1 MAPLima

F789 Aula 15

Auladehoje:PotenciaisetransformaçõesdeGaugeEm Mecanica Classica o ponto zero de energia nao tem significado fısico. A

evolucao temporal de variaveis dinamicas, tais como, r(t),p(t) ou L(t), nao e

alterada se usarmos V (r) ou V (r) + V0 (constante no espaco e no tempo).

Isso porque F = �rV (r) e ) V0 uma constante aditiva, nao e importante.

Como sera a situacao analoga em Mecanica Quantica?

Suponha

8><

>:

|↵, t0; ti ! V (R)

^|↵, t0; ti ! V (R) = V (R) + V0

ambas com |↵i em t = t0.

Nos sabemos evoluir kets no tempo

^|↵, t0; ti = exp�� i

⇥ p2

2m+ V (R) + V0

⇤ (t� t0)

~ |↵i =

= exp⇥� iV0(t� t0)

~⇤|↵, t0; ti.

A presenca de V0 faz com que os estados |↵, t0; ti e ^|↵, t0; ti fiquem

diferentes! Eles diferem por uma fase global exp⇥� iV0(t� t0)

~⇤.

<latexit sha1_base64="D+Nu/5aP6G/FqrEAf+SxiaGqfpA=">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</latexit><latexit sha1_base64="D+Nu/5aP6G/FqrEAf+SxiaGqfpA=">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</latexit><latexit sha1_base64="D+Nu/5aP6G/FqrEAf+SxiaGqfpA=">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</latexit><latexit sha1_base64="D+Nu/5aP6G/FqrEAf+SxiaGqfpA=">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</latexit>

2 MAPLima

F789 Aula 15

representação das coordenadas

Potencialconstante(independentedotempoedaposição)Para um estado estacionario, somar um potencial constante V0 na

Hamiltoniana implica em uma mudanca na dependencia temporal

do tipo

8><

>:

Se V (R) �! exp⇥� iE(t�t0)

~⇤

Se V (R) + V0 �! exp⇥� i(E+V0)(t�t0)

~⇤

ou seja, mudar de V (R) ! V (R) + V0 implica em E ! E + V0.

Efeitos observaveis, como evolucao temporal de valores esperados

(exemplos: hRi, hPi, etc.), dependem das chamadas frequencias de

Bohr (diferencas em energia) e portanto nao mudam mediante a

adicao de um potencial constante na Hamiltoniana.

Esse de fato, e um primeiro exemplo de transformacao de Gauge:

V (R) ! V (R) + V0

8><

>:

|↵, t0; ti ! exp⇥� iV0(t�t0)

~⇤|↵, t0; ti

E ! E + V0

(r0, t) ! exp⇥� iV0(t�t0)

~⇤ (r0, t)

<latexit sha1_base64="WoO3yAN+vcUbKDSl3guGL4iq6Ds=">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</latexit><latexit sha1_base64="WoO3yAN+vcUbKDSl3guGL4iq6Ds=">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</latexit><latexit sha1_base64="WoO3yAN+vcUbKDSl3guGL4iq6Ds=">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</latexit><latexit sha1_base64="WoO3yAN+vcUbKDSl3guGL4iq6Ds=">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</latexit>

3 MAPLima

F789 Aula 15

Potencialdependentesódotempo

⇤O pacote de onda de cada partıcula e dividido em dois.

O que aconteceria, se V0 fosse espacialmente uniforme, mas dependente do

tempo? |↵, t0; ti = U(t, t0)|↵i, com U(t, t0) = exp⇥� i

~

Z t

t0

H(t0)dt

0⇤. Mas,

se H(t) = H + V0(t), entao

|↵, t0; ti = exp⇥� i

~H(t� t0)⇤exp

⇥� i

~

Z t

t0

V0(t0)dt

0⇤|↵i

e ) terıamos |↵, t0; ti| {z } �! exp⇥� i

~

Z t

t0

V0(t0)dt

0⇤|↵, t0; ti| {z }

V (R) V (R) + V0(t)

Pergunta importante: Sera que a adicao de um potencial espacialmente

constante, mas variavel no tempo, causa algum efeito mensuravel no sistema?

Para responder isso, imagine um experimento com um feixe de partıculas que

e separado⇤em dois: Parte passa por uma regiao de potencial V1(t) e a outra

parte por um potencial V2(t) que variam de forma diferente no tempo. Em

seguida juntamos os feixes e realizamos uma experiencia de interferencia.

