❊st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡
❇♦s❡✲❊✐♥st❡✐♥
❍❡♥r✐q✉❡ ❋❛❜r❡❧❧✐ ❋❡rr❡✐r❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆r♥❛❧❞♦ ●❛♠♠❛❧
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ♠❡str❡ ❡♠ ❈✐ê♥❝✐❛s✳
❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛
Pr♦❢✳ ❉r✳ ❆r♥❛❧❞♦ ●❛♠♠❛❧ ✭■❋❯❙P✮
Pr♦❢✳ ❉r✳ ❊♠❛♥✉❡❧ ❆❧✈❡s ❞❡ ▲✐♠❛ ❍❡♥♥ ✭■❋❙❈✴❯❙P✮ Pr♦❢✳ ❉r✳ ❱❛❧❡r② ❙❤❝❤❡s♥♦✈✐❝❤ ✭❯❋❆❇❈✮
do Instituto de Física da Universidade de São Paulo
Ferreira, Henrique Fabrelli
Estabilidade de vórtices em condensados de Bose-Einstein.
São Paulo, 2016.
Dissertação (Mestrado) – Universidade de São Paulo. Instituto de Física.
Depto. de Física Experimental.
Orientador: Prof. Dr. Arnaldo Gammal.
Área de Concentração: Física.
Unitermos: 1. Condensado de Bose-Einstein; 2. Vórtices dos gases;
3. Métodos numéricos.
❙t❛❜✐❧✐t② ♦❢ ✈♦rt✐❝❡s ✐♥ ❇♦s❡✲❊✐♥st❡✐♥
❝♦♥❞❡♥s❛t❡s
❍❡♥r✐q✉❡ ❋❛❜r❡❧❧✐ ❋❡rr❡✐r❛
❙✉♣❡r✈✐s♦r✿ Pr♦❢✳ ❉r✳ ❆r♥❛❧❞♦ ●❛♠♠❛❧
❉✐ss❡rt❛t✐♦♥ s✉❜♠✐tt❡❞ t♦ t❤❡ P❤②s✐❝s ■♥st✐t✉t❡ ♦❢ ❯♥✐✈❡rs✐t② ♦❢ ❙ã♦ P❛✉❧♦ t♦ ♦❜t❛✐♥ t❤❡ ❞❡❣r❡❡ ♦❢ ▼❛st❡r ✐♥ ❙❝✐❡♥❝❡✳
❊①❛♠✐♥✐♥❣ ❇♦❛r❞
Pr♦❢✳ ❉r✳ ❆r♥❛❧❞♦ ●❛♠♠❛❧ ✭■❋❯❙P✮
Pr♦❢✳ ❉r✳ ❊♠❛♥✉❡❧ ❆❧✈❡s ❞❡ ▲✐♠❛ ❍❡♥♥ ✭■❋❙❈✴❯❙P✮ Pr♦❢✳ ❉r✳ ❱❛❧❡r② ❙❤❝❤❡s♥♦✈✐❝❤ ✭❯❋❆❇❈✮
❖ ♠❛✐s ❥✉st♦ ❛ s❡ ❢❛③❡r ❛q✉✐ s❡r✐❛ ❛❣r❛❞❡❝❡r ❛ t♦❞♦s ♦s ❢❛t♦s ❡ ♣❡ss♦❛s q✉❡✱ ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛✱ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✳ ■♥❢❡❧✐③♠❡♥t❡ s❡r❡✐ ♣r❛❣♠át✐❝♦ ❡ t❛❧✈❡③ ✐♥❥✉st♦ ♥♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✱ ♣♦✐s ❞❡✈♦ r❡s♣❡✐t❛r ♦ ❡s♣❛ç♦✳ ❉❡✈♦✱ ♣♦rt❛♥t♦✱ ❧✐♠✐t❛r✲♠❡ ❛♦s ❢❛t♦s ❡ ♣❡ss♦❛s q✉❡ ❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✳ ❆ t♦❞♦s ♦s ♦✉tr♦s✱ r❡s✉✲ ♠✐❞❛♠❡♥t❡✱ ♠❛♥✐❢❡st♦ ♠✐♥❤❛ ♠❛✐s ♣r♦❢✉♥❞❛ ❣r❛t✐❞ã♦✳
Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢❡ss♦r ❆r♥❛❧❞♦ ●❛♠♠❛❧✱ ♣♦r t♦❞❛ ❛ ❛t❡♥çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❡♥s✐♥❛♠❡♥t♦s ♣❛ss❛❞♦s✳ ❊✉ ♥ã♦ t❡♥❤♦ ♣❛❧❛✈r❛s ♣❛r❛ ❡①♣r❡ss❛r ♠✐♥❤❛ ❣r❛t✐❞ã♦ ♣♦r s✉❛ ❣❡♥❡r♦s✐❞❛❞❡ ❡♠ ♠❡ ❛❝♦❧❤❡r ❡♠ s❡✉ ❣r✉♣♦✳ ❉✉r❛♥t❡ ♦ ár❞✉♦ ❝❛♠✐♥❤♦ ❞❛ ❣r❛❞✉❛çã♦ ❢♦r❛♠ ♦s tr❛❜❛❧❤♦s ❞❡ ■♥✐❝✐❛çã♦ ❈✐❡♥tí✜❝❛ ❡ s❡✉s ❝♦♥s❡❧❤♦s s❡♠♣r❡ ❜❡♠✲✈✐♥❞♦s q✉❡ ♠❡ ♠♦t✐✈❛r❛♠ ❛ s❡❣✉✐r ❡♠ ❢r❡♥t❡ ❡ ♥ã♦ ♠❡ ❛❜❛t❡r ♣❡r❛♥t❡ ❛♦s ❞❡s❛✜♦s✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♠✐♥❤❛ ♠ã❡✱ ▲❡❞❛ ❙❛❧❡tt❡ ❋❛❜r❡❧❧✐✱ ♣♦r t♦❞♦ ❛♠♦r✱ ❛✉①í❧✐♦ ❡ ♣❛❝✐ê♥❝✐❛ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦s ❡ss❡s ❛♥♦s✳ ❆ t♦❞♦s ♦s t✐♦s ❡ ✐♥ú♠❡r♦s ♣r✐♠♦s ♣♦r t♦❞❛s ❛s ❛❧❡❣r✐❛s ❡ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ t❛♥t♦ ♥♦s ❜♦♥s q✉❛♥t♦ ♥♦s ♠❛✉s ♠♦♠❡♥t♦s✳
❆♦ ♠❡✉ ✈❡❧❤♦ ❛♠✐❣♦ ❉r✳ ❏♦sé ◆♦r❜❡rt♦ ❘❛♠♦s ▲❡✐t❡ ♣❡❧❛ s✐❣♥✐✜❝❛t✐✈❛ ✐♥✢✉ê♥❝✐❛ ❡①❡r✲ ❝✐❞❛ ♣❛r❛ q✉❡ ❡✉ ✜r♠❛ss❡ ♠✐♥❤❛ ♦♣çã♦ ❡♠ ❝✉rs❛r ❋ís✐❝❛ ❡ t❛♠❜é♠ ♣❡❧❛s ❛♥✐♠❛❞❛s ❡ ♣r♦✈❡✐t♦s❛s ❝♦♥✈❡rs❛s s♦❜r❡ ❋ís✐❝❛✱ ❊♣✐st❡♠♦❧♦❣✐❛✱ Ps✐q✉✐❛tr✐❛✱ ◆❡✉r♦❝✐ê♥❝✐❛✱ ❍✐stór✐❛✱ ❡t❝✳ ❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❣r✉♣♦✱ ❊❞✉❛r❞♦ ●❡♦r❣❡ ❑❤❛♠✐s✱ ❑❛r✐♥❡ P✐❛❝❡♥t✐♥✐ ❈♦❡❧❤♦ ❞❛ ❈♦st❛✱ ❑✐s❤♦r ❑✉♠❛r ❘❛♠❛✈❛r♠❛r❛❥❛ ❡ ❘♦❜✐♥s♦♥ ❏✉♥✐♦r P✐♥❤♦ P♦♠♣❡✉✱ ♣♦r t♦❞❛ ❛❥✉❞❛ ❡ ✐♥ú♠❡r♦s ❡♥s✐♥❛♠❡♥t♦s ♣❛ss❛❞♦s✳ ❆ t♦❞♦s ♦s ❞❡♠❛✐s ❛♠✐❣♦s ❞♦ ■❋❯❙P✱ ❞♦s q✉❛✐s ♥ã♦ ❝✐t❛r❡✐ ♥♦♠❡s ♣❛r❛ ❡✈✐t❛r ❛ ✐♥❥✉st✐ç❛ ❞❡ ❡sq✉❡❝❡r ❛❧❣✉♠✳
❆♦s ♣r♦❢❡ss♦r❡s ❆♥tô♥✐♦ ❋❡r♥❛♥❞♦ ❘✐❜❡✐r♦ ❞❡ ❚♦❧❡❞♦ P✐③❛ ❡ ❊♠❡rs♦♥ ❏♦sé ❱❡❧♦s♦ ❞❡ P❛ss♦s ♣❡❧❛s s❡♠❛♥❛✐s ❞✐s❝✉ssõ❡s ❡♠ ♥♦ss♦ ❏♦✉r♥❛❧ ❈❧✉❜✱ ❛ ♣❛rt✐r ❞❛s q✉❛✐s ✈❡♥❤♦ ♦❜t❡♥❞♦ ❜♦❛ ♣❛rt❡ ❞♦ ♠❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✳ ❙✐♥t♦✲♠❡ ♠✉✐t♦ ❤♦♥✲ r❛❞♦ ♣♦r t❡r ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❝❡❜❡r t❛✐s ❡♥s✐♥❛♠❡♥t♦s ❞❡ss❡s ♣r♦❢❡ss♦r❡s✳
❆♦s ❞❡♠❛✐s ❢✉♥❝✐♦♥ár✐♦s ❞♦ ■❋❯❙P✱ ❡s♣❡❝✐❛❧♠❡♥t❡ à ❙✐♠♦♥❡ ❡ à ❙✐❧✈✐❛✱ ♣r♦♣r✐❡tár✐❛s ❞❛ ❝❛♥t✐♥❛✱ ♣♦r t♦❞❛ ❛ ❛❧❡❣r✐❛ ❡ ♣❡❧♦s ❞❡❧✐❝✐♦s♦s ❞♦❝❡s✱ s❛❧❣❛❞♦s ❡ ❜♦❧❛❝❤❛s✱ q✉❡ ♠❡ ♠❛♥t✐✈❡✲ r❛♠ ❜❡♠ ❛❧✐♠❡♥t❛❞♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦ ♣r♦❢❡ss♦r ◆✐❦♦❧❛♦s P✳ Pr♦✉❦❛❦✐s ♣♦r ♠❡ r❡❝❡❜❡r ❡♠ s❡✉ ❣r✉♣♦ ♥❛ ◆❡✇❝❛st❧❡ ❯♥✐✈❡r✲ s✐t②✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ♣❛ss❛❞♦s✱ s✉❣❡stõ❡s ❛ ❡ss❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ✐♥✐❝✐❛r ✉♠❛ ❝♦❧❛❜♦r❛çã♦✳ ➚ ❋❛❜r✐③✐♦ ▲❛r❝❤❡r ❡ P❛♦❧♦ ❈♦♠❛r♦♥ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛✉①í❧✐♦ ❞✉r❛♥t❡ ♠✐♥❤❛ ❡st❛❞❛ ❡♠ ◆❡✇❝❛st❧❡✳ ●r❛③✐❡ ♣❡r t✉tt♦✦
P♦r ú❧t✐♠♦✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ❛❣r❛❞❡ç♦ à ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮ ❡ ❛♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥✲ tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✭❈◆Pq✮ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦✳
