Estudo analítico e soluções exatas da equação de spin

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛

❊st✉❞♦ ❆♥❛❧ít✐❝♦ ❡ ❙♦❧✉çõ❡s ❊①❛t❛s

❞❛ ❊q✉❛çã♦ ❞❡ ❙♣✐♥

▼❛r✐♦ ❈❡s❛r ❇❛❧❞✐♦tt✐

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉♠✐tr✐ ▼✳ ●✐t♠❛♥ ✭■❋✴❯❙P✮

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■❋❯❙P ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s✳

❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❉♠✐tr✐ ▼✳ ●✐t♠❛♥ ✭■❋✴❯❙P✮ Pr♦❢✳ ❉r✳ ❏♦s✐❢ ❋r❡♥❦❡❧ ✭■❋✴❯❙P✮ Pr♦❢✳ ❉r✳ ❈❡❧s♦ ▲✉✐③ ▲✐♠❛ ✭■❋✴❯❙P✮

Pr♦❢✳ ❉r✳ ❇r✉t♦ ▼❛① P✐♠❡♥t❡❧ ❊s❝♦❜❛r ✭■❋❚✴❯◆❊❙P✮ Pr♦❢✳ ❉r✳ ❏é❢❡rs♦♥ ❞❡ ▲✐♠❛ ❚♦♠❛③❡❧❧✐ ✭❋●❊✴❯◆❊❙P✮

❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ s❡ ❞❡st✐♥❛ ❛ ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❛ ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✱ ❛ q✉❛❧ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s✳ P❛r❛ ❝❛♠♣♦s ❡①t❡r♥♦s ❞❛❞♦s ♣♦r ❢✉♥çõ❡s r❡❛✐s✱ ❡st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠❛ r❡❞✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ P❛✉❧✐ ♣❛r❛ ♦ ❝❛s♦0+1❞✐♠❡♥s✐♦♥❛❧✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❞❡♠♦♥str❛r❡♠♦s

❆❜str❛❝t

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱

❏♦sé ❉❛♥✐❡❧ ❇❛❧❞✐♦tt✐✱

❘✐t❛ ❞❡ ❈áss✐❛ ❇❛❧❞✐♦tt✐

❡✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ❛♦ ♠❡✉ ✐r♠ã♦

❏♦sé ❈❛r❧♦s ❇❛❧❞✐♦tt✐✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❉♠✐tr✐ ▼✳ ●✐t♠❛♥❀ ❛♦ Pr♦❢✳ ❉r✳ ❱✳●✳ ❇❛❣r♦✈ ❡ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❙P q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛✳

❆♦ ❏♦ã♦ ▲✉✐s ▼❡❧♦♥✐ ❆ss✐r❛t✐✱ ❝✉❥❛ ❝♦♥tr✐❜✉✐çã♦ ❡♠ ♠✐♥❤❛ ❝❛rr❡✐r❛ ❢♦✐ ♥❛❞❛ ♠❡♥♦s q✉❡ ❡ss❡♥❝✐❛❧✳

❆♦s ✐♥ú♠❡r♦s ❛♠✐❣♦s q✉❡ ✜③ ♥♦ ■❋❯❙P✱ ❞❡♥tr❡ ♦s q✉❛✐s ❣♦st❛r✐❛ ❞❡ ❞❡st❛❝❛r✱ ❘♦❞r✐❣♦ ❋r❡s♥❡❞❛✱ ❈❛r❧♦s ▼♦❧✐♥❛ ▼❡♥❞❡s✱ ❚❤✐❛❣♦ ❞♦s ❙❛♥t♦s P❡r❡✐r❛✱ ❏♦s❡ ❈❧❡r✐st♦♥ ❈❛♠♣♦s ❞❡ ❙♦✉③❛✱ ❋❛❜✐♦ P❛♦❧✐♥✐✱ ❘♦♥❛❧❞♦ ❈❛r❧♦tt♦ ❇❛t✐st❛✱ ❱✐♥✐❝✐✉s ❞❡ ❙♦✉③❛ ❋❡r♥❛♥❞❡s✱ ▼❛r❝❡❧♦ ❖❧✐✈❡✐r❛ ❞❛ ❈♦st❛ P✐r❡s✱ ▼✐❧t♦♥ ❆❧❡①❛♥❞r❡ ❞❛ ❙✐❧✈❛ ❏✉♥✐♦r✱ ❆❧❡♥❝❛r ❏♦s❡ ❞❡ ❋❛r✐❛✱ ■✈❛♥ ❨❛s✉❞❛ ❡ ❆❞r✐❛♥❛ ❘❛♠♦s ❞❡ ▼✐r❛♥❞❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❈❛r❧♦s P❡❞r♦ ❞❛ ❙✐❧✈❛✱ ❏✉❧✐♦ ❈❡s❛r ❞❡ ▲✐♠❛ ❡ ❘♦❣ér✐♦ ▼♦r❡❧❧✐✳

❆♦ ♠❡✉ ♣r✐♠♦ ❍❛♥s ❏❡✛❡rs♦♥ ❘❛❞❦❡✳

➚ ❋❆P❊❙P ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳

❮♥❞✐❝❡

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❈❛♠♣♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❆♣❧✐❝❛çõ❡s r❡❝❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳✶ ❈♦♠♣✉t❛❞♦r❡s q✉â♥t✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸✳✷ ❋❛s❡ ❣❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦s ❢♦r♠❛✐s ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✶✷

✷✳✶ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♥❥✉❣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❘❡❞✉çã♦ ❞♦ ❝❛♠♣♦ ❡①t❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❘❡♣r❡s❡♥t❛çã♦ ✈❡t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ ❊q✉❛çõ❡s r❡❧❛❝✐♦♥❛❞❛s ✶✾

✸✳✵✳✶ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✵✳✷ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹ ❙♦❜r❡ ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✷✸

✹✳✶ ❙♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷ ❙♦❧✉çõ❡s ❡st❛❝✐♦♥ár✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✸ ❆ ♠❛tr✐③ ❞❡ tr❛♥s❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✹ ❖ ♦♣❡r❛❞♦r ❞❡ ❡✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✺ ❖ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✷✾

✻ ❊q✉❛çã♦ ❞❡ s♣✐♥ s✐♠étr✐❝❛ ✸✶

✻✳✶ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❡ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✻✳✷ ❋♦r♠❛s ❧❛❣r❛♥❣✐❛♥❛ ❡ ❤❛♠✐❧t♦♥✐❛♥❛ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ s✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✼ ❙♦❧✉çõ❡s ❡①❛t❛s ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✸✺

✼✳✶ ▲✐st❛ ❞❛s s♦❧✉çõ❡s ❡①❛t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✽ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❉❛r❜♦✉① ✹✹

✽✳✶ ❈♦♥str✉çã♦ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❡♥tr❡❧❛ç❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✽✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ❉❛r❜♦✉① ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✽✳✸ ❊①❡♠♣❧♦s ❞♦ ♠ét♦❞♦ ❞❡ ❉❛r❜♦✉① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✽✳✸✳✶ Pr✐♠❡✐r♦ ❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✽✳✸✳✷ ❙❡❣✉♥❞♦ ❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✽✳✸✳✸ ❚❡r❝❡✐r♦ ❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✾ ❈♦♥❝❧✉sã♦ ✺✺

❆ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ✺✼

❇ ❖ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ✺✾

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◆♦ tr❛t❛♠❡♥t♦ ❞❛ ✐♥t❡r❛çã♦ ❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s ❝♦♠ ❝❛♠♣♦s ❡❧❡tr♦♠❛❣♥ét✐❝♦s ✐♥t❡♥s♦s✱ é ✉♠ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ ❬✶❪ q✉❡ ✉♠❛ ❞❡s❝r✐çã♦ s❡♠✐❝❧áss✐❝❛✱ ♥❛ q✉❛❧ ♦ ❝❛♠♣♦ é tr❛t❛❞♦ ❝❧❛ss✐❝❛✲ ♠❡♥t❡✱ ❢♦r♥❡❝❡ r❡s✉❧t❛❞♦s ❡q✉✐✈❛❧❡♥t❡s ❛♦s ♦❜t✐❞♦s ♣♦r ✉♠❛ q✉❛♥t✐③❛çã♦ t♦t❛❧ ❞♦ ♣r♦❜❧❡♠❛✱ ❞❡s❞❡ q✉❡ ❛ ✢✉t✉❛çã♦ ♥♦ ♥ú♠❡r♦ ❞❡ ❢ót♦♥s ♣♦ss❛ s❡r ♥❡❣❧✐❣❡♥❝✐❛❞❛✳ ❆❧é♠ ❞✐st♦✱ s✐st❡♠❛s q✉â♥t✐❝♦s ❝♦♠♣❧❡①♦s✱ ❝♦♠ ✉♠ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛ ❞✐s❝r❡t♦✱ ❢r❡qü❡♥t❡♠❡♥t❡ ❡♥❝♦♥tr❛♠✲s❡ ❡♠ s✐t✉❛çõ❡s ❞✐♥â♠✐❝❛s ❡s♣❡❝✐❛✐s✱ ♥❛s q✉❛✐s ❛♣❡♥❛s ❞♦✐s ❡st❛❞♦s ❡st❛❝✐♦♥ár✐♦s ♣♦ss✉❡♠ ♣❛rt✐❝✐♣❛çã♦ s✐❣♥✐✜❝❛t✐✈❛✳ ◆❡st❡s ❝❛s♦s✱ ♣♦❞❡♠♦s r❡str✐♥❣✐r ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉❡ ❞❡s✲ ❝r❡✈❡ t❛❧ s✐st❡♠❛ ❛ ✉♠ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ❯♠ ót✐♠♦ ❡①❡♠♣❧♦ ❞❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ é ♦ tr❛t❛♠❡♥t♦ ❞❛ ♣♦❧❛r✐③❛çã♦ ❞❛ ♠♦❧é❝✉❧❛ ❞❡ ❛♠ô♥✐❛ ❬✷❪✳ ❚❛✐s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ❡♥❝♦♥tr❛♠ ✉♠❛ ✈❛st❛ ❣❛♠❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❡♠ ❢ís✐❝❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ t❡♦r✐❛ s❡♠✐❝❧áss✐❝❛ ❞♦ ❧❛s❡r ❬✸❪❀ ♥❛ ❞❡s❝r✐çã♦ ❞❡ ❡①♣❡r✐♠❡♥t♦s ❞❡ ❛❜s♦rçã♦ r❡ss♦♥❛♥t❡ ❡ ✐♥❞✉çã♦ ♥✉❝❧❡❛r ❬✹❪✱ ♦✉ ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ♠♦❧é❝✉❧❛s ❛tr❛✈és ❞❡ ✉♠❛ ❝❛✈✐❞❛❞❡ ✐♠❡rs❛ ❡♠ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ♦✉ ♠❛❣♥ét✐❝♦ ❬✺❪✳ ❆♣❧✐❝❛çõ❡s ♠❛✐s r❡❝❡♥t❡s ❞❡st❡ s✐st❡♠❛ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s✱ ❝♦♠ ✉♠ ❝❡rt♦ ♥í✈❡❧ ❞❡ ❞❡t❛❧❤❛♠❡♥t♦✱ ♥♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳ ❊st❡ tr❛❜❛❧❤♦ s❡ ❞❡st✐♥❛ ❛ ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞♦s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❛ ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✱ ❜❡♠ ❝♦♠♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❡st❛ ❡ ♦✉tr❛s ❡q✉❛çõ❡s ❡♥❝♦♥tr❛❞❛s ❡♠ ❞✐❢❡r❡♥t❡s ♣r♦❜❧❡♠❛s ❡♠ ❢ís✐❝❛✳ ❙❡rá t❛♠❜é♠ ❛♣r❡s❡♥t❛❞❛ ✉♠❛ sér✐❡ ❞❡ ♥♦✈❛s s♦❧✉çõ❡s ❡①❛t❛s ♣❛r❛ ❡st❛ ❡q✉❛çã♦✱ ❛❧é♠ ❞❡ ✉♠ ♠ét♦❞♦ q✉❡ ♣❡r♠✐t❡ ❛ ♦❜t❡♥çã♦ ❞❡

