nível de energia é 0 ou 1. Para descrever férmions reais, basta introduzir um fator de degenerescência que leve em consideração o número de partículas permitido em cada nível de energia. Como as partículas são não-interagentes, o Hamiltoniano do sistema de N partículas é a soma de N Hamiltonianos de uma partícula, Ĥ(q, p) = NX ĥ i (q i,p i ) onde os operadores ĥ i são formalmente idênticos, e diferem apenas nas coordenadas e momentos q i, p i.!5 i=1

A equação de Schrödinger independente do tempo para esse sistema é onde " E (q) são as autofunções de energia e E os correspondentes autovalores. Pela forma do hamiltoniano, um estado estacionário do sistema de N partículas pode ser descrito em função dos autoestados do hamiltoniano de uma partícula. A função de onda tem a forma Y NY E(q) = i (q i ), Eq. (*) e os autovalores ficam da forma Ĥ(q, p) E (q) =E E (q) onde φ ε i e εi são as autofunções e autovalores do hamiltoniano de uma partícula, i.e. ĥ i φ # i(q i ) = ε i φ # i(q i ).!6 i=1 XE = NX i, i=1

Adicionalmente, devemos levar em consideração a indistinguibilidade das partículas: Todos os microestados resultantes das permutações de N partículas devem ser considerados como um único estado. Por isso, a função de onda de um sistema de N partículas deve ser simétrica ou anti-simétrica pela troca das coordenadas de qualquer par de partículas (i.e ao fazermos ""* obteremos a mesma densidade de probabilidade). As partículas que são descritas por uma função de onda simétrica são denominadas bósons e as descritas por uma função de onda anti-simétrica são denominadas férmions.!7

Portanto, a função de onda apresentada anteriormente [na Eq. (*)] deve ser simetrizada ou anti-simetrizada em relação às coordenadas q i, ou, equivalentemente, em relação aos índices i. A função de onda resultante é da forma onde P é um operador que permuta os índices da funções de onda de uma partícula. X A soma deve ser realizada sobre as N! permutações possíveis. O fator $ P é E(q 1,...,q N )= 1 p N! X P P P [ 1 (q 1 )... N (q N )], P = 1 (para bósons), ( 1 se P é uma permutação par P = 1 se P é uma permutação impar (para férmions).!8

No caso dos férmions, a função de onda adota a forma do determinante de Slater E(q 1,...,q N )= p 1 2 (q 1 ) 2 (q 2 )... 2 (q N ) N!...... 1(q 1) 1(q 2)... 1(q N) N (q 1 ) N (q 2 )... N (q N ). Note que cada fila contém sempre a mesma função de onda de uma partícula, enquanto cada coluna contém o mesmo argumento na função de onda de uma partícula. A partir desta expressão, fica evidente que a função de onda " E é identicamente nula se duas funções de onda de uma partícula forem iguais, já que o determinante teria duas filas idênticas. Este é o princípio de exclusão de Pauli.!9

Base de Fock Construiremos um formalismo que nos permitirá levar em conta o princípio de Pauli sem precisar rotular as partículas indistinguíveis. O Hamiltoniano de partículas não-interagentes no espaço de Hilbert de N partículas, E H (N), tem a forma de de uma soma de operadores idênticos que se referem respectivamente às partículas i = 1,2,...,N. N!10

<latexit sha1_base64="ojhefi3tz9zi46of31lo9wbklxm=">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</latexit> Para evitar as complicações devidas à simetrização ou antissimetrização das funções de onda, trabalharemos no espaço de Fock deixando o número de partículas N arbitrário. Lembremos que estamos lidando com o espaço de Fock 1M E H = N=0 E (N) H que é a soma direta de espaços de Hilbert de N partículas indistinguíveis (bósons ou férmions). Usaremos a mesma notação para esses dois tipos de partículas, mesmo que os espaços correspondentes não sejam equivalentes (exceto no caso N 2).!11

Single-Particle States: We start from the Hilbert space E (1) H of the single-particle states. In contrast to the complete Fock space, this space has the same structure for fermions as for bosons: the nature of the particles does not manifest itself until there are several particles present. We choose for the base {% i } of the space E H (1) the set of eigenvectors of the single-particle Hamiltonian ĥ. We indicate the set of quantum numbers which characterize each of these vectors by % i, and the value of the corresponding energy eigenvalue by & i.!12

Estados de Partículas Individuais:: Analisemos o espaço de Hilbert E H (1) dos estados de uma única partícula. Em contraste com o espaço Fock completo, este espaço tem a mesma estrutura para férmions e bósons: a natureza das partículas não se manifesta até que haja várias partículas presentes. Para a base {% i } do espaço E (1) H escolhemos o conjunto de autovetores do Hamiltoniano ĥ de uma única partícula. Cada um dos elementos da base tem associado um conjunto de números quânticos. O autovalor de energia correspondente é & i.!13

