teoria de perturbação dependente do tempo. - ifi.unicamp.brmaplima/f789/2018/aula22.pdf · teoria...
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1 MAPLima
F789 Aula 22
TeoriadePerturbaçãodependentedotempo.Acoplamentocomestadosdoespectrocontínuo
• Se Ef pertence ao espectro contınuo de H0, isto e, se o estado final e
caracterizado por um ındice que varia continuamente, nos nao podemos
medir a probabilidade de encontrar o sistema em um estado bem definido
|'f i, no instante t.
• No Capıtulo III (os postulados - revisao nas primeiras aulas desta disciplina)
aprendemos que��h'f | (t)i
��2 e, neste caso, uma densidade de probabilidade.
O resultado de uma medida envolve uma integracao desta densidade de
probabilidade sobre um certo grupo de estados finais.
• Na aula de hoje, adaptaremos a teoria de perturbacao dependente do tempo,
desenvolvida na aula passada, para que possa ser aplicada em casos como esse.
• Veremos que esse caso permitira resolver e entender o problema de uma
probabilidade de transicao infinita (fisicamente inaceitavel) para o caso discreto,
resultado obtido com a perturbacao oscilante, quando ! ! !fi.
• Comecaremos por lembrar como integrar um contınuo de estados finais a partir
de densidades de estados. Para tanto, usaremos um exemplo de espalhamento
de uma partıcula, sem spin, por um potencial W (r).<latexit 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2 MAPLima
F789 Aula 22
TeoriadePerturbaçãodependentedotempo.Acoplamentocomestadosdoespectrocontínuo
• O estado | (t)i da partıcula no instante t pode ser expandido como uma
combinacao de estados |pi com momentos lineares bem definidos e energias
E =p2
2m. Tanto energia (H0) como momento linear P sao observaveis com
espectro contınuo no caso de uma partıcula livre e seus auto-estados formam
um C.C.O.C.
• As auto-funcoes de H0 =P2
2msao as ondas planas hr|pi = 1
(2⇡~)3/2eip·r
~ .
• Estando o sistema no estado | (t)i, quanto vale a densidade de probabilidade
associada com a medida do momento linear? Vimos que e��hp| (t)i
��2.• E a probabilidade de encontrar a partıcula com momento linear pf? Vimos
que por forca de ser um contınuo de possibilidades, essa probabilidade e dada
por �P(pf , t) =
Z
pf2Df
d3p��hp| (t)i
��2 onde Df e uma regiao centrada em pf ,
que respeita
(As direcoes de pf estao dentro de um angulo solido d⌦f ;
Seu modulo associado a energias entre Ef � �Ef
2 e Ef +�Ef
2
• Nestas condicoes �P(pf , t) e probabilidade de obter um sinal no detector.
• Se estado final e caracterizado pela energia ) mudanca de variavel.<latexit 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3 MAPLima
F789 Aula 22
TeoriadePerturbaçãodependentedotempo.Acoplamentocomestadosdoespectrocontínuo
• Mudanca de variavel: Lembre que d3p = p
2dpd⌦, e o elemento de volume
que aparece em �P(pf , t) =
Z
pf2Df
d3p��hp| (t)i
��2. Nada impede de troca-lo
por d3p = ⇢(E)dEd⌦, uma vez que E e p estao relacionados por E =p2
2m.
Basta exigir que ⇢(E)dEd⌦ = p2dpd⌦ ) ⇢(E) = p
2 dp
dE=
p2
dEdp
=p2
pm
= mp,
ou seja ⇢(E) = m
p2mE.
• E assim, podemos escrever que �P(pf , t) =
Z
E2�Ef ;⌦2�⌦f
d⌦ ⇢(E)dE��hp| (t)i
��2, e
a probabilidade de uma partıcula, ao colidir com W, sair com momento pf ,
dentro de uma faixa de energia (�Ef ) e uma faixa de angulo solido d⌦f .
• Isso que fizemos e generalizavel.
