MARCOS AURÉLIO LEANDRO DA COSTA FORMULAÇÃO DA MECÂNICA QUÂNTICA POR INTEGRAIS DE CAMINHO DE FEYNMAN

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▼❆❘❈❖❙ ❆❯❘➱▲■❖ ▲❊❆◆❉❘❖ ❉❆ ❈❖❙❚❆

❋❖❘▼❯▲❆➬➹❖ ❉❆ ▼❊❈➶◆■❈❆ ◗❯➶◆❚■❈❆ P❖❘

■◆❚❊●❘❆■❙ ❉❊ ❈❆▼■◆❍❖ ❉❊ ❋❊❨◆▼❆◆

  ❋❖❘❚❆▲❊❩❆ ✲ ❈❊ ✷✵✶✸

  ▼❆❘❈❖❙ ❆❯❘➱▲■❖ ▲❊❆◆❉❘❖ ❉❆ ❈❖❙❚❆ ❋❖❘▼❯▲❆➬➹❖ ❉❆ ▼❊❈➶◆■❈❆ ◗❯➶◆❚■❈❆ P❖❘ ■◆❚❊●❘❆■❙ ❉❊ ❈❆▼■◆❍❖ ❉❊ ❋❊❨◆▼❆◆

  ▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ ❋ís✐❝❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦✲ ❣✉❡✐r❛ ❞❛ ❈♦st❛ ❋✐❧❤♦

  ❋❖❘❚❆▲❊❩❆ ✲ ❈❊ ✷✵✶✸

  

Dados Internacionais de Catalogação na Publicação

Universidade Federal do Ceará

Biblioteca do Curso de Física

C874f Costa, Marcos Aurélio Leandro da. 60 f.: il., enc.; 30 cm. Marcos Aurélio Leandro da Costa. – Fortaleza, 2014. Formulação da Mecânica Quântica por Integrais de Caminho de Feynman / Orientador: Prof. Dr. Raimundo Nogueira da Costa Filho. Departamento de Física, Fortaleza, 2014. Monografia (graduação) - Universidade Federal do Ceará, Centro de Ciências, Título.

  1. Mecânica Quântica. 2. Integrais de Caminho. 3. Propagador de Feynman. I. CDD 530.12

  ▼❆❘❈❖❙ ❆❯❘➱▲■❖ ▲❊❆◆❉❘❖ ❉❆ ❈❖❙❚❆ ❋❖❘▼❯▲❆➬➹❖ ❉❆ ▼❊❈➶◆■❈❆ ◗❯➶◆❚■❈❆ P❖❘ ■◆❚❊●❘❆■❙ ❉❊ ❈❆▼■◆❍❖ ❉❊ ❋❊❨◆▼❆◆

  ▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ ❋ís✐❝❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚í✲ t✉❧♦ ❞❡ ❇❛❝❤❛r❡❧ ❡♠ ❋ís✐❝❛✳

  ❆♣r♦✈❛❞❛ ❡♠✿ ✵✸✴✵✶✴✷✵✶✹ Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆✳ ❞❛ ❈♦st❛ ❋✐❧❤♦

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮ ❖r✐❡♥t❛❞♦r

  Pr♦❢✳ ▼s✳ ❏♦ã♦ P❤✐❧✐♣❡ ▼❛❝❡❞♦ ❇r❛❣❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❛ ■♥t❡❣r❛çã♦ ■♥t❡r♥❛❝✐♦♥❛❧

  ❞❛ ▲✉s♦❢♦♥✐❛ ❆❢r♦✲❇r❛s✐❧❡✐r❛ ✭❯◆■▲❆❇✮ ▼s✳ ❆♥❞❡rs♦♥ ▼❛❣♥♦ ❈❤❛✈❡s ❈✉♥❤❛

  

❘❡s✉♠♦

  ❆ ❢♦r♠✉❧❛çã♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣♦r ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ s✉r❣❡ ❝♦♠♦ ✉♠❛ t❡r❝❡✐r❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✱ ❛♥á❧♦❣❛ ❛s ♦✉tr❛s ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s✱ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ❞❡ ❢♦r♠❛ q✉❡ sã♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✳ ❊♥✲ tr❡t❛♥t♦✱ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❋❡②♥♠❛♥ ♦ ❢♦r♠❛❧✐s♠♦ ❤❛♠✐❧t♦♥✐❛♥♦ é s✉❜st✐t✉í❞♦ ♣❡❧♦ ❢♦r♠❛❧✐s♠♦ ❧❛❣r❛♥❣✐❛♥♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ ✉t✐❧✐③❛❞❛ ♥❛ ❝♦♥str✉çã♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✳ ◆❛ ✐♥t❡❣r❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♦s ♦♣❡r❛❞♦r❡s ❞❡✐①❛♠ ❞❡ s❡r ♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t♦s ❞❡ ❡st✉❞♦✱ t❛❧ ❝♦♠♦ sã♦ ♥❛s ♦✉tr❛s ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s✱ ❞❛♥❞♦ ❧✉❣❛r ❛♦s ♣r♦♣❛❣❛❞♦r❡s✱ q✉❡ t❡♠ ✉♠❛ r❡❧❛çã♦ ✐♥trí♥s❡❝❛ ❝♦♠ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧✱ s❡♥❞♦ r❡s♣♦♥sá✈❡❧ ❞✐r❡t♦ ♣❡❧❛ ❡✈♦❧✉çã♦ ❞♦ ❡st❛❞♦ q✉â♥t✐❝♦✳ ❊st❡ ❡st✉❞♦ ✐♥✐❝✐❛✲s❡ ❝♦♠ ✉♠❛ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛ ❞♦ ❢♦r♠❛❧✐s♠♦ ❧❛❣r❛♥❣✐❛♥♦✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛ ♣♦r ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥✳ ❙❡❣✉❡✲s❡ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞♦ ♣r♦♣❛❣❛❞♦r q✉â♥t✐❝♦ ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣❛r❛ ✉♠❛ ❤❛♠✐❧t♦♥✐❛♥❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ✭s✐st❡♠❛ ❛✉tô♥♦♠♦ ✐s♦❧❛❞♦✮✳ ❆ ♣❛rt✐r ❞❡ss❡s ❝♦♥❝❡✐t♦s✱ ✐♥tr♦❞✉③✲s❡ ❛ ✐❞é✐❛ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ❝❛♠✐♥❤♦✱ ♦♥❞❡ ❛❣♦r❛ t❡♠✲s❡ q✉❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s sã♦ ♣♦ssí✈❡✐s✱ ❛♦ ❝♦♥trár✐♦ ❞❛ ❛çã♦ ❝❧áss✐❝❛ ♥❛ q✉❛❧ ♦ ú♥✐❝♦ ❝❛♠✐♥❤♦ ♣♦ssí✈❡❧ s❡r✐❛ ❛q✉❡❧❡ ❡♠ q✉❡ ❛ ❛çã♦ é ✉♠ ♠í♥✐♠♦✳ ❖ ❡st✉❞♦ ❝♦♥t✐♥✉❛ ❝♦♠ ❛ ✐♥t❡❣r❛❧ ❞❡ tr❛❥❡tór✐❛ ❡ s✉❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛ ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥✳ ❊♠ s❡❣✉✐❞❛✱ ✐♥✈❡st✐❣❛✲s❡ ❛s r❡❣r❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s✱ ❡ t❛♠❜é♠✱ ❛s r❡❣r❛s ♣❛r❛ ✈ár✐♦s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s✳ P♦r ✜♠✱ ✉t✐❧✐③❛✲s❡ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♦❜t✐❞♦s ❞❛ ■♥t❡❣r❛❧ ❞❡ ❈❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ ♣❛r❛ r❡s♦❧✈❡r ♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ❡ ♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❋♦rç❛❞♦✱ ❝❛❧❝✉❧❛♥❞♦✲s❡ s✉❛s ❛✉t♦❢✉♥çõ❡s ❡ ❛✉t♦✈❛❧♦r❡s ❛tr❛✈és ❞♦ ♣r♦♣❛❣❛❞♦r q✉â♥t✐❝♦✱ ♦❜t❡♥❞♦✲s❡ r❡s✉❧t❛❞♦s s❡♠❡❧❤❛♥t❡s ❛♦s ❡♥❝♦♥✲ tr❛❞♦s ♣♦r ❙❝❤rö❞✐♥❣❡r ❡ ❍❡✐s❡♥❜❡r❣✱ ❝♦♠♣r♦✈❛♥❞♦✲s❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ss❛ r❡♣r❡s❡♥t❛çã♦ ❝♦♠ ❛s ♦✉tr❛s ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ P❛❧❛✈r❛s✲❝❤❛✈❡✿ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✳ ■♥t❡❣r❛✐s ❞❡ ❈❛♠✐♥❤♦✳ Pr♦♣❛❣❛❞♦r ❞❡ ❋❡②♥♠❛♥✳

  

❆❜str❛❝t

  ❚❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ❜② ❋❡②♥♠❛♥ ♣❛t❤ ✐♥t❡❣r❛❧s ❛r✐s❡s ❛s ❛ t❤✐r❞ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ q✉❛♥t✉♠ t❤❡♦r②✱ s✐♠✐❧❛r❧② t❤❡ ♦t❤❡r t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s✱ ❙❝❤rö❞✐♥❣❡r ❛♥❞ ❍❡✐s❡♥❜❡r❣✱ s♦ t❤❛t t❤❡② ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t✳ ❍♦✇❡✈❡r✱ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❋❡②♥♠❛♥ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r♠❛❧✐s♠ ✐s r❡♣❧❛❝❡❞ ❜② t❤❡ ▲❛❣r❛♥❣✐❛♥ ❢♦r♠❛❧✐s♠ ❛s ❛ ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧ ✉s❡❞ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ q✉❛♥t✉♠ t❤❡♦r②✳ ■♥ ❢✉♥❝t✐♦♥❛❧ ✐♥t❡❣r❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✱ ♦♣❡r❛t♦rs ♥♦ ❧♦♥❣❡r t❤❡ ♠❛✐♥ ♦❜❥❡❝ts ♦❢ st✉❞②✱ s✉❝❤ ❛s t❤❡② ❛r❡ ✐♥ t❤❡ ♦t❤❡r t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s✱ ❣✐✈✐♥❣ ♣❧❛❝❡ t♦ t❤❡ ♣r♦♣❛❣❛t♦rs✱ ✇❤✐❝❤ ❤❛s ❛♥ ✐♥tr✐♥s✐❝ r❡❧❛t✐♦♥s❤✐♣ ✇✐t❤ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r✱ ❜❡✐♥❣ ❞✐r❡❝t❧② r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ q✉❛♥t✉♠ st❛t❡✳ ❚❤✐s st✉❞② ❜❡❣✐♥s ✇✐t❤ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ▲❛❣r❛♥❣✐❛♥ ❢♦r♠❛❧✐s♠✱ ❢✉♥❞❛♠❡♥t❛❧ t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ q✉❛♥t✉♠ t❤❡♦r② ♦❢ ❋❡②♥♠❛♥ ♣❛t❤ ✐♥t❡✲ ❣r❛❧s✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ t❤❡ q✉❛♥t✉♠ ♣r♦♣❛❣❛t♦r ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r ❢♦r ❛ t✐♠❡ ✐♥❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥ ✭✐s♦❧❛t❡❞ s②st❡♠✮✳ ❋r♦♠ t❤❡s❡ ❝♦♥❝❡♣ts ✱ ✐t ✐♥tr♦❞✉❝❡s t❤❡ ✐❞❡❛ ♦❢ ♣❛t❤ ✐♥t❡❣r❛❧✱ ✇❤❡r❡ ♥♦✇ ❤❛s t❤❛t ❛❧❧ ♣❛t❤s ❛r❡ ♣♦ss✐❜❧❡✱ ✉♥❧✐❦❡ t❤❡ ❝❧❛ss✐❝❛❧ ❛❝t✐♦♥ ✐♥ ✇❤✐❝❤ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✇❛② ✇♦✉❧❞ ❜❡ ♦♥❡ ✐♥ ✇❤✐❝❤ t❤❡ ❛❝t✐♦♥ ✐s ❛ ♠✐♥✐♠✉♠✳ ❚❤❡ st✉❞② ❝♦♥t✐♥✉❡s ✇✐t❤ t❤❡ ♣❛t❤ ✐♥t❡❣r❛❧ ❛♥❞ ✐ts ❛♥❛❧♦❣② ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧✳ ❚❤❡♥✱ ✐t ✐♥✈❡st✐❣❛t❡s t❤❡ r✉❧❡s ❢♦r t✇♦ s✉❝❝❡ss✐✈❡ ❡✈❡♥ts✱ ❛♥❞ ❛❧s♦✱ t❤❡ r✉❧❡s ❢♦r s❡✈❡r❛❧ s✉❝❝❡ss✐✈❡ ❡✈❡♥ts✳ ❋✐♥❛❧❧②✱ ✐t ✉s❡s t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❋❡②♥♠❛♥ ♣❛t❤ ✐♥t❡❣r❛❧ t♦ s♦❧✈❡ t❤❡ ❙✐♠♣❧❡ ❍❛r♠♦♥✐❝ ❖s❝✐❧❧❛t♦r ❛♥❞ ❋♦r❝❡❞ ❍❛r♠♦♥✐❝ ❖s❝✐❧❧❛t♦r ❜② ❝❛❧❝✉❧❛t✐♥❣ ✐ts ❡✐❣❡♥✈❛❧✉❡s ❄❄❛♥❞ ❡✐❣❡♥❢✉♥❝t✐♦♥s t❤r♦✉❣❤ q✉❛♥t✉♠ ♣r♦♣❛❣❛✲ t♦r✱ ♦❜t❛✐♥✐♥❣ s✐♠✐❧❛r r❡s✉❧ts t♦ t❤♦s❡ ❢♦✉♥❞ ❜② ❙❝❤rö❞✐♥❣❡r ❛♥❞ ❍❡✐s❡♥❜❡r❣✱ ♣r♦✈✐♥❣ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥ ✇✐t❤ t❤❡ ♦t❤❡r t✇♦ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛✲ ♥✐❝s✳ ❑❡②✇♦r❞s ✿ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✳ P❛t❤ ■♥t❡❣r❛❧s✳ ❋❡②♥♠❛♥ Pr♦♣❛❣❛t♦r✳

  ❉❡❞✐❝❛tór✐❛

  ❆♦s ♠❡✉s ♣❛✐s✳ ❆ ♠✐♥❤❛ ❡s♣♦s❛✳ ❆ ♠✐♥❤❛ ✜❧❤❛✳

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ s❛ú❞❡ ❡ ❝♦r❛❣❡♠ ♣❛r❛ ❡♥❢r❡♥t❛r ♦s ❞❡s❛✜♦s q✉❡ ❛ ✈✐❞❛ ♠❡ ♣r♦♣♦r❝✐♦♥♦✉✳

  ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❆❞r✐❛♥❛ ▲❡❛♥❞r♦ ❡ ❛ ♠✐♥❤❛ ✜❧❤❛ ▼❛r✐❛♥❛ ▲❡❛♥❞r♦ ♣❡❧♦ ❛♠♦r✱ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♠✐③❛❞❡ ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ❛♥♦s✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❏♦sé ❊♥❡❞✐❧t♦♥✱ ❏♦sé ●❛❞❡❧❤❛✱ ❏♦r❣❡ ▲✉✐③ ❡ ❆♥tô♥✐♦ ❏♦❡❧ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❛❥✉❞❛ ♥♦s ❡st✉❞♦s ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❞✉r❛♥t❡ ❡ss❛ ❞✐❢í❝✐❧ ❥♦r♥❛❞❛✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❊♠❛♥✉❡❧ ❲❡♥❞❡❧❧ ❡ ❲❡♥❞❡❧ ▼❡♥❞❡s ♣❡❧❛s ✐♠♣♦rt❛♥t❡s s✉❣❡stõ❡s ♥♦ ♠♦♠❡♥t♦ ❞❡ ❞❡✜♥✐çã♦ ❞♦ t❡♠❛ ❞❡st❛ ♠♦♥♦❣r❛✜❛ ❡ ♥❛ ❡s❝♦❧❤❛ ❞❛ ❜✐❜❧✐♦❣r❛✜❛✳ ❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❈✉rs♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ♣❡❧❛ q✉❛❧✐❞❛❞❡ ❞❡ ❡♥s✐♥♦ ♦❢❡r❡❝✐❞❛ ♣♦ss✐❜✐❧✐t❛♥❞♦ ♦ ♠❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✐♥t❡❧❡❝t✉❛❧ ♥♦s ú❧t✐♠♦s

  ❛♥♦s✳ ❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢✳ ❉r✳ ❘❛✐♠✉♥❞♦ ◆♦❣✉❡✐r❛ ❞❛ ❈♦st❛ ❋✐❧❤♦ ♣❡❧❛ ✈❛❧♦r♦s❛

  ♦r✐❡♥t❛çã♦ ❞❛ ♠♦♥♦❣r❛✜❛✱ ✐♥❞✐s♣❡♥sá✈❡❧ à ❝♦♥❝❧✉sã♦ ❞♦ ❝✉rs♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ s♦♥❤♦✳

  ✏◗✉❡♠ ❡♥tr❛ ❡♠ ❝♦♥t❛t♦ ❝♦♠ ❛ ❋ís✐❝❛ ◗✉â♥t✐❝❛ s❡♠ s❡ ❡s♣❛♥t❛r✱ s❡♠ ✜❝❛r ♣❡r✲ ♣❧❡①♦✱ é ♣♦rq✉❡ ♥❛❞❛ ❡♥t❡♥❞❡✉✳✑

  ❆❧❜❡rt ❊✐s♥t❡✐♥

  

❙✉♠ár✐♦

  ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✶ ■♥tr♦❞✉çã♦

  ♣✳ ✶✶ ✷ ❋♦r♠✉❧❛çã♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ♣✳ ✶✹

  ✷✳✶ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✹ ✷✳✷ ❆ ❢♦r♠✉❧❛çã♦ ❧❛❣r❛♥❣✐❛♥❛ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✺

  ✷✳✷✳✶ ❆ ❛çã♦ ❝❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✺ ✷✳✷✳✷ ❖ Pr✐♥❝í♣✐♦ ❞❛ ▼í♥✐♠❛ ❆çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✺ ✷✳✷✳✸ ▲❡✐s ❞❡ ❈♦♥s❡r✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✻

  ✷✳✸ ❆ ❢♦r♠✉❧❛çã♦ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽ ✷✳✸✳✶ ❆s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✽

  ✷✳✹ ❆ r❡❧❛çã♦ ❝♦♠ ❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✶✾ ✸ ❖ ♣r♦♣❛❣❛❞♦r ❞❡ ❋❡②♥♠❛♥

  ♣✳ ✷✷ ✸✳✶ ❖ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✷ ✸✳✷ ❖ ♣r♦♣❛❣❛❞♦r ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽

  ✸✳✷✳✶ ❆ ♦r✐❣❡♠ ❞♦ ♣r♦♣❛❣❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✽ ✸✳✷✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦♣❛❣❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✷✾ ✸✳✷✳✸ ❖ ♣r♦♣❛❣❛❞♦r ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✶

  ✸✳✸ ❖ ♣r♦♣❛❣❛❞♦r ❝♦♠♦ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✷ ✹ ■♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥

  ♣✳ ✸✺

  ✹✳✶ ❆ ♦r✐❣❡♠ ❞❛ ■♥t❡❣r❛❧ ❞❡ ❈❛♠✐♥❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✺ ✹✳✷ ❆ r❡❧❛çã♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼ ✹✳✸ ❆♥❛❧♦❣✐❛ ❝♦♠ ❛ ■♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾ ✹✳✹ ❆s r❡❣r❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶ ✹✳✺ ❆s r❡❣r❛s ♣❛r❛ ✈ár✐♦s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✷

  ✺ ❖ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ♣✳ ✹✹

  ✺✳✶ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✹ ✺✳✷ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❋♦rç❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✺✵

  ✻ ❈♦♥❝❧✉sõ❡s ♣✳ ✺✻

  ❘❡❢❡rê♥❝✐❛s ♣✳ ✺✽

  ▲✐st❛ ❞❡ ❋✐❣✉r❛s

  ✶ ❈❛♠✐♥❤♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝❧áss✐❝❛ ❞♦ s✐st❡♠❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦✳ ♣✳ ✶✺ ✷ ❆♠♣❧✐t✉❞❡s ❞❡ tr❛♥s✐çã♦ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ♣♦♥t♦ ✐♥✐❝✐❛❧ q = q ✱ N −1

  ♥✉♠ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t = 0✱ ❛ ✉♠ ♣♦♥t♦ ✜♥❛❧ q = q ✱ ♥✉♠ ✐♥st❛♥t❡ ✜♥❛❧ t = T ✱ ❝♦♠ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ✐♥✜♥✐t❡s✐♠❛❧ δt✱ ♦♥❞❡ ✉t✐❧✐③❛♠♦s i i ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s q ❡♠ s✉❜st✐t✉✐çã♦ ❛s ❝♦♦r❞❡♥❛❞❛s x ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✼

  ✸ ■♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥ s♦❜ ✉♠❛ ❝✉r✈❛ f(x) ❡ s♦❜r❡ ♦ ❡✐①♦ ❞❛s ❛❜❝✐ss❛s x✱ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦ h ❡♥tr❡ x i ❡ x i+1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✸✾

  ✹ ❆ s♦♠❛ s♦❜r❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ❧✐♠✐t❡✱ ❡♠ q✉❡ ❛ ♣r✐♥❝í♣✐♦ ♦ ❝❛♠✐♥❤♦ é ❡s♣❡❝✐✜❝❛❞♦✱ ❞❛❞♦ ❛♣❡♥❛s ♣❡❧❛ s✉❛ ❝♦♦r❞❡♥❛❞❛ x

  ♥✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❞❡ t❡♠♣♦s ❡s♣❡❝✐✜❝❛❞♦s s❡♣❛r❛❞♦s ♣♦r ✐♥t❡r✈❛❧♦s ♠✉✐t♦ ♣❡q✉❡♥♦s ε✳ ❆ s♦♠❛ ❞♦s ❝❛♠✐♥❤♦s é✱ ❡♥tã♦✱ ✉♠❛ ✐♥t❡❣r❛❧ s♦❜r❡ t♦❞❛s ❡ss❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❡❝í✜❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♣✳ ✹✶

  ✺ ❆♠♣❧✐t✉❞❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♣♦♥t♦ a ♣❛r❛ ✉♠ ♣♦♥t♦ ✐♥t❡r♠❡❞✐ár✐♦ c✱ ❡♠ s❡❣✉✐❞❛ ♣❛r❛ ♦ ♣♦♥t♦ b✳ ♣✳ ✹✷

  ✶ ■♥tr♦❞✉çã♦

  ❆ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ é ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ r❛♠♦ ❞❛ ❢ís✐❝❛ q✉❡ ❡st✉❞❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♠❛tér✐❛ ❡ ❞❛ ❡♥❡r❣✐❛ ❛ ♥í✈❡❧ ❛tô♠✐❝♦✱ s❡♥❞♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ❢♦rç❛s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♥❛t✉r❡③❛✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❋♦rç❛ ●r❛✈✐t❛❝✐♦♥❛❧✱ ♣♦rt❛♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ❛s ✐♥t❡r❛çõ❡s ♦❝♦rr✐❞❛s ♣❡❧❛ ❋♦rç❛ ❊❧❡tr♦♠❛❣♥ét✐❝❛✱ ❋♦rç❛ ◆✉❝❧❡❛r ❋♦rt❡ ❡ ❋♦rç❛ ◆✉❝❧❡❛r ❢r❛❝❛✳

  ◆♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳ ❡ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ❛❧❣✉♥s ❡①♣❡r✐♠❡♥t♦s t♦r♥❛r❛♠ ❡✈✐❞❡♥t❡s q✉❡ ❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❡r❛ ❢❛❧❤❛ ♦✉ ❛✐♥❞❛ ♥ã♦ ❡st❛✈❛ ❝♦♠♣❧❡t❛✱ ♥❛ ❡s❝❛❧❛ ♠❛❝r♦s❝ó♣✐❝❛ ❞♦s ❝♦r♣♦s ♠✉✐t♦ ♠❛ss✐✈♦s ♦✉ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s ♠✉✐t♦ ❣r❛♥❞❡s ♦✉ ♥❛ ❡s❝❛❧❛ ♠✐❝r♦s❝ó♣✐❝❛ ❞❛s ♣❛rtí❝✉❧❛s s✉❜❛tô♠✐❝❛s ❛ ♥❛t✉r❡③❛ ❝♦♠♣♦rt❛✈❛✲s❡ ❞❡ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ❞❛ ♣r❡✈✐sã♦ t❡ór✐❝❛ ❞❛ ❢ís✐❝❛ ❝❧áss✐❝❛✳ ❋♦✐ ♥❡ss❡ ❝♦♥t❡①t♦ q✉❡ s✉r❣✐r❛♠ ❛ t❡♦r✐❛ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡ ●❡r❛❧✱ q✉❡ r❡s♣♦♥❞✐❛ ❛s q✉❡stõ❡s ♥❛ ❡s❝❛❧❛ ♠❛❝r♦s❝ó♣✐❝❛✱ ❡ ❛ t❡♦r✐❛ ◗✉â♥t✐❝❛✱ q✉❡ r❡s♦❧✈✐❛ ❛s q✉❡stõ❡s ♥❛ ❡s❝❛❧❛ ♠✐❝r♦s❝ó♣✐❝❛✳