Números de ocupação: Para descrever um estado de N bósons ou de N férmions adotamos a seguinte estratégia. - vamos olhar para o conjunto de estados de uma partícula {% 1, % 2, % 3,,% } e especificar qual deles está ocupado e qual deles está vazio. - isto definirá para cada estado de uma partícula, % i, um número de ocupação n i, que pode ser zero. Descrição do vácuo: O vácuo é um estado (único) do espaço de Hilbert de zero partículas E H (0). No formalismo de números de ocupação, ele é descrito como o estado no qual todos os n i são nulos.!14

Descrição de estados de uma única partícula: Como descrevemos os estados de uma partícula, pertencentes ao espaço de Hilbert E H (1)? Para expressar que a partícula está no estado q, usamos a seguinte linguagem: - dizemos que o número de ocupação n i do estado % i é igual a 1 e que os números de ocupação n j dos outros estados % j (j i) são todos zero. - Cada estado de uma partícula da base {%} será então denotado por um conjunto de números de ocupação {n i } que são todos iguais a 0, exceto um deles que é igual a 1. Portanto, o ket % i que representa a partícula no estado % i, é escrito como: 0, 0,, {n q }=1, 0, 0, 0 onde organizamos os estados de uma única partícula, % i, em uma ordem padrão e onde os números de ocupação são escritos nessa ordem padrão.!15

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Para partículas não interagentes, o Hamiltoniano é da forma: Ĥ (2) = ĥ 1 + ĥ 2 Se desconsiderarmos a indistinguibilidade, as autofunções de Ĥ (2) são simplesmente o produto de duas funções, %(1) para a partícula 1 e %'(2) para a partícula 2, e a energia é & + &. No caso de férmions, o espaço de Hilbert E H (2) é o espaço de Hilbert das funções anti-simétricas. A função de onda tem a forma 1 p 2 [ (1) 0 (2) 0 (1) (2)] e é representada por 0,, 0,n =1, 0,, 0,n 0 =1, 0, i!16

