sistema de partículas idênticas na mecânica quânticamaplima/f789/2018/aula25.pdf · quando um...
TRANSCRIPT
1 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuânticasimetrizadoreseanti-simetrizadores
• O Postulado de Simetrizacao.
Quando um sistema inclui partıculas identicas, somente certos kets do seu
espaco de estados podem descrever seus estados fısicos. Os estados fısicos sao,
dependendo da natureza das partıculas identicas, completamente simetricos ou
completamente anti-simetricos com respeito a permutacao destas partıculas.
• Chamaremos
8><
>:
Bosons, as partıculas identicas que tem estados simetricos;
Fermions, as partıculas identicas que tem estados anti-simetricos.
• O estado | i nao e qualquer ket de E (espaco construıdo pelo produto tensorial
de espacos de uma partıcula). Isto e, | i 2 ES ou | i 2 EA, sub-espacos de E .
• Regra empırica:
8><
>:
spin semi-inteiro ! fermions: e�, e+, p+, n, muons;
spin inteiro ! bosons: fotons, mesons.
• Essa regra vale para composicao de partıculas. Exemplo
8><
>:
3He ! fermion;
4He ! boson.<latexit sha1_base64="VWZul7wgzjjZQ2CMZVW89/EL9IU=">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</latexit><latexit sha1_base64="VWZul7wgzjjZQ2CMZVW89/EL9IU=">AAAM53ictVdNjxtFEO2EBBLbwAaOXFpZgUAYy04iEZZLIAoip2yAJZHW3lXPuO3t7HzR3bPJxrHyA7hwgCjX/CVu3JA4oCDllD9A9Zte74y/4qwUW9PbXV1d9epVdY03yCJlbLv916nTb505+/Y7587X6o1333t/7cIHP5s016HcCtMo1XcDYWSkErlllY3k3UxLEQeRvBPsX3f7dw6kNipNfrKHmezFYpiogQqFJVF64exj9gnrsoDlLKKvZJZWnB5L8wc0jrA7INktejZZygxJnbZgfVpxGiWNPzLFYpzXNHtIu/+yf2i/xcZkoUteat5y+Tn2cptsCpZ4mznZ4uRJwZuklaC1ov2QPOc04ywjmabdFyQr8Bjo9Nl/pGdpFkLWhKUU6BJY47QjcTbFmX3gNoglhbYki6vhlqSdIdpjLiRQF/wYIE2xE3sNQ/41/T0AiiN/Zs7JAUVnEIlbt5CFV+s5i4775ooxOEwZ2On7sYjFcZ7Q6PLt8D6EpO+Zfh3+Q/Cf+QoTlVwYVM5zXzlHEaQrZ2CZbeGRfLHAS3GWIzoDDhSqgrO/EaHLT0xI7KSejzJol7BQrflld+s62wNiAQQxUI2hE9B6iIofeR4l9mqV8+52uhgiejTbpmdIJ3skbxOKdunbhOTKjORqSTJm37KXuCsJfDXB4OvkmrNfaN8xb329T1fqvDx87fniGN9shN+Rb5dT9UajXF53RX0UN62a3dWq5taUR04nH5FGBm9d8uT8D3G6etLd56KKn/soBGkV0ehJJzzqY2OwH6MD7AFnRNIbwFm2+elUFww9ty7mnNgsMLrbFPmZRkfM/V2z4MIgwwpeVu9bvOLbTLDn/p0xm9PP0EdvAl/BRBOx8oUcdnEP+UI2duntV2XEda9lWXm1xW+AqmrVkNWAamp+xIuz1Vqxrn5Az9HgTaKnvvBVK9gGcrJaV3LoivgM7Lh7oHxPVsg9hzVne0hYLXLk5PdnYh5M3dcNnJVsh2w2gXOHfY5ZNpklc7hzXbywsKzbHCM/Kd6g1EE3gP8l6izxb8IYjBXrlrd78k5wg3QN8qVLuTtA7uWk+kXpDZyiGsvvskXvMFc7D3wduHu7PP877HIF2fc0W42zaobHlJ9ydnaoo5/M7nEmihswj+Xa7tp6u9XGh89OOn6yzvxnc3ftz24/DfNYJjaMhDHbnXZmeyOhrQojOa51cyMzEe6LodymaSJiaXoj/FAf849J0ueDVNOTWA5p+cRIxMYcxgFpxsLumek9J5y3t53bwdXeSCVZbmUSFo4GecRtyt2vft5XWoY2OqSJCLUirDzcE1qElv43cCR0pkOenWxdan3V6ty+sn7tkmfjHPuIXaTu32FfsmuUmU22xcK6rP9a/73+R+Ne47fGk8bTQvX0KX/mQ1b5NJ79D/3Aplo=</latexit><latexit sha1_base64="VWZul7wgzjjZQ2CMZVW89/EL9IU=">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</latexit><latexit sha1_base64="VWZul7wgzjjZQ2CMZVW89/EL9IU=">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</latexit>
2 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuânticasimetrizadoreseanti-simetrizadores
• Removendo a degenerescencia de troca.