♥♦✈❛s s♦❧✉çõ❡s✳

✶✳✶ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥

P❛r❛ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧✱ ♥♦ q✉❛❧ ♦ s✐st❡♠❛ ❡stá s✉❥❡✐t♦ ❛ ✐♥t❡r❛çõ❡s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦✱ ❛ ❞❡s❝r✐çã♦ ❞♦ s✐st❡♠❛ ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❛tr❛✈és ❞❡ ✉♠ ❡s♣✐♥♦rV (t)❝♦♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s

❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦ ❡ ❛ ❞✐♥â♠✐❝❛ s❡rá ❞❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡♠

0 + 1 ❞✐♠❡♥sã♦ ✭~_{=c= 1}✮

idV

dt = [F0I+ (˜σF)]V , V =   v1 v2  ,

(˜σF) = F1σ1+F2σ2+F3σ3, ✭✶✳✶✮

♦♥❞❡ Ié ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ 2_×2✱ ˜σ= (σ1, σ2, σ3)sã♦ ❛s ♠❛tr✐③❡s ❞❡ P❛✉❧✐

σ1 =  

0 1

1 0



, σ₂ =  

0 ₋i i 0



, σ₃ =  

1 0

0 ₋1



, ✭✶✳✷✮

F0(t) ❡ F(t) = (F1, F2, F3) q✉❛tr♦ ❢✉♥çõ❡s✱ ❡♠ ❣❡r❛❧ ❝♦♠♣❧❡①❛s✱ ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ♦ ❝❛s♦ ❝♦♠ ❢✉♥çõ❡s Fi(t) ❝♦♠♣❧❡①❛s ❞❡s❝r❡✈❡ ✉♠ ♣♦ssí✈❡❧ ❛♠♦rt❡❝✐♠❡♥t♦

❞♦ s✐st❡♠❛✳

❙❡ V (t) é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮ ❝♦♠F0 = 0✱ ✉♠❛ s♦❧✉çã♦ V′_{(t) ♣❛r❛ F0 6= 0}

♣♦❞❡ s❡r ♦❜t✐❞❛ ❢❛③❡♥❞♦

V′(t) = exp

−i Z t

0

F0(τ) dτ

V (t) ,

♦ q✉❡ ♣❡r♠✐t❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❢❛③❡r F0 = 0 ❡♠ ✭✶✳✶✮✳ ❈♦♠ ✐st♦✱ ♦ ♣r♦❜❧❡♠❛

❣❡r❛❧ ❛ss✉♠❡ ❛ ❢♦r♠❛

iV˙ = (˜σF)V , V =

 

v1(t)

v2(t)



, V˙ =dV /dt . ✭✶✳✸✮

◆❡st❡ tr❛❜❛❧❤♦✱ ✐r❡♠♦s ♥♦s r❡❢❡r✐r ❛ ✭✶✳✸✮ ❝♦♠♦ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❡ ❛ F(t)❝♦♠♦ ♦ ❝❛♠♣♦

❡①t❡r♥♦✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛

F=K+iG, K= ReF, G= ImF,

K= (Kk), G= (Gk) , k= 1,2,3, ✭✶✳✹✮ ♦♥❞❡ K(t) ❡G(t) sã♦ ✈❡t♦r❡s r❡❛✐s✳

❯♠❛ r❡❛❧✐③❛çã♦ ❝♦♥❝r❡t❛ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✸✮ s❡r✐❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r t♦t❛❧ J = 1/2 ✐♥s❡r✐❞❛ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✭♦✉ ❡❧étr✐❝♦✮✱ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✱ ❞❡

✐♥t❡♥s✐❞❛❞❡ B(t) (E(t))✱ ♥❡st❡ ❝❛s♦ ✐❞❡♥t✐✜❝❛♠♦s

F=₋µB (F=₋µE),

♦♥❞❡ µ é ♦ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ✭❡❧étr✐❝♦✮ ❞❛ ♣❛rtí❝✉❧❛✳ ◆❡st❡ ❝❛s♦✱ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ P❛✉❧✐ ❬✻❪ ♣❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ✜①❛ ♥♦ ❡s♣❛ç♦✳ ❆❧é♠ ❞✐st♦✱ ❝♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❡♠ ❬✷❪❀ ✑♥ã♦ ✐♠♣♦rt❛ q✉❛❧ s❡❥❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞♦✐s ♥í✈❡✐s ♦r✐❣✐♥❛❧✱ ❡❧❡ s❡♠♣r❡ ♣♦❞❡rá s❡r ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❡❧étr♦♥✑✳

▼❡s♠♦ ♥♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s✱ q✉❛♥❞♦ F é ✉♠❛ ❝♦♥st❛♥t❡✱ ♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛ ♣♦ss✉✐

✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s✳ P♦r ❡①❡♠♣❧♦✱ ❢r❡qü❡♥t❡♠❡♥t❡✱ ❡♠ q✉í♠✐❝❛✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ót✐❝❛s ♣r♦✈❡♥✐❡♥t❡s ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ ❞♦✐s ♥í✈❡✐s ❞❡ ✉♠❛ ❝❡rt❛ ♠♦❧é❝✉❧❛ ♣♦r ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡stát✐❝♦E0 sã♦ ❡st✉❞❛❞❛s ❢❛③❡♥❞♦Fopt = (µE0,0,∆E)✱ ♦♥❞❡∆E = (E0−E1)/2é

❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♦ ❡①❝✐t❛❞♦✳ ❖✉tr❛ ❛♣❧✐❝❛çã♦ ✐♥t❡♥s❛✲ ♠❡♥t❡ ❡①♣❧♦r❛❞❛ ❞♦s ♠♦❞❡❧♦s ❞❡ ❞♦✐s ♥í✈❡✐s ❡stát✐❝♦s é ♦ ❡st✉❞♦ ❞♦s ❡❢❡✐t♦s ❞❡ t✉♥❡❧❛♠❡♥t♦ ❡♠ ✉♠ ❞✉♣❧♦ ♣♦ç♦ ❞❡ ♣♦t❡♥❝✐❛❧ ❬✼❪❀ ♥❡st❡ ❝❛s♦ é ❝♦♥✈❡♥✐❡♥t❡ ❡①♣r❡ss❛r♠♦s ♦ ❝❛♠♣♦ ❝♦♠♦

Ftun = −1/2 (∆0,0, ε0)✱ ♦♥❞❡ ε0 é ✉♠❛ ❛ss✐♠❡tr✐❛ ♥❛ ❡♥❡r❣✐❛ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡

❝❛❞❛ ♣♦ç♦ ❡ ∆0 ❛ ❡♥❡r❣✐❛ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ❡st❡s ❡st❛❞♦s✳ ❊st❡s ❞♦✐s ♣r♦❜❧❡♠❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ♣❡❧♦ ♦♣❡r❛❞♦r ✉♥✐tár✐♦ R = exp (iπσ2/4)✱ ♣♦✐s ❢❛③❡♥❞♦ ε0 = 2µE0 t❡r❡♠♦s R(σ˜Ftun)R−1 _{= (˜σFopt)✳}

❊①✐st❡♠ ✈ár✐❛s ❡q✉❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s✱ ♦✉ ❞❡ ❝❡rt❛ ❢♦r♠❛ r❡❧❛❝✐♦♥❛❞❛s✱ ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❝♦♥❤❡❝✐❞❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r q✉❡ s✉r❣❡ ♥❛ t❡♦r✐❛ ❞♦ ❣✐r♦s❝ó♣✐♦ ♦✉

♥❛ t❡♦r✐❛ ❞❛ ♣r❡❝❡ssã♦ ❞❡ ✉♠ ❣✐r♦♠❛❣♥❡t♦ ❝❧áss✐❝♦ ❡♠ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❬✺❪✳ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ♦♥❞❡ Fi(t), i = 1,2 sã♦ ♣✉r❛♠❡♥t❡ ✐♠❛❣✐♥ár✐♦s ❡ F3 é

❝♦♥st❛♥t❡ ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠♦ ✉♠ ❝❛s♦ ❞❡❣❡♥❡r❛❞♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❩❛❦❛r♦✈✲❙❤❛❜❛t✱ ❛ q✉❛❧ ♣♦ss✉✐ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ t❡♦r✐❛ ❞♦s só❧✐t♦♥s ❬✽❪✳

✶✳✷ ❈❛♠♣♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦

❖ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s s✉❥❡✐t♦s ❛ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ♣♦ss✉✐ ✉♠❛ ✈❛st❛ ❡ ❧♦♥❣❛ ❤✐stór✐❛✳ ❉❡♥tr❡ ♦s tr❛❜❛❧❤♦s ♣✐♦♥❡✐r♦s ♥❡st❛ ár❡❛✱ ♣♦❞❡♠♦s ❝✐t❛r ❘❛❜✐ ❬✾❪✱ ♦♥❞❡ s❡ ❝♦♥s✐❞❡r❛ ✉♠ s✐st❡♠❛ ❞❡ ❞♦✐s ♥í✈❡✐s ✐♠❡rs♦ ❡♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❝✐r❝✉❧❛r✲ ♠❡♥t❡ ♣♦❧❛r✐③❛❞♦✳ Pr♦✈❛✈❡❧♠❡♥t❡✱ ❡st❛ é ✉♠❛ ❞❛s ♣r✐♠❡✐r❛s s♦❧✉çõ❡s ❡①❛t❛s ❛♣r❡s❡♥t❛❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✳ ❘❡s✉❧t❛❞♦s ♠❛✐s r❡❝❡♥t❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬✶✵❪✳

P❛r❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ s♣✐♥ J = 1/2 s✉❥❡✐t❛ ❛ ✉♠ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❝♦♥st❛♥t❡ B0 ♥❛ ❞✐r❡çã♦z ❡ ✉♠ ❝❛♠♣♦ ❝✐r❝✉❧❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ❞❡ ✐♥t❡♥s✐❞❛❞❡ B1 ♥♦ ♣❧❛♥♦ x, y t❡♠♦s