<latexit sha1_base64="oemftyr9kccbngkma9tterokmxw=">aaaco3icbvdlsgmxfm34rpvvdekmwaqxpcyiobuh6mzlffuati2z9e4bmskmsuyo4/yxg3/cnrs3lhrx6950wkhbxkg4oedcbu7xis6utu0xa2fxaxllnbewx9/y3nou7ozwvrhlcjua8la2pakamwe1ztshzisbbb6hhje4homne5ckhejwdynob6qnmm8o0ybqfg4esf3clu2gwpvguhqsn+qz9nzjnnoko3nlfkd66xlbscuj6hhanulrlttz4vngteartarakty73zdgaqhnovgq5dirbideaky5phk3vharoia9abkosacqnws7p/jqmf3sh9icoxhg/u1iskdumpcmmyc6r6a1etlpa8xap2sntesxbkhhg/yyyx3iuzc4yyrqzycgecqz+sumfsij1sbuvanbmv55ftspy45ddq5pipwlsrw5ti8o0bfy0cmqocturtve0sn6re/ow3qy3qxp62tsxbampxvox1nfp03dqiy=</latexit> <latexit sha1_base64="oemftyr9kccbngkma9tterokmxw=">aaaco3icbvdlsgmxfm34rpvvdekmwaqxpcyiobuh6mzlffuati2z9e4bmskmsuyo4/yxg3/cnrs3lhrx6950wkhbxkg4oedcbu7xis6utu0xa2fxaxllnbewx9/y3nou7ozwvrhlcjua8la2pakamwe1ztshzisbbb6hhje4homne5ckhejwdynob6qnmm8o0ybqfg4esf3clu2gwpvguhqsn+qz9nzjnnoko3nlfkd66xlbscuj6hhanulrlttz4vngteartarakty73zdgaqhnovgq5dirbideaky5phk3vharoia9abkosacqnws7p/jqmf3sh9icoxhg/u1iskdumpcmmyc6r6a1etlpa8xap2sntesxbkhhg/yyyx3iuzc4yyrqzycgecqz+sumfsij1sbuvanbmv55ftspy45ddq5pipwlsrw5ti8o0bfy0cmqocturtve0sn6re/ow3qy3qxp62tsxbampxvox1nfp03dqiy=</latexit> <latexit sha1_base64="oemftyr9kccbngkma9tterokmxw=">aaaco3icbvdlsgmxfm34rpvvdekmwaqxpcyiobuh6mzlffuati2z9e4bmskmsuyo4/yxg3/cnrs3lhrx6950wkhbxkg4oedcbu7xis6utu0xa2fxaxllnbewx9/y3nou7ozwvrhlcjua8la2pakamwe1ztshzisbbb6hhje4homne5ckhejwdynob6qnmm8o0ybqfg4esf3clu2gwpvguhqsn+qz9nzjnnoko3nlfkd66xlbscuj6hhanulrlttz4vngteartarakty73zdgaqhnovgq5dirbideaky5phk3vharoia9abkosacqnws7p/jqmf3sh9icoxhg/u1iskdumpcmmyc6r6a1etlpa8xap2sntesxbkhhg/yyyx3iuzc4yyrqzycgecqz+sumfsij1sbuvanbmv55ftspy45ddq5pipwlsrw5ti8o0bfy0cmqocturtve0sn6re/ow3qy3qxp62tsxbampxvox1nfp03dqiy=</latexit> <latexit sha1_base64="oemftyr9kccbngkma9tterokmxw=">aaaco3icbvdlsgmxfm34rpvvdekmwaqxpcyiobuh6mzlffuati2z9e4bmskmsuyo4/yxg3/cnrs3lhrx6950wkhbxkg4oedcbu7xis6utu0xa2fxaxllnbewx9/y3nou7ozwvrhlcjua8la2pakamwe1ztshzisbbb6hhje4homne5ckhejwdynob6qnmm8o0ybqfg4esf3clu2gwpvguhqsn+qz9nzjnnoko3nlfkd66xlbscuj6hhanulrlttz4vngteartarakty73zdgaqhnovgq5dirbideaky5phk3vharoia9abkosacqnws7p/jqmf3sh9icoxhg/u1iskdumpcmmyc6r6a1etlpa8xap2sntesxbkhhg/yyyx3iuzc4yyrqzycgecqz+sumfsij1sbuvanbmv55ftspy45ddq5pipwlsrw5ti8o0bfy0cmqocturtve0sn6re/ow3qy3qxp62tsxbampxvox1nfp03dqiy=</latexit> <latexit