Considere |ui, um ket do espaco de N partıculas identicas (ignore, por hora,
simetrizacao). Se |ui e um estado fısico, P↵|ui tambem seria. Considere Euformado por todos os {P↵|ui}.
� A dimensao deste espaco seria algo entre 1 e N ! (numero de estados ortogonais).
� Tal dimensao e justamente a degenerescencia de troca.
� O novo postulado diz que, de fato, o estado fısico e ou totalmente simetrico
(bosons) ou totalmente anti-simetrico (fermions).
• Temos que mostrar que Eu contem um unico ket de ES ou um unico ket de EA.� Para isso, basta usar as relacoes S = SP↵ e A = ✏↵AP↵, pois se |ui e P↵|ui sao
ortogonais
8><
>:
S|ui = SP↵|ui ) o que faz deles colineares.
A|ui = ✏↵AP↵|ui ) o que faz deles colineares.
• Todos os kets de Eu = {P↵|ui} quando simetrizados ficam colineares, isso indica
que a simetrizacao de qualquer membro deste sub-espaco gera um unico ket.
• Adegenerescencia de troca foi removida. Vale tambem para os anti-simetrizados.
• Como veremos, existirao situacoes em que a simetrizacao ou a
anti-simetrizacao de |ui gerarao kets nulos.<latexit sha1_base64="/cfmhVH6MjQY7mY8nRCncVZqgaY=">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</latexit><latexit sha1_base64="/cfmhVH6MjQY7mY8nRCncVZqgaY=">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</latexit><latexit sha1_base64="/cfmhVH6MjQY7mY8nRCncVZqgaY=">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</latexit><latexit sha1_base64="/cfmhVH6MjQY7mY8nRCncVZqgaY=">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</latexit>
3 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuânticaConstruindo os kets físicos
⌅ Regra de Construcao para N Partıculas.
F Enumere as partıculas arbitrariamente e construa um ket |ui, correspondente
ao estado fısico considerado e aos numeros dados (arbitrados) as partıculas;
F Aplique S ou A em |ui, dependendo se as partıculas sao bosons ou fermions;
F Normalize o que obtido com a estrategia acima.
⌅ Aplicacao a sistemas de duas partıculas identicas.
� Suponha que uma possa ser caracterizada por um ket (de uma partıcula) |'ie a outra pelo ket (tambem de uma partıcula) |�i.
� Suponha que as partıculas ocupem kets distintos, |'i 6= |�i.F Primeira regra: |ui = |1: '; 2: �i
F Segunda regra:
8><
>:
S|ui =�12!
P↵ P↵
�|ui = 1
2
�|1: '; 2: �i+ |1: �; 2: 'i
�
A|ui =�12!
P↵ ✏↵P↵
�|ui = 1
2
�|1: '; 2: �i � |1: �; 2: 'i
�
F Para
(|';�i = cS|ui|';�i = cA|ui
) h';�|';�i = c2
4(1 + 1) = 1 ! c =
p2.