  ❆ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ t❡✈❡ ♦r✐❣❡♠ ♣♦r ✈♦❧t❛ ❞♦ ❛♥♦ ❞❡ ✶✾✵✵❬✶❪✱ q✉❛♥❞♦ ▼❛① P❧❛♥❝❦ ♣✉❜❧✐❝♦✉ ♥❛ r❡✈✐st❛ ❛❧❡♠ã ❆♥♥❛❧❡♥ ❞❡r P❤②s✐❦ ✉♠ ❛rt✐❣♦ ❝❤❛♠❛❞♦ ❙♦❜r❡ ❛ ❚❡♦r✐❛ ❞❛ ▲❡✐ ❞❡ ❉✐str✐❜✉✐çã♦ ❞❡ ❊♥❡r❣✐❛ ♥♦ ❊s♣❡❝tr♦ ◆♦r♠❛❧❬✷❪✳ ❆ s♦❧✉çã♦ ❞❡ P❧❛♥❝❦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❘❛❞✐❛çã♦ ❞♦ ❈♦r♣♦ ◆❡❣r♦ ❝♦♥t✐❞❛ ♥❡ss❡ ❛rt✐❣♦ ✐♥tr♦❞✉③✐✉ ♦ ❝❤❛♠❛❞♦ q✉❛♥t✉♠ ❞❡ ❡♥❡r❣✐❛✱ ❞❛❞♦ ♣♦r ε = h × ν✱ ❡♠ q✉❡ P❧❛♥❝❦ s✉❣❡r✐✉ q✉❡ ❛ tr♦❝❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦s ❡❧étr♦♥s ❞❛ ♣❛r❡❞❡ ❞❛ ❝❛✈✐❞❛❞❡ ❞♦ ❈♦r♣♦ ◆❡❣r♦ ❡ ❞❛ r❛❞✐❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛ só ❞❡✈❡r✐❛♠ ♦❝♦rr❡r ❡♠ q✉❛♥t✐❞❛❞❡s ❞✐s❝r❡t❛s✱ ♦✉ s❡❥❛✱ E = n × ε✱ ❝♦♠ n = 0, 1, 2, ... r❡s♦❧✈❡♥❞♦ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ❛❧t❛s ❢r❡q✉ê♥❝✐❛s ♦♥❞❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ✐r✐❛ ♣❛r❛ ♦ ✐♥✜♥✐t♦✱ ♣r♦❜❧❡♠❛ q✉❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛ ❝❛tástr♦❢❡ ❞♦ ✉❧tr❛✈✐♦❧❡t❛✳ ❊♠ ✶✾✵✺✱ ❆❧❜❡rt ❊✐♥st❡✐♥ ❛♦ ♣✉❜❧✐❝❛r ✉♠ ❛rt✐❣♦ ♥❛ r❡✈✐st❛ ❛❧❡♠ã ❆♥♥❛❧❡♥ ❞❡r P❤②s✐❦ s♦❜r❡ ♦ ❊❢❡✐t♦ ❋♦t♦❡❧étr✐❝♦❬✸❪ ❝♦♠ ♦ tít✉❧♦ ❙♦❜r❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❤❡✉ríst✐❝♦ ❛ r❡s♣❡✐t♦ ❞❛ ♣r♦❞✉çã♦ ❡ tr❛♥s❢♦r♠❛çã♦ ❞❛ ❧✉③❬✹❪✱ ♣r♦♣ôs q✉❡ ❛ ❧✉③ ✭♦♥❞❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛✮ ❡r❛ ❝♦♥st✐t✉í❞❛ ❞❡ ♣❛rtí❝✉❧❛s ❝❤❛♠❛❞❛s ❢ót♦♥s h ✭q✉❛♥t✉♠ ❞❡ ❧✉③✮ ❝♦♠ ❡♥❡r❣✐❛ ❞❛❞❛ ♣♦r E = h×ν ❡ ♠♦♠❡♥t♦ ❧✐♥❡❛r p = ♦♥❞❡ ν ❡ λ sã♦✱ λ

  −34

  r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❢r❡q✉ê♥❝✐❛ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ❡ h = 6, 6260693×10 ❏✳s é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦✳ ❊♠ ✶✾✶✸✱ ◆✐❡❧s ❇♦❤r❬✺❪ ❡st✉❞❛♥❞♦ ♦ ❡s♣❡❝tr♦ ❞❡ ❧✐♥❤❛ ❞♦s át♦♠♦s ✈❡r✐✜❝♦✉ q✉❡ ❛ss✐♠ ❝♦♠♦ ♥♦s ♦s❝✐❧❛❞♦r❡s ❞❡ P❧❛♥❝❦ ♦s át♦♠♦s só ♣♦❞❡♠ tr♦❝❛r ❡♥❡r❣✐❛ ❡♠ q✉❛♥t✐❞❛❞❡s ❞✐s❝r❡t❛s✳ ❇♦❤r ♣♦st✉❧♦✉ ❡♥tã♦ q✉❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞♦ ❡❧étr♦♥ ♥♦ át♦♠♦ h

  é ❞❛❞♦ ♣♦r L = n × ¯h ❝♦♠ n = 1, 2, ... ❡ ¯h = ✳ ❊ ♣♦st✉❧♦✉ t❛♠❜é♠✱ q✉❡ ❛ ♠✉❞❛♥ç❛ i f ❞❡ ✉♠ ❡st❛❞♦ ❞❡ ❡♥❡r❣✐❛ ✐♥✐❝✐❛❧ E ♣❛r❛ ✉♠ ❡st❛❞♦ ❞❡ ❡♥❡r❣✐❛ ✜♥❛❧ E ♣r♦✈♦❝❛ ❛ ❡♠✐ssã♦ i f ❞❡ ✉♠ ❢ót♦♥ ❞❡ ❡♥❡r❣✐❛ ε = h × ν✱ t❛❧ q✉❡ hν = E ✳ ❊♠ ✶✾✷✹✱ ▲♦✉✐s ❞❡ ❇r♦❣❧✐❡

  − E ♣r♦♣ôs q✉❡ ❛s r❡❧❛çõ❡s ❞❡ P❧❛♥❝❦✲❊✐♥st❡✐♥ ❢♦ss❡♠ ❡st❡♥❞✐❞❛s ♣❛r❛ ♣❛rtí❝✉❧❛s ♠❛t❡r✐❛✐s✱ ❝♦♠ ✐ss♦ t❡r✐❛✲s❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ ❡♥❡r❣✐❛ E✱ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ♦♥❞❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ E ❛♥❣✉❧❛r ω✱ ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦ ω = ✱ ❡ ♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ~p ❛ss♦❝✐❛❞♦ ❛♦ ❝♦♠♣r✐♠❡♥t♦ h ¯ h ❞❡ ♦♥❞❛ λ ❞❛❞♦ ♣❡❧❛ r❡❧❛çã♦ λ = ✳ ❊ss❛ ♣r♦♣♦st❛ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ❍✐♣ót❡s❡ p ❞❡ ❉❡ ❇r♦❣❧✐❡❬✻❪✳ ❖✉tr♦s ❡♠✐♥❡♥t❡s ❝✐❡♥t✐st❛s✱ t❛✐s ❝♦♠♦ ❆r♥♦❧❞ ❙♦♠♠❡r❢❡❧❞ ❡ ❍❡♥❞r✐❦ ❑r❛♠❡rs✱ ❞❡r❛♠ ❝♦♥tr✐❜✉✐çõ❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❝❤❛♠❛❞❛ ✏❱❡❧❤❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✑✳ ▼❛s ❢♦✐ ❛ ♣❛rt✐r ❞❡ ✶✾✷✺ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡ ▼❛① ❇♦r♥✱ ❊r✇✐♥ ❙❝❤rö❞✐♥❣❡r✱ ❲❡r♥❡r ❍❡✐s❡♥❜❡r❣ ❡ P❛✉❧ ❉✐r❛❝✱ ❞❡♥tr❡ ♦✉tr♦s ❢ís✐❝♦s ✐♠♣♦rt❛♥t❡s✱ q✉❡ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❝♦♠❡ç♦✉ ❛ s❡ ❢♦rt❛❧❡❝❡r ❝♦♠♦ t❡♦r✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡ ❙❝❤rö❞✐♥❣❡r ❡ ❍❡✐s❡♥❜❡r❣ q✉❡ ❜✉s❝❛r❛♠ ✉♠❛ s✐st❡♠❛t✐③❛çã♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡s❡♥✈♦❧✈❡♥❞♦✱ ♣❛r❛❧❡❧❛ ❡ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ r❡♣r❡s❡♥t❛çõ❡s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❞❛♥❞♦ ✐♥í❝✐♦ ❛ ❝❤❛♠❛❞❛ ✏◆♦✈❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✑✳ ❈♦♥t✉❞♦✱ ❡♠❜♦r❛ ❢♦ss❡♠ ❛❜♦r❞❛❣❡♥s ❞✐❢❡r❡♥t❡s tr❛t❛✈❛✲s❡ ❞❛ ♠❡s♠❛ t❡♦r✐❛✱ ❛♣r❡s❡♥t❛♥❞♦ r❡s✉❧t❛❞♦s ✐❞ê♥t✐❝♦s✳

  ❆ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣❬✼❪ t✐♥❤❛ ❡♠ s❡✉ ❛r❝❛❜♦✉ç♦ ✉♠❛ ❡str✉t✉r❛ ♠❛tr✐❝✐❛❧ ❡♠ q✉❡ ♦s ♦♣❡r❛❞♦r❡s sã♦ ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦ ❡ ♦s ❡st❛❞♦s q✉â♥t✐❝♦s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦❬✽❪✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r❬✾❪ ❜❛s❡❛✈❛✲s❡ ♥❛s ❡✈✐❞ê♥❝✐❛s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛s ♣❛rtí❝✉❧❛s ♥♦s s✐st❡♠❛s ♠✐❝r♦s❝ó♣✐❝♦s t❡♠ ♥❛t✉r❡③❛ ♦♥❞✉❧❛tór✐❛ ❡✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❞❡ ❙❝❤rö❞✐♥❣❡r ♦s ♦♣❡r❛❞♦r❡s sã♦ ❝♦♥st❛♥t❡s ❡ ❛❣♦r❛ sã♦ ♦s ❡st❛❞♦s q✉â♥t✐❝♦s q✉❡ ❡✈♦❧✉❡♠ ❝♦♠ ♦ t❡♠♣♦✳

  ❖❜s❡r✈❡✲s❡ q✉❡✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❛ ❡s♣❡❝tr♦s❝♦♣✐❛ q✉❡ ❧❡✈❛r❛♠ ♦s ❢ís✐❝♦s ❛ ❡st✉❞❛r❡♠ ❛s ❞✐❢❡r❡♥ç❛s ❞♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ♥♦ át♦♠♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❛ss♦❝✐❛çã♦ ❢❡✐t❛ ❡♥tr❡ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ s✉❛ ❡♥❡r❣✐❛ t♦t❛❧✱ ✜③❡r❛♠ ❞♦ ❢♦r♠❛❧✐s♠♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✉♠❛ ♣r❡❢❡rê♥❝✐❛ t❛♥t♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣ q✉❛♥t♦ ♥❛ ❞❡ ❙❝❤rö❞✐♥❣❡r✳

  ❚♦❞❛✈✐❛✱ ♥♦ ❛♥♦ ❞❡ ✶✾✹✷ ♦ ❢ís✐❝♦ ❛♠❡r✐❝❛♥♦ ❘✐❝❤❛r❞ ❋❡②♥♠❛♥ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❡ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦❬✶✵❪ ❛❞♦t♦✉ ♦ ❢♦r♠❛❧✐s♠♦ ❧❛❣r❛♥❣✐❛♥♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ ♥❛

  ❛♥á❧✐s❡ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✱ ❝r✐❛♥❞♦ ✉♠❛ ♦✉tr❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✳ ❋❡②♥♠❛♥ ❜❛s❡♦✉✲s❡ ♥✉♠ ❛rt✐❣♦ ❞❡ P❛✉❧ ❉✐r❛❝ ❞❡ ✶✾✸✸ q✉❡ t✐♥❤❛ ❝♦♠♦ tít✉❧♦ ❆ ▲❛❣r❛♥❣✐❛♥❛ ♥❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛❬✶✶❪✳ ❆ ✐❞❡✐❛ ❞❡ ❋❡②♥♠❛♥ ❢♦✐ ❢❛③❡r ✉♠❛ ❞❡s❝r✐çã♦ ❞♦s s✐st❡♠❛s q✉â♥t✐❝♦s ❛ ♣❛rt✐r ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❞♦s s✐st❡♠❛s ❝❧áss✐❝♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✱ ❡st❡♥❞❡♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦ ♣❛r❛ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❛tr❛✈és ❞❡ ✉♠❛ ❛♥❛❧♦❣✐❛ ❢❡✐t❛ ♣♦r ❉✐r❛❝ ❡♠ s❡✉ ❛rt✐❣♦ ❞❡ ✶✾✸✸✳ ■♥tr♦❞✉③✐♥❞♦ ❛ ✐❞❡✐❛ ❞❡ ♣r♦♣❛❣❛❞♦r❡s✱ ❋❡②♥♠❛♥ ✉t✐❧✐③❛✲s❡ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ q✉❡ ❞❡s❝r❡✈❡♠ t♦❞❛s ❛s tr❛❥❡tór✐❛s i ♣♦ssí✈❡✐s ♣❛r❛ q✉❡ ✉♠❛ ♣❛rtí❝✉❧❛ s❛✐♥❞♦ ❞❡ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ x ♥✉♠ ❝❡rt♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t f f

  ❝❤❡❣❛r ❛ ✉♠ ♣♦♥t♦ ✜♥❛❧ x ♥✉♠ ✐♥st❛♥t❡ ✜♥❛❧ t ✱ ♦♥❞❡ ♦ ♣r♦♣❛❣❛❞♦r s✐❣♥✐✜❝❛ ✉♠❛ >

  ❛♠♣❧✐t✉❞❡ ❞❡ tr❛♥s✐çã♦ ♣❛r❛ ♦ s✐st❡♠❛ q✉â♥t✐❝♦ ❡✈♦❧✉✐r ❞❡ ✉♠ ❡st❛❞♦ ✐♥✐❝✐❛❧ |Ψ, t ♣❛r❛ ; t >

  ✉♠ ❡st❛❞♦ ✜♥❛❧ |Ψ, t ✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❢❛r❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❛ ♣r♦♣♦st❛ ❞❡ ❋❡②♥♠❛♥ ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❛

  ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣♦r ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦✱ ♦♥❞❡ ♥♦ ❝❛♣ít✉❧♦ ✷ ❢❛r❡♠♦s ✉♠❛ r❡✈✐sã♦ ❞❛ ❢♦r♠✉❧❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛✱ ♣r♦ss❡❣✉✐♥❞♦ ♥♦ ❝❛♣ít✉❧♦ ✸ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞♦ ♣r♦♣❛❣❛❞♦r ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ ❢❛r❡♠♦s ✉♠❛ ❡①♣❧✐❝❛çã♦ ♠❛✐s ❞❡t❛✲ ❧❤❛❞❛ ❞❛s ■♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥✱ s❡❣✉✐♥❞♦✲s❡ ♥♦ ❝❛♣ít✉❧♦ ✺ ❝♦♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛s ■♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❞♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ❡ ❞♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❋♦rç❛❞♦✳ ❊ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ❛ ❝♦♥❝❧✉sã♦✱ ❞✐s❝✉t✐♥❞♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ❋❡②♥♠❛♥ ♣❛r❛ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳

  ✷ ❋♦r♠✉❧❛çã♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ✷✳✶ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❋✉♥❝✐♦♥❛❧

  ◆♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❜✉s❝❛✲s❡ ✉♠ ✈❛❧♦r ❞❡ x ♣❛r❛ ♦ q✉❛❧ ❛ ❢✉♥çã♦ y(x) t❡♠ ✉♠ ✈❛❧♦r ❞❡ ♠á①✐♠♦ ♦✉ ❞❡ ♠í♥✐♠♦✳ ❏á ♥♦ ❝á❧❝✉❧♦ ❞❛s ✈❛r✐❛çõ❡s ♦ q✉❡ q✉❡r❡♠♦s é ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ y(x)

  ♣❛r❛ ♦ q✉❛❧ ♦ ❢✉♥❝✐♦♥❛❧ I[y(x)] ❛❧❝❛♥ç❛ ✉♠ ✈❛❧♦r ❡①tr❡♠♦ ✭♠á①✐♠♦ ♦✉ ♠í♥✐♠♦✮✳ ❖ ❢✉♥❝✐♦♥❛❧ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝✉❥♦ ❞♦♠í♥✐♦ é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s❬✶✷❪✳

  ❉✐③❡r q✉❡ I[y(x)] é ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❛ ❢✉♥çã♦ y(x) s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ I é ✉♠ ♥ú♠❡r♦ ❝✉❥♦ ✈❛❧♦r ❞❡♣❡♥❞❡ ❞❛ ❢♦r♠❛ ❞❛ ❢✉♥çã♦ y(x)✱ ❡♠ q✉❡ x é ❛♣❡♥❛s ✉♠ ♣❛râ♠❡tr♦ ✉s❛❞♦ ♣❛r❛ ❞❡✜♥✐r y(x)✳ P♦rt❛♥t♦✱ Z x 2 I[y(x)] = F (y(x), x)dx. x 1

  ✭✷✳✶✮ I[y(x)]

  é ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ y(x) ✉♠❛ ✈❡③ q✉❡ ❡❧❡ ❛ss♦❝✐❛ ✉♠ ♥ú♠❡r♦ ❛ ❝❛❞❛ ❡s❝♦❧❤❛ ❞❛ ❢✉♥çã♦ y(x)✱ ♦✉ s❡❥❛✱ s✉❛ ✐♥t❡❣r❛❧✳ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛ ár❡❛ s♦❜ ✉♠❛ ❝✉r✈❛ é ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❛ ❢✉♥çã♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❝✉r✈❛✱ ❤❛❥❛ ✈✐st❛ q✉❡ ♣❛r❛ ❝❛❞❛ ♥ú♠❡r♦ q✉❡ ❞❡✜♥❡ ♦ ❢✉♥❝✐♦♥❛❧ t❡♠♦s ✉♠❛ ár❡❛ ❛ss♦❝✐❛❞❛✳

  ❯♠ ❢✉♥❝✐♦♥❛❧ ♣♦❞❡ t❡r ❡♠ s❡✉ ❛r❣✉♠❡♥t♦ ♠❛✐s ❞❡ ✉♠ ♣❛râ♠❡tr♦✱ ❛ss✐♠ t❡♠♦s q✉❡✿ Z Z s 2 t 2 I[x(t, s), y(t, s)] = x(t, s)y(t, s)dtds s 1 t 1 ✭✷✳✷✮ ❯♠ ❢✉♥❝✐♦♥❛❧ I[q(t)] ♣♦❞❡ t❛♠❜é♠ s❡r ❞❛❞♦ ❝♦♠♦ ✉♠❛ sér✐❡ ❞❡ ❢✉♥çõ❡s ❞❡ ✉♠ ♥ú♠❡r♦

  ✐♥✜♥✐t♦ ❞❡ ✈❛r✐á✈❡✐s✱ ♦♥❞❡ ❛s ✈❛r✐á✈❡✐s sã♦ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ q(t) ❡♠ ❝❛❞❛ ✈❛❧♦r ❞❡ t ❬✶✷❪✳

  ✷✳✷ ❆ ❢♦r♠✉❧❛çã♦ ❧❛❣r❛♥❣✐❛♥❛ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ✷✳✷✳✶ ❆ ❛çã♦ ❝❧áss✐❝❛

  ❊♠ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❛ ❛çã♦ é ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ L ❞❡ ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s q(t) ❡ ✈❡❧♦❝✐❞❛❞❡s ❣❡♥❡r❛❧✐③❛❞❛s ˙q(t) ❞❛❞❛ ♣♦r ❬✶✷❪ Z t 2 S = L q(t), ˙q(t) dt, t 1

  ✭✷✳✸✮ ♦♥❞❡ L é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛ q✉❡ ♣♦❞❡ s❡r ♦❜t✐❞❛ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❡♥❡r❣✐❛s ❝✐♥ét✐❝❛ ❡ ♣♦t❡♥❝✐❛❧ ❞♦ s✐st❡♠❛ ❬✶✸❪✱ ❞❡ ❢♦r♠❛ q✉❡✿

  L = T − V ✭✷✳✹✮ ❯t✐❧✐③❛♥❞♦✲s❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❡ ✈❡❧♦❝✐❞❛❞❡s ❣❡♥❡r❛❧✐③❛❞❛s ❡ ♣❛r❛ ✉♠ ♣♦t❡♥❝✐❛❧ V (q)

  ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❧❛❣r❛♥❣✐❛♥❛ ❝♦♠♦✿ 2 m ˙q L =

  − V (q) ✭✷✳✺✮

  2

  ✷✳✷✳✷ ❖ Pr✐♥❝í♣✐♦ ❞❛ ▼í♥✐♠❛ ❆çã♦

  ❖ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦ ❞✐③ q✉❡ ❛ ❡✈♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ 1 ♣❛r❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ 2✱ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s✱ ♦❝♦rr❡ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛ ❛çã♦ é ✉♠ ♠í♥✐♠♦ ❬✶✹❪✳ ◆❛ ✜❣✉r❛ ✶✱ t❡♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛ tr❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝❧áss✐❝❛✳

  ❋✐❣✉r❛ ✶✿ ❈❛♠✐♥❤♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝❧áss✐❝❛ ❞♦ s✐st❡♠❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦✳

  ❆ ❝♦♥❞✐çã♦ ❞❡ ❡①tr❡♠♦ s✉❣❡r✐❞❛ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥ é ♦❜t✐❞❛ ♣❡❧♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ❡♠ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  δS = S[q + δq] − S[q] = 0 Z t 2 Z t 2 " ! !# S[q + δq] = L(q + δq, ˙q + δ ˙q, t)dt t 1 ∂L ∂L

  S[q + δq] = L(q, ˙q, t) + δ ˙q + δq dt t 1 Z t 2 " ! !# ∂ ˙q ∂q ∂L ∂L

  S[q + δq] = S[q] + δ ˙q + δq dt t 1 t 2 Z t 2 " ! # ∂ ˙q ∂q ∂L d ∂L ∂L S[q + δq] = S[q] + δq δq dt.

  ✭✷✳✻✮ − − t 1

  ∂ ˙q dt ∂ ˙q ∂q t 1 ❊♥tã♦ t❡♠♦s q✉❡✿ t 2 Z t 2 " ! #

  ∂L d ∂L ∂L δS = δq δq dt,

  ✭✷✳✼✮ − − t 1

  ∂ ˙q dt ∂ ˙q ∂q t 1 1 ) = δq(t ) = 0 2 ♠❛s ♥♦s ❡①tr❡♠♦s ❛ ❝♦♥❞✐çã♦ é q✉❡ δq(t ✭♣♦♥t♦s ✜①♦s✮❬✶✺❪✱ ❧♦❣♦✿ Z t 2 " ! # d ∂L ∂L

  δq dt, δS = − − ✭✷✳✽✮ t 1 dt ∂ ˙q ∂q

  ♠❛s ❡♥tr❡ ♦s ❡①tr❡♠♦s δq ♣♦❞❡ ❛ss✉♠✐r ✈❛❧♦r❡s ❛r❜✐trár✐♦s✱ ❛ss✐♠ ♣❛r❛ t❡r♠♦s δS = 0✱ t❡♠♦s q✉❡✿ ! d ∂L ∂L = 0,

  − ✭✷✳✾✮ dt ∂ ˙q ∂q ❡st❛ ❡q✉❛çã♦ ❢♦✐ ♦❜t✐❞❛ ♣♦r ❊✉❧❡r s❡♥❞♦ ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ♣♦r ✉t✐❧✐③❛r ❛ ❢♦r♠✉❧❛çã♦ ❧❛❣r❛♥❣✐❛♥❛✳

  ✷✳✷✳✸ ▲❡✐s ❞❡ ❈♦♥s❡r✈❛çã♦

  ❙❡ ❛ ❝♦♦r❞❡♥❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ q ♥ã♦ ❛♣❛r❡❝❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ♥❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛✱ ♦♥❞❡ q é ❝❤❛♠❛❞❛ ❞❡ ❝♦♦r❞❡♥❛❞❛ ❝í❝❧✐❝❛ ❬✶✷❪✱ t❡♠♦s✿

  ∂L d ∂L ∂L = 0 = 0 = cte

  → → ✭✷✳✶✵✮ ∂q dt ∂ ˙q ∂ ˙q

  ∂L

  ❧♦❣♦ t♦r♥❛✲s❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥t❡r❡ss❡ ❢ís✐❝♦✱ ♣♦✐s ❞✐❛♥t❡ ❞❡ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ❡❧❛ ∂ ˙q s❡ ❝♦♥s❡r✈❛✳ ❉❡✈✐❞♦ ❛ ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛ ♠❡❝â♥✐❝❛ ◆❡✇t♦♥✐❛♥❛ ❡ss❛ ❣r❛♥❞❡③❛ s❡rá ❝❤❛♠❛❞❛ ❞❡ ♠♦♠❡♥t♦ ❣❡♥❡r❛❧✐③❛❞♦ ❝♦♥❥✉❣❛❞♦ à ✈❛r❛✐á✈❡❧ q✱ ♦✉ s✐♠♣❧❡♠❡♥t❡✱ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦✳ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ q✉❛♥❞♦ ❛ ❝♦♦r❞❡♥❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ q t❡♠ ❞✐♠❡♥sã♦ ❞❡ ∂L ❝♦♠♣r✐♠❡♥t♦ s❡rá ♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡ q✉❛♥❞♦ t❡♠ ❞✐♠❡♥sã♦ ❞❡ â♥❣✉❧♦ ❡ss❛ q✉❛♥t✐❞❛❞❡ ∂ ˙q s❡rá ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ❡ss❛ ❣r❛♥❞❡③❛ ❢ís✐❝❛ ♣♦r✿

  ∂L p i = .