◆❡st❡ tr❛❜❛❧❤♦ ❞❡ ♠❡str❛❞♦ é ❡st✉❞❛❞❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❡♥tr❡ ♦s át♦♠♦s ❛tr❛✈és ❞❛ s♦❧✉çã♦ ♥✉♠ér✐❝❛ ❞❛ ❡q✉❛✲ çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐✳ ■♥✐❝✐❛❧♠❡♥t❡ sã♦ r❡♣r♦❞✉③✐❞♦s r❡s✉❧t❛❞♦s ❞❛ ❧✐t❡r❛t✉r❛✱ ♥♦s q✉❛✐s sã♦ ❡st✉❞❛❞♦s ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❛tr❛t✐✈♦s ❝♦♠ ♣♦t❡♥❝✐❛❧ ✐♥t❡r❛tô✲ ♠✐❝♦ ❤♦♠♦❣ê♥❡♦ ❡♠ t♦❞♦ ♦ ❝♦♥❞❡♥s❛❞♦✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ t❛✐s s✐st❡♠❛s é ✐♥❢❡r✐❞❛ ❛tr❛✈és ❞❛ s♦❧✉çã♦ ♥✉♠ér✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s ❡ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s✳ ❉❡♠♦♥str❛✲s❡ q✉❡ ❡ss❡s ✈órt✐❝❡s sã♦ ❡stá✈❡✐s✱ ❛té ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❝rít✐❝♦ ❞❡ át♦♠♦s✱ ❛♣❡♥❛s ♣❛r❛ ✈❛❧♦r❡s ❞❡ ✈♦rt✐❝✐❞❛❞❡ S = 1✳ ❊♠ s❡❣✉✐❞❛✱ ❢♦✐ ♣r♦♣♦st♦ ✉♠ ♠♦❞❡❧♦
♥♦ q✉❛❧ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s át♦♠♦s é ❡s♣❛❝✐❛❧♠❡♥t❡ ♠♦❞✉❧❛❞♦✳ ◆❡st❡ ❝❛s♦✱ é ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ ✈órt✐❝❡s ❝♦♠ ✈❛❧♦r❡s ❞❡ ✈♦rt✐❝✐❞❛❞❡ ❞❡✱ ♣❡❧♦ ♠❡♥♦s✱ S = 6 sã♦
❡stá✈❡✐s✳ ❋✐♥❛❧♠❡♥t❡✱ é ❡st✉❞❛❞❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s tr✐❞✐♠❡♥s✐♦✲ ♥❛✐s ❛tr❛t✐✈♦s✱ ♥♦✈❛♠❡♥t❡ ❝♦♠ ♣♦t❡♥❝✐❛❧ ✐♥t❡r❛tô♠✐❝♦ ❤♦♠♦❣ê♥❡♦ ❡♠ t♦❞♦ ♦ ❝♦♥❞❡♥s❛❞♦✳ ❆ss✐♠ ❝♦♠♦ ♥♦ ❝❛s♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ♦❜s❡r✈❛✲s❡ q✉❡ t❛✐s ✈órt✐❝❡s sã♦ ❡stá✈❡✐s ♣❛r❛ ✈❛❧♦r❡s ❞❡ ✈♦rt✐❝✐❞❛❞❡ ❞❡S = 1✳ ❊s♣❡r❛✲s❡ ❡♠ ❜r❡✈❡ ❡st✉❞❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡s❛❞♦s
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ st❛❜✐❧✐t② ♦❢ ✈♦rt✐❝❡s ✐♥ ❛ttr❛❝t✐✈❡ ❇♦s❡✲❊✐♥st❡✐♥ ❝♦♥❞❡♥s❛t❡s ❜② s♦❧✈✐♥❣ ♥✉♠❡r✐❝❛❧❧② t❤❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡q✉❛t✐♦♥✳ ■♥✐t✐❛❧❧② ✇❡ r❡♣r♦❞✉❝❡ s♦♠❡ r❡s✉❧ts ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐♥ ✇❤✐❝❤ ✈♦rt✐❝❡s ✐♥ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛ttr❛❝t✐✈❡ ❇♦s❡✲❊✐♥st❡✐♥ ❝♦♥❞❡♥✲ s❛t❡s ✇✐t❤ ❤♦♠♦❣❡♥❡♦✉s ✐♥t❡r❛t♦♠✐❝ ♣♦t❡♥t✐❛❧ ❛r❡ st✉❞✐❡❞✳ ❚❤❡ st❛❜✐❧✐t② ♦❢ t❤❡s❡ s②st❡♠s ✐s ❞❡t❡r♠✐♥❡❞ ❜② s♦❧✈✐♥❣ ♥✉♠❡r✐❝❛❧❧② t❤❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s ❡q✉❛t✐♦♥s ❛♥❞ ❜② st✉❞②✐♥❣ t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡s❡ ✈♦rt✐❝❡s✳ ❲❡ ❞❡♠♦♥str❛t❡ t❤❛t t❤❡s❡ ✈♦rt✐❝❡s ❛r❡ st❛❜❧❡✱ ✉♣ t♦ ❛ ❝❡rt❛✐♥ ❝r✐t✐❝❛❧ ♥✉♠❜❡r ♦❢ ❛t♦♠s✱ ❥✉st ❢♦r t❤❡ ✈❛❧✉❡ ♦❢ ✈♦rt✐❝✐t② S = 1✳ ❆❢t❡r✱ ✇❡
♣r♦♣♦s❡ ❛ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ t❤❡ ✐♥t❡r❛t♦♠✐❝ ♣♦t❡♥t✐❛❧ ✐s s♣❛t✐❛❧❧② ♠♦❞✉❧❛t❡❞✳ ■♥ t❤✐s ❝❛s❡✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✈❡r✐❢② t❤❛t ✈♦rt✐❝❡s ✇✐t❤ ✈❛❧✉❡s ♦❢ ✈♦rt✐❝✐t②✱ ❛t ❧❡❛st✱S= 6❛r❡ st❛❜❧❡✳ ❋✐♥❛❧❧②✱
✇❡ st✉❞② t❤❡ st❛❜✐❧✐t② ♦❢ ✈♦rt✐❝❡s ✐♥ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❛ttr❛❝t✐✈❡ ❝♦♥❞❡♥s❛t❡s✱ ❛❣❛✐♥ ✇✐t❤ ❛ ❤♦♠♦❣❡♥❡♦✉s ✐♥t❡r❛t♦♠✐❝ ♣♦t❡♥t✐❛❧✳ ❆s ✐♥ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✱ ✇❡ s❤♦✇ t❤❛t ✈♦rt✐❝❡s ✐♥ t❤❡s❡ s②st❡♠s ❛r❡ st❛❜❧❡ t♦ ✈❛❧✉❡s ♦❢ ✈♦rt✐❝✐t② S = 1✳ ❚❤❡ ♥❡①t st❡♣ ✐♥ t❤✐s ✇♦r❦ ✐s
st✉❞② t❤❡ st❛❜✐❧✐t② ♦❢ ✈♦rt✐❝❡s ✐♥ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥❞❡♥s❛t❡s ✇✐t❤ s♣❛t✐❛❧❧② ♠♦❞✉❧❛t❡❞ ✐♥t❡r❛t♦♠✐❝ ♣♦t❡♥t✐❛❧✳
✶ ■♥tr♦❞✉çã♦ ✶ ✶✳✶ ❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❙♦❜r❡ ❡st❡ tr❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷ ❈♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✼
✷✳✶ ❊q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❙ó❧✐t♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸ ❱órt✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸✳✷ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ P❡q✉❡♥❛s ♦s❝✐❧❛çõ❡s ❡♠ ✉♠ ❝♦♥❞❡♥s❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✻ ❘❡ss♦♥â♥❝✐❛ ❞❡ ❋❡s❤❜❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ ▼ét♦❞♦s ♥✉♠ér✐❝♦s ✷✶
✸✳✶ ❙♣❧✐t✲❙t❡♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❆❧❣♦r✐t♠♦ ❞❡ r❡❧❛①❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✸ ▼ét♦❞♦s ❞❡ ❈r❛♥❦✲◆✐❝♦❧s♦♥ ❡ ❉✬②❛❦♦♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✹ ❉✐s❝r❡t✐③❛çã♦ ❞❛s ❡q✉❛çõ❡s ❇♦❣♦❧✐✉❜♦✈ ❞❡✲●❡♥♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✹ ❘❡s✉❧t❛❞♦s ✸✶
✹✳✶ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❛tr❛t✐✈♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✲
g ❝♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶✳✶ P♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❡ ❡♥❡r❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶✳✷ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✶✳✸ ❊✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❛tr❛t✐✈♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✲
g ❡s♣❛❝✐❛❧♠❡♥t❡ ♠♦❞✉❧❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✷✳✶ P♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❡ ❡♥❡r❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✷✳✷ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷✳✸ ❊✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✸ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❛tr❛t✐✈♦s tr✐❞✐♠❡♥s✐♦♥❛✐s ✲