) os kets normalizados sao |';�i = 1p2
�|1: '; 2: �i+ ✏|1: �; 2: 'i
�
com ✏ = 1 ! bosons e ✏ = �1 ! fermions.<latexit sha1_base64="z0OdW52Bgytryd6Wh67ZxhSCFl4=">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</latexit><latexit sha1_base64="z0OdW52Bgytryd6Wh67ZxhSCFl4=">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</latexit><latexit sha1_base64="z0OdW52Bgytryd6Wh67ZxhSCFl4=">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</latexit><latexit sha1_base64="z0OdW52Bgytryd6Wh67ZxhSCFl4=">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</latexit>
4 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuânticaOperadoresdePermutação-SistemadeNpartículas
F Suponha agora que |'i = |�iNeste caso, a primeira regra fornece |ui = |1: '; 2: 'i ) ja simetrizado.
A simetrizacao nos leva a
8><
>:
S|ui = 12
�|1: '; 2: 'i+ |1: '; 2: 'i
�= |ui;
A|ui = 12
�|1: '; 2: 'i � |1: '; 2: 'i
�= 0.
• O que permite concluir: Nao existe ket em EA para descrever um estado fısico
na qual dois fermions estao no mesmo estado individual |'i. Familiar? Esse
e o princıpio de exclusao de Pauli.
⌅ Generalizacao para um numero arbitrario de partıculas.
• Comece com N = 3. Suponha que os tres estados normalizados sejam dados
por |'i, |�i e |!i.F Regra 1: |ui = |1: '; 2: �; 3: !iF Regra 2: caso bosons.
S|ui = 1
3!
X
↵
P↵|ui =1
6
⇣|1: '; 2: �; 3: !i+ |1: !; 2: '; 3: �i+ |1: �; 2: !; 3: 'i+
+|1: '; 2: !; 3: �i+ |1: �; 2: '; 3: !i+ |1: !; 2: �; 3: 'i⌘
F Regra 3: Se os estados |'i, |�i e |!i sao ortonormais, c =p3!.
<latexit sha1_base64="LrEHjvGEaJ0LhWddHVoxHFMP1sc=">AAAQHnicxVfPjxs1FHbLj5YAmy3c4GJYUYG6jZJNBVWritKq0FNZumxbaROtJo6TuDuZmXo8C0vIX8KFf4ULBxBC4kRvIA6oSD1x48TnN85kMvnR7HYLiTzxPD8/f+/5e89OK/JVbKrVhydOPvf8Cy+eOv1S6eVXXl0pr5557U4cJlrIbRH6ob7X8mLpq0BuG2V8eS/S0uu3fHm3tXfdjt/dlzpWYfC5OYhks+91A9VRwjMQhWdOvcHOsgZrMcW6LGaGeUzjnaMZJtmXeA7YFktYxEIWsB7GOVoXb5r6DzAm8TtkX2POPs2PoKfwZjUC6PrQuELjYsZIA9+zrORWzbcxglvoxfTOYcNDP2TrhIRjNQ2bfYwph0mj33X9DiENIBEZzmQGthpkb1G7VPDjMmZtzB0t+nKbItlzkdRY/YuCL/fZ75DEDrMh9F9Buw3dCvAtE4+PZlr4k/0BGxxoQoxyQrRPUfiNPLf7bCOjoDFwcbRxHWK1rZlxaSB+ViKgX4PeAJEYZnx599jidu4YdyDF9h7tatGjyxRdTs8Sovh/+nz+mfhcBYcsdyRG24VdHjOrBb99mmtmsOvTXF5HeGrwTGXZZ+uAwNyE8u2SY9Z4doN0OXI2ZaOVqyx799yaEjY5RbeBngdZj7D6kN6AfJc4nrfLM8sRRcLyuk2eCcr4fULKgatP9tNq1iYMHfaY8sWiXy7DRmsFWZVLsVl7irKrwx65yNiIxNma4xzklJ8xnuEUIkX7o4Ba4Xdk/UlVtDIVk48xajH4aHbWhy6m1lOfat8OWhc73oTcsqOa+66T5MKU5GJOMiyseAN+WEYNF0TyWSJ4BPthVvctFx+jr0jWJo5ZzZShsduNVL6JCCUUqUoBfQtSm/F7mJHutaa8moz2J5RVknbEL9TcMSdT/gXsH9p9TaN2rEWM0qj+egJtOtPAC0HobL5WDpGt12Grn51vgt4sk26hGtRnMKZ4mo8yPT0zLMK/cmxO+Ro7RmvK1ZHvoxHLhvvEQ+vTWL5cpkVk98ncX597gyh6OL6PpJHpQreYR/noLrr73M7dJWqu2j39DUJk8vqEfB7eo6DdILTj+xLH3L+pF0wxbHJ3lr0L1IHaWomJ87voWW5EjlebBclyNt8ni9eOdNYeNqrTt46x5uLzd3qNWbycti/mIJ9ctz533aL9+fm1uJ0j64eL7iKMT+f94sgez+7N48aiW9U1d6s6SvbVXfZt5WprsaL+lxWvOGN0MoZ09oS56m5vN+uEfYi1r1B+PyCtUcZXdlfXqpUqffh0p+Y6a8x9NndXf2m0Q5H0ZWCE78XxTq0amebA00YJXw5LjSSWkSf2vK7cQTfw+jJuDujv7ZC/A0mbd0KNFhhO0vyMgdeP44N+C5p9z/Ti4pgVzhrbSUznYnOggigxMhDpQp3E5ybk9r8ybysthfEP0PGEVsDKRc/TnjD4R11CEGpFl6c7dzYqtWql9tmFtasbLhyn2ZvsbRS3GvuAXWU3USi3mVj5ZuW7lR9Wfix/W/6+/FP551T15Ak353U28Sn/+i/JPxFP</latexit><latexit sha1_base64="LrEHjvGEaJ0LhWddHVoxHFMP1sc=">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</latexit><latexit sha1_base64="LrEHjvGEaJ0LhWddHVoxHFMP1sc=">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</latexit><latexit sha1_base64="LrEHjvGEaJ0LhWddHVoxHFMP1sc=">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</latexit>
5 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuânticaF Comentarios adicionais: caso bosons
� Suponha que duas partıculas ocupem o mesmo estado, por exemplo, |'i = |�i
Neste caso, |';';!i = 1p3
⇣|1: '; 2: '; 3: !i+ |1: '; 2: !; 3: 'i+ |1: !; 2: '; 3: 'i
⌘
� Se os tres kets sao iguais, |'i = |�i = |!i, entao |ui = |1: '; 2: '; 3: 'iF Regra 2: caso fermions.
A|ui = 1
3!
X
↵
✏↵P↵|ui =1
6
X
↵
✏↵P↵|1: '; 2: �; 3: !i(Perceba que os sinais sao definidos segundo
as mesma regras que um determinante.
Esse e o chamado determinante de Slater:
A|ui = 1
3!
������
|1: 'i |1: �i |1: !i|2: 'i |2: �i |2: !i|3: 'i |3: �i |3: !i
������
� Note que respeita o princıpio de exclusao de Pauli: A|ui e zero, sempre que
duas partıculas ocupam o mesmo estado, isso e
8><
>:
|'i = |�i, ou
|�i = |!i, ou
|'i = |!i
F Regra 3: Normalizacao c =p3!. Mais partıculas e direto.
<latexit sha1_base64="/gtA1+XlENJGpi4ianSLXvmpDtQ=">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</latexit><latexit sha1_base64="/gtA1+XlENJGpi4ianSLXvmpDtQ=">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</latexit><latexit sha1_base64="/gtA1+XlENJGpi4ianSLXvmpDtQ=">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</latexit><latexit sha1_base64="/gtA1+XlENJGpi4ianSLXvmpDtQ=">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</latexit>
6 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuântica
F Comentarios adicionais: caso fermions
� Note que poderıamos ter dois estados com a mesma parte espacial, mas com
spins diferentes. Isso seria suficiente para garantir que o determinante nao
fosse nulo.
� A hipotese de uma solucao como um unico produto direto de kets de uma
partıcula e uma aproximacao forte. E a hipotese inicial (que trata partıculas
interagentes como se fossem independentes) da chamada aproximacao
Hartree-Fock). Precisamos definir bases que atuem nos espacos de estados
fısicos (A ou S).