Sera que perceberıamos diferencas nos resultados deste novo experimento,

conforme variassemos a dependencia temporal dos potenciais?<latexit sha1_base64="kpd++pUG3AsymsclFRj3T61AS6M=">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</latexit><latexit sha1_base64="kpd++pUG3AsymsclFRj3T61AS6M=">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</latexit><latexit sha1_base64="kpd++pUG3AsymsclFRj3T61AS6M=">AAAOFniczVdPjxs1FH8tUNqQwBYkLlwsVtAWtlGyQgIVVa2EkHpBLNDdVtpsV86Mk5idf4w92y5hPgUXvgoXDiDEFXHjiDhUIPU78PxsZ2YyyQq0BRHJGc/z83u/9/Ozn2ecRVLpweDXc+efefa5C89fvNR5odt78aWNyy/vqbTIA7EbpFGa3x9zJSKZiF0tdSTuZ7ng8TgS98ZHH5jxe8ciVzJN7uqTTBzEfJrIiQy4RlF6+cKr8CaMQIOAR/g/h4+BwRdQ4DsDDgGkkNBogC0HibItHFE0XmLbg0MY4LNug8EE5ymnJfCZkS0zO4IYJdYmQz8JSo12TnJjO0Ythc8Q3zPSDWszQtQt0dsIcTe9ml6MM1K45bB14CvUMD4zmBFyTWjfx+cIPXK0OsVRATdhF66i1GtcW5rZ1N5qxWt4ip3XZUs3UddoZqg5gjHGO8VnhL2cWDnCOddRMnHvc9QoSXeGOhylJl5JDBxSnMZuCQ+wd4d8XUEvIT0r+3nNfgl95KKJ+CNieWsNl359O/hvfZg47sDbbr2tpM2DXaffkQ22sGw8szO0f7KKZ2faRnt9sXrrGP331rRiuFrXcg2O07L0rOw3M8bvdyOb0Vkg3L4VrSzQNP4EscR0DthMGtFuD2nMRyBQ/++urvUe0Yk0pbNoijM0sZmj9KHDvNrL/3G9Tov16eydp9s6GOtVx+AE3z9FnNf+Qz/N06dT46jKvh1a+SllgckNhqtgq0JO77aO3ED5Z6T5W6PemRbijMfuDAtdlTJnu7Gh6YTzley02rb6XLX1VNWQ1CveMWWaREzHOGLsJ4SiqmxbVGs4IlIObUSxxoRlgk2idko2BXkqKMamPYVaylnlWC3LNUxyyklGe1yRf7+zGFlQDpEkO1PaHxVfdr/ZW4PnJYWqVloti/mRq+2CeOa0Vk9Qs0DUlhu7RqtZ/QPsjSRziO0d4QG81TqZBPk04wb/jcWJVmnsOO8eiXJM2wyyqD0nJuJ6nqzKEH9HGrqsZSsw2cxL0bYm/BbVaOlmljWQVWjW+TVetxdemz59xvuM444XAWxxG+MkkbRCee0GtpyTfXz7EPvrbxImQyXaMhY/X+xMXxvsf5UHynFifJpYJHxZ0/b817Prz0XkPgbp0OYOfaXRX4OzfRpY66Z6jFv1rMnLY5ejiRu1+8XkrnbZ6O+zqsbhMfG4epdU97L26VHdlusnhr1te3yeCX9/rjNUrVzucsUjXM4js0du1bKxc7ixOegP6MfanaHrbIL77Rxu/DIK06CIRaKDiCu1Pxxk+mDOcy2DSJSdUaFExoMjPhX72E14LNTBnD5rSvYGSkI2SXNsiWYkrc+Y81ipk3iMmjHXM7U8ZoSrxvYLPXnvYC6TrNAiCayjSRExnTLzjcRCmYtARyfY4UEuESsLZjzngcYvKUPCcDnkdmdvuz8c9IefvLN5e9vRcRFeg9dxQw7hXbiNF80d/FQIul93v+1+3/2h903vu96PvZ+s6vlzbs4r0Pj1fv4LKrG9Kw==</latexit><latexit sha1_base64="kpd++pUG3AsymsclFRj3T61AS6M=">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</latexit>

4 MAPLima

F789 Aula 15

A interferencia sera do tipo cos (�1 � �2) e/ou sin (�1 � �2), onde

�1 � �2 =1

~

Z t2

t1

dt�V2(t)� V1(t)

�. Embora nenhuma forca atue na

partıcula, pois rV (t) = 0, existe um efeito mensuravel. Efeito

puramente quantico, pois quando ~ ! 0

(cos (�1 � �2)

sin (�1 � �2)oscilam

muito, destruindo qualquer efeito interessante.

V1(t)

V2(t)

Dimensao do

pacote menor

que a gaiola

Pode criar diferenca

de potencialzona de

interferencia

Separa o feixe

O Experimento

Gailola metalica: gerando V1(t) e V2(t) depois que a

partıcula esta dentro dela e desligando antes dela sair

Experimentodecorrelação.Potencialdependentesódotempo

5 MAPLima

F789 Aula 15

GravidadeemMecânicaQuânticaSera que existe algum efeito gravitacional quantico? Veremos que sim, mas

antes exploremos as diferencas entre mecanica classica e quantica.

Considere um corpo em queda livre

minercialr = �mgravr�grav = �mgravgz

como minercial = mgrav ! r = �gz

Uma vez que a massa nao aparece na equacao acima, gravidade em mecanica

classica e considerada uma teoria puramente geometrica.

A situacao em Mecanica Quantica nao e bem assim:

⇥�~22m

r2 +m�grav

⇤ = i~@

@t! a massa nao cancela!

De fato, a massa aparece junto com ~, na forma~m. Isso tambem pode ser

visto na formulacao de integrais de caminho de Feynman

hrn, tn|rn�1, tn�1i =r

m

2⇡i~�texp

⇥i

Z tn

tn�1

dtLclassica(r, r)

~⇤com

Lclassica(r, r)

~ =12mz2 �mgz

~ ! note ~ e m juntos e na formam

~ .<latexit sha1_base64="j/4hw2QGileIezjj/Oh/y7Vw1Cg=">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</latexit><latexit sha1_base64="j/4hw2QGileIezjj/Oh/y7Vw1Cg=">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</latexit><latexit sha1_base64="j/4hw2QGileIezjj/Oh/y7Vw1Cg=">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</latexit><latexit sha1_base64="j/4hw2QGileIezjj/Oh/y7Vw1Cg=">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</latexit>

6 MAPLima

F789 Aula 15

GravidadeemMecânicaQuânticaNo limite nao quantico, caımos na trajetoria classica definida pelo

princıpio de mınima acao

�

Z t2

t1

dtLclassica(t)

~ = �

Z t2

t1

dt⇥ 12mz2 �mgz

~⇤= 0 e m se cancela.

Lembra do teorema de Ehrenfest?

md2

dt2hRi = d

dthPi = �hrV (R)i se rV (R) = mgz entao

md2

dt2hRi = �mgz (nao depende de ~ nem de m)

Para ter efeito quantico nao trivial precisamos ter dependencia com

~ e consequentemente com m tambem.

•Ate 1975, nao se sabia de nenhum experimento direto que permitisse

usar o termo m�grav na equacao de Schrodinger.

•Queda livre de partıculas elementares ! equacao de Ehrenfest e

suficiente para mostrar que nao teremos efeitos quanticos.

• Experimento sobre “peso de protons” de V. Pound et al. tambem

nao estabeleceu efeitos gravitacionais quanticos (eles mediram

deslocamentos de frequencias que nao dependem de ~ explicitamente).<latexit sha1_base64="Mw9JY24YljE8hpPcUGc4wEnRl7I=">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</latexit><latexit sha1_base64="Mw9JY24YljE8hpPcUGc4wEnRl7I=">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</latexit><latexit sha1_base64="Mw9JY24YljE8hpPcUGc4wEnRl7I=">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</latexit><latexit sha1_base64="Mw9JY24YljE8hpPcUGc4wEnRl7I=">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</latexit>

7 MAPLima

F789 Aula 15

Efeito gravitacional na escala microscopica e difıcil de detectar. Porque?

