O acoplamento spin-órbita no estudo de fases topológicas em uma rede hexagonal de...

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛

❖ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛ ♥♦ ❊st✉❞♦ ❞❡

❋❛s❡s ❚♦♣♦❧ó❣✐❝❛s ❡♠ ✉♠❛ ❘❡❞❡ ❍❡①❛❣♦♥❛❧

❞❡ ❇❛r✐❝❡♥tr♦s

❈❛r❧♦s ❆✉❣✉st♦ ▼❡r❛ ❆❝♦st❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆❞❛❧❜❡rt♦ ❋❛③③✐♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s

❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❆❞❛❧❜❡rt♦ ❋❛③③✐♦ ✭■❋✲❯❙P✮ Pr♦❢❛✳ ❉r❛✳ ❍❡❧❡♥❛ ▼❛r✐❛ P❡tr✐❧❧✐ ✭■❋✲❯❙P✮ Pr♦❢✳ ❉r✳ ❘♦❜❡rt♦ ❇❡❝❤❛r❛ ▼✉♥✐③ ✭■❋✲❯❋❋✮

FICHA CATALOGRÁFICA

Preparada pelo Serviço de Biblioteca e Informação

do Instituto de Física da Universidade de São Paulo

Acosta, Carlos Augusto Mera

O acoplamento Spin-Órbita no estudo de fases topológicas em uma rede hexagonal de baricentros. – São Paulo, 2013.

Dissertação (Mestrado) – Universidade de São Paulo. Instituto de Física – Depto. de Física dos Materiais e Mecânica Orientador: Prof. Dr. Adalberto Fazzio

Área de Concentração: Grupo SAMPA

Unitermos: 1. Física da Matéria Condensada; 2. Física do estado Sólido; 3. Estrutura dos Sólidos.

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦s ♠❡✉s ♣❛✐s✱ ❈❡s❛r ▼❡r❛ ❡ ▼❛rt❤❛ ❆❝♦st❛ ♣♦r s❡r❡♠ ✉♠❛ ❢♦♥t❡ ❞❡ ❢♦rç❛ ❡ ❡s♣❡r❛♥③❛✳

➚ ●✐♥❛ P♦❧♦✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ♣♦r t❛♥t♦ ❛♠♦r✱ ❝♦♠♣r❡❡♥sã♦✱ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦ ♥❡st❛ ❡t❛♣❛ ❞❡ ♥♦ss❛s ✈✐❞❛s✳

❆♦s ♠❡✉s ✐r♠ã♦s ❡ ♠✐♥❤❛ s♦❜r✐♥❤❛ ❆♠❛r✐❧❧❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s q✉❡ s❡♠♣r❡✱ s❡♠ ✐♠♣♦rt❛r ❛ ❞✐stâ♥❝✐❛✱ ♠❡ tr❛♥s♠✐t✐r❛♠ ❛ ♠❡❧❤♦r ❞❛s ❡♥❡r❣✐❛s✳

❆♦ Pr♦❢❡ss♦r ❆❞❛❧❜❡rt♦ ❋❛③③✐♦ ♣♦r ♠❡ ♣❡r♠✐t✐r tr❛❜❛❧❤❛r ♥♦ ❣r✉♣♦ ❙❆▼P❆ ❡ ♣❡❧❛s ❞❡♥s❛s ❞✐s❝✉ssõ❡s ❛❝❛❞ê♠✐❝❛s✳

❆♦ ▼❛t❤❡✉s P❛❡s ▲✐♠❛ ♣❡❧♦ ❛♣♦✐♦ ❛❝❛❞ê♠✐❝♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ❛♠✐③❛❞❡ ❡ ❡①❡♠♣❧♦✳ ❆♦ Pr♦❢❡ss♦r ❏♦sé ❘♦q✉❡ ❉❛ ❙✐❧✈❛ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ❡ ♣❡rt✐♥❡♥t❡s s✉❣❡stõ❡s✳

❆♦s ♠❡♠❜r♦s ❞♦ ❣r✉♣♦ ❙❆▼P❆ ❡ ❛♠✐❣♦s ❞♦ ✐♥st✐t✉t♦ ❞❡ ❢ís✐❝❛ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛♣♦✐♦✳

❘❊❙❯▼❖

◆❡st❡ tr❛❜❛❧❤♦✱ ❢♦r❛♠ ❡st✉❞❛❞❛s ❛s ❢❛s❡s t♦♣♦❧ó❣✐❝❛s ♥ã♦✲tr✐✈✐❛✐s ♣r❡s❡♥t❡s ❡♠ s✐st❡♠❛s ❢♦r♠❛❞♦s ♣❡❧❛ ❞❡♣♦s✐çã♦ ❞❡ át♦♠♦s ♥♦ ❣r❛❢❡♥♦✳ ❊♥❝♦♥tr❛♠♦s q✉❡ q✉❛♥❞♦ ✉♠ át♦♠♦ ❤✐❜r✐❞✐③❛ ❢♦rt❡♠❡♥t❡ ❝♦♠ ♦ ❣r❛❢❡♥♦✱ ❛♣r❡s❡♥t❛ ✉♠ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❡ ✉♠ ❢♦rt❡ s♣✐♥✲ór❜✐t❛ é ♣♦ssí✈❡❧ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠❛ ✏r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❜❛r✐❝❡♥tr♦s✑ q✉❡ ❡❢❡t✐✈❛♠❡♥t❡ ❣❡r❛ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✉♠ ❡❢❡✐t♦ ❤❛❧❧ q✉â♥t✐❝♦ ❛♥ô♠❛❧♦✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ❞❡t❡r♠✐♥❛♠♦s q✉❡ ♦ ❘✉ s❛t✐s❢❛③ ❡st❛s ❝❛r❛❝t❡ríst✐❝❛s✳ ◗✉❛♥❞♦ ❡st❡ ♠❡t❛❧ é ❞❡♣♦s✐t❛❞♦ ❡♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ tr✐❛♥❣✉❧❛r ♥♦ ❣r❛❢❡♥♦ ♦❝♦rr❡♠ ♣✐❝♦s ♥❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❧♦❝❛❧✐③❛❞♦s ♥♦ ❝❡♥tr♦ ❣❡♦♠étr✐❝♦ ✭❜❛r✐❝❡♥tr♦✮ ❞♦s tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧♦s ❘✉✳ ❊st❡s ♣✐❝♦s ❡stã♦ ❞✐str✐❜✉í❞♦s ❞❡ ❢♦r♠❛ ❤❡①❛❣♦♥❛❧ ❡ ❡❢❡t✐✈❛♠❡♥t❡ ❣❡r❛♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s q✉❡ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐ ❛♣r❡s❡♥t❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ s♣✐♥ ❝❛r❛t❡ríst✐❝❛ ❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❛♥ô♠❛❧♦✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❡♥❝♦♥tr❛♠♦s q✉❡ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ♣❡❧❛ ❛❞s♦rçã♦ ❞❡ ❇❛ ♦✉ ❙r ♥♦ ❣r❛❢❡♥♦ ❢❛✈♦r❡❝❡ ❛ ❢♦r♠❛çã♦ ❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❞❡ s♣✐♥✳ ◆❡st❡ s✐st❡♠❛✱ ♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛ ✭❙❖❈✮ ❣❡r❛ ✉♠ ❣❛♣ ♠❛✐s ❞❡ ✶✵✵✵ ✈❡③❡s ♠❛✐♦r ❛♦ ♣r❡❞✐t♦ ♥♦ ❣r❛❢❡♥♦ ♣r✐st✐♥♦✳ P❛r❛ ♦ ❡st✉❞♦ ❞❡st❡s s✐st❡♠❛s✱ ✐♠♣❧❡♠❡♥t❛♠♦s ♥♦ ❝ó❞✐❣♦ ❙■❊❙❚❆ ❛ ❛♣r♦①✐♠❛çã♦ ✏♦♥✲s✐t❡✑ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ ór❜✐t❛ ✈✐❛ ♦ ❢♦r♠❛❧✐s♠♦ ❞♦s ♣s❡✉❞♦♣♦t❡♥❝✐❛✐s r❡❧❛t✐✈íst✐❝♦s ❞❡ ♥♦r♠❛ ❝♦♥s❡r✈❛❞❛✳ ◆♦ss❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❢♦✐ t❡st❛❞❛ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞❡ ❢❡♥ô♠❡♥♦s ❥á ❝♦♥❤❡❝✐❞♦s✿ ✐✮ ♦ ✏str♦♥❣ s♣✐♥✲s♣❧✐tt✐♥❣✑ ❣❡r❛❞♦ ♥♦ ❣r❛❢❡♥♦ ♣❡❧❛ ❛❞s♦rçã♦ ❞❡ ❆✉✱ ii✮ ♦ ❡❢❡✐t♦ ❤❛❧❧ q✉â♥t✐❝♦ ❞❡ s♣✐♥ ♥♦ ♣♦ç♦ q✉â♥t✐❝♦ ❞❡ ❍❣❚❡✴❈❞❚❡ ❡✱ iii✮ ❛ ❢♦r♠❛çã♦ ❞❡ ❡st❛❞♦s t♦♣♦❧ó❣✐❝♦s ♥❛ s✉♣❡r❢í❝✐❡ ❞♦ ❇✐2❙❡3 ❡ ❛s ❢❛s❡s ♠❛❣♥ét✐❝❛s ❞❡st❡ ♠❛t❡r✐❛❧ ❝♦♠ át♦♠♦s ❞❡ ▼♥ ❛❞s♦r✈✐❞♦s✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ❆❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✱ ❊❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❞❡ s♣✐♥✱ ❊❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❛♥ô♠❛❧♦✱ ■s♦❧❛♥t❡s t♦♣♦❧ó❣✐❝♦s✱ ●r❛❢❡♥♦✳

