Repositório Institucional UFC: Parametrização de potenciais para a obtenção de grafeno artificial em uma superfície de hélio líquido

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❋❮❙■❈❆ ●❘❆❉❯❆➬➹❖ ❊▼ ❋❮❙■❈❆

▲❆❯❘❆ ❆◆❚❖◆■❆ ❇❆❘❚❍ ▼❆❘❚■◆❊❩

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

▲❆❯❘❆ ❆◆❚❖◆■❆ ❇❆❘❚❍ ▼❆❘❚■◆❊❩

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

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

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥❞r❡② ❈❤❛✈❡s✳ ❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦ã♦ ▼✐❧t♦♥ P❡r❡✐r❛ ❏r✳✳

▲❆❯❘❆ ❆◆❚❖◆■❆ ❇❆❘❚❍ ▼❆❘❚■◆❊❩

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

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

❆♣r♦✈❛❞❛ ❡♠ ✵✸✴✵✼✴✷✵✶✺✳

❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆

Pr♦❢✳ ❉r✳ ❏♦ã♦ ▼✐❧t♦♥ P❡r❡✐r❛ ❏r✳ ✭❈♦♦r✐❡♥t❛❞♦r✮ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

Pr♦❢✳ ❏❡❛♥❧❡① ❙♦❛r❡s ❞❡ ❙♦✉s❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará

Biblioteca do Curso de Física

Martinez, Laura Antonia Barth

Parametrização de potenciais para obtenção de grafeno artificial em uma superfície de hélio líquido / Laura Antonia Barth Martinez. – Fortaleza, 2015.

46 f. : il. algumas color. enc.; 30 cm.

Monografia (Graduação em Física) – Universidade Federal do Ceará, Centro de Ciências, Departamento de Física, Curso de Bacharelado em Física, Fortaleza, 2015.

Orientação: Prof. Dr. Andrey Chaves.

Coorientação: Prof. Dr. João Milton Pereira Júnior. Inclui bibliografia.

1. Grafeno. 2. Hélio líquido. 3. Modelo Tight-Binding. I. Chaves, Andrey. II. Pereira Júnior, João Milton. III. Título.

❆♦ ♠❡✉ ♣❛✐✳ ✏❚❡♥❤❛ ❞♦❝❡s s♦♥❤♦s✱ ♠❛s

♥ã♦ tã♦ ❞♦❝❡s✦ ❉✉r♠❛ ❜❡♠ ❡ ❛té

❆●❘❆❉❊❈■▼❊◆❚❖❙

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❡ ♣r♦❢❡ss♦r ❆♥❞r❡② ❈❤❛✈❡s ♣♦r t❡r ❛❝❡✐t♦ ♠❡ ♦r✐❡♥t❛r ❡ t❡r ♠❡ ❞❛❞♦ ❡ss❡ tr❛❜❛❧❤♦ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❉❛✈✐ ❙♦❛r❡s ❉❛♥t❛s✱ q✉❡ t❛♥t♦ ♣❡rt♦ ❝♦♠♦ ❧♦♥❣❡ ♠❡ ❛❥✉❞♦✉ ♠✉✐t♦ ♥❛ r❡❛❧✐③❛çã♦ ❞♦ ♠❡s♠♦✳

❆♦ ♠❡✉ ❝♦♦r✐❡♥t❛❞♦r ❡ ♣r♦❢❡ss♦r ▼✐❧t♦♥ P❡r❡✐r❛ ❏r✳ ♣♦r t❡r ❛❝❡✐t♦ ♠❡ ❝♦♦r✐❡♥t❛r ♥❛ ❡s❝r✐t❛ ❡ ✜♥❛❧✐③❛çã♦ ❞❛ ♠♦♥♦❣r❛✜❛✳

❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ♠✉✐t♦ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡✱ ❱✐❝t♦r✐❛ ❞❡ ❇❛rt❤✱ ♣❡❧♦ ❝❛✲ r✐♥❤♦✱ ❛t❡♥çã♦ ❡ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❡st✉❞♦✳ ❆❞♠✐r♦ ❛ s✉❛ ❢♦rç❛ ❡ ❝♦r❛❣❡♠✳ ❆ ❊❞✉❛r❞♦ ❇❛r❜♦s❛ ❆r❛ú❥♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❛♠♦r✱ ♣❛❝✐ê♥❝✐❛✱ ❛❥✉❞❛ ❡ ❛t❡♥çã♦ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❛♥♦s✳ ❆♦s ♣r♦❢❡ss♦r❡s ❚✐t♦ ▲í✈✐♦ ❡ ❘♦❣ér✐❛ P❡r❡✐r❛✱ ♣♦r t❡r❡♠ ❞❛❞♦ s✉♣♦rt❡ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ❯❋❈✱ ♣♦r ❝♦♠♣❛rt✐❧❤❛r❡♠ ❛❧❡❣r✐❛s ❡ ❞✐✜❝✉❧❞❛❞❡s ❛♦ ❧♦♥❣♦ ❞❡ss❡s ✹ ❛♥♦s✿ ❉❛♥✐❡❧ ▲✐♥❤❛r❡s✱ ▼❛t❤❡✉s ❋❛❧❝ã♦✱ ❲❛❣♥❡r ❙❡♥❛✱ ❏♦♥❛t❤❛♥ ❙❛❧❡s✱ P❡❞r♦ ❍❡♥r✐q✉❡✱ ❱✐t♦r ◆♦❝r❛t♦✱ ❏♦ã♦ P❛✉❧♦ ◆♦❜r❡✱ ❘❛❢❛❡❧ ❋❛r✐❛s✱ ❘♦♥❞✐♥❡❧❧② ❖❧✐✈❡✐r❛✱ ❆r✐❧♦ P✐♥❤❡✐r♦✱ ▲✉❝❛s ▼✐r❛♥❞❛✱ ❉❛♥✐❡❧ ❇r✐t♦✱ ❇✐❛♥❝❛ ●♦❞✐♠✱ ❆❞r✐❛♥❛ ●♦♠❡s✱ ❊♠❛♥✉❡❧ ❋♦♥✲ t❡❧❧❡s✱ P❛❜❧♦ ❘❛♠ó♥✱ ▼✐❝❤❡❧ ❘♦❞r✐❣✉❡s✱ ●❛❜r✐❡❧ ❖❧✐✈❡✐r❛✱ ▲✉❛♥ ❱✐❡✐r❛✱ ❉✉❛rt❡ ❏♦sé✱ ❏♦ã♦ P❛✉❧♦ ◆♦❣✉❡✐r❛✱ ❉✐❡❣♦ ❋é❧✐①✱ ❘❛✈❡♥♥❛ ❘♦❞r✐❣✉❡s✱ ❲✐❧❧✐❛♠ ▼❡sq✉✐t❛✱ ❏♦r❣❡ ▼♦t❛ ❡ ▲✉❝❛s ❚❡✐①❡✐r❛✳

❆❣r❛❞❡ç♦ ❛♦s ❛❧✉♥♦s ❞❛ ♣ós✲❣r❛❞✉❛çã♦ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✿ ❉✐❡❣♦ ▲✉❝❡♥❛✱ ❉✐❡❣♦ ❘❛❜❡❧♦✱ ❍②❣♦r P✐❛❣❡t✱ ❙❛✉❧♦ ❘❡✐s✱ ❚❛t✐❛♥❛ ❆♠♦r✱ ▲❡✈✐ ▲❡✐t❡✱ ❱❛❣♥❡r ❇❡ss❛✱ ❍❡✐t♦r ❈❡r❞✐❞♦✱ ❱✐t♦r ❙❛♥t♦s✱ ❏❛♥❛í♥❛ ❙♦❜r❡✐r❛✱ ❈és❛r ❱✐❡✐r❛ ❡ ❘✐❧❞❡r P✐r❡s✳

P❡❧❛ ❧♦❣íst✐❝❛ ❞❡ ❡st✉❞♦ ❛❣r❛❞❡ç♦ ❛ P❛✉❧✐♥♦ ▼❛❝✐❡❧✱ ❏♦s❡♥✐❧❞♦ ▼❛rr❡✐r❛ ❡ ▼ár❝✐♦ ✭♠♦t♦r✐st❛ ❞♦ ✐♥t❡r❝❛♠♣✉s ▲❛❜♦♠❛r✲❯❋❈✮✳

❆♦ ❈◆Pq✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❘❊❙❯▼❖

◆ós s✉❣❡r✐♠♦s ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ ♣r♦❞✉③✐r ❣r❛❢❡♥♦ ❛rt✐✜❝✐❛❧ ❛tr❛✈és ❞❡ ❡❧étr♦♥s ❝♦♥✜♥❛❞♦s s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✳ ◆♦ s✐st❡♠❛ ♣r♦♣♦st♦✱ ❡❧étr♦♥s ♣r❡s♦s à s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✱ ❞❡✈✐❞♦ ❛ ✐♥❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛s ❝♦♥st❛♥t❡s ❞✐❡❧étr✐❝❛s ❞♦ ✈á❝✉♦ ❡ ❞♦ ❍é❧✐♦✱ ❡stã♦ s✉❜♠❡t✐❞♦s ❛ ✉♠ ❝♦♥✜♥❛♠❡♥t♦ ❧❛t❡r❛❧ ❡①tr❛ ❞❡✈✐❞♦ ❛ ❡❢❡✐t♦s ❞❡ ❝❛r❣❛✲✐♠❛❣❡♠ ✐♥❞✉③✐❞♦ ♣♦r ❡s❢❡r❛s ❞❡ ♦✉r♦ ❞❡♣♦s✐t❛❞❛s ❡♠ ✉♠ ❢♦r♠❛t♦ ❞❡ r❡❞❡ ❢❛✈♦✲❞❡✲ ♠❡❧ ❧♦❣♦ ❛❝✐♠❛ ❞♦ s✉❜str❛t♦ ♥♦ q✉❛❧ ♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❡stá ❞❡♣♦s✐t❛❞♦✳ ◆♦ss❛ ✐♥✈❡st✐❣❛çã♦ t❡ór✐❝❛ ♠♦str❛ q✉❡✱ ❛❧é♠ ❞❡ s❡r ✉♠ s✐st❡♠❛ ❛❧t❛♠❡♥t❡ ♣✉r♦✱ ✐❞❡❛❧ ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❡①✲ ♣❡r✐♠❡♥t❛❧ ❞♦s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ú♥✐❝♦s q✉❡ ♦❝♦rr❡♠ ♥♦ ❣r❛❢❡♥♦✱ ♦ s✐st❡♠❛ ♣r♦♣♦st♦ é ❛❧t❛♠❡♥t❡ ❝♦♥tr♦❧á✈❡❧✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❡❢❡t✐✈❛ ❞❡ ❋❡r♠✐✱ ♣❛râ♠❡tr♦ ❞❡ ❤♦♣♣✐♥❣ ❡ ❡s♣❛ç❛♠❡♥t♦ ❞❡ r❡❞❡ ♣♦❞❡♠ s❡r ❛❥✉st❛❞♦s ❛♣❡♥❛s ✈❛r✐❛♥❞♦ ♣❛râ♠❡tr♦s ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ q✉❡ sã♦ ❢❛❝✐❧♠❡♥t❡ ❛❞❛♣t❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳

❆❇❙❚❘❆❈❚

❲❡ s✉❣❣❡st ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② ❢♦r ♣r♦❞✉❝✐♥❣ ❛rt✐✜❝✐❛❧ ❣r❛♣❤❡♥❡ ❜② ✉s✐♥❣ tr❛♣♣❡❞ ❡❧❡❝tr♦♥s ♦♥ ❛ ❧✐q✉✐❞ ❤❡❧✐✉♠ s✉r❢❛❝❡✳ ■♥ t❤❡ ♣r♦♣♦s❡❞ s②st❡♠✱ ❡❧❡❝tr♦♥s ❛tt❛❝❤❡❞ t♦ t❤❡ ♣❧❛♥❛r ❧✐q✉✐❞ ❤❡❧✐✉♠ s✉r❢❛❝❡✱ ❞✉❡ t♦ ❞✐❡❧❡❝tr✐❝ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ ✈❛❝✉✉♠ ❛♥❞ ❤❡❧✐✉♠✱ ❛r❡ s✉❜♠✐tt❡❞ t♦ ❛♥ ❡①tr❛ ❧❛t❡r❛❧ ❝♦♥✜♥❡♠❡♥t ❞✉❡ t♦ ✐♠❛❣❡ ❝❤❛r❣❡ ❡✛❡❝ts ✐♥❞✉❝❡❞ ❜② ❣♦❧❞ s♣❤❡r❡s ❞❡♣♦s✐t❡❞ ❛s ❛ ❤♦♥❡②❝♦♠❜ ❧❛tt✐❝❡ ❥✉st ❛❜♦✈❡ t❤❡ s✉❜str❛t❡ ✐♥ ✇❤✐❝❤ t❤❡ ❧✐q✉✐❞ ❤❡❧✐✉♠ ✐s ❞❡♣♦s✐t❡❞✳ ❖✉r t❤❡♦r❡t✐❝❛❧ ✐♥✈❡st✐❣❛t✐♦♥ s❤♦✇s t❤❛t✱ ❜❡s✐❞❡s ❜❡✐♥❣ ❛ ❤✐❣❤❧② ♣✉r❡ s②st❡♠✱ ✐❞❡❛❧ ❢♦r ❡①♣❡r✐♠❡♥t❛❧ ✈❡r✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✉♥✐q✉❡ ♣❤②s✐❝❛❧ ♣❤❡♥♦♠❡♥❛ t❤❛t ♦❝❝✉r ✐♥ ❣r❛♣❤❡♥❡✱ t❤❡ ♣r♦♣♦s❡❞ s②st❡♠ ✐s ❤✐❣❤❧② ❝♦♥tr♦❧❧❛❜❧❡✱ s♦ t❤❛t ✐ts ❡✛❡❝t✐✈❡ ❋❡r♠✐ ✈❡❧♦❝✐t②✱ ❤♦♣♣✐♥❣ ❡♥❡r❣✐❡s ❛♥❞ ❧❛tt✐❝❡ s♣❛❝✐♥❣ ❝❛♥ ❜❡ ❛❞❥✉st❡❞ ❥✉st ❜② ✈❛r②✐♥❣ ❧✐q✉✐❞ ❤❡❧✐✉♠ ♣❛r❛♠❡t❡rs t❤❛t ❛r❡ ❡❛s✐❧② ❡①♣❡r✐♠❡♥t❛❧❧② t✉♥❛❜❧❡✳

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙

❋✐❣✉r❛ ✶ ✕ ■♠❛❣❡♠ ✐❧✉str❛t✐✈❛ ❞❡ ✉♠ ❣ás ❞❡ ❡❧étr♦♥s ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✳ ✭❋♦♥t❡✿ ♣ró♣r✐❛ ❛✉t♦r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❋✐❣✉r❛ ✷ ✕ ❋✉♥çã♦ ❞❡ ♦♥❞❛ ♣❛r❛ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✉♠ ❡❧étr♦♥ s♦❜r❡ ❛ s✉♣❡r✲

❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✳ ✭❋♦♥t❡✿ ❬✷✶❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❋✐❣✉r❛ ✸ ✕ P♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦a✳ ✭❋♦♥t❡✿ ❬✸✸❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ❋✐❣✉r❛ ✹ ✕ ❙✉❝❡ssã♦ ❞❡ ♣♦ç♦s q✉❛❞r❛❞♦s✳ ✭❋♦♥t❡✿ ❬✸✸❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ❋✐❣✉r❛ ✺ ✕ ❆✉t♦✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ❝♦♠❡ç❛♠ ❛ ❢♦r♠❛r ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛ ❝♦♥tí♥✉❛ à

♠❡❞✐❞❛ q✉❡ t ❛✉♠❡♥t❛ ❛ ♣❛rt✐r ❞❡ ③❡r♦✳ ✭❋♦♥t❡✿ ❬✸✸❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❋✐❣✉r❛ ✻ ✕ ❖s ✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ♣❡r♠✐t✐❞❛ ❢♦r♠❛♠ ✉♠❛ ❜❛♥❞❛ ❝♦♥tí♥✉❛ ❡♥tr❡E0−2t

❡ E0+ 2t✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❩♦♥❛ ❞❡ ❇r✉❧❧♦✉✐♥✳ ✭❋♦♥t❡✿ ❬✸✸❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

❋✐❣✉r❛ ✼ ✕ ●rá✜❝♦ ❞❛ ❡♥❡r❣✐❛ ❊ ❝♦♥t❛ ♦ ✈❡t♦r ❞❡ ♦♥❞❛ ❦ ♣❛r❛ ✉♠ ❡❧étr♦♥ ❧✐✈r❡✳ ✭❋♦♥t❡✿ ❬✸✹❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❋✐❣✉r❛ ✽ ✕ ❛✮ ❊str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❞❡ ✉♠❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✱ ❝✉❥♦ ✈❡t♦r❡s ❞❡

r❡❞❡ sã♦ ~aj✱ ♠♦str❛♥❞♦ ❛ s✉♣❡r♣♦s✐çã♦ ❞❛s ❞✉❛s s✉❜✲r❡❞❡s ❆ ❡ ❇✳ ❜✮

❊s♣❛ç♦ r❡❝í♣r♦❝♦ ❞❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ✭❋♦♥t❡✿ ❬✸✺❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ❋✐❣✉r❛ ✾ ✕ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❡ ✉♠❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ✭❋♦♥t❡✿ ❬✸✺❪✮ ✳ ✳ ✳ ✸✷ ❋✐❣✉r❛ ✶✵ ✕❈é❧✉❧❛ ❞♦ ❣r❛❢❡♥♦ ❛rt✐✜❝✐❛❧✱ ♦♥❞❡ds✱R ❡H sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ ❞✐stâ♥✲

❝✐❛ ❡♥tr❡ ❛s ❣♦t❛s ❞❡ ♦✉r♦✱ ♦ r❛✐♦ ❞❛s ❣♦t❛s ❡ ❛❧t✉r❛ ❞♦ ❜❛♥❤♦ ❞❡ ❍é❧✐♦✳ ✭❋♦♥t❡✿ ♣ró♣r✐❛ ❛✉t♦r❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❋✐❣✉r❛ ✶✶ ✕✭❛✮ ❙❡❝çã♦ tr❛♥s✈❡rs❛❧✱ ❡♠ y = 0✱ ❞♦ ♣♦t❡♥❝✐❛❧ ❛tr❛t✐✈♦ ❝❛r❣❛✲✐♠❛❣❡♠

❡♥tr❡ ♦ ❡❧étr♦♥ ❡①tr❛ ♥❛ s✉♣❡r❢í❝✐❡ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❡ ❛s ❡s❢❡r❛s ❞❡ ♦✉r♦ s❡♣❛r❛❞❛s ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛ ds✳ ✭❜✮ ❉✐stâ♥❝✐❛ ♠í♥✐♠❛ ❡♥tr❡ ♦s ♣♦ç♦s

❞♦ ♣♦t❡♥❝✐❛❧ ❡♠ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ ds ❡♥tr❡ ❛s ❡s❢❡r❛s✳ ✭❝✮ ❆❧t✉r❛ ❞❛

❜❛rr❡✐r❛ ❡♥❡r❣ét✐❝❛✱ q✉❡ s❡♣❛r❛ ❛s ❞✉❛s ❝❛✈✐❞❛❞❡s ❞♦ ♣♦t❡♥❝✐❛❧✱ ❝♦♥tr❛ ❛ ❞✐stâ♥❝✐❛ ds✳ ❚rês ✈❛❧♦r❡s ❞❡ ❜❛♥❤♦ ❞❡ ❍é❧✐♦ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞❛s✱ ❍ ❂ ✹✵

♥♠ ✭❧✐♥❤❛ ✈❡r♠❡❧❤❛ ❝❤❡✐❛✮✱ ✺✵ ♥♠ ✭❧✐♥❤❛ ❛③✉❧ tr❛❝❡❥❛❞❛✮ ❡ ✻✵ ♥♠ ✭❧✐♥❤❛ ❛♠❛r❡❧❛ ♣♦♥t✐❧❤❛❞❛✮ ❡ ♦ r❛✐♦ ❞❛ ❡s❢❡r❛ é ❘ ❂ ✽ ♥♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❋✐❣✉r❛ ✶✷ ✕❖ ♠❡s♠♦ ❞❛ ❋✐❣✉r❛ ✶✶✱ ♠❛s ❝♦♠ ❍ ❂ ✹✵ ♥♠ ❡ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ r❛✐♦✱

❋✐❣✉r❛ ✶✸ ✕❆ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛s é r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❧✐♥❤❛ ❜r❛♥❝❛ ♣♦♥t✐❧❤❛❞❛✳ ❖ ❢✉♥❞♦ ❛③✉❧ s✐❣♥✐✜❝❛ ❛ r❡❣✐ã♦ ♦♥❞❡ ♦s ❞♦✐s ❡st❛❞♦s ❡stã♦ ❛❜❛✐①♦ ❞❛ ❜❛rr❡✐r❛ ♣♦t❡♥❝✐❛❧ q✉❡ s❡♣❛r❛ ♦s ❞♦✐s ♣♦ç♦s✳ ❖ ❢✉♥❞♦ ✈❡r♠❡❧❤♦ ✐♥❞✐❝❛ ✉♠❛ r❡❣✐ã♦ ♥ã♦ ❞❡s❡❥❛❞❛✱ ♦♥❞❡ ♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡①❝✐t❛❞♦ ❡stá ❛❝✐♠❛ ❞❛ ❜❛rr❡✐r❛ ♣♦t❡♥❝✐❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❋✐❣✉r❛ ✶✹ ✕❉✐❢❡r❡♥ç❛ ❡♥❡r❣ét✐❝❛ E12 ❝♦♥tr❛ ❛ ✈❛r✐❛çã♦ ❞❛ ❛❧t✉r❛ ❞♦ ❜❛♥❤♦ ❞❡ ❍é❧✐♦

♣❛r❛ ✉♠❛ ❞✐stâ♥❝✐❛ds ❂ ✼✺ ♥♠ ✜①❛ ❡♥tr❡ ❛s ❡s❢❡r❛s✳ ❆s ❧✐♥❤❛s ❝❤❡✐❛s ❝♦♠

sí♠❜♦❧♦s✱ ❝ír❝✉❧♦s ✭✈❡r♠❡❧❤♦✮✱ q✉❛❞r❛❞♦s ✭❛♠❛r❡❧♦✮ ❡ tr✐â♥❣✉❧♦s ✭❛③✉❧✮ r❡♣r❡s❡♥t❛♠ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❘ ❂ ✽✱ ✾ ❡ ✶✵ ♥♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❋✐❣✉r❛ ✶✺ ✕❉✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❝♦♠ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛

❡♥tr❡ ❛s ❡s❢❡r❛s✳ ✭❛✲❝✮ ❆ s✐♠✉❧❛çã♦ ❞♦ ❡rr♦ ❡①♣❡r✐♠❡♥t❛❧ é ❞❡±10%♣❛r❛ ❛ ❛❧t✉r❛ ❞♦ ❜❛♥❤♦ ❞❡ ❍é❧✐♦ ✭H = 40✱ ✺✵ ❡ ✻✵ ♥♠✮✳ ✭❞✲❢✮ ❆ s✐♠✉❧❛çã♦ ❞♦ ❡rr♦ ❡①♣❡r✐♠❡♥t❛❧ é ❞❡ ±10% ♣❛r❛ ♦ r❛✐♦ ❞❛s ❡s❢❡r❛s ✭R = 8✱ ✾ ❡ ✶✵ ♥♠✮✳ ❆ ❧✐♥❤❛ ❛③✉❧ ❝❤❡✐❛ ✭✈❡r♠❡❧❤❛✮ ❝♦♠ tr✐â♥❣✉❧♦s ♣❛r❛ ❜❛✐①♦ ✭❝✐♠❛✮ r❡♣r❡s❡♥t❛♠ ✉♠ ❡rr♦ ❞❡ ✲✶✵✪ ✭✰✶✵✪✮ ❡ ❛ ❧✐♥❤❛ ❛♠❛r❡❧❛ ❝❤❡✐❛ ❝♦♠ ❝ír❝✉❧♦s r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❝♦rr❡t♦ ♣❛r❛ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛râ♠❡tr♦✳ ❆s ❧✐♥❤❛s tr❛❝❡❥❛❞❛s s❡♣❛r❛♠ ❛s r❡❣✐õ❡s ♥❛s q✉❛✐s ❛s ❞✉❛s ❡♥❡r❣✐❛s ❡stã♦ ❝♦♥✜♥❛❞❛s ✭❛ ❞✐r❡✐t❛✮ ❡ r❡❣✐ã♦ ♦♥❞❡✱ ♣❡❧♦ ♠❡♥♦s✱ E2 ♥ã♦ ❡stá ❝♦♥✜♥❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