⌅ Construcao de uma base no espaco de estados fısicos.
• Considere um sistema de N partıculas identicas.
� Comece com uma base {|uii} no espaco de estados de uma partıcula e
construa {|1: ui; 2: uj ; ...;N: upi} ⌘ E ! E base. Aprendemos isso em F689.
� O espaco fısico de partıculas identicas nao e E e sim ES ou EA.• Como determinar uma base do espaco fısico?
� Aplicando S ou A em todos os kets de E = {|1: ui; 2: uj ; ...;N: upi} podemos
obter um conjunto de vetores que definem ES ou EA.� Sera que eles formam uma base?
<latexit sha1_base64="mfUkS9qJ9Nt4xuHivTGVZi9MIh0=">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</latexit><latexit sha1_base64="mfUkS9qJ9Nt4xuHivTGVZi9MIh0=">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</latexit><latexit sha1_base64="mfUkS9qJ9Nt4xuHivTGVZi9MIh0=">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</latexit><latexit sha1_base64="mfUkS9qJ9Nt4xuHivTGVZi9MIh0=">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</latexit>
7 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuântica
• Seja |'i um ket arbitrario de ES (caso EA e semelhante).
� Como E e uma base, podemos sempre escrever
|'i =X
i,j,...,p
ai,j,...p|1: ui; 2: uj ; ...;N: upi
� Mas |'i 2 ES , entao S|'i = |'i. E isso implica em
S|'i = |'i =X
i,j,...,p
ai,j,...pS|1: ui; 2: uj ; ...;N: upi
� Note que nem sempre os S|1: ui; 2: uj ; ...;N: upi sao independentes!
� De fato, os kets nao simetrizados (por exemplo, |1: ui; 2: uj ; ...;N: upi), que
possuem as mesmas funcoes (ui, uj , ..., up) de uma partıcula (mesmas em
numero e em especie), quando simetrizados dao o mesmo ket simetrizado
(ja vimos isso).
� O mesmo ocorreria se tivessemos antissimetrizando.
� Tal discussao introduz o conceito de Numero de Ocupacao: por definicao
para o ket |1: ui; 2: uj ; ...;N: upi, o numero de ocupacao nk do estado
individual |uki e igual ao numero de vezes que o estado |uki aparece na
sequencia |uii, |uji, ..., |upi. Isso e o numero de partıculas no estado |uki.
Temos, obviamente,
X
k
nk = N.<latexit sha1_base64="P+yfPEdpNO8LFvhAWMWG3awZ7HI=">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</latexit><latexit sha1_base64="P+yfPEdpNO8LFvhAWMWG3awZ7HI=">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</latexit><latexit sha1_base64="P+yfPEdpNO8LFvhAWMWG3awZ7HI=">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</latexit><latexit sha1_base64="P+yfPEdpNO8LFvhAWMWG3awZ7HI=">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</latexit>
8 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuântica
� Dois kets diferentes, com os mesmos numeros de ocupacao, quando sujeitos
ao operador S, dao a mesma coisa. Daqui para frente sera denotado por
|n1, n2, ..., nk, ...i = cS| 1: u1; 2: u1; ...;n1: u1| {z }n1 em |u1i
;n1+1: u2, ..., n1+n2: u2| {z }n2 em |u2i
; ...i.
Genericamente, nesta notacao, temos nk partıculas em |uki.� Para o caso de fermions basta trocar S por A.
⌅ Algumas propriedades dos estados |n1, n2, ..., nk, ...i.• O que devemos esperar do produto escalar de dois kets, |n1, n2, ..., nk, ...i
e |n01, n
02, ..., n
0k, ...i?
� Serao diferentes de zero, somente se os numeros de ocupacao forem iguais,
isto e: nk = n0k, 8 k. Se n1 > n0
1, vai sobrar |u1i que e ortogonal a 8 |ujipara j 6= i.
• Se as partıculas estudadas sao bosons, os kets |n1, n2, ..., nk, ...i, com nk
arbitrario, mas restrito a condicao
X
k
nk = N, formam uma base.