g ❝♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸✳✶ P♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❡ ❡♥❡r❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸✳✷ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❆ ❘❡❡s❝❛❧♦♥❛♠❡♥t♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡ ♦r❞❡♥s ❞❡ ❣r❛♥❞❡③❛ ✺✶ ❆✳✶ ❘❡❡s❝❛❧♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ❆✳✷ ❖r❞❡♥s ❞❡ ❣r❛♥❞❡③❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ❇ ▲✐♠✐t❡ ❛ss✐♥tót✐❝♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ✺✺
❈ ▼ét♦❞♦ ✈❛r✐❛❝✐♦♥❛❧ ✺✾
❈✳✶ ❈á❧❝✉❧♦s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ✈órt✐❝❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ❈✳✷ ❈á❧❝✉❧♦s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ✈órt✐❝❡s tr✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ❈✳✸ ▼ét♦❞♦ ♥✉♠ér✐❝♦ ♣❛r❛ s♦❧✉çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ♥ã♦✲❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✻✸
■♥tr♦❞✉çã♦
✶✳✶ ❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥
◆♦ ❛♥♦ ❞❡ ✷✵✶✺ ❢♦r❛♠ ❝♦♠❡♠♦r❛❞♦s ♦s ❛♥✐✈❡rsár✐♦s ❞❡ ✾✵ ❛♥♦s ❞♦ ✐♥í❝✐♦ ❞❛ ❢♦r♠✉❧❛çã♦ t❡ór✐❝❛ ❡ ❞❡ ✷✵ ❛♥♦s ❞❛s ♣r✐♠❡✐r❛s r❡❛❧✐③❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ✉♠❛ ❞❛s ♠❛✐s ❛t✐✈❛s ❡ ♣r♦♠✐ss♦r❛s ár❡❛s ❞❡ ♣❡sq✉✐s❛ ❞❛ ❋ís✐❝❛ ♥❛ ❛t✉❛❧✐❞❛❞❡✿ ❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✳❆ ❝♦♥❝❡♣çã♦ ❞❡st❛ ár❡❛ s❡ ❞á ❡♠ ♠❡❛❞♦s ❞❡ ✶✾✷✹✱ q✉❛♥❞♦ ♦ ❢ís✐❝♦ ✐♥❞✐❛♥♦ ❙❛t②❡♥❞r❛ ◆❛t❤ ❇♦s❡ ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡r✐✈❛çã♦ ❛❧t❡r♥❛t✐✈❛ à ❡st❛tíst✐❝❛ ❞❡ ❢ót♦♥s ❡ à ❢ór♠✉❧❛ ❞❡ r❛❞✐❛çã♦ ❞❡ ❝♦r♣♦ ♥❡❣r♦✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❞❡❞✉③✐❞❛ ♣❡❧♦ ❢ís✐❝♦ ❛❧❡♠ã♦ ▼❛① P❧❛♥❝❦✳ ❙❡♥❞♦ s❡✉ tr❛❜❛❧❤♦ r❡❥❡✐t❛❞♦ ♣❛r❛ ❛ ♣✉❜❧✐❝❛çã♦ ♥❛ P❤✐❧♦s♦♣❤✐❝❛❧ ▼❛❣❛③✐♥❡✱ ❇♦s❡ r❡❝♦rr❡ ❛♦ ❢ís✐❝♦ ❛❧❡♠ã♦ ❆❧❜❡rt ❊✐♥st❡✐♥ ♣❛r❛ q✉❡ ❡st❡ ✐♥t❡r❝❡❞❛ ♥❛ ♣✉❜❧✐❝❛çã♦ ♥♦ ♣❡r✐ó❞✐❝♦ ❩❡✐ts❝❤r✐❢t ❢ür P❤②s✐❦✳ ❘❡❝♦♥❤❡❝❡♥❞♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ tr❛❜❛❧❤♦✱ ❊✐♥st❡✐♥ ❛t❡♥❞❡ ❛♦ ♣❡❞✐❞♦ ❞❡ ❇♦s❡ ❡ ❛❝r❡s❝❡♥t❛ ♥❛s ❝♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s q✉❡ ✏❛ ❞❡❞✉çã♦ ❞❡ ❇♦s❡ ♣❛r❛ ❛ ❢ór♠✉❧❛ ❞❡ P❧❛♥❝❦ r❡♣r❡s❡♥t❛ ✉♠ ✐♠♣♦rt❛♥t❡ ❛✈❛♥ç♦✳ ❖ ♠ét♦❞♦ ❛q✉✐ ✉t✐❧✐③❛❞♦ ♣r♦❞✉③ t❛♠❜é♠ ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞♦ ❣ás ✐❞❡❛❧✱ ❝♦♠♦ ❞✐s❝✉t✐r❡✐ ♠❛✐s ❞❡t❛❧❤❛❞❛♠❡♥t❡ ❡♠ ♦✉tr♦ ❛rt✐❣♦✏ ❬✶✱✷❪✳
❊♠ ✶✾✷✺✱ ❊✐♥st❡✐♥ ❡s❝r❡✈❡ ✉♠❛ sér✐❡ ❞❡ três ❛rt✐❣♦s ♥♦ q✉❛❧ ❢❛③ ✉s♦ ❞❛s té❝♥✐❝❛s ❛♣r❡✲ s❡♥t❛❞❛s ♣♦r ❇♦s❡ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❡st❛tíst✐❝♦✲q✉â♥t✐❝♦ ❞❡ ❣❛s❡s ✐❞❡❛✐s✳ ❊♠ s❡✉s tr❛❜❛✲ ❧❤♦s✱ ❊✐♥st❡✐♥ ✉t✐❧✐③❛ ❛ ✐❞❡✐❛ ❞❡ ♦♥❞❛s ❞❡ ♠❛tér✐❛✱ ✐♥tr♦❞✉③✐❞❛s ♣❡❧♦ ❢ís✐❝♦ ❢r❛♥❝ês ▲♦✉✐s ❞❡ ❇r♦❣❧✐❡✱ ❛r❣✉♠❡♥t❛♥❞♦ q✉❡✱ t❡♥❞♦ ♦s ❝♦♥st✐t✉✐♥t❡s ❞♦ ❣ás ❝♦♠♣♦rt❛♠❡♥t♦ ♦♥❞✉❧❛tór✐♦ t❛❧ ❝♦♠♦ ♦s ❢ót♦♥s✱ ❛♦s ♠❡s♠♦s s❡ ❛♣❧✐❝❛ ❛ ❡st❛tíst✐❝❛ ❞❡ ❇♦s❡✳ ❈✉r✐♦s❛♠❡♥t❡✱ ♦ ❢ís✐❝♦ ❛✉strí❛❝♦ ❊r✇✐♥ ❙❝❤rö❞✐♥❣❡r t♦♠❛ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s ✐❞❡✐❛s ❞❡ ❞❡ ❇r♦❣❧✐❡✱ ❝r✉❝✐❛✐s ♥❛ s✉❛ ❢♦r♠✉❧❛çã♦ ❞❛ ❡q✉❛çã♦ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡✱ ❛tr❛✈és ❞❡ss❛ sér✐❡ ❞❡ ❛rt✐❣♦s ❞❡ ❊✐♥st❡✐♥ ❬✸❪✳
◆♦ s❡❣✉♥❞♦ tr❛❜❛❧❤♦ ❞❛ sér✐❡✱ ❊✐♥st❡✐♥ ❞❡♠♦♥str❛ q✉❡ ♦ ❛✉♠❡♥t♦ ❞❛ ❞❡♥s✐❞❛❞❡✱ à t❡♠♣❡r❛t✉r❛ ❝♦♥st❛♥t❡✱ ❡♠ ✉♠ ❣ás s✉❥❡✐t♦ à ❡st❛tíst✐❝❛ ❞❡ ❇♦s❡ ♦❝❛s✐♦♥❛ ✉♠❛ t❡♥❞ê♥❝✐❛ ❡♠ s❡✉s ❝♦♥st✐t✉✐♥t❡s ❞❡ ♦❝✉♣❛r ♦ ❡st❛❞♦ q✉â♥t✐❝♦ ❢✉♥❞❛♠❡♥t❛❧✳ ❊st❛✱ ❡♠ s✉❛s ♣❛❧❛✈r❛s✱ ❝♦rr❡s♣♦♥❞❡ à ✉♠❛ ❢r❛çã♦ ❝♦♥❞❡♥s❛❞❛ ❞♦ ❣ás✱ ❡♥q✉❛♥t♦ ♦s ♦✉tr♦s ❝♦♥st✐t✉✐♥t❡s q✉❡ ❡stã♦ ❞✐str✐❜✉í❞♦s ❡♠ ♦✉tr♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❝♦rr❡s♣♦♥❞❡♠ à ✉♠❛ ❢r❛çã♦ s❛t✉r❛❞❛ ❞♦ ❣ás ❬✷❪✳ P♦rt❛♥t♦✱ ❞❡ss❡ tr❛❜❛❧❤♦ s✉r❣❡ ♦ ❝♦♥❝❡✐t♦ ❞❛ ❈♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✳
❆✐♥❞❛ ❡♠ s❡✉ s❡❣✉♥❞♦ tr❛t❛❞♦✱ ❊✐♥st❡✐♥ ❞❡♠♦♥str♦✉ q✉❡ ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✱ ✐st♦ é✱ ♣❛r❛ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥st✐t✉✐♥t❡s ❡ ✈♦❧✉♠❡ ❞♦ s✐st❡♠❛ t❡♥❞❡♥❞♦ ❛♦ ✐♥✜♥✐t♦✱ ♠❛♥t❡♥❞♦✲s❡ ❛ ❞❡♥s✐❞❛❞❡ ❝♦♥st❛♥t❡✱ ♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❡♠ ✉♠ ❣ás ❞❡ ❜ós♦♥s ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡s❝♦♥t✐♥✉✐✲ ❞❛❞❡ ♥❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛✱ ♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❊♥tr❡t❛♥t♦ ♦ ❢ís✐❝♦ ❤♦❧❛♥❞ês ●❡♦r❣❡ ❊✉❣❡♥❡ ❯❤❧❡♥❜❡❝❦ ❝♦♥t❡st♦✉ t❛❧ r❡s✉❧t❛❞♦✱ ❛r❣✉♠❡♥✲ t❛♥❞♦ q✉❡ ♦ ♠❡s♠♦ ♥ã♦ s❡r✐❛ ✈á❧✐❞♦ ♣❛r❛ s✐st❡♠❛s ✜♥✐t♦s✳ ❚r❛♥s✐çõ❡s ❞❡ ❢❛s❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❛✐♥❞❛ ♥ã♦ ❡r❛♠ ❜❡♠ ❝♦♠♣r❡❡♥❞✐❞❛s ❡✱ ❛♣❛r❡♥t❡♠❡♥t❡✱ ❛té ♠❡s♠♦ ❊✐♥st❡✐♥ ❛❝❡✐✲ t♦✉ ♦s ❛r❣✉♠❡♥t♦s ❞❡ ❯❤❧❡♥❜❡❝❦ ❬✸❪✳ ❊♠❜♦r❛ t❡♥❤❛ s✐❞♦ ❛ ♦r✐❣❡♠ ❞♦ ❝❡t✐❝✐s♠♦ ❞❛ t❡♦r✐❛✱
♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞❡ ✉♠ ❣ás ❞❡ ❜ós♦♥s ✐♥s♣✐r♦✉ ♦ ❢ís✐❝♦ ❛❧❡♠ã♦ ❋r✐t③ ▲♦♥❞♦♥ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ t❡♦r✐❛ s♦❜r❡ ❛ s✉♣❡r✢✉✐❞❡③ ❞♦4❍❡ ❡♠ ✶✾✸✽✳
❖4❍❡ ❢♦✐ ❧✐q✉❡❢❡✐t♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣❡❧♦ ❢ís✐❝♦ ❤♦❧❛♥❞ês ❍❡✐❦❡ ❑❛♠❡r❧✐♥❣❤ ❖♥♥❡s ❡♠