As forcas envolvidas sao muito pequenas. Para ilustrar, considere o estado

ligado do atomo de hidrogenio. Sabemos que o tamanho tıpico do atomo

e estimado por a0 ⌘ raio de Bohr =~2

e2me. Como seria “a0” se a forca de

ligacao fosse gravitacional? Para calcular isso, troque o proton por um

neutron, o que equivale a trocar � e2

r2por � Gmemn

r2. Obterıamos com

isso, um raio de Bohr gravitacional “a0” =~2

Gm2emn

. A forca gravitacional

entre o neutron e eletron e cerca de 1039vezes menor que a forca eletrica

entre o proton e o eletron, poise2

Gmemn⇡ 2.3⇥ 10

39.

Isto fornece um raio de Bohr gravitacional de 1031cm ⇡ 10

13anos-luz.

(Tamanho algumas vezes maior que o raio estimado do universo - 1028cm)

GravidadeemMecânicaQuântica

8 MAPLima

F789 Aula 15

InterferênciaQuânticaInduzidaporGravidadeExperimentodecorrelaçãocomnêutrons(massamn)

C

Pacote de onda

de um neutron

D

A

B Pacote e reunido em

Medida de Interferencia

`1

`2

Pacote e dividido

em A: pedaco por ABD

outro por ACD

dois casos

8>>>>>><

>>>>>>:

1) ABCD em um plano horizontal: potencial gravitacional igual

em ambos os percursos.

2) ABCD rodado de � em torno de AC. Potencial gravitacional

no percurso AB e igual ao do CD, mas no percurso BD e

diferente do potencial do percurso AC.

�A

CB

D

`1

`2

� grav=mng2sin�

9 MAPLima

F789 Aula 15

PartículacarregadasujeitaàumcampoeletromagnéticoUm pouco sobre cargas interagindo com campos eletromagneticos

1) Comeca com a Forca de Lorentz md2r

dt2= F = e

⇥E+

v ⇥B

c

⇤.

2) Das Equacoes de Maxwell, percebemos a importancia de � e A e

temos

8>>>>>><

>>>>>>:

r.B = 0 ! B = r⇥A

r⇥E = � 1c@B@t ! r⇥

�E+ 1

c@A@t

�= 0

) E+ 1c@A@t = �r� ! E = �r�� 1

c@A@t

3) Equacoes de Movimento de Newton mx = e�Ex + y

Bz

c� z

By

c

�!

mx = e�� @�

@x� 1

c

@Ax

@t+

y

c

⇥@Ay

@x� @Ax

@y

⇤� z

c

⇥@Ax

@z� @Az

@x

⇤

4) Lagrangeana: construıda de tal forma qued

dt

� @L@qi

�=

@L

@qiforneca a

equacao de movimento. Verifique que a expressao serve:

L =1

2mr2 +

e

cr.A(r, t)� e�(r, t)

<latexit sha1_base64="yL14Ew2oeo0swa6flyKb+O99LMA=">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</latexit><latexit sha1_base64="yL14Ew2oeo0swa6flyKb+O99LMA=">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</latexit><latexit sha1_base64="yL14Ew2oeo0swa6flyKb+O99LMA=">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</latexit><latexit sha1_base64="yL14Ew2oeo0swa6flyKb+O99LMA=">AAAOvHicvVfNbxtFFH9tobQLNmk5chkRgRK1sexQqaVSUT9IxSGgIpG0Uhyi8Xocj7If7s448Yf8T/bGFQ6IA0fuvHk7+71bh1ZgaWdn39e893tv3owHE08q3e3+euXqtQ8+vP7RjZvOx5+02p9u3Lp9qMJp5IoDN/TC6NWAK+HJQBxoqT3xahIJ7g888XJw9szwX56LSMkw+FnPJ+LY56eBHEmXaySFt65L+Ar6oEHADMclHIAPDCYQwhRcHBkoHAcQoQRDCsfZKY4KvyQEpBnh9yl9DUnD6PlW2idbRlqAh49GacM1GgH8id+S5BWs0I8+euMU/OnBNuo+Ix0Bf6BeZt/Mn+M8svQh+bhPFEG+LfDb2GXpw1CzDyPy2UX7Q/gFdvHdxxhHyI1QfkV0TZwV0h7l+M8txcRjaBLj6GNkA2vxDDmZ9B5K36GVsxUz7jn5ZhAwsamC5lPrh0syyTpRbp0O2e3bsYzbLuH2nc3UHrzGfHLE6W+7UozVD0ibwQXlxoO7lHmTTxfHAT6+zR2nbMe5jND+b4iui5QMdePlBMZIYwU/mOXmY3tC3/VSdbHo1BMj4ZAdYStuSVWmKKqYm63TRz7HuYfjCvEqomuy2CWpiNAd4zqmuk19XlRy4VDOm2zns1iMM8tQs2f12nvWx51S/fRydZFRDfqccmOy4tVUUlmC4WzVGP/lvE0qc2tNzV/O53KNNHudrLtNOcxjbLwbUw0LtBPavvX/emcqZacRP1bYKeuw3yN7l7NmVv0v4zLoCuryxV3X3IW+pi7U3H1C7IFxLZluHaacH3G8IEpAfcGhrj2k80UjZVbpwVu4ygnS75BcLDUv5fkpSixSTJjdWYn0okZ6XunB241Zy+Jnlafs/aOG00Og/s6abCX5rs/TzEb2LnXwhBB8W69wCN+8jQzr6mlVPBXXrz1fE9NlbDT5P288R6uIZxXx/jE1+bP4FzEt3oJLU0x1dFG7R+9RTe/TjczIBfgW9ObwEOLbVoC71tzcpvAXopOd+tp6Endanzhmrwu7a4s3reRelXm9tSb6/YbIsxy9Rnxk6UR4N5uJpfLNJI4tSG+fPL2pFiVF2uV+z3Uyv9LjOtSpD+mMkmhdpoglb06dbYYeRtQvlbWoSOccx4eF23K137zPs19CMO4e5ibsp7gX78vmnsxKnUHk9k6Sq0wr1svfyMzJs1Wyexe1tu0uEWnfq5dyUjyck43NbqdLP1ad9OxkE+zvxcnGm/4wdKe+CLTrcaWOet2JPl7ySEvXEyunP1Viwt0zfiqOcBpwX6jjJf35WrEvkTJkozDCJ9CMqHmNJfeVmvsDlPS5HqsyzxDreEdTPXpwvJTBZKpF4MYLjaYe0yEz/+TYUEbC1d4cJ9yNJPrK3DGPuKvx/54BoVcOuTo53O30up3eT/c2H+9aOG7A5/AFgtyD+/AYvocX+JfQbX3TOmmNW7L9bXvYPmv7sejVK1bnMyj82uf/AOmS5Jo=</latexit>