❋✐❣✉r❛ ✶✻ ✕❆ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❋❡r♠✐✱ ❧✐♥❤❛ ❛③✉❧ ❝❤❡✐❛✱ ♣❧♦t❛❞❛ ❝♦♥tr❛ ❛ ❞✐stâ♥❝✐❛ ds

❡♥tr❡ ❛s ❡s❢❡r❛s✱ ♣❛r❛ três ✈❛❧♦r❡s ❞✐st✐♥t♦s ❞❡ r❛✐♦ ✭R = 8✱ ✾ ❡ ✶✵ ♥♠✮ ❡ ❞♦✐s ✈❛❧♦r❡s ❞❡ ❛❧t✉r❛ ✭H = 40 ❡ ✺✵ ♥♠✮✳ ❆ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ ❛③✉❧ ❞❡✜♥❡ ❛ tr❛♥s✐çã♦ ❞♦ s❡❣✉♥❞♦ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ ❞❡ ❢♦r❛ ✭à ❡sq✉❡r❞❛✮ ♣❛r❛ ❞❡♥tr♦ ✭à ❞✐r❡✐t❛✮ ❞♦ ♣♦ç♦ ❞✉♣❧♦✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛ ❧✐♥❤❛ ✈❡r♠❡❧❤❛ ❝❤❡✐❛ é ✉s❛❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❋❡r♠✐ ♦❜t✐❞❛ ✉s❛♥❞♦ ❛ ❞✐stâ♥❝✐❛ ds ❡♥tr❡ ❛s ❡s❢❡r❛s ❛♦ ✐♥✈és ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ♣♦ç♦s ❞♦ ♣♦t❡♥❝✐❛❧ dr✳

❙❯▼➪❘■❖

✶✶

✶ ■◆❚❘❖❉❯➬➹❖

✶✳✶ ❍é❧✐♦ ▲✐q✉✐❞♦

❖ ❍é❧✐♦ ♣♦ss✉✐ três ✐sót♦♣♦s✿ 3_❍❡✱ 4_{❍❡ ❡} 6_{❍❡✳ ❖s ❞♦✐s ♣r✐♠❡✐r♦s tê♠ ♠❡♥♦r❡s}

♣♦♥t♦s ❞❡ ❡❜✉❧✐çã♦ ❡ s♦❧✐❞✐✜❝❛çã♦ ❝♦♠♣❛r❛❞♦ ❝♦♠ ♦✉tr❛s s✉❜stâ♥❝✐❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❤✐❞r♦❣ê♥✐♦✱ ♥❡ô♥✐♦ ❡ ❡t❝ ❬✶❪✳ ❉❡♥tr❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❛ t❛❜❡❧❛ ♣❡r✐ó❞✐❝❛✱ ♦ át♦♠♦ ❞❡ ❍é❧✐♦ ❢♦✐ ♦ ú❧t✐♠♦ ❛ s❡r ❧✐q✉❡❢❡✐t♦ ❡ t❛♠❜é♠ s♦❧✐❞✐✜❝❛❞♦✳ ❊♠ ✶✾✵✽✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛❝❤❛r ♦s ♣♦♥t♦s ❞❡ ❧✐q✉❡❢❛çã♦ ❡ s♦❧✐❞✐✜❝❛çã♦ ❞♦ ✐sót♦♣♦ 4_{He✱ ❑❛♠❡r❧✐♥❣❤ ❖♥♥❡s ❞❡s❡♥✈♦❧✈❡✉}

té❝♥✐❝❛s ❞❡ r❡❢r✐❣❡r❛çã♦ ♣✐♦♥❡✐r❛s✱ ♦❜t❡♥❞♦ ❝♦♠ s✉❝❡ss♦ ♦ ❍é❧✐♦ ❡♠ s❡✉ ❡st❛❞♦ ❧íq✉✐❞♦ ❛♦ ❛❧❝❛♥ç❛r ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❞❡ ❛♣❡♥❛s ✺✳✷ ❑ ❬✷❪✳ ❆♣❡s❛r ❞❛ ❢r✉str❛çã♦ ❞❡ ♥ã♦ t❡r ♦❜t✐❞♦ ê①✐t♦ ♥❛ t❡♥t❛t✐✈❛ ❞❡ s♦❧✐❞✐✜❝❛r ♦ ❍é❧✐♦✱ ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ❛♣❡♥❛s s♦❜ ♣r❡ssõ❡s ❡❧❡✈❛❞❛s✱ ❛ ♠❛r❝❛ ❛t✐♥❣✐❞❛ ♣♦r ❖♥♥❡s ❞❡✐①♦✉ ✉♠ ✐♠♣♦rt❛♥t❡ ❧❡❣❛❞♦ ♣❛r❛ ❋ís✐❝❛ ❞❡ ❜❛✐①❛s t❡♠♣❡r❛t✉r❛s✳ P♦r ❡①❡♠♣❧♦✱ ♦ ✉s♦ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❝♦♠♦ ✉♠ ✢✉✐❞♦ ❝r✐♦❣ê♥✐❝♦ ♣♦ss✐❜✐❧✐t♦✉ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❛ r❡s✐stê♥❝✐❛ ❞♦s ♠❛t❡r✐❛✐s s♦❜ t❡♠♣❡r❛t✉r❛s ❡①tr❡♠❛♠❡♥t❡ ❜❛✐①❛s✳ ❊ss❡ ❡st✉❞♦ ❝✉❧♠✐♥♦✉ ♥❛ ❞❡s❝♦❜❡rt❛ ❞♦ ❢❡♥ô♠❡♥♦ ❞❛ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡ ❡♠ ✶✾✶✶✱ r❡♥❞❡♥❞♦✲ ❧❤❡ ♦ Prê♠✐♦ ◆♦❜❡❧ ❡♠ ❋ís✐❝❛ ❞❡ ✶✾✶✸ ❬✸❪✳ ❊♠ ✶✾✷✻ s❡✉ ❞✐s❝í♣✉❧♦ ❲✐❧❧❡♠ ❍❡♥❞r✐❦ ❑❡❡s♦♠ ❝♦♥s❡❣✉✐✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ s♦❧✐❞✐✜❝❛r ♦ ❤é❧✐♦ ❬✹❪✳

◗✉❛s❡ três ❞é❝❛❞❛s ❞❡♣♦✐s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❡♠ ✶✾✸✽✱ P✳▲✳ ❑❛♣✐t✐③❛ ❞❡s❝♦✲ ❜r✐✉ q✉❡ q✉❛♥❞♦ s✉❜♠❡t✐❞♦ ❛ t❡♠♣❡r❛t✉r❛s ✐♥❢❡r✐♦r❡s ❛ ✷✳✶✼ ❑✱ ♦ ❍é❧✐♦ ❧íq✉✐❞♦ s♦❢r✐❛ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❡ ♣❛ss❛✈❛ ❛ ❡①✐❜✐r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✢✉✐r ♣♦r t✉❜♦s ❡ ❝❛♣✐❧❛r❡s s❡♠ r❡s✐s✲ tê♥❝✐❛ ❛♦ ♠♦✈✐♠❡♥t♦✳ ❆ ❡ss❛ ❝❛♣❛❝✐❞❛❞❡ ❢♦✐ ❞❛❞♦ ♥♦♠❡ ❞❡ s✉♣❡r✢✉✐❞❡③ ❬✺❪✳ ❆❞✈✐♥❞♦ ❞❛ ♥❛t✉r❡③❛ q✉â♥t✐❝❛ ❞❛ ♠❛tér✐❛✱ ♦ ❢❡♥ô♠❡♥♦ ❞❛ s✉♣❡r✢✉✐❞❡③ ❢❡③ ♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❞❡✐①❛r ❞❡ s❡r ❛♣❡♥❛s ✉♠ ✢✉✐❞♦ ❝r✐♦❣ê♥✐❝♦✱ ♣❛ss❛♥❞♦ ❛ s❡r t❛♠❜é♠ ❛❧✈♦ ❞❡ ❞✐✈❡rs♦s ❡st✉❞♦s t❡ór✐✲ ❝♦s✳ ❉❡ ❢❛t♦✱ ❛ ❡①♣❧✐❝❛çã♦ ❞♦ ❢❡♥ô♠❡♥♦ ❞❛ s✉♣❡r✢✉✐❞❡③ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ r❡♥❞❡✉ t❛♠❜é♠✱ ❡♠ ✶✾✻✷✱ ♦ Prê♠✐♦ ◆♦❜❡❧ ❞❡ ❋ís✐❝❛ ❛ ▲❡✈ ▲❛♥❞❛✉ ❬✻✱ ✼❪✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♣r♦♣♦st❛ ✉t✐❧✐✲ ③❛❞❛ ❡♠ ✶✾✸✽ ♣♦r ❋✳ ▲♦♥❞♦♥ ❬✽✱ ✾❪ ♣❛r❛ ❡①♣❧✐❝❛r ❛ s✉♣❡r✢✉✐❞❡③ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❝♦♠♦ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❞✐st♦r❝✐❞♦ ♣❡❧❛s ❢♦rt❡s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s át♦♠♦s ❞❡ ❍é❧✐♦✱ ❢♦✐ ✉t✐❧✐③❛❞❛ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ♠✐❝r♦s❝ó♣✐❝❛ ❞❛ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡ ♣❡❧♦s ❢ís✐❝♦s ❇❛r❞❡❡♥✱ ❈♦♦♣❡r ❡ ❙❝❤r✐✛❡r ✭❚❡♦r✐❛ ❇❈❙✮ ❬✶✵❪✳ ✶✳✶✳✶ ❊❧étr♦♥s ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦

✶✷

❡st❛❞♦s ❞❡ s✉♣❡r❢í❝✐❡s ♦r✐✉♥❞♦s ❞❡ ❡❢❡✐t♦s ❞❡ ❝❛r❣❛ ✐♠❛❣❡♠ ❡ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s át♦♠♦s ❞❡ ❍é❧✐♦✱ ❡❧étr♦♥s ❝♦♥✜♥❛❞♦s ❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦ t❡♠ s✐❞♦ ✉♠ tó♣✐❝♦ ❛♠♣❧❛✲ ♠❡♥t❡ ✐♥✈❡st✐❣❛❞♦ t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦ q✉❛♥t♦ ❡①♣❡r✐♠❡♥t❛❧✱ s❡♥❞♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ s✐♠✉❧❛r ❡❧❡♠❡♥t♦s ❡❧❡trô♥✐❝♦s ❝♦♠♦ ♣♦♥t♦s✱ ✜♦s ❡ ❛♥é✐s q✉â♥t✐❝♦s ❬✶✹❪✳ ❆❧é♠ ❞✐ss♦✱ s✐st❡♠❛s ❞❡ ❡❧étr♦♥s ❝♦♥✜♥❛❞♦s ❡♠ ♣♦♥t♦s q✉â♥t✐❝♦s s♦❜r❡ s✉♣❡r❢í❝✐❡s ❞❡ ❍é❧✐♦ ❧✐q✉✐❞♦ sã♦ ❡①❛✉st✐✈❛♠❡♥t❡ ♣r♦♣♦st♦s ❝♦♠♦ ✉♠ ❞♦s ♠❡❧❤♦r❡s ❝❛♥❞✐❞❛t♦s ♣❛r❛ ♦s q✉❜✐ts ❞♦s ❢✉✲ t✉r♦s ❝♦♠♣✉t❛❞♦r❡s q✉â♥t✐❝♦s✱ ♦♥❞❡ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡①❝✐t❛❞♦ ❢✉♥❝✐♦♥❛r✐❛♠ ❝♦♠♦ ♦ ✵ ❡ ♦ ✶ ❞♦s ❜✐ts ❡♠ ❝♦♠♣✉t❛❞♦r❡s ❛t✉❛✐s ❬✶✺✱ ✶✻❪✳