✶✾✵✽ ❬✹❪✳ ❊♠ ✶✾✶✶✱ ❖♥♥❡s ❡ s❡✉s ❝♦❧❛❜♦r❛❞♦r❡s ❞❡s❝♦❜r✐r❛♠ ♦ ❢❡♥ô♠❡♥♦ ❞❛ s✉♣❡r❝♦♥❞✉t✐✲ ✈✐❞❛❞❡ ♥♦ ♠❡r❝úr✐♦ ❬✺❪ ❡ t❛♠❜é♠ ♦❜s❡r✈❛r❛♠ q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❛t✐♥❣✐❛ ✉♠ ✈❛❧♦r ♠á①✐♠♦ ❡♠ ✉♠❛ t❡♠♣❡r❛t✉r❛ ✉♠ ♣♦✉❝♦ ❛❝✐♠❛ ❞❡ ✷❑ ❬✻❪✳
❊♠ ✶✾✷✹ ❖♥♥❡s ♦❜s❡r✈♦✉ ♦ ❛✉♠❡♥t♦ ❛❜r✉♣t♦ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❡♠ t♦r♥♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✷✳✶✼❑ ✳ ❊st❛ ♠❡❞✐❞❛ ❢♦✐ ❝♦♥✜r♠❛❞❛✱ ❡♠ ✶✾✸✵✱ ♣❡❧♦ ❢ís✐❝♦ ❤♦❧❛♥❞ês ❲✐❧❧✐❛♠ ❍❡♥❞r✐❦ ❑❡❡s♦♥✱ q✉❡ ♦❜s❡r✈♦✉ ✉♠❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ♥❛ ❝✉r✈❛ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ♥❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✷✳✶✼❑✳ ❑❡❡s♦♥ s✉❣❡r✐✉ q✉❡ t❛❧ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❝♦rr❡s♣♦♥❞❡r✐❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ tr✐♣❧♦ ♥♦ ❤é❧✐♦✱ ✐st♦ é✱ ❛ ❝♦❡①✐stê♥❝✐❛ ❞❛s ❢❛s❡s só❧✐❞❛✱ ❧íq✉✐❞❛ ❡ ❣❛s♦s❛ ❛ ❡st❛ t❡♠♣❡r❛t✉r❛✳ ❊st❛ ❤✐♣ót❡s❡✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ ❢♦✐ ❝♦♥✜r♠❛❞❛ q✉❛♥❞♦ ♦ ❢ís✐❝♦ ❤♦❧❛♥❞ês ❑r✐❥♥ ❲r❥❜r❡♥ ❚❛❝♦♥✐s✱ ❡♠ ✶✾✸✷✱ s✉❜♠❡t❡✉ ♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❛❜❛✐①♦ ❞❡ ✷✳✶✼❑ ❛ r❛✐♦s ❳ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜s❡r✈❛r ❡str✉t✉r❛s ❝r✐st❛❧✐♥❛s✳ ◆ã♦ s❡♥❞♦ ♦❜s❡r✈❛❞❛s t❛✐s ❡str✉t✉r❛s✱ ❑❡❡s♦♥ ♣r♦♣ôs ❝♦♠♦ ❛❧t❡r♥❛t✐✈❛ ❛ ❝♦❡①✐stê♥❝✐❛ ❞❡ ❞✉❛s ❢❛s❡s ❧íq✉✐❞❛s ♥♦ ❤é❧✐♦✱ s❡♥❞♦ ❛s ♠❡s♠❛s ❞❡♥♦♠✐♥❛❞❛s ❍❡ ■ ❡ ❍❡ ■■ ❬✼❪✳
◆♦t❛✲s❡ q✉❡ ❛ ❝✉r✈❛ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❛ss❡♠❡❧❤❛✲s❡ ❛ ❧❡tr❛ ❣r❡❣❛ ❧❛♠❜❞❛ ✭λ✮✳ P♦r ❡st❡ ♠♦t✐✈♦✱ ♦ ❢ís✐❝♦ ❛✉strí❛❝♦ P❛✉❧ ❊❤r❡♥❢❡st✱ ❡♠ ✉♠ tr❛❜❛❧❤♦ ❞❡ ✶✾✸✸ ❬✽❪✱ ♥♦♠❡♦✉ ❛ tr❛♥s✐çã♦ ❡♥tr❡ ❛s ❢❛s❡s ❍❡ ■ ❡ ❍❡ ■■ ❞❡ ♣♦♥t♦✲λ✳
❊♠ ✶✾✸✻ ❡ ✶✾✸✼ ♦ ❢ís✐❝♦ s♦✈✐ét✐❝♦ ▲❡✈ ❉❛✈✐❞♦✈✐❝❤ ▲❛♥❞❛✉ ❛♣r❡s❡♥t♦✉ s✉❛s ✐❞❡✐❛s ❛❝❡r❝❛ ❞❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❬✾✕✶✶❪✳ ❊♠ s❡✉s tr❛❜❛❧❤♦s✱ ▲❛♥❞❛✉ ❛r❣✉♠❡♥t❛ q✉❡ ❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞♦ ♣♦♥t♦✲λ é ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ♦♥❞❡ ♦ ❡st❛❞♦ ❡♥tr❡ ❛s ❢❛s❡s é ♦ ♠❡s♠♦ ❡ s✉❛s t❡♠♣❡r❛t✉r❛s sã♦ ❞✐❢❡r❡♥t❡s ❬✼❪✳
◆♦ ❛♥♦ ❞❡ ✶✾✸✽ ♦s ❢ís✐❝♦s ❝❛♥❛❞❡♥s❡s ❏♦❤♥ ❋r❛♥❦ ❆❧❧❡♥ ❡ ❉♦♥ ▼✐s❡♥❡r ❡✱ ✐♥❞❡♣❡♥✲ ❞❡♥t❡♠❡♥t❡✱ ♦ ❢ís✐❝♦ s♦✈✐ét✐❝♦ P②♦tr ▲❡♦♥✐❞♦✈✐❝❤ ❑❛♣✐t③❛ ♣✉❜❧✐❝❛r❛♠ ♥❛ ♠❡s♠❛ ❡❞✐çã♦ ❞❛ r❡✈✐st❛ ◆❛t✉r❡ ❬✶✷✱✶✸❪ s❡✉s r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s s♦❜r❡ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ❍❡ ■■✳ ❙❡❣✉♥❞♦ ♦s ❛✉t♦r❡s✱ ❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ❍❡ ■■ é ❞❛ ♦r❞❡♠ ❞❡ 10−9P✱ ❡♠ ❝♦♥tr❛st❡ ❝♦♠ ♦ ❤✐❞r♦❣ê♥✐♦
♥♦ ❡st❛❞♦ ❣❛s♦s♦ q✉❡✱ ♥❛ é♣♦❝❛✱ ❡r❛ ♦ ✢✉✐❞♦ ❝♦♠ ♠❡♥♦r ✈❛❧♦r ❞❡ ✈✐s❝♦s✐❞❛❞❡ ❝♦♥❤❡❝✐❞♦✱ ❞❛ ♦r❞❡♠ ❞❡10−5P✳ ❊♠ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛s ❝♦rr❡♥t❡s ❡♠ ✉♠ s✉♣❡r❝♦♥❞✉t♦r✱ ❑❛♣✐t③❛ ♥♦♠❡✐❛ ♦ ❢❡♥ô♠❡♥♦ ❞❛ ❜❛✐①❛ ✈✐s❝♦s✐❞❛❞❡ ❞♦ ❍❡ ■■ ❞❡ s✉♣❡r✢✉✐❞❡③✳
❆✐♥❞❛ ❡♠ ✶✾✸✽✱ ▲♦♥❞♦♥✱ t❡♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❞✐✈❡rs♦s ❡①♣❡r✐♠❡♥t♦s r❡❛❧✐③❛❞♦s ❝♦♠ ♦ ❤é❧✐♦ ❧íq✉✐❞♦✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s r❡s✉❧t❛❞♦s q✉❡ ❞❡♠♦str❛✈❛♠ ✉♠ ♣✐❝♦ ♥❛ ❝✉r✈❛ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦✱ ♣r♦♣ôs ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝♦♥❡①ã♦ ❡♥tr❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❍❡ ■■ ❡ ♦ ❢❡♥ô♠❡♥♦ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✳ ❊♠ s❡✉s ❝á❧❝✉❧♦s✱ ▲♦♥❞♦♥ ❞❡♠♦♥str♦✉ q✉❡ ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❞❡ ✉♠ ❣ás ❞❡ ❇♦s❡ ✐❞❡❛❧ ❢♦r♥❡❝❡ ✉♠❛ ❜♦❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ♣♦♥t♦✲λ✭T0 = 3.09K✮ ❬✶✹✕✶✻❪✳
▲♦♥❞♦♥ ❡①♣ôs s✉❛s ✐❞❡✐❛s ❛♦ ❢ís✐❝♦ ❤✉♥❣❛r♦ ▲ás③❧ó ❚✐s③❛✱ ♦ q✉❡ ✐♥s♣✐r♦✉ ♦ ♠❡s♠♦ ♥❛ ❝r✐❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❞♦✐s ✢✉✐❞♦s ♣❛r❛ ❡①♣❧✐❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❝♦♥s✐✲ ❞❡r❛♥❞♦ ❛s ❢❛s❡s ♥♦r♠❛❧ ✭❍❡ ■✮ ❡ s✉♣❡r✢✉✐❞❛ ✭❍❡ ■■✮✱ ❡ s✉♣♦♥❞♦ q✉❡ ❛ ❢❛s❡ ❍❡ ■■ é ❞❡s❝r✐t❛ ♣❡❧❛ ❡st❛tíst✐❝❛ ❞❡ ✉♠ ❣ás ❞❡ ❇♦s❡ ✐❞❡❛❧✳ ❚✐s③❛ ❛✐♥❞❛ ♣r❡✈✐✉ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♦s❝✐❧❛çõ❡s ❞❡ t❡♠♣❡r❛t✉r❛ ❡♠ ✉♠ s✉♣❡r✢✉✐❞♦✱ s❡♥❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ❞❡♥♦♠✐♥❛❞❛s ♣♦r ▲❛♥❞❛✉ ❞❡ s❡❣✉♥❞♦ s♦♠ ❬✸✱✶✼✱✶✽❪✳
❊♠ ✉♠ ❛rt✐❣♦ ❞❡ ✶✾✹✶ ❬✶✾❪✱ ▲❛♥❞❛✉ ❝r✐t✐❝❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ ❚✐s③❛ ❛♦ ♣r♦❜❧❡♠❛ ❛❧❡❣❛♥❞♦ q✉❡✱ ❞❡✈✐❞♦ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s át♦♠♦s ❞❡ ❤é❧✐♦✱ ❛ ❡st❛tíst✐❝❛ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❞❡ ✉♠ ❣ás ✐❞❡❛❧ ♥ã♦ ❞❡✈❡ s❡r ❛♣❧✐❝❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ ♠♦❞❡❧♦ ❞❡ ❚✐s③❛ ♦s át♦♠♦s ♥❛ ❢❛s❡ ❍❡ ■ s❡ ♠♦✈❡♠ s❡♠ ❢r✐❝çã♦✳ ❈♦♠ ✐ss♦✱ ♦ ♠♦❞❡❧♦ ❞❡ ❞♦✐s ✢✉✐❞♦s ♣r♦♣♦st♦ ♣♦r ▲❛♥❞❛✉ ❝♦♥s✐❞❡r❛ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s át♦♠♦s ❡ tr❛t❛ ♦ ♣r♦❜❧❡♠❛ ✐♥tr♦❞✉③✐♥❞♦ ♦♣❡r❛❞♦r❡s ❞❡ ♠♦♠❡♥t♦ ❡ ❞❡♥s✐❞❛❞❡✳