F=₋µ(2B1cos (Ωt),−2B1sin (Ωt), B0) . ✭✶✳✺✮ ❊st❡ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r ❞r❛st✐❝❛♠❡♥t❡ s✐♠♣❧✐✜❝❛❞♦ ♣❡❧♦ ✉s♦ ❞❛s ❝❤❛♠❛❞❛s ❝♦♦r❞❡♥❛❞❛s ❣✐r❛♥t❡s ✭r♦t❛t✐♥❣ ❝♦♦r❞❡♥❛t❡s✮ ❬✹❪✱ s✐t✉❛çã♦ ❡♠ q✉❡ s❡ ✉t✐❧✐③❛ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛✲ ❞❛s q✉❡ ❣✐r❛ ❥✉♥t♦ ❝♦♠ ♦ ❝❛♠♣♦✱ ♣❛ss❛♥❞♦ ❛ s❡r ✉♠ ♣r♦❜❧❡♠❛ ❡stát✐❝♦✱ ❡ ♦ r❡s✉❧t❛❞♦ é tr❛♥s❢♦r♠❛❞♦ ♥♦✈❛♠❡♥t❡ ♣❛r❛ ♦ s✐st❡♠❛ ✜①♦ ❞♦ ❧❛❜♦r❛tór✐♦✳ ❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ s❡ ♠♦s✲ tr♦✉ ❡①tr❡♠❛♠❡♥t❡ út✐❧ ♥❛ s♦❧✉çã♦ ❞❡ ✈ár✐♦s ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❝❛♠♣♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦ ♣❡r♠✐t✐♥❞♦✱ ♥♦ ❝❛s♦ ❣❡r❛❧✱ r❡❞✉③✐r ♦ ❝❛♠♣♦F ❛ ❛♣❡♥❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ♥ã♦

♥✉❧❛s✳

❖✉tr♦ ♣r♦❜❧❡♠❛ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛✱ ♠❛s q✉❡ ♥ã♦ ♣♦❞❡✱ ❡♠ ❣❡r❛❧✱ s❡r tr❛✲ t❛❞♦ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ♦ ❛♥t❡r✐♦r✱ é ♦ ❝❛s♦ ❞♦ ❝❛♠♣♦ ❧✐♥❡❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ❬✶✶❪✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡ s♣✐♥ 1/2 ❝♦♠ ✉♠ ❝❛♠♣♦ ❝♦♥st❛♥t❡

B0 ♥❛ ❞✐r❡çã♦ z ❡ ✉♠ ❝❛♠♣♦ ♦s❝✐❧❛♥t❡ B1 ♥❛ ❞✐r❡çã♦ x✱

F=₋µ(2B1cos (Ωt),0, B0) . ✭✶✳✻✮ ▼♦❞❡❧♦s ❝♦♠ ❡st❡ ❝❛♠♣♦ sã♦ ❢r❡qü❡♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✲ ✈❡♥❞♦ r❡ss♦♥â♥❝✐❛ ❡❧étr♦♥✲s♣✐♥✱ r❡ss♦♥â♥❝✐❛ ♠❛❣♥ét✐❝❛ ♥✉❝❧❡❛r ❡ ❡s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❢❡✐①❡ ❞❡ át♦♠♦s✳ P❛r❛ ♦ ❝❛♠♣♦ F ❞❛❞♦ ❛❝✐♠❛✱ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✸✮

♥ã♦ é ✉♠❛ t❛r❡❢❛ s✐♠♣❧❡s✳ ❊♥tr❡t❛♥t♦✱ ♣❛r❛ ❝❛♠♣♦s ❝♦♠ B1 ♥ã♦ ♠✉✐t♦ ✐♥t❡♥s♦s✱ ❝♦♠ r❡❧❛çã♦ ❛ B0✱ s♦❧✉çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♣❛r❛ r❡❣✐♠❡s ♣ró①✐♠♦s à ❢r❡qüê♥❝✐❛ ❞❡ r❡ss♦♥â♥❝✐❛ ❞♦ s✐st❡♠❛✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ Ω é ♠✉✐t♦ ♣ró①✐♠♦ ❛ µB0✱ ✉t✐❧✐③❛♥❞♦ ♦ ❝❤❛✲

♠❛❞♦ ❘❲❆ ✭r♦t❛t✐♥❣✲✇❛✈❡ ❛♣♣r♦①✐♠❛t✐♦♥✮ ❬✶✷❪✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛♠♣♦ ❧✐♥❡❛r♠❡♥t❡ ♣♦✲ ❧❛r✐③❛❞♦ ❡♠ ✭✶✳✻✮ ❝♦♠♦ ✉♠❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❞♦✐s ❝❛♠♣♦s ❝✐r❝✉❧❛r♠❡♥t❡ ♣♦❧❛r✐③❛❞♦s✱

cos (Ωt) = [exp (₋iΩt) + exp (iΩt)]/2✱ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ exp (₋iΩt) r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡

r♦t❛t✐♥❣✲✇❛✈❡✱ ❡♥q✉❛♥t♦ ♦ t❡r♠♦ exp (iΩt) é ❝❤❛♠❛❞♦ ❞❡ ❛♥t✐✲r♦t❛t✐♥❣✲✇❛✈❡✳ ❆ ❛♣r♦①✐✲

♠❛çã♦ ❘❲❆ ❝♦♥s✐st❡ ❡♠ ♥❡❣❧✐❣❡♥❝✐❛r ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ ❛♥t✐✲r♦t❛t✐♥❣✲✇❛✈❡ ♥❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✱ ❞❡s♣r❡③❛♥❞♦ t❡r♠♦s ♣r♦♣♦r❝✐♦♥❛✐s ❛ exp [i(Ω +µB0)]✳

❉❡✈✐❞♦ à ✐♠♣♦rtâ♥❝✐❛ ❞♦ ♣r♦❜❧❡♠❛ ❡ à ❞✐✜❝✉❧❞❛❞❡ ♥❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çõ❡s ❡①❛t❛s✱ ✈ár✐♦s ♠ét♦❞♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❞❡ ✭✶✳✸✮✳ ❊♠ ❬✶✸❪ é ❞❡s❡♥✈♦❧✈✐❞❛ ✉♠❛ sér✐❡ ♣❡rt✉r❜❛t✐✈❛ ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ✉♠ ❝❛♠♣♦ ♥❛ ❢♦r♠❛

F= (F1,0, F3), F1 = const, ✭✶✳✼✮

❝♦♠F3(t)✉♠❛ ❢✉♥çã♦ ♣❡r✐ó❞✐❝❛✳ P❛r❛ ❝❛♠♣♦sF(t)♣❡r✐ó❞✐❝♦s ♦✉ q✉❛s✐✲♣❡r✐ó❞✐❝♦s✱ s✐st❡✲

♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ❢♦r❛♠ ✐♥t❡♥s❛♠❡♥t❡ ❡st✉❞❛❞♦s ♣♦r ✐♥ú♠❡r♦s ❛✉t♦r❡s✱ ✉t✐❧✐③❛♥❞♦ ✈ár✐❛s ❛♣r♦①✐♠❛çõ❡s ❞✐❢❡r❡♥t❡s✱ ❡✳❣✳✱ ❡①♣❛♥sõ❡s ♣❡rt✉r❜❛t✐✈❛s ❬✶✹❪ ❡✱ ❝♦♠♦ ❞❡s❝r✐t♦ ❛❝✐♠❛✱ ♦ ♠é✲ t♦❞♦ ❘❲❆✳ P❛r❛ ❡st❡s ❝❛♠♣♦s ♣❡r✐ó❞✐❝♦s✱ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ❋❧♦q✉❡t ❬✶✺❪ ❢♦r♥❡❝❡ ✉♠ ♣♦❞❡✲ r♦s♦ ❢❡rr❛♠❡♥t❛❧ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞♦ ♣r♦❜❧❡♠❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ♥✉♠ér✐❝♦s✳ P❛r❛ ✉♠❛ r❡✈✐sã♦ ♠♦❞❡r♥❛ ❞❡st❡s ♠ét♦❞♦s ✈❡❥❛ ❬✼❪ ❡ s✉❛s r❡❢❡rê♥❝✐❛s✳

P❛r❛ s✐st❡♠❛s ❡♠ q✉❡ F ♥ã♦ é ♣❡r✐ó❞✐❝♦✱ ♥❡♠ q✉❛s❡✲♣❡r✐ó❞✐❝♦✱ ♦ ♣r♦❜❧❡♠❛ t♦r♥❛✲s❡

❜❛st❛♥t❡ ❝♦♠♣❧✐❝❛❞♦✳ ❊♥tr❡t❛♥t♦✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s ❡s♣❡❝✐❛✐s✱ s♦❧✉çõ❡s ❡①❛t❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s✳ ❊♠ ❬✶✻❪✱ sã♦ ❛♣r❡s❡♥t❛❞❛s s♦❧✉çõ❡s ❡①❛t❛s ♣❛r❛ ✉♠ ❝❛♠♣♦ ♥❛ ❢♦r♠❛ ✭✶✳✼✮

♣❛r❛ ❢✉♥çõ❡s F3(t) ♥ã♦ ♣❡r✐ó❞✐❝❛s ❞❛❞❛s ♣♦r

F3(t) = r0

coshτ , F3(t) = r0

T tanhτ+ r1 T , τ =

t

T , ✭✶✳✽✮

♦♥❞❡ r0, r1 ❡ T sã♦ ❝♦♥st❛♥t❡s r❡❛✐s✳ ❊st❛s ❢✉♥çõ❡s ✈ã♦ ❛ ③❡r♦ ♥♦ ✐♥✜♥✐t♦✱ ♦ q✉❡ ❛s t♦r♥❛ ❝♦♥✈❡♥✐❡♥t❡s ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✳ ❊♠ ❬✶✼❪ t❡♠♦s ❛ ❝♦♥str✉çã♦ ❞❡ s♦❧✉çõ❡s ❡①❛t❛s ♣❛r❛ três ♥♦✈❛s ❢✉♥çõ❡s F3✱ ❡♥tr❡ ❛s q✉❛✐s ❛ ♠❛✐s s✐♠♣❧❡s t❡♠ ❛ ❢♦r♠❛

F3 =c0 +2 (c

2 1−c20) Q+c0 , Q=

  

c1coshϕ , c2 1 > c20

c1cosϕ , c2 1 < c20

,

ϕ= 2

t

q

|c2

1−c20|+c2

, c0,1,2 = const. , ✭✶✳✾✮

❛s ❞❡♠❛✐s ❡①♣r❡ssõ❡s ❡♥✈♦❧✈❡♠ ❝♦♠♣❧✐❝❛❞❛s ❝♦♠❜✐♥❛çõ❡s ❞❡ ❢✉♥çõ❡s ❡s♣❡❝✐❛✐s✳

✶✳✸ ❆♣❧✐❝❛çõ❡s r❡❝❡♥t❡s

✶✳✸✳✶ ❈♦♠♣✉t❛❞♦r❡s q✉â♥t✐❝♦s

❆❧é♠ ❞❛s ✈ár✐❛s ❛♣❧✐❝❛çõ❡s ❝✐t❛❞❛s ❛❝✐♠❛✱ ❛t✉❛❧♠❡♥t❡ ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s t❡♠ ❛tr❛í❞♦ ✉♠❛ ❛t❡♥çã♦ ❛✐♥❞❛ ♠❛✐♦r ❞❡✈✐❞♦ ❛ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝❤❛♠❛❞♦s ❝♦♠♣✉t❛❞♦r❡s q✉â♥t✐❝♦s✳