❊♠ ❛❞✐çã♦ às ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ♣❛r❛ ♦ ❝♦♠♣✉t❛❞♦ q✉â♥t✐❝♦✱ ❡ss❡s s✐st❡♠❛s sã♦ ❡①❝❡❧❡♥t❡s ♣❛r❛ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s q✉â♥t✐❝♦s q✉❡ ♦❝♦rr❡♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ♠❛t❡r✐❛✐s s❡♠✐❝♦♥❞✉t♦r❡s✳ ❉❡ ❢❛t♦✱ ❤á ♠✉✐t♦s ❛♥♦s✱ ♦ ❍é❧✐♦ ❧íq✉✐❞♦ é ❝♦♥s✐✲ ❞❡r❛❞♦ ✉♠ ❜♦♠ ❛♠❜✐❡♥t❡ ♣❛r❛ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ ❞❡✈✐❞♦ à ❛✉sê♥❝✐❛ ❞❡ ✐♠♣✉r❡③❛s✱ ❡s❝❛❧❛❜✐❧✐❞❛❞❡✱ ❛❧t❛ ❝♦♥tr♦❧❛❜✐❧✐❞❛❞❡✱ ❛❧t❛ ♠♦❜✐❧✐❞❛❞❡ ❡❧❡trô♥✐❝❛ ❡ t❛♠❜é♠ ❛ ✉♠ ❡❧❡✈❛❞♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦✱ ♦ ♠❛✐♦r ❝♦♥❤❡❝✐❞♦ ❡♠ t♦❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ ❬✶✼❪✳ ❊♠ ❣❡r❛❧✱ ❞❡✈✐❞♦ à ❜❛✐①❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✱ s✐st❡♠❛s ❞❡ ❡❧étr♦♥s s♦❜r❡ s✉♣❡r❢í❝✐❡s ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦ sã♦ r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❞❡ s❡ ♠♦❞❡❧❛r✱ ♦ q✉❡ t❡♠ ✐♠♣✉❧s✐♦♥❛❞♦ ❞✐✈❡r✲ s❛s ♣r♦♣♦st❛s ♣❛r❛ ✈❡r✐✜❝❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ✐♠♣♦rt❛♥t❡s ❢❡♥ô♠❡♥♦s ♦❜s❡r✈❛❞♦s ♥❛ ♥❛t✉r❡③❛ q✉â♥t✐❝❛ ❞❛ ♠❛tér✐❛✳ ❆s ❞❡s✈❛♥t❛❣❡♥s ❞♦ ❤é❧✐♦ sã♦✱ ❜❛✐①❛ ❡❧❡tr♦♥❡❣❛t✐✈✐❞❛❞❡ ❡ ❛❧t♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♦♥✐③❛çã♦✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ❡❧étr♦♥s ❝♦♥✜♥❛❞♦s ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧✐q✉✐❞♦ ❢♦r❛♠ ♣r♦♣♦st♦s ♣❛r❛ s✐♠✉❧❛r ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ♦♥❞❡ ❡s❢❡r❛s ❞❡ ♦✉r♦ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♣❛r❛ ✐♠♣♦r ✉♠ ❝♦♥✜♥❛♠❡♥t♦ ❧❛t❡r❛❧ ♣❡r✐ó❞✐❝♦ ❛♦s ❡❧étr♦♥s ❝♦♥✜♥❛❞♦s s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❬✶✽❪✳

✶✳✶✳✷ ❊❧étr♦♥s ❡♠ ✉♠ s✐st❡♠❛ ✷❉ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦

❖ ❝♦♥✜♥❛♠❡♥t♦ ❞❡ ❡❧étr♦♥s s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ ♣♦❞❡ s❡r ❡①✲ ♣❧✐❝❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♦ ❡❧étr♦♥ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ♣♦❧❛r✐③❛ ♦ ❍é❧✐♦ ❧íq✉✐❞♦✱ ❣❡r❛♥❞♦ ✉♠ ♣♦t❡♥❝✐❛❧ ❛tr❛t♦r ❝❛r❣❛✲✐♠❛❣❡♠ q✉❡ ❛♣r✐s✐♦♥❛ ♦ ❡❧étr♦♥ ♥❛ s✉♣❡r❢í❝✐❡✳ ❖ ♣r✐♥❝í♣✐♦ ❞❛ ❡①❝❧✉sã♦ ❞❡ P❛✉❧✐♥❣ ❣❛r❛♥t❡ q✉❡ ♦ ❡❧étr♦♥ ♥ã♦ ♣❡♥❡tr❛ ♥♦ ❍é❧✐♦ ❧íq✉✐❞♦✱ ♦r✐❣✐♥❛♥❞♦ ✉♠❛ ❜❛rr❡✐r❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❞❡ ✶ ❡❱ ❬✶✾❪✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ❍é❧✐♦ ♦❝✉♣❛ ✉♠❛ r❡❣✐ã♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❛❧t✉r❛z _≤0✱ ♦ ♣♦t❡♥❝✐❛❧ ❞♦ ❡❧étr♦♥ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ z ❞❛ s✉♣❡r❢í❝✐❡ é ❞❛❞❛ ♣♦r✿

V(z) = (

−Λ/z ✱ s❡ z >0 V0 ✱ s❡ z <0,

✭✶✳✶✮

♦♥❞❡ Λ = e2_{(ǫ−1)/4(ǫ + 1)✱ ❝♦♠ ǫ = 1,057 é ❛ ❝♦♥st❛♥t❡ ❞✐❡❧étr✐❝❛ ❞♦ ❍é❧✐♦ s♦❜r❡ ❛}

✶✸

hélio líquido

+ + + + + + + + + +

e

-❋✐❣✉r❛ ✶✿ ■♠❛❣❡♠ ✐❧✉str❛t✐✈❛ ❞❡ ✉♠ ❣ás ❞❡ ❡❧étr♦♥s ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✳ ✭❋♦♥t❡✿ ♣ró♣r✐❛ ❛✉t♦r❛✮

❆ ❜❛rr❡✐r❛ ♣♦t❡♥❝✐❛❧ r❡♣✉❧s✐✈❛ V0 é ♠✉✐t♦ ❣r❛♥❞❡ ❡♠ ❝♦♠♣❛r❛çã♦ ❛♦ ♣♦t❡♥❝✐❛❧

❛tr❛t♦r ❝❛r❣❛✲✐♠❛❣❡♠ ❡ ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s t♦♠❛r V0 = ∞✳ ▲♦❣♦✱ ♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛

♥❛ ❞✐r❡çã♦z é s✐♠✐❧❛r ❝♦♠ ❛♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦✱

En =−

Λ2

2n2Ry, ✭✶✳✷✮

♦♥❞❡n = 1,2. . . é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡Ry = 13.6 ❡❱ é ❛ ❝♦♥st❛♥t❡ ❞❡ ❘②❞❜❡r❣✳

❆ s❡♣❛r❛çã♦ ❡♥tr❡ ♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡①❝✐t❛❞♦ ❡ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ é ❞❡E2−E1 = 5.6❑✱

♣❛r❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣❡r❛t✉r❛ ❞❡0.1❑ ❛ 1.0❑✳ ❖ ❡❧étr♦♥ ♥♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ t❡♠ ❡♥❡r❣✐❛ ❞❡ E1 =−7.5 ❑✳ ❆s ❡♥❡r❣✐❛s ♣♦ss✉❡♠ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✿ T =E/kB✱

♦♥❞❡kB é ❛ ❝♦♥st❛♥t❡ ❞❡ ❇♦❧③♠❛♥♥✳ ❯♠❛ ✈❡③ q✉❡ ❛ s❡♣❛r❛çã♦ ❡♥tr❡ ♦s ♥í✈❡✐s ❡♥❡r❣ét✐❝♦s

é ❣r❛♥❞❡ ❡♠ ❝♦♠♣❛r❛çã♦ à ❡♥❡r❣✐❛ ❞♦ ❡❧étr♦♥ ♥❡ss❛s t❡♠♣❡r❛t✉r❛s✱ ♦ s✐st❡♠❛ t♦r♥❛✲s❡ q✉❛s❡ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ❆ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡ss❡ s✐st❡♠❛ é ❞❛❞❛ ♣♦r✿

Φ1(z) =

2 a3/2ze

−z/a_{, ✭✶✳✸✮}

♦♥❞❡✱ a=a0/Λ = 7.6 ♥♠ é ♦ r❛✐♦ ❞❡ ❇♦❤r ❡❢❡t✐✈♦ ❡ a0 = 0.0529 ♥♠ é ♦ r❛✐♦ ❞❡ ❇♦❤r✳ ❆

❞✐stâ♥❝✐❛ ♠é❞✐❛ ❡♥tr❡ ♦ ❡❧étr♦♥ ♥♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❛ s✉♣❡r❢í❝✐❡ ❞♦ ❍é❧✐♦ ❧íq✉✐❞♦ é ❞❡

hz1i= 3a/2 = 11.4 ♥♠✳ P❛r❛ ♦ s❡❣✉♥❞♦ ❡st❛❞♦ ❡①❝✐t❛❞♦ éhz2i= 45.5 ♥♠✳ ◆❛ ❋✐❣✉r❛ ✷ é

❛♣r❡s❡♥t❛❞❛ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♣❛r❛ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✉♠ ❡❧étr♦♥ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦ ❬✷✵❪✳

✶✹

❋✐❣✉r❛ ✷✿ ❋✉♥çã♦ ❞❡ ♦♥❞❛ ♣❛r❛ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✉♠ ❡❧étr♦♥ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ❍é❧✐♦ ❧íq✉✐❞♦✳ ✭❋♦♥t❡✿ ❬✷✶❪✮

❡①tr❛s ♥♦ ❍é❧✐♦ ❧íq✉✐❞♦ ❞❡✈❡ s✐♠✉❧❛r ❛ ✐♥t❡r❛çã♦ ♣r❡s❡♥t❡ ♥♦ ❣r❛❢❡♥♦✱ ❡♥tr❡ ♦s ❡❧étr♦♥s ♥❛ ❜❛♥❞❛ ❞❡ ❝♦♥❞✉çã♦ ❡ ♦s át♦♠♦s ❞❡ ❝❛r❜♦♥♦✳

✶✳✷ ●r❛❢❡♥♦

❖ ❣r❛❢❡♥♦ é ✉♠❛ ♠♦♥♦❝❛♠❛❞❛ ♣❧❛♥❛ ❞❡ át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ❛rr❛♥❥❛❞♦s ❡♠ ✉♠❛ ❡str✉t✉r❛ ❞♦ t✐♣♦ ❤❡①❛❣♦♥❛❧ ❝♦♠ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ 0,142 ♥♠ ❡♥tr❡ ♦s át♦♠♦s✳ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ▲✐♥✉s P❛✉❧✐♥❣ ❞♦ ❝❛r❜♦♥♦ é 1s2_,2s2_,2p2_{✱ ♠❛s ♥❛ ❢♦r♠❛ ❞❡ ❣r❛❢❡♥♦✱ ♦s}

át♦♠♦s ❞❡ ❝❛r❜♦♥♦ ♣♦ss✉❡♠ ✉♠❛ ❤✐❜r✐❞✐③❛çã♦ ❞♦ t✐♣♦sp2_{✳ ■ss♦ ❝❛r❛❝t❡r✐③❛ três ❧✐❣❛çõ❡s}

s✐❣♠❛ ✭σ✮ ❝♦♠ â♥❣✉❧♦ ❞❡ ✶✷✵◦ _{❡♥tr❡ s✐ ❡ ✉♠❛ ❧✐❣❛çã♦ ♣✐ ✭π✮✳ ❆s ❧✐❣❛çõ❡s σ ❢♦r♠❛♠ ✉♠} ♣❧❛♥♦ ❡ ❛ ❧✐❣❛çã♦π é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♠❡s♠♦✱ s❡♥❞♦ ❡st❛ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ tr❛♥s♣♦rt❡ ❞❡ ❡❧étr♦♥s ❛♦ ❧♦♥❣♦ ❞❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦ ❣❡r❛♥❞♦ ❛s ❜❛♥❞❛s ❞❡ ❝♦♥❞✉çã♦✳ P♦rt❛♥t♦✱ ♦ ❣r❛❢❡♥♦ ♣♦ss✉✐ ✉♠❛ ❣❡♦♠❡tr✐❛ ❞♦ t✐♣♦ tr✐❣♦♥❛❧ ♣❧❛♥❛✱ ♦ q✉❡ ❥✉st✐✜❝❛ ♦ s❡✉ ❢♦r♠❛t♦ ❤❡①❛❣♦♥❛❧ ✭❢❛✈♦✲❞❡✲♠❡❧✮✳