  ✭✷✳✶✶✮ ∂ ˙q i

  ❖❜s❡r✈❡✲s❡ q✉❡ ♦❜t✐✈❡♠♦s ❡ss❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ s✐♠❡tr✐❛ ❞❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛✱ ♣♦✐s q✉❛♥❞♦ ❛ ❧❛❣r❛♥❣✐❛♥❛ ♥ã♦ ❞❡♣❡♥❞❡✱ ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ❞❡ ✉♠❛ i ❞❡t❡r♠✐♥❛❞❛ ❝♦♦r❞❡♥❛❞❛ q ❡❧❛ ❛♣r❡s❡♥t❛ ✉♠❛ s✐♠❡tr✐❛ ♣❡r❛♥t❡ ❛ ❛çã♦ ❞❡st❛ ❝♦♦r❞❡♥❛❞❛✳

  ❆❣♦r❛ ♦❜s❡r✈❡♠♦s ✉♠❛ ♦✉tr❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ q✉❡ s✉r❣❡ ❛ ♣❛rt✐r ❞❛ ❞❡r✐✈❛çã♦ ❞❛ ❧❛❣r❛♥❣✐❛♥❛ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦ ❬✶✹❪✱ ♦✉ s❡❥❛✱ X X dL ∂L ∂L ∂L i i + = + ˙q q ¨ dt ∂q ∂ ˙q ∂t X i i i i ! dL d ∂L ∂L ∂L i i = q ¨ + + ˙q dt dt ∂ ˙q ∂ ˙q ∂t i i i X dL ∂L i i ✭✷✳✶✷✮ + = p ˙q dt ∂t i

  ♦♥❞❡ ♥❛ s❡❣✉♥❞❛ ♣❛ss❛❣❡♠ ❢♦✐ ✉t✐❧✐③❛❞❛ ❛ ❡q✉❛çã♦ ✭✷✳✾✮ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡ ♥❛ t❡r❝❡✐r❛ ♣❛ss❛❣❡♠ ❛ ❞❡✜♥✐çã♦ ❞♦ ♠♦♠❡♥t♦ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✶✶✮✳ ▼❛s✱ ❛❣♦r❛ ✈❡❥❛♠♦s ♦ ❝❛s♦ ∂L

  = 0 ❡♠ q✉❡ ❛ ❧❛❣r❛♥❣✐❛♥❛ ♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞♦ t❡♠♣♦✱ t❛❧ q✉❡ ✱ ❡ ♣♦rt❛♥t♦✿ X ! X ∂t d p i ˙q i = 0 p i ˙q i

  ✭✷✳✶✸✮ − L → − L = cte dt i i

  ❡♥tã♦✱ ❛q✉✐ s✉r❣❡ ♠❛✐s ✉♠❛ q✉❛♥t✐❞❛❞❡ q✉❡ s❡ ❝♦♥s❡r✈❛✳ ❆ss✐♠✱ q✉❛♥❞♦ ❛ ❧❛❣r❛♥❣✐❛♥❛ ♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞♦ t❡♠♣♦✱ ❛ ❣r❛♥❞❡③❛ ❞❛❞❛ ♣♦r✿ X H = p i ˙q i

  ✭✷✳✶✹✮ i − L(q, ˙q), q✉❡ s❡rá ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❢✉♥çã♦ ❤❛♠✐❧t♦♥✐❛♥❛ ❬✶✹❪✱ s❡ ❝♦♥s❡r✈❛✳ ❊ ✈❡r❡♠♦s q✉❡ ♣❛r❛ ❛❧❣✉♥s ❝❛s♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ✉♠ ♣♦t❡♥❝✐❛❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ s❡rá ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛✳

  

✷✳✸ ❆ ❢♦r♠✉❧❛çã♦ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛

✷✳✸✳✶ ❆s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥

  ◆❛ ❢♦r♠✉❧❛çã♦ ❧❛❣r❛♥❣✐❛♥❛ ❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s ❡r❛♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛✲ ❧✐③❛❞❛s q✱ ❛s ✈❡❧♦❝✐❞❛❞❡s ❣❡♥❡r❛❧✐③❛❞❛s ˙q ❡ ♦ t❡♠♣♦ t✳ ❆ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦ s✐st❡♠❛ ❢ís✐❝♦ ❡r❛ ❞❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ▲❛❣r❛♥❣❡✳ ❆❣♦r❛✱ ✐♥tr♦❞✉③✐r❡♠♦s ✉♠❛ ❢♦r♠✉❧❛çã♦ ♦♥❞❡ ❛s ✈❛r✐á✈❡✐s ❞❛❞❛s sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✱ ♦ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦ ❡ ♦ t❡♠♣♦✳ ❊ss❛ ♥♦✈❛ ❢♦r♠✉❧❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❤❛♠✐❧t♦♥✐❛♥❛ ❬✶✻❪✳ ❚♦♠❛♥❞♦ ❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛ L(q, ˙q, t)

  ✱ t❛❧ q✉❡✿ X X ∂L ∂L ∂L dL = + dq + d ˙q dt i i i i X ∂q ∂ ˙q ∂t i i X

  ∂L ∂L i i i + + dL = dq p d ˙q dt ✭✷✳✶✺✮ i i ∂q i ∂t

  ♦♥❞❡ ✉t✐❧✐③❛♠♦s ❛ ❡q✉❛çã♦ ✭✷✳✶✶✮ ♣❛r❛ s✉❜st✐t✉✐r♠♦s ♦ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦ ♥❛ ú❧t✐♠❛ ❡q✉❛✲ çã♦✳ ❆❣♦r❛ ✉t✐❧✐③❛♥❞♦ ♦ ❛rt✐❢í❝✐♦ ♠❛t❡♠át✐❝♦ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦✱ ❡♠ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r p d ˙q = d(p ˙q dp i i i i i i

  ) − ˙q ✱ ❡ ❡♠ s❡❣✉✐❞❛ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✭✷✳✶✺✮✱ t❡r❡♠♦s✿ X ! X X ∂L ∂L d p ˙q = ˙q dp dq dt, i i i i i

  − L − − ✭✷✳✶✻✮ i i i ∂q ∂t i ❞♦♥❞❡ s✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✹✮ ♥❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ t❡♠♦s q✉❡✿ X X

  ∂L ∂L dH = ˙q dp dq dt, i i i − − ✭✷✳✶✼✮ i i ∂q i ∂t

  ♦♥❞❡ ✈❡♠♦s ❛❣♦r❛ q✉❡ H é ✉♠❛ ❢✉♥çã♦ ❞❡ q✱ p ❡ t✱ ❝♦♥❢♦r♠❡ ✜❝❛ ❝❧❛r♦ ♣❡❧❛s ❞✐❢❡r❡♥❝✐❛✐s ∂L dq i i i ✱ dp ❡ dt✳ ❆❣♦r❛ s❡ r❡❡s❝r❡✈❡r♠♦s ❛ ❡q✉❛çã♦ ✭✷✳✶✼✮ s✉❜st✐t✉✐♥❞♦✲s❡ ♣♦r ˙p ✱ t❡♠♦s✿ X X ∂q i

  ∂L dH = ˙q dp p ˙ dq dt, i i i i − − ✭✷✳✶✽✮ i i ∂t

  ♠❛s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ H = H(q, p, t) ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ X X ∂H ∂H ∂L dH = + dq dp dt, i i

  • i i

  ✭✷✳✶✾✮

  ∂q ∂p ∂t i i ❛❣♦r❛ ❝♦♠♣❛r❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✷✳✶✽✮ ❡ ✭✷✳✶✾✮✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r q✉❡✿

  ∂H ∂H ,

  ˙p = − ❡ ˙q = ✭✷✳✷✵✮ ∂q ∂p

  ♦♥❞❡ ❛s ❡q✉❛çõ❡s ❛❝✐♠❛ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ❬✶✻❪✳

  • ∂A ∂p i dp i dt !
  • ∂A

  ∂H ∂p i

  {A + B, C} = {A, C} + {B, C} ✭✷✳✷✼✮ {AB, C} = {A, C}B + A{B, C} ✭✷✳✷✽✮

  {A, B} = −{B, A} ✭✷✳✷✻✮

  é ❝❤❛♠❛❞♦ ❞❡ ♣❛rê♥t❡s❡ ❞❡ P♦✐ss♦♥ ❬✶✷❪ ❞❡ A ❡ H✳ ❖s ♣❛rê♥t❡s❡s ❞❡ P♦✐ss♦♥ ❛♣r❡s❡♥t❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♥t❡r❡ss❛♥t❡s✱ ❛ s❛❜❡r✿

  ∂q i ! ✭✷✳✷✺✮

  ∂p i ∂H

  − ∂A

  ∂H ∂p i

  ∂A ∂q i

  ♦♥❞❡ {A, H} = X i

  ∂t ✭✷✳✷✹✮

  = {A, H} + ∂A

  ∂t ✭✷✳✷✸✮ dA dt

  ∂H ∂q i !

  − ∂A ∂p i

  = X i ∂A ∂q i

  {A, {B, C}} + {C, {A, B}} + {B, {C, A}} = 0 ✭✷✳✷✾✮

  ∂A ∂t dp +

  ❊♠ t❡r♠♦s ❞❛ ❢✉♥çã♦ ❤❛♠✐❧t♦♥✐❛♥❛ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛çã♦ ❝♦♠♦ ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ q✭t✮ ❡ ♣✭t✮ ❞❛❞♦ ♣♦r✿

  S = Z t 2 t 1 [p(t) ˙q(t) − H(q(t), p(t), t)] dt

  ✭✷✳✷✶✮ s❡♥❞♦ q✉❡ ❡ss❛ é ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❛ ❛çã♦✳

  ✷✳✹ ❆ r❡❧❛çã♦ ❝♦♠ ❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛

  ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ A(q, p, t) ❝♦♠ q ❡ p s❡♥❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ❝♦♦r❞❡♥❛❞❛ ❣❡✲ ♥❡r❛❧✐③❛❞❛ ❡ ♦ ♠♦♠❡♥t♦ ❝❛♥ô♥✐❝♦ ❝♦♥❥✉❣❛❞♦ ❡ t é ♦ t❡♠♣♦✳ ❙✉❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❬✶✷❪ s❡rá ❞❛❞❛ ♣♦r✿ dA =

  ∂A ∂t dq +

  ∂A ∂t dt dA dt

  ❛❣♦r❛ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ú❧t✐♠❛ ❡q✉❛çã♦ ❝♦♠♦✿ dA dt

  = X i ∂A ∂q i dq i dt

  ∂t dA dt

  = X i ∂A ∂q i

  ˙q i + ∂A ∂p i

  ˙p i ! + ∂A

  ∂t ✭✷✳✷✷✮

  • ∂A

  ◆❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ✉t✐❧✐③❛♠♦s ♦♣❡r❛❞♦r❡s ❤❡r♠✐t✐♥✐❛♥♦s ❛t✉❛♥❞♦ s♦❜r❡ ✉♠ ❢✉♥çã♦ ❞❡ ❡st❛❞♦ ♣❛r❛ ♦❜t❡r♠♦s ♦s ♦❜s❡r✈á✈❡✐s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❛✉t♦✈❛❧♦r ❬✶✼❪✿

  ∂A ∂t

  Haψ ✭✷✳✸✸✮

  ❡ s✉❜st✐✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✷✸✮ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ t❡♠♦s✿ ∂A

  ∂t ψ +

  1 i¯h AHψ = da dt

  ψ +

  1 i¯h HAψ

  ✭✷✳✸✹✮

  1 i¯h (AH − HA)ψ +

  ψ = da dt

  ψ +

  ψ ✭✷✳✸✺✮

  1 i¯h [A, H] +

  ∂A ∂t !

  ψ = dA dt

  ψ ✭✷✳✸✻✮ dA dt

  =

  1 i¯h [A, H] +

  ∂A ∂t

  ✭✷✳✸✼✮ ♦♥❞❡ [A, H] = AH − HA é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠✉t❛❞♦r ❡♥tr❡ ♦s ♦♣❡r❛❞♦r❡s A ❡ H✳

  1 i¯h ˆ

  A ˆ Hψ = da dt

  ˆ Aψ = aψ

  ( ˆ Aψ) = ∂

  ✭✷✳✸✵✮ ❆ ❝♦♥❞✐çã♦ ♣❛r❛ ♦ ♦♣❡r❛❞♦r ˆ

  A s❡r ❤❡r♠✐t✐❛♥♦ ❞❡❝♦rr❡ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❛✉t♦✈❛❧♦r a s❡r ✉♠❛ q✉❛♥t✐❞❛❞❡ r❡❛❧✳ ◆❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❬✶✼❪ ❞♦ ❡st❛❞♦ ψ ♦❝♦rr❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ i¯h

  ∂ ∂t

  ψ = ˆ Hψ ✭✷✳✸✶✮

  ♦♥❞❡ ˆ H

  é ♦ ♦♣❡r❛❞♦r ❤❡r♠✐t✐❛♥♦✳ ❯t✐❧✐③❛♥❞♦ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦ ♦♣❡r❛❞♦r ˆ

  A ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦✿

  ∂ ∂t

  ∂t (aψ)

  1 i¯h ˆ

  ∂ ˆ A ∂t

  ψ + ˆ A ∂ψ

  ∂t = da dt

  ψ + a ∂ψ

  ∂t ✭✷✳✸✷✮

  ♠❛s t❡♠♦s ❞❛ ❡q✉❛çã♦ ✭✷✳✷✹✮ q✉❡ ∂ψ ∂t = 1 i¯ h

  ˆ Hψ

  ✱ ❧♦❣♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ✜❝❛✿ ∂ ˆ A

  ∂t ψ +

  ❖s ❝♦♠✉t❛❞♦r❡s t❛♠❜é♠ ♦❜❡❞❡❝❡♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❬✶✽❪✱ t❛❧ q✉❡✿ [A, B] = − [B, A] ✭✷✳✸✽✮

  [A + B, C] = [A, C] + [B, C] ✭✷✳✸✾✮

  [AB, C] = [A, C] B + A [B, C] ✭✷✳✹✵✮

  [A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0 ✭✷✳✹✶✮

  ❖❜s❡r✈❡ ❛ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❝♦♠✉t❛❞♦r❡s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♣❛rê♥t❡s❡s ❞❡ P♦✐ss♦♥✳ ■st♦ s✉❣❡r❡ q✉❡ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞❛ s✉❛ ❝♦r✲ r❡s♣♦♥❞❡ ❝❧áss✐❝❛ ❢❛③❡♥❞♦✲s❡ ❛ s❡❣✉✐♥t❡ s✉❜st✐t✉✐çã♦✿

  {A, B} −→

  1 i¯h [A, B]

  ✭✷✳✹✷✮ ❊ss❡ ♣r♦❝❡ss♦ ❞❡ q✉❛♥t✐③❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r P❛✉❧ ❉✐r❛❝ ❡ r❡❝❡❜❡✉ ♦ ♥♦♠❡ ❞❡ q✉❛♥t✐③❛çã♦ ❝❛♥ô♥✐❝❛ ❬✶✷❪✳

  ✸ ❖ ♣r♦♣❛❣❛❞♦r ❞❡ ❋❡②♥♠❛♥ ✸✳✶ ❖ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧

  ◆❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♦ t❡♠♣♦ é ❛♣❡♥❛s ✉♠ ♣❛râ♠❡tr♦✱ ♦✉ s❡❥❛✱ ♥ã♦ é ✉♠ ♦❜s❡r✈á✈❡❧✱ ♦♥❞❡ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♦ t❡♠♣♦ é ✉♠ ♣❛râ♠❡tr♦ ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ Ψ(x, t) q✉❡ ♥♦s ❢♦r♥❡❝❡ ♦ ❡st❛❞♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ ♣♦❞❡♥❞♦ t❛♠❜é♠ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ❡st❛❞♦ ❦❡t |Ψ > ♥❛ ♥♦t❛çã♦ ❞❡ ❉✐r❛❝✳ ❆ss✐♠✱ ♣❛r❛ ❡✈✐❞❡♥❝✐❛r ♦ t❡♠♣♦ ❝♦♠♦ ✉♠ ♣❛râ♠❡tr♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ✉♠ ❡st❛❞♦ ❦❡t ❛r❜✐trár✐♦ ❝♦♠♦✿

  > |Ψ, t ✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ t = t ✳

  ; t > >

  |Ψ, t ✱ ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ t > t ✱ q✉❡ ❡st❛✈❛ ❡♠ |Ψ, t ✱ q✉❛♥❞♦ t = t ✳ >

  P♦❞❡♠♦s✱ ❡♥tã♦✱ ♥♦s ♣❡r❣✉♥t❛r ❝♦♠♦ ♣❛ss❛♠♦s ❞❡ ✉♠ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧ |Ψ, t ♣❛r❛ ; t >

  ✉♠ ❡st❛❞♦ ❦❡t ❢✉t✉r♦ |Ψ, t ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ❙❛❜❡♠♦s q✉❡ ♥❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❜❛st❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✱ ♠❛s ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ s❡❣✉♥❞♦ ❉✐r❛❝✱ é ✐♠♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❛♥á❧♦❣❛ ❛ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❬✶✵❪✳ ❊ ❝♦♠♦ s❡ ❞á ❡ss❛ ❡✈♦❧✉çã♦ ❞♦ ❡st❛❞♦ ❦❡t ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛❄ ❆ r❡s♣♦st❛ ♥ã♦ ♣♦❞❡r✐❛ s❡r ♦✉tr❛ s❡♥ã♦ ❛tr❛✈és ❞❡ ✉♠ ♦♣❡r❛❞♦r ❛t✉❛♥❞♦ ♥♦ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧✳ ❊ q✉❡♠ s❡r✐❛ ❡ss❡ ♦♣❡r❛❞♦r❄

  ) ❖ ❖♣❡r❛❞♦r ❞❡ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧ ❯(t, t ✱ ❞❡ ❢♦r♠❛ q✉❡✿ ; t >= > .

  |Ψ, t ❯(t, t )|Ψ, t ✭✸✳✶✮ ❖ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ t❡♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❬✶✾❪ ✐♠♣♦rt❛♥t❡s q✉❡ sã♦✿ ✶✮ ❯♥✐t❛r✐❡❞❛❞❡✿ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ é ✉♥✐tár✐♦✳ ❉❡ ❢♦r♠❛ q✉❡ t❡♠♦s✿ (t, t ) ) = 1.

  ❯ ❯(t, t ✭✸✳✷✮

  ❆ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❡stá ♥♦ ❢❛t♦ ❞❡ ❝♦♥s❡r✈❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❞❡ ✉♠ ❡st❛❞♦ ❦❡t✱ ❞❡✈❡♥❞♦ ✉♠ ❡st❛❞♦ ❦❡t q✉❡ ❡st❛✈❛ ♥♦r♠❛❧✐③❛❞♦ ❡♠ t = t ♣❡r♠❛♥❡❝❡r ♥♦r♥❛❧✐③❛❞♦ ♥✉♠ t❡♠♣♦ ❢✉t✉r♦ t > t ✳

  ✷✮ ❈♦♠♣♦s✐çã♦✿ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♦❜❡❞❡❝❡ ❛ ❧❡✐ ❞❛ ❝♦♠♣♦s✐çã♦✱ ❛ss✐♠✿ , t ) = , t ) , t ), > t > t .

  ❯(t 2 ❯(t 2 1 ❯(t 1 ♣❛r❛ t 2 1 ✭✸✳✸✮ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ ♥♦s ❞✐③ q✉❡ ♣❛r❛ ♦❜t❡r♠♦s ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧

  > ; t 2 > ♣❛r❛ ✉♠ ❡st❛❞♦ ❦❡t |Ψ, t ❜❛st❛ ❛♣❧✐❝❛r ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧

  |Ψ, t ; t > 1

  ❞✉❛s ✈❡③❡s✱ ♣r✐♠❡✐r♦ ♣❛r❛ ♦❜t❡r ♦ ❡st❛❞♦ ❦❡t |Ψ, t ❡ ♥♦✈❛♠❡♥t❡ ♣❛r❛ ♦❜t❡r ♦ ❡st❛❞♦ ; t 2 >

  ❦❡t |Ψ, t ✳ ✸✮ ■♥✜♥✐t❡s✐♠❛❧✿ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❞❡ ❢♦r♠❛

  ✐♥✜♥✐t❡s✐♠❛❧✱ t❛❧ q✉❡✿ ; t + dt >= + dt, t > .

  |Ψ, t ❯(t )|Ψ, t ✭✸✳✹✮ P♦❞❡♠♦s ✈❡r✜✜❝❛r ❝♦♠♦ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♥♦ ❧✐♠✐t❡ ❡♠ q✉❡ dt → 0✱ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ s❡ tr❛♥s❢♦r♠❛ ♥✉♠❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ♦♣❡r❛❞♦r ✉♥✐tár✐♦✱ t❛❧ q✉❡✿ lim + dt, t ) = 1. dt→0 ❯(t

  ✭✸✳✺✮ ❆❣♦r❛ q✉❡ ❥á s❛❜❡♠♦s q✉❛✐s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣♦❞❡✲

  ♠♦s ♥♦s ♣❡r❣✉♥t❛r ❝♦♠♦ ❝❛❧❝✉❧❛r ❡ss❡ ♦♣❡r❛❞♦r✳ ❇❡♠✱ ♣❛rt✐♥❞♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r H

  ♥✉♠ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦ ♦♥❞❡ t❡♠♦s ✉♠ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ˆ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ q✉❡ ❛t✉❛♥❞♦ ♥♦ ❡st❛❞♦ ❦❡t ❛r❜✐trár✐♦ ♥♦s ❢♦r♥❡❝❡ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ E ❞❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ q✉❡ ♣❡r♠❛♥❡❝❡ ❝♦♥st❛♥t❡✱ t❛❧ q✉❡✿

  ∂ i¯h |Ψ >= ˆ H|Ψ >, ✭✸✳✻✮ ¯ h ∂ 2 2 2 + V (x) ∂t

  ❝♦♠ ˆ H = − ✳ 2m ∂x P❛r❛ r❡s♦❧✈❡r♠♦s ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✭✸✳✻✮ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ✉♠❛ té❝♥✐❝❛ ❜❛st❛♥t❡

  ❝♦♥❤❡❝✐❞❛ ❞♦s ❢ís✐❝♦s✱ ❛ té❝♥✐❝❛ ❞❛ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❬✷✵❪✱ ♦♥❞❡ ❢❛③❡♠♦s Ψ(x, t) = φ(t)ψ(x)

  ✱ ♦✉ s❡❥❛✱ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❛q✉✐ ❛♣❛r❡❝❡ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ❞✉❛s s♦❧✉çõ❡s φ(t)✱ q✉❡ ❞❡♣❡♥❞❡ só ❞♦ t❡♠♣♦✱ ❡ ψ(x)✱ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❛ ♣♦s✐çã♦✳ ❆❣♦r❛ s✉❜st✐t✉✐♥❞♦

  Ψ(x, t) = φ(t)ψ(x) ♥❛ ❡q✉❛çã♦ ✭✸✳✻✮✱ t❡♠♦s q✉❡✿ 2 2 dφ ¯h d ψ i¯hψ φ + V (x)ψφ

  = − 2 ✭✸✳✼✮ dt 2m dx ❛❣♦r❛ ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♣♦r Ψ(x, t) = φ(t)ψ(x)✱ t❡♠♦s✿ 2 2 1 dφ ¯h d ψ

  1 i¯h + V (x) ✭✸✳✽✮

  = − 2 φ dt 2m dx ψ

  ❡♥tr❡t❛♥t♦ ❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ só ♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r❛ s❡ ♦s ❞♦✐s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❢♦r❡♠ ❝♦♥st❛♥t❡s✳ ❆ss✉♠✐♥❞♦ ❝♦♠♦ ✈❡r❞❛❞❡✐r❛ ❛ ✐❣✉❛❧❞❛❞❡✱ ❝❤❛♠❛r❡♠♦s ❡ss❛ ❝♦♥st❛♥t❡ ❞❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❞❡ E✱ ❧♦❣♦ t❡♠♦s ♣❛r❛ ❛ ♣❛rt❡ q✉❡ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❬✶✽❪✿ 1 dφ i¯h = E

  ✭✸✳✾✮ φ dt dφ i

  Edt = − ✭✸✳✶✵✮ Z Z φ ¯h dφ i

  E dt = −

  ✭✸✳✶✶✮ φ ¯h i

  Et ln φ = − ✭✸✳✶✷✮ ¯h i

  Et] ✭✸✳✶✸✮

  φ(t) = exp[− ¯h

  = 0 ♦♥❞❡ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡ ♦♣t❛♠♦s ♣♦r ❢❛③❡r t ❡ ♦♠✐t✐♠♦s ❛ ❝♦♥st❛♥t❡ ❛r❜✐trár✐❛ q✉❡ s✉r❣❡ ♥❛ ú❧t✐♠❛ ❡q✉❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ♣♦❞❡ s❡r ❛❜s♦r✈✐❞❛ ♣❡❧❛ ♣❛rt❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♥❛ ❡q✉❛çã♦ ✭✸✳✻✮✳

  ❆ ❝♦♥st❛♥t❡ E é ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛✱ q✉❡ s✉r❣❡ ❝♦♠♦ ✉♠ ❛✉t♦✈❛❧♦r ❞♦ ♦♣❡r❛❞♦r H

  ❤❛♠✐❧t♦♥✐❛♥♦ ˆ ❛♣❧✐❝❛❞♦ ♥♦s ❛✉t♦❦❡ts ❞❡ ❡♥❡r❣✐❛✱ ♣❡r♠❛♥❡❝❡♥❞♦ ✐♥❛❧t❡r❛❞❛ ♥♦ ❝❛s♦ ❡st❛✲ H )

  ❝✐♦♥ár✐♦✳ P♦❞❡♠♦s ❡♥tã♦ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✭✸✳✻✮ ❡♠ t❡r♠♦s ❞♦s ♦♣❡r❛❞♦r❡s ˆ ❡ ❯(t, t ✱ t❛❧ q✉❡✿ ∂ i¯h ) = ˆ H ),

  ❯(t, t ❯(t, t ✭✸✳✶✹✮ i ∂t ˆ

  )] ♦♥❞❡ ❯(t, t é ♦ ❖♣❡r❛❞♦r ❞❡ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧✳

  ) = exp[− H(t − t ¯ h >

  ❆❣♦r❛✱ s❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❡q✉❛çã♦ ✭✸✳✶✹✮ ♣♦r |Ψ, t ♣❡❧♦ ❧❛❞♦ ❞✐r❡✐t♦ ♥♦s ❞♦✐s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❬✷✶❪✱ ♦❜t❡♠♦s✿

  ∂ i¯h >= ˆ H >, ❯(t, t )|Ψ, t ❯(t, t )|Ψ, t ✭✸✳✶✺✮

  ∂t >

  ♠❛s ✈❡❥❛ q✉❡ |Ψ, t é ♦ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧ ♣❛r❛ t = t ✱ ❡♥tã♦ ❝♦♠ ❛ ❛t✉❛çã♦ ❞♦ ♦♣❡r❛❞♦r

  ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♥❡st❡ ❡st❛❞♦ ❦❡t✱ t❡♠♦s✿ ∂ i¯h ; t >= ˆ ; t >,

  |Ψ, t H|Ψ, t ✭✸✳✶✻✮ ∂t q✉❡ é ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♣❛r❛ ✉♠ ❡st❛❞♦ ❦❡t ❛r❜✐trár✐♦✳

  ❆ ❡q✉❛çã♦ ✭✸✳✶✹✮ ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞❛ ❢❛③❡♥❞♦✲s❡ ✉♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ✐♥✜♥✐t❛❬✷✷❪ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧✱ t❛❧ q✉❡✿ ❙❛❜❡♥❞♦ q✉❡ 2 3 n x x x x + e = 1 + x + + . . . + + . . .

  2! 3! n! ❊♥tã♦ t❡♠♦s ♣❛r❛ − H(t−t ) h ¯ i ˆ

  ) = e ❯(t, t q✉❡✿   2 ! 2

  ˆ ˆ H )   (−i) H(t − t ) + + . . . ❯(t, t

  ✭✸✳✶✼✮ ) = 1 − i (t − t

  ¯h 2 ¯h − H(t−t ) ¯ h i ˆ ) = e

  ❆❣♦r❛ t♦♠❛♥❞♦ ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧❬✽❪ ❞❡ ❯(t, t t❡♠♦s q✉❡✿ h i ˆ ˆ i   ! 2 ∂ H H − H(t−t ) ˆ 2 e ¯ h   ) + . . .

  • ∂t ¯h ¯h

  ✭✸✳✶✽✮ = −i (−i) (t − t

  ❛❣♦r❛ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ♣♦r i¯h✱ t❡♠♦s✿ h i i " # ∂ ˆ i ˆ ) − H(t−t ) H(t − t ¯ h i¯h e = ˆ H + . . .