❆✐♥❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✱ ▲❛♥❞❛✉ ❞❡s❡♥✈♦❧✈❡ ✉♠❛ ❞❡s❝r✐çã♦ q✉❛♥t✐t❛t✐✈❛ ❞♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛ ❞♦ ❍❡ ■■ ✭♠❛✐s ❞❡t❛❧❤❛❞❛ ❡♠ ✉♠ tr❛❜❛❧❤♦ ♣♦st❡r✐♦r ❞❡ ✶✾✹✼ ❬✷✵❪✮✱ ♥❛ q✉❛❧ ♣r♦♣õ❡ q✉❡ ❛s ❡①❝✐t❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❞❡ ❜❛✐①❛ ❡♥❡r❣✐❛ ❞♦ ✢✉✐❞♦ sã♦ ❢ô♥♦♥s✳ P❛r❛ ❡♥❡r❣✐❛s ♠❛✐♦r❡s✱ ♦ ❡s♣❡❝tr♦✱ ❛té ❡♥tã♦ ❧✐♥❡❛r ❞❡✈✐❞♦ ❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ❞♦s ❢ô♥♦♥s✱ ❡①✐❜❡ ✉♠ ♠í♥✐♠♦ ❛♦ q✉❛❧ ▲❛♥❞❛✉ ❛tr✐❜✉✐ ❛ ♦✉tr♦ t✐♣♦ ❞❡ ❡①❝✐t❛çã♦ ❡❧❡♠❡♥t❛r✱ ❞❡♥♦♠✐♥❛❞❛ rót♦♥✳ ❚❛❧ ❡①❝✐t❛çã♦ ❡❧❡♠❡♥t❛r ❢♦✐ ❡rr♦♥❡❛♠❡♥t❡ ❛ss♦❝✐❛❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✈órt✐❝❡s ♥♦ ❤é❧✐♦ ❧íq✉✐❞♦✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ s❡❣✉♥❞♦ s♦♠✱ ❛♥t❡r✐♦r♠❡♥t❡ ❞❡s❝r✐t♦ ♣♦r ❚✐s③❛✱ t❛♠❜é♠ é ❛❜♦r❞❛❞♦ ♥♦ tr❛❜❛❧❤♦ ❞❡ ▲❛♥❞❛✉✳
❊♥tr❡ ❛s ❞é❝❛❞❛s ❞❡ ✶✾✹✵ ❡ ✶✾✺✵✱ ▲❛♥❞❛✉ ❡ ♦ ❢ís✐❝♦ s♦✈✐ét✐❝♦ ■s❛❛❦ ▼❛r❦♦✈✐❝❤ ❑❤❛❧❛t✲ ♥✐❦♦✈ ❞❡r❛♠ s❡❣♠❡♥t♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❛ s✉♣❡r✢✉✐❞❡③✳ ❆✐♥❞❛ ❛t✉❛❧♠❡♥t❡ ❬✸❪ ❛ t❡♦r✐❛ ❞❡ ▲❛♥❞❛✉✲❑❤❛❧❛t♥✐❦♦✈ é ❝♦♠✉♠❡♥t❡ ✉s❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡r ❛ s✉♣❡r✢✉✐❞❡③ ❬✷✶✱✷✷❪✳ ❊♠❜♦r❛ ❛ t❡♦r✐❛ ❞❡ ▲❛♥❞❛✉✲❑❤❛❧❛t♥✐❦♦✈ t❡♥❤❛ s❡✉ s✉❝❡ss♦✱ ❛ ♠❡s♠❛ ♥ã♦ ❡r❛ ❝❛♣❛③ ❞❡ ❞❡s❝r❡✈❡r ♦ ❢❡♥ô♠❡♥♦ ❞❛ t✉r❜✉❧ê♥❝✐❛ ❡ ❢♦r♠❛çã♦ ❞❡ ✈órt✐❝❡s ♥♦ ❍❡ ■■✱ ♦❜s❡r✈❛❞♦s ♣♦r ❑❛♣✐t③❛ ❡♠ ✶✾✹✶ ❬✷✸❪✱ ✉♠❛ ✈❡③ q✉❡ ❛ t❡♦r✐❛ ♣r❡❞✐③ q✉❡ s✉♣❡r✢✉✐❞♦s sã♦ ✐rr♦t❛❝✐♦♥❛✐s✳ ❉❡✈✐❞♦ à ❛♣❛r❡♥t❡ ✐♥❝♦♥s✐stê♥❝✐❛ ♥♦ ♠♦❞❡❧♦✱ ♦ q✉í♠✐❝♦ ♥♦r✉❡❣✉ês ▲❛rs ❖♥s❛❣❡r ♣r♦♣ôs✱ ❡♠ ✶✾✹✾✱ q✉❡ ❛s ❧✐♥❤❛s ❞❡ ✈órt✐❝❡s ❡♠ ✉♠ s✉♣❡r✢✉✐❞♦ sã♦ q✉❛♥t✐③❛❞❛s ❡♠ ✉♥✐❞❛❞❡s ❞❡
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h/m❬✼✱✷✹❪✳
❊♠ ✶✾✹✼✱ ♦ ❢ís✐❝♦ s♦✈✐ét✐❝♦ ◆✐❦♦❧❛✐ ◆✐❦♦❧❛✐❡✈✐❝❤ ❇♦❣♦❧✐✉❜♦✈ ❛♣r❡s❡♥t❛ ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ q✉❡ ❞❡s❝r❡✈❡ ✉♠ ❣ás ❞❡ ❜ós♦♥s ❢r❛❝❛♠❡♥t❡ ✐♥t❡r❛❣❡♥t❡s ❬✷✺❪✱ ❞♦ q✉❛❧ s❡ ♦❜té♠ ✉♠❛ ❡①✲ ♣r❡ssã♦ ❞❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ♣❛r❛ ❡①❝✐t❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❬✷✻❪✳ ❊♥tr❡t❛♥t♦✱ t❛❧ t❡♦r✐❛ ♥ã♦ ❝♦♥q✉✐st♦✉ r❡❝♦♥❤❡❝✐♠❡♥t♦ ✐♠❡❞✐❛t♦✱ ✉♠❛ ✈❡③ q✉❡ s❡ ✉t✐❧✐③❛ ❞♦ ❢♦r♠❛❧✐s♠♦ ❞❡ s❡❣✉♥❞❛ q✉❛♥t✐③❛çã♦✱ ♣♦✉❝♦ ❢❛♠✐❧✐❛r ❡♥tr❡ ♦s ❢ís✐❝♦s ❞❛ ♠❛tér✐❛ ❝♦♥❞❡♥s❛❞❛ ❞❛ é♣♦❝❛✳ ❆♣❡♥❛s ❡♠ ✶✾✺✼✱ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❇❈❙ ❞❛ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡ ❬✷✼❪✱ ❛ q✉❛❧ t❛♠❜é♠ s❡ ✉t✐❧✐③❛ ❞♦ ❢♦r♠❛❧✐s♠♦ ❞❛ s❡❣✉♥❞❛ q✉❛♥t✐③❛çã♦ ❡ ❛♣r❡s❡♥t❛ ❛♥❛❧♦❣✐❛s ❝♦♠ ♦ tr❛t❛♠❡♥t♦ ❞❡ ❇♦❣♦❧✐✉❜♦✈✱ s❡✉ tr❛❜❛❧❤♦ ❝♦♠❡ç♦✉ ❛ ❛❧❝❛♥ç❛r ♣♦♣✉❧❛r✐❞❛❞❡ ❬✸❪✳
❊♥tr❡ ♦s ❛♥♦s ❞❡ ✶✾✺✸ ❡ ✶✾✺✽✱ ♦ ❢ís✐❝♦ ♥♦rt❡✲❛♠❡r✐❝❛♥♦ ❘✐❝❤❛r❞ P❤✐❧❧✐♣s ❋❡②♥♠❛♥ ❞❡✲ s❡♥✈♦❧✈❡ ✉♠❛ sér✐❡ ❞❡ tr❛❜❛❧❤♦s s♦❜r❡ ♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❬✷✽❪✱ ♥♦s q✉❛✐s ❝♦♥s✐❞❡r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❧✐♥❤❛s ❞❡ ✈órt✐❝❡s q✉❛♥t✐③❛❞❛s ♥♦ ❍❡ ■■✱ t❛❧ ❝♦♠♦ s✉❣❡r✐❞♦ ♣♦r ❖♥s❛❣❡r✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❧✐♥❤❛s ❞❡ ✈órt✐❝❡ s❡♠❡❧❤❛♥t❡s ❛s ✐❞❡♥t✐✜❝❛❞❛s ♥♦ ❍❡ ■■ t❛♠❜é♠ sã♦ ♦❜s❡r✈❛❞❛s ♥♦s ❝❤❛♠❛❞♦s s✉♣❡r❝♦♥❞✉t♦r❡s ❞❡ t✐♣♦✲■■ ❬✷✾❪✳
◆♦ ❛♥♦ ❞❡ ✶✾✺✻✱ ❖♥s❛❣❡r ❡ ♦ ❢ís✐❝♦ ✐♥❣❧ês ❖❧✐✈❡r P❡♥r♦s❡ ❛♣r❡s❡♥t❛♠ ✉♠ ❛rt✐❣♦ ♦♥❞❡ ♣r♦♣õ❡♠ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✳ ❙❡❣✉♥❞♦ ❡ss❡ ❝r✐tér✐♦✱ ❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ♦❝♦rr❡ ❡♠ ✉♠ s✐st❡♠❛ s❡ ♦ ♠❛✐♦r ❛✉t♦✈❛❧♦r ❞❡ s✉❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❢♦r ❞❛ ♦r❞❡♠ ❞❡ s❡✉ ♥ú♠❡r♦ ❞❡ ❝♦♥st✐t✉✐♥t❡s ❬✸✵❪✳ ◆❡ss❡ tr❛❜❛❧❤♦ ❛r❣✉♠❡♥t❛♠ q✉❡ ❛ ❝♦♥❞❡♥s❛çã♦ ♥ã♦ ❞❡✈❡ s❡r ❡s♣❡r❛❞❛ ❡♠ ✉♠ só❧✐❞♦✱ ♠❡s♠♦ ❡♠T = 0K✳ ❊♥tr❡t❛♥t♦✱ ❞❡♠♦♥str❛♠ ❛tr❛✈és ❞❡ ✉♠ ❝á❧❝✉❧♦ s✐♠♣❧❡s ❞❡ ✉♠ ❣ás ❞❡ ❡s❢❡r❛s ❞✉r❛s q✉❡✱ ❡♠ t❡♠♣❡r❛t✉r❛ ♥✉❧❛✱ ❛ ❢r❛çã♦ ❞❡ át♦♠♦s ❝♦♥❞❡♥s❛❞♦s ♥♦ ❤é❧✐♦ ❧íq✉✐❞♦ é ❞❡ ❝❡r❝❛ ❞❡8% ❬✸✶❪✳
❆t✉❛❧♠❡♥t❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❝á❧❝✉❧♦s à t❡♠♣❡r❛t✉r❛ ❡ ♣r❡ssã♦ ♥✉❧❛s ♣r❡❞✐③❡♠ ✉♠ ✈❛❧♦r ❞❡ ❝❡r❝❛ ❞❡10%♣❛r❛ ❛ ❢r❛çã♦ ❝♦♥❞❡♥s❛❞❛ ♥♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❬✷✻❪✳
◆♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✺✵ ❛ ♠❡❛❞♦s ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✻✵ ♦✉tr♦s tr❛❜❛❧❤♦s ❛ r❡s♣❡✐t♦ ❞❛ ❡①✐stê♥❝✐❛ ❡ ♥❛t✉r❡③❛ ❞❡ ✈órt✐❝❡s ♥♦ ❤é❧✐♦ ♦✉✱ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✱ ❡♠ ❣❛s❡s ❞❡ ❇♦s❡ ♥ã♦✲ ✐❞❡❛✐s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♣❡❧♦s ❢ís✐❝♦s s♦✈✐ét✐❝♦s ❱✐t❛❧② ▲❛③❛r❡✈✐❝❤ ●✐♥③❜✉r❣ ❡ ▲❡✈ P❡tr♦✈✐❝❤ P✐t❛❡✈s❦✐✐✱ ❡ ♣❡❧♦ ❢ís✐❝♦ ♥♦rt❡✲❛♠❡r✐❝❛♥♦ ❊✉❣❡♥❡ ●r♦ss✳
❊♠ ✶✾✻✶ ●r♦ss ❡ P✐t❛❡✈s❦✐✐ ❛♣r❡s❡♥t❛♠✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ❡♠ s❡✉s ❡st✉❞♦s ❞❡ ✈órt✐✲ ❝❡s ❡♠ ❤é❧✐♦ ❧íq✉✐❞♦✱ ✉♠❛ ❡q✉❛çã♦ ❞❡r✐✈❛❞❛ ❞❡ ✉♠ ❢♦r♠❛❧✐s♠♦ ❞❡ ❝❛♠♣♦ ♠é❞✐♦ q✉❡ ❞❡s❝r❡✈❡ ✉♠ ❣ás ❢r❛❝❛♠❡♥t❡ ✐♥t❡r❛❣❡♥t❡ à t❡♠♣❡r❛t✉r❛ ♥✉❧❛ ❬✸✷✱ ✸✸❪✳ ❊ss❛ ❡q✉❛çã♦✱ ❤♦❥❡ ❝♦♥❤❡❝✐❞❛ ❝♦♠ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❬✷✻❪✱ é ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❢❡rr❛♠❡♥t❛s t❡ór✐❝❛s ❡♠ ✉s♦ ♥❛ ❛t✉❛❧✐❞❛❞❡ ♥❛ ❞❡s❝r✐çã♦ ❞❡ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥✱ ✉♠❛ ✈❡③ q✉❡ ❛♣r❡s❡♥t❛ ❜♦❛