❆ ❝♦♠♣✉t❛çã♦ ❝❧áss✐❝❛ s❡ ❜❛s❡✐❛ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ❝♦❞✐✜❝❛❞❛s ❡♠ ✉♠ s✐st❡♠❛ ❜✐♥ár✐♦✱ ♦✉ s❡❥❛✱ t♦❞❛ ✐♥❢♦r♠❛çã♦ ❛ s❡r ♣r♦❝❡ss❛❞❛ é ❛r♠❛③❡♥❛❞❛ ❡♠ ✉♠❛ ❝❛❞❡✐❛ ❞❡ ③❡r♦s ❡ ✉♥s ❡ ❛ ❝♦♠♣✉t❛çã♦ s❡ ❞á ♣❡❧❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡st❛s ❝❛❞❡✐❛s✳ ❈❛❞❛ ❞í❣✐t♦ ❞❡st❛s ❝❛❞❡✐❛s r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❜✐t✳ ❉❡ ♦✉tr❛ ❢♦r♠❛✱ t❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ n ❜✐ts q✉❡✱ ❛♣ós s❡r ♠❛♥✐♣✉❧❛❞❛✱ ❢♦r♥❡❝❡ ❝♦♠♦ r❡s✉❧t❛❞♦ ✉♠❛ ♥♦✈❛ s❡qüê♥❝✐❛ ❞❡ m ❜✐ts✳ ❊st❛ ♠❛♥✐♣✉❧❛çã♦ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ♣♦rt❛ ❧ó❣✐❝❛ ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦rt❛✮ ❞❡ n ❡♥tr❛❞❛s ❡ m s❛í❞❛s✳ ❆❧é♠ ❞✐st♦✱ ❛ s❛í❞❛ ❞❡st❛ ♣♦rt❛ ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞❛ ❡♠ ✉♠❛ ♦✉tr❛ ♣♦rt❛ ❞❡ m ❡♥tr❛❞❛s ❡ s♦❢r❡r ✉♠❛ ♥♦✈❛ ♠❛♥✐♣✉❧❛çã♦✳ ❆ ❡st❡ ❡♥❝❛❞❡❛♠❡♥t♦ ❞❡ ♣♦rt❛s ❧ó❣✐❝❛s ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ r❡❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ r❡❞❡✮✳ ❆ss✐♠✱

❛ ✐♥❢♦r♠❛çã♦ ♠❛✐s s✐♠♣❧❡s q✉❡ ♣♦❞❡♠♦s tr❛t❛r ❡stá ❛r♠❛③❡♥❛❞❛ ❡♠ ✉♠ ú♥✐❝♦ ❜✐t ❡ ❛s ú♥✐❝❛s ♦♣❡r❛çõ❡s ♣♦ssí✈❡✐s sã♦ ❛ ✐❞❡♥t✐❞❛❞❡✱ ♥❛ q✉❛❧ ♦ ❜✐t é ♠❛♥t✐❞♦ ✐♥❛❧t❡r❛❞♦✱ ❡ ❛ ✐♥✈❡rsã♦✱ q✉❛♥❞♦ ✐♥✈❡rt❡♠♦s s❡✉ ✈❛❧♦r ♦r✐❣✐♥❛❧✳ ❊st❛ ♦♣❡r❛çã♦ ❞❡ ✐♥✈❡rsã♦ é ❝❤❛♠❛❞❛ ♣♦rt❛ ◆❖❚✳ ❊♠ s❡❣✉✐❞❛✱ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ✉♠❛ ♣♦rt❛ ❝♦♠ ❞♦✐s ❜✐ts ❞❡ ❡♥tr❛❞❛ ❡ ✉♠ ❞❡ s❛í❞❛✱ ♥❡st❡ ❝❛s♦ ❡①✐st❡♠ 42 _{= 16 ♦♣❡r❛çõ❡s ♣♦ssí✈❡✐s✳ ❊♥tr❡t❛♥t♦✱ ♥❡♠ t♦❞❛s ❡st❛s} ♦♣❡r❛çõ❡s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❡✱ ♥❛ ✈❡r❞❛❞❡✱ t♦❞❛s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❝♦♠ ✉♠❛ r❡❞❡ ❞❡ três ♣♦rt❛s ❢✉♥❞❛♠❡♥t❛✐s✿ ❛ ◆❖❚✱ ✐♥tr♦❞✉③✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ♣♦rt❛ ❆◆❉✱ ❝✉❥♦ r❡s✉❧t❛❞♦ é ♦ ♣r♦❞✉t♦ ❞♦s ❜✐ts ❞❡ ❡♥tr❛❞❛ ❡ ❛ ♣♦rt❛ ❖❘✱ ❝✉❥❛ s❛í❞❛ é ③❡r♦ ❛♣❡♥❛s s❡ ❛s ❡♥tr❛❞❛s ❢♦r❡♠ ❛♠❜❛s ③❡r♦✳ ➱ ✉♠ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ q✉❡ q✉❛❧q✉❡r ♠❛♥✐♣✉❧❛çã♦ ❞❡ n✲❜✐ts ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞❛ ❛tr❛✈és ❞❡ ✉♠❛ r❡❞❡ q✉❡ ✉t✐❧✐③❡ ❛♣❡♥❛s ✉♠❛ ❞❛❞❛ ❝♦♠❜✐♥❛çã♦ ❞❡st❛s três ♣♦rt❛s✱ ♦✉ s❡❥❛✱ q✉❛❧q✉❡r ♣♦rt❛ ❧ó❣✐❝❛ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ❝♦♠♦ ✉♠ ❝❡rt♦ ❡♥❝❛❞❡❛♠❡♥t♦ ❞❛s ♣♦rt❛s ◆❖❚✱ ❆◆❉ ❡ ❖❘✶_{✳ P♦r ❝♦♥t❛ ❞✐ss♦✱ ❡st❛s r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ ♣♦rt❛s ✉♥✐✈❡rs❛✐s✳}

◆❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛✱ ♦ ❜✐t é s✉❜st✐t✉í❞♦ ♣❡❧♦ ❝❤❛♠❛❞♦ ❜✐t q✉â♥t✐❝♦ ♦✉ q✉❜✐t✱ q✉❡ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ♦ ❡st❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❞♦✐s ♥í✈❡✐s✳ ❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ ❜✐t ❝❧áss✐❝♦✱ r❡str✐t♦ ❛♦s ✈❛❧♦r❡s ③❡r♦ ❡ ✉♠✱ ♦ q✉❜✐t ♣♦❞❡ s❡ ❡♥❝♦♥tr❛r ♥❛ s✉♣❡r♣♦s✐çã♦ ❞♦s ♥í✈❡✐s ❢✉♥❞❛♠❡♥t❛✐s✱ ✐✳❡✳✱ ✉♠ q✉❜✐t ♣♦❞❡ ❛ss✉♠✐r ♦ ✈❛❧♦r✿

|ψ_i=α_|0_i+β_|1_i , _|α_|2+_|β_|2 = 1,

♦♥❞❡ |0_i ❡ _|1_i sã♦ ♦s ♥í✈❡✐s ❞♦ s✐st❡♠❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❡st❛❞♦s ❞♦ s♣✐♥ ❞❡ ✉♠ ❡❧étr♦♥✳

❯♠ s✐st❡♠❛ ❞❡ n✲q✉❜✐ts é ♦❜t✐❞♦ ❢❛③❡♥❞♦

|ψ_i= X x∈{0,1}n

αx|xi ,

X

x∈{0,1}n

|αx|2 = 1,

♦♥❞❡ |x_ié ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ n ❡st❛❞♦s ❞❡ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ❞♦✐s q✉❜✐ts ♣♦❞❡rí❛♠♦s t❡r |ψ_i= 1/√2 (_|01_i+_|10_i) ❝♦♠ _|01_i=_|0_{i ⊗ |}1_i✳

❆ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛ é ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ❡✈♦❧✉çã♦ ✉♥✐tár✐❛ ❞♦ ❡st❛❞♦ ❞❡ ❡♥tr❛❞❛✱ ♦✉ s❡❥❛✱ ❛s ♣♦rt❛s ❧ó❣✐❝❛s ❝❧áss✐❝❛s ❞❡ n ❡♥tr❛❞❛s sã♦ s✉❜st✐t✉í❞❛s ♣♦r ♦♣❡r❛❞♦r❡s ✉♥✐tár✐♦s ❛❣✐♥❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♣r♦❞✉t♦ H⊗n_{❀ ❡st❡s ♦♣❡r❛❞♦r❡s sã♦ ❝❤❛♠❛❞♦s ♣♦rt❛s ❧ó❣✐❝❛s}

✶_{◆❛ ✈❡r❞❛❞❡✱ ❛♣❡♥❛s ❞✉❛s ♣♦rt❛s sã♦ ♥❡❝❡ssár✐❛s✿ ◆❖❚✱ ❆◆❉ ♦✉ ◆❖❚✱ ❖❘✳}

q✉â♥t✐❝❛s✳ ❖ ♣r♦❝❡ss♦ é ❡♥❝❡rr❛❞♦ ❝♦♠ ❛ ♠❡❞✐çã♦ ❞♦ ❡st❛❞♦ ✜♥❛❧✱ ♦♣❡r❛çã♦ ❡st❛✱ ❡♠ ❣❡r❛❧✱ ♥ã♦ ✉♥✐tár✐❛ ❡ s✉❥❡✐t❛ à ✐♥t❡r♣r❡t❛çã♦ ♣r♦❜❛❜✐❧íst✐❝❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ❉✐❢❡r❡♥t❡ ❞❡ ♦♣❡r❛çõ❡s ❝❧áss✐❝❛s ✐rr❡✈❡rsí✈❡✐s ❝♦♠♦ ❛s ❆◆❉ ❡ ❖❘✱ ❡♠ q✉❡ ❛ ❡♥tr❛❞❛ ♥ã♦ ♣♦❞❡ s❡r s❡♠♣r❡ r❡❝♦♥str✉í❞❛ ❛ ♣❛rt✐r ❞❛ s❛í❞❛✱ ❛ ✉♥✐t❛r✐❡❞❛❞❡ ❞♦s ♦♣❡r❛❞♦r❡s ❡①✐❣❡ q✉❡ t♦❞♦ ♣r♦❝❡ss♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ q✉â♥t✐❝♦ s❡❥❛ r❡✈❡rsí✈❡❧✱ ❞❡ ❢♦r♠❛ q✉❡ t♦❞❛s ❛s ♣♦rt❛s q✉â♥t✐❝❛s ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡♥tr❛❞❛s ❡ s❛í❞❛s✳ ❊♥tr❡t❛♥t♦✱ s❡♠♣r❡ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛❧❣✉♥s ❞♦s q✉❜✐ts ❞❡ s❛í❞❛ ♣❛r❛ s✐♠✉❧❛r q✉❛❧q✉❡r ♣♦rt❛ ❝❧áss✐❝❛ ❬✶✽❪✳