  1 − ✭✸✳✶✾✮ ∂t ¯h

  ) ❛ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ❯(t, t q✉❡ ❛❣♦r❛ ❛♣❛r❡❝❡ ♥❛s ❞✉❛s ❢♦r♠❛s✱ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❡①♣❛♥❞✐❞♦✱ t❡♠♦s✿

  ∂ i¯h ) = ˆ H ) ❯(t, t ❯(t, t ✭✸✳✷✵✮

  ∂t q✉❡ é ❡①❛t❛♠❡♥t❡ ❛ ❡q✉❛çã♦ ✭✸✳✶✹✮✳ A

  H ◗✉❛♥❞♦ ✉♠ ♦♣❡r❛❞♦r ˆ ❝♦♠✉t❛ ❝♦♠ ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ˆ ✱ ❞❡ ❢♦r♠❛ q✉❡ ♦s

  A H ❛✉t♦❦❡ts ❞❡ ˆ sã♦ t❛♠❜é♠ ❛✉t♦❦❡ts ❞❡ ˆ ✱ ❝❤❛♠❛❞♦s ❞❡ ❛✉t♦❦❡ts ❞❡ ❡♥❡r❣✐❛✱ ❝✉❥♦s ❛✉t♦✲ ✈❛❧♦r❡s sã♦ ❞❡♥♦t❛❞♦s ♣♦r E ✿ a ′ ′

  ˆ >= E >

  H|a a |a ✭✸✳✷✶✮ ) P♦❞❡♠♦s ❡①♣❛♥❞✐r ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❯(t, t ❡♠ t❡r♠♦s ❞♦s ❛✉t♦❦❡ts ′ ′

  > >< a

  |a ✱ ❛tr❛✈és ❞♦ ❡♠♣r❡❣♦ ❞♦ ♦♣❡r❛❞♦r ❞❡ ♣r♦❥❡çã♦❬✷✶❪✱ ❞❛❞♦ ♣♦r |a | q✉❡ ♦❜❡❞❡❝❡

  = 0 ❛ r❡❧❛çã♦ ❞❡ ❝♦♠♣❧❡t❡③❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ P♦rt❛♥t♦ t♦♠❛♥❞♦ t ✱ t❡♠♦s q✉❡✿ ! ! X X ′′ ′′ ′ ′ i ˆ Ht i ˆ Ht exp = >< a >< a

  − |a | exp − |a | ¯h ′ ′′ ¯h ! X a a X ′′ ′′ ′ ′ i ˆ Ht iE t a exp = >< a >< a

  − |a | exp − |a | ¯h ′ ′′ ¯h ! a a X X ′′ ′′ ′ ′ i ˆ Ht iE t a exp = exp >< a >< a

  − − |a |a | ′′ ′ ′ ′′ ¯h ¯h a a ′ ′′ >= δ

  ♦♥❞❡ < a |a ✱ ❛ss✐♠✿ a a ! ! X ′ ′ i ˆ Ht i ˆ Ht exp = exp >< a

  − − |a | ✭✸✳✷✷✮ ¯h ¯h a

  ❖ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❡s❝r✐t♦ ❞❡ss❛ ❢♦r♠❛ ♥♦s ♣❡r♠✐t❡ r❡s♦❧✈❡r q✉❛❧q✉❡r >

  ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡①♣❛♥sã♦ ❞♦ ❦❡t ✐♥✐❝✐❛❧ ❡♠ t❡r♠♦s ❞❡ |a é ❝♦♥❤❡❝✐❞❛✳ ❊①❡♠♣❧♦✿ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡①♣❛♥sã♦ ❞♦ ❦❡t ✐♥✐❝✐❛❧ s❡❥❛ X ′ ′ ′ X

  = 0 >= >< a >= c > ✭✸✳✷✸✮

  |Ψ, t |a |Ψ, t a |a a a ′ ′ ♦✉ ❛✐♥❞❛✱ ! i ˆ Ht

  = 0; t >= exp = 0 > |Ψ, t − |Ψ, t X ! ¯h ′ ′ i ˆ Ht

  = 0; t >= exp >< a > |Ψ, t − |a |Ψ, t X X ′′ ′′ ′ ′ a ′ ¯h ′′ iE t a

  = 0; t >= exp >< a >< a > |Ψ, t − |a |a |Ψ, t a a ′ ′′ ¯h X ′ = 0; t >= c > .

  |Ψ, t a (t)|a ✭✸✳✷✹✮ a ◆❡st❡ ❝❛s♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❡①♣❛♥sã♦ ✈❛r✐❛ ❝♦♠ ♦ t❡♠♣♦✱ t❛❧ q✉❡✿ ′ ′ iE a t a (t) = c (t = 0) exp , a a − ✭✸✳✷✺✮

  ¯h ❝♦♠ s❡✉ ♠ó❞✉❧♦ ✐♥❛❧t❡r❛❞♦✳ ◆♦t❡ q✉❡ ❛s ❢❛s❡s r❡❧❛t✐✈❛s ❡♥tr❡ ❛s ✈ár✐❛s ❝♦♠♣♦♥❡♥t❡s ✈❛r✐❛♠

  /¯h ❝♦♠ ♦ t❡♠♣♦ ♣♦rq✉❡ ❛s ❢r❡q✉ê♥❝✐❛s ❞❡ ♦s❝✐❧❛çõ❡s✱ E ✱ sã♦ ❞✐❢❡r❡♥t❡s✳ ′ ′ a

  > ◗✉❛♥❞♦ ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ é ✉♠ ❞♦s {|a >}✱ ♦✉ s❡❥❛✱ t❡♠♦s |Ψ, t = 0 >= |a ✱ ♣♦❞❡♠♦s

  ❡s❝r❡✈❡r ♣❛r❛ t❡♠♣♦s ❢✉t✉r♦s q✉❡✿ ′ ′ iE t a , t > exp . |Ψ = a = 0; t >= |a − ✭✸✳✷✻✮

  ◗✉❛♥❞♦ ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✱ ♠❛s ❝♦♠✉t❛ ❡♥tr❡ s✐ ♣❛r❛ t❡♠♣♦s H(t) H(t )

  ❞✐st✐♥t♦s✱ ♦✉ s❡❥❛✱ ˆ ❝♦♠✉t❛ ❝♦♠ ˆ ✱ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧❬✷✸❪ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿ Z t ′ ′ i

  ˆ ) = exp dt H(t ) . ❯(t, t − ✭✸✳✷✼✮

  ¯h t ◗✉❛♥❞♦ ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✱ ♠❛s✱ ❛♦ ❝♦♥trár✐♦ ❞♦ ❝❛s♦ ❛♥✲ t❡r✐♦r✱ ♥ã♦ ❝♦♠✉t❛ ❡♥tr❡ s✐ ❡♠ t❡♠♣♦s ❞✐st✐♥t♦s✱ ❡s❝r❡✈❡r❡♠♦s ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❝♦♠ ✉♠❛ sér✐❡✱ t❛❧ q✉❡✿ X n Z Z Z t t 1 t n−1 i

  ˆ ) = 1 + dt dt . . . dt H(t ) ˆ H(t ) . . . ˆ H(t ),

  ❯(t, t 1 2 n 1 2 n ✭✸✳✷✽✮t t t n ¯h q✉❡ é ❝❤❛♠❛❞❛ ❞❡ sér✐❡ ❞❡ ❉②s♦♥❬✷✶❪✱ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛♦ s❡✉ ❝r✐❛❞♦r✳

  H ❆❣♦r❛ ✈♦❧t❛♥❞♦ ❛♦ ❝❛s♦ ❞♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ˆ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ ♣♦❞❡♠♦s

  ) H ❝❛❧❝✉❧❛r ❯(t, t ❡♠ t❡r♠♦s ❞♦s ❛✉t♦❦❡ts ❞❡ ˆ ✱ t❛❧ q✉❡✿ X ′′ ′′ i ˆ Ht ′ ′ h ¯

  = 0) = >< a >< a ❯(t, t |a |e |a | ✭✸✳✷✾✮

  ♥❡st❡ ❝❛s♦ ♦ ❡s♣❡❝tr♦ é ❞✐s❝r❡t♦ ✭❛✉t♦❦❡ts ❞❡ ❡♥❡r❣✐❛✮✳ ❊ ❝♦♠♦✱ ′′ i ˆ Ht ′ a′ − − h h ¯ ¯ ′′ ′ iE t < a >= δ e

  |e |a a a ✭✸✳✸✵✮ t❡♠♦s ❡♥tã♦✱ q✉❡✿ X a′ ′ ′ iE t h ¯ = 0) = e >< a

  ❯(t, t |a | ✭✸✳✸✶✮ 1 , q , . . . , q ), 2 j ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ❛❣♦r❛ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ♦♥❞❡ t❡♠♦s ~q ≡ (q s❡♥❞♦ j

  ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞♦ s✐st❡♠❛✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❝♦♠♦ q > |~ ✳ ❈♦♠ ✐ss♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ ♣❛rtí❝✉❧❛ ♥♦ ❡st❛❞♦ |~q > s❡rá ❞❛❞❛ ♣♦r✿ 2 2 P (~q, ~ q q q

  ; t) = | < ~q|❯(t, t )|~ > | = |K(~q, ~ ; t)| ✭✸✳✸✷✮ ♦♥❞❡✱

  K(~q, t; ~ q q > ✭✸✳✸✸✮

  , 0) =< ~q|❯(t, t )|~ ❉❡✜♥✐❞♦ ❞❡ss❛ ❢♦r♠❛✱ ✈❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡ ♦ ❦❡r♥❡❧ ✭♥ú❝❧❡♦✮ ❑ é ✉♠ ♣r♦♣❛❣❛❞♦r❬✷✶❪

  > ❞♦ s✐st❡♠❛✱ ♣❛ss❛♥❞♦✲♦ ❞♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ |~q ♥♦ ✐♥st❛♥t❡ t ♣❛r❛ ♦ ❡st❛❞♦ ♣♦st❡r✐♦r |~q > ♥♦ ✐♥st❛♥t❡ ♣♦st❡r✐♦r t > t ✱ ❧♦❣♦ ♣❡r❝❡❜❡✲s❡ q✉❡ ♦ ❦❡r♥❡❧ ✭♥ú❝❧❡♦✮ K r❡♣r❡s❡♥t❛ ✉♠❛

  > ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧ |~q ❛✈❛♥ç❛r ❛♦ ❡st❛❞♦ ❦❡t ♣♦st❡r✐♦r

  > ✱

  |~q >✳ P♦❞❡♠♦s t❛♠❜é♠ r❡♣r❡s❡♥t❛r ♦ ❦❡r♥❡❧ ✭♥ú❝❧❡♦✮ K ❡♠ t❡r♠♦s ❞♦s ❛✉t♦❦❡ts |a ❜❛st❛ q✉❡ t♦♠❡♠♦s ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❛ ❡q✉❛çã♦ ✭✸✳✸✶✮ s✉❜st✐t✉✐♥❞♦✲♦ ♥❛

  ❡q✉❛çã♦ ✭✸✳✸✸✮✱ ♦ q✉❡ ♥♦s ❞á✿ X a′ ∗ − iE t h ¯ K(~q, t; ~ q , 0) = Ψ (~q)Ψ (~ q )e n a ✭✸✳✸✹✮

  ♦♥❞❡ t❡♠♦s q✉❡✿ Ψ > a (~q) =< ~q|a ✭✸✳✸✺✮

  ✸✳✷ ❖ ♣r♦♣❛❣❛❞♦r ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ✸✳✷✳✶ ❆ ♦r✐❣❡♠ ❞♦ ♣r♦♣❛❣❛❞♦r

  ❈♦♠♦ ✈✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r ♣♦❞❡♠♦s s♦❧✉❝✐♦♥❛r ♦ ♣r♦❜❧❡♥❛ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ H

  ❞♦ s✐st❡♠❛ ❝♦♠ ✉♠ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ˆ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ❛tr❛✈és ❞❛ ❡①♣❛♥sã♦ A H

  ❞♦ ❡st❛❞♦ ❦❡t ✐♥✐❝✐❛❧ ❡♠ t❡r♠♦s ❞♦s ❛✉t♦❦❡ts ❞❡ ✉♠ ♦♣❡r❛❞♦r ˆ q✉❡ ❝♦♠✉t❡ ❝♦♠ ˆ ✱ ❞❡ ♠♦❞♦ q✉❡✿ " # i ˆ )

  H(t − t ; t >= exp >

  |Ψ, t − |Ψ, t ′ ′ " # ¯h iE ) a (t − t >< a > exp ,

  |Ψ, t ; t >= |a |Ψ, t − ✭✸✳✸✻✮ ¯h

  ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ ♣❡❧❛ ❡sq✉❡r❞❛✱ ♣♦r < x | t❡♠♦s✿ X ′ ′ ′ " # iE ) a (t − t < x ; t >= < x >< a > exp ,

  |Ψ, t |a |Ψ, t − ✭✸✳✸✼✮ a ¯h q✉❡ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ♣♦r✿ ′ ′ ′ ′ X ′ ′ " # iE ) a (t − t Ψ(x , t) = c (t )u (x ) exp , a a − ✭✸✳✸✽✮ a ¯h

  ❝♦♠ ′ ′ ′ ′ u (x ) =< x >, a |a A s❡♥❞♦ ❛ ❛✉t♦❢✉♥çã♦ ❞❡ ˆ ❝♦♠ ❛✉t♦✈❛❧♦r a ✳

  P♦❞❡♠♦s ❛✐♥❞❛ ❝❛❧❝✉❧❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❡①♣❛♥sã♦ c a ✱ t❛❧ q✉❡✿ ′ ′ ′ ′ ′ Z < a >= dx < a >< x >,

  ✭✸✳✸✾✮ |Ψ, t |x |Ψ, t >= c

  ♦♥❞❡ < a |Ψ, t a ✱ t❛❧ q✉❡✿ ′ ′ Z ′ ′ ′ c (t ) = dx u (x )Ψ(x , t ). a a ✭✸✳✹✵✮

  ❆❣♦r❛✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ❡q✉❛çã♦ ✭✸✳✸✼✮ ✉t✐❧✐③❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✸✳✸✾✮✱ ♦♥❞❡ ✈❡r❡♠♦s q✉❡ ❛♣❛r❡❝❡ ✉♠ ♦♣❡r❛❞♦r ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ❛t✉❛♥❞♦ ♥❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ✐♥✐❝✐❛❧ ♦✉ ❦❡t ✐♥✐❝✐❛❧✱ t❛❧ q✉❡✿ ′′ X ′′ ′ ′ ′ ′ ′ Z " # iE ) a (t − t

  Ψ(x , t) = < x > dx < a >< x > exp |a |x |Ψ, t − ′′ ′ a Z X ′′ ′ ′ ′ ′ ′ " # ¯h iE ) a (t − t

  Ψ(x , t) = dx < x >< a > exp < x > |a |x − |Ψ, t ′′ ′ ′′ ′ ′ ′ ′ Z a X ′ ′ " # ¯h iE ) a (t − t

  Ψ(x , t) = dx < x >< a > exp Ψ(x , t ) |a |x − a ¯h ♦♥❞❡ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❝♦♠♦✱ ′′ ′ ′′ ′ ′ Z

  Ψ(x , t) = dx K(x , t; x , t )Ψ(x , t ), ✭✸✳✹✶✮

  ✈❡♠♦s ❛q✉✐ q✉❡ ♦ ❦❡r♥❡❧ K s✉r❣❡✱ ♥❛t✉r❛❧♠❡♥t❡✱ ❝♦♠♦ ♦ ♥ú❝❧❡♦ ❞♦ ♦♣❡r❛❞♦r ✐♥t❡❣r❛❧✱ s❡♥❞♦ ♦ r❡❢❡r✐❞♦ ♥ú❝❧❡♦ ♦ ♥♦ss♦ ♣r♦♣❛❣❛❞♦r q✉â♥t✐❝♦✱ q✉❡ é ❞❛❞♦ ♣♦r✿ ′′ ′ ′′ ′ ′ ′ X " # iE ) a (t − t K(x , t; x , t ) = < x >< a > exp

  ✭✸✳✹✷✮ a ′ ¯h |a |x −

  ✸✳✷✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♣r♦♣❛❣❛❞♦r

  ❖ ♣r♦♣❛❣❛❞♦r ❛♣r❡s❡♥t❛ ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ✐♠♣♦rt❛♥t❡s ❞❛s q✉❛✐s ❞✐s❝✉t✐r❡♠♦s ❛❜❛✐①♦✿ ′′ ′

  , t; x , t )

  • P❛r❛ ✉♠ ✐♥st❛♥t❡ t > t ✱ K(x s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❡♣❡♥❞❡♥t❡ ′′ ′

  ❞♦ t❡♠♣♦ ♥❛s ✈❛r✐á✈❡✐s x ❡ t✱ ❝♦♠ x ❡ t ✜①♦s✳ ❉❡♠♦♥str❛çã♦✿ ❆ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ✭✸✳✹✷✮✱ ′′ ′ ′ ′ ′ X ′ " # iE ) a (t − t

  K(x , t : x , t ) = >< a > exp ✭✸✳✹✸✮ a ′ ¯h |a |x − q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✱ ′′ ′ X ′ ′′ ′ ′ iE t iE t a ∗ a ∗ ′ ′

  K(x , t : x , t ) = exp u (x ) exp u (x ) a ′ ¯h ¯h ′′ ′ a a X ′ ′′ K(x , t : x , t ) = u (x , t )u (x , t) ′′ a a a ✭✸✳✹✹✮

  (x , t) ❝♦♠♦ u s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✱ ❞❡s❞❡ a q✉❡ x ❡ t s❡❥❛♠ ✜①♦s✱ t❛♠❜é♠ s❛t✐s❢❛③✳ P♦t❛♥t♦✱ K t❛♠❜é♠ s❛t✐s❢❛③ ❡ss❛ ❡q✉❛çã♦ ❞❡

  ′′ ′

  • ❖ ❧✐♠✐t❡ ❞❡ K(x ′′ ′ ′′ ′
  • 3 lim K(x , t; x , t ) = δ (x ) t→t − x ❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❝♦♠♣❧❡t❡③❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣❛r❛ ♦s ❛✉t♦❦❡ts

      , t : x , t ) q✉❛♥❞♦ t → t s❡rá✿

      > |a ♥❛ ❡q✉❛çã♦ ′′ ′ ′′ ′ ′ ′ X ′ " # iE ) a (t − t

      K(x , t; x , t ) = < x >< a > exp |a |x − a ¯h r❡❞✉③✐♠♦s ❛ ′′ ′ X ′′ ′ " # iE ) a (t − t

      K(x , t; x , t ) = < x > exp a ′ ¯h |x − ❡ ❛❣♦r❛ ✉s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ♦rt♦❣♦♥❛❧ ❞♦s {|x ′′ ′ ′′ ′ >}✱ t❡♠♦s✿ " # 3 a (t − t iE )

      K(x , t; x , t ) = δ (x ) exp − x − ✭✸✳✹✺✮

      ¯h ❛❣♦r❛ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ t → t t❡♠✲s❡✿ ′′ ′ ′′ ′ 3 lim K(x , t; x , t ) = δ (x ). t→t − x ✭✸✳✹✻✮

      ❉❡✈✐❞♦ ❛ ❡ss❛s ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s ♦ ♣r♦♣❛❣❛❞♦r✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❢✉♥çã♦ ′′ ❞❡ x é s✐♠♣❧❡s♠❡♥t❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛✱ ♥♦ ✐♥st❛♥t❡ t✱ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ q✉❡ ❡st❛✈❛ ♣r❡❝✐s❛♠❡♥t❡ ❧♦❝❛❧✐③❛❞❛ ♥♦ ♣♦♥t♦ x ♥✉♠ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t ✳ ❉❡ ❢❛t♦✱ ❡st❛ ✐♥t❡r♣r❡t❛çã♦ s❡❣✉❡ ❞❛ ♦❜s❡r✈❛çã♦ q✉❡ ❛ ❡q✉❛çã♦ ✭✸✳✹✷✮ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿ ′′ ′ ′′ ′ " # iE ) a (t − t

      K(x , t; x , t ) =< x >, | exp − |x ✭✸✳✹✼✮

      ¯h >

      ♦♥❞❡ ♦ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❛t✉❛♥❞♦ s♦❜r❡ |x é ❥✉st❛♠❡♥t❡ ♦ ❡st❛❞♦ ❦❡t ♥♦ < t

      ✐♥st❛♥t❡ t ❞❡ ✉♠ s✐st❡♠❛ q✉❡ ❡st❛✈❛ ❧♦❝❛❧✐③❛❞♦ ❡♠ x ♥♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t ✳ ❙❡ q✉✐s❡r♠♦s r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ♠❛✐s ❣❡r❛❧ ♦♥❞❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♥ã♦ ❡stá ❧♦❝❛✲

      ❧✐③❛❞❛✱ ♠❛s ❞✐str✐❜✉✐✲s❡ ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ r❡❣✐ã♦ ✜♥✐t❛✱ ❞❡✈❡♠♦s ♠✉❧t✐♣❧✐❝❛r ❛ ❢✉♥çã♦ ❞❡ ′ ′ ′′ ′ ′ , t ) , t; x , t )

      ♦♥❞❛ Ψ(x ♣❡❧♦ ♣r♦♣❛❣❛❞♦r K(x ❡ ✐♥t❡❣r❛r s♦❜r❡ t♦❞♦ ♦ ❡s♣❛ç♦ ❞❡ x ✱ t❛❧ q✉❡✿ ′′ ′ ′′ ′ ′ Z 3 Ψ(x , t) = d x K(x , t; x , t )Ψ(x , t )

      ✭✸✳✹✽✮

      ✸✳✷✳✸ ❖ ♣r♦♣❛❣❛❞♦r ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥

      ◆❡st❛ s❡çã♦ ♥ã♦ ❞✐s❝✉t✐r❡♠♦s ♦s ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ♦✉ s✉❛s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❢ís✐❝❛ ❡ ♥❛ ♠❛t❡♠át✐❝❛✳ ❖ q✉❡ ♠♦str❛r❡♠♦s ❛q✉✐ s❡rá ❝♦♠♦ ❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ♣r♦♣❛❣❛❞♦r ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳

      ❆ ❢✉♥çã♦ ❞❡ ●r❡❡♥❬✷✵❪ s✉r❣❡ ❝♦♠ ❛ ❞❡r✐✈❛çã♦ ❞❡ ✉♠ ♣♦t❡♥❝✐❛❧ ❡❧❡tr♦stát✐❝♦ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ r❡❣✐ã♦ ❢❡❝❤❛❞❛ ♥♦ ❡s♣❛ç♦ ❡♠ t❡r♠♦s ❞❡ s❡✉ ✈❛❧♦r ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ r❡❣✐ã♦✳ ❉❛í✱ ♣♦❞❡✲s❡ ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ✉♠❛ ❝❧❛ss❡ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦ ❤♦♠♦❣ê♥❡❛s ❝♦♠ ❢♦♥t❡s ♣♦♥t✉❛✐s✳

      ❆s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦ ❤♦♠♦❣ê♥❡❛s ❝♦♠ ♦✉tr♦s t✐♣♦s ❞❡ ❢♦♥t❡s ♣♦❞❡♠ s❡r ❞❡t❡r♠✐♥❛❞❛s ✉♠❛ ✈❡③ q✉❡ s❡❥❛♠ ❝♦♥❤❡❝✐❞❛s ❛s s♦❧✉çõ❡s ❝♦♠ ❢♦♥t❡s ♣♦♥t✉❛✐s✱ ❝♦♠ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ✐♠♣♦st❛s ❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧✳

      ❊♠ ♠❡❝â♥✐❝❛ ♦♥❞✉❧❛tór✐❛ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠❛ ❢r❡♥t❡ ❞❡ ♦♥❞❛ ♣♦❞❡ s❡r ❞❡t❡r✲ ♠✐♥❛❞❛ ✉♠❛ ✈❡③ q✉❡ ❢♦r ❝♦♥❤❡❝✐❞❛ ❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ q✉❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ✐♠♣♦st❛s ❛♦ ♣r♦❜❧❡♠❛✳

      ❙❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ é ❝♦♥❤❡❝✐❞❛ ♥✉♠ ♣❛rt✐❝✉❧❛r ✐♥st❛♥t❡ t ♥✉♠ ♣♦♥t♦ ~x ❞♦ ❡s♣❛ç♦✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♥✉♠ ✐♥st❛♥t❡ t > t ♥✉♠ ♣♦♥t♦ ~x é ❞❛❞❛ ♣♦r✿ Z

      Ψ(~x, t) = dx G(~x, t; ~ x , t )Ψ( ~ x , t ) ✭✸✳✹✾✮

      , t ) s❡♥❞♦ G(~x, t; ~x ❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥✱ q✉❡ ♥❡ss❡ ❝❛s♦ é ❝❤❛♠❛❞❛ ❞❡ ♣r♦♣❛❣❛❞♦r✳ ❖❜s❡r✈❡✲ s❡ ❛ s❡♠❡❧❤❛♥❛ç❛ ❞❛ ❡q✉❛çã♦ ✭✸✳✹✽✮ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✸✳✹✾✮✳

      ❆ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ❞❡s❝r❡✈❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ❢✉♥çã♦ ❡♠ ~x ♥♦ ✐♥st❛♥t❡ t s♦❜r❡ ❛ ❢✉♥çã♦ ❡♠ ~x ♥♦ ✐♥st❛♥t❡ t > t ✳

      ❖ ♣r♦♣❛❣❛❞♦r ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✿ " ! # 2 ′′2 ′′ ′′ ′ ′′ ′

      ¯h ∂ 3

    • V (x K(x , t; x , t (x )

      ✭✸✳✺✵✮ − ∇ ) − i¯h ) = −i¯hδ − x )δ(t − t

      2m ∂t ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦✱ ′′ ′

      K(x , t; x , t ) = 0, t < t ❧♦❣♦ t❡♠♦s q✉❡ ♦ ♣r♦♣❛❣❛❞♦r ✈❛✐ ❞❡♣❡♥❞❡r ❞♦ t✐♣♦ ❞❡ ♣♦t❡♥❝✐❛❧ ❛ q✉❛❧ ♦ s✐st❡♠❛ s❡rá s✉❜♠❡t✐❞♦✳

      ✸✳✸ ❖ ♣r♦♣❛❣❛❞♦r ❝♦♠♦ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

      ❆ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❞✐❢❡r❡ ❞❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛✱ ❞❡♥tr❡ ♦✉tr♦s ❛s♣❡❝t♦s ❢ís✐❝♦s✱ ♣❡❧♦ ❢❛t♦ ❞❡ s❡ tr❛❜❛❧❤❛r ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ♦❜✈✐❛♠❡♥t❡ q✉❡ ✐ss♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦s s❡✉s ♣♦st✉❧❛❞♦s ❡ ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥❝❡rt❡③❛ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ❡ ♣♦rt❛♥t♦ ♥ã♦ ♣♦❞❡♠♦s ♠❡❞✐r ❝❡rt❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s q✉❡ só ❡①✐st❡♠ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ❝♦♠♦ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛✱ q✉❡ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❛❜str❛t❛ ❞♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛✳ ❊❧❛ s♦♠❡♥t❡ t❡♠ s✐❣♥✐✜❝❛❞♦ ♥♦ ❝♦♥t❡①t♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✳ ❆ ♣❛rt✐r ❞❛ ✐♥t❡r♣r❡t❛çã♦ ❡st❛tíst✐❝❛ ❞❡ ▼❛① ❇♦r♥ q✉❡ q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s ❝♦♠♦ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♣❛ss❛r❛♠ ❛ ❢❛③❡r s❡♥t✐❞♦ ♥♦ ♥♦ss♦ ♠✉♥❞♦ r❡❛❧✳

      ◆❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❡✱ ✉t✐❧✐③❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ ❉✐r❛❝❬✶✼❪✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ✉♠ ❜r❛ < x | ✜①♦

      ; t > ❝♦♠ ✉♠ ❦❡t |α, t q✉❡ ❡✈♦❧✉✐ ❝♦♠ ♦ t❡♠♣♦✱ t❛❧ q✉❡✿ ′ ′

      Ψ (x , t) =< x ; t > α |α, t

      P♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐✲❧❛ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ >

      ❞❡ ✉♠ ❜r❛ ❞❡ ♣♦s✐çã♦ < x , t| ♠♦✈❡♥❞♦ ♥♦ t❡♠♣♦ ❡♠ s❡♥t✐❞♦ ♦♣♦st♦ ❝♦♠ ✉♠ ❦❡t |α, t ✜①♦ ♥♦ t❡♠♣♦✱ t❛❧ q✉❡✿ ′ ′