❆❇❙❚❘❆❈❚

■♥ t❤✐s ✇♦r❦✱ ✇❡r❡ st✉❞✐❡❞ t❤❡ ♥♦♥✲tr✐✈✐❛❧ t♦♣♦❧♦❣✐❝❛❧ ♣❤❛s❡s ♣r❡s❡♥t ✐♥ s②st❡♠s ❢♦r♠❡❞ ❜② ❞❡♣♦s✐t✐♦♥ ♦❢ ❛t♦♠s ✐♥ ❣r❛♣❤❡♥❡✳ ❲❡ ❢♦✉♥❞ t❤❛t ✇❤❡♥ ❛♥ ❛t♦♠ ❤②❜r✐❞✐③❡s str♦♥❣❧② ✇✐t❤ ❣r❛♣❤❡♥❡✱ ❤❛s ❛ ♠❛❣♥❡t✐❝ ♠♦♠❡♥t ❛♥❞ ❛ str♦♥❣ s♣✐♥✲♦r❜✐t ✐t ✐s ♣♦ss✐❜❧❡ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ ❛ ✧❤❡①❛❣♦♥❛❧ ♥❡t✇♦r❦ ♦❢ ❜❛r②❝❡♥tr❡s✧t❤❛t ❡✛❡❝t✐✈❡❧② ❣❡♥❡r❛t❡s ❛ str✉❝t✉r❡ ❜❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❛ q✉❛♥t✉♠ ❛♥♦♠❛❧♦✉s ❍❛❧❧ ❡✛❡❝t✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❞❡t❡r♠✐♥❡❞ t❤❛t ❘✉ s❛t✐s✜❡s t❤❡s❡ ❝❤❛r❛❝t❡r✐st✐❝s✳ ❲❤❡♥ t❤✐s ♠❡t❛❧ ✐s ❞❡♣♦s✐t❡❞ ✐♥ ❛ tr✐❛♥❣✉❧❛r ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ ❣r❛♣❤❡♥❡✱ ♣❡❛❦s ♦❝❝✉r ✐♥ t❤❡ ❞❡♥s✐t② ♦❢ ❧♦❝❛❧✐③❡❞ st❛t❡s ✐♥ t❤❡ ❣❡♦♠❡tr✐❝ ❝❡♥t❡r ✭❝❡♥tr♦✐❞✮ ♦❢ t❤❡ tr✐❛♥❣❧❡s ❢♦r♠❡❞ ❜② ❘✉✳ ❚❤❡s❡ ♣❡❛❦s ❛r❡ ❞✐str✐❜✉t❡❞ ✐♥ ❛ ❤❡①❛❣♦♥❛❧ str✉❝t✉r❡ ❛♥❞ ❡✛❡❝t✐✈❡❧② ❣❡♥❡r❛t❡s ❛ ❜❛♥❞ str✉❝t✉r❡ t❤❛t ♥❡❛r t❤❡ ❋❡r♠✐ ❧❡✈❡❧ ❤❛s ❛ s♣✐♥ ❝♦♥✜❣✉r❛t✐♦♥ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ s♣✐♥ q✉❛♥t✉♠ ❍❛❧❧ ❡✛❡❝t ❛♥♦♠❛❧♦✉s✳ ❆❞❞✐t✐♦♥❛❧❧②✱ ✇❡ ❢♦✉♥❞ t❤❛t t❤❡ s②st❡♠ ❝♦♠♣♦s❡❞ ❜② t❤❡ ❛❞s♦r♣t✐♦♥ ♦❢ ❇❛ ♦r ❙r ✐♥ ❣r❛♣❤❡♥❡✱ ♣r♦♠♦t❡s t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ s♣✐♥ q✉❛♥t✉♠ ❍❛❧❧ ❡✛❡❝t✳ ■♥ t❤✐s s②st❡♠✱ t❤❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ✭❙❖❈✮ ❣❡♥❡r❛t❡s ❛ ❣❛♣ ♠♦r❡ t❤❛♥ ✶✵✵✵ t✐♠❡s ❣r❛t❡r t❤❛t ♣r❡❞✐❝t❡❞ ✐♥ ♣r✐st✐♥❡ ❣r❛♣❤❡♥❡✳ ❚♦ st✉❞② t❤❡s❡ s②st❡♠s✱ ✇❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❝♦❞❡ ❙■❊❙❚❆ t❤❡ ✏ ♦♥✲s✐t❡✑ ❛♣♣r♦❛❝❤ ♦❢ t❤❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ t❤r♦✉❣❤ t❤❡ ❢♦r♠❛❧✐s♠ ♦❢ ♥♦r♠ ❝♦♥s❡r✈❡❞ r❡❧❛t✐✈✐st✐❝ ♣s❡✉❞♦✲ ♣♦t❡♥t✐❛❧s✳ ❖✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✇❛s t❡st❡❞ ❢r♦♠ t❤❡ st✉❞② ♦❢ ♣❤❡♥♦♠❡♥❛ ❛❧r❡❛❞② ❦♥♦✇♥✿ ✐✮ t❤❡ ✏ str♦♥❣✑ s♣✐♥✲s♣❧✐tt✐♥❣ ❣❡♥❡r❛t❡❞ ✐♥ ❣r❛♣❤❡♥❡ ❜② ❛❞s♦r♣t✐♦♥ ♦❢ ❆✉✱ ii✮ t❤❡ q✉❛♥t✉♠ s♣✐♥ ❍❛❧❧ ❡✛❡❝t ✐♥ q✉❛♥t✉♠ ✇❡❧❧ ♦❢ ❍❣❚❡ ✴ ❈❞❚❡ ❛♥❞ ✱ ✐✐✐✮ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ t♦♣♦❧♦❣✐❝❛❧ st❛t❡s ✐♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❇✐2❙❡3 ❛♥❞ t❤❡ ♠❛❣♥❡t✐❝ ♦❢ t❤✐s ♠❛t❡r✐❛❧ ✇✐t❤ ▼♥ ❛t♦♠s ❛❞s♦r❜❡❞✳

❑❡② ✇♦r❞s✿ ❙♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣✱ ◗✉❛♥t✉♠ ❙♣✐♥ ❍❛❧❧ ❡✛❡❝t✱ ❆♥♦♠❛❧♦✉s ◗✉❛♥t✉♠ ❍❛❧❧ ❡✛❡❝t✱ t♦♣♦❧♦❣✐❝❛❧ ✐♥s✉❧❛t♦rs✱ ●r❛♣❤❡♥❡✳

❈♦♥t❡ú❞♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❖ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛ ✺

✷✳✶ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷ ❖ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛ ♥❛ ❚❡♦r✐❛ ❞❡ ❉✐r❛❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷✳✸ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝ ♣❛r❛ ✉♠ ♣♦t❡♥❝✐❛❧ ❝❡♥tr❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸ ❊str✉t✉r❛ ❡❧❡trô♥✐❝❛ ✷✵

✸✳✶ ❆♣r♦①✐♠❛çã♦ ❞❡ ❇♦r♥ ❖♣♣❡♥❤❡✐♠❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✷ ❚❡♦r✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✸ ❙✐st❡♠❛s P❡r✐ó❞✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✸✳✹ Ps❡✉❞♦♣♦t❡♥❝✐❛❧ ❞❡ ◆♦r♠❛ ❈♦♥s❡r✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛ ♥♦ ❈ó❞✐❣♦ ❙■❊❙❚❆ ✸✾

✹✳✶ ❋♦r♠❛❧✐s♠♦ ❞♦ Ps❡✉❞♦♣♦t❡♥❝✐❛❧ ❘❡❧❛t✐✈íst✐❝♦ ❞❡ ◆♦r♠❛ ❈♦♥s❡r✈❛❞❛ ✳ ✹✶

✹✳✶✳✶ ▲❈❆❖ ❡ ❛♣r♦①✐♠❛çã♦ ✏♦♥✲s✐t❡✑ ❞❛ ✐♥t❡r❛çã♦ s♣✐♥✲ór❜✐t❛ ✳ ✳ ✳ ✳ ✹✺

✹✳✶✳✷ ▼❛tr✐③ ❞❡ ❉❡♥s✐❞❛❞❡ ❡ ❊♥❡r❣✐❛ ❚♦t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✹✳✶✳✸ ❋♦rç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✹✳✷ ❈ó❞✐❣♦ ❙✐❡st❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺ ❆♣❧✐❝❛çõ❡s ✺✶

✺✳✶ ●❛♣ ❙❖❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✺✳✷ ❊❢❡✐t♦ ❍❛❧❧ ◗✉â♥t✐❝♦ ❞❡ ❙♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✺✳✷✳✶ ❋❡♥♦♠❡♥♦❧♦❣✐❛ ❞♦ ❡❢❡✐t♦ ◗❙❍ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✺✳✷✳✷ ■♥✈❡rsã♦ ❞❡ ❜❛♥❞❛ ❡ ❡str✉t✉r❛ ❡❧❡trô♥✐❝❛ ❞♦ ❍❣❚❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

✺✳✷✳✸ P♦ç♦s q✉â♥t✐❝♦ ❡♠ ❍❣❚❡✴❈❞❚❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✺✳✸ ■s♦❧❛♥t❡s t♦♣♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✺✳✸✳✶ ❊st❛❞♦s t♦♣♦❧ó❣✐❝♦s ❡♠ ❇✐2❙❡3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

✺✳✸✳✷ ❖r❞❡♠ ♠❛❣♥ét✐❝❛ ♥♦ ❇✐2❙❡3 ❝♦♠ át♦♠♦s ❞❡ ▼♥ ❛❞s♦r✈✐❞♦s ✳ ✳ ✼✸

✺✳✹ ✏❙tr♦♥❣ ❙♣✐♥✲❙♣❧✐tt✐♥❣✑ ❡♠ ●r❛❢❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✻ ❋❛s❡s t♦♣♦❧ó❣✐❝❛s ❡♠ ✉♠❛ r❡❞❡ ❞❡ ✏❜❛r✐❝❡♥tr♦s✑ ❝♦♠ s✐♠❡tr✐❛

❤❡①❛❣♦♥❛❧ ✽✵

✻✳✶ ❘❡❞❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❜❛r✐❝❡♥tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

✻✳✶✳✶ ➪t♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ♥♦ ❜❛r✐❝❡♥tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✻✳✶✳✷ ❙❚▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✻✳✶✳✸ ❘❡❞❡ tr✐❛♥❣✉❧❛r ❞❡ ❘✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

✻✳✶✳✹ ❘❡❞❡ r❡t❛♥❣✉❧❛r ❞❡ ❘✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶

✻✳✷ ❊❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❛♥ô♠❛❧♦ ❡♠ ❜❛r✐❝❡♥tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷

✈✐✐✐

✻✳✸ ◗❙❍ ❡♠ ❜❛r✐❝❡♥tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼

✼ ❈♦♥❝❧✉sõ❡s ✶✶✶

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❊str✉t✉r❛ ✜♥❛ ❞♦s ❡st❛❞♦s ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ♥♦ ❡s♣❡❝tr♦ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✺✳✶ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ●❛❆s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✺✳✷ ❊sq✉❡♠❛ ❞❛ ♠✉❞❛♥ç❛ ❞❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ♥♦ ♣♦♥t♦Γ❡♠ s✐st❡♠❛s

❝♦♠ s✐♠❡tr✐❛ Td ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✺✳✸ ❱❛r✐❛çã♦ ❞♦ ∆0 ❡♠ r❡❧❛çã♦ à ❢♦rç❛ ❞♦ ❙❖❈ ♥♦ ●❛❆s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✹ ❈♦♠♣❛r❛çã♦ ❞♦s ❡st❛❞♦s ❞❡ ❜♦r❞❛ ❡♠ s✐st❡♠❛s q✉❡ ❛♣r❡s❡♥t❛♠ ♦

❡❢❡✐t♦ ◗❍ ❡ ◗❙❍✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺✳✺ ❋♦r♠❛çã♦ ❞♦ ◗❙❍ ❛ ♣❛rt✐r ❞❡ ❞✉❛s ❝♦♣✐❛s ❞♦ ❡st❛❞♦ ◗❍✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✺✳✻ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❍❣❚❡ ❡ ❈❞❚❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

✺✳✼ ❊st❛❞♦s ❞❡ ✐♥t❡r❢❛❝❡ ♥♦ ♣♦ç♦ q✉â♥t✐❝♦ ❞❡ ❍❣❚❡✴❈❞❚❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✺✳✽ ❈♦♠♣❛r❛çã♦ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ❋❡r♠✐ ♣❛r❛ ✉♠ ✐s♦❧❛♥t❡ t♦♣♦❧ó❣✐❝♦ ❢♦rt❡ ❡ ✉♠ ✐s♦❧❛♥t❡ t♦♣♦❧ó❣✐❝♦ ❢r❛❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✺✳✾ ❉✐s♣❡rsã♦ ❡❧❡trô♥✐❝❛ ❡♥tr❡ ❞♦s ♣♦♥t♦s ❞❡ s✉♣❡r❢í❝✐❡ ❑r❛♠❡rs ❞❡❣❡♥❡✲ r❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