❈♦♠♦ ♥❛ ❝♦♠♣✉t❛çã♦ ❝❧áss✐❝❛✱ ❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛ ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ r❡❞✉③✐❞♦ ❞❡ ♣♦rt❛s ✉♥✐✈❡rs❛✐s✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ é ♣♦ssí✈❡❧ ♠♦str❛r ❬✶✽❪ q✉❡ q✉❛❧✲ q✉❡r ♦♣❡r❛çã♦ ✉♥✐tár✐❛ ❛❣✐♥❞♦ ❡♠ n q✉❜✐ts ♣♦❞❡ s❡r ✐♠♣❧❡♠❡♥t❛❞❛ ❛tr❛✈és ❞❛s ❝❤❛♠❛❞❛s ♣♦rt❛ ❞❡ ❍❛❞❛♠❛r❞ ✭H✮ ❡ ❞❛ ♣♦rt❛ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ❢❛s❡ ❝♦♥tr♦❧❛❞❛ ✭B✮✱ ❞❡ ✉♠ ❡ ❞♦✐s q✉❜✐ts✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ r❡♣r❡s❡♥t❛❞❛s ♥❛ ❜❛s❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ {|00_i,_|01_i,_|10_i,_|11_i}

♣❡❧♦s ♦♣❡r❛❞♦r❡s ✉♥✐tár✐♦s

H = √1

2

 

1 1

1 ₋1



 , B(φ) =        

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 eiφ

        .

❆ ✐♠♣❧❡♠❡♥t❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❞❡st❛s ♣♦rt❛s ❧ó❣✐❝❛s q✉â♥t✐❝❛s ❡①✐❣❡ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡st❡s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ♣❡❧❛ ❛çã♦ ❞❡ ❝❛♠♣♦s ❡①t❡r♥♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❡♠ ❬✶✾❪ é ❛♥❛❧✐s❛❞❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠❛ ♣♦rt❛ q✉â♥t✐❝❛ ❞❡ ✉♠ q✉❜✐t ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ♥❛ ❢♦r♠❛ ✭✶✳✺✮✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ❛ ❡✈♦❧✉çã♦ ❞❡st❡s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞❡st❡s ❝❛♠♣♦s é ❞❡s❝r✐t❛ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✳ ❆ss✐♠✱ ❛ ❛♥á❧✐s❡ ❞❡st❛ ❡q✉❛çã♦ é ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛✳

✶✳✸✳✷ ❋❛s❡ ❣❡♦♠étr✐❝❛

❉❛❞♦ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH❞❡ ❡st❛❞♦s q✉â♥t✐❝♦s✱ ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♣r♦❥❡t✐✈♦ ❛ss♦❝✐❛❞♦ P =_H/U(1)✱ ♦✉ ❡s♣❛ç♦ ❞♦s r❛✐♦s✱ é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ✈❡t♦r❡s ♠ú❧t✐♣❧♦s ✭♣♦r ✉♠

♥ú♠❡r♦ ❝♦♠♣❧❡①♦ q✉❛❧q✉❡r✮ ❡♥tr❡ s✐✳ ❆ss✐♠✱ ❛ q✉❛❧q✉❡r ❡st❛❞♦ V _{∈ H} ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❝❧❛ss❡ Π (V)❞❡ t♦❞♦s ♦s ❡st❛❞♦s V′ _{♠ú❧t✐♣❧♦s ❞❡ V✱}

Π (V) =_{V′ :V′ =cV, ❝♦♠ c✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦_}

❯♠❛ ❝❧❛ss❡ Π (V) é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ _P ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝❧❛ss❡s Π (V)✱ q✉❛♥❞♦V ♣❡r❝♦rr❡ t♦❞♦ ♦ H✱ ❝♦♥st✐t✉✐ ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♣r♦❥❡t✐✈♦ P✳ ❯♠❛ tr❛❥❡tór✐❛ V (t) ❡♠

H t❡♠✱ ❛ss✐♠✱ ✉♠❛ ✐♠❛❣❡♠ Π (V (t))♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ _P✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ♥♦tá✈❡❧

❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s é ❛ ❝❛♣❛❝✐❞❛❞❡ ❞♦ ❡st❛❞♦ V ❞❡ ♠❛♥t❡r ✉♠❛ ♠❡♠ór✐❛ ❞❛ tr❛❥❡tó✲ r✐❛ Π (V (t)) ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ _P q✉❛♥❞♦ Π (V (t)) é ♣❡r✐ó❞✐❝❛✳ ❉❛❞❛ ✉♠❛ ❡✈♦❧✉çã♦

t❡♠♣♦r❛❧

iV˙ =HV, ✭✶✳✶✵✮

❝♦♠ ✉♠ ♦♣❡r❛❞♦r ❤❡r♠✐t✐❛♥♦H(t)✱ ❛ ♥♦r♠❛ ❞❡ V s❡ ❝♦♥s❡r✈❛ ❡✱ ♣♦rt❛♥t♦✱ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❡♠ P s✐❣♥✐✜❝❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❢❛s❡ ❡♠ H✱ ♦✉ s❡❥❛✱ s❡ ♦ ♣❡rí♦❞♦ é τ✱

V (τ) = exp (iΓ)V (0),

❝♦♠ ✉♠❛ ❢❛s❡ Γ r❡❛❧✳ ❉❡✜♥✐♥❞♦ V˜(t) = exp [₋if(t)]V✱ ❝♦♠ f(τ)₋f(0) = Γ✱ ♦❜t❡♠♦s ˜

V (0) = ˜V (τ)❡✱ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✶✵✮✱

df dt =i

˜

V , d dtV˜

−(V, HV). ✭✶✳✶✶✮ ■♥t❡❣r❛♥❞♦ ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ♦❜t❡♠♦s

Γ =i Z τ

0

˜

V , d dtV˜

dt₋

Z τ 0

(V, HV) dt . ✭✶✳✶✷✮ ◆❡st❛ ❡①♣r❡ssã♦✱ ✐❞❡♥t✐✜❝❛♠♦s ❛ ❢❛s❡ ❞✐♥â♠✐❝❛ γD ❡ ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ ❬✷✵❪ γ✱

γD =−

Z τ 0

(V, HV) dt , γ =i Z τ

0

˜

V , d dtV˜

dt . ✭✶✳✶✸✮

❉✐❢❡r❡♥t❡ ❞❛ ❢❛s❡ ❞✐♥â♠✐❝❛✱ ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ é ✐♥✈❛r✐❛♥t❡ ♣♦r ✉♠❛ r❡♣❛r❛♠❡tr✐③❛çã♦ ❞♦ t❡♠♣♦t′ _{=f(t)✱ ♦ q✉❡ ♠♦str❛ q✉❡ ❡st❛ ❢❛s❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ❞✐❢❡r❡♥t❡s}

♣❛rt❡s ❞❛ tr❛❥❡tór✐❛✱ t❛♥t♦ ❡♠ H q✉❛♥t♦ ❡♠ P✱ sã♦ ♣❡r❝♦rr✐❞❛s✳ ❊st❛ ❝❛r❛❝t❡ríst✐❝❛ ✈❡♠

❞♦ ❢❛t♦ ❞❡ q✉❡ t♦❞♦s ♦s ❡st❛❞♦s ♦❜t✐❞♦s ♣♦r ❡st❛ r❡♣❛r❛♠❡tr✐③❛çã♦ r❡♣r❡s❡♥t❛♠ tr❛❥❡tór✐❛s ❡♠ H q✉❡ ❞✐❢❡r❡♠ ❛♣❡♥❛s ♣♦r ✉♠❛ ❢❛s❡✱ ❡♥q✉❛♥t♦ ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ s❡ r❡❧❛❝✐♦♥❛ ❛♣❡♥❛s

❝♦♠ ❛ ár❡❛ ❧✐♠✐t❛❞❛ ♣❡❧❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ Π (V (t))✱ t_∈[0, τ]✱ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ _P✳

P❛r❛ ♦ ❝❛s♦ ❞❡ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s✱ ♦s ✈❡t♦r❡s ❞❡ ❡st❛❞♦ ♥♦ ❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ P

♣♦❞❡♠ s❡r ❛ss♦❝✐❛❞♦s ❝♦♠ ♣♦♥t♦s ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛S2_{✱ ❝❤❛♠❛❞❛ ❡s❢❡r❛ ❞❡ ❇❧♦❝❤✳} ■st♦ ♣♦❞❡ s❡r ❝♦♥❝r❡t✐③❛❞♦✱ ❡s❝r❡✈❡♥❞♦ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♦ ✭✈❡❥❛ ❡q✉❛çã♦ ✭✷✳✸✷✮✮✿

V =

 

exp

−2iϕ(t)

cosθ(t)/2

exp_i 2ϕ(t)

sinθ(t)/2



 , ✭✶✳✶✹✮

❡ ✐❞❡♥t✐✜❝❛♥❞♦ ❡st❡ ❡st❛❞♦ ❝♦♠♦ ♦ ♣♦♥t♦ ❝♦♠ ❝♦♦r❞❡♥❛❞❛ ♣♦❧❛r θ ❡ ❛③✐♠✉t❛❧ ϕ ♥❛ ❡s❢❡r❛ ❞❡ ❇❧♦❝❤✳ P❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ s♣✐♥ 1/2✱ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ❡st❡s ♣♦♥t♦s

❝♦♠♦ ❛ ❞✐r❡çã♦ ❞♦ s♣✐♥✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛s tr❛❥❡tór✐❛s ❞♦ s✐st❡♠❛ ♣♦❞❡♠ s❡r tr❛ç❛❞❛s ❞✐r❡t❛♠❡♥t❡ ♥❛ s✉♣❡r❢í❝✐❡ ❞❡st❛ ❡s❢❡r❛✳ P❛r❛ ❡st❡ s✐st❡♠❛ ❞❡ ❞♦✐s ♥í✈❡✐s✱ é ❢á❝✐❧ ♠♦str❛r q✉❡ ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ γ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ❛ s❡❣✉✐♥t❡ ✐♥t❡❣r❛❧ ❡♠S2