❝♦♥❝♦r❞â♥❝✐❛ ❝♦♠ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❞❡ ❣❛s❡s ❝♦♥✜♥❛❞♦s ❡♠ ❛r♠❛❞✐❧❤❛s ó♣t✐❝❛s✳ ❆ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦✲❧✐♥❡❛r ❝♦♠ ✉♠ t❡r♠♦ ❝ú❜✐❝♦✱ ♣♦ss✉✐♥❞♦ ❛♥á❧♦❣♦s ❡♠ ó♣t✐❝❛ ♥ã♦✲❧✐♥❡❛r ❬✸✹❪ ❡ ❡♠ s✉♣❡r❝♦♥❞✉t♦r❡s✱ ♥♦ ❝♦♥t❡①t♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❬✷✾❪✳
❊♠❜♦r❛ ❛♣ós ❛ ❢♦r♠✉❧❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ♦s ❡st✉❞♦s ❛❝❡r❝❛ ❞❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❞♦ ❤é❧✐♦ ❧íq✉✐❞♦ ❡ ❞❡ ❣❛s❡s ❜♦sô♥✐❝♦s t❡♥❤❛♠ s❡❣✉✐❞♦✱ ♣♦✉❝♦s tr❛❜❛❧❤♦s t❡ór✐❝♦s ❛❧❝❛♥ç❛r❛♠ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❛ ♥í✈❡❧ ❞❡ s❡✉s ❛♥t❡❝❡ss♦r❡s✳
❆ ♣❛rt✐r ❞❛ ❞é❝❛❞❛ ❞❡ ✶✾✽✵ s✉r❣✐r❛♠ ♦s ♣r✐♥❝✐♣❛✐s ❛✈❛♥ç♦s ❡①♣❡r✐♠❡♥t❛✐s ❬✸✺✕✹✶❪ q✉❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ❛ r❡❛❧✐③❛çã♦ ❞❡ ❝♦♥❞❡♥s❛❞♦s ❛tô♠✐❝♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❡♠ ❣❛s❡s ❛tô♠✐❝♦s ❞✐❧✉í❞♦s ❛tr❛✈és ❞♦ r❡s❢r✐❛♠❡♥t♦ ❞❡ át♦♠♦s ♥❡✉tr♦s✳
❊♠ t❛✐s té❝♥✐❝❛s✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦s át♦♠♦s ❡♠ ✉♠ ❣ás é r❡❞✉③✐❞❛ ♣❡❧❛ ✐♥t❡r❛çã♦ ❝♦♠ ❢ót♦♥s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ❧❛s❡rs✳ ❆❧é♠ ❞✐ss♦✱ sã♦ ✉t✐❧✐③❛❞♦s ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡ ♦✉tr♦s ❛rr❛♥❥♦s ❞❡ ❧❛s❡rs ♣❛r❛ ❝♦♥✜♥❛r✱ ❝♦♠♣r✐♠✐r ❡ ❡✈❛♣♦r❛r ❣❛s❡s ❛tô♠✐❝♦s✳ ❆ ❞✐s❝✉ssã♦ ❞❡ té❝♥✐❝❛s ❞❡ r❡s❢r✐❛♠❡♥t♦ ❞❡ ❣❛s❡s ❛tô♠✐❝♦s ❢♦❣❡ ❛♦ ❡s❝♦♣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❊♥tr❡t❛♥t♦ ❛ ❞❡s❝r✐çã♦ ❞❡t❛❧❤❛❞❛ ❞❡ t❛✐s té❝♥✐❝❛s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛s r❡❢❡rê♥❝✐❛s ❬✹✷✱ ✹✸❪✳
❖s ♣r✐♠❡✐r♦s ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❡♠ ❣❛s❡s ❛tô♠✐❝♦s ❞✐❧✉í❞♦s ❢♦r❛♠ r❡❛❧✐③❛✲ ❞♦s ❡♠ ✶✾✾✺ ♣❡❧♦s ❣r✉♣♦s ❞♦ ❏♦✐♥t ■♥st✐t✉t❡ ❢♦r ▲❛❜♦r❛t♦r② ❆str♦♣❤②s✐❝s ✭❏■▲❆✮✱ ❧✐❞❡r❛❞♦ ♣❡❧♦s ❢ís✐❝♦s ♥♦rt❡✲❛♠❡r✐❝❛♥♦s ❊r✐❝ ❆❧❧✐♥ ❈♦r♥❡❧❧ ❡ ❈❛r❧ ❲✐❡♠❛♥✱ ❡ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♣❡❧♦ ❣r✉♣♦ ❞♦ ▼❛ss❛❝❤✉s❡tts ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣② ✭▼■❚✮✱ ❧✐❞❡r❛❞♦ ♣❡❧♦ ❢ís✐❝♦ ❛❧❡♠ã♦ ❲♦❧❢❣❛♥❣ ❑❡tt❡r❧❡✳
❖ ❝♦♥❞❡♥s❛❞♦ ♣r♦❞✉③✐❞♦ ♣❡❧♦ ❏■▲❆ ❡r❛ ❝♦♥st✐t✉í❞♦ ❞❡ ✉♠ ❣ás ❞❡ 87❘❜ ❝♦♠ ❞❡♥s✐❞❛❞❡
❞❡✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ✷✳✺×1012 át♦♠♦s✴❝♠3 ❡ r❡s❢r✐❛❞♦ ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✶✼✵♥❑ ❬✹✹❪✳
❖ ❣r✉♣♦ ❞♦ ▼■❚ ♣r♦❞✉③✐✉ ✉♠ ❝♦♥❞❡♥s❛❞♦ ❝♦♠ 23◆❛ ❝✉❥❛ ❞❡♥s✐❞❛❞❡ ❡r❛ ❞❛ ♦r❞❡♠ ❞❡ 1014 át♦♠♦s✴❝♠3✱ ♥❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✷µ❑ ❬✹✺❪✳ P♦r ❡ss❡s tr❛❜❛❧❤♦s ❈♦r♥❡❧❧✱ ❑❡tt❡r❧❡ ❡
❲✐❡♠❛♥ ❢♦r❛♠ ♦s ✈❡♥❝❡❞♦r❡s ❞♦ Prê♠✐♦ ◆♦❜❡❧ ❞❡ ✷✵✵✶✳ ❉❡s❞❡ ❡♥tã♦✱ ❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✈❡♠ s❡♥❞♦ r❡❛❧✐③❛❞❛ ✉t✐❧✐③❛♥❞♦ ❣❛s❡s ❞✐❧✉í❞♦s ❞❡ ❞✐❢❡r❡♥t❡s ❡s♣é❝✐❡s ❬✹✻✕✹✾❪ ❡ ❡♥ú♠❡r♦s sã♦ ♦s ❡s❢♦rç♦s t❡ór✐❝♦s ❡ ❡①♣❡r✐♠❡♥t❛✐s ♣❛r❛ ❡❧✉❝✐❞❛r s❡✉s ❢❡♥ô♠❡♥♦s ♣❡❝✉❧✐❛r❡s ❡ s✉r♣r❡❡♥❞❡♥t❡s✳
✶✳✷ ❱órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥
❆ ❢♦r♠❛çã♦ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❢♦✐ ❞❡♠♦♥str❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✶✾✾✾ ♣♦r ▼❛tt❤❡✇s ❡t ❛❧✳ ❬✺✵❪ ❡♠ ✉♠ ❣ás ❢♦r♠❛❞♦ ♣♦r ❞✉❛s ♠✐st✉r❛s ❤✐♣❡r✜♥❛s ❞❡87❘❜✱ ♦♥❞❡
♦❜s❡r✈♦✉✲s❡ ❛ ❢♦r♠❛çã♦ ❞❡ ❛♣❡♥❛s ✉♠ ✈órt✐❝❡ ♥♦s ❡①♣❡r✐♠❡♥t♦s✳
◆♦ ❛♥♦ s❡❣✉✐♥t❡✱ ▼❛❞✐s♦♥ ❡t ❛❧✳ ❬✺✶❪ r❡❧❛t❛r❛♠ ❛ ♦❜s❡r✈❛çã♦ ❞❡ ✈órt✐❝❡s ❡♠ ✉♠ ❝♦♥❞❡♥✲ s❛❞♦ ❞❡87❘❜✱ ❞❡st❛ ✈❡③ ❝♦♥st✐t✉í❞♦ ❞❡ ❛♣❡♥❛s ✉♠❛ ❝♦♠♣♦♥❡♥t❡✳ ◆❡st❡ ❡①♣❡r✐♠❡♥t♦✱ ❢♦r❛♠
♣r♦❞✉③✐❞♦s ❝♦♥❞❡♥s❛❞♦s ❡♠ r♦t❛çã♦ ❛tr❛✈és ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ ❛r♠❛❞✐❧❤❛ ❛ss✐♠étr✐❝❛ ❡ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❧❛s❡r ❡①t❡r♥♦ ♣❛r❛ ♣r♦❞✉③✐r ✉♠❛ ❛❣✐t❛çã♦ ♥♦ ❝♦♥❞❡♥s❛❞♦ ✭❞♦ ✐♥❣❧ês ❧❛s❡r st✐rr✐♥❣✮ ❬✺✷❪✳ ❖❜s❡r✈♦✉✲s❡ q✉❡ ♦ ❛✉♠❡♥t♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ r♦t❛çã♦ ❞♦ ❝♦♥❞❡♥s❛❞♦ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ♦✉ ♠❛✐s ✈órt✐❝❡s✳ ❖ ❛rr❛♥❥♦ ❞❡ t❛✐s ✈órt✐❝❡s ♥♦ ❝♦♥❞❡♥s❛❞♦ é s❡♠❡❧❤❛♥t❡ àq✉❡❧❡ ♦❜s❡r✈❛❞♦ ♥♦4❍❡ s✉♣❡r✢✉✐❞♦ ❡♠ r♦t❛çã♦ ❬✺✸❪✳
P♦st❡r✐♦r♠❡♥t❡✱ ❆❜♦✲❙❤❛❡❡r ❡t ❛❧✳ ❬✺✹❪ ♦❜s❡r✈❛r❛♠ ❛ ❢♦r♠❛çã♦ ❞❡ r❡❞❡s tr✐❛♥❣✉❧❛r❡s ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s✱ s❡♠❡❧❤❛♥t❡s às r❡❞❡s ❢♦r♠❛❞❛s ♣❡❧♦ ✢✉①♦ ❞❡ s✉♣❡r❝♦rr❡♥t❡s ❡♠ ✉♠ s✉♣❡r❝♦♥❞✉t♦r ❞❡ t✐♣♦✲■■ ✭❝♦♥❤❡❝✐❞❛s ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ r❡❞❡s ❞❡ ❆❜r✐❦♦s♦✈✮ ❬✺✺❪✳ ❆✐♥❞❛ ♥♦ ❝♦♥t❡①t♦ ❞❡ r❡❞❡s tr✐❛♥❣✉❧❛r❡s ❞❡ ✈órt✐❝❡s ❈♦❞❞✐♥❣t♦♥ ❡t ❛❧✳ ❬✺✻❪ ♦❜s❡r✈❛r❛♠ ♦s❝✐❧❛çõ❡s ❞❡ t❛✐s r❡❞❡s✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♦s❝✐❧❛çõ❡s ❞❡ ❚❦❛❝❤❡♥❦♦✱ ❛s q✉❛✐s ❢♦r❛♠ ♣r✐♠❡✐r❛♠❡♥t❡ ❞✐s❝✉t✐❞❛s ♥♦ ❝♦♥t❡①t♦ ❞❡ ✢✉✐❞♦s ✐♥❝♦♠♣r❡ssí✈❡✐s ✐rr♦t❛❝✐♦♥❛✐s ❬✺✼✱✺✽❪✳
❆t✉❛❧♠❡♥t❡ ✉♠❛ ✐♥t❡♥s❛ ❛t✐✈✐❞❛❞❡ t❡ór✐❝❛ ❡ ❡①♣❡r✐♠❡♥t❛❧ é r❡❛❧✐③❛❞❛ ❡♥✈♦❧✈❡♥❞♦ ✈ór✲ t✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❝♦♠ ✉♠❛ ♦✉ ♠❛✐s ❝♦♠♣♦♥❡♥t❡s ❡ s❡✉s ♠♦❞♦s ❞❡ ♦s❝✐❧❛çã♦ ❝♦❧❡t✐✈❛✳ ❯♠❛ ❜♦❛ r❡✈✐sã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺✾❪✳ ❱órt✐❝❡s t❛♠❜é♠ tê♠ s❡ ♠♦str❛❞♦ ✐♥❣r❡❞✐✲ ❡♥t❡ ❡ss❡♥❝✐❛❧ ♥♦ ❡st✉❞♦ ❞❡ t✉r❜✉❧ê♥❝✐❛ q✉â♥t✐❝❛ ❬✻✵❪✳