sha1_base64="eny1f5yynu0bilbv8dkpkznfmek=">aaacmxicbvdns8mwhe39npor6tflcagbwmihomehlx0nua9o60izdatl2pqkwij9l7z4n4ixhrtx6j9huvwgmw9chu+9h8nv+tgjulnwzfhb39jc2i7tlhf39g8ozapjrowsgukhrywsfr9jwmhioooqrvqxiij7jpt8yw3u956ikdqk79u0jh5ho5agfcolpyhzcgobcgpnqssfhuobwqydnx7tql2d+f2qurggngtvrg1elelfrlvewkxydwsouersglragfbafhwheu44crvmserhtmllpugoihnjym4isyzwbi2io2miojfeot84g+dagcigevqecs7v3xmp4ljoua+thkmxxpzy8t/psvrw7au0jbnfqrx4kegyvbhm64ndkghwbkojwolqv0i8rrpcpusu6xls5zvxsbdrt626fxdzad4udztaktgdvwcdk9aeldaghydbm3gd7+ddedfmxqfxtyiugcxmcfgd4/shlpqogw==</latexit> <latexit sha1_base64="eny1f5yynu0bilbv8dkpkznfmek=">aaacmxicbvdns8mwhe39npor6tflcagbwmihomehlx0nua9o60izdatl2pqkwij9l7z4n4ixhrtx6j9huvwgmw9chu+9h8nv+tgjulnwzfhb39jc2i7tlhf39g8ozapjrowsgukhrywsfr9jwmhioooqrvqxiij7jpt8yw3u956ikdqk79u0jh5ho5agfcolpyhzcgobcgpnqssfhuobwqydnx7tql2d+f2qurggngtvrg1elelfrlvewkxydwsouersglragfbafhwheu44crvmserhtmllpugoihnjym4isyzwbi2io2miojfeot84g+dagcigevqecs7v3xmp4ljoua+thkmxxpzy8t/psvrw7au0jbnfqrx4kegyvbhm64ndkghwbkojwolqv0i8rrpcpusu6xls5zvxsbdrt626fxdzad4udztaktgdvwcdk9aeldaghydbm3gd7+ddedfmxqfxtyiugcxmcfgd4/shlpqogw==</latexit> <latexit sha1_base64="eny1f5yynu0bilbv8dkpkznfmek=">aaacmxicbvdns8mwhe39npor6tflcagbwmihomehlx0nua9o60izdatl2pqkwij9l7z4n4ixhrtx6j9huvwgmw9chu+9h8nv+tgjulnwzfhb39jc2i7tlhf39g8ozapjrowsgukhrywsfr9jwmhioooqrvqxiij7jpt8yw3u956ikdqk79u0jh5ho5agfcolpyhzcgobcgpnqssfhuobwqydnx7tql2d+f2qurggngtvrg1elelfrlvewkxydwsouersglragfbafhwheu44crvmserhtmllpugoihnjym4isyzwbi2io2miojfeot84g+dagcigevqecs7v3xmp4ljoua+thkmxxpzy8t/psvrw7au0jbnfqrx4kegyvbhm64ndkghwbkojwolqv0i8rrpcpusu6xls5zvxsbdrt626fxdzad4udztaktgdvwcdk9aeldaghydbm3gd7+ddedfmxqfxtyiugcxmcfgd4/shlpqogw==</latexit> <latexit sha1_base64="eny1f5yynu0bilbv8dkpkznfmek=">aaacmxicbvdns8mwhe39npor6tflcagbwmihomehlx0nua9o60izdatl2pqkwij9l7z4n4ixhrtx6j9huvwgmw9chu+9h8nv+tgjulnwzfhb39jc2i7tlhf39g8ozapjrowsgukhrywsfr9jwmhioooqrvqxiij7jpt8yw3u956ikdqk79u0jh5ho5agfcolpyhzcgobcgpnqssfhuobwqydnx7tql2d+f2qurggngtvrg1elelfrlvewkxydwsouersglragfbafhwheu44crvmserhtmllpugoihnjym4isyzwbi2io2miojfeot84g+dagcigevqecs7v3xmp4ljoua+thkmxxpzy8t/psvrw7au0jbnfqrx4kegyvbhm64ndkghwbkojwolqv0i8rrpcpusu6xls5zvxsbdrt626fxdzad4udztaktgdvwcdk9aeldaghydbm3gd7+ddedfmxqfxtyiugcxmcfgd4/shlpqogw==</latexit> No caso de bósons, o espaço de Hilbert E H (2) é o espaço de Hilbert das funções simétricas. A função de onda tem a forma 1 p 2 [ (1) 0 (2) + 0 (1) (2)] e é representada por 0,, 0,n =1, 0,, 0,n 0 =1, 0, i O uso da mesma notação para o estado antissimétrico e para o estado simétrico não deve levar a qualquer confusão, desde que deixemos claro desde o início se estamos lidando com férmions ou com bósons.!17