①

✺✳✶✵ ■♥✈❡rsã♦ ❞❡ ❜❛♥❞❛ ♥♦ ❜✉❧❦ ❞❡ ❇✐2❙❡3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

✺✳✶✶ ❙✉♣❡r❝é❧✉❧❛ ❤❡①❛❣♦♥❛❧ ❞❡ ❇✐2❙❡3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

✺✳✶✷ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ♣❛r❛ ❞✐❢❡r❡♥t❡s ◗❧✬s ❞❡ ❇✐2❙❡3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

✺✳✶✸ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞✸ ✹◗▲✬s ❇✐2❙❡3 ❝♦♠ ▼♥ ❛❞s♦r✈✐❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

✺✳✶✹ ❋✐❣✉r❛ ❡sq✉❡♠át✐❝❛ ❞♦s sít✐♦s ❞❡ ❛❞s♦rçã♦ ♥♦ ❣r❛❢❡♥♦ ❡st✉❞❛❞♦s✳ ✳ ✳ ✳ ✼✼

✺✳✶✺ ❆✉ ❛❞s♦r✈✐❞♦ ♥❛ ♣♦s✐çã♦ ❇ ❡♠ ✉♠❛ s✉♣❡r❝é❧✉❧❛ ✹❳✹ ❞❡ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✼✽

✺✳✶✻ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ Pt q✉❛♥❞♦ ❛❞s♦r✈✐❞♦ ♥❛ ♠♦♥♦✲✈❛❝â♥❝✐❛ ❞❡ ✉♠❛ s✉♣❡r❝é❧✉❧❛ ✻❳✻ ❞❡ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾

✻✳✶ ❈♦♥str✉çã♦ ❞❛ r❡❞❡ ❞❡ ❜❛r✐❝❡♥tr♦s ♣❡❧❛ ❞❡♣♦s✐çã♦ ❞❡ ♠♦❧é❝✉❧❛s ❞❡ ❈❖✱ ✈✐❛ ❙❚▼✱ ♥❛ s✉♣❡r❢í❝✐❡ ❞❡ ❈✉✭✶✶✶✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶

✻✳✷ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ✧❜✉❧❦✧❞❡ ❘✉tê♥✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

✻✳✸ ❊str✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ s✐st❡♠❛ ❢♦r♠❛❞♦ ♣❡❧❛ ❛❞s♦rçã♦ ❞♦ ❘✉ ❡♠ ✉♠❛ s✉♣❡r❝é❧✉❧❛ ✺❳✺ ❞❡ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺

✻✳✹ ❙❡q✉ê♥❝✐❛ ❞❛ ✈❛r✐❛çã♦ ❞❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✻✳✺ ❩♦♥❛ ❞❡ ❇r✐❧❧♦✉✐♥ ❞❛s s✉♣❡r❝é❝❧✉❧❛s ✸❳✸✱ ✹❳✹ ❡ ✺❳✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

✻✳✻ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❛ s✉♣❡r❝é❧✉❧❛ ✻❳✻ ❞❡ ❣r❛❢❡♥♦ ♣r✐st✐♥♦✳ ✳ ✳ ✳ ✳ ✳ ✾✵

✻✳✼ ▲❉❖❙ ♣♦r s♣✐♥ ♣❛r❛ ❝❛❞❛ ❝♦♥❡ ❞❛ ❡str✉t✉r❛ ❞❛ ❞❡❜❛♥❞❛s ❞❛ s✉♣❡r❝é❧✉❧❛ ✺❳✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

✻✳✽ ▲❉❖❙ ♣❛r❛ ♦s ❝♦♥❡s ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐ ❞❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❛ s✉♣❡r❝é❧✉❧❛ ✹❳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷

✻✳✾ ❋✐❣✉r❛ ❡sq✉❡♠át✐❝❛ ❞❛ ♠♦❧é❝✉❧❛ ❇▼❘✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✻✳✶✵ ❋✉♥çã♦ ❞❡ ♦♥❞❛ ❞♦s ♦r❜✐t❛✐s ♠♦❧❡❝✉❧❛r❡s ❍❖▼❖ ✭dxz ❡dyz✲↑✮ ❡ ❞♦ ▲❖▼❖

①✐

✻✳✶✶ ❈♦♠♣❛r❛çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♥tr❡ ♦ ❍❖▼❖✭✉♣ ❞②③✮ ❡ ♦ ▲❯▼❖✭❞♦✇♥ ❞②③✮ ❡♠ r❡❧❛çã♦ à ✈❛r✐❛çã♦ ❞❛ ✐s♦s✉♣❡r❢í❝✐❡ ✉s❛❞❛ ♣❛r❛ ✏♣❧♦t❛r✑ ❛ ❢✉♥çã♦ ❞❡

♦♥❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺

✻✳✶✷ ▲❉❖❙ ❞❛ ❢❛♠í❧❛ ✹❳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

✻✳✶✸ ➪t♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ♥♦ ❜❛r✐❝❡♥tr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✻✳✶✹ ■♠❛❣❡♥ ❙❚▼ ❞❛ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❜❛r✐❝❡♥tr♦s ♣❛r❛ ✉♠❛ s✉♣❡r❝é❧✉❧❛ ✹❳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

✻✳✶✺ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦ ❘✉ ❞❡♣♦s✐t❛❞♦s ❡♠ ❝♦♥✜❣✉r❛çõ❡s tr✐❛♥❣✉❧❛r❡s ❡ ❤❡①❛❣♦♥❛✐s ♥♦ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵

✻✳✶✻ ❊str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞❛ r❡❞❡ r❡t❛♥❣✉❧❛r ❞❡ ❘✉ ❞❡♣♦s✐t❛❞♦s ❡♠ ❣r❛❢❡♥♦✳✶✵✷

✻✳✶✼ ✭❛✮❡str✉t✉r❛ ❞❡ ❜❛♥❞❛ ❞♦ ❣r❛❢❡♥♦ ♣r✐st✐♥♦ ❝♦♠ ❝♦♥❡s ❞❡ ❉✐r❛❝ ❝❡♥tr❛❧✐③❛❞♦s ♥♦s ♣♦♥t♦sK❡K′_{♥♦ ❡s♣❛ç♦ r❡❝í♣r♦❝♦✳ ✭❜✮◗✉❛♥❞♦ ✉♥✐❝❛♠❡♥t❡ ♦ ♣♦t❡♥❝✐❛❧}

❞❡ tr♦❝❛ é ❛♣❧✐❝❛❞♦✳ ✭❝✮ ◗✉❛♥❞♦ só ♦ ❙❖❈ ❘❛s❤❜❛ é ❛♣❧✐❝❛❞♦✱ ♦s ❡st❛❞♦s ❞❡

s♣✐♥ sã♦ ♠✐st✉r❛❞♦s ❡ ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞❡ s♣✐♥ é ❧❡✈❛♥t❛❞❛✳ ✭❞✮ ❯♠ ❣❛♣

❞❡ ✏❜✉❧❦✑ é ❛❜❡rt♦ q✉❛♥❞♦ t❛♥t♦ ♦ ❙❖❈ ❘❛s❤❜❛ ❡ ♦ ❝❛♠♣♦ ❞❡ ✐♥t❡r❝â♠❜✐♦

❡stã♦ ♣r❡s❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸

✻✳✶✽ ❊❢❡✐t♦ ❍◗❆ ♥♦ ❘✉✴●r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹

✻✳✶✾ ❱❛r✐❛çã♦ ❞♦ ●❛♣ ❙❖❈ ❡♠ r❡❧❛çã♦ ❛♦ t❛♠❛♥❤♦ ❞❛s s✉♣❡r❝é❧✉❧❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺

✻✳✷✵ ❱❛r✐❛çã♦ ❞❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❡♠ r❡❧❛çã♦ à ♠✉❞❛♥ç❛ ❞❛ ❞✐stâ♥❝✐❛ ❘✉✲❣r❛❢❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻

✻✳✷✶ ❚r❛♥s❢❡rê♥❝✐❛ ❞♦ ❙❖❈ ❞♦ ❇❛ ❛♦ ❣r❛❢❡♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✷✳✶ ❙♦❧✉çõ❡s ♣❛r❛ ♦s ❞♦✐s ♣♦ssí✈❡✐s ❛✉t♦✈❛❧♦r❡s ❞♦ ♦♣❡r❛❞♦r K ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷✳✷ ❊st❛❞♦s ❞❡ ♠❡♥♦r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧ j ❡ ♥ú♠❡r♦ q✉â♥t✐❝♦ ❙❖❈ ✭κ=±(j+1₂)✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷✳✸ ❊str✉t✉r❛ ✜♥❛✿ ❊st❛❞♦s ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦✳ ❆ ❡♥❡r❣✐❛ é ♠❡♥♦r ♣❛r❛ ♠❡♥♦r n ❡ ❧♦❣♦ ♣❛r❛ ♠❡♥♦rj✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✺✳✶ ❱❛❧♦r❡s ❞♦s ❣❛♣ ❙❖❈ ❡♠ ❡❱ ❝❛❧❝✉❧❛❞♦s ✉s❛♥❞♦ ♦ ❙■❊❙❚❆✲❙❖❈ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦s ✈❛❧♦r❡s ❡①♣❡r✐♠❡♥t❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✺✳✷ ❱❛❧♦r❡s ❞♦s ❣❛♣ ❙❖❈ ❡♠ ❡❱ ❝❛❧❝✉❧❛❞♦s ✉s❛♥❞♦ ♦ ❙■❊❙❚❆✲❙❖❈ ❝♦♠ ❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❛ ❜❛s❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

Cap´ıtulo

1

■♥tr♦❞✉çã♦

❖ ❡st❛❞♦ ✐s♦❧❛♥t❡✱ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✉♠ ❣❛♣ ❞❡ ❡♥❡r❣✐❛ q✉❡ s❡♣❛r❛ ❛s ❜❛♥❞❛s ❡❧❡trô♥✐❝❛s t♦t❛❧♠❡♥t❡ ♣r❡❡♥❝❤✐❞❛s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❜❛♥❞❛s ❝♦♠♣❧❡t❛♠❡♥t❡ ✈❛③✐❛s é ✉♠ ❞♦s ❡st❛❞♦s ♠❛✐s ❡❧❡♠❡♥t❛r❡s ❞❛ ♠❛tér✐❛✳ ❆ ❢ís✐❝❛ ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ ❞♦ sé❝✉❧♦ ❳❳ r❡✈♦❧✉❝✐♦♥♦✉✲s❡ ❝♦♠ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ✉♠ ♥♦✈♦ ❡st❛❞♦✱ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✏❜✉❧❦✑ ✐s♦❧❛♥t❡ ❝♦♠ ❡st❛❞♦s ♠❡tá❧✐❝♦s ❡stá✈❡✐s ♥❛ ❜♦r❞❛ r♦❜✉st♦s à ❞❡s♦r❞❡♠✱ ❞❡♥♦♠✐♥❛❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ✭❍◗✮✳ ❊st❡ ❡❢❡✐t♦ é ❝❛r❛t❡r✐③❛❞♦ ♣♦r ❛♣r❡s❡♥t❛r ✉♠❛ ♠❛❣♥❡t♦✲❝♦♥❞✉tâ♥❝✐❛ tr❛♥s✈❡rs❛❧ ✭σxy✮ q✉❛♥t✐③❛❞❛ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❡①tr❡♠❛❞❛♠❡♥t❡