γ =₋1

2∆Ω, ∆Ω =

Z

S

n.ds. ✭✶✳✶✺✮

♦♥❞❡ S é ❛ s✉♣❡r❢í❝✐❡ ❡♠S2 _{❧✐♠✐t❛❞❛ ♣❡❧❛ tr❛❥❡tór✐❛ ❢❡❝❤❛❞❛ ❡ ✭✈❡❥❛ ❡q✉❛çã♦ ✭✷✳✷✾✮✮}

n= (V,˜σV) = (sinθcosϕ,sinθsinϕ,cosθ) . ✭✶✳✶✻✮ ◆❛ ❡①♣r❡ssã♦ ✭✶✳✶✺✮ ∆Ω é ♦ â♥❣✉❧♦ só❧✐❞♦ ❡♥❝❡rr❛❞♦ ♣❡❧❛ tr❛❥❡tór✐❛ ❡♠ S2_{✳ ❖❜✈✐❛♠❡♥t❡✱} ❡st❛s ❡①♣r❡ssõ❡s só ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❣r❛ç❛s ❛♦ ❝❛rát❡r ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞♦ s✐st❡♠❛✳ ■st♦ ♠♦str❛ ♣♦rq✉❡ ♦s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s sã♦ ♦❜❥❡t♦s ✐♥t❡r❡ss❛♥t❡s ♣❛r❛ ♦s ❡st✉❞♦s ❡ ❛♣❧✐❝❛çõ❡s ❞❛ ❢❛s❡ ❣❡♦♠étr✐❝❛✳ ❆❧é♠ ❞✐st♦✱ ❡①✐st❡ t❛♠❜é♠ ✉♠❛ ❢♦rt❡ r❡❧❛çã♦ ❡♥tr❡ ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ ♣❛r❛ s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ❡ ❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛✳ ❯♠❛ ár❡❛ ❞❡♥♦♠✐♥❛❞❛ ❝♦♠♣✉t❛çã♦ q✉â♥t✐❝❛ ❣❡♦♠étr✐❝❛ ❬✷✶❪ ♣r♦♣õ❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡st❛ ❢❛s❡ ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ✉♠❛ ♣♦rt❛ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ❢❛s❡ ❝♦♥tr♦❧❛❞❛ ❡✱ ❝♦♠♦ ✈✐♠♦s✱ ❛ ❝♦♠❜✐♥❛çã♦ ❞❡st❛ ♣♦rt❛ ❝♦♠ ❛ ♣♦rt❛ ❞❡ ❍❛❞❛♠❛r❞ ❢♦r♥❡❝❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦rt❛s q✉â♥t✐❝❛s ✉♥✐✈❡rs❛✐s✳

❖ ❝á❧❝✉❧♦ ❞❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ ❞❡♣❡♥❞❡ ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ❞❛ ❡q✉❛✲ çã♦ ❞❡ s♣✐♥✳ ◆❡st❡ r❡s♣❡✐t♦✱ ♦s r❡s✉❧t❛❞♦s ❛ s❡r❡♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♥♦ ❝❛♣ít✉❧♦ ✺ s♦❜r❡ ✑❖

Pr♦❜❧❡♠❛ ✐♥✈❡rs♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❙♣✐♥✑ s❡rã♦ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❡①✐❣✐r q✉❡ ♥♦ss♦ ❡s♣✐♥♦r t❡♥❤❛ ❛ ❢♦r♠❛

V = exp [iα]

 

exp [₋iβ] cosϕ

exp [iβ] sinϕ 

 , (V, V) = const.

❝♦♠ α, β, ϕ ❢✉♥çõ❡s r❡❛✐s ❞♦ t❡♠♣♦✳ ❊st❡ ❡s♣✐♥♦r s❡rá s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ♣❛r❛ ♦ ❝❛♠♣♦ ❡①t❡r♥♦

F1 =₋ϕ˙sin 2β₋α˙ cos 2βsin 2ϕ , F2 = ˙ϕcos 2β₋α˙ sin 2βsin 2ϕ , F3 = ˙β₋α˙ cos 2ϕ .

❆ss✐♠✱ ♣♦❞❡♠♦s ❣❛r❛♥t✐r ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ❢❛s❡✱ ❡①✐❣✐♥❞♦ ❛ ♣❡r✐♦❞✐❝✐❞❛❞❡ ❞❛s ❢✉♥çõ❡s β(t) ❡ ϕ(t) ❡ ❝❛❧❝✉❧❛r ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ ❞❡st❛s s♦❧✉çõ❡s

✉t✐❧✐③❛♥❞♦ ❛ ❡①♣r❡ssã♦ ✭✶✳✶✺✮✳ ■st♦ ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ s✐♠♣❧❡s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❢❛s❡ ❣❡♦♠étr✐❝❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✈ár✐♦s ❝❛♠♣♦s ❡①t❡r♥♦s ❞✐❢❡r❡♥t❡s✳ ❊st❡ ❝♦♥tr♦❧❡ s♦❜r❡ ❛ ❢♦r♠❛ ❞❛s s♦❧✉çõ❡s ♣♦❞❡ t❛♠❜é♠ s❡r ❛♠♣❧❛♠❡♥t❡ ❡①♣❧♦r❛❞♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ ♣♦rt❛s ❧ó❣✐❝❛s q✉â♥t✐❝❛s✳

❈❛♣ít✉❧♦ ✷

❉❡s❡♥✈♦❧✈✐♠❡♥t♦s ❢♦r♠❛✐s ❞❛ ❡q✉❛çã♦

❞❡ s♣✐♥

❊st❡ ❝❛♣ít✉❧♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ❝❡rt❛s ♥♦t❛çõ❡s ❡ ♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ❛ s❡r❡♠ ❡♠♣r❡❣❛❞❛s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦s✲ tr❛r❡♠♦s ❝♦♠♦ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ✉♠ ❡s♣✐♥♦r ❣❡♥ér✐❝♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✳ ❊st❡ r❡s✉❧t❛❞♦ s❡rá ✉t✐❧✐③❛❞♦ q✉❛♥❞♦ tr❛t❛r♠♦s ❞❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❡st❛ ❡q✉❛çã♦ ♥❛ s❡çã♦ ✹✳✶✳

❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rsí✈❡❧ ❝♦♠ ❛ q✉❛❧ ♣♦❞❡♠♦s r❡❞✉③✐r ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❞♦ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❛❧é♠ ❞❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ♣❛r❛ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ❝❛♠♣♦✱ ❛ ♣❛rt✐r ❞❡ ❝❡rt♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞❡st❛ tr❛♥s❢♦r♠❛çã♦✳

◆♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✈❡t♦r✐❛❧ ❛ss♦❝✐❛❞❛ ❛ ✉♠ ♣❛r ❞❡ ❡s♣✐♥♦r❡s✱ ❜❡♠ ❝♦♠♦ ✉♠❛ sér✐❡ ❞❡ r❡❧❛çõ❡s ❛ss♦❝✐❛❞❛s ❛ ❡st❡ ✈❡t♦r✳ ❱ár✐❛s ❞❡st❛s r❡❧❛çõ❡s r❡♣r❡s❡♥t❛♠ ❛♣❡♥❛s ♠❛♥✐♣✉❧❛çõ❡s ❛❧❣é❜r✐❝❛s✱ ♠❛s s✉❛ ❛♠♣❧❛ ✉t✐❧✐③❛çã♦ ♥❡st❡ tr❛❜❛❧❤♦ ❥✉st✐✜❝❛ s✉❛ ❡①♣❧✐❝✐t❛çã♦✳

✷✳✶ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♥❥✉❣❛❞❛

❆ ❡q✉❛çã♦ ❝♦♥❥✉❣❛❞❛ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✭✶✳✸✮ t❡♠ ❛ ❢♦r♠❛

iV˙+ =₋V+(σ˜F∗) , V+ = v∗

1(t) v2∗(t)

. ✭✷✳✶✮

❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ❞♦✐s ❡s♣✐♥♦r❡s U ❡ V é ❞❡✜♥✐❞♦ ❝♦♠♦✿

(U, V) = U+V = (u₁∗v1 +u∗₂v2) . ✭✷✳✷✮ ❊①♣❧✐❝✐t❛♥❞♦✲s❡ ❝❛❞❛ ✉♠❛ ❞❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s✱ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❡ s✉❛ ❝♦♥❥✉❣❛❞❛ ❢♦r♥❡❝❡♠ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❛❝♦♣❧❛❞❛s

iv1˙ =F3v1+ (F1₋iF2)v2, iv2˙ =₋F3v2+ (F1 +iF2)v1, ✭✷✳✸✮ iv˙₁∗ =₋F₃∗v₁∗₋(F₁∗ +iF₂∗)v₂∗, iv˙₂∗ =F₃∗v₂∗₋(F₁∗ ₋iF₂∗)v∗₁. ✭✷✳✹✮ ■♥tr♦❞✉③✐♥❞♦ ♦ ❡s♣✐♥♦r ❛♥t✐❝♦♥❥✉❣❛❞♦ _V¯

¯

V =₋iσ2V∗ =  

−v∗

2

v∗

1 

 , ✭✷✳✺✮

♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ s✐st❡♠❛ ✭✷✳✹✮ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✭✶✳✸✮ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ ❝♦♠♣❧❡①♦ ❝♦♥❥✉❣❛❞♦ F∗_✱

i

·

¯

V = (σ˜F∗) ¯V . ✭✷✳✻✮

❆s s❡❣✉✐♥t❡s r❡❧❛çõ❡s sã♦ ✈á❧✐❞❛s ♣❛r❛ t♦❞♦ ❡s♣✐♥♦r V ❡ s❡✉ ❛♥t✐❝♦♥❥✉❣❛❞♦ V¯ ✭✷✳✺✮✿

¯

V

=₋V , V ,¯ V¯

= (V, V) , V , V¯

= V,V¯

= 0. ✭✷✳✼✮

❯♠❛ ✈❡③ q✉❡ V ₆= 0 é ♦rt♦❣♦♥❛❧ ❛ V¯✱ ❡st❡s ✈❡t♦r❡s sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ P♦r✲ t❛♥t♦✱ q✉❛❧q✉❡r ❡s♣✐♥♦r U ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦

U = (V, V)−1

(V, U)V + ¯V , U ¯ V

. ✭✷✳✽✮

◆❛ ✈❡r❞❛❞❡✱ ✐st♦ ✐♠♣❧✐❝❛ ♥❛ r❡❧❛çã♦ ❞❡ ❝♦♠♣❧❡t❡③❛

♦♥❞❡

V U+=

 

υ1u∗

1 υ1u∗2 υ2u∗

1 υ2u∗2 

,detV U+= 0. ✭✷✳✶✵✮

✷✳✷ ❘❡❞✉çã♦ ❞♦ ❝❛♠♣♦ ❡①t❡r♥♦

❉❛❞❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦ F✱ ❡♠ ❣❡r❛❧ ❝♦♠ três ❝♦♠♣♦♥❡♥t❡s

♥ã♦ ♥✉❧❛s✱ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rsí✈❡❧ _Tˆ_{(t) q✉❡ ♣❡r♠✐t❡}

❡❧✐♠✐♥❛r ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ F✱ ❣❡r❛♥❞♦✱ ❛ss✐♠✱ ✉♠❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❡q✉✐✈❛❧❡♥t❡

❝♦♠ ✉♠ ❝❡rt♦ ❝❛♠♣♦ ❡①t❡r♥♦ r❡❞✉③✐❞♦ F′ ❞❡ ❛♣❡♥❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s✳

❱❛♠♦s ✐♥tr♦❞✉③✐r ❛ tr❛♥s❢♦r♠❛çã♦

V′ = ˆT V , Tˆ(t) = exp [iα(t) (˜σl)] , ✭✷✳✶✶✮ ♦♥❞❡ V é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✭✶✳✸✮ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ F✱ l ✉♠ ✈❡t♦r

❝♦♠♣❧❡①♦ ❛r❜✐trár✐♦ ❡ α(t)✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❛r❜✐trár✐❛✳

❙❡l2 ₆= 0♣♦❞❡♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❢❛③❡rl2 = 1✳ ❈♦♠ ✐st♦✱ ❛ ♠❛tr✐③ ✭✷✳✶✶✮

t♦r♥❛✲s❡

ˆ

T = cosα+i(˜σl) sinα .