      Ψ (x , t) =< x > α , t|α, t

      ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ♣r♦♣❛❣❛❞♦r ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ✐❞❡✐❛ ❞♦s ♣r♦❞✉t♦s ✐♥t❡r♥♦s ❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♠♣❧❡t❡③❛ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✱ ❞❡ ❢♦r♠❛ q✉❡✿ ′′ ′ ′′ ′ ′ ′ X " # iE ) a (t − t

      K(x , t; x , t ) = < x >< a > exp ′′ ′ ′′ ′ ′ ′ X ′ ′ a ′ ¯h |a |x − iE t iE t a a K(x , t; x , t ) = < x >< a >

      | exp − |a | exp |x a ¯h ¯h ′′ ′ ′′ ′ K(x , t; x , t ) =< x , t > ′ ′′

      , t|x ✭✸✳✺✶✮ , t >

      ♦♥❞❡ |x ❡ < x , t| sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❛✉t♦❦❡t ❡ ♦ ❛✉t♦❜r❛ ❞♦ ♦♣❡r❛❞♦r ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣✳ ′ ′ >

      ◆❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛❞♦ ♣♦r < b , t|a r❡♣r❡s❡♥t❛ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ s✐st❡♠❛✱ ✐♥✐❝✐❛❧♠❡♥t❡ ♣r❡♣❛r❛❞♦ ♥♦ ❡st❛❞♦ ❦❡t ❞❡

      A ✉♠ ♦♣❡r❛❞♦r ˆ ❝♦♠ ❛✉t♦✈❛❧♦r a ♥✉♠ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t ✱ s❡r ❡♥❝♦♥tr❛❞♦ ♥✉♠ ✐♥st❛♥t❡

      B ♣♦st❡r✐♦r t ♥♦ ❡st❛❞♦ ❦❡t ❞♦ ♦♣❡r❛❞♦r ˆ ❝♦♠ ❛✉t♦✈❛❧♦r b ✱ q✉❡ s❡ ❞❡♥♦♠✐♥♦✉ ❞❡ ❛♠♣❧✐t✉❞❡ ′ ′

      > > tr❛♥s✐çã♦ ♣❛r❛ ✐r ❞♦ ❡st❛❞♦ ❦❡t |a ♣❛r❛ ♦ ❡st❛❞♦ ❦❡t |b ✳

      ❆❣♦r❛ ♣♦❞❡♠♦s ❜✉s❝❛r ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛ ❡①♣r❡ssã♦ q✉❡ s✉r❣❡ ♣❛r❛ ♦ ♣r♦♣❛❣❛❞♦r✱ ′′ ′ ′′ ′ K(x , t; x , t ) =< x , t >

      , t|x ♦♥❞❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ ♦ ❛♥t❡r✐♦r ❡stá ♥♦ ❢❛t♦ ❞♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t = 0

      ✱ t♦❞❛✈✐❛ ♥♦ ❝❛s♦ ❞♦ ♣r♦♣❛❣❛❞♦r ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦ ♥❛ ❞✐❢❡r❡♥ç❛ t − t ✳ P♦rt❛♥t♦✱ ′′ ′ , t >

      ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r < x ❝♦♠♦ s❡♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ ✐♥✐❝✐❛❧♠❡♥t❡ , t|x

      ♣r❡♣❛r❛❞❛ ♥♦ ✐♥st❛♥t❡ t ❝♦♠ ♦ ❛✉t♦✈❛❧♦r ❞❛ ♣♦s✐çã♦ x s❡r ❡♥❝♦♥tr❛❞❛ ♥✉♠ ✐♥st❛♥t❡ ′′ ′′ ′ , t; x , t )

      ♣♦st❡r✐♦r t ❝♦♠ ❛✉t♦✈❛❧♦r ❞❛ ♣♦s✐çã♦ x ✳ ❊♥tã♦✱ ❡♠ r❡s✉♠♦✱ ♦ ♣r♦♣❛❣❛❞♦r K(x é ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ✉♠ ❡st❛❞♦ ❧♦❝❛❧✐③❛❞♦ ♥✉♠ ♣♦♥t♦ ′′

      , t ) , t)

      ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ (x ♣❛r❛ ✉♠ ♦✉tr♦ ♣♦♥t♦ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ (x ✳ ❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ✐♥t❡r♣r❡t❛r ♦ ♣r♦♣❛❣❛❞♦r ♣❛r❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱

      ❝✉❥♦ ♦ ❡st❛❞♦ ♥ã♦ ❡✈♦❧✉✐ ❝♦♠ ♦ t❡♠♣♦ ❡ ❛ ❜❛s❡ ✭❛✉t♦ ❡st❛❞♦s ❞♦ ♦♣❡r❛❞♦r✮ ❡✈♦❧✉✐✱ ♦♥❞❡ , t >

      |x é ✉♠ ❛✉t♦❦❡t ❞❛ ♣♦s✐çã♦ ♥♦ ✐♥st❛♥t❡ t ❝♦♠ ♦ s❡✉ ❛✉t♦✈❛❧♦r x ✳ ❈♦♠♦ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ♥✉♠ ✐♥st❛♥t❡ q✉❛❧q✉❡r ♦s ❛✉t♦❦❡ts ❞❡ ♦♣❡r❛❞♦r ✭♦❜s❡r✈á✈❡❧✮ ❝♦♠♦ ❦❡ts ❞❡ ❜❛s❡✱ ❡♥tã♦ ′′ ′

      < x , t > , t|x

      ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ ❢❛③ ❛ ❝♦♥❡①ã♦ ❡♥tr❡ ♦s ❞♦✐s ❛✉t♦ ❡st❛❞♦s ✭❜❛s❡s✮ ❡♠ t❡♠♣♦s ❞✐st✐♥t♦s✳

      ❯s❛♥❞♦ ✉♠❛ ♥♦t❛çã♦ ♠❛✐s s✐♠étr✐❝❛ ♣❛r❛ ♦s ❛✉t♦❦❡ts ❞❡ ♣♦s✐çã♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ q✉❡ ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❡♠ q✉❛❧q✉❡r ✐♥st❛♥t❡ ❛r❜✐trár✐♦✳ P♦❞❡♠♦s ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❝♦♠♦✿ Z 3 ′′ ′′ ′′ ′′ ′′ d x , t >< x , t

      |x | = 1. ✭✸✳✺✷✮ ❆❣♦r❛✱ ♣♦❞❡♠♦s ✉s❛r ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ♣❛r❛ ❡s❝r❡✈❡r ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠ ′ ′′′ s✐st❡♠❛ q✉â♥t✐❝♦ ❞♦ ✐♥st❛♥t❡ t ♣❛r❛ ♦ ✐♥st❛♥t❡ t ✳ P❛r❛ ✐ss♦✱ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ✐♥t❡r✈❛❧♦ ′ ′′′ ′ ′′ ′′ ′′′

      (t , t ) , t ) , t ) ❡♠ ❞♦✐s ✐♥t❡r✈❛❧♦s ♠❡♥♦r❡s (t ❡ (t ✳ ❆ss✐♠✱ ′′′ ′′′ ′ ′ ′′ ′′′ ′′′ ′′ ′′ ′′ ′′ ′ ′ Z 3

      < x , t , t >= d x < x , t , t >< x , t , t >, ′′′ ′′ ′ |x |x |x ✭✸✳✺✸✮ > t > t

      ❝♦♠ t ✳ ❊st❛ é ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♠♣♦s✐çã♦ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ ❞❡ tr❛♥s✐çã♦❬✷✶❪✳

      P♦❞❡♠♦s ❞✐✈✐❞✐r ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❡♠ s✉❜✐♥t❡r✈❛❧♦s ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s ♥❛ q✉❛♥t✐✲

      ❞❛❞❡ q✉❡ ❞❡s❡❥❛r♠♦s✳ ▲♦❣♦✿ ′ ′ ′′′ ′′ ′′′′ ′′′′ ′′′ ′′′ ′′′ ′′′ ′′ ′′ ′′ ′′ ′ ′ Z Z 3 3 , t >= d x d x < x , t , t >< x , t , t >< x , t , t >

      < x, t|x |x |x |x ′′′′ ′′′ ′′ ′ ✭✸✳✺✹✮

      > t > t > t ❝♦♠ t ✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ′′ ′′ ′ ′

      , t , t > ❊ s❡ ❞❡ ❛❧❣✉♠ ♠♦❞♦ s♦✉❜éss❡♠♦s ❛ ❢♦r♠❛ ❞❡ < x ♣❛r❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ′ ′ |x ′′ ′′ ′ ′

    • dt , t , t > t❡♠♣♦ ✐♥✜♥✐t❡s✐♠❛❧ t ❡ t ✱ ♣♦❞❡rí❛♠♦s ♦❜t❡r ❛ ❛♠♣❧✐t✉❞❡ < x |x ♣❛r❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ✜♥✐t♦ ❝♦♠♣♦♥❞♦ ❛♠♣❧✐t✉❞❡s ❞❡ tr❛♥s✐çã♦ ❛♣r♦♣r✐❛❞❛s ♣❛r❛ ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ ❡q✉❛çã♦ ✭✸✳✺✹✮✳ ❊ ❢♦✐ ❡①❛t❛♠❡♥t❡ ❡ss❡ ♦ r❛❝✐♦❝í♥✐♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❘✐❝❤❛r❞ ❋❡②♥♠❛♥ q✉❡ ♦ ❧❡✈♦✉ ❛ ✉♠❛ ♥♦✈❛ ❢♦r♠✉❧❛çã♦ ♣❛r❛ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❞❡♥♦♠✐♥❛❞❛ ❞❡ ■♥t❡❣r❛✐s ❞❡ ❈❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥✳

      ✹ ■♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ ✹✳✶ ❆ ♦r✐❣❡♠ ❞❛ ■♥t❡❣r❛❧ ❞❡ ❈❛♠✐♥❤♦

      ❆ ♣r♦♣♦st❛ ❞❡ ❋❡②♥♠❛♥ s✉r❣✐✉ ❛ ♣❛rt✐r ❞❡ ✉♠ ❛rt✐❣♦ ❞❡ P❛✉❧ ❉✐r❛❝ ❞❡ ✶✾✸✸✱ ❆ ▲❛❣r❛♥❣✐❛♥❛ ♥❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛❬✶✶❪✱ ♥♦ q✉❛❧ ❉✐r❛❝ ♠♦str❛ q✉❡ ♦ ❛♥á❧♦❣♦ q✉â♥t✐❝♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ▼í♥✐♠❛ ❆çã♦❬✶✵❪ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❞✐♥â♠✐❝❛ q✉❡ ❡✈♦❧✉✐ ❡♥tr❡ ♦s t❡♠♣♦s T ❡ t é ❞❛❞♦ ♣♦r✿ ZZ Z

      (q t T ) = . . . (q t m )dq m (q m m−1 )dq m−1 . . . (q 2 1 )dq 1 (q 1 T ) ✭✹✳✶✮

      |q |q |q |q |q t T N N ) , t , t > 1 1 ♦♥❞❡✱ ♥❛ ♥♦t❛çã♦ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✱ (q |q ❡q✉✐✈❛❧❡ ❛ < x |x ♣❛r❛ ♦s N 1 t❡♠♣♦s t = t ❡ T = t ✳

      ❙❡❣✉♥❞❛ ♦ ❛rt✐❣♦ ❞❡ ❉✐r❛❝✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ✉♠❛ s❡qû❡♥❝✐❛ ❞❡ t❡♠♣♦s ✐♥t❡r♠❡❞✐ár✐♦s ❡♥tr❡ T ❡ t✱ t❛❧ q✉❡✿

      T < t , t . . . t , t < t 1 2 m−1 m j j−1 = δt ❞❛ q✉❛❧ ❢❛③❡♥❞♦✲s❡ t − t ✱ ✐st♦ r❡s✉❧t❛ ❡♠ 1 q t+δt t

      − q (q t+δt t L , q t+δt δt

      |q ) ∼ exp ¯h δt

      ❉✐r❛❝ ♠♦str❛ q✉❡ L é✱ ❞❡ ❢❛t♦✱ ❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛✱ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝♦rr❡s♣♦♥✲ ❞ê♥❝✐❛ ❞❡ ❇♦❤r✱ ♣♦✐s q✉❛♥❞♦ ¯h → 0✱ Z Z Z Z Z t t t m 2 t 1 t t m t m−1 t Ldt + Ldt + . . . + Ldt + Ldt = Ldt = S, 1 T T

      ♦♥❞❡ S é ❛ ❛çã♦ ❝❧áss✐❝❛ ❞♦ s✐st❡♠❛ ❝❧áss✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡❬✶✵❪✳

      ❉✐❛♥t❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ❝♦♥t✐❞❛s ♥♦ tr❛❜❛❧❤♦ ❞❡ ❉✐r❛❝✱ ❋❡②♥♠❛♥ ♣❛ss❛ ❛ ❛♥❛❧✐s❛r ♦ ❡♠♣r❡❣♦ ❞❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛ ♥❛s s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ❡♠ s✐st❡♠❛s q✉â♥t✐❝♦s ❡ ❝❤❡❣❛ ❛ ❝♦♥❝❧✉sã♦ q✉❡ ♥ã♦ s♦♠❡♥t❡ 1 q t+δt t

      − q (q L , q δt , t+δt t t+δt

      |q ) ∼ exp ¯h δt

      ♠❛s ♥❛ ✈❡r❞❛❞❡✱ ❢r❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s q✉❡ sã♦ ✐❣✉❛✐s✱ t❛❧ q✉❡✿ 1 q t+δt t − q

      (q ) = exp L , q δt t+δt t t+δt |q

      ¯h δt ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❡ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛

      ❞❡✈❡✲s❡ ❛ ❛ss♦❝✐❛çã♦ ❞♦s ❝❛♠✐♥❤♦s ❡♥tr❡ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❡ ✉♠ ♣♦♥t♦ ✜♥❛❧ ❞❛ tr❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✳ ❊♥q✉❛♥t♦ ♥❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❡ss❡ ❝❛♠✐♥❤♦ é ú♥✐❝♦✱ ❞❡✈✐❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦✱ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s sã♦ ❝♦♥s✐❞❡r❛❞♦s✳ ❊ ♥♦ ❧✐♠✐t❡ ¯h → 0 ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ r❡✈❡rt❡✲s❡ ♥❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛✳ ❙❡❣✉♥❞♦ ❙t❡♣❤❡♥ ❍❛✇❦✐♥❣❬✷✹❪✱ ✧❋❡②♥♠❛♥ ❞❡s❛✜♦✉ ♦ ♣r❡ss✉♣♦st♦ ❝❧áss✐❝♦ ❜ás✐❝♦ ❞❡ q✉❡ ❛ ♣❛rtí❝✉❧❛ ♣♦ss✉✐ ✉♠❛ ❤✐st♦r✐❛ ♣❛rt✐❝✉❧❛r✳ ❊♠ ✈❡③ ❞✐ss♦✱ ❡❧❡ ♣r♦♣ôs q✉❡ ❛s ♣❛rtí❝✉❧❛s s❡ ❞❡s❧♦❝❛ss❡♠ ❞❡ ✉♠ ❧♦❝❛❧ ♣❛r❛ ♦ ♦✉tr♦ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞❛s ❛s tr❛❥❡tór✐❛s ♣♦ssí✈❡✐s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭✳✳✳✮ ❈♦♥t✉❞♦✱ ♥♦ ❞✐❛✲❛✲❞✐❛✱ ♣❛r❡❝❡✲♥♦s q✉❡ ♦s ♦❜❥❡t♦s s❡❣✉❡♠ ✉♠❛ ú♥✐❝❛ tr❛❥❡tór✐❛ ❡♥tr❡ s✉❛ ♦r✐❣❡♠ ❡ s❡✉ ❞❡st✐♥♦ ✜♥❛❧✳ ■ss♦ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ✐❞é✐❛ ❞❛s ❤✐st♦r✐❛s ♠ú❧t✐♣❧❛s ❞❡ ❋❡②♥♠❛♥✱ ♣♦rq✉❡ ♣❛r❛ ♦❜❥❡t♦s ❣r❛♥❞❡s✱ s✉❛ r❡❣r❛ ❞❡ ❛tr✐❜✉✐r ♥ú♠❡r♦s ❛ ❝❛❞❛ tr❛❥❡tór✐❛✱ ❛ss❡❣✉r❛ q✉❡ t♦❞❛s ❛s tr❛❥❡tór✐❛s✱ ❡①❝❡t♦ ✉♠❛✱ ❛♥✉❧❛♠✲s❡ q✉❛♥❞♦ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s s❡ ❝♦♠❜✐♥❛♠✧✳

      ❆❞♦t❛r❡♠♦s ✉♠ tr❛t❛♠❡♥t♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♣❛r❛ ❛ ✐♥t❡❣r❛çã♦ ❢✉♥❝✐♦♥❛❧ ❡ s✉❜st✐t✉✐✲ ′′′′′ ′′′′ N vezes ... N r❡♠♦s ❛ ♥♦t❛çã♦ ✉t✐❧✐③❛❞❛ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r x ♣♦r x ✳ ❆❣♦r❛✱ ❝♦♠ ❡st❛ ♥♦t❛çã♦ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❛♠♣❧✐t✉❞❡ ❞❡ tr❛♥s✐çã♦ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞❛❞♦ 1 , t 1 ) N , t N ) ♣♦r (x ❛♦ ♣♦♥t♦ ✜♥❛❧ ❞❛❞♦ ♣♦r (x ✱ ❡♠ q✉❡ ❞✐✈✐❞✐r❡♠♦s ♦ ✐♥t❡r✈❛❧♦ t♦t❛❧ ❡♥tr❡ t 1 N

      ❡ t ❡♠ N − 1 ♣❛rt❡s ✐❣✉❛✐s ✭✜❣✳ ✷✮✱ t❛❧ q✉❡✿ t N 1 − t t j j−1 = δt =

      − t N − 1

      ♦♥❞❡✱ ❡①♣❧♦r❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦♠♣♦s✐çã♦✱ t❡♠♦s✿ Z Z Z < x , t , t > = dx dx . . . dx < x , t , t > N N 1 1 N −1 N −2 2 N N N −1 N −1

      |x |x

      < x , t , t > . . . < x , t , t > N −1 N −1 N −2 N −2 2 2 1 1 |x |x ✭✹✳✷✮ N N , t ; x , t ) 1 1 q✉❡ é ✐❣✉❛❧ ❛♦ ♣r♦♣❛❣❛❞♦r K(x ✳

      ❋✐❣✉r❛ ✷✿ ❆♠♣❧✐t✉❞❡s ❞❡ tr❛♥s✐çã♦ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ♣♦♥t♦ ✐♥✐❝✐❛❧ q = q ✱ ♥✉♠ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t = 0✱ ❛ ✉♠ ♣♦♥t♦ ✜♥❛❧ q = q N −1 ✱ ♥✉♠ ✐♥st❛♥t❡ ✜♥❛❧ t = T ✱ ❝♦♠ ✐♥t❡r✈❛❧♦s i ❞❡ t❡♠♣♦ ✐♥✜♥✐t❡s✐♠❛❧ δt✱ ♦♥❞❡ ✉t✐❧✐③❛♠♦s ❝♦♦r❞❡♥❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s q ❡♠ s✉❜st✐t✉✐çã♦ i ❛s ❝♦♦r❞❡♥❛❞❛s x ✳

      P♦st✉❧❛❞♦s ❞❡ ❋❡②♥♠❛♥ N N , t , t > 1 1 ▲❡♠❜r❛♥❞♦ q✉❡ < x |x é ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ✐r 1 1 N N

      ❞❡ x ♥✉♠ ✐♥st❛♥t❡ t ❛ x ♥✉♠ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦ t ✱ t❡♠♦s ❡♥tã♦ q✉❡✿ √

      < x , t , t > N N 1 1 |x é ❛ s♦♠❛ ❞❡ ✐♥✜♥✐t❛s ❛♠♣❧✐t✉❞❡s ♣❛r❝✐❛✐s✱ s❡♥❞♦ ✉♠❛ ♣❛r❛ ❝❛❞❛ 1 , t ) , t ) 1 N N

      ✉♠ ❞♦s ❝❛♠✐♥❤♦s q✉❡ s❛✐ ❞❡ (x ❡ ❝❤❡❣❛ ❛ (x ❀ N N , t , t > 1 1 Γ √ ❆ ❛♠♣❧✐t✉❞❡ ♣❛r❝✐❛❧ < x |x ❛ss♦❝✐❛❞❛ ❛ ✉♠ ❞❡ss❡s ❝❛♠✐♥❤♦s Γ é ❞❡t❡r♠✐✲ Γ ♥❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡❥❛ S ❛ ❛çã♦ ❝❧áss✐❝❛ ❝❛❧❝✉❧❛❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❝❛♠✐♥❤♦ Γ✱ t❛❧ q✉❡ Z

      S = dtL (x, ˙x) Γ cl (Γ) iS Γ cl N N , t , t > 1 1 Γ ♦♥❞❡ L é ❛ ❧❛❣r❛♥❣✐❛♥❛ ❝❧áss✐❝❛✳ ▲♦❣♦✱ s❡ < x |x ∼ exp ❡♥tã♦✱ X X ¯ h iS Γ

      < x , t , t > = < x , t , t > exp N N 1 1 N N 1 1 Γ |x |x ∼ Γ Γ ¯h

      ✹✳✷ ❆ r❡❧❛çã♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

      ❊♠ s✉❛ t❡s❡❬✶✵❪✱ ❋❡②♥♠❛♥ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❡①♣r❡ssã♦ ❣❡♥ér✐❝❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❛ ♣❛rt✐r ❞❛ r❡❧❛çã♦ ♣r♦♣♦st❛ ♣♦r ❉✐r❛❝ q✉❡ ❡♥✈♦❧✈❡ ❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛ ❞❡ ✉♠ s✐st❡♠❛ ❝❧áss✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ s✐st❡♠❛ q✉â♥t✐❝♦✳ P❛r❛ t❛♥t♦✱ ✐♥s❡r✐♥❞♦ q g(q)dq

      ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♥♦r♠❛❧✐③❛çã♦✱ A✱ ♥♦ ❧✐♠✐t❡ ❡♠ q✉❡ δt s❡ ❛♣r♦①✐♠❛ ❞❡ ③❡r♦✱ ❝♦♠ s❡♥❞♦ ♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ ♥♦ ❡s♣❛ç♦✲q✱ t❡♠♦s q✉❡✿ ′ ′ ′ ′ ′ Z q Ψ(q , t + δt) = (q )Ψ(q , t) g(q )dq t+δt t+δt |q t t t t Z √ iδt t+δt− ′ q q ′ ′ t ¯ h δt t+δt L( ,q ) gdq

      = e Ψ(q , t) t ✭✹✳✸✮ A(δt)

      ♣♦rt❛♥t♦✱ ❡ss❛ ❡①♣r❡ssã♦ s❡r✈❡ ♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❝♦♠♦ ❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö✲ ❞✐♥❣❡r✱ ❡ ❞❡ ❢❛t♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡ss❛ ❡q✉❛çã♦ s❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ♥♦r♠❛❧✐③❛çã♦ A(δt)✱ ✉♠❛ ❢✉♥çã♦ ❞❡ δt✱ ❢♦r ❡s❝♦❧❤✐❞❛ ❛❞❡q✉❛❞❛♠❡♥t❡✳ P❛r❛ ✈❡r ❝♦♠♦ ✐ss♦ ❛❝♦♥t❡❝❡✱ t♦♠❛♠♦s ♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♠❛ss❛ m ♠♦✈❡♥❞♦✲s❡ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦ ❡♠ ✉♠ ❝❛♠♣♦ ❞❡ 1 2 m ˙x

      ❢♦rç❛ ❞❛❞♦ ♣❡❧♦ ♣♦t❡♥❝✐❛❧ V (x)✳ ❆ss✐♠✱ ❛ ❢✉♥çã♦ ❧❛❣r❛♥❣✐❛♥❛ ❝❧áss✐❝❛ é L = 2 − V (x)✱ ♦♥❞❡ s✉❜st✐t✉✐r❡♠♦s δt → ε✱ ♣❛r❛ ε ✐♥✜♥✐t❡s✐♥❛❧✱ t❛❧ q✉❡ ❛ ❡q✉❛çã♦ s❡rá ❞❛❞❛ ♣♦r✿ Z iε m x−2 2 ¯ h ε { −V (x)} 2 ( ) dy

      Ψ(x, t + ε) = e Ψ(y, t) ✭✹✳✹✮

      A ❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ y = η + x ♥❛ ✐♥t❡❣r❛❧✱ t❡♠♦s✿ Z i mη 2 ¯ h { −εV (x)} 2 ε

      Ψ(x, t + ε) = e Ψ(x + η, t) ✭✹✳✺✮

      A ❙♦♠❡♥t❡ ♦s ✈❛❧♦r❡s ❞❡ η ♣ró①✐♠♦ ❞❡ ③❡r♦ ✐rã♦ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ✐♥t❡❣r❛❧✱ ✉♠❛ ✈❡③ q✉❡✱ ♣❛r❛ ε

      ♣❡q✉❡♥♦✱ ♦✉tr♦s ✈❛❧♦r❡s ❞❡ η ❢❛③❡♠ ❛ ❡①♣♦①❡♥❝✐❛❧ ♦s❝✐❧❛r tã♦ r❛♣✐❞❛♠❡♥t❡ q✉❡ s✉r❣✐r✐❛♠ ♣❡q✉❡♥❛s ❝♦♥tr✐❜✉✐çõ❡s ♣❛r❛ ❛ ✐♥t❡❣r❛❧✳ ❙♦♠♦s✱ ♣♦rt❛♥t♦✱ ❧❡✈❛❞♦s ❛ ❡①♣❛♥❞✐r Ψ(x + η, t) ♥✉♠❛ sér✐❡ ❞❡ ❚❛②❧♦r ❡♠ t♦r♥♦ ❞❡ η = 0✱ ♦❜t❡♥❞♦✱ ❞❡♣♦✐s ❞❡ r❡♦r❣❛♥✐③❛r ❛ ✐♥t❡❣r❛❧✿ − V (x) Z iε " h ¯ i m 2 2 2 # e ∂Ψ(x, t) η ∂ Ψ(x, t) η Ψ(x, t + ε) = e Ψ(x, t) + η + . . . dη. ¯ h 2 ε 2 ✭✹✳✻✮

    • R ¯ h ε i m
    • 2 η 2π¯ hεi 2 A ∂x q 2 ∂x e dη = ❆❣♦r❛✱ −∞ ✭✈❡r P✐❡r❝❡s ✐♥t❡❣r❛❧ t❛❜❧❡s ✹✽✼✮✱ ❡ ❢❛③❡♥❞♦ ❛ ❞✐❢❡r❡♥❝✐❛çã♦ m ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ❝♦♠ r❡s♣❡✐t♦ ❛ m✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡✿ Z i m 2 s 2 η h ¯ 2 ε 2π¯hεi ¯hεi η e dη = . −∞