❆♥t❡r✐♦r♠❡♥t❡ à s✉❛ ❞❡s❝♦❜❡rt❛✱ ❛❝r❡❞✐t❛✈❛✲s❡ q✉❡ ❡str✉t✉r❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ❝♦♠♦ ♦ ❣r❛❢❡♥♦✱ ♥ã♦ ♣♦❞❡r✐❛♠ ❡①✐st✐r✱ ♣♦✐s✱ ♥❛ ♥❛t✉r❡③❛✱ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ss❡s ❝r✐st❛✐s ❡①✐❣✐✲ r✐❛ ❛❧t❛s t❡♠♣❡r❛t✉r❛s ❬✷✷❪✳ ▲♦❣♦✱ ❡str✉t✉r❛s ❞❡ ❜❛✐①❛s ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡✱ ❝♦♠♦ ♦ ❣r❛❢❡♥♦✱ ❡r❛♠ ❝♦♥s✐❞❡r❛❞♦s ✐♥stá✈❡✐s ♣♦r s✉❛s ❛❣✐t❛çõ❡s tér♠✐❝❛s ❡ s❡ ❞❡❝♦♠♣♦r✐❛♠ ❡♠ ❡str✉t✉r❛s tr✐❞✐♠❡♥s✐♦♥❛✐s✱ t❡r♠♦❞✐♥❛♠✐❝❛♠❡♥t❡ ♠❛✐s ❢❛✈♦rá✈❡✐s✳ ❆❧é♠ ❞✐ss♦✱ ♥ã♦ s❡ ❛❝r❡❞✐t❛✈❛ q✉❡ t❛✐s ❡str✉t✉r❛s✱ s❡ ❡①✐st✐ss❡♠✱ ♣♦❞❡r✐❛♠ s❡r ♦❜s❡r✈❛❞❛s ✭❚❡♦r❡♠❛ ❞❡ ▼❡r♠✐♥✲❲❛❣♥❡r ❬✷✸❪✮✳ ❈♦♥t✉❞♦✱ ❞❡s❝♦❜r✐✉✲s❡ q✉❡ ♦ ❣r❛❢❡♥♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♣♦r ♠✐❝r♦s❝♦♣✐❛ ó♣t✐❝❛ ❝♦♠✉♠✱ ♣♦r ✉♠ ❡❢❡✐t♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ❞❡♣♦s✐t❛❞♦ s♦❜r❡ ✉♠ s✉❜str❛t♦ ❞❡ s✐❧í❝✐♦ ❬✷✷❪✳

✶✺

✶✾✹✼✱ ♣❡❧♦ ❝✐❡♥t✐st❛ P✳❘✳ ❲❛❧❧❛❝❡✳ ❊❧❡ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝❛❧❝✉❧❛r✱ ♣❡❧♦ ♠♦❞❡❧♦ ❚✐❣❤t✲ ❇✐♥❞✐♥❣✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❣r❛✜t❡ ✭❛rr❛♥❥♦ ❞❡ ❣r❛❢❡♥♦s ❡♠♣✐❧❤❛❞♦s✮ ❬✷✹❪✳

❊♠ ♦✉t✉❜r♦ ❞❡ ✷✵✶✵✱ ♦ Prê♠✐♦ ◆♦❜❡❧ ❞❡ ❋ís✐❝❛ ❢♦✐ ❝♦♥❝❡❞✐❞♦ ❛♦s ❢ís✐❝♦s ❑♦♥s✲ t❛♥t✐♥ ◆♦✈♦s❡❧♦✈ ❡ ❆♥❞r❡ ●❡✐♠ ♣❡❧❛ s✐♥t❡t✐③❛çã♦ ❞❡ ♠♦♥♦❝❛♠❛❞❛s ❞❡ ❣r❛✜t❡ ❬✷✺❪✳ ❊ss❛s ♠♦♥♦❝❛♠❛❞❛s ❞❡ ❣r❛✜t❡ ♣❛ss❛r❛♠ ❛ s❡r ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❣r❛❢❡♥♦✳ ❆ ♣❛rt✐r ❞❡ss❡ tr❛✲ ❜❛❧❤♦✱ t❡♠✲s❡ r❡❛❧✐③❛❞♦ ❞✐✈❡rs♦s ❡st✉❞♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✱ ♠❡❝â♥✐❝❛s✱ ó♣t✐❝❛s✱ tér♠✐❝❛s ❡ q✉í♠✐❝❛s ❞♦ ❣r❛❢❡♥♦✱ ❛ ✜♠ ❞❡ ✉t✐❧✐③á✲❧♦ ❡♠ ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s✳

❯♠❛ ♣r♦♣r✐❡❞❛❞❡ t❡ór✐❝❛ ✐♥t❡r❡ss❛♥t❡ q✉❡ ♦ ❣r❛❢❡♥♦ ♣♦ss✉✐ é ❛ ❞❡ s❡✉s ❡❧étr♦♥s s❡ ❝♦♠♣♦rt❛r❡♠ ❝♦♠♦ ♣❛rtí❝✉❧❛s r❡❧❛t✐✈íst✐❝❛s ❞❡ s♣✐♥ ✶✴✷✱ s❡♠ ♠❛ss❛ ❡ ♦❜❡❞❡❝❡♥❞♦ ❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝✳ ■ss♦ ❛❥✉❞❛ ❛ ❡♥t❡♥❞❡r ❢❡♥ô♠❡♥♦s ❝♦♠♦ ♦ ❊❢❡✐t♦ ❍❛❧❧ ◗✉â♥t✐❝♦ ❆♥ô♠❛❧♦ ❡ ❚✉♥❡❧❛♠❡♥t♦ ❞❡ ❑❧❡✐♥ ❬✷✷✱ ✷✻✱ ✷✼❪✳

✶✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❣r❛❢❡♥♦

❋❛r❡♠♦s✱ ❛ s❡❣✉✐r✱ ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❣r❛❢❡♥♦ ❡✱ ♠❛✐s ❛❞✐❛♥t❡✱ ❞❡ s✉❛s ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s✳

✶✳ Pr♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s✿ ♣♦ss✉✐ ❛❧t❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡❧étr✐❝❛ q✉❛♥❞♦ ❛♣❧✐❝❛❞❛ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❡①t❡r♥❛ ♦✉ q✉❛♥❞♦ ✐♥t❡r❛❣❡ ❝♦♠ ♦✉tr❛s ♠♦❧é❝✉❧❛s ♥❛ s✉❛ s✉♣❡r❢í❝✐❡✳ ❆ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ♥♦ ❣r❛❢❡♥♦ s✉r❣❡ ❛tr❛✈és ❞❛s ❧✐❣❛çõ❡s σ✳ ❆ ❛❧t❛ ♠♦❜✐❧✐❞❛❞❡ ❡❧❡trô♥✐❝❛ s❡ ❞á ❛tr❛✈és ❞❛s ❧✐❣❛çõ❡s π✱ ❝♦♠ ✉♠ ✈❛❧♦r ♠á①✐♠♦ ❞❡ 200.000 ❝♠2_✴❱s

❬✷✽❪✳

✷✳ Pr♦♣r✐❡❞❛❞❡ ♠❡❝â♥✐❝❛s✿ é ✉♠ ♠❛t❡r✐❛❧ ♠✉✐t♦ ❢♦rt❡ ❡ r❡s✐st❡♥t❡✳ ❙✉❛ r❡s✐stê♥❝✐❛ à t❡♥sã♦ é ❞❡ 130 ●P❛✳ ❖ ❣r❛❢❡♥♦ t❛♠❜é♠ ♣♦ss✉✐ ❣r❛♥❞❡ ♠ó❞✉❧♦ ❞❡ ❡❧❛st✐❝✐❞❛❞❡ s❡♥❞♦ ❝❛♣❛③ ❞❡ ♠❛♥t❡r s❡✉ t❛♠❛♥❤♦ ♦r✐❣✐♥❛❧ ❛♣ós s❡r t❡♥s✐♦♥❛❞♦ ❬✷✽❪✳

✸✳ Pr♦♣r✐❡❞❛❞❡s ó♣t✐❝❛s✿ ✉♠❛ ❢♦❧❤❛ ❞❡ ❣r❛❢❡♥♦ ❛❜s♦r✈❡ ✷✱✸✪ ❞❛ ❧✉③ ✐♥❝✐❞❡♥t❡✳ ❆ ❛❜✲ s♦rçã♦ ❛✉♠❡♥t❛ ❧✐♥❡❛r♠❡♥t❡ ❝♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❝❛♠❛❞❛s✳❬✷✾❪

✹✳ Pr♦♣r✐❡❞❛❞❡ tér♠✐❝❛s✿ t❡♠ ❛❧t❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛ ♣♦r ❝❛✉s❛ ❞❛s ❧✐❣❛çõ❡s ❝♦✈❛✲ ❧❡♥t❡s ❡♥tr❡ ♦s át♦♠♦s ❞❡ ❝❛r❜♦♥♦✳ ❋♦✐ ♠❡❞✐❞♦ ❝♦♠ ♠✐❝r♦✲❘❛♠❛♥ ❝♦♥❢♦❝❛❧ ✉♠ ✈❛❧♦r ❞❡ ❛té ✺✸✵✵ ❲✴♠❑ ❬✷✻❪✳

✶✻

✶✳✷✳✷ ❆♣❧✐❝❛çõ❡s ❞♦ ❣r❛❢❡♥♦

❆❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐t❛s ❛♥t❡r✐♦r♠❡♥t❡ ♣♦❞❡♠ ❧❡✈❛r ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❢✉t✉r❛✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦ ❬✷✽❪✿

✶✳ ♥❛ ♦♣t♦❡❧❡trô♥✐❝❛✱ ❡♠ ✉♠ ❢✉t✉r♦ ♣ró①✐♠♦ ♣♦❞❡r❡♠♦s ✉s❛r ❣r❛❢❡♥♦ ❡♠ t♦✉❝❤✲s❝r❡❡♥✱ ▲❈❉✱ ❞✐♦❞♦s ❡ ❡t❝✳

✷✳ ❡♠ ♠❛t❡r✐❛✐s ❛❡r♦❡s♣❛❝✐❛✐s✱ ♣♦r s❡r ✉♠ ♠❛t❡r✐❛❧ ❧❡✈❡✱ ❢♦rt❡ ❡ r❡s✐st❡♥t❡✳ ✸✳ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ❝♦♠♦ ❝❛♣❛ ♣r♦t❡t♦r❛ ❝♦♥tr❛ r❡❧â♠♣❛❣♦s✳

✹✳ ❝♦♠♦ ❞✐s♣♦s✐t✐✈♦ ❞❡ s❡❣✉r❛♥ç❛ ❝♦♠ ✉♠ s❡♥s♦r ❞❡ ❣ás✳ ✺✳ ♥❛ ♣r♦❞✉çã♦ ❞❡ ❝é❧✉❧❛s s♦❧❛r❡s✳

✶✳✸ ●r❛❢❡♥♦ ❆rt✐✜❝✐❛❧

❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡st✉❞❛r ❣r❛❢❡♥♦ ❛rt✐✜❝✐❛❧ s✉r❣❡ ♣❛r❛ ❡①♣❧♦r❛r s✐st❡♠❛s q✉❡ ❛✐♥❞❛ ♥ã♦ ♣♦❞❡♠ s❡r ❡①❡❝✉t❛❞♦s ❝♦♠ ❣r❛❢❡♥♦ ♦✉ ♣❛r❛ ❞❡s❝♦❜r✐r ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦ ❣r❛❢❡♥♦ ♥ã♦ ♣♦ss✉✐✳ ●r❛❢❡♥♦ ❛rt✐✜❝✐❛❧ ❥á ❢♦✐ ♣r♦❞✉③✐❞♦ ♣♦r ❞✐❢❡r❡♥t❡s ♠ét♦❞♦s✳ ❈✐t❛♠♦s✱ ❝♦♠♦ ❡①❡♠♣❧♦✿

• P♦r ❝♦♥✜♥❛♠❡♥t♦ ❞❡ ❢ót♦♥s ❞❡ ♠✐❝r♦♦♥❞❛s ❡♠ ✉♠❛ r❡❞❡ ❤❡①❛❣♦♥❛❧✳ ❯t✐❧✐③❛♥❞♦

❞✐s❝♦s ✭❝♦♠ ❛❧t♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦✮ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ♠❡tá❧✐❝❛ ❡ ♥♦ t♦♣♦ ❞♦s ❞✐s❝♦s ✉♠❛ s❡❣✉♥❞❛ s✉♣❡r❢í❝✐❡ ❢♦✐ ❝♦❧♦❝❛❞❛✱ ❛s ♦♥❞❛s ♥♦ ❛r ❛❝♦♣❧❛❞♦s ❝♦♠ ♦s ❞✐s❝♦s sã♦ ❡✈❛♥❡s❝❡♥t❡s✱ ❧❡✈❛♥❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❚✐❣❤t✲❇✐♥❞✐♥❣ ✜♥✐t♦ ❬✸✵❪✳