P♦r s✉❜st✐t✉✐çã♦ ❞✐r❡t❛✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♦ ❡s♣✐♥♦r V′ _{é t❛♠❜é♠ ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦}

❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ r❡❞✉③✐❞♦F′✱

F′ = [F₋l(Fl)] cos 2α+ [F_×l] sin 2α+l(Fl₋α˙). ✭✷✳✶✷✮ ❙❡♥❞♦ ❛ ♠❛tr✐③ _Tˆ _{✐♥✈❡rsí✈❡❧✱ ❛s ❡q✉❛çõ❡s ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ F❡ ❝♦♠ ♦ ❝❛♠♣♦}

❡①t❡r♥♦F′ ✭✷✳✶✷✮ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ P♦❞❡♠♦s ❛❣♦r❛ ❡s❝♦❧❤❡r ✉♠❛ ❢✉♥çã♦ α(t)q✉❡ r❡s♣❡✐t❡

❛ r❡❧❛çã♦

˙

α=Fl=_⇒F′l= 0. ✭✷✳✶✸✮

❈♦♠ ✐st♦✱ ❛ ♣r♦❥❡çã♦ ❞❡ F′ ♥❛ ❞✐r❡çã♦ ❞❡ l s❡ ❛♥✉❧❛ ❡ ♦ ❝❛♠♣♦ r❡❞✉③✐❞♦ F′ ♣❛ss❛ ❛ t❡r

❛♣❡♥❛s ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ♣❡rt❡♥❝❡♥t❡s ❛♦ ♣❧❛♥♦ ♦rt♦❣♦♥❛❧ ❛ l✳

P❛r❛ ♦ ✈❡t♦r l❝♦♠ l2 = 0✱ t❡♠♦s ˆ

T = 1 +iα(˜σl) , ♥❡st❡ ❝❛s♦✱

F′ =F+ 2α[F_×l] +l2α2(Fl)₋α˙

. ✭✷✳✶✹✮

P❛r❛ ✉♠❛ ❡s❝♦❧❤❛ ❛♣r♦♣r✐❛❞❛ ❞♦ ✈❡t♦r ❝♦♠♣❧❡①♦l✱ ♣♦❞❡✲s❡ s❡♠♣r❡ ❡❧✐♠✐♥❛r ✉♠❛ ❞❛s ❞✉❛s

❝♦♠♣♦♥❡♥t❡sK= ReF′ _{♦✉G= ImF}′_{✳ ❊♥tr❡t❛♥t♦✱ ♥❡st❡ ❝❛s♦✱ ♥ã♦ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛rF}′

❝♦♠♦ ✉♠ ✈❡t♦r ❡♠ ✉♠ ♣❧❛♥♦ ✜①♦✱ ❞✐❢❡r❡♥t❡ ❞♦ ❝❛s♦ ❝♦♠ lr❡❛❧✳

❱❛♠♦s ❡s❝♦❧❤❡r l ❝♦♠♦ ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥❛ ❞✐r❡çã♦ z✱ l = (0,0,1)✱ ❡ α s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦

˙

α=F3.

❊♥tã♦✱ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ r❡❞✉③✐❞♦ F′ ❛ss✉♠❡ ❛ ❢♦r♠❛

F′ = (F₁′, F₂′,0) , ✭✷✳✶✺✮ ♦♥❞❡

F₁′ =F1cos 2α+F2sin 2α, F₂′ =F2cos 2α₋F1sin 2α , F1 =F₁′cos 2α₋F₂′sin 2α, F2 =F₂′cos 2α+F₁′sin 2α . ❊s❝♦❧❤❡♥❞♦ l= (0,0,1) ❡ s❡❧❡❝✐♦♥❛♥❞♦ α ♣❛r❛ s❡r ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦

F1 =F₁′cos 2α , F2 =F₁′sin 2α , F3 =F₃′ + ˙α ,

♦❜t❡♠♦s

F₁′ =F1cos 2α+F2sin 2α , F₃′ =F3₋α ,˙

F₂′ =F2cos 2α₋F1sin 2α = 0, ✭✷✳✶✻✮

❡ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ r❡❞✉③✐❞♦F′ ❛ss✉♠❡ ❛ ❢♦r♠❛

F′ = (F₁′,0, F₃′) . ✭✷✳✶✼✮ ❆❧é♠ ❞✐st♦✱ ♣♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡✱ s❡ V′ _{é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦}

❡①t❡r♥♦ ✭✷✳✶✼✮✱ ❡♥tã♦✿

✶✳ U = (2)−1/2(1 +iσ1)V′é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦

F′ = (F₁′, F₃′,0) ; ✭✷✳✶✽✮

✷✳ U =σ1V′ _{é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦}

F′ = (F₁′,0,₋F₃′) ; ✭✷✳✶✾✮

✸✳ U =σ3V′ _{é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦}

F′ = (₋F1′,0, F3′) ; ✭✷✳✷✵✮ ✹✳ U =σ2V′ é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦

F′ = (₋F₁′,0,₋F₃′) ; ✭✷✳✷✶✮

✺✳ U = (2)−1/2(σ1+σ3)V′ _{é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦}

F′ = (F₃′,0, F₁′). ✭✷✳✷✷✮

✷✳✸ ❘❡♣r❡s❡♥t❛çã♦ ✈❡t♦r✐❛❧

❆ s❡❣✉✐♥t❡ r❡❧❛çã♦ é ✈á❧✐❞❛ ♣❛r❛ q✉❛✐sq✉❡r ❡s♣✐♥♦r❡sU, V ❡ s❡✉s ❛♥t✐❝♦♥❥✉❣❛❞♦sU ,¯ V¯ ✭✷✳✺✮✿

(U,V¯) =₋(V,U¯), ( ¯U , V) = ₋( ¯V , U), ( ¯U ,V¯) = (V, U),

(U,V¯)( ¯V , U) = (U, U)(V, V)₋(U, V)(V, U) = ( ¯U , V)(V,U¯)_≥0. ✭✷✳✷✸✮ ❉❛❞♦s ❞♦✐s ❡s♣✐♥♦r❡s U ❡ V ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ ✈❡t♦r ❝♦♠♣❧❡①♦LU,V✿

LU,V = (U,σV˜ ) = (u∗₁v2+u∗₂v1, iu₂∗v1₋iu∗₁v2, u∗₁v1₋u∗₂v2), ✭✷✳✷✹✮

♦ q✉❛❧ r❡s♣❡✐t❛ ❛s ♣r♦♣r✐❡❞❛❞❡s

i) LU,V∗ =LV,U, LU ,¯V¯ =₋LV,U,

ii)LU,VLU′,V′ = 2 (U, V′) (U′, V)₋(U, V) (U′, V′) ,

iii)LV,VLV,V = (V, V)2 , LV ,V¯ LV ,V¯ =LV,V¯LV,V¯ = 0,

iv)LV ,V¯ LV,V¯ = 2 (V, V)2 , LV,VLV ,V¯ =LV,VLV,V¯ = 0,

v) hLV,V¯ _×LV ,V¯ i = 2i(V, V)LV,V , hLV ,V¯ _×LV,Vi =i(V, V)LV ,V¯ , h

LV,V _×LV,V¯i=i(V, V)LV,V¯ ,

vii)LU,V = (V, V)−1h(U, V)LV,V + U,V¯

LV ,V¯ i . ✭✷✳✷✺✮

❯s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❛ s❡❣✉✐r ✭✷✳✽✮ U = (V, V)−1

(V, U)V + ¯V , U _¯ V

,

❞❛❞♦ ✉♠ ✈❡t♦r p ❡ ✉♠ ❡s♣✐♥♦r V ♣♦❞❡♠♦s ♦❜t❡r✱

(˜σp)V = (V, V)−1h LV,VpV +LV ,V¯ pV¯i . ✭✷✳✷✻✮ ❆s r❡❧❛çõ❡s ✭✷✳✷✺✮ ✐♠♣❧✐❝❛♠ q✉❡ t♦❞♦ ❡s♣✐♥♦r V ❣❡r❛ três ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡✲ ♣❡♥❞❡♥t❡s✱

LV,V, LV ,V¯ ,LV,V¯ . ✭✷✳✷✼✮ ❚♦❞♦ ✈❡t♦r ❝♦♠♣❧❡①♦ a ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ♥❡st❡s ✈❡t♦r❡s

a=a1LV,V +a2LV ,V¯ +a3LV,V¯ , a1 = aL

V,V

(V, V)2 , a2 =

aLV,V¯

2 (V, V)2 , a3 =

aLV ,V¯

2 (V, V)2 . ✭✷✳✷✽✮

❈♦♠ ❛ ❛❥✉❞❛ ❞♦s ✈❡t♦r❡s ✭✷✳✷✼✮✱ ♣♦❞❡♠♦s ❞❡✜♥✐r três ✈❡t♦r❡s ♦rt♦❣♦♥❛✐s r❡❛✐s

e1 =

LV,V¯ +LV ,V¯

2 (V, V) , e2 =i

LV,V¯ ₋LV ,V¯

2 (V, V) , n=

LV,V

(V, V), ✭✷✳✷✾✮

❡st❡s ♦❜❡❞❡❝❡♠ às r❡❧❛çõ❡s

♦♥❞❡ ǫijk é ♦ sí♠❜♦❧♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ✭ǫ123= 1✮✳ ❆s r❡❧❛çõ❡s ✐♥✈❡rs❛s tê♠ ❛ ❢♦r♠❛

LV,V = (V, V)n, LV ,V¯ = (V, V) (e1+ie2), LV, ¯

V _{= (V, V) (e}

1 −ie2) . ✭✷✳✸✶✮ ❯♠ ❡s♣✐♥♦r V s❡♠♣r❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦

V =N eiα2  

e−iϕ

2 cosθ 2 eiϕ

2 sinθ 2



, N2 = (V, V) , ✭✷✳✸✷✮ ♦♥❞❡ N, α, θ, ❡ϕ sã♦ ♥ú♠❡r♦s r❡❛✐s✳ ◆❡st❛ r❡♣r❡s❡♥t❛çã♦✱ ♦ ❡s♣✐♥♦r ❛♥t✐❝♦♥❥✉❣❛❞♦ t❡♠ ❛ ❢♦r♠❛