10 MAPLima

F789 Aula 15

Partículacarregadasujeitaàumcampoeletromagnético5) Momento canonico (direcao x) px ⌘ @L

@x= mx+

e

cAx(r, t)

Ou seja, p = mr+e

cA(r, t) (nao e simplesmente mv)

6) Construa a Hamiltoniana Classica

H = p.r� L =1

2m

�p� e

cA�2

+ e�

7) Utilize as ferramentas as desenvolvidas para estudar o analogo quantico.

Para ilustrar, suponha inicialmente que E e B possam ser derivados de A

e � independentes do tempo. Isto e:

�(r, t) = �(r) e A(r, t) = A(r), com

8><

>:

E = �r�

B = r⇥A

H = 12m

�p� e

cA�2

+ e�

Na mecanica quantica, A e � sao funcoes do operador posicao R da partıcula.

Sendo assim, como [Ri, Pj ] = i~�ij ! [Pi, Aj ] 6= 0. Isso exige cuidado para

lidar com H. Por exemplo, a parte envolvendo o operador

�P� e

cA�2

= P2 � e

c

�A.P+P.A| {z }

�+

e2

c2A2

note simetria em A e P e que H e Hermiteano<latexit sha1_base64="2OPuE1i4uThZ8f7XmoFTJaQCbV4=">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</latexit><latexit sha1_base64="2OPuE1i4uThZ8f7XmoFTJaQCbV4=">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</latexit><latexit sha1_base64="2OPuE1i4uThZ8f7XmoFTJaQCbV4=">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</latexit><latexit sha1_base64="2OPuE1i4uThZ8f7XmoFTJaQCbV4=">AAAOBHicrVdLjxtFEO4ECGGxYQNHLiNWoKyyWPYqAYRkKZsVkiPxMI9NIu0sVnvctic7L+axZLF84MJf4cIBhLjyI7hxBA4oB25w56vqHs+MPZNFCEs97q6urq/6q6rumXHkuUna7f5y6fJTTz9z5dmrz20932q/8OL2tZfuJWEWO+rICb0wfjCWifLcQB2lbuqpB1GspD/21P3x6SHN3z9TceKGwafpeaROfDkL3KnryBSi8NqVtnhdbAlbpEKJR3guxC2xKyzxvgiFD1kAWYixIyT6f6G56JPkupigH0PnD/E7S5awQGtttEiMMLIx+7nIoHfGKFPoS6xfoB+hF8O6i38PK97D+jq5DZwQYwv2lnj20fyK1BI3KrYVW3LwPGAvrrPdMTQs6CzFHtbtQmKbvVtrreDiQ/huiQTjh7C9x3u0StaiGo+qWM2+le0cYFzvZdUfYj0wbD9GI99cYEdgSqGfR0yxp37J4hnG5T0XNt9klEPYDGAhBXYGXy1uAzx9IHicBQHHJOCZQ8h+hX7C+SBhvYnPAfipMtYpRW+drTc4EzSnZd56zNs+vFnyzBi4swpnkVlfz3aV63z9rvgMNilGFnQp9+aQW7U8vcU8HXFmemhfMssSDFiwq4AZM1s6AlqunxOODcnPsGsPTxcyPaezXTI+sZ/xTIxxyOsDsOyhP+Mx1dJvbF9XYcewYTf4PFxZJ58zE1+S7XHuZMCnuM6Njq5uXXnlXCJctZH97xr0cn5uat2p0SJUyh3iS9dXbHiK+ayQnB85d1ZN/GivVas5cjmKZUzaHVmLeF+T1e40CrFLI59962B0l9nSlfYO2y4Y1gj1FdtvnN+s5k22mk+C/gVauxzTqn2HT/El+zyGfMYsLLhiE947za3HtM91VMhsrvkxckLXeblKqqvvmNXNa1M+rwi5upvc2oD5q6/7/7/qi5rf4ruK8qLKTl1VfcDVQrtwuBr1CbhenXX3xcETKqYpbxNz3k9hPcBd+3clZ0PO59hUTLyqLbd0Kxf4H9fgT9j3/Nb9E55nHK1OKefLKz4xPIXmfNM30N4q2zQmMeYBVcfklLFH0CS9IXoP2Y+4otHnKrXBwpi9sblKPT5NR0B2sWpp1lEc5zxD524ovoB8E3O4wjxowKT8pAh0eb9VZu7y7kKO0SNGVLzLzJzfmoP8BC9OpWq2eEY7XjFE/Axq8IYmgiRR5lYP2ftyhBRrFHdJEY36fGi+mYu2WVfD/1xXfSBV7ehqa7ZW4Nuc4/oeyKOkoFXF66zZv7E27qzplz3Ud33VD/JP+5L3yqtJdjGDF7VynAN+91Eif3tTfCu75i3AN+fBvzs31iO2qVO+uwcbGo9ZSnzTex7NSPYPFTza3ul2uvyzNjs909kR5jccbf9sT0In81WQOp5MkuNeN0pPFjJOXcdTyy07S1QknVM5U8foBtJXycmCP2KW1muQTKxpGKMFqcXS8oqF9JPk3B9D05fpPFmfI2Hd3HGWTt8+WbhBlKUqcDTQNPOsNLToi8iauLFyUu8cHenELny1nLmMpZPiu4lI6K1vebNzb7/T63Z6H93cub1v6LgqXhGvIqV7eHG8DYqHeHV0Wl+1vml91/q+/XX72/YP7R+16uVLZs3LovJr//QP8cO1CA==</latexit>

11 MAPLima

F789 Aula 15

A Hamiltoniana pode ser reescrita (formalismo de Heisenberg) como

H =1

2m

�P� e

cA�2

+ e� =⇧2

2m+ e�

Ja sabemos que mdR

dt= ⇧ = P� e

cA. Quanto vale m

d2R

dt2=

d⇧

dt?