• P♦s✐❝✐♦♥❛♥❞♦ ♠♦❧é❝✉❧❛s ❞❡ ♠♦♥ó①✐❞♦ ❞❡ ❝❛r❜♦♥♦ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ ♣❛r❛ s❡r❡♠ ❛❜✲

s♦r✈✐❞❛s ♣♦r ❝♦❜r❡ ♣✉r♦✱ ✉t✐❧✐③❛♥❞♦ ❞✐❣✐t❛❧✐③❛çã♦ ❞❡ t✉♥❡❧❛♠❡♥t♦ ♠✐❝r♦s❝ó♣✐❝♦✳ ❖ ❡①♣❡r✐♠❡♥t♦ ❢♦✐ ❢❡✐t♦ ❡♠ ✉♠❛ t❡♠♣❡r❛t✉r❛ ❞❡ 4,2 ❑✳ ❈♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ♣♦✲ t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ ♥❛ s✉♣❡r❢í❝✐❡ s✉r❣✐✉ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ♥❛s ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛✱ ❢♦r♠❛♥❞♦ ✉♠ ♣♦♥t♦ ❞❡ ❉✐r❛❝ ❡ ✉♠❛ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ❬✸✶❪✳

• P❡❧♦ ❛♣r✐s✐♦♥❛♠❡♥t♦ ❞❡ ❢ér♠✐♦♥s ✉❧tr❛❢r✐♦s ❡♠ r❡❞❡s ó♣t✐❝❛s ❤❡①❛❣♦♥❛❧✳ ❖ ❣r❛❢❡♥♦

✶✼

✷ ▼➱❚❖❉❖❙ ❚❊Ó❘■❈❖❙

P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♥ós ❡♠♣r❡❣❛♠♦s ♦ ♠♦❞❡❧♦ ❚✐❣❤t✲ ❇✐♥❞✐♥❣✱ ♥♦ q✉❛❧ ❛♣❡♥❛s ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s sã♦ ❝♦♥t❛❜✐❧✐③❛❞❛s✳ P❛r❛ t❛♥t♦✱ ✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❛♣r❡s❡♥t❛r ❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ♠ét♦❞♦ ♣❛r❛ ✉♠❛ r❡❞❡ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✳ ❆ s❡❣✉✐r✱ ❡①♣❛♥❞✐♠♦s ♦ ♠ét♦❞♦ ♣❛r❛ ✉♠❛ r❡❞❡ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠♦ ❛ ❞♦ ❣r❛❢❡♥♦✳

✷✳✶ ▼♦❞❡❧♦ ❚✐❣❤t✲❇✐♥❞✐♥❣ ♣❛r❛ ✉♠ s✐st❡♠❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧

❈♦♥s✐❞❡r❡ ✉♠ ♣♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦V ❡♠ ✉♠❛ ❞✐♠❡♥sã♦✱ ❝♦♠ ♣❡rí♦❞♦ a✿ V(x_± a) =V(x)✳ P❛r❛ t♦r♥❛r ♦ ♣r♦❜❧❡♠❛ ♠❛✐s r❡❛❧íst✐❝♦ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ❡❧étr♦♥ ❡♠ ✉♠❛ ❝❛❞❡✐❛ ❞❡ í♦♥s ♣♦s✐t✐✈♦s ❡s♣❛ç❛❞♦s ❡♠a ✭✈❡r ❋✐❣✉r❛ ✸✮✳

❋✐❣✉r❛ ✸✿ P♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦a✳ ✭❋♦♥t❡✿ ❬✸✸❪✮

❆ tr❛♥s❧❛çã♦ é r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ♦♣❡r❛❞♦r τ(l)✱ s❡♥❞♦ l ❛r❜✐trár✐♦✳ ❆s s✉❛s ♣r♦♣r✐❡❞❛❞❡s sã♦✿

✶✳ ➱ ✉♥✐tár✐♦✿

τ†_{(l)τ(l) = 1. ✭✷✳✶✮}

✷✳ ❈♦♠♣♦s✐çã♦ ❞❡ tr❛♥s❧❛çõ❡s✿

τ(l′_{)τ(l) = τ(l}′_{+l). ✭✷✳✷✮}

✸✳ ❚r❛♥s❧❛çã♦ ♥❛ ❞✐r❡çã♦ ♦♣♦st❛ é ♦ ♠❡s♠♦ q✉❡ ♦ ✐♥✈❡rs♦ ❞❛ tr❛♥s❧❛çã♦ ♦r✐❣✐♥❛❧✿

τ(₋l) = τ−1_{(l). ✭✷✳✸✮}

✹✳ ◗✉❛♥❞♦ l t❡♥❞❡ ❛ ③❡r♦✱ ❛ tr❛♥s❧❛çã♦ s❡ r❡❞✉③ ❛♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡✿

lim

✶✽

❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ τ(l) ❡ ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ s❡❥❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡♠ l✳ ❆ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❞❛ tr❛♥s❧❛çã♦ é✿

τ†(l)xτ(l) = x+l,

τ(l)_|x_i=_|x+l._i ✭✷✳✺✮

❋❛③❡♥❞♦l ❝♦✐♥❝✐❞✐r ❝♦♠ ♦ ❡s♣❛ç❛♠❡♥t♦ ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s✿ ✭l =a✮

τ†_{(a)V(x)τ(a) = V(x+a) = V(x). ✭✷✳✻✮} ❖ ❍❛♠✐❧t♦♥✐❛♥♦✱ H = p2_{/2m+V(x)✱ ❡♠ ❣❡r❛❧✱ ✈❛r✐❛ ♣❛r❛ ✉♠❛ tr❛♥s❧❛çã♦✳}

▼❛s ♥♦ ❝❛s♦ ❡♠ q✉❡stã♦✱ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ sã♦ ✐♥✈❛r✐❛♥t❡s ♣❛r❛ ❛ tr❛♥s❧❛çã♦ ❡♠ q✉❛❧q✉❡r ❞❡s❧♦❝❛♠❡♥t♦✳ ❆ss✐♠✱ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ s❛t✐s❢❛③✿

τ†(a)Hτ(a) =H. ✭✷✳✼✮

P❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♦♣❡r❛❞♦r tr❛♥s❧❛çã♦✱τ(a)é ✉♥✐tár✐♦✱ ✐ss♦ q✉❡r ❞✐③❡r✱ s❡✉ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ tr❛♥s♣♦st♦ ✭❤❡r♠✐t✐❛♥♦✮ é ✐❣✉❛❧ ❛♦ ✐♥✈❡rs♦✿ τ†_{(a) = τ}−1_{(a)✳ ❆♣❧✐❝❛♥❞♦ ❛ ♣r♦✲}

♣r✐❡❞❛❞❡ ❞❡ ✉♥✐❝✐❞❛❞❡ ♥❛ ❡q✉❛çã♦ ✷✳✼ t❡♠♦s✿

τ†(a)Hτ(a) = H τ−1_{(a)Hτ(a) = H}

τ(a)τ−1_{(a)Hτ(a) = τ(a)H}

Hτ(a) = τ(a)H Hτ(a)₋τ(a)H = 0,

[H, τ(a)] = 0. ✭✷✳✽✮

❖ ❍❛♠✐❧t♦♥✐❛♥♦ ❝♦♠✉t❛ ❝♦♠ ♦ ♦♣❡r❛❞♦r ❞❡ tr❛♥s❧❛çã♦✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♣♦✲ ❞❡♠ s❡r ❞✐❛❣♦♥❛❧✐③❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✉s❛♥❞♦ ❛ ♠❡s♠❛ ❜❛s❡✳ ▼❛s ❛❣♦r❛ t❡♠✲s❡ ✉♠ ♣r♦❜❧❡♠❛✿ τ(a) ♥ã♦ é ❤❡r♠✐t✐❛♥♦✱ q✉❡r ❞✐③❡r✱ s❡✉ ❝♦♥❥✉❣❛❞♦ ❝♦♠♣❧❡①♦ tr❛♥s♣♦st♦ ♥ã♦ é ✐❣✉❛❧ ❛♦ ♣ró♣r✐♦ ♦♣❡r❛❞♦r ❬τ†_{(a) = τ(a)❪✳ ❉❡st❛ ❢♦r♠❛✱ s❡✉ ❛✉t♦✈❛❧♦r é ✉♠ ♥ú♠❡r♦} ❝♦♠♣❧❡①♦ ❞❡ ♠ó❞✉❧♦ ✉♠✳

P❛r❛ ❞❡s❝♦❜r✐r ♦ ❛✉t♦✈❛❧♦r ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❡ ❞♦ ♦♣❡r❛❞♦r tr❛♥s❧❛çã♦ ✈❛♠♦s t♦r♥❛r ❛ ❜❛rr❡✐r❛ ❡♥tr❡ ❞♦✐s sít✐♦s ✈✐③✐♥❤♦s ✐♥✜♥✐t❛♠❡♥t❡ ❛❧t❛✳ ▼✉❞❛♠♦s✱ ❛ss✐♠✱ ♦ ♣♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ ♣❛r❛ ✉♠❛ s✉❝❡ssã♦ ❞❡ ♣♦ç♦s q✉❛❞r❛❞♦s ✐♥✜♥✐t♦s ❝♦♠ ❡s♣❛ç❛♠❡♥t♦a ✭❋✐❣✉r❛ ✹✮✳ ◆❡ss❡ ❝❛s♦ ♥ã♦ ♦❝♦rr❡ ♦ t✉♥❡❧❛♠❡♥t♦✳

✶✾

❋✐❣✉r❛ ✹✿ ❙✉❝❡ssã♦ ❞❡ ♣♦ç♦s q✉❛❞r❛❞♦s✳ ✭❋♦♥t❡✿ ❬✸✸❪✮

♥✲és✐♠♦ sít✐♦✱ ❝♦♠ ❢✉♥çã♦ ❞❡ ♦♥❞❛ |ψni✳ ◆❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✱H|ψni =E0|ψni✱ ♦

❛✉t♦✈❛❧♦r ❞❡|ψ_nié ❡♥❡r❣✐❛ E0✳ ❙✉❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛hψx|ψni é ✜♥✐t❛ ♥♦ ♥✲és✐♠♦ sít✐♦✳ ❯♠

❡st❛❞♦ s✐♠✐❧❛r ❝♦♠ ❛ ♠❡s♠❛ ❡♥❡r❣✐❛E0 é ❡♥❝♦♥tr❛❞♦ ❡♠ ♦✉tr♦ sít✐♦✱ s❡♥❞♦ ❛ss✐♠✱ t❡♠✲s❡

✉♠ ♥ú♠❡r♦ ✐♥✜♥✐t♦ ❞❡ ❡st❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ψn✱ q✉❡ ✈ã♦ ❞❡ −∞ ❛té +∞✳

❆♣❧✐❝❛♥❞♦|ψni ♥♦ ♦♣❡r❛❞♦r ❞❡ tr❛♥s❧❛çã♦ t❡♠♦s✿

τ(a)_|ψ_ni=_|ψn+1i, ✭✷✳✾✮

♦♥❞❡ ♥♦t❛♠♦s q✉❡ |ψni é ❛✉t♦❢✉♥çã♦ ❞❡ H✱ ♠❛s ♥ã♦ é ❞❡ τ(a)✱ ❣❡r❛♥❞♦ ✉♠❛ ❞❡❣❡♥❡r❡s✲

❝ê♥❝✐❛ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛✱ ♣♦✐s✿

hψn+1|H|ψn+1i=hψn|τ†(a)Hτ(a)|ψni

=_hψn|H|ψni,

=E0. ✭✷✳✶✵✮

❉❛❞❛ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❡s♣❡❝í✜❝❛✱ ✈❛♠♦s ❛✜r♠❛r q✉❡ ❡❧❛ é ✉♠ ❛✉t♦✈❡t♦r s✐♠✉❧tâ♥❡♦ ❞❡H ❡ τ(a)✿

|ψθi=

∞ X

n=−∞

einθ_|ψni, ✭✷✳✶✶✮

♦♥❞❡ θ é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧ q✉❡ ✈❛✐ ❞❡ ₋π ❛té +π✳ _|ψ_θi é ❛✉t♦✈❡t♦r ❞❡ H✱ ♣♦✐s _|ψ_ni é ❛✉t♦✈❡t♦r ❝♦♠ ❛✉t♦✈❛❧♦r E0✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ n✳ |ψθi é ❛✉t♦✈❡t♦r ❞♦ ♦♣❡r❛❞♦r