� Ja vimos que 8 |ui simetrico, e combinacao de S|1: ui; 2: uj ; ...;N: upie cada um desses kets e por definicao um |n1, n2, ..., nk, ...i.
<latexit sha1_base64="Bkq2CUfFyZGfyBax27nhEoDcshE=">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</latexit><latexit sha1_base64="Bkq2CUfFyZGfyBax27nhEoDcshE=">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</latexit><latexit sha1_base64="Bkq2CUfFyZGfyBax27nhEoDcshE=">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</latexit><latexit sha1_base64="Bkq2CUfFyZGfyBax27nhEoDcshE=">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</latexit>
9 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuântica
continuacao: ainda para bosons.
� Assim, 8 |ui simetrico, ele sera uma combinacao de |n1, n2, ..., nk, ...i,o que caracteriza {|n1, n2, ..., nk, ...i} como sendo uma base.
� Vimos tambem que |n1, n2, ..., nk, ...i e |n01, n
02, ..., n
0k, ...i sao ortogonais,
salvo se n1 = n01, n2 = n0
2, ..., nk = n0k (e nesse caso a norma e 1). O
caracteriza {|n1, n2, ..., nk, ...i} como sendo uma base ortonormal.
� Para caracterizar que temos de fato uma base a partir da simetrizacao de
um conjunto completo de kets de E , precisamos apenas mostrar que o ket
nulo nao pode ser obtido nesse processo. Se conseguirmos o ket nulo, a
base seria linearmente dependente.
|n1, n2, ..., nk, ...i = cS| 1: u1; 2: u1; ...;n1: u1| {z }n1 em |u1i
;n1+1: u2, ..., n1+n2: u2| {z }n2 em |u2i
; ...i =
= c1
n!P↵|1: u1; 2: u1; ...;n1: u1;n1+1: u2, ..., n1+n2: u2; ...i ) troca entre
iguais gera kets colineares com coeficientes positivos e iguais. Troca entre
distintos gera kets ortogonais. Apos juntar os iguais (soma direta pois
todos tem o mesmo coeficiente) temos uma soma de vetores ortogonais
e portanto, nunca igual a zero.<latexit sha1_base64="h9sqLHWTMNKM7aH3Aac5iZkVsg4=">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</latexit><latexit sha1_base64="h9sqLHWTMNKM7aH3Aac5iZkVsg4=">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</latexit><latexit sha1_base64="h9sqLHWTMNKM7aH3Aac5iZkVsg4=">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</latexit><latexit sha1_base64="h9sqLHWTMNKM7aH3Aac5iZkVsg4=">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</latexit>
10 MAPLima
F789 Aula 25
SistemadePartículasIdênticasnaMecânicaQuântica
• E se as partıculas forem fermions?
A base do estado fısico seria obtida escolhendo o conjunto de kets
|n1, n2, ..., nk, ...i nos quais, os numeros de ocupacao sao nk = 1 ou
nk = 0. Continua valendo
X
k
nk = N.
� Conforme vimos nos slides 2 e 5, para os caso de duas e tres partıculas,
respectivamente, quando mais de uma partıcula ocupam o mesmo estado
individual, a antisimetrizacao anula o ket fısico resultante. Ou seja, se um
dos numeros de ocupacao for maior ou igual a 2 em |n1, n2, ..., nk, ...i, esse
ket fica automaticamente igual a zero devido a antissimetrizacao.
� Se todos os numeros de ocupacao forem 1 ou 0, entao |n1, n2, ..., nk, ...i 6= 0.
Isso porque (escolhi os primeiros N estados individuais como ocupados)
|1, 1, ..., 1, 0, 0, 0...i| {z }N partıculas em estados distintos
= cX
↵
✏↵ P↵|1: u1; 2: u2; ...;N: uki| {z }kets ortogonais
• Quanto vale c, em
|n1, n2, ..., nk, ...i = cS| 1: u1; 2: u1; ...;n1: u1| {z }n1 em |u1i
;n1+1: u2, ..., n1+n2: u2| {z }n2 em |u2i
; ...i?
Mostre que c =pN ! para fermions e c =
pN !/n1!n2!... para bosons.