♣r❡❝✐s❛ ❡♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ e2_{/~✱ ❛ ❞❡s♣❡✐t♦ ❞❡ t❡r♠♦s ✉♠ ❣❛♣ ❞❡ ❡♥❡r❣✐❛✳ ❖} ❡❢❡✐t♦ ❍◗ ♣r❡s❡♥t❡ ❡♠ ✉♠ ❣ás ❞❡ ❡❧étr♦♥s ❜✐❞✐♠❡♥s✐♦♥❛❧ ❛ ❜❛✐①❛s t❡♠♣❡r❛t✉r❛s s♦❜ ❛ ❛çã♦ ❞❡ ✉♠ ❢♦rt❡ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦ ♣❛r❡❝✐❛ ✈✐♦❧❛r ❛ ✐♠❛❣❡♠ ♥❛ q✉❛❧ ✉♠ s✐st❡♠❛ ❝♦♠ ❣❛♣ ❞❡ ❡♥❡r❣✐❛ ♥ã♦ ❛♣r❡s❡♥t❛ tr❛♥s♣♦rt❡ ❡❧❡trô♥✐❝♦✳ ❊st❡ ❛♣❛r❡♥t❡ ♣❛r❛❞♦①♦ ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r ▲❛✉❣❤❧✐♥❬✶❪✱ ♦❜s❡r✈❛♥❞♦ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ t❡♠ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ tr❛♥s♣♦rt❡✱ ♦✉ s❡❥❛✱ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❝♦❡rê♥❝✐❛ ❞❡ ❢❛s❡ ❞❡ ❢✉♥çõ❡s ❞❡ ♠✉✐t♦s ❝♦r♣♦s✳ ❆ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ❞❡ ❝♦♠♦ ❡st❛ ❢❛s❡ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ σxy ❢♦✐ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡r✐✈❛❞❛ ♣♦r ❚❤♦✉❧❡ss✱ ❑♦❤♠♦t♦✱ ◆✐❣❤t✐♥❣ ❡ ◆✐❥s ✭❚❑◆◆✮❬✷❪✱

tr❛③❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ t♦♣♦❧♦❣✐❛ ♣❛r❛ ❛ ❝❛r❛t❡r✐③❛çã♦ ❞❡ ❡st❛❞♦s ✐s♦❧❛♥t❡s ♥❛ ♠❛tér✐❛✳ ❊♥q✉❛♥t♦ ♦s ♠❡t❛✐s ❛♣r❡s❡♥t❛♠ ✉♠❛ ❝♦rr❡♥t❡ ✏❖❤♠✐❝❛✑✱ ♦r✐✉♥❞❛ ❞♦ ❞❡s✈✐♦ ❞♦ ❡q✉✐❧í❜r✐♦ ♥❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❋❡r♠✐✲❉✐r❛❝ s♦♠❡♥t❡ ♣❛r❛ ❡st❛❞♦s ♣ró①✐♠♦s

❞❡ ♥í✈❡❧ ❞❡ ❋❡r♠✐✱ ❛ ❝♦rr❡♥t❡ ❍❛❧❧ é ✉♠❛ ❝♦rr❡♥t❡ t♦♣♦❧ó❣✐❝❛ q✉❡ ❡♥✈♦❧✈❡ t♦❞♦s ♦s ❡st❛❞♦s ❛❜❛✐①♦ ❞♦ ♥í✈❡❧ ❞❡ ❋❡r♠✐✳ ❆ ❝♦♥❞✉tâ♥❝✐❛ ❍❛❧❧ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞♦ ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ❞❡ ❜♦r❞❛✱ ♦s q✉❛✐s sã♦ ✉♥✐❞✐r❡❝✐♦♥❛✐s ❞❡✈✐❞♦ à q✉❡❜r❛ ❞❡ s✐♠❡tr✐❛ ❞❡ r❡✈❡rsã♦✲t❡♠♣♦r❛❧ q✉❡ r❡s✉❧t❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❡①t❡r♥♦✳

◆❛ ♣r✐♠❡✐r❛ ❞é❝❛❞❛ ❞♦ sé❝✉❧♦ ❳❳■ ❛ ❢ís✐❝❛ ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ ✈♦❧t❛ à ❡①❛❧t❛çã♦ ♣❡❧❛ ❞❡s❝♦❜❡rt❛ ❞❡ ✉♠ ❛♥á❧♦❣♦ ❞♦ ◗❍ ❞❡♥♦♠✐♥❛❞♦ ❡❢❡✐t♦ ❍❛❧❧ ◗✉â♥t✐❝♦ ❞❡ ❙♣✐♥ ✭◗❙❍✮✳ ◆❡st❡ ❡❢❡✐t♦✱ ♣r❡❞✐t♦ ♣♦r ❑❛♥❡ ❡ ▼❡❧❡❬✸❪ ♦ ♣❛♣❡❧ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ é ❢♦r♥❡❝✐❞♦ ♣❡❧♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✱ ❞❡ ❢♦r♠❛ q✉❡ ❛ s✐♠❡tr✐❛ ❞❡ r❡✈❡rsã♦ t❡♠♣♦r❛❧ é ♠❛♥t✐❞❛✳ ❊st❛ s✐♥❣✉❧❛r ❝❛r❛t❡ríst✐❝❛ r❡s✉❧t❛ ❡♠ ✉♠ s✐st❡♠❛ ✐s♦❧❛♥t❡ ♥♦ ✏❜✉❧❦✑ ❝♦♠ ✉♠ ♣❛r ❞❡ ❡st❛❞♦s ❞❡ ❜♦r❞❛s ❝♦♠ s♣✐♥ ♣r♦♣❛❣❛♥❞♦✲s❡ ❡♠ ❞✐r❡çõ❡s ♦♣♦st❛s✳ ❊st❡s ❡st❛❞♦s ❛♣r❡s❡♥t❛♠ ✉♠❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❧✐♥❡❛r ✲ ❝♦♥❡ ❞❡ ❉✐r❛❝✱ ♦ q✉❛❧ ♠♦str❛ ✉♠❛ t❡①t✉r❛ ❞❡ s♣✐♥ ❝♦♠ ❝♦rr❡♥t❡s ❞❡ s♣✐♥ s❡♠ ❞✐ss✐♣❛çã♦✱ s✉♣r✐♠✐♥❞♦ ♦ r❡tr♦❡s♣❛❧❤❛♠❡♥t♦✳ ❊♠ ❝♦♥s❡q✉ê♥❝✐❛✱ ♦s ❡st❛❞♦s ♠❡tá❧✐❝♦s ❡stá✈❡✐s ♥❛ ❜♦r❞❛ sã♦ r♦❜✉st♦s à ❞❡s♦r❞❡♠ s❡♥❞♦ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ✏♣r♦t❡çã♦✑ r❡❣✐❞♦ ♣❡❧❛ s✐♠❡tr✐❛ ❞❡ r❡✈❡rsã♦ t❡♠♣♦r❛❧✳ ❖ ♣r✐♠❡✐r♦ ♠❛t❡r✐❛❧ q✉❡ ❛♣r❡s❡♥t❛ ♦ ◗❙❍ ❢♦✐ ♣r❡❞✐t♦ ♣r✐♠❡✐r❛♠❡♥t❡ ♣♦r ❩❤❛♥❣ ❡t ❛❧✳❬✹❪✱ ❡ ♦❜s❡r✈❛❞♦ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♣♦r ❑ö♥✐❣ ❡t ❛❧✳❬✺❪✱ ♥♦ ♣♦ç♦ q✉â♥t✐❝♦ ✭◗❲✮ ❞❡ ❍❣❚❡✴❈❞❚❡✳ ❊♥tr❡t❛♥t♦✱ ❡①✐st❡♠ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ✸❉ ❞♦ ◗❙❍ ❞❡♥♦♠✐♥❛❞♦ ✐s♦❧❛♥t❡ t♦♣♦❧ó❣✐❝♦ ✭❚■✮✳ ❖ ❚■ ❢♦✐ ♣r❡❞✐t♦ ♦❝♦rr❡r ♥❛ ❧✐❣❛ ❇✐1−x❙❜x ♣❛r❛ ✉♠❛ ❝❡rt❛ ❝♦♠♣♦s✐çã♦ ❡✱ r❡❝❡♥t❡♠❡♥t❡ ❍s✐❡❤ ❡t

❛❧✳❬✻❪✱ ❛tr❛✈és ❞❡ ❊s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❋♦t♦❡♠✐ssã♦ ❝♦♠ ❘❡s♦❧✉çã♦ ❆♥❣✉❧❛r ✭❆❘P❊❙✮ ♦❜s❡r✈❛r❛♠ ♦s ❡st❛❞♦s ❞❡ s✉♣❡r❢í❝✐❡ ♥ã♦✲tr✐✈✐❛✐s ♥❡ss❡ s✐st❡♠❛✳ ❊st❛ ❝❧❛ss❡ ❞❡ ♠❛t❡r✐❛✐s✱ ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ❡①ót✐❝❛s✱ ♦❢❡r❡❝❡ ✉♠ ❣r❛♥❞❡ ♣♦t❡♥❝✐❛❧ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ♥♦ ❝❛♠♣♦ ❞❛ s♣✐♥trô♥✐❝❛✳ ❆ ❞❡s❝♦❜❡rt❛ ❞❡ ♠❛t❡r✐❛✐s q✉❡ ❛♣r❡s❡♥t❛♠ ❡st❛❞♦s t♦♣♦❧ó❣✐❝♦s ♥ã♦ tr✐✈✐❛✐s t♦r♥♦✉✲s❡ ✉♠ ❞♦s ❛t✉❛✐s ❢♦❝♦s ❞❡ ✐♥t❡r❡ss❡ ❞♦s ❢ís✐❝♦s t❡ór✐❝♦s ❡ ❡①♣❡r✐♠❡♥t❛✐s ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛✳

◆❛ ♣r❡❞✐çã♦ ❞❡ ♥♦✈♦s ♠❛t❡r✐❛✐s✱ s❡✉ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ♦ ❡st✉❞♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠ ❞✐❢❡r❡♥t❡s ❝♦♥❞✐çõ❡s✱ ❛s s✐♠✉❧❛çõ❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣♦ss✉❡♠

✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ❞❛❞♦ q✉❡ ♣❡r♠✐t❡♠ ♦ ❝♦♥tr♦❧❡ ❞❡ ❝♦♥❞✐çõ❡s✱ ♣❛râ♠❡tr♦s ❡ ❡❧❡♠❡♥t♦s✳ ❈♦♥t✉❞♦✱ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❡❢❡✐t♦ ❍❛❧❧ q✉â♥t✐❝♦ ❞❡ s♣✐♥ ✈✐❛ ❝á❧❝✉❧♦s ❞❡ ♣r✐♠❡✐r♦s ♣r✐♥❝í♣✐♦s ♣r❡❝✐s❛✲s❡ ❞♦ ✉s♦ ❞❡ ✉♠ ❝ó❞✐❣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ q✉❡ ❧❡✈❡ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✳ ◆♦ ❝♦♥t❡①t♦ ❞❡st❡ tr❛❜❛❧❤♦ é ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ♦s ♠ét♦❞♦s ✏❛❜ ✐♥✐t✐♦✑ ❜❛s❡❛❞♦s ♥❛ t❡♦r✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ✭❉❋❚✮ t❡♠ s✐❞♦ ♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❡st❛❞♦s t♦♣♦❧ó❣✐❝♦s✱ ♣r❡❞✐çã♦ ❞❡ ♦✉tr♦s ♠❛t❡r✐❛✐s✱ ❡❢❡✐t♦s ❞❡ ✐♠♣✉r❡③❛s ♠❛❣♥ét✐❝❛s✱ tr❛♥s♣♦rt❡ ❡❧❡trô♥✐❝♦ ❡t❝✳ ◆♦ss♦ ❣r✉♣♦✱ ❞❡♥tr♦ ❞♦s ❝ó❞✐❣♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❞❡ ❝á❧❝✉❧♦s ❞❡ ♣r✐♠❡✐r♦s ♣r✐♥❝í♣✐♦s ✉t✐❧✐③❛ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♦ ❝ó❞✐❣♦ ❙■❊❙❚❆✱ ❡ t❡♠ ❢❡✐t♦ ✐♠♣❧❡♠❡♥t❛çõ❡s ❞❡ ❢♦r♠❛ ❛ ❡♥r✐q✉❡❝❡r ♦ tr❛t❛♠❡♥t♦ t❡ór✐❝♦✳ ❈♦♠♦ ❡①❡♠♣❧♦ ✐♠♣❧❡♠❡♥t❛♠♦s ❛ ✭i✮ ❝♦rr❡çã♦ ❞❡ ❛✉t♦✲✐♥t❡r❛çã♦ ✭❙■❈✮ ❡ ♦ ✭ii✮ tr❛♥s♣♦rt❡ ❡❧❡trô♥✐❝♦ ❜❛❧íst✐❝♦ ✈✐❛ ❢✉♥çõ❡s ❞❡ ●r❡❡♥ ❢♦r❛ ❞♦ ❡q✉✐❧í❜r✐♦ ✭❚r❛♥s❛♠♣❛✲❝♦❞❡✮✱ ❡t❝✳ ◆❡st❡ tr❛❜❛❧❤♦ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ♣s❡✉❞♦♣♦t❡♥❝✐❛✐s r❡❧❛t✐✈íst✐❝♦s ❞❡ ♥♦r♠❛ ❝♦♥s❡r✈❛❞❛ ✐♠♣❧❡♠❡♥t❛♠♦s ♥♦ ❝ó❞✐❣♦ ❙■❊❙❚❆ ❛s ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❝♦♠♦ ♦ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡trô♥✐❝❛s ❞♦s ❡st❛❞♦s t♦♣♦❧ó❣✐❝♦s ♥ã♦✲tr✐✈✐❛✐s✳ ❊st❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❢♦✐ t❡st❛❞❛ ✈✐❛ ♦ ❡st✉❞♦ ❞♦s s❡❣✉✐♥t❡s ❢❡♥ô♠❡♥♦s✿

✐✳ ✏❙♣✐♥✲s♣❧✐tt✐♥❣✑ ♥♦ ●❛❆s✳

✐✐✳ ◗❙❍ ♥♦ ♣♦ç♦ q✉â♥t✐❝♦ ❞❡ ❍❣❚❡✴❈❞❚❡✳

✐✐✐✳ ❆ ♣r❡s❡♥ç❛ ❞❡ ❡st❛❞♦s ♠❡tá❧✐❝♦s ♣r♦t❡❣✐❞♦s ♣❡❧❛ s✐♠❡tr✐❛ ❞❡ r❡✈❡rsã♦ t❡♠♣♦r❛❧ ♥❛ s✉♣❡r❢í❝✐❡ ❞♦ ❚■ ❇✐2❙❡3 ❡ ♦s ❡❢❡✐t♦s ❞❛ ♦r❞❡♠ ♠❛❣♥ét✐❝❛ ♥❡st❡ ♠❛t❡r✐❛❧ ❝♦♠ át♦♠♦s ❞❡ ▼♥ ❛❞s♦r✈✐❞♦s ♥❛ s✉♣❡r❢í❝✐❡✳

✐✈✳ ❊❢❡✐t♦s ❞♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛ ♥❛ ✐♥t❡r❛çã♦ ❞❡ ♠❡t❛✐s ❞❡ tr❛♥s✐çã♦ q✉❛♥❞♦ ❛❞s♦r✈✐❞♦s ❡♠ ❣r❛❢❡♥♦✳

❊st❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❢♦✐ ✉s❛❞❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ❢❛s❡s t♦♣♦❧ó❣✐❝❛s ♣r❡s❡♥t❡s ❡♠ s✐st❡♠❛s ❢♦r♠❛❞♦s ♣❡❧❛ ❞❡♣♦s✐çã♦ ❞❡ át♦♠♦s ♥♦ ❣r❛❢❡♥♦✳ ▼♦t✐✈❛❞♦s ♣❡❧❛

❡♥❣❡♥❤❛r✐❛ ❞❡ s✉♣❡r❢í❝✐❡s ♥♦ ❣r❛❢❡♥♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❡st❛❞♦s ❛ss♦❝✐❛❞♦s ❝♦♠ ♦s ❡❢❡✐t♦s✿ ❍❛❧❧ q✉â♥t✐❝♦ ❞❡ s♣✐♥ ❡ ❍❛❧❧ q✉â♥t✐❝♦ ❛♥ô♠❛❧♦❬✻✹✱✼✺❪✱ ♣r♦❝✉r❛♠♦s át♦♠♦s q✉❡ ♣❡r♠✐t❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡st❡s ❡❢❡✐t♦s q✉❛♥❞♦ ❛❞s♦r✈✐❞♦ ♥♦ ❣r❛❢❡♥♦✳ ❊♥tr❡t❛♥t♦ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❡♥❢♦❝❛❞♦ ♥♦ ❡st✉❞♦ ❞❛s ❢❛s❡s t♦♣♦❧ó❣✐❝❛s ❞❡ ✉♠❛ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❜❛r✐❝❡♥tr♦s✳ ❊st❡ s✐st❡♠❛ r❡s✉❧t❛ ❞❛ ❞❡♣♦s✐çã♦ ❞❡ ❘✉ ❡♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ tr✐❛♥❣✉❧❛r ♥♦ ❣r❛❢❡♥♦✱ ♦✉ s❡❥❛✱ ❡st❛ s✐♥❣✉❧❛r ❝♦♥✜❣✉r❛çã♦ ❛♣r❡s❡♥t❛ ♣✐❝♦s ♥❛ ❞❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s ❧♦❝❛❧✐③❛❞♦s ♥♦ ❝❡♥tr♦ ❣❡♦♠étr✐❝♦ ❞♦s tr✐â♥❣✉❧♦s ❢♦r♠❛❞♦s ♣❡❧♦s ❘✉✳ ❊st❡s ❝❡♥tr♦s ❣❡♦♠étr✐❝♦s ✭❜❛r✐❝❡♥tr♦s✮ ❢♦r♠❛♠ ✉♠❛ r❡❞❡ ❝♦♠ s✐♠❡tr✐❛ ❤❡①❛❣♦♥❛❧ q✉❡ ❞❡ ❢♦r♠❛ ❡❢❡t✐✈❛ ❣❡r❛ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❝♦♠ q✉❛tr♦ ❝♦♥❡s ❞❡ ❉✐r❛❝✳ ◆♦ss❛ ✐♠♣❧❡♠❡♥t❛çã♦ ♣❡r♠✐t✐✉ ✐♥❢❡r✐r ❛ ❝❧❛ss❡ t♦♣♦❧ó❣✐❝❛ ❞❡st❡ s✐st❡♠❛✳

❆ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ✐♥t❡r❛çã♦ s♣✐♥✲ór❜✐t❛ ♥♦s ❝á❧❝✉❧♦s ❞❡ ❡str✉t✉r❛ ❡❧❡trô♥✐❝❛ ✉s❛♥❞♦ t❡♦r✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ❡①✐❣❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ♦r✐❣❡♠ ❞❡st❡ ❢❡♥ô♠❡♥♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡①♣❧✐❝❛❞♦ ❝♦♠♦ ♦ t❡r♠♦ ❛ss♦❝✐❛❞♦ ❛♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛ ❞❡r✐✈❛ ♥❛t✉r❛❧♠❡♥t❡ ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ t❡♦r✐❛ r❡❧❛t✐✈íst✐❝❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❞❡ ❉✐r❛❝✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞♦ ♦ ♣r♦❜❧❡♠❛ ❡❧❡trô♥✐❝♦ ❡ ❝♦♠♦ ❛ t❡♦r✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ❞❡♥s✐❞❛❞❡ ♣❡r♠✐t❡ r❡s♦❧✈ê✲❧♦ ❢❛③❡♥❞♦ ✈✐á✈❡❧ ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❝♦♠♣♦st♦s ♣♦r ♠✉✐t♦s át♦♠♦s ❝♦♠ ❛ ✐♥❝❧✉sã♦ ❞♦s ♣s❡✉❞♦♣♦t❡♥❝✐❛✐s ❞❡ ♥♦r♠❛ ❝♦♥s❡r✈❛❞❛✳ ◆♦ ❝❛♣ít✉❧♦ q✉❛tr♦ ❛♣r❡s❡♥t❛✲s❡ ❛ ❢♦r♠❛ ❞❡ ❡①tr❛✐r ♦ ♣s❡✉❞♦♣♦t❡♥❝✐❛❧ ❞❡ s♣✐♥✲ór❜✐t❛ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧✱ ❛s ❛♣r♦①✐♠❛çõ❡s ❢❡✐t❛s ♣❛r❛ ❛ ✐♥❝❧✉sã♦ ❞❡st❡ t❡r♠♦ ♥♦ ❢♦r♠❛❧✐s♠♦ ❞❛ t❡♦r✐❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❞❛ ❞❡♥s✐❞❛❞❡✳ ◆❛ s❡q✉ê♥❝✐❛✱ ♥♦ ❝❛♣ít✉❧♦ ❝✐♥❝♦ sã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♦s ❡st✉❞♦s q✉❡ ✈❡r✐✜❝❛♠ ♦ ❜♦♠ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡st❛ ✐♠♣❧❡♠❡♥t❛çã♦✳ ◆♦ ❝❛♣ít✉❧♦ s❡✐s✱ ♣r❡❞✐③✲s❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ r❡❞❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❜❛r✐❝❡♥tr♦s ❣❡r❛❞❛ ♣❡❧❛ ❞❡♣♦s✐çã♦ ❞❡ át♦♠♦s ♥♦ ❣r❛❢❡♥♦ ❡ ❡st✉❞❛♠✲s❡ ❛s ♣♦ssí✈❡✐s ❢❛s❡s t♦♣♦❧ó❣✐❝❛s ❞❡st❡ s✐st❡♠❛✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ♠♦str❛✲s❡ q✉❡ ❡①✐st❡♠ ♠❡t❛✐s ✭♥ã♦ r❡♣♦rt❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✮ q✉❡ q✉❛♥❞♦ ❛❞s♦r✈✐❞♦s ♥♦ ❣r❛❢❡♥♦ ❡st❛❜✐❧✐③❛♠ ❢❛s❡s t♦♣♦❧ó❣✐❝❛s ❛ss♦❝✐❛❞❛s ❝♦♠ ❡st❛❞♦s ◗❙❍✳