❋♦✐ ❞❡♠♦♥str❛❞♦ t❡♦r✐❝❛♠❡♥t❡ q✉❡ ❡♠ s✐st❡♠❛s ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ♦s ✈órt✐❝❡s ♣♦✲ ❞❡♠ s❡ t♦r♥❛r ✐♥stá✈❡✐s ♣❛r❛ ✉♠ ♥ú♠❡r♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡ át♦♠♦s✳ ❊ss❛ ✐♥st❛✲ ❜✐❧✐❞❛❞❡ ♣r♦❞✉③ ✉♠ ♠♦❞♦ ❞❡ r❡s♣✐r♦ ✭❜r❡❛t❤✐♥❣ ♠♦❞❡✮ ❝♦♠ ❛ q✉❡❜r❛ ❞♦ ✈órt✐❝❡ ❡♠ ❞✉❛s ♣❛rt❡s ❡ r❡❝♦♥❡①ã♦ s✉❝❡ss✐✈❛ ❬✻✶❪✳ ❊♠ ✉♠ tr❛❜❛❧❤♦ ♣♦st❡r✐♦r✱ ♠♦str♦✉✲s❡ q✉❡ ❛ ♦r✐❣❡♠ ❞❛ ✐♥st❛❜✐❧✐❞❛❞❡ ❡r❛ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ✉♠ ♠♦❞♦ ❞❡ ❇♦❣♦❧✐✉❜♦✈ ✐♥stá✈❡❧ ❝♦♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r L = 2 ❬✻✷❪✳ ◆♦ ♥♦ss♦ ❣r✉♣♦ t❛♠❜é♠ ❢♦r❛♠ ❡st✉❞❛❞♦s s✐st❡♠❛s ❞❡ ✈órt✐❝❡s ❝♦♥✲
tr❛❣✐r❛♥t❡s ❡♠ ❞✉❛s ❡s♣é❝✐❡s ✐♥t❡r❛❣❡♥t❡s ❬✻✸❪✳ ◆❡st❡ ú❧t✐♠♦ tr❛❜❛❧❤♦ ❢♦✐ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠ ❞✐❛❣r❛♠❛ ❞❡ ✐♥st❛❜✐❧✐❞❛❞❡ ♦♥❞❡ ✜❝♦✉ ❡✈✐❞❡♥❝✐❛❞♦ q✉❡ ♠♦❞♦s ❝♦♠ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r L= 1,2,3 ♦✉ ✹ ❞♦♠✐♥❛♥t❡ ❝♦rr❡s♣♦♥❞✐❛♠ ❛ ✐♥st❛❜✐❧✐❞❛❞❡ ❝♦♠ q✉❡❜r❛ ❡♠ ✉♠❛✱ ❞✉❛s✱ três
♦✉ q✉❛tr♦ ♣❛rt❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✶✳✸ ❙♦❜r❡ ❡st❡ tr❛❜❛❧❤♦
❖ ♦❜❥❡t✐✈♦ ✐♥✐❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ♦ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❡♥tr❡ ♦s át♦♠♦s✱ r❡♣r♦❞✉③✐♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❛ r❡❢❡rê♥❝✐❛ ❬✻✷❪✳ P❛r❛ ✐ss♦ ❢♦r❛♠ ❡st✉❞❛❞❛s ♥✉♠❡r✐❝❛♠❡♥t❡ ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲ P✐t❛❡✈s❦✐✐ ♣❛r❛ ✉♠ ❝♦♥❞❡♥s❛❞♦ ❡♠ ❝♦♥✜♥❛♠❡♥t♦ ❤❛r♠ô♥✐❝♦✱ s❡♥❞♦ ❛ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ ❝♦♥❞❡♥s❛❞♦✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ss❡ s✐st❡♠❛ ❢♦✐ ❞❡t❡r♠✐♥❛❞❛ ❛ ♣❛rt✐r ❞❛ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s ❡ ♣❡❧❛ ❛♥á❧✐s❡ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s✳ ❚❛✐s r❡s✉❧t❛❞♦s ❞❡♠♦♥str❛♠ q✉❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❛tr❛t✐✈♦s sã♦ ❡st❛✈❡✐s ♣❛r❛ ✉♠ ❝❡rt♦ ♥ú♠❡r♦ ❞❡ át♦♠♦s ♥♦ s✐st❡♠❛ ❛♣❡♥❛s s❡ ♦ ✈❛❧♦r ❞❡ s✉❛ ✈♦rt✐❝✐❞❛❞❡✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ❝❛r❣❛✱ ❢♦r ✐❣✉❛❧ ❛ ✉♠ ✭S= 1✮✳ ◆❡ss❡s s✐st❡♠❛s ❞❡♠♦♥str❛✲s❡
q✉❡ ✈órt✐❝❡s ❝♦♠S >1sã♦ ✐♥stá✈❡✐s ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ❞❡ át♦♠♦s✳
❊♠ s❡❣✉✐❞❛✱ ❢♦✐ ♣r♦♣♦st♦ ✉♠ ♠♦❞❡❧♦ ♥♦ q✉❛❧ ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s át♦♠♦s é t❛♠❜é♠ ❛tr❛t✐✈❛✱ ♣♦ré♠ ❡s♣❛❝✐❛❧♠❡♥t❡ ♠♦❞✉❧❛❞❛ ❡♠ ✉♠ ❢♦r♠❛t♦ ❣❛✉ss✐❛♥♦ s❡♥❞♦✱ ♣♦rt❛♥t♦✱ ♠❛✐s ✐♥t❡♥s❛ ♥♦ ❝❡♥tr♦ ❞♦ ✈órt✐❝❡ ❡ ♠❡♥♦r ❡♠ s✉❛ ❜♦r❞❛✳ ❈♦♠ ✐ss♦✱ ❢♦✐ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ t❛✐s ❝♦♥❞❡♥s❛❞♦s sã♦ ❡stá✈❡✐s ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s ♣❛r❛ S = 6✱ ♥♦ ♠í♥✐♠♦✳ ❊ss❡s
r❡s✉❧t❛❞♦s ♥✉♥❝❛ ❢♦r❛♠ ♦❜s❡r✈❛❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ♦✉ ♣r❡❞✐t♦s t❡♦r✐❝❛♠❡♥t❡✱ ❞❡ ♠♦❞♦ q✉❡ ♦s ♠❡s♠♦s ❢♦r❛♠ r❡❝❡♥t❡♠❡♥t❡ ♣✉❜❧✐❝❛❞♦s ♥♦ ♣❡r✐ó❞✐❝♦ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆ ❬✻✹❪✳
P♦r ú❧t✐♠♦✱ ❡st✉❞♦✉✲s❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s tr✐❞✐♠❡♥s✐♦♥❛✐s ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❡ ❤♦♠♦❣ê♥❡❛✱ r❡♣r♦❞✉③✐♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❛ r❡❢❡rê♥❝✐❛ ❬✻✺❪✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥❛s ❡t❛♣❛s ❛♥t❡r✐♦r❡s ❞❡ss❡ tr❛❜❛❧❤♦✱ ❢♦r❛♠ ❡♥❝♦♥tr❛❞❛s ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛✲ çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❢♦✐ ❡st✉❞❛❞❛ ❛tr❛✈és ❞❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❛♥á❧✐s❡ ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s ❡♠ três ❞✐♠❡♥sõ❡s ❡stá ❡♠ ❛♥❞❛♠❡♥t♦✳
❊st❡ tr❛❜❛❧❤♦ é ❞✐✈✐❞✐❞♦ ❝♦♠♦ s❡ s❡❣✉❡✳ ◆♦ ❈❛♣ít✉❧♦ ✷ é ❛♣r❡s❡♥t❛❞❛ ❛ ❞❡r✐✈❛çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡ s✉❛s s♦❧✉çõ❡s ❞♦ t✐♣♦ só❧✐t♦♥ ❡ ✈órt✐❝❡ sã♦ ❞✐s❝✉t✐❞❛s✳ ❖✉tr♦ tó♣✐❝♦ ❞♦ ❝❛♣ít✉❧♦ ❞✐③ r❡s♣❡✐t♦ às ♣❡q✉❡♥❛s ♦s❝✐❧❛çõ❡s ❡♠ ✉♠ ❝♦♥❞❡♥s❛❞♦✱ q✉❡ tê♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❡ sã♦ ❞❡s❝r✐t❛s ♣❡❧❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s✳ ❋✐♥❛❧♠❡♥t❡✱ é ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ s♦❜r❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡♥tr❡ ♦s át♦♠♦s ❡ r❡ss♦♥â♥❝✐❛ ❞❡ ❋❡s❤❜❛❝❤✳
◆♦ ❈❛♣ít✉❧♦ ✸ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ♠ét♦❞♦s ♥✉♠ér✐❝♦s ❡♠♣r❡❣❛❞♦s ♥❛ s♦❧✉çã♦ ❞❛ ❡q✉❛✲ çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡ ❛ ❞✐s❝r❡t✐③❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s✳