✷✵

τ(a)_|ψθi=

∞ X

n=−∞

einθ_|ψn+1i

= ∞ X

n=−∞

ei(n−1)θ_|ψni

= ∞ X

n=−∞

einθe−iθ_|ψni,

=e−iθ_|ψθi. ✭✷✳✶✷✮

❆❣♦r❛ |ψθi é ❛✉t♦✈❛❧♦r t❛♥t♦ ❞❡ H q✉❛♥t♦ ❞❡ τ(a)✱ ♣❛r❛♠❡tr✐③❛❞♦ ♣♦r ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♥✲

tí♥✉❛θ✳ ❊ ♦ ❛✉t♦✈❡t♦r ❞❡❧❡s✱E0✱ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ θ✳

❱♦❧t❛♥❞♦ á r❡❛❧✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛✱ ♠❛s ❞❡st❛ ✈❡③ ❝♦♠ t✉♥❡❧❛♠❡♥t♦ q✉â♥t✐❝♦ ♣❛r❛ ♦s sít✐♦s ✈✐③✐♥❤♦s✱ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ hψx|ψni t❡♠ ❛❣♦r❛ ✉♠❛ ❝❛✉❞❛ q✉❡ s❡ ❡st❡♥❞❡

♣❛r❛ ♦✉tr♦ sít✐♦s✳

❆ ♠❛tr✐③ ❍❛♠✐❧t♦♥✐❛♥❛ ♥❛ ❜❛s❡ |ψni t❡♠ ♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧

s❡♥❞♦ t♦❞♦s ✐❣✉❛✐s✱ ❞❡✈✐❞♦ à ✐♥✈❛r✐â♥❝✐❛ ♥❛ tr❛♥s❧❛çã♦✳ ❈♦♠♦ t❡♠♦s t✉♥❡❧❛♠❡♥t♦ ♣♦❞❡✲ ♠♦s ❞❡s❝♦♥s✐❞❡r❛r ♦s sít✐♦s ♠✉✐t♦ ❞✐st❛♥t❡s ❡ ❝♦♥s✐❞❡r❛r s♦♠❡♥t❡ ♦s sít✐♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ✈✐③✐♥❤♦s✳ ❊ss❛ ❤✐♣ót❡s❡ é ❝❤❛♠❛❞❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ❞❡ ❧✐❣❛çã♦ ❢♦rt❡ ♦✉ ❛♣r♦①✐♠❛çã♦ ❚✐❣❤t✲ ❇✐♥❞✐♥❣✳

hψn′|H|ψ_ni= (

E0 ✱ s❡ n′ =n

−t ✱ s❡ n′ _=n

±1. , ✭✷✳✶✸✮

♦♥❞❡✱ t é ♦ ♣❛râ♠❡tr♦ ❞❡ ✐♥t❡r❛çã♦ ❡♥❡r❣ét✐❝❛ ❡♥tr❡ ♦s sít✐♦s✳ ❊❧❡ é ♥❡❣❛t✐✈♦✱ ♣♦✐s ❡①✐st❡ ❛tr❛çã♦ ❡♥tr❡ ♦s sít✐♦s✳ ❆ss✐♠✱ P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r |ψni ❡ |ψn′i ❝♦♠♦ s❡♥❞♦ ♦rt♦❣♦♥❛✐s ♣❛r❛ n₆=n′_{✱ ♦❜t❡♠♦s✿}

H_|ψni=E0|ψni −t|ψn+1i −t|ψn−1i=E|ψni. ✭✷✳✶✹✮

■ss♦ ❡q✉✐✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♦♣❡r❛çã♦ ♠❛tr✐❝✐❛❧          

E0 −t 0 0 . . .

−t E0 −t 0 . . .

0 ₋t E0 −t . . .

0 0 ₋t E0 . . .

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳                    

|ψ1i

|ψ2i

|ψ3i

|ψ4i

✳✳✳           =E          

|ψ1i

|ψ2i

|ψ3i

|ψ4i

✳✳✳           , ✭✷✳✶✺✮

s❡♥❞♦ ❛ ♠❛tr✐③ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ✉♠❛ ♠❛tr✐③ ❡s♣❛rs❛ tr✐❞✐❛❣♦♥❛❧✳

✷✶

r✐♦✱ ✈❛♠♦s s✉❜st✐t✉✐r ❛ ❡q✉❛çã♦ ✷✳✶✶ ♥❛ ❡q✉❛çã♦ ✷✳✶✹✿

H ∞ X

n=−∞

einθ_|ψni=E0

∞ X

n=−∞

einθ_|ψni −t

∞ X

n=−∞

einθ_|ψn+1i −t

∞ X

n=−∞

einθ_|ψn−1i

=E0

∞ X

n=−∞

einθ_|ψni −t

∞ X

n=−∞

ei(n−1)θ

|ψni −t

∞ X

n=−∞

ei(n+1)θ_|ψni

=E0

∞ X

n=−∞

einθ_|ψni −t

∞ X

n=−∞

einθ(e−iθ_+eiθ₎ |ψni,

H_|ψθi= (E0−2tcosθ)|ψθi. ✭✷✳✶✻✮

❆ s✐t✉❛çã♦ ❛❣♦r❛ ♠✉❞♦✉✱ ♣♦✐s ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞❡s❛♣❛r❡❝❡✉ q✉❛♥❞♦ t s❡ t♦r♥♦✉ ✜♥✐t♦✳ ❖s ❛✉t♦❡st❛❞♦s ❞❡♣❡♥❞❡♠ ❞❡ ✉♠ ♣❛râ♠❡tr♦ ❝♦♥tí♥✉♦ ❡ r❡❛❧ ❡ ♦s ❛✉t♦✈❛❧♦r❡s sã♦ ❝♦♥tí♥✉♦s ❡♥tr❡E0−2t ❡ E0+ 2t✳ ❈♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✺✿

0

t

❋✐❣✉r❛ ✺✿ ❆✉t♦✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ❝♦♠❡ç❛♠ ❛ ❢♦r♠❛r ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛ ❝♦♥tí♥✉❛ à ♠❡❞✐❞❛ q✉❡ t ❛✉♠❡♥t❛ ❛ ♣❛rt✐r ❞❡ ③❡r♦✳ ✭❋♦♥t❡✿ ❬✸✸❪✮

P❛r❛ ❡♥t❡♥❞❡r ♦ s✐❣♥✐✜❝❛❞♦ ❢ís✐❝♦ ❞♦ ♣❛râ♠❡tr♦θ✱ ✈❡♠♦s q✉❡ ♣❛r❛ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛hψx|ψθi tr❛♥s❧❛❞❛❞❛ ♥❛ r❡❞❡✱ τ(a)✿

✶✳ ❖♣❡r❛♥❞♦ τ(a)❡♠ _hψx|

hψx|τ(a)|ψθi=hψx−a|ψθi. ✭✷✳✶✼✮

✷✳ ❖♣❡r❛♥❞♦ τ(a)❡♠ _|ψ_θi

✷✷

❱❛♠♦s s✉♣♦r q✉❡✿

hψx|ψθi=eikxuk(x), ✭✷✳✶✾✮

s❡♥❞♦ k = θ/a ❡ ♦♥❞❡ uk(x) é ✉♠❛ ❢✉♥çã♦ ♣❡r✐ó❞✐❝❛ ❝♦♠ ♣❡rí♦❞♦ a✱ ❝♦♠♦ ✈❡r✐✜❝❛❞♦

❛❜❛✐①♦✿

eik(x−a)_u

k(x−a) =eikxuk(x)eika. ✭✷✳✷✵✮

❊ss❛ ❝♦♥❞✐çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❇❧♦❝❤✱ ❡❧❛ é ✈á❧✐❞❛ ♠❡s♠♦ s❡ ♥ã♦ ❤♦✉✈❡r ❚✐❣❤t✲❇✐♥❞✐♥❣✳ ■❣✉❛❧❛♥❞♦ ❛s ❡q✉❛çõ❡s ✷✳✶✼ ❡ ✷✳✶✽✱ t❡♠♦s✿

hψx−a|ψθi=e−iθhψx|ψθi. ✭✷✳✷✶✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✷✳✶✾ ❡♠ ✷✳✷✶✱ ✈❛♠♦s ♣r♦✈❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❡♠a✿

hψx−a|ψθi=ei(kx−θ)uk(x)

=eik(x−a)_u

k(x),

=eik(x−a)uk(x−a).

❆❣♦r❛ ✈❛♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ❞❡ ❇❧♦❝❤ ♥❛ ❡q✉❛çã♦ ✷✳✶✻✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦rhψx|✿

hψx|H|ψθi=hψx|E0−2tcos(θ)|ψθi

Ek =E0−2tcos(ka), ✭✷✳✷✷✮

♦♥❞❡ −π/a_≤ K _≤π/a✳ ❊ss❛ ❡q✉❛çã♦ ❞❡✜♥❡ ✉♠❛ ❝✉r✈❛ ❞❡ ❞✐s♣❡rsã♦ ❡ ♦ s❡✉ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ❚✐❣❤t✲❇✐♥❞✐♥❣✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡ss❛ ❡q✉❛çã♦ s❡❣✉❡ ❛❜❛✐①♦ ✭❋✐❣✉r❛ ✻✮ ❬✸✸❪✳

❈♦♥s✐❞❡r❛♥❞♦ ❛ ❡q✉❛çã♦ ✷✳✷✷✱ ❛ ❡①♣❛♥sã♦ ❞❡ ❚❛②❧♦r ❞❡cos(ka)é✿

cos(ka) = ∞ X

n=0

(₋1)n(ka)

2n

(2n)! = 1− (ka)2

2! + (ka)4

4! . . . ✭✷✳✷✸✮ ❊①♣❛♥❞✐♥❞♦ s♦♠❡♥t❡ ❛té ❛ s❡❣✉♥❞❛ ♦r❞❡♠ ❡ ❞❡s♣r❡③❛♥❞♦ ♦s t❡r♠♦s ❞❡ ♠❛✐♦r ♦r❞❡♠✱ ♦❜t❡♠♦s

E(k) =E0′ +ta2k2, ✭✷✳✷✹✮

❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ q✉❡ E0−2t =E0′✳

◆♦ ♠♦❞❡❧♦ ❞♦ ❡❧étr♦♥ ❧✐✈r❡✱ ♦s ✈❛❧♦r❡s ♣❡r♠✐t✐❞♦s ❞❛ ❡♥❡r❣✐❛ ❡stã♦ ❞✐str✐❜✉í❞♦s ❝♦♥t✐♥✉❛♠❡♥t❡ ❞❡ ③❡r♦ ❛♦ ✐♥✜♥✐t♦✿

E(k) = ~

2_k2

✷✸

❋✐❣✉r❛ ✻✿ ❖s ✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ♣❡r♠✐t✐❞❛ ❢♦r♠❛♠ ✉♠❛ ❜❛♥❞❛ ❝♦♥tí♥✉❛ ❡♥tr❡E0−2t ❡

E0+ 2t✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❩♦♥❛ ❞❡ ❇r✉❧❧♦✉✐♥✳ ✭❋♦♥t❡✿ ❬✸✸❪✮

❆s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ♣❛r❛ ✉♠❛ ❝♦♥❞✐çã♦ ❞♦ ❡❧étr♦♥ ❧✐✈r❡ sã♦ψk(~r) = ei~k~r✱ ❡❧❛s r❡♣r❡s❡♥t❛♠

♦♥❞❛s ♣r♦❣r❡ss✐✈❛s ❡ tr❛♥s♣♦rt❛♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦✈✐♠❡♥t♦~p=~~k✳

❯♠ ❡❧étr♦♥ ❧✐✈r❡ ♥✉♠ ♣♦t❡♥❝✐❛❧ ♣❡r✐ó❞✐❝♦ é ❛❝❡❧❡r❛❞♦✱ ❡♠ r❡❧❛çã♦ ❛ r❡❞❡✱ ♥✉♠ ❝❛♠♣♦ ❛♣❧✐❝❛❞♦✳ ◗✉❛♥❞♦ ✐ss♦ ❛❝♦♥t❡❝❡ ♦ ❡❧étr♦♥ ♣❛ss❛ ❛ t❡r ✉♠❛ ♠❛ss❛ ❡❢❡t✐✈❛✱ me✱ q✉❡