<latexit sha1_base64="lal/jaopfft/9xz5h1p4irigbfu=">aaacghicbzbns8mwgmdtx+d8q3r0ehychzhbiehfghrxomg9wfpkmqzbwjqwjbvg3cfw4lfx4kerr7v5buy7hntzgcav///zkdx/p2fuksv6nlzw19y3nitb1e2d3b198+cwk+nuynlbmytf30esmmpjr1hfsd8rbeu+iz1/fjv7vuciji35g5okxi3qknoqyqs05jnnt9cqqwchszl1hlmxocmitq+bxbw0ocmqhzicpbnmnayi4dlyjdraww3pndlbjnoiciuzknjgw4lymyquxyxmq04qsylwga3jqcnhezfuviw2hadacwayc324gox6eyjdkzstynedevijuejl4n/eifxhlztrnqskcdx/kewzvdhmu4ibfqqrntgaskd6rxcpkeby6syrogr7cevl6dybttww7y9qrzsyjgo4bifgdnjgertahwiddsdggbycd/bhvbhvxqfxnw9dmcqzi/cnjnkpxcydiq==</latexit> <latexit sha1_base64="lal/jaopfft/9xz5h1p4irigbfu=">aaacghicbzbns8mwgmdtx+d8q3r0ehychzhbiehfghrxomg9wfpkmqzbwjqwjbvg3cfw4lfx4kerr7v5buy7hntzgcav///zkdx/p2fuksv6nlzw19y3nitb1e2d3b198+cwk+nuynlbmytf30esmmpjr1hfsd8rbeu+iz1/fjv7vuciji35g5okxi3qknoqyqs05jnnt9cqqwchszl1hlmxocmitq+bxbw0ocmqhzicpbnmnayi4dlyjdraww3pndlbjnoiciuzknjgw4lymyquxyxmq04qsylwga3jqcnhezfuviw2hadacwayc324gox6eyjdkzstynedevijuejl4n/eifxhlztrnqskcdx/kewzvdhmu4ibfqqrntgaskd6rxcpkeby6syrogr7cevl6dybttww7y9qrzsyjgo4bifgdnjgertahwiddsdggbycd/bhvbhvxqfxnw9dmcqzi/cnjnkpxcydiq==</latexit> <latexit sha1_base64="lal/jaopfft/9xz5h1p4irigbfu=">aaacghicbzbns8mwgmdtx+d8q3r0ehychzhbiehfghrxomg9wfpkmqzbwjqwjbvg3cfw4lfx4kerr7v5buy7hntzgcav///zkdx/p2fuksv6nlzw19y3nitb1e2d3b198+cwk+nuynlbmytf30esmmpjr1hfsd8rbeu+iz1/fjv7vuciji35g5okxi3qknoqyqs05jnnt9cqqwchszl1hlmxocmitq+bxbw0ocmqhzicpbnmnayi4dlyjdraww3pndlbjnoiciuzknjgw4lymyquxyxmq04qsylwga3jqcnhezfuviw2hadacwayc324gox6eyjdkzstynedevijuejl4n/eifxhlztrnqskcdx/kewzvdhmu4ibfqqrntgaskd6rxcpkeby6syrogr7cevl6dybttww7y9qrzsyjgo4bifgdnjgertahwiddsdggbycd/bhvbhvxqfxnw9dmcqzi/cnjnkpxcydiq==</latexit> <latexit sha1_base64="lal/jaopfft/9xz5h1p4irigbfu=">aaacghicbzbns8mwgmdtx+d8q3r0ehychzhbiehfghrxomg9wfpkmqzbwjqwjbvg3cfw4lfx4kerr7v5buy7hntzgcav///zkdx/p2fuksv6nlzw19y3nitb1e2d3b198+cwk+nuynlbmytf30esmmpjr1hfsd8rbeu+iz1/fjv7vuciji35g5okxi3qknoqyqs05jnnt9cqqwchszl1hlmxocmitq+bxbw0ocmqhzicpbnmnayi4dlyjdraww3pndlbjnoiciuzknjgw4lymyquxyxmq04qsylwga3jqcnhezfuviw2hadacwayc324gox6eyjdkzstynedevijuejl4n/eifxhlztrnqskcdx/kewzvdhmu4ibfqqrntgaskd6rxcpkeby6syrogr7cevl6dybttww7y9qrzsyjgo4bifgdnjgertahwiddsdggbycd/bhvbhvxqfxnw9dmcqzi/cnjnkpxcydiq==</latexit> <latexit sha1_base64="ajby9qq/zoqzvj7zqxpgmnqp8oc=">aaab+xicbvbns8naej3ur1q/oh69lbahvzskchosevfywbsfnptndtmu3wzc7qzqqv+jfw+kepwfeppfue1z0nyhwzzem2fnx5bwprtjffulre2d3b3yfuxg8oj4xd4966g4lyr6joax7avyuc4e9tttnpyssxeucnonpvdlvzujurfypol5qv0ijwulgchasepbhiqtvnprko/nohravafh5ecbxc1ifqq0h/bxybstnkjce46v6rtoov0ms80ip4vkifu0wwskx7rvqmarvx6wx75av0yzotcwporgufp7i8oruvmomjmr1ho17i3f/7x+qsnbp2mistuvzpvqmhkky7smay2ypetzusgysgzurwscjsbahfuxibjrx94knwbddrru43w1dvfeuyyluiqauhadlxiannhayabp8apvvma9wo/wx2q0zbu75/ah1ucpyp6ryg==</latexit> <latexit sha1_base64="ajby9qq/zoqzvj7zqxpgmnqp8oc=">aaab+xicbvbns8naej3ur1q/oh69lbahvzskchosevfywbsfnptndtmu3wzc7qzqqv+jfw+kepwfeppfue1z0nyhwzzem2fnx5bwprtjffulre2d3b3yfuxg8oj4xd4966g4lyr6joax7avyuc4e9tttnpyssxeucnonpvdlvzujurfypol5qv0ijwulgchasepbhiqtvnprko/nohravafh5ecbxc1ifqq0h/bxybstnkjce46v6rtoov0ms80ip4vkifu0wwskx7rvqmarvx6wx75av0yzotcwporgufp7i8oruvmomjmr1ho17i3f/7x+qsnbp2mistuvzpvqmhkky7smay2ypetzusgysgzurwscjsbahfuxibjrx94knwbddrru43w1dvfeuyyluiqauhadlxiannhayabp8apvvma9wo/wx2q0zbu75/ah1ucpyp6ryg==</latexit> <latexit sha1_base64="ajby9qq/zoqzvj7zqxpgmnqp8oc=">aaab+xicbvbns8naej3ur1q/oh69lbahvzskchosevfywbsfnptndtmu3wzc7qzqqv+jfw+kepwfeppfue1z0nyhwzzem2fnx5bwprtjffulre2d3b3yfuxg8oj4xd4966g4lyr6joax7avyuc4e9tttnpyssxeucnonpvdlvzujurfypol5qv0ijwulgchasepbhiqtvnprko/nohravafh5ecbxc1ifqq0h/bxybstnkjce46v6rtoov0ms80ip4vkifu0wwskx7rvqmarvx6wx75av0yzotcwporgufp7i8oruvmomjmr1ho17i3f/7x+qsnbp2mistuvzpvqmhkky7smay2ypetzusgysgzurwscjsbahfuxibjrx94knwbddrru43w1dvfeuyyluiqauhadlxiannhayabp8apvvma9wo/wx2q0zbu75/ah1ucpyp6ryg==</latexit> <latexit sha1_base64="ajby9qq/zoqzvj7zqxpgmnqp8oc=">aaab+xicbvbns8naej3ur1q/oh69lbahvzskchosevfywbsfnptndtmu3wzc7qzqqv+jfw+kepwfeppfue1z0nyhwzzem2fnx5bwprtjffulre2d3b3yfuxg8oj4xd4966g4lyr6joax7avyuc4e9tttnpyssxeucnonpvdlvzujurfypol5qv0ijwulgchasepbhiqtvnprko/nohravafh5ecbxc1ifqq0h/bxybstnkjce46v6rtoov0ms80ip4vkifu0wwskx7rvqmarvx6wx75av0yzotcwporgufp7i8oruvmomjmr1ho17i3f/7x+qsnbp2mistuvzpvqmhkky7smay2ypetzusgysgzurwscjsbahfuxibjrx94knwbddrru43w1dvfeuyyluiqauhadlxiannhayabp8apvvma9wo/wx2q0zbu75/ah1ucpyp6ryg==</latexit> No caso de dois bósons, existe ainda a possibilidade de colocar as duas partículas no mesmo estado %. A função de onda tem a forma e é representada por (1) (2) 0,, 0,n =2, 0, i!18

<latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> Podemos estender facilmente esses argumentos a um número de partículas arbitrário. Para N férmions, cada microestado pode estar ocupado por, no máximo, uma partícula, devido ao princípio de exclusão de Pauli. Assim, um estado qualquer tem a forma: n 1,,n i, i onde os números de ocupação podem adotar apenas os valores 0 ou 1. O número de partículas do sistema é dado por: N = X i n i e a energia da configuração caracterizada por {n i } é: E = X i n i i!19

<latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> Para N bósons, cada microestado pode estar ocupado por mais de uma partícula. Assim, um estado qualquer tem a forma: n 1,,n i, i onde os números de ocupação podem adotar qualquer valor inteiro (ou zero): n i = 0, 1, 2,.. O número de partículas do sistema é dado por: N = X i n i e a energia da configuração caracterizada por {n i } é: E = X i n i i!20

<latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit sha1_base64="uygffwnssq7n0o/wty0c8o6og+w=">aaacexicbvc7tsmwfhv4lvikmljyvegdqipbsdbwsdawit6knoocx2mtok5k3ybvob/awq+wmiaqkxsbf4pbzocwi1k+pudexd8tpijrcjxva2v1bx1js7rv3t7z3du3dw7boskuzs2aier1a6kz4jk1ging3vqxegecdylr9dtv3doleslvyjwylyydysnocrjjt6spwppudfdpmicuyfpk+ctchyl7isibyni3k07dmqeve7cgfvsg6dtf/tchwcwkueg07rlocl5ofhaq2ktczzrlcr2raeszkknmtjfpnprgu6oeoequorlwtp3dkzny63ecmmqywfavelpxp6+xqxtp5vymgtbj54oitgbi8dqehhlfkiixiyqqbv6k6zaoqsgewdyhuisrl5p2wd116u7teavxvcrrqsfobfwriy5qa92gjmohih7rm3pfb9at9wk9wx/z0hwr6dlcf2b9/gazypto</latexit> <latexit 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sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="hozyrwkcbeehlcierokqmr3q34g=">aaacbhicbvbns8naen3ur1q/oh57wsycp5kiobehkilhcvydmha220m7dlmjuxuhhb68+fe8efdeqz/cm//gbzudtj4yelw3w8y8movmacf5tkorq2vrg+xnytb2zu6evx/qvkkmkbrowhpzdykczgs0nnmcuqkeeoccouhoeup3hkaqloh7pu7bj8lasihroo0u2nubfik9lcvbzizybax7kcrgjcccu+bunrnwmnelukmfmoh95futmsugnoveqz7rpnrpidsmcphuvexbsuiidkbnqcaxkd+fpthbx0bp4yirpotgm/x3re5ipczxadpjoodq0zuk/3m9tecxfs5emmkqdl4oyjjwcz4mgvtmatv8baihkplbmr0ssag2uvvmco7iy8ukfvp3nbp7d1zrxbvxlfevhaet5kjz1ec3qilaikjh9ixe0zv1zl1y79bhvlvkftoh6a+szx+z7zea</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> <latexit sha1_base64="kndihhn++n12j0ubaagokp5dgc8=">aaab+xicbvbns8naej3ur1q/oh69lbbbu0le0itq9ojjktgpaepybdft0t1n2n0usug/8ejbea/+e2/+g7dtdtr6yodx3gwz86kum20879spra1vbg6vtys7u3v7b+7huusnmsk0srkeqe6enevm0qzhhtnoqigweaftahq389tjqjrl5jozpdqqecbzzag2vgpd9whdoj7orjizkzihc92qv/pmqkvel0gvcjrc96vxt0gmqdsey627vpeaimfkmmlptnllne0xgeeb7voqsaa6yoext9gzvfootpqtadbc/t2ry6h1res2u2az1mvetpzp62ymvg5yjtpmuekwi+kmi5ogwqyozxqlhk8swuqxeysiq6wwmtasig3bx355lbquar5x8x8vq/xbio4ynmapnimpv1che2haewim4rle4c3jnrfn3flytjacyuyy/sd5/afqqzld</latexit> Para N bósons, cada microestado pode estar ocupado por mais de uma partícula. Assim, um estado qualquer tem a forma: n 1,,n i, i onde os números de ocupação podem adotar qualquer valor inteiro (ou zero): n i = 0, 1, 2,.. O número de partículas do sistema é dado por: N = X i n i e a energia da configuração caracterizada por {n i } é: E = X i n i i!21

Partindo dos estados de uma partícula no espaço de Hilbert E H (1), conseguimos construir uma base completa para o espaço de Fock de estados com um número arbitrário de partículas E H = E H (0) E H (1) E H (2) E H (3) +. em dois casos diferentes: quando as funções de onda devem ser antisimétricas (férmions) ou quando devem ser simétricas (bósons). Cada estado desta base, é caracterizado através dos números quânticos n i (números de ocupação), cada um dos quais pode assumir os valores 0,1 para férmions, ou 0,1,2,... para bósons. A base é chamada base de Fock associada à base {% i } dos estados de uma única partícula.!22