Facamos a conta para a componente i = 1, isto e:

d⇧1

dt=

1

i~ [⇧1, H] =1

i~ [⇧1,⇧2

2m+ e�]

Mostre que

8>>><

>>>:

[⇧1,⇧21] = 0

[⇧1,⇧22] =

i~ec

�m

dR2dt B3 +B3m

dR2dt

�

[⇧1,⇧23] = � i~e

c

�m

dR3dt B2 +B2m

dR3dt

�

[⇧1, e�] = �i~e @�@R1

= i~eE1

e obtenha

para a componente 1 :d⇧1

dt= eE1 +

e

2c

��dRdt

⇥B�1��B⇥ dR

dt

�1

Enfim temos: md2R

dt2=

d⇧

dt= e

⇥E+

1

2c

�dRdt

⇥B�B⇥ dR

dt

�]

A versao quantica da forca de Lorentz (teorema de Ehrenfest).<latexit sha1_base64="g4vcpEKEVbCbVG0ro34prEPYfrU=">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</latexit><latexit sha1_base64="g4vcpEKEVbCbVG0ro34prEPYfrU=">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</latexit><latexit sha1_base64="g4vcpEKEVbCbVG0ro34prEPYfrU=">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</latexit><latexit sha1_base64="g4vcpEKEVbCbVG0ro34prEPYfrU=">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</latexit>

Partículacarregadasujeitaàumcampoeletromagnético

12 MAPLima

F789 Aula 15

PartículacarregadasujeitaàumcampoeletromagnéticoComo fica a equacao de Schrodinger com � e A?

hr0|H|↵, t0; ti = i~ @@t

hr0|↵, t0; ti

Comecemos por

1

2mhr0|

�P� e

cA(R)

�2|↵, t0; ti =

=1

2m

�� i~r0 � e

cA(r0)

�hr0|

�P� e

cA(R)

�|↵, t0; ti =

=1

2m

�� i~r0 � e

cA(r0)

��� i~r0 � e

cA(R0)

�)hr0|↵, t0; ti

Note que o gradiente da esquerda “pega”todo mundo (inclusive o A(r0)

da direita). Com isso, a equacao de Schrodinger e dada por:

1

2m

�� i~r0 � e

cA(r0)

��� i~r0 � e

cA(r0)

�hr0|↵, t0; ti+ e�(r0)hr0|↵, t0; ti =

= i~ @@t

hr0|↵, t0; ti

Como seria a nova definicao de j, o fluxo de probabilidade? Lembram

de como definı-lo? Use@⇢

@t+r.j = 0, ⇢ =

⇤ , com = hr0|↵, t0; ti

<latexit sha1_base64="BaRj9syhmD8hZClocGUg71sC+R0=">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</latexit><latexit sha1_base64="BaRj9syhmD8hZClocGUg71sC+R0=">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</latexit><latexit sha1_base64="BaRj9syhmD8hZClocGUg71sC+R0=">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</latexit><latexit sha1_base64="BaRj9syhmD8hZClocGUg71sC+R0=">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</latexit>

13 MAPLima

F789 Aula 15

PartículacarregadasujeitaàumcampoeletromagnéticoPara calcular j, o fluxo de probabilidade, comece por

@⇢

@t=@ ⇤

@t + ⇤ @

@t,

e use a equacao de Schrodinger do slide anterior para obter:

@⇢

@t=

1

i~⇥� hr0|⇧

2

2m|↵, t0; ti⇤hr0|↵, t0; ti+ hr0|↵, t0; ti⇤hr0|

⇧2

2m|↵, t0; ti.

Note que a parte em � cancelou, pois �⇤ = �. A tarefa se resume em colocar

a expressao acima na forma:@⇢

@t+r0j = 0. Depois de algum trabalho, temos:

j =~mIm( ⇤r0 )� e

mcA| |2

que poderia ter sido obtido, com

8><

>:

troca p ! p� emcA

~ir

0 � emcA(r0) ! representacao r

Quando escrevemos a funcao de onda na forma (r0, t) =p⇢ exp

� iS~�,

S real, obtivemos Im( ⇤r0 ) =⇢

~r0S. Assim, o j, pode ser escrito por:

j =~m

⇢

~r0S � e

mcA⇢ =

⇢

m

�r0S � e

cA�

Mostre, seguindo o que fizemos para A = 0, que

Zd3r0j =

h⇧im

.<latexit sha1_base64="JJ+eB6u5tUJ2ZFhLuBLrJvKTQa4=">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</latexit><latexit sha1_base64="JJ+eB6u5tUJ2ZFhLuBLrJvKTQa4=">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</latexit><latexit sha1_base64="JJ+eB6u5tUJ2ZFhLuBLrJvKTQa4=">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</latexit><latexit sha1_base64="JJ+eB6u5tUJ2ZFhLuBLrJvKTQa4=">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</latexit>

14 MAPLima

F789 Aula 15

TransformaçãodeGaugeEstamos prontos para estudar transformacao de Gauge. Considere primeiro:

� ! �+ � e A ! A, onde � e constante - nao depende de r e t.

Esta transformacao ja foi estudada antes (mudanca no zero da energia).

Uma transformacao mais interessante e:

8><

>:

� ! �� 1c@�@t

A ! A+r�(r, t)

Cujos campos

8>><

>>:

E = �r(�� 1c@�@t )�

1c

@@t (A+r�) = �r�� 1

c@@tA

B = r⇥ (A+r�) = r⇥A+r⇥r�| {z } = r⇥A

0

ficam inalterados e ) a trajetoria classica da partıcula carregada sob

acao destes campos nao se altera (a forca de Lorenz nao se altera).

Queremos estudar os efeitos quanticos da transformacao de Gauge.