Cap´ıtulo

2

❖ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛

◆❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❙❝❤rö❞✐♥❣❡r ♦❜té♠✲s❡ ✉♠❛ ❞❡s❝r✐çã♦ ❛♣r♦①✐♠❛❞❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s s✐st❡♠❛s ❛tô♠✐❝♦s✳ ❊♠❜♦r❛ ❡st❛ ❞❡s❝r✐çã♦ ♣❡r♠✐t❛ ✉♠ ❡♥t❡♥❞✐♠❡♥t♦ ♣r♦❢✉♥❞♦ ❞❛ ♠❛tér✐❛✱ ♥ã♦ ✐♥❝❧✉✐ ♦s ❡❢❡✐t♦s ❛ss♦❝✐❛❞♦s às ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s✳

❖ ❡s♣❡❝tr♦ ❛tô♠✐❝♦ é ❢♦rt❡♠❡♥t❡ ❛❢❡t❛❞♦ ♣❡❧♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛ ✭❙❖❈✮❬✽❪✱ ♦ q✉❛❧ ❡✈✐❞❡♥❝✐❛✲s❡ ♥❛ ❡str✉t✉r❛ ✜♥❛ ❞♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛✳ ❖ ❙❖❈ é ✉♠ ❡❢❡✐t♦ q✉â♥t✐❝♦ r❡❧❛t✐✈íst✐❝♦ s❡♠ ❛♥á❧♦❣♦ ❝❧áss✐❝♦ ❛ss♦❝✐❛❞♦ à ✐♥t❡r❛çã♦ ❞♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ✐♥trí♥s❡❝♦ ❞♦ ❡❧étr♦♥ ❝♦♠ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧ ❞❡st❡✳ ◆♦s át♦♠♦s✱ ❛s ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s ❛✉♠❡♥t❛♠ ❞❡ r❡❧❡✈â♥❝✐❛ ❡♥q✉❛♥t♦ ❛✉♠❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ♦r❜✐t❛✐s ❛tô♠✐❝♦s ♦❝✉♣❛❞♦s✶_{✳ ❆ss✐♠✱ ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s q✉❡ ❛♣r❡s❡♥t❛♠ ♦s s✐st❡♠❛s} q✉❡ ♣♦ss✉❡♠ át♦♠♦s ♣❡s❛❞♦s✱ t❛❧ ❝♦♠♦ ♦ ❢♦rt❡ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✱ ♥ã♦ ♣♦❞❡♠ s❡r ❡♥t❡♥❞✐❞♦s ♥❛ ✈✐sã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳

P♦❞❡✲s❡ ❡♥t❡♥❞❡r ❛ ♦r✐❣❡♠ ❞♦ ❙❖❈ ♠❡❞✐❛♥t❡ ♦ ❡st✉❞♦ ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛

✶_{❖ t❛♠❛♥❤♦ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ❡stá ❞❛❞♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦✿}

a0=

~

αmc =⇒p∼αmc,

❝♦♠α= e2 4πǫ0

1

~c✳ ❆s ❝♦♥tr✐❜✉✐çõ❡s ❞♦s ❡❢❡✐t♦s r❡❧❛t✐✈íst✐❝♦s ♥❛ ❡♥❡r❣✐❛E 2

=p2

c2

+m2

c4 _{❝r❡s❝❡♠}

❝♦♠ ❛ ♠❛ss❛✱ ❡♠ q✉❛♥t♦ q✉❡ ♦ ❝♦❝✐❡♥t❡ pc

mc2 ∼ α ❞✐♠✐♥✉✐✳ ❆ss✐♠✱ ❛♦ ❛✉♠❡♥t❛r ♦ t❛♠❛♥❤♦ ❞♦

át♦♠♦ ❛s ❝♦♥tr✐❜✉✐çõ❡s ❞♦s ❡❢❡✐t♦s ❛✉♠❡♥t❛♠✳

r❡❧❛t✐✈íst✐❝❛ ♣r♦♣♦st❛ ♣♦r P❛✉❧ ❉✐r❛❝✱ ❛ q✉❛❧ ❡♠ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦r❞❡♠ ③❡r♦ ❝♦rr❡s♣♦♥❞❡ à ❞❡s❝r✐çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ◆♦ ❡♥t❛♥t♦✱ ✐♥tr♦❞✉③ ♦ ❙❖❈ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❝♦♠♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥❛ ❢ís✐❝❛ ❛tô♠✐❝❛✱ ❛ ✐♥t❡r❛çã♦ s♣✐♥✲ór❜✐t❛ ❡♥tr❛ ♥♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♥ã♦✲r❡❧❛t✐✈íst✐❝❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝✳ ❊st❛ ❛♣r♦①✐♠❛çã♦ ❞á ♦r✐❣❡♠ ❛♦ t❡r♠♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡r♠♦ ❙❖ ❞❡ P❛✉❧✐✿

HSO =−

~

4m2 0c2

σ·pˆ×(∇V0), ✭✷✳✶✮ ♦♥❞❡~é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦✱_m0 é ❛ ♠❛ss❛ ❞❡ ✉♠ ❡❧étr♦♥ ❧✐✈r❡✱_cé ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛

❧✉③✱pˆé ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❧✐♥❡❛r✱ V0 é ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ❈♦✉❧♦♠❜ ❞♦ ♥ú❝❧❡♦ ❛tô♠✐❝♦ ❡ σ = (σx, σy, σz) é ♦ ✈❡t♦r ❞❛s ♠❛tr✐③❡s ❞❡ P❛✉❧✐✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ❡♥t❡♥❞❡r

❝❧❛r❛♠❡♥t❡ ❝♦♠♦ ♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞♦ s♣✐♥ ❛❢❡t❛♠✲s❡ ♣❡❧♦ ❙❖❈ ♥♦s s✐st❡♠❛s ❛tô♠✐❝♦s✳

❊♠ ✉♠ só❧✐❞♦ ❝r✐st❛❧✐♥♦✱ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❡❧étr♦♥s ❡stá ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛ ❡♥❡r❣✐❛ ❞❡ ❜❛♥❞❛sEn(k)❝♦♠ í♥❞✐❝❡ ❞❡ ❜❛♥❞❛n❡ ✈❡t♦r ❞❡ ♦♥❞❛k✳ ■❣✉❛❧♠❡♥t❡✱ ♥❡st❡

❝❛s♦ ♦ ❙❖❈ ❛♣r❡s❡♥t❛ ✉♠ ♣r♦❢✉♥❞♦ ❡❢❡✐t♦ ♥❛s ❡♥❡r❣✐❛sEn(k)❛ss♦❝✐❛❞❛s à ❡str✉t✉r❛

❞❡ ❜❛♥❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ❛ s❡♣❛r❛çã♦ ❩❡❡♠❛♥ ❞❡ ❡❧étr♦♥s ❡ ❜✉r❛❝♦s❬✾❪✳

❖ ❙❖❈ t❡♠ ❝♦♥s❡q✉ê♥❝✐❛s ✐♠♣♦rt❛♥t❡s ♥❛ ❡str✉t✉r❛ ❞❡ ❜❛♥❞❛s ❞♦s s❡♠✐✲ ❝♦♥❞✉t♦r❡s❬✽❪✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ●❛❆s ❞á ♦r✐❣❡♠ ❛ ✉♠❛ q✉❡❜r❛ ❞❡ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ♥♦ t♦♣♦ ❞❛ ❜❛♥❞❛ ❞❡ ✈❛❧ê♥❝✐❛ ❝✉❥♦s ❡st❛❞♦s ❡❧❡trô♥✐❝♦s tê♠ ❝❛rát❡rp✭♠♦♠❡♥t♦ ♦r❜✐t❛❧ ❛♥❣✉❧❛rl = 1✮✳ ◆❛ ♣r❡s❡♥ç❛ ❞♦ ❙❖❈ ♦s ❡st❛❞♦s ❝♦♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧j = 3/2

❡ j = 1/2 ❡stã♦ s❡♣❛r❛❞♦s ❡♠ ❡♥❡r❣✐❛ ♣♦r ✉♠ ❣❛♣ ❙❖❈ ∆0✳ ❊♠❜♦r❛✱ ♦ ❡①❡♠♣❧♦

❛♥t❡r✐♦r ✐❧✉str❛ ❝♦♠♦ ♦ ♠♦✈✐♠❡♥t♦ ♦r❜✐t❛❧ ❞♦s ❡❧étr♦♥s ❡♠ ✉♠ ❝r✐st❛❧ é ❛❢❡t❛❞♦ ♣❡❧♦ ❙❖❈✱ ❞❡ ❢♦r♠❛ ❣❡r❛❧ ♥ã♦ é ó❜✈✐♦ ❝♦♠ ♦s ❡❢❡✐t♦s ❞♦ ❙❖❈ ❡✈✐❞❡♥❝✐❛♠✲s❡ ♥❛ ❡str✉t✉r❛ ❡❧❡trô♥✐❝❛ ❞♦s só❧✐❞♦s ❝r✐st❛❧✐♥♦s✳

P❛r❛ ❡♥t❡♥❞❡r ♦ ❢♦r♠❛❧✐s♠♦ q✉❡ ♣❡r♠✐t❡ ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❙❖❈

♥♦ ❡st✉❞♦ ❞❛ ❡str✉t✉r❛ ❡❧❡trô♥✐❝❛ ❞❡ ✉♠ só❧✐❞♦ ❝r✐st❛❧✐♥♦ é ♥❡❝❡ssár✐♦ r❡✈✐s❛r ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❛ss♦❝✐❛❞♦s à ✐♥❝❧✉sã♦ ❞❡st❡ ❡❢❡✐t♦ ♥❛ ❢ís✐❝❛ ❛tô♠✐❝❛✳

✷✳✶ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝

❖ ❢❛t♦ ❞❡ q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦ ❞❡s❝r❡✈❛ ❡❢❡✐t♦s ❛ss♦❝✐❛❞♦s ❝♦♠ ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s✱ ♦✉ s❡❥❛✱ ♥ã♦ s❡❥❛ ✐♥✈❛r✐❛♥t❡ ▲♦r❡♥t③✱ ♠❛♥✐❢❡st❛✲s❡ ❝❧❛r❛♠❡♥t❡ ♥❛ ❞✐❢❡r❡♥ç❛ ❞❛ ♦r❞❡♠ ❞❛s ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s ❡ t❡♠♣♦r❛✐s ❛ss♦❝✐❛❞❛s às ❝♦♠♣♦♥❡♥t❡s ❞♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡ à ❡♥❡r❣✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿

ˆ

H0ψn(r, t) =

− ~

2m∇

2_+V(r)

