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=">AAAC+niclVJda9swFJW9r877SrbHvYiFlQ6y4JhAW0qgYyvssYNlLUROkBU5EbVlT7oeDUJ/ZS972MZe90v2tn8zxXZhbfeyKwSHc3V07z3cpMyEhjD87fk3bt66fWfrbnDv/oOHjzrdxx90USnGJ6zICnWaUM0zIfkEBGT8tFSc5knGT5Kz15v8ySeutCjke1iXPM7pUopUMAqOmne97na6Q4qcL+mLMUkVZWZkDVklVM2ihp9FlmghZ5Fp8g2LwZrIWpLwpZCGuRa0DWrdeHiACYbx/gEhmJCAvOEZ0EbVlohIKawBOw77u4TLRasnZDtwkv86htQmTNUyiU04COvoh4PRBdirgSXAz8G8+liJPraXmjmapy+P5qKd2tp5p3fxD74Ohi3ooTaO551fZFGwKucSWEa1ng7DEmJDFQiWcedKpXlJ2Rld8qmDkuZcx6Zu2+LnjlngtFDuSsA1+7fC0FzrdZ64lzmFlb6a25D/yk0rSPdiI2RZAZesKZRWGYYCbxYBL4TiDLK1A5Qp4XrFbEWdIeDWJXAmDK+OfB1MosH+IHw36h1GrRtb6Cl6hnbQEO2iQ/QWHaMJYt6599n76n3zrf/F/+7/aJ76Xqt5gi6F//MPG5fnig==</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