        ✭✹✳✼✮ m m ❡♥tr❡t❛♥t♦✱ ❛ ✐♥t❡❣r❛❧ ❝♦♠ η ♥♦ ✐♥t❡❣r❛♥❞♦ é ③❡r♦✱ ✉♠❛ ✈❡③ q✉❡ é ❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ í♠♣❛r✳ P♦rt❛♥t♦✱ q 2π¯ hεi ( ) 2 m − V (x) ¯ h ¯hεi ∂ Ψ 2

        Ψ(x, t + ε) = .e Ψ(x, t) + ) 2 ❖(ε ✭✹✳✽✮

      • A m ∂x

        ❛♥❛❧✐s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✹✳✽✮ ✈❡r✐✜❝❛✲s❡ q✉❡ q✉❛♥❞♦ ε → 0 ❞❡✈❡♠♦s t❡r Ψ(x, t+ε) → Ψ(x, t)✱ q 2π¯ hεi ♠❛s ♣❛r❛ q✉❡ ✐ss♦ s❡❥❛ ✈❡r❞❛❞❡ ❞❡✈❡♠♦s t❡r A(ε) = ✳ ■ss♦ ❞❡✈❡✲s❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛ m ♠❡❞✐❞❛ ❞❡ ✐♥t❡❣r❛çã♦ ♥♦ ❡s♣❛ç♦ ❝♦♠♣❧❡①♦ ❞❛s tr❛❥❡tór✐❛s ♣♦ssí✈❡✐s ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ♥ã♦ é ♣♦s✐t✐✈♦✲❞❡✜♥✐❞❛✳

        ❊①♣❛♥❞✐♥❞♦✲s❡ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ ✭✹✳✽✮ ❡♠ t❡r♠♦s ❞❡ ♣♦tê♥❝✐❛ ❞❡ ε ❡♠ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❡♥❝♦♥tr❛♠♦s✿ 2

        ∂Ψ(x, t) iε ¯hiε ∂ Ψ(x, t) Ψ(x, t) + ε V (x)Ψ(x, t) + ,

        ✭✹✳✾✮ = Ψ(x, t) − 2

        ❡ ♣♦rt❛♥t♦✱ 2 2 ¯h ∂Ψ ¯h ∂ Ψ

      • V (x)Ψ − = −
      • 2 ✭✹✳✶✵✮ i ∂t 2m ∂x q✉❡ é ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♣❛r❛ ♦ s✐st❡♠❛ ❡♠ q✉❡stã♦✳

          ✹✳✸ ❆♥❛❧♦❣✐❛ ❝♦♠ ❛ ■♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥

          ❙❛❜❡♠♦s q✉❡ ♥❛ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ♦ q✉❡ ✐♠♣♦rt❛ é ♦ ♣❛rt✐❝✉❧❛r ✈❛❧♦r ❞❛ tr❛❥❡tór✐❛ ♥✉♠ ❡①tr❡♠♦ ✭♠í♥✐♠♦✮ q✉❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❛çã♦ ❝❧áss✐❝❛✱ ❛❣♦r❛ ❛ ✐❞❡✐❛ é s♦♠❛r t♦❞❛s ❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ❝❛❞❛ tr❛❥❡tór✐❛ q✉❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ t♦t❛❧ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ✉♠ ♣♦♥t♦ x ❛♦ ♣♦♥t♦ x N ✳ ❈❛❞❛ tr❛❥❡tór✐❛ ♣❛rt✐❝✉❧❛r ❛♣r❡s❡♥t❛ ✉♠❛ ❢❛s❡ ❞❡ ❝♦♥tr✐❜✉✐çã♦ q✉❡ é ❛ ❛çã♦ S ♣ró♣r✐❛ ❛ ❡ss❛ tr❛❥❡tór✐❛✱ ❞❛❞❛ ❡♠ ✉♥✐❞❛❞❡s ❞❡ q✉❛♥t✉♠ ❞❡ ❛çã♦ ¯h✳ X K(b, a) = φ[x(t)] traj. a→b ✭✹✳✶✶✮

          ❆ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ q✉❡ ♣♦ss✉✐ ✉♠❛ ❢❛s❡ ♣r♦♣♦r❝✐♦♥❛❧ à ❛çã♦ S é ❞❛❞❛ ♣♦r φ[x(t)] = A exp {iS[x(t)]/¯h}✱ ♦♥❞❡ ❡st❛ ❛çã♦ é ❛ ❞♦ s✐st❡♠❛ ❝❧áss✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❆q✉✐ ✈❡♠♦s q✉❡ s✉r❣❡ ✉♠❛ ❝♦♥st❛♥t❡ A q✉❡ s❡rá ❝❛❧❝✉❧❛❞❛ ❞❡ ❢♦r♠❛ q✉❡ s❡ ♣♦ss❛ ♥♦r♠❛❧✐③❛r K(b, a)

          ✳ ❊♠ s❡✉ ❧✐✈r♦ s♦❜r❡ ■♥t❡❣r❛✐s ❞❡ ❈❛♠✐♥❤♦❬✶✺❪✱ ❋❡②♥♠❛♥ ♣❛rt❡ ❞❛ ❞❡✜♥✐çã♦ ❞❛ ■♥t❡❣r❛❧

          ❞❡ ❘✐❡♠❛♥♥✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛çã♦ ❛❜❛✐①♦ ✭✜❣✳ ✷✮✱ ♦♥❞❡ ✉♠❛ ár❡❛ A s♦❜ ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ f(x) é✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ❛ s♦♠❛ ❞❡ t♦❞❛s ❛s ♦r❞❡♥❛❞❛s ❞❛ ❢✉♥çã♦ t❛❧ q✉❡ X f (x ), i

          A ∼ i ❋✐❣✉r❛ ✸✿ ■♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥ s♦❜ ✉♠❛ ❝✉r✈❛ f(x) ❡ s♦❜r❡ ♦ ❡✐①♦ ❞❛s ❛❜❝✐ss❛s x✱ ❝♦♠ i i+1 ❡s♣❛ç❛♠❡♥t♦ h ❡♥tr❡ x ❡ x ✳ i+1 i ) = h

          q✉❡ (x ✳ ❆❣♦r❛ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ h t❡♥❞❡ ❛ 0 ❡ ❝❛❧❝✉❧❛♥❞♦ A ♣❛r❛ − x

          ❡ss❡ ♣❛ss♦ h ❝♦♥st❛♥t❡✱ t❡♠♦s q✉❡✿ X ! A = lim h f (x ) . i h→0 i ✭✹✳✶✷✮ ❆❣♦r❛✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛ ✐♥t❡❣r❛❧ ❞❡ ❘✐❡♠❛♥♥ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ s♦♠❛ s♦❜r❡ t♦❞❛s

          ❛s tr❛❥❡tór✐❛s ♣♦ssí✈❡✐s ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ♣♦♥t♦ x ♥✉♠ ✐♥st❛♥t❡ t ❛♦ ♣♦♥t♦ ❢✉✲ N N t✉r♦ x ♥✉♠ ✐♥st❛♥t❡ ❢✉t✉r♦ t ✳ ■♥✐❝✐❛❧♠❡♥t❡ t♦♠❛♠♦s ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ tr❛❥❡tór✐❛s ❡ ❞✐✈✐❞✐♠♦s ♦ t❡♠♣♦ ❡♠ s✉❜✐♥t❡r✈❛❧♦s ✐❣✉❛❧♠❡♥t❡ ❡s♣❛ç❛❞♦s✱ t❛❧ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞♦✐s i ✐♥st❛♥t❡s s✉❜s❡q✉❡♥t❡s s❡rá ε✳ ❆ ❝❛❞❛ t❡♠♣♦ t s❡❧❡❝✐♦♥❛♠♦s ❡s♣❡❝✐❛❧♠❡♥t❡ ✉♠ ♣♦♥t♦ x i

          ✱ ❡ ❡♠ s❡❣✉✐❞❛ ❝♦♥str✉í♠♦s ✉♠❛ tr❛❥❡tór✐❛✱ ✐♥t❡r❧✐❣❛♥❞♦ t♦❞♦s ❡ss❡s ♣♦♥t♦s s❡❧❡❝✐♦♥❛❞♦s ❛tr❛✈és ❞❡ s❡❣♠❡♥t♦s ❞❡ ❧✐♥❤❛s r❡t❛s✳ ➱ ♣♦ssí✈❡❧ ❡♥tã♦ ❞❡✜♥✐r ✉♠❛ s♦♠❛ s♦❜r❡ t♦❞❛s ❛s i tr❛❥❡tór✐❛s ❛ss✐♠ ❝♦♥str✉í❞❛s✱ t♦♠❛♥❞♦ ✉♠❛ ✐♥t❡❣r❛❧ ♠ú❧t✐♣❧❛ s♦❜r❡ t♦❞♦s ♦s ✈❛❧♦r❡s x ✱ ❝♦♠ i ✈❛r✐❛♥❞♦ ❡♥tr❡ 1 ❡ N − 1✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ N.ε = t , ε = t , t = t , t = t , x = x , x = x . b a i+1 1 a N b a N b

          − t − t ❆ ❡q✉❛çã♦ s❡rá ZZ Z . . . φ[x(t)]dx dx . . . dx . 1 2 N −1 N K(b, a) ∼

          ✭✹✳✶✸✮ ❖s ❡①tr❡♠♦s x ❡ x ♥ã♦ ❢❛③❡♠ ♣❛rt❡ ❞❛ s♦♠❛✱ ♣♦r s❡ tr❛t❛r❡♠ ❞❛s ❡①tr❡♠✐❞❛❞❡s ✜①❛s x a b

          ❡ x ✳ 1 2 m ˙x ❆❣♦r❛ ❛♥❛❧✐s❛♥❞♦ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❧❛❣r❛♥❣✐❛♥❛ é ❞❛❞❛ ♣♦r L = − V (x)✳ 2 −N

          ❈♦♠♦ ❥á ✈✐♠♦s ♥❛ s❡çã♦ ✹✳✷ ♦ ❢❛t♦r ❞❡ ♥♦r♠❛❧✐③❛çã♦ ♣❛r❛ ❡st❡ ❝❛s♦ s❡rá A ✱ ♦♥❞❡ 1/2 A = (2πi¯hε/m)

          ✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ ZZ Z 1 dx dx dx 1 2 N −1 K(b, a) = lim . . . exp [(iS [b, a]) /¯h] . . . , ε→0

          ✭✹✳✶✹✮ R t b A A A A L( ˙x, x, t)dt i

          ♦♥❞❡ ❛ ❛çã♦ S[b, a] = t ❛♦ ❧♦♥❣♦ ❞♦ ❝❛♠✐♥❤♦ ❞❛❞♦ ♣❡❧♦s ♣♦♥t♦s x é ❛ ✐♥t❡❣r❛❧ a ❞❡ ❧✐♥❤❛ ❡①❡❝✉t❛❞❛ s♦❜r❡ ❛ tr❛❥❡tór✐❛ ♣♦❧✐❣♦♥❛❧ ❝♦♠♣♦st❛ ♣❡❧♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ q✉❡ i ❝♦♥❡❝t❛♠ t♦❞❛s ♦s ♣♦♥t♦s x ✳ ❆ ✜❣✉r❛ ✸ ✐❧✉str❛ ♦ ❧✐♠✐t❡ ❞❛❞♦ ♥❛ ❡q✉❛çã♦ ✭✹✳✶✹✮✳

          ❚❛♠❜é♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ s♦♠❛ s♦❜r❡ t♦❞❛s ❛s tr❛❥❡tór✐❛s✱ ❛❣♦r❛ ✉s❛♥❞♦ ✉♠❛ ♥♦t❛çã♦ ♠❛✐s ❝♦♠♣❛❝t❛✱ t❛❧ q✉❡✱ Z b i

          K(b, a) = exp S[b, a] a ❉[x(t)], ✭✹✳✶✺✮ ¯h q✉❡ s❡rá ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❦❡r♥❡❧✭♥ú❝❧❡♦✮ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ❝❛♠✐♥❤♦✱ s✐♠♣❧✐✜❝❛❞❛ ♣❡❧❛ ♥♦t❛çã♦

          ❉[x(t)] q✉❡ r❡♣r❡s❡♥t❛ ❛ ♠❡❞✐❞❛ ❞❡ ✐♥t❡❣r❛çã♦ ❛❞❡q✉❛❞❛✱ t❛❧ q✉❡✱ " # N −1 Y

          1 dx k ❉[x(t)] = lim ε→0 N

          A k=1 ❆ ❡q✉❛çã♦ ✭✹✳✶✺✮ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ■♥t❡❣r❛❧ ❞❡ ❈❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥❬✶✺❪✳ ❋✐❣✉r❛ ✹✿ ❆ s♦♠❛ s♦❜r❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ❧✐♠✐t❡✱ ❡♠ q✉❡ ❛ ♣r✐♥❝í♣✐♦ ♦ ❝❛♠✐♥❤♦ é ❡s♣❡❝✐✜❝❛❞♦✱ ❞❛❞♦ ❛♣❡♥❛s ♣❡❧❛ s✉❛ ❝♦♦r❞❡♥❛❞❛ x ♥✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❞❡ t❡♠♣♦s ❡s♣❡❝✐✜❝❛❞♦s s❡♣❛r❛❞♦s ♣♦r ✐♥t❡r✈❛❧♦s ♠✉✐t♦ ♣❡q✉❡♥♦s ε✳ ❆ s♦♠❛ ❞♦s ❝❛♠✐♥❤♦s é✱ ❡♥tã♦✱ ✉♠❛ ✐♥t❡❣r❛❧ s♦❜r❡ t♦❞❛s ❡ss❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❡❝í✜❝❛s✳

          ✹✳✹ ❆s r❡❣r❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s c

          P❛r❛ ❞♦✐s ❡✈❡♥t♦s ❡♠ s✉❝❡ssã♦✱ ❛♥❛❧✐s❛r❡♠♦s ♦ ❝❛s♦ ❞❡ ✉♠ ♣♦♥t♦ x ✱ ❞❛❞♦ ♥✉♠ ✐♥st❛♥t❡ t c a b ✱ ❧♦❝❛❧✐③❛❞♦ ❡♥tr❡ ♦s ♣♦♥t♦s x ❡ x ✱ ❡♠ q✉❡ ❛ ❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ tr❛❥❡tór✐❛ s❡rá a

          ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛s ❛çõ❡s ❞♦s ❞♦✐s ❡✈❡♥t♦s ♦❝♦rr✐❞♦s ❡♥tr❡ ♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t ❡ ♦ b ✐♥st❛♥t❡ ✜♥❛❧ t ✱ t❛❧ q✉❡✿

          S[b, a] = S[b, c] + S[c, a] ✭✹✳✶✻✮

          ❆❣♦r❛ s✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✭✹✳✶✻✮ ♥❛ ❡q✉❛çã♦ ✭✹✳✶✺✮✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ♣r♦♣❛❣❛✲ ❞♦r ✭❦❡r♥❡❧✮ ❝♦♠♦✿ Z i i

          K(b, a) = exp S[b, c] + S[c, a] ❉x(t). ✭✹✳✶✼✮

          ¯h ¯h ♣♦rt❛♥t♦✱ ❛ tr❛❥❡tór✐❛ ✐♥✐❝✐❛❧ ♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ♣❛rt❡s✱ s❡♥❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ a c t❡rá ❝♦♠♦ ❡①tr❡♠✐❞❛❞❡s ♦s ♣♦♥t♦s x ❡ x ✱ ❡♥q✉❛♥t♦ ❛ s❡❣✉♥❞❛ t❡rá ❝♦♠♦ ❡①tr❡♠✐❞❛❞❡s c b

          ♦s ♣♦♥t♦s x ❡ x ✳ P❛r❛ ♦❜t❡r♠♦s ❛ ❛♠♣❧✐t✉❞❡ t♦t❛❧ ❞❡✈❡♠♦s ✐♥t❡❣r❛r s♦❜r❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s✭tr❛❥❡tór✐❛s✮ ❞❡ a ♣❛r❛ c ❡ ❡♠ s❡❣✉✐❞❛ ❞❡ c ♣❛r❛ b✱ ❡ ✜♥❛❧♠❡♥t❡ ✐♥t❡❣r❛r s♦❜r❡ c c t♦❞♦s ♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ❞❡ x ✱ ♥✉♠ ✐♥st❛♥t❡ t ✳ a b

          ❆ss✐♠ t❡r❡♠♦s ❛ ❛♠♣❧✐t✉❞❡ t♦t❛❧ ❝❛❧❝✉❧❛❞❛ ❡♥tr❡ ♦s ✐♥st❛♥t❡s t ❡ t ✱ ♣❛ss❛♥❞♦ ♣❡❧♦ c

          ✐♥st❛♥t❡ t ✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ♣r♦♣❛❣❛❞♦r❬✶✺❪ ♣♦r✿ Z K(b, a) = K(b, c)K(c, a)dx . c x c ✭✹✳✶✽✮

          ♦♥❞❡ t❡♠♦s q✉❡ ❡ss❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r r❡s✉♠✐❞♦ ♣❡❧❛s s❡❣✉✐♥t❡s r❡❣r❛s✿ ✶✳ ❖ ♣r♦♣❛❣❛❞♦r ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ ✉♠ ♣♦♥t♦ a ♣❛r❛ ✉♠ ♣♦♥t♦ b s❡rá ❞❛❞♦ ♣❡❧❛ s♦♠❛ ❞❡ t♦❞❛s ❛s ❛♠♣❧✐t✉❞❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ a ♣❛r❛ c ❡ ❞❡ c ♣❛r❛ b✱ s♦❜r❡ t♦❞♦s ♦s c

          ✈❛❧♦r❡s ♣♦ssí✈❡✐s ❞❡ x ❀ ✷✳ ❆ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ t♦t❛❧ ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ✐r ❞❡ a ♣❛r❛ c ❡ ❡♠ s❡❣✉✐❞❛ ❞❡ c ♣❛r❛ b s❡rá ♦❜t✐❞❛ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ ♣r♦♣❛❣❛❞♦r ❞❡ a ♣❛r❛ c✱ K(c, a)✱ ♣❡❧♦ ♣r♦❣❛❣❛❞♦r ❞❡ c ♣❛r❛ b✱ K(b, c)✳ ❆ ✜❣✉r❛ ✸✱ ✐❧✉str❛ ❛s tr❛❥❡tór✐❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s✳ ❋✐❣✉r❛ ✺✿ ❆♠♣❧✐t✉❞❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♣♦♥t♦ a ♣❛r❛ ✉♠ ♣♦♥t♦ ✐♥t❡r♠❡❞✐ár✐♦ c✱ ❡♠ s❡❣✉✐❞❛ ♣❛r❛ ♦ ♣♦♥t♦ b✳

          ✹✳✺ ❆s r❡❣r❛s ♣❛r❛ ✈ár✐♦s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s

          ❆❣♦r❛ q✉❡ s❛❜❡♠♦s q✉❛✐s ❛s r❡❣r❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❛✐s sã♦ ❛s r❡❣r❛s ♣❛r❛ N ❡✈❡♥t♦s s✉❝❡ss✐✈♦s✳ P❛r❛ t❛♥t♦ s❡rá ♥❡❝❡ssár✐♦ ❛♠♣❧✐❛r♠♦s ♦ ❝♦♥❝❡✐t♦ q✉❡ ✈✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳

          ■♥✐❝✐❛r❡♠♦s ❜✉s❝❛♥❞♦ ✉♠ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ♣r♦♣❛❣❛❞♦r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ q✉❡ ✈❛✐ ❞❡ ✉♠ ♣♦♥t♦ a ♣❛r❛ ✉♠ ♣♦♥t♦ b✱ ♠❛s ❛❣♦r❛ ♣❛ss❛ ♣♦r ❞♦✐s ♣♦♥t♦s ✐♥t❡r♠❡❞✐ár✐♦s✱ ♣r✐♠❡✐r♦ c✱ ❡ ❡♠ s❡❣✉✐❞❛ d✳ ▲♦❣♦✱ ♣❡❧♦ q✉❡ ✈✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ ❡ss❡ ♣r♦♣❛❣❛❞♦r s❡rá ❞❛❞♦ ♣♦r✿ Z Z K(b, a) = K(b, d)K(d, c)K(c, a)dx dx . c d x c x d

          ✭✹✳✶✾✮ ❖❜s❡r✈❡✲s❡ q✉❡ ❛ ♣❛rtí❝✉❧❛✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ✈❛✐ ❞❡ a ♣❛r❛ c✱ ❞❡♣♦✐s ❞❡ c ♣❛r❛ d ❡ ♣♦r c d

          ú❧t✐♠♦ ✈❛✐ ❞❡ d ♣❛r❛ b✱ ✐♥t❡❣r❛♥❞♦✲s❡ s♦❜r❡ t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ x ❡ x ✳

          ✐♥✐❝✐❛❧ a ❡ ✜♥❛❧ b✱ ♥♦ q✉❛❧ t❡rí❛♠♦s ♣❛r❛ ♦ ♣r♦♣❛❣❛❞♦r ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ Z Z Z K(b, a) = 1 dx . . . dx , N −1 2 N −1 x 1 x 2 x K(b, N − 1)K(N − 1, N − 2) . . . K(1, a)dx ✭✹✳✷✵✮

          ♦♥❞❡ ✈❡r✐✜❝❛♠♦s q✉❡ ❛s ❛♠♣❧✐t✉❞❡s ♣❛r❛ ❡✈❡♥t♦s q✉❡ ♦❝♦rr❡♠ ❡♠ s✉❝❡ssã♦ ❝♦♠ ❛ ❡✈♦❧✉✲ çã♦ t❡♠♣♦r❛❧ ♠✉❧t✐♣❧✐❝❛♠✲s❡✱ s❡♥❞♦ ♣♦rt❛♥t♦✱ ✉♠❛ ❡①t❡♥sã♦ ❞❛s r❡❣r❛s ♣❛r❛ ❞♦✐s ❡✈❡♥t♦s s✉❝❡ss✐✈♦s✳

          ✺ ❖ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦

          ❯t✐❧✐③❛r❡♠♦s ❛q✉✐ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ❍❛❣❡♥ ❑❧❡✐♥❡rt❬✷✸❪ ♥❛ s♦❧✉çã♦ ❞♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ♣♦r ■♥t❡❣r❛✐s ❞❡ ❈❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥✳

          ✺✳✶ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s

          ❆ ❧❛❣r❛♥❣✐❛♥❛ ♣❛r❛ ♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦ é ❞❛❞❛ ♣♦r✿

          1 2

          1 2 2 L(x, ˙x) = µ ˙x µω x .

          − ✭✺✳✶✮

          2

          2 ❉❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✱ t❡♠♦s✿

          ∂L d ∂L 2 = 0 ¨ x + ω x = 0

          − ⇒ ✭✺✳✷✮ ∂x dt ∂ ˙x

          ❝✉❥❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r✱ x(t) = Acosωt + Bsenωt.

          ✭✺✳✸✮ ❆♣❧✐❝❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✱ t ❂ ✵ ❡ t ❂ ❚✱ x(t = 0) = x = A

          ✭✺✳✹✮ x(t = T ) = x = x cosωT + BsenωT T x cosωT T − x . ⇒ B = ✭✺✳✺✮ senωT

          ❆ss✐♠✱ x senωt + x T senω(T − t) x(t) =

          ✭✺✳✻✮ senωT x T

          ωcosωt − x ωcosω(T − t) ˙x(t) =

          ✭✺✳✼✮ senωT

          ❊♥tã♦✱ S cl = Z T dtL(x, ˙x) = Z T dt

          ∂ (ε) − ω 2

          µ

          2 ( ˙

          X 2 − ω 2 X 2 )

          ✭✺✳✶✵✮ ❡ ♦ ❝á❧❝✉❧♦ ❡①♣❧í❝✐t♦ é ❞❛❞♦ ♣❡❧❛s ♠ú❧t✐♣❧❛s ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦✱ ❝♦♠ N ❛♣r♦①✐♠❛çõ❡s ❞❡ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ❞✐♠❡♥sã♦ ε✳ ❊♥tã♦✱

          φ N (T ) = r µ

          2πi¯hε N Y j=1 Z −∞ r µ

          2πi¯hε dX j exp i ¯h

          µ

          2 ε N +1 X j,k=0

          X j [−∂ (ε)

          ] jk

          )cosωT − 2x x T ] ✭✺✳✾✮

          X k ✭✺✳✶✶✮

          ❆q✉✐ ✉s❛♠♦s ♦s ♦♣❡r❛❞♦r❡s ∂ (ε) ❡ ∂ (ε) ❞❡✜♥✐❞♦s ♣♦r ∂ (ε) x(t) ≡

          1 ε

          [x(t + ε) − x(t)] ✭✺✳✶✷✮

          ∂ (ε) x(t) ≡

          1 ε

          [x(t) − x(t − ε)] ✭✺✳✶✸✮

          ♥♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦✱ t❡♠♦s✿ lim ε→0(ε) x(t), ∂ (ε) x(t) → ∂ t .

          ✭✺✳✶✹✮ ❊♥tã♦✱ s❡ ❡ss❡s ♦♣❡r❛❞♦r❡s ❛t✉❛♠ ❝♦♠♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s

          ∂ (ε) x j

          1 ε

          ❖ ❢❛t♦r ❞❡ ✢✉t✉❛çã♦ é ❞❡✜♥✐❞♦ ♣❡❧❛ ♣❛rt❡ ✢✉t✉❛♥t❡ ❞❛ ❛çã♦✱ S (f l) [X] = Z T dt

          2senωT [(x 2 T + x 2

          1

          1

          2 µ ˙x 2

          −

          1

          2 µω 2 x 2

          = Z T dt "

          1

          2 µ d dt

          (x ˙x) −

          1

          2 µx¨ x −

          2 µω 2 x 2 #

          = µω

          = −

          1

          2 µ Z T dtx[¨ x + ω 2 x] +

          µ

          2 x ˙x| T ✭✺✳✽✮ ♠❛s ¨x + ω 2 x = 0

          ✱ ❧♦❣♦✿ =

          µ

          2 [x(T ) ˙x(T ) − x(0) ˙x(0)]

          = µ

          2 x T ω senωT (x T cosωT − x ) − x ω senωT

          (x T − x cosωT )

          (x j+1 − x j ) 0 ≤ j ≤ N ✭✺✳✶✺✮

          ∂ (ε) x j

          = − N X j,k=0 x j (∂ (ε)(ε) )x k ❡♠ q✉❡ ❛ ♠❛tr✐③ (N + 1)x(N + 1)✱ ❝♦♠ ∂ (ε)

          2 ✭✺✳✶✾✮

          −1

          2 −1 . . . . . . . . .

          . . . . . . . . . −1

          2 −1

          2 −1 . . . . . . . . . −1 2 . . . . . . . . . ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ . . . . . . . . .

          −1

          2 −1 . . . . . . . . .

          1 ε 2

          ∂ (ε)(ε) = ∂ (ε)(ε) =

          ∂ (ε) ✱ s✐♠étr✐❝❛✱ é ❞❛❞❛ ♣♦r

          ✭✺✳✶✽✮ q✉❡ ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞♦s ❝♦♠♦ ✉♠❛ ♠❛tr✐③✲♣❛❞rã♦ − N X j=0 x j(ε)(ε) x j

          1 ε

          ∂ (ε) x j 2 = N +1 X j=1(ε) x j 2 = − N X j=0 x j(ε)(ε) x j .