ψn(r, t) =i~

∂

∂tψn(r, t), ✭✷✳✷✮ s❡♥❞♦ _Hˆ_{0 ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❞❡ ✉♠ s✐st❡♠❛ ♥❛ ♣r❡s❡♥ç❛ ❞❡}

✉♠ ♣♦t❡♥❝✐❛❧V(x) ❡ ψn(x, t)❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛✳

P❛✉❧ ❉✐r❛❝ ♣r♦♣ôs ✉♠❛ ❡q✉❛çã♦ ♥❛ q✉❛❧ ❛♣❛r❡❝❡r❛♠ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛ç♦ t❡♠♣♦r❛✐s ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❛♣r❡s❡♥t❛r❛ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ♣♦ssí✈❡✐s ♣❛r❛ ❛ r❡❧❛çã♦ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦✳ ❆t✉❛❧♠❡♥t❡ ❡st❛ ❡q✉❛çã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝✳ ❖ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ ♣r♦♣♦st♦ ♣♦r ❉✐r❛❝ ❡s❝r❡✈❡✲s❡ ❝♦♠♦✿

ˆ

H =cα·pˆ+βmc2 =

3 X

i=1

cαipi+βmc2, ✭✷✳✸✮

♦♥❞❡ c é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦✱ m é ❛ ♠❛ss❛ ❡♠ r❡♣♦✉s♦ ❞❛ ♣❛rtí❝✉❧❛✱ pˆ ♦

♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❧✐♥❡❛r ❡α ❡ β sã♦ ❝♦❡✜❝✐❡♥t❡s✳

P♦❞❡✲s❡ ♦❜t❡r ❛ ❢♦r♠❛ ❞♦s ❝♦❡✜❝✐❡♥t❡s ✐♠♣♦♥❞♦ q✉❡ s❡❥❛ s❛t✐s❢❡✐t❛ ❛ r❡❧❛çã♦ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ r❡❧❛t✐✈íst✐❝❛E2 _=p2_c2_+m2_c4_{✳ ❆ss✐♠✱ ❡♠ t❡r♠♦s ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦}

❞❡ ❉✐r❛❝ ❡st❛ r❡❧❛çã♦ r❡❡s❝r❡✈❡✲s❡ ❝♦♠♦✿

ˆ

H2 =c2 3 X

i,k=1

1

2(αiαk+αkαi)p

i

pk+mc3 3 X

i=1

(αiβ+βαi)pi+β2m2c4. ✭✷✳✹✮

▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❡i❡k sã♦ í♥❞✐❝❡s ♠✉❞♦s ❡pi_❡pk _{❝♦♠✉t❛♠✱ ❛s r❡❧❛çõ❡s✿}

αiαk+αkαi = 2δik; αiβ+βαi = 0; β2 = 1 ✭✷✳✺✮

sã♦ ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ❡ ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❉✐r❛❝ ❡st❡❥❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ r❡❧❛çã♦ ❡♥❡r❣✐❛✲♠♦♠❡♥t♦ r❡❧❛t✐✈íst✐❝❛✳

P❛r❛ s❛t✐s❢❛③❡r ❡st❛s ❝♦♥❞✐çõ❡s ♦s ❝♦❡✜❝✐❡♥t❡s αi ❡ β ❞❡✈❡♠ s❡r ♠❛tr✐③❡s✳

❈♦♠♦ ❝♦♥❞✐çã♦ ❛❞✐❝✐♦♥❛❧ ❛s ♠❛tr✐③❡♠ ❞❡✈❡♠ s❡r ❤❡r♠✐t✐❛♥❛s ♣❛r❛ q✉❡ ♦ ❤❛♠✐❧t♦♥✐✲ ❛♥♦ t❛♠❜é♠ ♦ s❡❥❛✳

❆s ♠❛tr✐③❡s ❞❛ ♠í♥✐♠❛ ❞✐♠❡♥sã♦ q✉❡ s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦ αiαk+αkαi =

2δik sã♦ ❛s ♠❛tr✐③❡s ❞❡ P❛✉❧✐ ✷✱ ❡s❝r✐t❛s ❝♦♠♦✿

σx=

  0 1

1 0

 

; σy =

  0 −i

i 0

 

; σz =

  1 0

0 −1

 

. ✭✷✳✻✮

◆♦ ❡♥t❛♥t♦✱ ♥❡st❛ ❞✐♠❡♥sã♦ ✭✷①✷✮ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠❛ ♠❛tr✐③ q✉❡ ❝✉♠♣r❛β2 _{= 1 ❡} βσi+σiβ = 0✳ P❛r❛ s❛t✐s❢❛③❡r ❡st❛s ❞✉❛s ❝♦♥❞✐çõ❡s é ♥❡❝❡ssár✐♦ ❝♦♥s✐❞❡r❛r ♠❛tr✐③❡s

❞❡ ❞✐♠❡♥sã♦ ✹✱ s❡♥❞♦ ✉♠❛ ♣♦ssí✈❡❧ ❡s❝♦❧❤❛✿

αi =

 

 0 σi σi 0

 

; β =

 

 L 0

0 −L

 

. ✭✷✳✼✮

♦♥❞❡ L é ❛ ♠❛tr✐③ ✉♥✐tár✐❛ ✷①✷✳ ◆❛ ❞❡s❝r✐çã♦ ❞❡ ❉✐r❛❝ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❛

✷_{❆s ♠❛tr✐③❡s ❞❡ P❛✉❧✐ s❛t✐s❢❛③❡♠✿}

σiσk+σkσi={σi, σk}= 2δik; [σi, σj] = 2i∈ijkσk

❡q✉❛çã♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ❡ ❛✉t♦✈❡t♦r❡s ♣❛r❛ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ t❡♠ q✉❛tr♦ ❝♦♠♣♦♥❡♥t❡s✳ ❆♥❛❧♦❣❛♠❡♥t❡ à ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ❞❛ ❢♦r♠❛✿

cα·pˆ+βmc2ψ(r, t) = i~∂

∂tψ(r, t), ✭✷✳✽✮ ♦♥❞❡ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ t❡♠ q✉❛tr♦ ❝♦♠♣♦♥❡♥t❡s✿

ψ(r, t) =

        

ψ1(r, t) ψ2(r, t) ψ3(r, t) ψ4(r, t)

        

. ✭✷✳✾✮

❉❡st❛ ❢♦r♠❛✱ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ♣r♦♣♦r❝✐♦♥❛ ✉♠❛ ❞❡s❝r✐çã♦ q✉â♥t✐❝❛ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ❝♦♠ ❛ ❡♥❡r❣✐❛ ❞❡ ✉♠ s✐st❡♠❛ ❞❡♥tr♦ ❞❛ t❡♦r✐❛ r❡❧❛t✐✈íst✐❝❛✳ ❊st❛ ❞❡s❝r✐çã♦ ❞❡♥♦♠✐♥❛✲s❡ t❡♦r✐❛ q✉â♥t✐❝❛ r❡❧❛t✐✈íst✐❝❛✳

❋✐♥❛❧♠❡♥t❡ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞❛s s♦❧✉çõ❡s ❞❡ ❡♥❡r❣✐❛ ❞❡✜♥✐❞❛ ψ(r, t) =

u(r)e−iEt~ ❛s q✉❛✐s ♣❡r♠✐t❡♠ ♦❜t❡r ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✿

cα·pˆ+βmc2u(r) = Eu(r). ✭✷✳✶✵✮

❋♦r♠❛ ❈♦✈❛r✐❛♥t❡ ❞❛ ❊q✉❛çã♦ ❞❡ ❉✐r❛❝

P❛r❛ ❢❛❝✐❧✐t❛r ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❡s❝r❡✈❡✲s❡ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ♥✉♠❛ ♥♦t❛çã♦ ❝♦✈❛r✐❛♥t❡ ♥❛ q✉❛❧ ❞❡✜♥✐♠♦s ♦s q✉❛❞r✐✈❡t♦r❡s ❛ss♦❝✐❛❞♦s à ♣♦s✐çã♦ ❡ ❛♦

♠♦♠❡♥t♦✸ _{✱ ❝♦♠♦✿}

xµ = (ct,r); pµ= (E

c,p), ✭✷✳✶✶✮

❞❡ ❢♦r♠❛ q✉❡ ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❡s❝r❡✈❡✲s❡ ♥❡st❛ ♥♦t❛çã♦ ❝♦♠♦✿

ˆ

pµ _=i~∇µ_=i~ ∂

∂xµ

. ✭✷✳✶✷✮

❯s❛♥❞♦ ❡st❡s ❡❧❡♠❡♥t♦s ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ❡①♣r❡ss❛✲s❡ ❞❛ ❢♦r♠❛✿

(γµpˆµ−mc)Ψ(x, t) = 0, ✭✷✳✶✸✮

♦♥❞❡ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ♦♣❡r❛❞♦r γµ = (γ0, γ1, γ2, γ3, γ4) sã♦ ❞❡✜♥✐❞❛s ❡♠ t❡r♠♦s ❞♦s ❝♦❡✜❝✐❡♥t❡s α ❡β✿

γ0 =β; γi =βαi. ✭✷✳✶✹✮

❈♦rr❡çõ❡s ❘❡❧❛t✐✈íst✐❝❛s

❯♠❛ ♣❛rtí❝✉❧❛ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ♣♦t❡♥❝✐❛❧ V(r) é ♦❜s❡r✈❛❞❛ ❡♠ ✉♠

❡st❛❞♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ψ(r) =

   φ χ  

, ♦♥❞❡ φ ❡ χ sã♦ ✈❡t♦r❡s ❝♦♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s✳ P❛r❛ ❡st❡ s✐st❡♠❛ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿

 

V(r) +mc

2 _cσ· ˆ

p

cσ·pˆ V(r)−mc2      φ χ   =E

  φ χ   . ✭✷✳✶✺✮

✸_{◆❛ ❢♦r♠❛ ❝♦♥tr❛✈❛r✐❛♥t❡ ❡s❝r❡✈❡✲s❡✿}

xµ= (ct,−r); pµ= (

E c,−p)

❋❛③❡♥❞♦ǫ=E−mc2 _{❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ r❡s✉❧t❛ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✿}

(σ·pˆ)χ= 1

c{ǫ−V (r)}φ, ✭✷✳✶✻✮

(σ·pˆ)φ= 1

c

ǫ−V (r) + 2mc2 χ, ✭✷✳✶✼✮ ❉❛ ❡q✉❛çã♦ ✭✷✳✶✼✮ t❡♠✲s❡ q✉❡✿

χ= c

ǫ−V (r) + 2mc2 (σ·pˆ)φ, ✭✷✳✶✽✮ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ s❡ ♣♦❞❡ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ ❡♠ ✉♠❛ ❢♦r♠❛ ❝♦♠♣❛❝t❛✿

(σ·pˆ) c

ǫ−V (r) + 2mc2 (σ·pˆ)φ=

1

c

ǫ−V (r)−mc2 φ. ✭✷✳✶✾✮ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❧✐♠✐t❡ ♥ã♦ r❡❧❛t✐✈íst✐❝♦ ǫ≪mc2_{✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ❛ ❡①♣❛♥sã♦✿}

1

ǫ−V (r) + 2mc2 =

1 2mc2

(

1 1 + ǫ−_2mcV(2r)

)