<latexit sha1_base64="UiMKegTRjDNqkQGISt5O4GchJ4s=">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</latexit><latexit sha1_base64="UiMKegTRjDNqkQGISt5O4GchJ4s=">AAAN+XicnVfNb1tFEN8WKCHYkJYjlw0RCERkbAtUIIpaVCHBgdKqhFaKo2i9fk62fu/t69t9gdTxn8KFAyCu/CfcuCH1gEDixI0TM7P7Pv0hE1v73uzszM7Mb2Z27WESKmO73d+uXH3u+Reuvbjx0ubLrfYrr25dv/G10VkqgwOpQ50+GgoThCoODqyyYfAoSQMRDcPg4XByB9cfngWpUTr+yp4nwVEkTmI1VlJYYOnr1zbYW2zAhixjIXwDZmHGYVigv4XnlH0KcwMzzgS8OUvgncLK30ySluOOmQZuwCKi/wQ6Ygp4MazeYjPYcQCWNv3u1VFa+gTmQ9oPrY1AmwNlYE342RisGthX0gzlUpgJoDVoWqBHNEMtlAnZKdCx18YhyafH4HkM8prsoLUJxY6RLPeVswvQOmY9tkvvPrw79HXzSTEfgF8CeCeEaTVGDlxNdp6ADwI8NqDFPS9m/wJuGJWbO980IY24/8Ge+cjdG711lvfBq7odDTqrosn1uuRvVfMOYYRoxuQjZ2fwDCtI4q4G1iLao9zrLu2V25OwQwrPZp7d/q5iIorwDCSjAgP3NmBREQI461NWOfuA0HI1KArcJFVNmc0R+W28jgXZv5bW7u6a1ZmSJwk8JWGDmESECcrsFhnNMYp8dnOfMuIs8qGa4cjXaUTWorkeWM9XRV4owtVhEZKHgobLrSHMA0JHsaeV6hKU99yzvDea3efwQDnr9wwg+5x9SXWHvfkY+LvF6YHxr+f96H92g6ukHHFHu+pX0IEues5+L6ooogq+XC/vNnoFMcD41ousRFJRzSLWGZ1D6Lv13LKq5iN46vFwaLj8ar9WZnZZbjs1P5f15wPfNbrIxGXy4XDuFbnoUi24yJ75U+Sy52lMlru102Y56p8TIpp6Dz3DPkX9t2v3hCriTAg1jFQVsd5t9KGh3DS7TPnTSPvOLbFxOu+srJP5gc+MrCDqQ0JBAj0F5BC1XgUxpLvFWITbDDCdUiT1Gl58o7v8NWMeUYzubrDEm8F3n6oovxEGVK8JYCqADoDCegzp1K+vLo/vXkPygu63bRofgw5WzR5UTJ3XB56LfA/irK9NFqJRR6L8FaAJEQ2y6LXL7KyWvVW/me77m8D9wshvT6x4SV3QPEMituqmvlyXYE4egO4yhNfBs1fB0/nQXHco5muLorrwkk3095Z6Vlp6l55Nq00EqrIOoaZ86WV/pZf9BV4uwvbWWufOF9Qfls5CXpw6WAP71C1PqMawI7cp+/M9Kdj8r2m+Ypf3PB45Dtve++W7D9k/dDbizii3eby10+106cPniZ4ndpj/3Dve+nUw0jKLgtjKUBhz2Osm9mgqUqtkGMw2B5kJEiEn4iQ4BDIWUWCOpvTXZcbfBM6Ij3UKI7acuFWNqYiMOY+GIBkJe2qaa8hctHaY2fGHR1MVJ5kNYukMjbOQW83xfxAfqTSQNjwHQshUga9cnopUSAv/lhCEXjPkeeKg3/mo07v//s7tvkdjg73O3oB7pcdustvsMzjFDphsfdP6rvVD68f2Rfv79k/tn53o1Ste5zVW+7R/+Q+o27HG</latexit><latexit sha1_base64="UiMKegTRjDNqkQGISt5O4GchJ4s=">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</latexit><latexit sha1_base64="UiMKegTRjDNqkQGISt5O4GchJ4s=">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</latexit>