Segunda quantização: A construção da base de Fock, a partir do espaço dos estados de um única partícula, é chamada de segunda quantização. Este nome é justificado pela existência de dois níveis de números quânticos, que são diferentes em caráter e no papel que desempenham: - Os vetores da base {% i } do espaço de Hilbert E H (1) dos estados de uma única partícula são caracterizados por certos números quânticos % de uma partícula (por exemplo m x, m y, m z = 1, 2,. para uma partícula livre numa caixa). - Quando mudamos para o espaço de Fock E H com um número arbitrário de bósons ou férmions, temos um novo conjunto de números quânticos que caracterizam os elementos da base de Fock os números de ocupação {n i }. Na representação de Fock, não é mais necessário considerar os números % como números quânticos; eles passam a desempenhar o papel de índices dos números de ocupação n.!23

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sha1_base64="shiqp+6shsy5p2ku0w4z1udpypy=">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</latexit> <latexit sha1_base64="shiqp+6shsy5p2ku0w4z1udpypy=">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</latexit> <latexit sha1_base64="shiqp+6shsy5p2ku0w4z1udpypy=">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</latexit> <latexit sha1_base64="shiqp+6shsy5p2ku0w4z1udpypy=">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</latexit> Função de partição grande canônica Calcularemos a função de partição grande canônica na base de Fock: Ĥ+ µ =Tr e ˆN = X Ĥ+ µ hn 1,,n 1 e ˆN n 1,,n 1 i n 1,,n 1 Como os estados da base são autoestados do hamiltoniano Ĥ e do operador número N, ˆ ambos operadores podem ser substituídos por seus autovalores E = i & i n i e N= i n i respectivamente: = X hn 1,,n 1 e n 1,,n 1 = X hn 1,,n 1 e n 1,,n 1 E+ µn n 1,,n 1 i P i in i + µ P i n i n 1,,n 1 i!24

<latexit sha1_base64="thfifhyxlcspar2+msgqetdleg8=">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</latexit> <latexit sha1_base64="thfifhyxlcspar2+msgqetdleg8=">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</latexit> <latexit sha1_base64="thfifhyxlcspar2+msgqetdleg8=">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</latexit> <latexit sha1_base64="thfifhyxlcspar2+msgqetdleg8=">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</latexit> Agora, a exponencial é um número e pode ser escrita como um produto de exponenciais: = X P hn 1,,n 1 e i [ ( i µ)n i ] n 1,,n 1 i n 1,,n 1 = X Y hn 1,,n 1 e [ ( i µ)n i ] n 1,,n 1 i n 1,,n 1 = X Y h i n i ( e i µ) n 1,,n 1 = X i h e ( 1 µ) i n 1 h e ( 1 µ) i n 1 n 1,,n 1 = X n 1 he ( 1 µ) i n 1! X h e ( 1 µ) i n 1! n 1!25

<latexit sha1_base64="aqjkqs6f6idhnd+ff7eueanarcs=">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</latexit> <latexit sha1_base64="aqjkqs6f6idhnd+ff7eueanarcs=">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</latexit> <latexit sha1_base64="aqjkqs6f6idhnd+ff7eueanarcs=">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</latexit> <latexit sha1_base64="aqjkqs6f6idhnd+ff7eueanarcs=">aaadfxicjvjda9swffxsfxtzv7o97uwy0jfsbbdgoh0plo1ljx0sbsbyjszliagsg0kubonfsx+2t/2vvwxy6kdnljelgqnzp86ruhehhbfb8kvn+q8epnq896t/9nnzfy8h+68utf5qxqcsl7mexdrwkrsfwmelnxwa0yyw/dk+/tlkl2+4nijx3+2q4gfgf0qkglhrqgi/94pmblyduwaiewphqeyzrzwk8glwigto+tnwqwqogmtcuhgrxhgh3qdccfk5hhqifoulda+a3s1tdiqlutxd4rurkrwrutx4r8sm9l6ltfkoiok3z2kfmxyaxa13zta9o1rqihc6tyibsimnsw0jggydsbao2aa4bupuxnk0+emsnjuzv5zjaswcb4unk6qtyjlxfviaxlb2trd87qcigtdhtd6egg4ck0caa3euhtv7t6oimtgrlhavgbvl08015h25ewntk7asqigtv+xwkc0l2byavyream6sxdlamrbok7al1zrzt7b99wm4++rtcpfxgomj/vzpepa5/y499aa9rsoe0te6q1/rozoi1vvtgtf23nt//ap/0j/clnq9tuc1+if8479jia9g</latexit> <latexit sha1_base64="s9xjhlhnsgpcto0wrxcyblw73ho=">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</latexit> <latexit sha1_base64="s9xjhlhnsgpcto0wrxcyblw73ho=">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</latexit> <latexit sha1_base64="s9xjhlhnsgpcto0wrxcyblw73ho=">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</latexit> <latexit sha1_base64="s9xjhlhnsgpcto0wrxcyblw73ho=">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</latexit> No caso de férmions, temos ni = 0, 1, logo = X n 1 =0,1 h e ( 1 µ) i n 1! ( = 1+e 1 µ) = Y ( 1+e i µ) i 1+e X n 1 =0,1 ( 1 µ) h e ( 1 µ) i n 1! A expressão anterior pode ser escrita na forma: = Y i 1+ze i onde z=exp('() é a fugacidade do sistema.!26