¯

V =N e−iα2  

−e−iϕ

2 sinθ 2

eiϕ

2 cosθ 2



 . ✭✷✳✸✸✮

❈♦♥s✐❞❡r❛♥❞♦ θ ❡ ϕ ❝♦♠♦ â♥❣✉❧♦s ❞❡ ✉♠ s✐st❡♠❛ ❞❡ r❡❢❡r❡♥❝✐❛❧ ❡s❢ér✐❝♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦s ✈❡t♦r❡s ✉♥✐tár✐♦s ♦rt♦❣♦♥❛✐seϕ,eθ,❡ n✱

eθ = (cosθcosϕ,cosθsinϕ,−sinθ) = [eϕ×n] ,

eϕ = (sinϕ,cosϕ,0) = [n×eθ] ,

n = (sinθcosϕ,sinθsinϕ,cosθ) = [eθ×eϕ] , ✭✷✳✸✹✮

❡♠ t❡r♠♦ ❞♦s q✉❛✐s✱ ♦s ✈❡t♦r❡s ✭✷✳✷✼✮ ❡ ✭✷✳✷✾✮ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦

LV,V =N2n, LV ,V¯ =N2(e_θ+ieϕ)eiα, LV,V¯ =N2(e_θ−ieϕ)e−iα,

e₁ =e_θcosα₋e_ϕsinα , e₂ =e_θsinα+e_ϕcosα . ✭✷✳✸✺✮ ❆❧é♠ ❞✐st♦✱ s❡❣✉❡ ❞❡ ✭✷✳✷✻✮ ❡ ✭✷✳✸✺✮ q✉❡

(˜σF)V = (nF)V + (Fe_θ+iFe_ϕ) exp (iα) ¯V . ✭✷✳✸✻✮

❈❛♣ít✉❧♦ ✸

❊q✉❛çõ❡s r❡❧❛❝✐♦♥❛❞❛s

❈♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ✈ár✐❛s ❡q✉❛çõ❡s ❡♥❝♦♥tr❛❞❛s ❡♠ ❞✐❢❡r❡♥t❡s ♣r♦❜❧❡♠❛s ❡♠ ❢ís✐❝❛ ❡stã♦ ♣r♦❢✉♥❞❛♠❡♥t❡ ❧✐❣❛❞❛s ❝♦♠ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✳ ❆ s❡❣✉✐r✱ ✈❛♠♦s ❡st❛❜❡❧❡✲ ❝❡r ❡st❛s ❧✐❣❛çõ❡s ❡①♣❧✐❝✐t❛♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡st❛s ❡q✉❛çõ❡s✳ ❆ ❝✐ê♥❝✐❛ ❞❡st❛s r❡❧❛çõ❡s ♣♦ss✐❜✐❧✐t❛ ❡st❡♥❞❡r ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦✲ ❜❧❡♠❛✱ ❛❧é♠ ❞❡ ♣❡r♠✐t✐r s✉❛ r❡❢♦r♠✉❧❛çã♦ ❛tr❛✈és ❞❡ ❡①♣r❡ssõ❡s ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡s✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ❡♠ ❬✶✻❪ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ✭✶✳✸✮ ❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤ö❞✐♥❣❡r ♥❛ ❢♦r♠❛ ✭✸✳✸✮ ❛❜❛✐①♦ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ♦❜t❡r ❛s s♦❧✉çõ❡s ✭✶✳✽✮✱ ❡♠ ❬✺❪ ❛ r❡❧❛çã♦ ❡♥tr❡ ✭✶✳✸✮ ❡ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r ✭✸✳✶✶✮ ♥♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦ ♣❡r♠✐t❡ ❛♥❛❧✐s❛r ✉♠❛ sér✐❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ♠♦❧é❝✉❧❛s ♥✉♠❛ ❝❛✈✐❞❛❞❡ s♦❜ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ❡❧étr✐❝♦ ♦s❝✐❧❛♥t❡ ❞❡ ❛♠♣❧✐t✉❞❡ ❝♦♥st❛♥t❡✳

✸✳✵✳✶ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

✶✳ ❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡♠ 0 + 1 ❞✐♠❡♥sã♦ ♣❛r❛ ✉♠ ❡s♣✐♥♦r Ψ (t) ❝♦♠

❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♠♣❧❡①❛s ❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦✳ ◆♦ ❝❛s♦ ❣❡r❛❧✱ ❡st❛ ❡q✉❛çã♦ ❛ss✉♠❡ ❛ ❢♦r♠❛

iΨ =˙ HΨ, ✭✸✳✶✮

♦♥❞❡ ♦ ❤❛♠✐❧t♦♥✐❛♥♦Hé ✉♠❛ ♠❛tr✐③2_×2❝♦♠♣❧❡①❛ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✳ ❆ ♠❛tr✐③

H ♣♦❞❡ s❡♠♣r❡ s❡r ❞❡❝♦♠♣♦st❛ ♥❛s ♠❛tr✐③❡s ❞❡ ❜❛s❡ H = F0I +˜σF✱ ❝♦♠ F0 =

F0(t)❡F= (Fk(t), k= 1,2,3)✳ ❆tr❛✈és ❞❛ tr❛♥s❢♦r♠❛çã♦Ψ =V exp −i

R F0dt

✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ♣❛r❛ ♦ ❡s♣✐♥♦rV✳

✷✳ ❆ ❡q✉❛çã♦ ❞❡ s♣✐♥ ♣♦❞❡ s❡r r❡❞✉③✐❞❛ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♠ ❣❡r❛❧ ❝♦♠ ♣♦t❡♥❝✐❛✐s ❝♦♠♣❧❡①♦s✳ ❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✷✳✸✮ ❛s ❢✉♥çõ❡s

vs=

p

Asψs, As=F1+ (−1)siF2, ✭✸✳✷✮

♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

¨

ψs−Vsψs = 0, s= 1,2, ✭✸✳✸✮

Vs = 3 4

˙

As

As

!2

− 1₂A_A¨s

s −

A1A2₋F₃2₋i(₋1)s F3A˙s As −

˙

F3 !

. ✭✸✳✹✮

■❞❡♥t✐✜❝❛♥❞♦ ❛s ❞❡r✐✈❛❞❛s ❡♠ ✭✸✳✸✮ ❝♦♠♦ ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s ✭_ψ¨ _{= dψ/dx}2_{✮✱ ❝❛❞❛} ✉♠❛ ❞❡st❛s ❡q✉❛çõ❡s ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡st❛✲ ❝✐♦♥ár✐❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ♣♦t❡♥❝✐❛✐s ❝♦♠♣❧❡①♦sVs✳

✸✳✵✳✷ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r

P❛r❛ ✉♠ ♣❛r ❞❡ ❡s♣✐♥♦r❡s U ❡V ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ✈❡t♦r ❝♦♠♣❧❡①♦ LU,V = (U,σV˜ ) ✭✈❡❥❛

✭✷✳✷✹✮✮✱ s❡V é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ s♣✐♥✱ ❡♥tã♦ ♦s ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s

LV,V, LV ,V¯ ❡ LV,V¯ ✭✷✳✷✼✮ ♦❜❡❞❡❝❡♠ às s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ˙

LV,V =i(F∗ ₋F) (V, V) +

(F+F∗)_×LV,V ,

˙

LV ,V¯ = 2hF_×LV ,V¯ i , L˙V,V¯ = 2hF∗_×LV,V¯i , ✭✸✳✺✮ ❡♥q✉❛♥t♦ ❛ ❡q✉❛çã♦ ❞❡ s♣✐♥ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ✭✷✳✷✺✮ ✐♠♣❧✐❝❛♠ ♥❛s r❡❧❛çõ❡s

LV,V˙ =₋i(V, V)F+

F_×LV,V , LV ,¯ V˙ =L

·

¯

V ,V ₌h_F

❆❧é♠ ❞✐st♦✱ ♦s ✈❡t♦r❡se₁,e₂,n ✭✷✳✷✾✮✱ e1 =

LV,V¯ _+LV ,V¯

2 (V, V) , e2 =i

LV,V¯ _−LV ,V¯

2 (V, V) , n=

LV,V (V, V),

♦❜❡❞❡❝❡♠ às s❡❣✉✐♥t❡s ❡q✉❛çõ❡s✿

˙e1 = 2e2(Kn)−2n(Kn+Ge1) ,

˙e2 = 2n(Ke1 −Ge2)−2e1(Kn) ,

˙

n= 2e1(Ke2 +Ge1) + 2e2(Ge2−Ke1) , ✭✸✳✼✮ ❝♦♠K❡G❞❛❞♦s ♣♦r ✭✶✳✹✮✳ ❈♦♠♦V ♦❜❡❞❡❝❡ à ❡q✉❛çã♦ ❞❡ s♣✐♥✱ ❛tr❛✈és ❞❛ r❡♣r❡s❡♥t❛çã♦ ✭✷✳✸✷✮✱

V =N eiα2  

e−iϕ

2 cosθ 2

eiϕ

2 sinθ 2



, N2 = (V, V) ,

❡ ❞♦s ✈❡t♦r❡s ♦rt♦❣♦♥❛✐s ✭✷✳✸✹✮✱

eθ = cosθ(cosϕ,sinϕ,−tanθ) = e1cosα+e2sinα

eϕ = (sinϕ,cosϕ,0) =e2cosα−e1sinα ,

♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦s ♣❛râ♠❡tr♦s N, α, θ, ❡ ϕ✳ ❚♦♠❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ❡①♣r❡ssã♦ ♣❛r❛ _V¯ _{✭✷✳✸✸✮ ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ✭✷✳✽✮ t❡♠♦s}

2 ˙V =2N−1N˙ +iα˙ ₋iϕ˙cosθV +θ˙+iϕ˙sinθexp (iα) ¯V . ✭✸✳✽✮ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❛s ❡①♣r❡ssõ❡s ✭✸✳✽✮✱ ✭✷✳✸✻✮ ❡ ✭✶✳✹✮ ♦❜t❡♠♦s✿

˙

θ = 2Ke_ϕ+ 2Ge_θ, ϕ˙sinθ= 2Ge_ϕ−2Ke_θ, ✭✸✳✾✮

˙

α= ˙ϕcosθ₋2Kn, N˙ =NGn. ✭✸✳✶✵✮ ❆s ❡q✉❛çõ❡s ✭✸✳✾✮ sã♦ ❛✉tô♥♦♠❛s✱ ♥♦ s❡♥t✐❞♦ ❞❡ ♥ã♦ ❞❡♣❡♥❞❡r❡♠ ❞❛s ❢✉♥çõ❡s N ❡ α✱ ❡ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ♥❛ ❢♦r♠❛ ❝♦♠♣❛❝t❛

˙

n= 2 [G₋(Gn)n] + 2 [K_×n] . ✭✸✳✶✶✮