♣♦❞❡ s❡r ♠❛✐♦r ♦✉ ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❛ss❛ ❞♦ ❡❧étr♦♥ ❧✐✈r❡✱ ♣♦❞❡♥❞♦ ✐♥❝❧✉s✐✈❡ s❡r ♥❡❣❛t✐✈❛✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ♠❛ss❛ ❡❢❡t✐✈❛ é✿

1 me

= 1

~2

∂2_E(k)

∂k2 . ✭✷✳✷✻✮

❆♣❧✐❝❛♥❞♦ ✷✳✷✹ ❡♠ ✷✳✷✻✱

1 me

= 1

~2

∂2_(ta2_k2₎

∂k2

= 1

~22ta 2_,

ta2 = ~

2

2me

. ✭✷✳✷✼✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✷✳✷✼ ♥❛ ✷✳✷✹✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ✉♠ ❡❧étr♦♥ ❧✐✈r❡ ❡♠ ✉♠ só❧✐❞♦ ❧✐♥❡❛r ❬✸✹❪ ✭❋✐❣✉r❛ ✼✮✿

E(k) =E′

0+ ~2_k2

2me

, ✭✷✳✷✽✮

✷✹

❋✐❣✉r❛ ✼✿ ●rá✜❝♦ ❞❛ ❡♥❡r❣✐❛ ❊ ❝♦♥t❛ ♦ ✈❡t♦r ❞❡ ♦♥❞❛ ❦ ♣❛r❛ ✉♠ ❡❧étr♦♥ ❧✐✈r❡✳ ✭❋♦♥t❡✿ ❬✸✹❪✮

✷✳✷ ▼♦❞❡❧♦ ❚✐❣❤t✲❇✐♥❞✐♥❣ ♣❛r❛ ✉♠ s✐st❡♠❛ ❜✐❞✐♠❡♥s✐♦♥❛❧

❙❡❥❛ ✉♠❛ r❡❞❡ ❝r✐st❛❧✐♥❛ q✉❛❧q✉❡r✱ ♦♥❞❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ é ❝♦♥st✐t✉í❞❛ ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s✿

Φj(~k, ~r) =

1

√

N

N

X

~ R

ei~k ~Rϕj(~r−R),~ j = 1,2,3, . . . , n ✭✷✳✷✾✮

♦♥❞❡

~

R =✈❡t♦r ❞❡ r❡❞❡ r❡❧❛t✐✈♦ ~k =✈❡t♦r ❞❡ ♦♥❞❛

ϕj =❢✉♥çã♦ ❞❡ ♦♥❞❛ ❛tô♠✐❝❛ ♥♦ ❡st❛❞♦ ❥,

◆=♥ú♠❡r♦ ❞❡ ❝é❧✉❧❛s ✉♥✐tár✐❛s.

❆ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡✈❡ s❛t✐s❢❛③❡r ♦ t❡♦r❡♠❛ ❞❡ ❇❧♦❝❤✿

Φj(~k, ~r+R) =~ ei~k ~Rϕj(~r, ~R). ✭✷✳✸✵✮

P❛r❛ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ♣❡r♠✐t✐❞♦s ❞♦s ✈❡t♦r❡s ❞❡ ♦♥❞❛~k ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦ ❞❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞❡✈❡ s❡r s❛t✐s❢❡✐t♦✱ ♦✉ s❡❥❛✱Φj(~k, ~r) = Φj(~k, ~r+R)✳ ■st♦~

✐♠♣❧✐❝❛✱ei~k ~R _{= 1✳ ❚❡♠♦s ❛❣♦r❛ ❡♥tã♦ ❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ♥♦ só❧✐❞♦ ❞❛❞♦s ♣❡❧❛ ❝♦♠❜✐♥❛çã♦}

❧✐♥❡❛r ❞❛s ❢✉♥çõ❡s ❞❡ ❇❧♦❝❤✿

Ψj(~k, ~r) = n

X

j′₌₁

✷✺

♦♥❞❡n❂♥ú♠❡r♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❇❧♦❝❤✳

❙✉❜st✐t✉✐♥❞♦ ✷✳✸✶ ♥❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♦s ❛✉t♦✈❛❧♦✲ r❡s ❞❡Ej(~k)✿

HΨj(~k, ~r) =Ej(~k)Ψj(~k, ~r), n

X

j′₌₁

cjj′(~k)HΦ_j′(~k, ~r) =E_j(~k)

n

X

j′₌₁

cjj′(~k)Φ_j′(~k, ~r).

✭✷✳✸✷✮ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ♣❡❧♦ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ ❞❡ Ψj(~k, ~r)✿

Ej(~k) =

Pn

j,j′₌₁c ∗

j,j′cj,jhΦj|H|Φj′i Pn

j,j′₌₁c∗_j,j′cj,jhΦj|Φj′i

. ✭✷✳✸✸✮

❋❛③❡♥❞♦✿

▼❛tr✐③ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦✿ hΦj|H|Φj′i=H_j,j′(~k), ▼❛tr✐③ ❖✈❡r❧❛♣✿ hΦj|Φj′i=S_j,j′(~k).

✭✷✳✸✹✮

Sj,j′(~k) sã♦ ❛s ✐♥t❡❣r❛✐s s✉♣❡r♣♦s✐çã♦ ❞♦s ❡st❛❞♦s Φ_j′s✱ q✉❡r ❞✐③❡r✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♥ã♦✲ ♦rt♦❣♦♥❛❧✐❞❛❞❡ ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❞♦ só❧✐❞♦✳

P❛r❛ ♦❜t❡r ❛ ❝♦♥❞✐çã♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✱ ❞❡r✐✈❛✲s❡ ❛ ❡①♣r❡ssã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ♣❡❧♦ ♣❛râ♠❡tr♦ c∗

j,j′✳ ❊♠ s❡❣✉✐❞❛✱ r❡❛rr❛♥❥❛♥❞♦ ♦s t❡r♠♦s ♦❜tê♠✲s❡✱ ∂Ej(~k)

∂c∗

j,j′

= 0, ✭✷✳✸✺✮

n

X

j′₌₁

cj,j′(~k)H_j,j′(~k) = Pn

j,j′₌₁c ∗

j,j′(~k)cj,j′(~k)H_j,j′(~k) Pn

j′₌₁cj,j′(~k)S_j,j′(~k) Pn

j,j′₌₁c∗_j,j′(~k)cj,j′(~k)S_j,j′(~k)

. ✭✷✳✸✻✮

❙✉❜st✐t✉✐♥❞♦ ✷✳✸✸ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♦❜t❡♠♦s✿

n

X

j′₌₁

cj,j′(~k)H_j,j′(~k) =E_j(~k)

n

X

j′₌₁

cj,j′(~k)S_j,j′(~k). ✭✷✳✸✼✮

P❛r❛ ✉♠ só❧✐❞♦ ❡♠ ❣❡r❛❧✱ ❛s ❜❛♥❞❛s ❞❡ ❡♥❡r❣✐❛ ❞❡ Ej(~k) sã♦ ♦❜t✐❞❛s ❛tr❛✈és

❞❡st❛ ❡q✉❛çã♦ s❡❝✉❧❛r✿

det(H₋ES) = 0. ✭✷✳✸✽✮

✷✻

s✉❣❡r❡ q✉❡ s❡rã♦ ❝♦♥s✐❞❡r❛❞❛s ❛♣❡♥❛s ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s✳ ❍á ❞♦✐s ❝❛s♦s✱ ❝♦♠ ♦✉ s❡♠ ❖✈❡r❧❛♣✱ q✉❡r ❞✐③❡r✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ♦✉ ♥ã♦ ❛ s✉♣❡r♣♦s✐çã♦ ❞♦s ♦r❜✐t❛✐s ❛tô♠✐❝♦s ❞♦s ❡❧étr♦♥sπ✱ q✉❡ sã♦ r❡s♣♦♥sá✈❡✐s ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❣r❛❢❡♥♦✳ ❆ ❝é❧✉❧❛ ✉♥✐tár✐❛ ❞♦ ❣r❛❢❡♥♦ é ❞❡✜♥✐❞❛ ♣♦r ❞✉❛s s✉❜✲r❡❞❡s ❆ ✭❛③✉❧✮ ❡ ❇ ✭✈❡r♠❡❧❤♦✮✳ ❖s t❛♥t♦ ♦s át♦♠♦s ❞♦ t✐♣♦ ❆ q✉❛♥t♦ ❞♦ ❇ tê♠ ✉♠ ✈❡t♦r ❞❡ tr❛♥s❧❛çã♦_T~ _{=n ~a1+m ~a2✱ s❡♥❞♦ n ❡} m ✐♥t❡✐r♦s✳

R

R

R

❋✐❣✉r❛ ✽✿ ❛✮ ❊str✉t✉r❛ ❝r✐st❛❧✐♥❛ ❞❡ ✉♠❛ ♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✱ ❝✉❥♦ ✈❡t♦r❡s ❞❡ r❡❞❡ sã♦~aj✱ ♠♦str❛♥❞♦ ❛ s✉♣❡r♣♦s✐çã♦ ❞❛s ❞✉❛s s✉❜✲r❡❞❡s ❆ ❡ ❇✳ ❜✮ ❊s♣❛ç♦ r❡❝í♣r♦❝♦ ❞❛

♠♦♥♦❝❛♠❛❞❛ ❞❡ ❣r❛❢❡♥♦✳ ✭❋♦♥t❡✿ ❬✸✺❪✮

❈♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✽✱ ♦s ✈❡t♦r❡s ❞❛ r❡❞❡ sã♦✿

~ a1 =a

√

3 2 ,

1 2

!

~ a2 =a

√

3 2 ,−

1 2

! ,

✭✷✳✸✾✮ ♦♥❞❡

a =_|a~1|=|a~2|

d= 0,142 ♥♠ ✭❛r❡st❛ ❞❛ ❝é❧✉❧❛ ✉♥✐tár✐❛✮✱ a =√3d= 2,46 ♥♠.

✷✼

❖s ✈❡t♦r❡s ❞❛ r❡❞❡ r❡❝í♣r♦❝❛ sã♦✿ ~ b1 =

2π √ 3a, 2π a ~ b2 =

2π

√

3a,− 2π

a

,

✭✷✳✹✶✮ ♦♥❞❡✱|b~1|=|b~2|= 4π/

√

3a✳

❈♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✽✱ ✉♠ át♦♠♦ ❞❡ ❝❛r❜♦♥♦ t✐♣♦ ❆ é ❝✐r❝✉♥❞❛❞♦ ♣♦r três ❞♦ t✐♣♦ ❇ ♠❛✐s ♣ró①✐♠♦s✳ ❙❡✉s ✈❡t♦r❡s ♣♦s✐çã♦ r❡❧❛t✐✈❛ sã♦ ❞❛❞♦s ♣♦r✿

~ R1 =

a

√

3,0

~

R2 =R~1−a~2 =

− a

2√3,− a 2

, ~

R3 =R~1−a~1 =

− a

2√3, a 2

.

✭✷✳✹✷✮ ◆♦ ❣r❛❢❡♥♦ ❤á ❞♦✐s át♦♠♦s ♣♦r ❝é❧✉❧❛ ✉♥✐tár✐❛✱ ❛ss✐♠ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❚✐❣❤t✲❇✐♥❞✐♥❣ é ❞❛❞❛ ❝♦♥❢♦r♠❡ ❛ ❡q✉❛çã♦ ✷✳✸✶✿

Ψ(~k, ~r) =cA(~k)ΦA(~k, ~r) +cB(~k)ΦB(~k, ~r), ✭✷✳✹✸✮

♦♥❞❡✱

ΦA(~k, ~r) =

1 √ N N X ~ RA

ei~k ~RA_ϕ

A(~r−R~A),

ΦB(~k, ~r) =

1 √ N N X ~ RB

ei~k ~RB_ϕ

B(~r−R~B).

✭✷✳✹✹✮ ❈♦♥❢♦r♠❡ ❛ ❡q✉❛çã♦ ✷✳✸✹✱ ❛ ♠❛tr✐③ ❍❛♠✐❧t♦♥✐❛♥❛ é✿

H= hΦA|H|ΦAi hΦA|H|ΦBi

hΦB|H|ΦAi hΦB|H|ΦBi

!

✭✷✳✹✺✮

♦♥❞❡✱

hΦ_A|H_|Φ_Ai=HAA =

1 N

N

X

~ RA, ~R′A

ei~k(RA−R′ A)hΦ