Para isso, considere um caso particular| {z }

8><

>:

� ! �

A ! A+r�(r)

�(r, t) = �(r) nao depende de t.<latexit sha1_base64="4FVudDJFXhYdu0xTk9S4PtzTLj4=">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</latexit><latexit sha1_base64="4FVudDJFXhYdu0xTk9S4PtzTLj4=">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</latexit><latexit sha1_base64="4FVudDJFXhYdu0xTk9S4PtzTLj4=">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</latexit><latexit sha1_base64="4FVudDJFXhYdu0xTk9S4PtzTLj4=">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</latexit>

15 MAPLima

F789 Aula 15

TransformaçãodeGauge

Considere B = Bz. Tal campo pode ser derivado de

8>>>>>><

>>>>>>:

Ax = �By2

Ay = Bx2

Az = 0

pois,

B = r⇥A =�@Az

@y� @Ay

@z

�x+

�@Ax

@z� @Az

@x

�y +

�@Ay

@x� @Ax

@y

�z = Bz

ou de

8>>>>>><

>>>>>>:

Ax = �By

Ay = 0

Az = 0

pela mesma razao, ou seja B = r⇥A = Bz.

Note que os dois A0s se relacionam por Anovo = Avelho +r��Bxy

2| {z }

�

�(r)<latexit sha1_base64="ROHTfGjc5B+yop5WIOtHUHYywEk=">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</latexit><latexit sha1_base64="ROHTfGjc5B+yop5WIOtHUHYywEk=">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</latexit><latexit sha1_base64="ROHTfGjc5B+yop5WIOtHUHYywEk=">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</latexit><latexit sha1_base64="ROHTfGjc5B+yop5WIOtHUHYywEk=">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</latexit>

16 MAPLima

F789 Aula 15

TransformaçãodeGauge

Para B = Bz, considere a forca de Lorentz

F = e�v ⇥B

c

�=

e

c

������

x y zvx vy vz0 0 B

������=

eB

c(vyx� vxy) =

=eB

cv?(sin'x� cos'y| {z }) = �eB

cv?(e')

� e'

@ forca na direcao z e a forca sempre aponta ? a v e no plano xy.

) a partıcula realiza uma espiral (movimento circular no plano xy).

'e'v?

vk

B = Bz

v

x

y

z

v?

e'

e''

'

'

x

y

17 MAPLima

F789 Aula 15

TransformaçãodeGauge

mostre isso usando a transformação de Gauge para redefinir a Lagrangeana e consequentemente p.

Considere agora a equacao de Hamiltondpx

dt= �@H

@x

onde H =1

2m

�p� e

cA�2

+ e �|{z} =1

2m

�p2 � e

cp.A� e

cA.p+

e2

c2A2

�

0

dpx

dt= �@H

@x

8><

>:

= 0 se usarmos A = (�By, 0, 0)

6= 0 se usarmos A = (�By2 ,

Bx2 , 0)

Isto mostra que em geral o momento canonico nao e invariante de Gauge.

E importante, entretanto que ⇧ seja invariante de Gauge, pois ele

e responsavel pela trajetoria que depende apenas das condicoes iniciais

e da forca de Lorentz (que nao depende de �).

Como ⇧ e p estao relacionados por ⇧ = p� e

cA ! p e

e

cA devem mudar

de forma a nao alterar ⇧| {z } .

18 MAPLima

F789 Aula 15

TransformaçãodeGaugeE razoavel esperar que os valores esperados de quantidades classicas

invariantes mediante transformacoes de Gauge tambem sejam invariantes.

Assim, esperamos que hRi e h⇧i nao mudem mediante transformacao de

Gauge, mas hPi mude. Em outras palavras:

Se

8><

>:

|↵i um estado na presenca de A

f|↵i um estado na presenca de eA = A+r�

entao, esperamos que:

8><

>:

h↵|R|↵i = he↵|R|e↵i

h↵|P� ecA|↵i = he↵|P� e

ceA|e↵i

com h↵|↵i = he↵|e↵i = 1.

Vamos construir um operador G tal que |e↵i = G|↵i e que quando inserido

em he↵|R|e↵i = h↵|G†RG|↵i = h↵|R|↵i e ) G†RG = R.

Ou, de forma semelhante, para o momento cinematico:

he↵|P� e

ceA|e↵i = h↵|G†�P� e

cA� e

cr�

�G|↵i = h↵|

�P� e

cA�|↵i

e ) G†�P� e

cA� e

cr�

�G =

�P� e

cA�

<latexit sha1_base64="KL0qgRq35mGA+0GthqD6vysNXHQ=">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</latexit><latexit sha1_base64="KL0qgRq35mGA+0GthqD6vysNXHQ=">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</latexit><latexit sha1_base64="KL0qgRq35mGA+0GthqD6vysNXHQ=">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</latexit><latexit sha1_base64="KL0qgRq35mGA+0GthqD6vysNXHQ=">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</latexit>

19 MAPLima

F789 Aula 15

TransformaçãodeGauge

(c.q.f.)

(c.q.f.)

Tiramos do chapeu que G = exp� ie~c�(R)

�e a solucao. Inspirada na

condicao h↵|↵i = he↵|e↵i = 1 ! G†G = 1

Podemos verificar se funciona para o caso

G†RG = exp�� ie

~c�(R)�R exp

� ie~c�(R)

�= R pois, [R,�(R)] = 0

De forma semelhante, podemos verificar:

G†�P� e

cA� e

cr�

�G = exp

�� ie

~c�(R)��P� e

cA� e

cr�

�exp

� ie~c�(R)

�

= exp�� ie

~c�(R)�P exp

� ie~c�(R)

��e

cA� e

cr�

| {z }[A,�] = 0 e [g(�),�] = 0

falta exp�� ie

~c�(R)�P exp

� ie~c�(R)

�que pode ser calculado com auxılio

da expressao

[P, G(R)] = �i~rG(R) ! PG(R)�G(R)P = �i~ exp� ie~c�(R)

� ie~cr�

ou seja PG(R)�G(R)P = Ge

cr� ! G†�PG(R)�G(R)P

�= G†G

e

cr�

G†PG = P+e

cr� ! G†�P� e

cA� e

cr�

�G = P� e

cA

<latexit sha1_base64="RU9loqQ5mF6cfVb2pphRMUzEnnM=">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</latexit><latexit sha1_base64="RU9loqQ5mF6cfVb2pphRMUzEnnM=">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</latexit><latexit sha1_base64="RU9loqQ5mF6cfVb2pphRMUzEnnM=">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</latexit><latexit sha1_base64="RU9loqQ5mF6cfVb2pphRMUzEnnM=">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</latexit>

20 MAPLima

F789 Aula 15

TransformaçãodeGaugeOlhemos a equacao de Schrodinger para |↵, t0; ti com o potencial vetor A. Se⇥ 1

2m

�P� e

cA�2

+ e�⇤|↵, t0; ti = i~ @

@t|↵, t0; ti e satisfeita, a expectativa e que

^|↵, t0; ti satisfaca a equacao:⇥ 1

2m

�P� e

cA� e

cr�

�2+ e�

⇤ ^|↵, t0; ti= i~ @

@t^|↵, t0; ti.