� Suponha que H0|↵i=↵|↵i, onde ↵ representa a parte contınua do espectro
de H0.
� Queremos, sabendo que no instante t, o sistema se encontra no estado | (t)i,calcular a probabilidade �P(↵f , t) de encontrar o sistema, em uma
medida, em um dado grupo de estados finais.<latexit 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4 MAPLima
F789 Aula 22
TeoriadePerturbaçãodependentedotempo.Acoplamentocomestadosdoespectrocontínuo
� Esse grupo e caracterizado pelo domınio Df de valores do parametro ↵,
centrado em ↵f (considere que seus valores formam um contınuo).
�P(↵f , t) =
Z
↵f2Df
d↵��h↵| (t)i
��2.
� Novamente, nada impede de trocarmos as variaveis para E,�, onde �
aparece, caso H0 nao forme um C.C.O.C..
� Exigimos que d↵ = ⇢(�, E) d� dE para escrever:
�P(↵f , t) =
Z
�2��f;E2�Ef
⇢(�, E)d� dE�� h↵||{z} (t)i
��2.
podıamos trocar h↵| por: h�, E|• Regra de ouro de Fermi.
Considere | (t)i, um estado normalizado do sistema no instante t.
Suponha que esse sistema estivesse inicialmente no estado |'ii, um
estado pertencente ao espectro discreto de H0. Poderıamos trocar a
notacao �P(↵f , t) por �P('i,↵f , t) para lembrar desta condicao inicial.
• O que foi feito para o caso discreto, sera extendido agora para o caso
do contınuo. Comecaremos por W constante.<latexit 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5 MAPLima
F789 Aula 22
RegradeourodeFermi.Acoplamentocomestadosdoespectrocontínuo
• Para W (t) = W cos!t com ! = 0 (caso de uma perturbacao constante no
tempo), obtivemos �Pif (t) =|Wfi|2
~2 F (t,!fi) =��h'f | (t)i
��2, com
F (t,!fi) =h sin [!fit/2]
!fi/2
i2.
���h'f | (t)i
��2 pode ser trocado por��h�, E| (t)i
��2 e por analogia:
|Wfi|2
~2 F (t,!fi) por|h�, E|W |'ii|2
~2 F (t,!fi), para escrevermos
�P('i,↵f , t) =1
~2
Z
�2��f;E2�Ef
d� dE ⇢(�, E)��h�, E|W | (t)i
��2F (t,!fi).
� A funcao F (t,!fi), varia rapidamente quando E = Ei (ver detalhes na aula
passada). Para facilitar, os proximos 2 slides trazem o comportamento tıpico.
� No limite, quando t vai para infinito, e possıvel mostrar que
limt!1
F (t,!fi) = ⇡t��E � Ei
2~�= 2⇡~t�(E � Ei).
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6 MAPLima
F789 Aula 22
0,0
5,0
10,0
15,0
20,0
25,0
30,0
-5 -4 -3 -2 -1 0 1 2 3 4 5
!1/w^2!
4sin^2(wt/2)!
produto:f(w)!
Teoria de Perturbação dependente do tempo: potencial constante
ω
Quase toda a acao esta entre � 2⇡t ! 2⇡
t .
Isso ocorre entre os primeiros zeros do sin2!t
2.
f(!) =4
~2!2sin2
!t
2
8><
>:
~ = 1; t = 5;
�! = 2⇡t = 1, 3
Aqui, ! =Ef � Ei
~<latexit sha1_base64="+t1nr5HnblIXFFBdvxRTQI61k/M=">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</latexit><latexit sha1_base64="+t1nr5HnblIXFFBdvxRTQI61k/M=">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</latexit><latexit sha1_base64="+t1nr5HnblIXFFBdvxRTQI61k/M=">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</latexit>
7 MAPLima
F789 Aula 22
0,0
20,0
40,0
60,0
80,0
100,0
120,0
-5 -4 -3 -2 -1 0 1 2 3 4 5
!1/w^2!