          ∂ (ε) p j x j . P❛r❛ ❢✉♥çõ❡s ❝✉❥♦s ♣♦♥t♦s ❡①tr❡♠♦s s❡ ❛♥✉❧❛♠✱ ❞❡ss❛s r❡❧❛çõ❡s ❛♥t❡r✐♦r❡s t❡♠♦s✱ N +1 X j=1

          ✱ t❡♠♦s N +1 X j=1 p j(ε) x j = − N +1 X j=1

          ❡ x = x N +1

          ✷· ❙❡ p = p N +1

          = − N X j=0(ε) p j x j .

          = x = 0 ✱ t❡♠♦s N +1 X j=1 p j(ε) x j

          ✭✺✳✶✼✮ ❆ss✐♠✱ t❡♠♦s ❛❧❣✉♥s ❝❛s♦s út❡✐s✿ ✶· ❙❡ x N +1

          ∂ (ε) p j x j .

          ε N +1 X j=1 p j(ε) x j = p j x j | N +1 − ε N X j=0

          ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡❧❡s s❛t✐s❢❛③❡♠ ❛ r❡❣r❛ ❞❡ s♦♠❛ ♣♦r ♣❛rt❡✱ q✉❡ é ❛♥á❧♦❣❛ ❛ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱

          (x j − x j−1 ) 1 ≤ j ≤ N + 1 ✭✺✳✶✻✮

          ❊ss❛ é ♦❜✈✐❛♠❡♥t❡ ❛ ✈❡rsã♦ ❞❛ ❞❡r✐✈❛❞❛ ∂ 2 t ♥♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦✳ ❯s❛♥❞♦ ❛s ❝♦♠♣♦♥❡♥t❡s

          ❞❛ ❡①♣❛♥sã♦ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ❋♦✉r✐❡r✱ ❡♠ q✉❡ ✉s❛♠♦s ❛ ❞❡❝♦♠♣♦s✐çã♦✱ Z −iωt x(t) = dωe x(ω), −∞ ✭✺✳✷✵✮

          ✈❡♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r sã♦ ❛✉t♦❢✉♥çõ❡s ❞❛s ❞❡r✐✈❛❞❛s ❞✐s❝r❡t❛s✱ (ε) −iωt −iωt i∂ e = Ωe (ε) −iωt −iωt ✭✺✳✷✶✮ i∂ e = Ωe ✭✺✳✷✷✮

          ❝♦♠ ♦s r❡s♣❡❝t✐✈♦s ❛✉t♦✈❛❧♦r❡s✱

          1 −iωε Ω = (e

          − 1) ✭✺✳✷✸✮ ε

          1 iωε (e

          Ω = − − 1). ✭✺✳✷✹✮ ε

          ❊stá ❝❧❛r♦ q✉❡ ♥♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦ lim ε→0 Ω, Ω → ω, t ✐st♦ é✱ t❛♥t♦ Ω q✉❛♥t♦ Ω t♦r♥❛♠✲s❡ ❛✉t♦✈❛❧♦r❡s ❞❡ i∂ ✳ ❉❡ ✭✺✳✷✸✮ ❡ ✭✺✳✷✹✮ ✈❡♠♦s q✉❡ ∗ (ε) (ε) (ε) (ε) Ω = Ω ∂ ∂

          ✱ t❛❧ q✉❡ ♦ ♦♣❡r❛❞♦r −∂ = −∂ é r❡❛❧ ❡ t❡♠ ✈❛❧♦r❡s ♥ã♦ ♥❡❣❛t✐✈♦s✱ (ε) (ε) −iωt −iωt

          1 ∂ e =

          ✭✺✳✷✺✮ −∂ [2 − 2cosωε]e 2

          ε ♦✉ s❡❥❛✱ [2 − 2cosωε] ≥ 0.

          ❱♦❧t❡♠♦s ❛❣♦r❛ à s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✺✳✶✶✮✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ❡ss❛ ❡q✉❛çã♦ ❝♦♠♦✱ r N r N Y Y Z N µ µ i µ 2 2 φ (T ) = da exp (Ω Ω )a , n n n

          − ω n ✭✺✳✷✻✮ n 2πi¯hε −∞ 2πi¯hε ¯h n=1 n=1

          2 ❡♠ q✉❡ a sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❡①♣❛♥sã♦ ❞❡ ❋♦✉r✐❡r ❞❡ X(T )✳

          ❈❛❧❝✉❧❛♥❞♦ ❡ss❛ ✐♥t❡❣r❛❧✱ t❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ❞✐st✐♥çã♦ ♣❛r❛ ♦s ❛✉t♦✈❛❧♦r❡s✱ 2 1 πn 2 Ω Ω = ε , n n

          − ω 2 − 2cos − ω ✭✺✳✷✼✮ 2 ε T

          ❞♦ ♦♣❡r❛❞♦r ♥❛ ❡①♣♦♥❡♥❝✐❛❧ ❞♦ ✐♥t❡❣r❛♥❞♦ ❞❛ ❡q✉❛çã♦ ✭✺✳✷✻✮✳ ❊❧❡ ♣♦❞❡rá s❡r ♥❡❣❛t✐✈♦ ♣❛r❛ ❝❛s♦s ❡♠ q✉❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ T = t − t ✱ s❡❥❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

          Pr✐♠❡✐r♦✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦s ❛✉t♦✈❛❧♦r❡s ✐♥t❡✐r❛♠❡♥t❡ ♣♦s✐t✐✈♦s✳ ❆ ❢ór♠✉❧❛ ❣❛✉ss✐❛♥❛

          ♥♦s ❞á r N Y N µ

          1 φ (T ) = , 2 ✭✺✳✷✽✮

          √ 2 2πi¯hε n=1 n n ε (Ω Ω ) − ω

          ω ✐♥tr♦❞✉③✐♥❞♦ ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ ♣❛r❛ ❛ ❢r❡q✉ê♥❝✐❛✱ ✱ q✉❡ s❛t✐s❢❛③

          ω ε εω sen = ,

          2

          2 ♦ ♣r♦❞✉t♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❡♠ r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ♦ ❢❛t♦r ❞❡ ✢✉t✉❛çã♦✳ ❆ss✐♠✱ Y N   2   sen b 1 sen(2(N + 1)b)

          = , ✭✺✳✷✾✮ 1 − 2 sen2b (N + 1) n=1 ε ω sen 2(N +1)

          ♥❛t✉r❛❧♠❡♥t❡✱ ✉s❛♥❞♦ b = ✱ t❡♠♦s Y Y Y N N N 2 " # 2 2 2 2 2 − ω ε (Ω Ω ) n n ε (Ω Ω ) = ε Ω Ω n n m m

          − ω 2 n=1 m=1 n=1 Y Y N N 2 ε ω  ∼  ε Ω Ω n n 2   sen 2 = ε Ω Ω . m m 1 − ✭✺✳✸✵✮ 2 m=1 n=1 sen 2(N +1) ❈♦♠♦ r❡s✉❧t❛❞♦✱ ❝❤❡❣❛♠♦s ❡♠ Y N 2 ε ε sen 2 2 2 ω T det (∂ ∂ )) = ε (Ω Ω ) = , N n n ∼ ✭✺✳✸✶✮

          (−ε − ω − ω ω n=1 senε

          ❡ ♦ ❢❛t♦r ✢✉t✉❛♥t❡ t♦t❛❧ s❡rá r u u ∼ v ω N t µ senε φ (T ) = .

          ✭✺✳✸✷✮ 2πi¯h

          ω T εsen

          ❊ss❡ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ♣♦s✐t✐✈♦s✱ ❡♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ q✉❡ s❛t✐s❢❛③ π ) < . T = (t − t ∼

          ω π ❙❡ ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ T = t − t ❛✉♠❡♥t❛r ❡ ✜❝❛r ♠❛✐♦r q✉❡ ✱ ♦ ♠❡♥♦r ❛✉t♦✈❛❧♦r 2 ω

          Ω 1 Ω 1 s❡ t♦r♥❛ ♥❡❣❛t✐✈♦ ❡ ❛ ❛♠♣❧✐t✉❞❡ r❡s✉❧t❛♥t❡ ❝❛rr❡❣❛rá ✉♠ ❢❛t♦r ❞❡ ❢❛s❡ ❡①tr❛ −i π 2 − ω e q✉❡ s❡rá ✈á❧✐❞♦ ❛té T = t − t t♦r♥❛r✲s❡ ♠❛✐♦r q✉❡ ✱ ❡♠ q✉❡ ♦ s❡❣✉♥❞♦ ❛✉t♦✈❛❧♦r ω −i π 2 s❡ t♦r♥❛ ♥❡❣❛t✐✈♦ t❛♠❜é♠ ❡ é ♥❡❝❡ssár✐♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♦✉tr♦ ❢❛t♦r ❞❡ ❢❛s❡ e ✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

          ◆♦ ❧✐♠✐t❡ ❝♦♥tí♥✉♦ ε → 0✱ ♦ ❢❛t♦r ❞❡ ✢✉t✉❛çã♦ s❡rá ❡♥tã♦ r r N µ ω φ (T ) = .

          ✭✺✳✸✸✮ 2πi¯h senωT

          ▲♦❣♦ ♦ ❦❡r♥❡❧ s❡rá✿ ¯ h i K(x , T ; x , 0) =< x , 0 >= e S φ(T ) T T cl r h i , T |x

          µω iµω 2 2 = exp x + x x , T cosωT − 2x ✭✺✳✸✹✮ T

          2πi¯hsenωT 2¯hsenωT q✉❡ é ❛ ❡①♣r❡ssã♦ ❞❛ ❛♠♣❧✐t✉❞❡ ❞❡ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ♣❛r❛ ✉♠ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦✳ ❆✉t♦❢✉♥çõ❡s ❡ ❆✉t♦✈❛❧♦r❡s ❞♦ ❖s❝✐❧❛❞♦r✿ ♣❛r❛ ♦❜t❡r ♦s ❛✉t♦❡st❛❞♦s ❡ ❛✉t♦✈❛❧♦r❡s ❞❡ ❡♥❡r✲ ❣✐❛✱ ♣r❡❝✐s❛♠♦s r❡❢♦r♠✉❧❛r ♦ ♣r♦♣❛❣❛❞♦r✱ ❡q✉❛çã♦ ✭✺✳✸✹✮✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♣❡r♠✐t❛ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❞✐r❡t❛ ❝♦♠ ❛ r❡♣r❡s❡♥t❛çã♦ ❡s♣❡❝tr❛❧ ❞♦ ♣r♦♣❛❣❛❞♦r ❞❡ ❋❡②♥♠❛♥ ❞❛❞♦ ♣♦r X ′ i ∗ − E n T ¯ h

          K(x, T ; x , 0) = η(T ) ϕ n (x)ϕ (x )e , n n −iωT ✭✺✳✸✺✮ ❡♠ q✉❡ η(T ) é ❛ ❢✉♥çã♦ ❞❡❣r❛✉ ❞❡ ❍❡❛✈✐s✐❞❡✳ ❙❡ ❞❡✜♥✐r♠♦s ❛ ✈❛r✐á✈❡❧ z = e ✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✱ 2

          1 1 − z senωT = ,

          2i z ❡ 2 1 + z cosωT = . q q µω µω 2z

          = x = x T T ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♠♦s ξ ❡ ξ ✳ ¯ h ¯ h ❆ss✐♠✱ ❡①♣r❡ss❛♠♦s ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❝♦♠♦✿ r 1 ( " !#) 2

          µ 2 − 2 1 1 + z 2 2 K(x T , T ; x , 0) = ) exp 2ξ ξ T + ξ ) zπ¯h(1 − z z − (ξ T 2 ω

          2 r 1 1 − z ( ) 2 2 2 µ 2 − 2 1 2ξ ξ T + ξ )z 2 2 z − (ξ T

          = ) exp (ξ + ξ ) exp , zπ¯h(1 − z − T 2 ω

          2 1 − z ✭✺✳✸✻✮

          ❡♠ q✉❡ ✉s❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ 2 2 1 + z 1 z 2 + = . 2

          )

          2 2(1 − z 1 − z

          ❆❣♦r❛ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢ór♠✉❧❛ ❞❡ ▼❡❤❧❡r 1 ( ) 2 2 2 n X 2 − 2xyz − (x 2 + y )z z ) exp = H (x)H (y) n n

          (1 − z 2 n (|z| < 1). ✭✺✳✸✼✮ 2 n! 1 − z n=0

          ❊♥tr❡t❛♥t♦✱ ❞❡✈❡♠♦s t♦♠❛r ❛❧❣✉♠ ❝✉✐❞❛❞♦ ❛♦ ✉s❛r ❛ ❡q✉❛çã♦ ✭✺✳✸✼✮ ♥❛ ❡q✉❛çã♦ ✭✺✳✸✻✮✱ ♣♦✐s |z| = 1 ❡ ❛ ❢ór♠✉❧❛ ❞❡ ▼❡❤❧❡r r❡q✉❡r |z| < 1✳ ❊ss❡ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r s♦❧✉❝✐♦♥❛❞♦ s❡ ❛❞✐❝✐♦♥❛r♠♦s ✉♠❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❡♠ ω✱ ♥♦♠✐♥❛❧♠❡♥t❡✱ s❡ ✜③❡r♠♦s ω → ω − iε✱ ❡ ❛♣ós ♦s ❝á❧❝✉❧♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ε → 0✳ ❆ss✐♠✱ s❡ ✉s❛r♠♦s ❛ ❡q✳ ✭✺✳✸✼✮✱ ❛ ❡q✳ ✭✺✳✸✻✮ t❡rá ❛ ❢♦r♠❛ r ∞ r r −iωT n+ 1 X ( ) 2

          µω µω µω µω e 2 2 K(x , T ; x , 0) = exp x + (x H x H x , T n T n − T n

          π¯h 2¯h ¯h ¯h n=0 2 n! q q µω µω ✭✺✳✸✽✮ T T = x = x ❡♠ q✉❡ ✉s❛♠♦s x = ξ ❡ y = ξ ✳ ¯ h ¯ h

          ❈♦♠♣❛r❛♥❞♦ ❛ ❡q✳ ✭✺✳✸✽✮ ❝♦♠ ❛ ❡q✳ ✭✺✳✸✺✮✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❡s♣❡❝tr❛❧✱ t❡♠♦s✿ 1 2 1 4 µω 2 r 1 µω µω − x 2 π ϕ n (x) = e H n x , n

          ✭✺✳✸✾✮ 2 n! π¯h ¯h ❡

          1 E n = n + ¯hω ✭✺✳✹✵✮

          2 q✉❡ sã♦ ♦s ❝♦♥❤❡❝✐❞♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❛s ❛✉t♦❢✉♥çõ❡s ✭❛ ♠❡♥♦s ❞❡ ✉♠ ❢❛t♦r ❞❡ ❢❛s❡✮ ❡ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦✳

          ✺✳✷ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❋♦rç❛❞♦

          ❆ ❧❛❣r❛♥❣✐❛♥❛ ♣❛r❛ ♦ ❖s❝✐❧❛❞♦r ❍❛r♠ô♥✐❝♦ ❋♦rç❛❞♦ ❞❡ ♠❛ss❛ µ ❡ ❢r❡q✉ê♥❝✐❛ ω✱ é ❞❛❞♦ ♣♦r

          1 2

          1 2 2 L = µ ˙x µω x + xf (t), −

          ✭✺✳✹✶✮

          2

          2 ❡♠ q✉❡ f(t) r❡♣r❡s❡♥t❛ ✉♠❛ ❢♦rç❛ ❡①t❡r♥❛ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ q✉❡ ♣♦❞❡ s❡r ❞❡✈✐❞❛ ❛ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ❛tr❛✈és ❞❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ ❞✐♣♦❧♦ ❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❝❛rr❡❣❛❞❛✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ❦❡r♥❡❧✱ ✉t✐❧✐③❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ s❡♠✐❝❧áss✐❝❛✱ ❝♦♠♦ Z h ¯ i S(x) K(x , t; x , 0) = D e f i x

          ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ♦❜t❡r ♦ ❦❡r♥❡❧ ♣❛r❛ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ♣♦s✲ i f sí✈❡✐s ❝♦♠❡ç❛♥❞♦ ❡♠ x q✉❛♥❞♦ t = 0 ❡ t❡r♠✐♥❛♥❞♦ ❡♠ x q✉❛♥❞♦ t = t✳ ➱ ❝♦♥✈❡♥✐❡♥t❡ ❞❡❝♦♠♣♦r ♦s ❝❛♠✐♥❤♦s ❡♠ x(s) = x cl (s) + ξ(s), cl (0) = x i , x cl (t) = ✭✺✳✹✷✮

          ❡♠ q✉❡ s é ✉♠❛ ✈❛r✐á✈❡❧ ❛✉①✐❧✐❛r t❡♠♣♦r❛❧✱ ♦♥❞❡ ♦ ❝❛♠✐♥❤♦ ❝❧áss✐❝♦ x x f ❡ ✉♠❛ ♣❛rt❡ ❞❛ ✢✉t✉❛çã♦ q✉â♥t✐❝❛ ξ ❞❡s❛♣❛r❡❝❡ ♥♦ ❝♦♥t♦r♥♦✱ ✐st♦ é✱ ξ(0) = ξ(t) = 0✳

          ❖ ❝❛♠✐♥❤♦ ❝❧áss✐❝♦ t❡♠ q✉❡ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ 2 µ¨ x + µω x = f (s), cl cl

          ✭✺✳✹✸✮ q✉❡ é ♦❜t✐❞❛ ❞❛ ❧❛❣r❛♥❣✐❛♥❛✳ ❆ ❛çã♦ é ❞❛❞❛ ♣♦r Z t S = dtL.

          ❙✉❜st✐t✉✐♥❞♦ ❛ ❧❛❣r❛♥❣✐❛♥❛ ♥❛ r❡❧❛çã♦ ❛❝✐♠❛ Z t µ 2

          1 2 2 S = ds ˙x µω x + xf (s) , −

          2

          2 ❡ ✉s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✺✳✹✷✮ Z t

          µ 2

          1 2 2 S = ds ˙x + µω x + x cl f (s) cl clZ t

          2

          2 2 ds µ ˙x cl x cl ξ + ξf (s) Z t ˙ξ − µω

        • µ
        • 2

            1 2 2 ˙ξ ds µω ξ .

            ✭✺✳✹✹✮ −

            2

            2 P❛r❛ ♦ ♥♦ss♦ ❝❛s♦✱ ❞❡ ✉♠ ♣♦t❡♥❝✐❛❧ ❤❛r♠ô♥✐❝♦✱ ♦ t❡r❝❡✐r♦ t❡r♠♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ i f ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ x ❡ x ❛ss✐♠ ❝♦♠♦ ❝♦♥❞✐çõ❡s ❡①t❡r♥❛s✳ ❖ s❡❣✉♥❞♦ t❡r♠♦ ❞❡s❛♣❛r❡❝❡ ❡♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❡①♣❛♥sã♦ ❡♠ t♦r♥♦ ❞♦ ❝❛♠✐♥❤♦ ❝❧áss✐❝♦✳ ■ss♦ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧❛ cl ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s ❡ ✉s❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ x é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦

            ❝❧áss✐❝❛✱ Z Z t t 2 2 ds µ ˙x x ξ + ξf (s) ds µ¨ x + µω x ξ = 0 cl cl cl cl ˙ξ − µω = − − f(s) ✭✺✳✹✺✮

            ❱❛♠♦s ❛❣♦r❛ ♣r♦❝❡❞❡r ❡♠ ❞✉❛s ❡t❛♣❛s✱ ♣r✐♠❡✐r♦ ❞❡t❡r♠✐♥❛♥❞♦ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ ❝❛✲ ♠✐♥❤♦ ❝❧áss✐❝♦ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❛❜♦r❞❛♥❞♦ ❛s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s✳ ❆ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❝❧áss✐❝❛ q✉❡ s❛t✐s❢❛③ às ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ é ❞❛❞❛ ♣♦r senωs

            1 senω(t − s) x (s) = x + x + cl f i Z Z s t senωt senωt µω senωs dusenω(s − u)f(u) − dusenω(t − u)f(u) ✭✺✳✹✻✮ senωt

            ❡♠ q✉❡ u t❛♠❜é♠ é ✉♠❛ ✈❛r✐á✈❡❧ ❛✉①✐❧✐❛r t❡♠♣♦r❛❧✳ ❆ ✐❞❡✐❛ ❞❡ ❛✈❛❧✐❛r ❛ ❛çã♦ ❞♦ ❝❛♠✐♥❤♦ ❝❧áss✐❝♦ ♣♦❞❡ s❡r s✐♠♣❧✐✜❝❛❞❛ ♣♦r ✉♠❛ ✐♥t❡❣r❛çã♦

            ♣♦r ♣❛rt❡s Z t µ 2

            1 2 2 S = ds ˙x µω x + x f (s) cl cl cl − cl t Z t

            2

            2 µ µ

            1 2 2 2 = x x ˙ ds x x ¨ µω x + f (s) cl cl cl cl

            − cl − x cl

            2

            2

            2 Z t µ

            1 = (x ˙x ˙x (0)) + dsx (s)f (s), f cl i cl cl

            (t) − x ✭✺✳✹✼✮

            2

            2 ❡♠ q✉❡ ✉s❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❝❧áss✐❝❛ ♣❛r❛ ♦❜t❡r ❛ t❡r❝❡✐r❛ ❧✐♥❤❛✳ ❉❛ s♦❧✉çã♦✱ ❡q✳ ✭✺✳✹✻✮✱ t❡♠♦s✿ Z t x cosωt f i

            1 − x

            ˙x cl (0) = ω ✭✺✳✹✽✮

            − dssenω(t − s)f(s) senωt µsenωt Z t x f i

            1 cosωt − x

          • ˙x cl (t) = ω dssenωsf (s).

            ✭✺✳✹✾✮ senωt µsenωt ■♥s❡r✐♥❞♦ ❛s ✈❡❧♦❝✐❞❛❞❡s ✐♥✐❝✐❛❧ ❡ ✜♥❛❧ ♥❛ ❡q✳ ✭✺✳✹✼✮✱ ❡♥❝♦♥tr❛♠♦s ❛ ❛çã♦ ❝❧áss✐❝❛ h i

            µω 2 2 S cl = x + x i x + f i f cosωt − 2x Z Z t t 2senωt x x f i dssenωsf (s) + dssenω(t − s)f(s) − senωt senωt Z Z t t

            1 ds dusenωusenω(t − s)f(s)f(u), ✭✺✳✺✵✮

            µωsenωt i f s❡♥❞♦ t ♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❞❡s❧♦❝❛r ❞❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧ x ♣❛r❛ ❛ ♣♦s✐çã♦ ✜♥❛❧ x ✳ ❈♦♠♦ ✉♠ s❡❣✉♥❞♦ ♣❛ss♦✱ t❡♠♦s q✉❡ ❝❛❧❝✉❧❛r ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s✱ q✉❡ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ t❡r❝❡✐r♦ t❡r♠♦ ❞❛ ❡q✳ ✭✺✳✹✹✮✳ ❉❡♣♦✐s ❞❡ ✉♠❛ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱

            ❡ss❡ t❡r♠♦ ✜❝❛✿ Z t t 2 Z ! ! 2 (2) µ ˙ξ 1 µ d 2 2 2 S = ds µω ξ ds ξ + ω ξ.