∼ 1

2mc2

1− ǫ−V (r)

2mc2 +· · ·

. ✭✷✳✷✵✮

❉❡ ♠♦❞♦ q✉❡ ❡♠ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ♦r❞❡♠ ③❡r♦ ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝ tr❛♥s❢♦r♠❛✲s❡ ♥❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ _Hˆ_{0φ = [}pˆ2

2m +

V (r)]φ = ǫφ. ❖ ❤❛♠✐❧t♦♥✐❛♥♦ ❞♦ s✐st❡♠❛ ❡♠ ✉♠❛ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❞❡ ❛♣r♦①✐♠❛çã♦ r❡❡s❝r❡✈❡✲s❡ ❡♠ t❡r♠♦s ❞❛ ❡①♣❛♥sã♦ ♥ã♦ r❡❧❛t✐✈íst✐❝❛ ✭✷✳✷✵✮ ❝♦♠♦ ❛ s♦♠❛ ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ♦r❞❡♠ ③❡r♦ _Hˆ_{0 ❡ ❛s ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ H}ˆ_1✱

❞❛ ❢♦r♠❛✿

ˆ

H = ˆH0+ ˆH1, ✭✷✳✷✶✮

♦♥❞❡ ❛s ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦✿

ˆ

H1φ=−(σ·pˆ)

ǫ−V (r)

4m2_c2 (σ·pˆ)φ. ✭✷✳✷✷✮

✬P♦ré♠✱ ♣❛r❛ ♣♦t❡♥❝✐❛✐s ❝❡♥tr❛✐s t❡♠✲s❡ q✉❡✿

(σ·pˆ)V (r) (σ·pˆ)φ=−~2_{V (}r)∇2_φ−~2∇_V(r)·∇_φ− ~

2

r

dV(r)

dr iσ·(r×∇)φ, ✭✷✳✷✸✮ q✉❡ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❞❛ ❢♦r♠❛✿

(σ·pˆ)V (r) (σ·pˆ)φ = 1 2 pˆ

2_{V (r) +V (r)} ˆ

p2φ+~

2

2 ∇

2_V(r)φ+ 2 r

dV(r)

dr L·Sφ. ✭✷✳✷✹✮ P♦rt❛♥t♦✱ ♦ t❡r♠♦ ❞❡ ❝♦rr❡çã♦ ❡①♣r❡ss❛✲s❡ ❝♦♠♦✿

ˆ

H1 =− ˆ

p4

8m3_c2 +

~2

8m2_c2∇

2_{V(r) +} 1

2m2_c2_r

dV(r)

dr L·S, ✭✷✳✷✺✮

♣♦r ❡①❡♠♣❧♦ ♣❛r❛ ♦ ♣♦t❡♥❝✐❛❧ ❈♦✉❧♦♠❜✐❛♥♦ ❣❡r❛❞♦ ♣♦r ✉♠ ♥ú❝❧❡♦ ❛tô♠✐❝♦ ❛s ❝♦rr❡çõ❡s ♣♦r ❡❢❡✐t♦s r❡❧❛t✐✈íst✐❝♦s ✜❝❛♠ ❝♦♠♦✿

ˆ

H1 =− ˆ

p4

8m3_c2 + π~2

2m2_c2 (Z~cα)δ

3_{(r) +} 1

2m2_c2

1

r

dV(r)

dr L·S. ✭✷✳✷✻✮

❉❡ ❛❝♦r❞♦ ❝♦♠ ❡st❛ ❡①♣r❡ssã♦✱ ❝❧❛ss✐✜❝❛♠✲s❡ ❛s ❝♦rr❡çõ❡s r❡❧❛t✐✈íst✐❝❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠♦✿

✐✳ ❈♦rr❡çã♦ à ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛

ˆ

Trel =−

p4

8m3_c2. ✭✷✳✷✼✮

✐✐✳ ❆❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛

ˆ

HSOC =

1 2m2_c2

1

r dV

drL·S. ✭✷✳✷✽✮

✐✐✐✳ ❚❡r♠♦ ❞❡ ❉❛r✇✐♥

ˆ

HDarwin =

π~2

2m2_c2 (Z~cα)δ

✷✳✷ ❖ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛ ♥❛ ❚❡♦r✐❛ ❞❡ ❉✐r❛❝

◆❛ ❞❡s❝r✐çã♦ ❞❡ ❉✐r❛❝ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❛♦ ❝❛❧❝✉❧❛r ♦ ❝♦♠✉t❛❞♦r ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦_Hˆ _{=cα·pˆ+βmc}2 _{❡ ❛ ♣r♦❥❡çã♦ ♥♦ ❡✐①♦z ❞♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧}

ˆ

L=r×pˆ✱ ♦❜t❡♠✲s❡✿

[ ˆH,Lˆz] =cαi[pˆi, x]pˆy−cαi[pˆi, y]pˆx =−i~c αxpˆy−αypˆx

=−i~_c(α×pˆ)

z.

✭✷✳✸✵✮ ◆❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ r❡❧❛t✐✈íst✐❝❛ ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ❞❡ s♣✐♥ é ❞❡✜♥✐❞♦ ❡♠ t❡r♠♦s ❞❛s ♠❛tr✐③❡s ❞❡ P❛✉❧✐ ❝♦♠♦ ✉♠ ♦♣❡r❛❞♦r q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❢♦r♠❛✿

X

i

=

   σi 0

0 σi

 

; Sˆi =

~

2

X

i

. ✭✷✳✸✶✮

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❛t✐s❢❛③ ❛s r❡❣r❛s ❞❡ ❝♦♠✉t❛çã♦ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ ♦✉ s❡❥❛✱ h_Sˆ_i,Sˆ_ji _{= i}~_ǫijkSˆ_k✳ ❈❛❧❝✉❧❛♥❞♦ ♦ ❝♦♠✉t❛❞♦r ❞❛ ♣r♦❥❡çã♦ ❞❡st❡

♦♣❡r❛❞♦r ♥♦ ❡✐①♦z ❝♦♠ ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❉✐r❛❝✱ ♦❜t❡♠✲s❡✿

h

ˆ

H,Sˆz

i

=

cα·pˆ+βmc2_,−i~

4(αxαy−αyαx)

=i~_{c αxpˆ}

y −αypˆx

=i~_c(α×pˆ)

z.

✭✷✳✸✷✮

P♦r ❝♦♥s❡❣✉✐♥t❡✱ t❛♥t♦ ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ♦r❜✐t❛❧ Lˆ q✉❛♥t♦ ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❞❡ s♣✐♥ Sˆ ♥ã♦ sã♦ ❜♦♥s ♥ú♠❡r♦s q✉â♥t✐❝♦s ❡✱ ♣♦rt❛♥t♦✱ ❡st❡s ♥ã♦ sã♦ ❝♦♥st❛♥t❡s ❞❡ ♠♦✈✐♠❡♥t♦✳ P♦ré♠✱ s❡ ❞❡✜♥✐♠♦s ♦ ♦♣❡r❛❞♦r Jˆ= Lˆ +Sˆ✱ ♦

❝♦♠✉t❛❞♦r ❝♦♠ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✜❝❛ ❝♦♠♦✿

h

ˆ

H,Jˆi= 0. ✭✷✳✸✸✮

❆ss✐♠✱ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r q✉❡ s❡ ❝♦♥s❡r✈❛ é ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧✿

ˆ

J =r×pˆ+ ~ 2

X

. ✭✷✳✸✹✮

❆ r❡❧❛çã♦ ✭✷✳✸✸✮ ✐♠♣❧✐❝❛ ❡♠ h_H,ˆ _Jˆ2i _{= 0 ❡} h_{H,ˆ Jˆ}

z

i

= 0. ◆♦ ❝❛s♦ r❡❧❛t✐✈íst✐❝♦ ♥ã♦ s❡ ♣♦❞❡♠ ❝❧❛ss✐✜❝❛r ♦s ❡st❛❞♦s q✉â♥t✐❝♦s ♣♦r ♠❡✐♦ ❞❡ três ♦♣❡r❛❞♦r❡s ❝♦♠♦ ♥♦ ❝❛s♦ ♥ã♦ r❡❧❛t✐✈íst✐❝♦✱ ♦♥❞❡ ♦s ❡st❛❞♦s q✉â♥t✐❝♦s ✜❝❛♠ ❞❡✜♥✐❞♦s ♣❡❧♦s ♦♣❡r❛❞♦r❡s _Hˆ_{✱ J}ˆ2 ❡ _Jˆ_{z✳ ❉❛❞♦ q✉❡ ♦ ❡s♣❛ç♦ é q✉❛❞r✐❞✐♠❡♥s✐♦♥❛❧ ♣r❡❝✐s❛✲s❡ ❞❡ ♠❛✐s ✉♠ ♦♣❡r❛❞♦r q✉❡}

❝♦♠✉t❡ ❝♦♠ ♦s ♦♣❡r❛❞♦r❡s_Hˆ_{✱ J}ˆ2 _{❡ J}ˆ

z✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞❡✜♥❡✲s❡ ♦ ♦♣❡r❛❞♦r✿

ˆ

K =βX·Jˆ− ~

2β =β

X

·Lˆ+~ˆ_1, ✭✷✳✸✺✮

q✉❡ s❛t✐s❢❛③ ❛s r❡❣r❛s ❞❡ ❝♦♠✉t❛çã♦ r❡q✉❡r✐❞❛s✳ ❖ ♦♣❡r❛❞♦r K q✉❡ é ❞❡♥♦♠✐♥❛❞♦ ✏❖♣❡r❛❞♦r ❞♦ ❆❝♦♣❧❛♠❡♥t♦ ❙♣✐♥✲Ór❜✐t❛✑ ❡ s❡✉s ❛✉t♦✈❛❧♦r❡s ♣♦❞❡♠ s❡r ❞❡t❡r♠✐♥❛❞♦s ❛ ♣❛rt✐r ❞❡✿

ˆ

K2 = ˆL2+~X_·Lˆ ₊3

4~

2₊ 1

4~

2 _{= ˆJ}2₊ 1

4~

2_{, ✭✷✳✸✻✮}

q✉❡ ❧❡✈❛ ❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦ q✉❛❞r❛❞♦ ❞♦ ❛✉t♦✈❛❧♦r κ ❞♦ ♦♣❡r❛❞♦r ❙❖❈ ❡ ♦ ❛✉t♦✈❛❧♦r j ❞♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧✿

~2_κ2 ₌~2_{j(j+ 1) +} 1

4~

2 _=⇒κ2 ₌

j+ 1 2

2

, ✭✷✳✸✼✮

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ♦♣❡r❛❞♦r ❙❖❈ ✜❝❛♠ ❞❡✜♥✐❞♦s ♣❡❧❛ ❡①♣r❡ssã♦✿

κ=±

j+1

2

. ✭✷✳✸✽✮

❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ❛ ❞❡s❝r✐çã♦ ❢❡✐t❛ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ♣r❡s❡♥ç❛ ❞♦ s♣✐♥ é ♥❡❝❡ssár✐♦ tr❛❜❛❧❤❛r ♥❛ ❜❛s❡ ❛❝♦♣❧❛❞❛ ❞❡ s♣✐♥ ❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✳ ❉❡❧✐♠✐t❛♥❞♦ ❛