4sin^2(wt/2)!
produto:f(w)!
Teoria de Perturbação dependente do tempo: potencial constante
ω
Quase toda a acao esta entre � 2⇡t ! 2⇡
t .
Isso ocorre entre os primeiros zeros do sin2!t
2.
f(!) =4
~2!2sin2
!t
2
8><
>:
~ = 1; t = 9;
�! = 2⇡t = 0, 7
Aqui, ! =Ef � Ei
~<latexit sha1_base64="RV0+E5iE1EfyUu0pL0NJkXbhl3A=">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</latexit><latexit sha1_base64="RV0+E5iE1EfyUu0pL0NJkXbhl3A=">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</latexit><latexit sha1_base64="RV0+E5iE1EfyUu0pL0NJkXbhl3A=">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</latexit>
8 MAPLima
F789 Aula 22
RegradeourodeFermi.Acoplamentocomestadosdoespectrocontínuo
• Para facilitar, suponha que o intervalo ��f e muito pequeno e o integrando nao
muda com �. Use isso para escrever
�P('i,↵f , t) = ��f2⇡
~ t��h�f , Ef = Ei|W |'ii
��2⇢(�f , Ef = Ei), se Ei 2 �Ef ,
�P('i,↵f , t) = 0 se Ei /2 �Ef .
� Esse resultado e familiar para o caso de uma perturbacao constante no tempo:
a perturbacao constante induz apenas transicoes entre estados degenerados.
• A probabilidade cresce linearmente com o tempo. Consequentemente a
probabilidade de transicao por unidade de tempo (a taxa de transicao) e dada
por: �W('i,↵f ) ⌘d
dt�P('i,↵f , t). Note que e independente do tempo.
Se definirmos W('i,↵f ) =�W('i,↵f )
��f
8><
>:
densidade de probabilidade de transicao
por unidade de tempo e por unidade de
intervalo da variavel �f
Com isso, podemos escrever a Regra de Ouro de Fermi:
W('i,↵f ) =2⇡
~��h�f , Ef = Ei|W |'ii
��2⇢(�f , Ef = Ei)<latexit sha1_base64="0/fb4aHnwn3CEvkEuwHYC8zz9Nc=">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</latexit><latexit sha1_base64="0/fb4aHnwn3CEvkEuwHYC8zz9Nc=">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</latexit><latexit sha1_base64="0/fb4aHnwn3CEvkEuwHYC8zz9Nc=">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</latexit>
9 MAPLima
F789 Aula 22
ComentáriossobreaRegradeourodeFermi.Acoplamentocomestadosdoespectrocontínuo
• E se W (t) for senoidal, W sin!t? O que muda?
W('i,↵f ) =2⇡
~��h�f , Ef = Ei + ~!|W |'ii
��2⇢(�f , Ef = Ei + ~!)
• Aplicacao para o problema de espalhamento.
� Onde a perturbacao W e local, isto e hr|W |r0i = W (r)�(r� r0).
� E o estado inicial do sistema e de uma partıcula livre: | (t=0)i= |pii.� Apos a colisao com o potencial, queremos a taxa de transicao para o
estado final |pf i. O que pode ser escrito como:
W(pi,pf ) =2⇡
~��hpf |W |pii
��2⇢(�f , Ef = Ei)
� Considerando
8<
:⇢(E) = m
p2mE
hr|pi =⇣
12⇡~
⌘3/2ei
r·p~
) podemos escrever:
W(pi,pf ) =2⇡
~ mp
2mEi
⇣1
2⇡~
⌘6���Z
d3r ei(pi�pf )·r
~ W (r)���2
Se dividirmos pela corrente de probabilidade inicial Ji =⇣
1
2⇡~
⌘3 ~kim
temos:W(pi,pf )
Ji= �(pi,pf ) =
m2
4⇡2~4���Z
d3r ei(pi�pf )·r
~ W (r)���2
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1º Termo de Born
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