Verifiquemos se a seguinte relacao e verdadeira ^|↵, t0; ti = exp� ie~c�(R)

�|↵, t0; ti?

Para tanto, insira esta expressao na equacao acima de Schrodinger para ^|↵, t0; ti,

e multiplique a equacao toda pela esquerda por exp�� ie

~c�(R)�.

Note que exp�� ie

~c�(R)�⇥�

P� e

cA� e

cr�

�2⇤exp

� ie~c�(R)

�=

exp�� ie

~c�(R)��P� e

cA� e

cr�

�exp

� ie~c�(R)

�⇥

⇥ exp�� ie

~c�(R)��P� e

cA� e

cr�

�exp

� ie~c�(R)

�e aplique

exp�� ie

~c�(R)��P� e

cA� e

cr�

�exp

� ie~c�(R)

�= P� e

cA duas vezes. A

equacao que sobra e a equacao de Schrodinger para |↵, t0; ti com A e isso

demonstra a relacao entre ^|↵, t0; ti e |↵, t0; ti.<latexit sha1_base64="tIr1CvHB3EfhoOWd3/zKf/8/Bwo=">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</latexit><latexit sha1_base64="tIr1CvHB3EfhoOWd3/zKf/8/Bwo=">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</latexit><latexit sha1_base64="tIr1CvHB3EfhoOWd3/zKf/8/Bwo=">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</latexit><latexit sha1_base64="tIr1CvHB3EfhoOWd3/zKf/8/Bwo=">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</latexit>

21 MAPLima

F789 Aula 15

TransformaçãodeGauge

Na representacao das coordenadas a relacao entre ^|↵, t0; ti e |↵, t0; ti fica

e ↵(r0, t) = exp

� ie~c�(r

0)� ↵(r

0, t), onde �(r0) e uma funcao real do vetor

posicao r0. Em termos de ⇢ e S, a funcao de onda pode ser escrita por

e ↵, (r0, t) = exp

� ie~c�(r

0)�p⇢ exp

� iS~�=

p⇢ exp

� i~ (S +

e

c�(r0)

�

Ou seja, basta

8><

>:

⇢! ⇢

S ! S +ec�

Como fica o fluxo de probabilidade j =⇢

m

�rS � e

cA�?

Mediante a transformacao de Gauge A ! A+r�, temos

j =⇢

m

�r(S +

e

c�)� e

c(A+r�)

�=

⇢

m

�rS � e

cA�e

) o fluxo de probabilidade e invariante mediante transformacao de Gauge.<latexit sha1_base64="kvM8C25ZQ/L9X4uupPqb5BodQ8M=">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</latexit><latexit sha1_base64="kvM8C25ZQ/L9X4uupPqb5BodQ8M=">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</latexit><latexit sha1_base64="kvM8C25ZQ/L9X4uupPqb5BodQ8M=">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</latexit><latexit sha1_base64="kvM8C25ZQ/L9X4uupPqb5BodQ8M=">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</latexit>

22 MAPLima

F789 Aula 15

Variáveldinâmicafundamental:AouB?Partículaprisioneiraemcabocoaxial

Dois Cilindros metalicos, um dentro do outro.z

⇢a⇢b

L

Partıcula confinada entre ⇢a e ⇢b

Condicoes de contorno para (z, ⇢,')

8>>>>>>>>>><

>>>>>>>>>>:

(0, ⇢,') = 0

(L, ⇢,') = 0

(z, ⇢a,') = 0

(z, ⇢b,') = 0

O espectro de energia e encontrado

resolvendo a equacao � ~22m

r2 = E

em coordenadas cilındricas e condicoes

de contorno abaixo.

23 MAPLima

F789 Aula 15

OEspectrodeEnergia

m ⌘ m-esima raiz

A equacao de Schrodinger � ~22m

r2 = E em coordenadas cilındricas fica:

@2

@⇢2+

1

⇢

@

@⇢+

1

⇢2@2

@'2+@2

@z2= �2mE

~2 ! com (z, ⇢,') = R(⇢)Q(')Z(z)

temos

8>>>>>><

>>>>>>:

d2Z(z)dz2 = �k20Z ! Z(z) = exp (±ik0z)

8><

>:

Z(0) = Z(L) = 0

Z(z) = sin k0z;

k0 = `⇡L , ` inteiro

d2Q(')d'2 = �⌫2Q ! Q(') = exp (±i⌫')

(Q(0) = Q(2⇡)

⌫ inteiro

Para ⇢ temos:@2R

@⇢2+

1

⇢

@R

@⇢� ⌫2

⇢2R+

�2mE

~2 � k20�R = 0

se tomarmos

(x = k⇢

k =q

2mE~2 � k20

! d2R

dx2+

1

x

dR

dx+ (1� ⌫2

x2)R = 0 com

solucoes conhecidas R = AJ⌫(k⇢) +BN⌫(k⇢). Aplique R(k⇢a) = R(k⇢b) = 0, e

obtenha

8><

>:

AJ⌫(k⇢a) +BN⌫(k⇢a) = 0

AJ⌫(k⇢b) +BN⌫(k⇢b) = 0

J⌫(k⇢a)N⌫(k⇢b)� J⌫(k⇢b)N⌫(k⇢a) = 0| {z }

Isso quantiza k e a energia E`m⌫ =~22m

�k2m⌫ + (

`⇡

L)2�

<latexit sha1_base64="VTkPyO8G6sVPEpNsT35l6LSEbjw=">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</latexit><latexit sha1_base64="VTkPyO8G6sVPEpNsT35l6LSEbjw=">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</latexit><latexit sha1_base64="VTkPyO8G6sVPEpNsT35l6LSEbjw=">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</latexit><latexit sha1_base64="VTkPyO8G6sVPEpNsT35l6LSEbjw=">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</latexit>

24 MAPLima

F789 Aula 15

CampomagnéticoconstantenointeriordocaboApartículanãovêoB.SóvêoA!