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sha1_base64="dg7wdj9sgozf+tqnh6mebvp6tea=">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</latexit> <latexit sha1_base64="dg7wdj9sgozf+tqnh6mebvp6tea=">aaac3hicfvjlbxmxepyurxjeay5crksgvcrrlkiql0ovxdgwibrb8sbyorojva93zc9wilyrfw4gxjufxo3fwr/am26kkhzgsvz5m8c3hjsttxiurb+c8nr1gzdv7dzu3ll77/6d7snhx66ormshlhrhr6lwqjxbisnsocotijzvejkevm38j2donsrmb1qwmoriblsmpcbptbu/+ujbczga4boz6gn3vt6tztq+ipbipebyvpdzuzizlvfqxo0bjzw8aj4icehzlj3svl7cchm1cyvgvs0xleyawpvtbltvw6xuy/9d87+sm9wrltfocw3wzrtbiwbr2uayifvqy60dtbs/+ayqvy6gpbbojeoopkqwlptuuorwymep5kmy49hdi3j0sb1+nbu888wmssl6zqjw7mwmwutolfpur+acfm7b15bx+cyvza+twpmyijtyxcirnfabzuvdtfmupjcecgmv7xxkqlghyf+hzgjx9puvg+oxgzgaxo9f9q7ftopyyu/yu9znmdtnh+wdo2jdjoopwafgs/a1nisfw2/h9/pqmghzhro/lpzxb5jx4gq=</latexit> <latexit 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sha1_base64="odbveaixsecxfinj1+r1bzjiapw=">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</latexit> <latexit sha1_base64="odbveaixsecxfinj1+r1bzjiapw=">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</latexit> <latexit sha1_base64="odbveaixsecxfinj1+r1bzjiapw=">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</latexit> <latexit sha1_base64="odbveaixsecxfinj1+r1bzjiapw=">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</latexit> <latexit sha1_base64="odbveaixsecxfinj1+r1bzjiapw=">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</latexit> No caso de bósons, o número de ocupação pode ser qualquer valor inteiro, n i = 0, 1,, portanto temos: X i! n 1 X i! n 1 ( 1 µ) ( 1 µ) = n 1 =0,1,,1 h e n 1 =0,1,,1 h e Logo, se a grandeza exp[ '(&i - ()] for menor do que 1 para todo i, teremos um conjunto de séries geométricas. Usando n q n = (1 q) 1 com q < 1, obtemos: = = Y i 1 1+e ( 1 µ) 1 1+e ( i µ) = Y i 1 1+e ( 1 µ) 1 1+ze i!27

No caso de bósons deve verificar-se exp[ '(&i - ()]<1 ou equivalentemente ze -'ε i < 1 para todo i, para que - não fique divergente. A condição exp[ '(&i - ()] <1 implica '(&i - () >0, portanto: ( < &i Isto é, o potencial químico deve ser menor que qualquer um dos autovalores de energia de uma única partícula. Ou seja, o para bósons o potencial químico deve ser menor que a energia do estado fundamental de uma única partícula: ( < &0 Em muitos casos temos &0=0, e portanto, ( < 0. Se essa condição não se verifica, a função de partição ficará divergente, não podendo representar um sistema físico.!28

De forma unificada, podemos escrever a função de partição macrocanônica para um gás ideal quântico como (z,v,t) = Y i (1 ± ze i ) ±1, onde o sinal (+) corresponde a férmions e o sinal ( ) a bósons.!29

<latexit sha1_base64="7jmdibjdcnbhauiai03qucn2uuu=">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</latexit> <latexit sha1_base64="7jmdibjdcnbhauiai03qucn2uuu=">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</latexit> <latexit sha1_base64="7jmdibjdcnbhauiai03qucn2uuu=">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</latexit> <latexit sha1_base64="7jmdibjdcnbhauiai03qucn2uuu=">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</latexit> Termodinâmica do gás ideal quântico Para estabelecer a conexão com a Termodinâmica, usamos (z,v,t) = PV = kt ln (z,v,t) PV kt = kt ln (z,v,t) =kt ln Y i PV kt = ±kt X i ln(1 ± ze X i ) (1 ± ze i ) ±1 Esta expressão pode ser escrita de forma genérica como PV kt =ln (z,v,t) =1 a X ln(1 + aze i i ), onde a = ( 8 1 > para bósons +1 para férmions!30

A partir de Ξ podemos obter o número médio de partículas N N = hni = z @ @z ln V,T P = z a X i Lembrando que N = i n i podemos identificar o número de ocupação P médio n i de cada nível de energia como ae i 1+aze i = X i 1 z 1 e i + a. hn i i = 1 z 1 e i + a. Finalmente, a energia interna U é dada por U = hei = @ @ ln z,v = X i 1 a az i e i 1+aze i = X i que, como era de esperar, ficou da forma U = i ε i n i. P i z 1 e i + a!31