            ✭✺✳✺✶✮ − = − 2

            2

            2 2 ds ❆q✉✐✱ ♦ ♥ú♠❡r♦ (2) ❛❝✐♠❛ ❞❡ S ✐♥❞✐❝❛ q✉❡ ❡ss❡ t❡r♠♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❡♠ ξ✳ ➱ ❛♣r♦♣r✐❛❞♦ ❡①♣❛♥❞✐r ❛ ✢✉t✉❛çã♦ ξ✱ X

            ξ(s) = a ξ (s), n=1 n n ✭✺✳✺✷✮ ❡♠ ❛✉t♦❢✉♥çõ❡s ❞❡ 2 ! d 2

          • ω ξ n = λ n ξ n
          • 2 ✭✺✳✺✸✮ ds

              ξ (0) = ξ (t) = 0 n n n ❝♦♠ ✳ ❈♦♠♦ ❛✉t♦❢✉♥çõ❡s ❞❡ ✉♠ ♦♣❡r❛❞♦r ❛✉t♦❛❞❥✉♥t♦✱ ξ ✱ sã♦ ❝♦♠♣❧❡t♦s✱ ❡ ♣♦❞❡♠♦s t❛♠❜é♠ ❞✐③❡r q✉❡ sã♦ ♦rt♦♥♦r♠❛✐s✳ ❘❡s♦❧✈❡♥❞♦ ❛ ❡q✳ ✭✺✳✺✸✮✱ t❡♠♦s ♣❛r❛ ❛s ❛✉t♦❢✉♥çõ❡s s 2 s

              ξ n (s) = sen πn ✭✺✳✺✹✮ t t

              ❡ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛✉t♦✈❛❧♦r❡s 2 πn 2

              λ + ω n = −

              ✭✺✳✺✺✮ t ❉❡✈❡♠♦s ❡♥❢❛t✐③❛r q✉❡ ❛ ✭✺✳✺✷✮ ♥ã♦ é ✉♠❛ sér✐❡ ❞❡ ❋♦✉r✐❡r ✉s✉❛❧✱ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ t✳

              ❯♠❛ ❡①♣❛♥sã♦ s❡♠❡❧❤❛♥t❡ ♣♦❞❡ s❡r ✉s❛❞❛ ♥❛ ❢♦r♠❛ s X 2 s s ξ(s) = a cos 2πn + sen 2πn , n

              − 1 ✭✺✳✺✻✮ t t t n=1 q✉❡ ♥♦s ❣❛r❛♥t❡ q✉❡ ❛s ✢✉t✉❛çõ❡s ❞❡s❛♣❛r❡❝❡♠ ♥♦s ❝♦♥t♦r♥♦s✳ s❡ r❡✜③❡r♠♦s ❛s ❝♦♥t❛s s✉❜st✐t✉✐♥❞♦ ♣♦r ✭✺✳✺✻✮✱ ✈❛♠♦s ♦❜s❡r✈❛r q✉❡ ♥♦ ✜♥❛❧ ❡♥❝♦♥tr❛♠♦s ♦ ♠❡s♠♦ ♣r♦♣❛❣❛❞♦r✳

              ❆ ✐♥t❡❣r❛çã♦ s♦❜r❡ ❛s ✢✉t✉❛çõ❡s ❛❣♦r❛ s❡ t♦r♥❛ ✉♠❛ ✐♥t❡❣r❛çã♦ s♦❜r❡ ♦s ❝♦❡✜❝✐❡♥t❡s n ❞❛ ❡①♣❛♥sã♦✱ a ✳ ■♥s❡r✐♥❞♦ ❛ ❡①♣❛♥sã♦ ✭✺✳✺✹✮ ♥❛ ❛çã♦✱ ❡♥❝♦♥tr❛♠♦s X ∞ ∞ X " # 2 (2) µ µ πn 2 2 2 S λ a = a . n

              = − n − ω n ✭✺✳✺✼✮

              2 2 t

              ❈♦♠♦ ❡ss❡ r❡s✉❧t❛❞♦ ♠♦str❛✱ ❛ ❛çã♦ ❝❧áss✐❝❛ é s♦♠❡♥t❡ ✉♠ ❡①tr❡♠♦ ❞❛ ❛çã♦✱ ♠❛s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠ ♠í♥✐♠♦✱ ❡♠❜♦r❛ ❡st❡ s❡❥❛ ♦ ❝❛s♦ ♣❛r❛ ♣❡q✉❡♥♦s ✐♥t❡r✈❛❧♦s ❞❡ π n = t❡♠♣♦ t < ✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝♦♥❥✉❣❛❞♦s ♥♦s t❡♠♣♦s T ♠♦str❛ ❛q✉✐ ✉♠ ω n n ω

              ❞❡s❛♣❛r❡❝✐♠❡♥t♦ ❞♦ ❛✉t♦✈❛❧♦r λ ✳ ❆ ❛çã♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ a ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ♣❛r❛ n cl + a n ξ n n ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ T t♦❞♦s ♦s ❝❛♠✐♥❤♦s x ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❛r❜✐trár✐♦s a sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❝❧áss✐❝❛✳

              ❉❡♣♦✐s ❞❛ ❡①♣❛♥sã♦ ❞❛s ✢✉t✉❛çõ❡s ❡♠ t❡r♠♦s ❞❛s ❛✉t♦❢✉♥çõ❡s ✭✺✳✺✹✮✱ ♦ ♣r♦♣❛❣❛❞♦r t❡♠ ❛ ❢♦r♠❛ Z Y ∞ ∞ ! " # X i i µ 2 K(x , t; x S da exp λ a . f i cl n n ✭✺✳✺✽✮ , 0) ≈ exp − n

              ¯h ¯h n=1 n=1

              2 ❈♦♠♦ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ❏❛❝♦❜✐ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦✱ ♣♦❞❡♠♦s

              ❝♦♠♣❛r❛r ❝♦♠ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡✳ ❘❡s♦❧✈❡♥❞♦ ❛s ✐♥t❡❣r❛✐s ❞❡ ✢✉t✉❛çã♦ ❣❛✉ss✐❛♥❛s✱ ❡♥❝♦♥✲ tr❛♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❢❛♦tr❡s ❞♦s ♣r♦♣❛❣❛❞♦r❡s K ω ❡ K ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❡ ❞❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡✱ − S ¯ h i cl,ω s

              K e D ω i = . − S ¯ h cl,0 ✭✺✳✺✾✮ D

              K e ❆q✉✐ ✐♥tr♦❞✉③✐♠♦s ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ✢✉t✉❛çã♦ ♣❛r❛ ♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ 2 ∞ ! Y d 2 D = det + ω = λ , 2 n ✭✺✳✻✵✮ ds n=1

              ❡ ♦ ❞❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ 2 ! Y d D = det = λ . 2 n ✭✺✳✻✶✮ ds n=1

              ❖s ❛✉t♦✈❛❧♦r❡s ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ 2 πn

              λ n = − t sã♦ ♦❜t✐❞♦s ❞♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❝♦❧♦❝❛♥❞♦ ❛ ❢r❡q✉ê♥❝✐❛ ω ✐❣✉❛❧ ❛ ③❡r♦ ❡♠ ✭✺✳✺✺✮✳ ❯s❛♥❞♦ ♦ ❢❛t♦r ❞♦ ♣r♦♣❛❣❛❞♦r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ i r − S ¯ h cl,0 µ

              K e = ✭✺✳✻✷✮

              2πi¯ht ❡ ❛ ❡q✉❛çã♦ ✭✺✳✺✾✮✱ ♦ ♣r♦♣❛❣❛❞♦r ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ✜❝❛ r s i

              µ D h ¯ S cl,0 K(x , t; x , 0) = e . f i ✭✺✳✻✸✮

              2πi¯ht D ❈❛❞❛ ✉♠ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ✭✺✳✻✵✮ ❡ ✭✺✳✻✶✮ ❞✐✈❡r❣❡ ♣♦r s✐ só✳ ❊♥tr❡t❛♥t♦✱ ❡st❛♠♦s

              ✐♥t❡r❡ss❛❞♦s ♥❛ r❡❧❛çã♦ ❡♥tr❡ ❡❧❡s✱ q✉❡ é ❜❡♠ ❞❡✜♥✐❞❛✱ Y " # 2 D ωt senωt = = 1 −

              D πn ωt n=1

              ❈♦❧♦❝❛♥❞♦ ❡ss❡ r❡s✉❧t❛❞♦ ❡♠ ✭✺✳✻✸✮✱ t❡♠♦s ❛ ❢♦r♠❛ ✜♥❛❧ ❞♦ ♣r♦♣❛❣❛❞♦r ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❢♦rç❛❞♦ r i

              µω h ¯ S cl K(x , t; x , 0) = e f i s 2πi¯hsenωt

              µω i π π = exp S + n , cl

              − i ✭✺✳✻✹✮ ¯h

              4

              2 2π¯h|senωt|

              ❝♦♠ ❛ ❛çã♦ ❝❧áss✐❝❛ ❞❡✜♥✐❞❛ ❡♠ ✭✺✳✺✵✮✳ ωt ❖ í♥❞✐❝❡ ❞❡ ▼♦rs❡ n ♥♦ ❢❛t♦r ❞❡ ❢❛s❡ é ❞❛❞♦ ♣❡❧❛ ♣❛rt❡ ✐♥t❡✐r❛ ❞❡ ✳ ➱ ✐♥t❡r❡ss❛♥t❡ −in 2 π π

              ♥♦t❛r q✉❡ ♦ ❢❛t♦r ❞❡ ❢❛s❡ e ❡♠ ✭✺✳✻✹✮ ✐♠♣❧✐❝❛ q✉❡✿ 2π

              K x , ; x , 0 = K(x , 0; x ), f i f i f i , 0) = −δ(x − x

              ω ♦✉ s❡❥❛✱ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡♣♦✐s ❞❡ ✉♠ ♣❡rí♦❞♦ ❞❡ ♦s❝✐❧❛çã♦ ❞✐❢❡r❡ ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♦r✐❣✐♥❛❧ ♣♦r ✉♠ ❢❛t♦r −1✳ ❖ ♦s❝✐❧❛❞♦r r❡t♦r♥❛ ❛♦ s❡✉ ❡st❛❞♦ ♦r✐❣✐♥❛❧ s♦♠❡♥t❡ ❛♣ós ❞♦✐s 1

              ♣❡rí♦❞♦s✱ s❡♥❞♦ ♣♦rt❛♥t♦ ♣❛r❡❝✐❞♦ ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ s♣✐♥ ✳ ❊ss❡ ❡❢❡✐t♦ ♣♦❞❡ s❡r 2 ♦❜s❡r✈❛❞♦ ♥♦ ❝❛s♦ ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ❞❡✐①❛♥❞♦ ✐♥t❡r❢❡r✐r ❛s ♦♥❞❛s ❞❡ ❞♦✐s ♦s❝✐❧❛❞♦r❡s ❝♦♠ ❢r❡q✉ê♥❝✐❛s ❞✐❢❡r❡♥t❡s✳

              ❊ss❡ r❡s✉❧t❛❞♦ t❡♠ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❛✈❛♥ç❛❞♦s✱ ❡♠ ❡s♣❡❝✐❛❧ q✉❛♥❞♦ ❛♣❧✐❝❛♠♦s ❡♠ ❡❧❡tr♦❞✐♥â♠✐❝❛ q✉â♥t✐❝❛✱ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞♦ ❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ♣♦❞❡r s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦s❝✐❧❛❞♦r❡s ❤❛r♠ô♥✐❝♦s ❢♦rç❛❞♦s✳

              ✻ ❈♦♥❝❧✉sõ❡s

              ❆♥❛❧✐s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s ♦❜t✐❞♦s ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥❝❧✉✐♠♦s q✉❡ ❛tr❛✈és ❞❛ ❢♦r♠✉❧❛çã♦ ❞❛s ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ ❡♥❝♦♥tr❛♠♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❢♦r♠✉❧❛❞❛ ♣♦r ❙❝❤rö❞✐♥❣❡r ❡ ❍❡✐s❡♥❜❡r❣✳ ❊♥q✉❛♥t♦ q✉❡ ♥❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r ❡ ❍❡✐s❡♥❜❡r❣ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♦♣❡r❛❞♦r❡s q✉❡ ❛t✉❛♠ s♦❜r❡ ✈❡t♦r❡s ❞❡ ❡st❛❞♦✱ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ❝❛♠✐♥❤♦ ❋❡②♥♠❛♥ tr❛❜❛❧❤❛r❡♠♦s ❛♣❡♥❛s ❝♦♠ ❢✉♥çõ❡s✳ ❚♦❞❛✈✐❛✱ ❛s três ❢♦r♠✉❧❛çõ❡s ✭❙❝❤rö❞✐♥❣❡r✱ ❍❡✐s❡♥❜❡r❣ ❡ ❋❡②♥♠❛♥✮ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

              P♦❞❡♠♦s ❝♦♥❝❧✉✐r t❛♠❜é♠ q✉❡ ❛ ❢♦r♠✉❧❛çã♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣♦r ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛✱ q✉❡ ❣❡♥❡r❛❧✐③❛ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❍❛♠✐❧t♦♥✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦✱ ❡①t❡♥❞❡♥❞♦✲ ♦ à ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❡♠ q✉❡ ❡ss❡ ❢♦r♠❛❧✐s♠♦ s✉❜st✐t✉✐ ❛ ♥♦çã♦ ❝❧áss✐❝❛ ❞❡ ✉♠❛ ú♥✐❝❛ tr❛❥❡tór✐❛✱ ❞❛❞♦ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ♠í♥✐♠❛ ❛çã♦✱ ♣♦r ✉♠ ❢✉♥❝✐♦♥❛❧ ✐♥t❡❣r❛❧✱ s♦❜r❡ ♦ q✉❛❧ ♣♦❞❡♠♦s t❡r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ tr❛❥❡tór✐❛s ♣♦ssí✈❡✐s ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ❛♠♣❧✐t✉❞❡ ❞❡ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡ ❞♦ ❝❛♠✐♥❤♦ q✉â♥t✐❝♦❬✷✺❪✳

              ❉✉r❛♥t❡ ♦s ❝á❧❝✉❧♦s ❞❛s ✐♥t❡❣r❛✐s ❞❡ ❝❛♠✐♥❤♦ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ♣r♦❜❧❡♠❛s q✉❡ ❛♣r❡s❡♥t❛r❛♠✱ ❡♠ s✉❛ ❢♦r♠✉❧❛çã♦✱ ❛❧❣✉♠❛s ❝❧❛ss❡s ❞❡ ✐♥t❡❣r❛✐s ❣❛✉ss✐❛♥❛s✱ s❡♥❞♦ q✉❡ ❡st❡ é ♦ ♠❛✐♦r ❞❡s❛✜♦ ♥❛ té❝♥✐❝❛✳ ◆❛ t❡♥t❛t✐✈❛ ❞❡ s♦❧✉❝✐♦♥❛r ❡ss❡s ♣r♦❜❧❡♠❛s ❜✉s❝❛♠♦s ❢❛③❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦✱ ❛tr❛✈és ❞❡ ✉♠❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❡ ❚❛②❧♦r ❞♦ ♣♦t❡♥❝✐❛❧ V (q, t) ❛té s❡❣✉♥❞❛ ♦r❞❡♠✱ t♦r♥❛♥❞♦✲♦ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✉♠ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦✳

              ❆ ✐♥t❡❣r❛❧ ❞❡ ❝❛♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥ r❡♣r♦❞✉③ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ ❛s ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ❡ ❛s r❡❧❛çõ❡s ❝❛♥ô♥✐❝❛s ❞❡ ❝♦♠✉t❛çã♦ ❡ ♠♦str❛ q✉❡ ❡❧❛s sã♦ ❝♦♠♣❛tí✈❡✐s ❝♦♠ ❛ r❡❧❛t✐✈✐❞❛❞❡✱ ♠❛s ❛❣♦r❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❥á ♥ã♦ é tã♦ ❡✈✐❞❡♥t❡✱ ♣♦ré♠ ♣♦❞❡✲s❡ ✈❡r✐✜❝á✲❧❛ ❛❧t❡r❛♥❞♦ ✈❛r✐á✈❡✐s ♣❛r❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❝❛♥ô♥✐❝❛✳

              ◆♦ ❝❛s♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♥ã♦✲r❡❧❛t✐✈íst✐❝❛✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s ❛♠♣❧✐t✉❞❡s ❞❡

              ❊♥tr❡t❛♥t♦✱ ❛ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛ ❢♦r♠✉❧❛çã♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❛♣❛r❡❝❡ ♥❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s✱ ♦♥❞❡ ❛s ❢✉♥çõ❡s ❞❡ ●r❡❡♥ sã♦ ❛s q✉❛♥t✐❞❛❞❡s q✉❡ ❜✉s❝❛♠♦s ❝❛❧❝✉❧❛r ❛tr❛✈és ❞❡ ♠ét♦❞♦s ♣❡rt✉r❜❛t✐✈♦s ❡ ♥ã♦✲♣❡rt✉r❜❛t✐✈♦s✳

              ❉❡✈✐❞♦ ❛ ❡♥❣❡♥❤♦s✐❞❛❞❡ ♠❛t❡♠át✐❝❛ ♥❛ s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ♣♦r ✐♥t❡❣r❛✐s ❞❡ ❝❛✲ ♠✐♥❤♦ ❞❡ ❋❡②♥♠❛♥✱ ❣❡r❛❧♠❡♥t❡✱ ❡ss❛ ❢♦r♠✉❧❛çã♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛ ♥ã♦ é ❡st✉❞❛❞❛ ♥♦s ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦ ❞❛s ✉♥✐✈❡rs✐❞❛❞❡s ❜r❛s✐❧❡✐r❛s✱ ♦♥❞❡ s❡ ❢❛③ ♦♣çã♦ ♣❡❧❛s ❢♦r♠✉❧❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r ❡ ❍❡✐s❡♥❜❡r❣ ♣❛r❛ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳

              

            ❘❡❢❡rê♥❝✐❛s

              ❬✶❪ ❊■❙❇❊❘●✱ ❘✳❀ ❘❊❙◆■❈❑✱ ❘✳ ❋✐s✐❝❛ q✉❛♥t✐❝❛✿ ❛t♦♠♦s ♠♦❧❡❝✉❧❛s✱ s♦❧✐❞♦s✱ ♥✉❝❧❡♦s ❡ ♣❛rt✐❝✉❧❛s✳ ❬❙✳❧✳❪✿ ❈❛♠♣✉s✱ ✶✾✽✽✳

              ❬✷❪ P▲❆◆❈❑✱ ▼✳ ❩✉r ❚❤❡♦r✐❡ ❞❡s ●❡s❡t③❡s ❞❡r ❊♥❡r❣✐❡✈❡rt❡✐❧✉♥❣ ✐♠ ◆♦r♠❛❧s♣❡❝tr✉♠✳ ❬❙✳❧✳❪✿ ❆♥♥❛❧❡♥ ❞❡r P❤②s✐❦✱ ✶✾✵✵✳

              ❬✸❪ ◆❯❙❙❊◆❩❱❊■●✱ ❍✳ ❈✉rs♦ ❞❡ ❢ís✐❝❛ ❜ás✐❝❛✿ ót✐❝❛✱ r❡❧❛t✐✈✐❞❛❞❡✱ ❢ís✐❝❛ q✉â♥t✐❝❛✳ ❬❙✳❧✳❪✿ ❊❞❣❛r ❇❧ü❝❤❡r✱ ✶✾✾✽✳ ✭❈✉rs♦ ❞❡ ❢ís✐❝❛ ❜ás✐❝❛✮✳ ■❙❇◆ ✾✼✽✽✺✷✶✷✵✶✻✸✷✳

              ❬✹❪ ❊■◆❙❚❊■◆✱ ❆✳ Ü❜❡r ❡✐♥❡♥ ❞✐❡ ❊r③❡✉❣✉♥❣ ✉♥❞ ❱❡r✇❛♥❞❧✉♥❣ ❞❡s ▲✐❝❤t❡s ❜❡tr❡✛❡♥❞❡♥ ❤❡✉r✐st✐s❝❤❡♥ ●❡s✐❝❤ts♣✉♥❦t✳ ❬❙✳❧✳❪✿ ❆♥♥❛❧❡♥ ❞❡r P❤②s✐❦✱ ✶✾✵✺✳

              ❬✺❪ ❇❖❍❘✱ ◆✳ ❖♥ t❤❡ ❈♦♥st✐t✉t✐♦♥ ♦❢ ❆t♦♠s ❛♥❞ ▼♦❧❡❝✉❧❡s✳ ❬❙✳❧✳❪✿ P❤✐❧♦s♦♣❤✐❝❛❧ ▼❛❣❛✲ ③✐♥❡✱ ✶✾✶✸✳

              ❬✻❪ ❈❆❘❯❙❖✱ ❋✳❀ ❖●❯❘■✱ ❱✳ ❋ís✐❝❛ ♠♦❞❡r♥❛✿ ♦r✐❣❡♥s ❝❧áss✐❝❛s ❡ ❢✉♥❞❛♠❡♥t♦s q✉â♥t✐❝♦s✳ ❬❙✳❧✳❪✿ ❊❧s❡✈✐❡r✱ ✷✵✵✻✳ ■❙❇◆ ✾✼✽✽✺✸✺✷✶✽✼✽✼✳

              ❬✼❪ ❍❊■❙❊◆❇❊❘●✱ ❲✳ Ü❜❡r ◗✉❛♥t❡♥t❤❡♦r❡t✐s❝❤❡ ❯♠❞❡✉t✉♥❣ ❑✐♥❡♠❛t✐s❝❤❡r ❯♥❞ ▼❡❝❤❛✲ ♥✐s❝❤❡r ❇❡③✐❡❤✉♥❣❡♥✳ ❬❙✳❧✳❪✿ ❩❡✐ts❝❤r✐❢t ❢✉r P❤②s✐❦✱ ✶✾✷✺✳ ■❙❇◆ ✾✼✽✶✶✺✼✻✹✻✼✷✸✳

              ❬✽❪ P■❩❆✱ ❆✳ ❉✳ ❚✳ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❱♦❧✳ ✺✶✳ ❬❙✳❧✳❪✿ ❊❉❯❙P✱ ✷✵✵✸✳ ■❙❇◆ ✾✼✽✽✺✸✶✹✵✼✹✽✷✳ ❬✾❪ ❙❈❍❘❖❉■◆●❊❘✱ ❊✳ ◗✉❛♥t✐s✐❡r✉♥❣ ❛❧s ❊✐❣❡♥✇❡rt♣r♦❜❧❡♠✳ ✭❊rst❡ ▼✐tt❡✐❧✉♥❣✳✮✳ ❆♥♥✳

              P❤②s✳ ✭▲❡✐♣③✐❣✮✱ ✈✳ ✼✾✱ ♣✳ ✸✻✶✕✸✼✻✱ ✶✾✷✻✳ ❬✶✵❪ ❋❊❨◆▼❆◆✱ ❘✳❀ ❇❘❖❲◆✱ ▲✳ ❋❡②♥♠❛♥✬s ❚❤❡s✐s✿ ❆ ◆❡✇ ❆♣♣r♦❛❝❤ t♦ ◗✉❛♥t✉♠ ❚❤❡✲

              ♦r②✳ ❬❙✳❧✳❪✿ ❲♦r❧❞ ❙❝✐❡♥t✐✜❝✱ ✶✾✹✷✳ ■❙❇◆ ✾✼✽✾✽✶✷✺✻✼✻✸✺✳ ❬✶✶❪ ❉■❘❆❈✱ P✳ ❆✳ ❚❤❡ ▲❛❣r❛♥❣✐❛♥ ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✳ ❬❙✳❧✳❪✿ P❤②s✐❦❛❧✐s❝❤❡ ❩❡✐ts❝❤r✐❢t

              ❞❡r ❙♦✇❥❡t✉♥✐♦♥✱ ✶✾✸✸✳ ❬✶✷❪ ◆❊❚❖✱ ❏✳ ▼❡❝â♥✐❝❛ ◆❡✇t♦♥✐❛♥❛✱ ▲❣r❛♥❣✐❛♥❛ ❡ ❍❛♠✐❧t♦♥✐❛♥❛✳ ❬❙✳❧✳❪✿ ❊❞✐t♦r❛ ▲✐✈r❛r✐❛

              ❞❛ ❋ís✐❝❛✱ ✷✵✵✹✳ ■❙❇◆ ✾✼✽✽✺✽✽✸✷✺✷✻✺✳ ❬✶✸❪ ▲■❇❖❋❋✱ ❘✳ ■♥tr♦❞✉❝t♦r② ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✳ ❬❙✳❧✳❪✿ ❆❉❉■❙❖◆ ❲❊❙▲❊❨ P✉✲

              ❜❧✐s❤✐♥❣ ❈♦♠♣❛♥② ■♥❝♦r♣♦r❛t❡❞✱ ✷✵✵✸✳ ■❙❇◆ ✾✼✽✵✽✵✺✸✽✼✶✹✽✳ ❬✶✹❪ ●❖▲❉❙❚❊■◆✱ ❍✳❀ P❖❖▲❊✱ ❈✳❀ ❙❆❋❑❖✱ ❏✳ ❈❧❛ss✐❝❛❧ ▼❡❝❤❛♥✐❝s✱ ✸❡✳ ❬❙✳❧✳❪✿ ❆❞❞✐s♦♥✲

              ❲❡s❧❡② ▲♦♥❣♠❛♥✱ ■♥❝♦r♣♦r❛t❡❞✱ ✷✵✵✷✳ ■❙❇◆ ✾✼✽✵✷✵✶✻✺✼✵✷✾✳ ❬✶✺❪ ❋❊❨◆▼❆◆✱ ❘✳❀ ❍■❇❇❙✱ ❆✳ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❛♥❞ P❛t❤ ■♥t❡❣r❛❧s✿ ❊♠❡♥❞❡❞ ❊❞✐✲ t✐♦♥✳ ❬❙✳❧✳❪✿ ❉♦✈❡r P✉❜❧✐❝❛t✐♦♥s✱ ■♥❝♦r♣♦r❛t❡❞✱ ✷✵✶✷✳ ■❙❇◆ ✾✼✽✵✹✽✻✶✸✹✻✸✺✳

              ❬✶✻❪ ▲❊▼❖❙✱ ◆✳ ▼❡❝â♥✐❝❛ ❆♥❛❧ít✐❝❛✳ ❬❙✳❧✳❪✿ ▲■❱❘❆❘■❆ ❉❆ ❋■❙■❈❆✱ ✷✵✵✼✳ ■❙❇◆ ✾✼✽✽✺✽✽✸✷✺✷✹✶✳

              ❬✶✼❪ ●❘■❋❋■❚❍❙✱ ❉✳ ▼❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ❬❙✳❧✳❪✿ P❡❛rs♦♥ Pr❡♥t✐❝❡ ❍❛❧❧✱ ✷✵✶✶✳ ■❙❇◆ ✾✼✽✽✺✼✻✵✺✾✷✼✶✳

              ❬✶✽❪ ❈❖❍❊◆✲❚❆◆◆❖❯❉❏■✱ ❇✳ ❈✳ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❱♦❧✉♠❡ ✶✳ ❬❙✳❧✳❪✿ ❍❡r♠❛♥♥✳ ■❙❇◆ ✾✼✽✷✼✵✺✻✽✸✾✷✹✳

              ❬✶✾❪ ▼❊❙❙■❆❍✱ ❆✳ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✳ ❬❙✳❧✳❪✿ ❉♦✈❡r P✉❜❧✐❝❛t✐♦♥s✱ ✶✾✾✾✳ ✭❉♦✈❡r ❜♦♦❦s ♦♥ ♣❤②s✐❝s✮✳ ■❙❇◆ ✾✼✽✵✹✽✻✹✵✾✷✹✺✳

              ❬✷✵❪ ❇❯❚❑❖❱✱ ❊✳ ❋ís✐❝❛ ♠❛t❡♠át✐❝❛✳ ❬❙✳❧✳❪✿ ▲✐✈r♦s ❚é❝♥✐❝♦s ❡ ❈✐❡♥tí✜❝♦s✱ ✶✾✽✽✳ ■❙❇◆ ✾✼✽✽✺✷✶✻✶✶✹✺✺✳

              ❬✷✶❪ ❙❆❑❯❘❆■✱ ❏✳ ❏✳ ❏✳❀ ◆❆P❖▲■❚❆◆❖✱ ❏✳ ▼♦❞❡r♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✳ ❬❙✳❧✳❪✿ ❆❉❉■✲ ❙❖◆ ❲❊❙▲❊❨ P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥② ■♥❝♦r♣♦r❛t❡❞✱ ✷✵✶✵✳ ■❙❇◆ ✾✼✽✵✽✵✺✸✽✷✾✶✹✳

              ❬✷✷❪ ❲❊❇❊❘✱ ❍✳❀ ❆❘❋❑❊◆✱ ●✳ ❊ss❡♥t✐❛❧ ▼❛t❤❡♠❛t✐❝❛❧ ▼❡t❤♦❞s ❢♦r P❤②s✐❝✐sts✳ ❬❙✳❧✳❪✿ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ✷✵✵✹✳ ■❙❇◆ ✾✼✽✵✶✷✵✺✾✽✼✼✾✳

              ❬✷✸❪ ❑▲❊■◆❊❘❚✱ ❍✳ P❛t❤ ■♥t❡❣r❛❧s ✐♥ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s✱ ❙t❛t✐st✐❝s✱ ❛♥❞ P♦❧②♠❡r P❤②s✐❝s✳ ❬❙✳❧✳❪✿ ❲♦r❧❞ ❙❝✐❡♥t✐✜❝ P✉❜❧✐s❤✐♥❣ ❈♦♠♣❛♥②✱ ■♥❝♦r♣♦r❛t❡❞✱ ✶✾✾✺✳ ■❙❇◆ ✾✼✽✾✽✶✵✷✶✹✼✷✷✳

              ❬✷✹❪ ❍❆❲❑■◆●✱ ❙✳ ❖ ✉♥✐✈❡rs♦ ♥✉♠❛ ❝❛s❝❛ ❞❡ ♥♦③✳ ❬❙✳❧✳❪✿ ❆r①✱ ✷✵✵✹✳ ■❙❇◆ ✾✼✽✽✺✼✺✽✶✵✶✼✵✳ ❬✷✺❪ ❘❖❉❘■●❯❊❙✱ ❆✳ ❋✳ ❆ ■♥t❡❣r❛❧ ❞❡ ❋❡②♥♠❛♥✿ ❉❛s ♦r✐❣❡♥s ❛s t❡♦r✐❛s ❞❡ ❝❛♠♣♦s ❛ t❡♠♣❡r❛t✉r❛s ✜♥✐t❛s✳ ❬❙✳❧✳❪✿ ❈❇P❋✱ ✷✵✶✵✳

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