Suponha agora um campo magnetico B = B0z no interior do cilindro ⇢ < ⇢a.

Quanto vale o potencial vetor para ⇢ < ⇢a e ⇢ > ⇢a?

Caso ⇢ > ⇢aZsuperfıciecircular deraio ⇢>⇢a

B.da = B0⇡⇢2a =

Zsuperfıciecircular deraio ⇢>⇢a

(r⇥A).da =

Ilinha

fechadasobre o anelde raio ⇢>⇢a

A.d`

mas, d` = ⇢d' e ) A.d` = ⇢A�d' ! B0⇡⇢2a = ⇢A�

I

linhafechada

sobre o anel

d' = 2⇡⇢A�

) A� =B0⇢2a2⇢

! Mostre que B = r⇥A = 0.

Caso ⇢ < ⇢aZsuperfıciecircular deraio ⇢>⇢a

B.da = B0⇡⇢2 =

Zsuperfıciecircular deraio ⇢<⇢a

(r⇥A).da =

Ilinha

fechadasobre o anelde raio ⇢<⇢a

A.d`

de novo, d` = ⇢d' e ) A.d` = ⇢A�d' ! B0⇡⇢2 = ⇢A�

I

linhafechada

sobre o anel

d' = 2⇡⇢A�

) A� =B0⇢

2! Mostre que B = r⇥A = B0z.

25 MAPLima

F789 Aula 15

NovoEspectrodeEnergiaApartículanãovêoB.SóvêoA!

A solucao do problema, com campo magnetico constante no interior do cilindro

⇢ < ⇢a, sera obtida com

(A⇢ = Az = 0 e A� = B0⇢

2 p/ ⇢ < ⇢a ! B = B0z

A⇢ = Az = 0 e A� = B0⇢2a

2⇢ p/ ⇢ > ⇢a ! B = 0

Note que, embora a partıcula nao sinta o campo B na regiao ⇢ > ⇢a, ela sente

A, pois A 6= 0 nesta regiao.

Para encontrar os autovalores para este novo problema, precisamos trocar

r por r� ie

~cA. Lembrando que r em coordenadas cilındricas e:

r = ⇢@

@⇢+ z

@

@z� '

1

⇢

@

@'temos que trocar

@

@'! @

@'� ie

~cB0⇢2a2⇢

Tal troca resulta em uma mudanca no espectro de energia! (surpreendente, pois

o campo e zero onde a partıcula pode estar.) A equacao de Schrodinger em

coordenadas cilındricas e:

@2

@⇢2+

1

⇢

@

@⇢+

1

⇢2@2

@'2+@2

@z2= �2mE

~2 e usando a troca sugerida com

a =e

~cB0⇢2a2

, temos@2

@⇢2+

1

⇢

@

@⇢+

1

⇢2@2

@'2+@2

@z2� a2

⇢2 � 2ai

⇢2@

@'= �2mE

~2

26 MAPLima

F789 Aula 15

OpotencialvetorAéavariáveldinâmicafundamental

E`m⌫0 =~22m

�k2m⌫0 + (

`⇡

L)2�.

m ⌘ m-esima raiz

@2

@⇢2+

1

⇢

@

@⇢+

1

⇢2@2

@'2+@2

@z2� a2

⇢2 � 2ai

⇢2@

@'= �2mE

~2 e = R(⇢)Q(')Z(z)

nos leva a:

8>>>>>><

>>>>>>:

d2Z(z)dz2 = �k20Z ! Z(z) = exp (±ik0z)

8><

>:

Z(0) = Z(L) = 0

Z(z) = sin k0z;

k0 = `⇡L , ` inteiro

d2Q(')d'2 = �⌫2Q ! Q(') = exp (±i⌫')

(Q(0) = Q(2⇡)

⌫ inteiro positivo

Para ⇢ agora temos:

@2R

@⇢2+

1

⇢

@R

@⇢� ⌫2

⇢2R� a2

⇢2R� 2ai

⇢2(±i⌫)R+

�2mE

~2 � k20�R = 0

@2R

@⇢2+

1

⇢

@R

@⇢� 1

⇢2(⌫ ± a)2R+

�2mE

~2 � k20�R = 0 se

(x = k⇢

k =q

2mE~2 � k20

! d2R

dx2+

1

x

dR

dx+ (1� (⌫ ± a)2

x2)R = 0 com R = AJ⌫0(k⇢) +BN⌫0(k⇢).

Note que ⌫0 = |⌫ ± a| nao e inteiro. Aplique R(k⇢a) = R(k⇢b) = 0,

!

8><

>:

AJ⌫0(k⇢a) +BN⌫0(k⇢a) = 0

AJ⌫0(k⇢b) +BN⌫0(k⇢b) = 0

J⌫0(k⇢a)N⌫0(k⇢b)� J⌫0(k⇢b)N⌫0(k⇢a) = 0| {z }<latexit sha1_base64="7ow9aJN2OYAgRr58Gj5PMDtgGC8=">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</latexit><latexit sha1_base64="7ow9aJN2OYAgRr58Gj5PMDtgGC8=">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</latexit><latexit sha1_base64="7ow9aJN2OYAgRr58Gj5PMDtgGC8=">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</latexit><latexit sha1_base64="7ow9aJN2OYAgRr58Gj5PMDtgGC8=">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</latexit>

27 MAPLima

F789 Aula 15

OefeitodeAharanov-BohmExperimentodecorrelação

CilindroB 6= 0

fonte de partıculasregiao de interferencia

Pacote de ondas antes de passar pelo cilindro impenetrável

Pacote dividido após passar pelo cilindro: Parte do pacote passou por cima. Parte passou por baixo.

B = 0Outro caminho

de Feynman

Um caminho de Feynman

FI

Queremos estudar como a probabilidade de encontrar a partıcula

na regiao de interferencia I depende do fluxo magnetico. O estudo

sera feito com integrais de Feynman. A partıcula nao pode penetrar

no cilindro, assim, de novo, ela percebe o potencial vetor, mas nao

vai onde B 6= 0. B = 0 fora do cilindro.

28 MAPLima

F789 Aula 15

OefeitodeAharanov-Bohm