❖s r❡s✉❧t❛❞♦s ❞♦ tr❛❜❛❧❤♦ sã♦ ❞✐s❝✉t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✹✳ ■♥✐❝✐❛❧♠❡♥t❡ sã♦ ❡①✐❜✐❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❡ ❡♥❡r❣✐❛ ❞♦ ❝♦♥❞❡♥s❛❞♦ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ át♦♠♦s ♣❛r❛ ✉♠ ❝♦♥❞❡♥s❛❞♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❡ ❤♦♠♦❣ê♥❡❛ ❡♥tr❡ ♦s át♦♠♦s✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ❝♦♥❞❡♥s❛❞♦ é ❞✐s❝✉t✐❞❛ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s ❡ ❝♦♠♣❛r❛❞❛ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ♣r♦✈❡♥✐❡♥t❡s ❞❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞♦s ✈órt✐❝❡s✳ ❊♠ s❡❣✉✐❞❛ ❛ ♠❡s♠❛ ❛♥á❧✐s❡ é ❢❡✐t❛ ♣❛r❛ ❝♦♥❞❡♥s❛❞♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❝♦♠ ✐♥t❡r❛çã♦ ❛tô♠✐❝❛ ❡s♣❛❝✐❛❧♠❡♥t❡ ♠♦❞✉❧❛❞❛✳ ◆♦ ❝❛s♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ♥♦✈❛♠❡♥t❡ ❝♦♠ ✐♥t❡r❛çã♦ ❛tr❛t✐✈❛ ❡ ❤♦♠♦❣ê♥❡❛ ❡♥tr❡ ♦s át♦♠♦s✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❡ ❡♥❡r❣✐❛ ❞♦ ❝♦♥❞❡♥s❛❞♦ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ át♦♠♦s ❡ ❛ ❛♥á❧✐s❡ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❛s ❡q✉❛çõ❡s❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s✳
❆ ❝♦♥❝❧✉sã♦ ❞♦ tr❛❜❛❧❤♦ s❡ ❞á ♥♦ ❈❛♣ít✉❧♦ ✺✱ ♦♥❞❡ é ❢❡✐t❛ ✉♠❛ ❞✐s❝✉ssã♦ ✜♥❛❧ ❛❝❡r❝❛ ❞♦s r❡s✉❧t❛❞♦s✱ ❞❛s ♣❡rs♣❡❝t✐✈❛s ❡ ♣r♦❥❡t♦s ❢✉t✉r♦s ♥♦ ❡st✉❞♦ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❛tr❛t✐✈♦s✳ ❆ ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧✱ ❡st✉❞♦ ❛ss✐♥tót✐❝♦ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ❡ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❡stã♦ ♣r❡s❡♥t❡s ♥♦s ❆♣ê♥❞✐❝❡s✳
❈♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥
◆❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛❞♦ ♦ ❢♦r♠❛❧✐s♠♦ t❡ór✐❝♦ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦✳ ◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ é ❞❡r✐✈❛❞❛✱ ❜❡♠ ❝♦♠♦ ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❞❡ ✉♠ ❝♦♥❞❡♥s❛❞♦✳ ❊♠ s❡❣✉✐❞❛✱ ♥❛ s❡❣✉♥❞❛ s❡çã♦✱ é ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ ❛ r❡s♣❡✐t♦ ❞❡ só❧✐t♦♥s ♥❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐✳ ❆ t❡♦r✐❛ ❞❡ ✈órt✐❝❡s ❡♠ ❝♦♥❞❡♥s❛❞♦s ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ é ❞❡s❝r✐t❛ ♥❛ t❡r❝❡✐r❛ s❡çã♦✱ ❜❡♠ ❝♦♠♦ ❛ ❢♦r♠❛ ❛❞✐♠❡♥s✐♦♥❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐ ✉t✐❧✐③❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✳ ◆❛ q✉❛rt❛ s❡çã♦ ❞✐s❝✉t❡✲s❡ ❛❝❡r❝❛ ❞❡ ♣❡q✉❡♥❛s ♦s❝✐❧❛çõ❡s ❡♠ ✉♠ ❝♦♥❞❡s❛❞♦ ❞❡ ❇♦s❡✲❊✐♥st❡✐♥ ❡ ❛s ❡q✉❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈✲❞❡ ●❡♥♥❡s ♣❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❞❡♥s❛❞♦ sã♦ ❞❡r✐✈❛❞❛s✳ ❋✐♥❛❧♠❡♥t❡✱ ♥❛ q✉✐♥t❛ ❡ ♥❛ s❡①t❛ s❡çã♦✱ ❡stá ♣r❡s❡♥t❡ ✉♠❛ ❜r❡✈❡ ❞✐s❝✉ssã♦ s♦❜r❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡ r❡ss♦♥â♥❝✐❛ ❞❡ ❋❡s❤❜❛❝❤✳
✷✳✶ ❊q✉❛çã♦ ❞❡ ●r♦ss✲P✐t❛❡✈s❦✐✐
❖ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❜ós♦♥s ❝♦♠ ✐♥t❡r❛çã♦ ❡♥tr❡ ♣❛r❡s é ❞❛❞♦✱ ♥❛ ❢♦r♠❛ ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ ❝❛♠♣♦✱ ♣♦r
ˆ
H =
Z
ˆ
Ψ†(r, t) ˆH0Ψ(ˆ r, t)dr+ 1 2
Z Z
ˆ
Ψ†(r, t) ˆΨ†(r′, t)V(r,r′) ˆΨ(r′, t) ˆΨ(r, t)dr′dr, ✭✷✳✶✮ ♦♥❞❡Hˆ0=−¯h2
2m∇2+Vext(r)é ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ s❡♥❞♦Vext(r)✉♠ ♣♦t❡♥❝✐❛❧
❡①t❡r♥♦✱ ❡V(r,r′) ✉♠ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ❞✉❛s ♣❛rtí❝✉❧❛s✳
P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ s❡❥❛ Ψ(ˆ r, t) ≡ Ψˆ✱ Ψˆ†(r, t) ≡ Ψˆ†✱ Ψ(ˆ r′, t) ≡ Ψˆ′ ❡ Ψˆ†(r′, t) ≡ Ψˆ′† ❞❡
❢♦r♠❛ q✉❡ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ é ❡s❝r✐t❛ ❝♦♠♦
ˆ
H =
Z
ˆ
Ψ†Hˆ0Ψˆdr+ 1 2
Z Z
ˆ
Ψ†Ψˆ′†V(r,r′) ˆΨ′Ψˆdr′dr. ✭✷✳✷✮ ❆s r❡❧❛çõ❡s ❞❡ ❝♦♠✉t❛çã♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❜♦sô♥✐❝♦s sã♦
[ ˆΨ,Ψˆ′†] =δ(r,r′), ✭✷✳✸✮ ❡
[ ˆΨ,Ψˆ′] = [ ˆΨ†,Ψˆ′†] = 0. ✭✷✳✹✮
P❡❧❛ ❡q✉❛çã♦ ❞❡ ❍❡✐s❡♥❜❡r❣
i¯h∂Ψˆ
∂t = [ ˆΨ,Hˆ]
= [ ˆΨ,Hˆ] + [ ˆΨ,Gˆ], ✭✷✳✺✮ ♦♥❞❡Hˆ = ˆH+ ˆG✱ ❝♦♠
ˆ
H=
Z
ˆ
Ψ†Hˆ0Ψˆdr, ✭✷✳✻✮
❡
ˆ
G= 1 2
Z
ˆ Ψ†
Z
ˆ
Ψ′†V(r,r′) ˆΨ′dr′
ˆ
Ψdr. ✭✷✳✼✮
❈❛❧❝✉❧❛♥❞♦ ♦ ❝♦♠✉t❛❞♦r ❞❡Ψˆ′ ❝♦♠ Hˆ t❡♠✲s❡ q✉❡
[ ˆΨ′,Hˆ] = ˆΨ′H −ˆ HˆΨˆ′ =
Z
ˆ
Ψ′Ψˆ†Hˆ0Ψˆdr−
Z
ˆ
Ψ†Hˆ0Ψ ˆˆΨ′dr=
Z
( ˆΨ′Ψˆ†−Ψˆ†Ψˆ′) ˆH0Ψˆdr =
Z
δ(r,r′) ˆH0Ψˆdr= ˆH0′Ψˆ′. ✭✷✳✽✮
❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ s✐st❡♠❛ s❡❥❛ ✉♠ ❣ás ❞❡ ❜ós♦♥s ❢r❛❝❛♠❡♥t❡ ✐♥t❡r❛❣❡♥t❡ ❡ ❡♠ ❜❛✐①❛ t❡♠♣❡r❛t✉r❛✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❡ ❞❡ ❇r♦❣❧✐❡ λdB =
q
2π¯h2/mkBT✱ ♦♥❞❡ m é
❛ ♠❛ss❛ ❞❡ ✉♠ ❜ós♦♥ ❡ T ❛ t❡♠♣❡r❛t✉r❛ ❞♦ s✐st❡♠❛✱ é ♠✉✐t♦ ♠❛✐♦r q✉❡ ♦ ❛❧❝❛♥❝❡ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s át♦♠♦s ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞♦ ❛ ✉♠ ♣♦t❡♥❝✐❛❧ ❞❡ ❝♦♥t❛t♦ ♥❛ ❢♦r♠❛ V(r,r′)≡gδ(r,r′)✱ ♦♥❞❡ g✱ ❞❡♥♦♠✐♥❛❞❛
❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦✱ é ❧✐♥❡❛r♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❡♥tr❡ ♦s át♦♠♦s✱ ❝♦♥❢♦r♠❡ ❞❡♠♦♥str❛❞♦ ♥❛ ❙❡çã♦ ✷✳✺✳ ❉❡st❡ ♠♦❞♦✱ ♥❛ ❡q✉❛çã♦ ✭✷✳✼✮✱
ˆ
G= g 2
Z
ˆ Ψ†
Z
ˆ
Ψ′†δ(r,r′) ˆΨ′dr′
ˆ Ψdr= g
2
Z
ˆ
Ψ†Ψˆ†Ψ ˆˆΨdr. ✭✷✳✾✮ ❈♦♠ ✐ss♦
[ ˆΨ′,Gˆ] = ˆΨ′G −ˆ GˆΨˆ′ = g 2
Z
( ˆΨ′Ψˆ†Ψˆ†Ψ ˆˆΨ−Ψˆ†Ψˆ†Ψ ˆˆΨ ˆΨ′)dr= g 2
Z
( ˆΨ′Ψˆ†Ψˆ†−Ψˆ†Ψˆ†Ψˆ′) ˆΨ ˆΨdr
= g 2
Z
( ˆΨ′Ψˆ†Ψˆ†−Ψˆ†Ψˆ†Ψˆ′+ ˆΨ†Ψˆ′Ψˆ†−Ψˆ†Ψˆ′Ψˆ†) ˆΨ ˆΨdr
= g 2
Z
([ ˆΨ′,Ψˆ†] ˆΨ†+ ˆΨ†[ ˆΨ′,Ψˆ†]) ˆΨ ˆΨdr
=g
Z
δ(r,r′) ˆΨ†Ψ ˆˆΨdr=gΨˆ′†Ψˆ′Ψˆ′. ✭✷✳✶✵✮ ❆ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ✭✷✳✽✮ ❡ ✭✷✳✶✵✮ ❡♠ ✭✷✳✺✮
i¯h∂Ψˆ
∂t = ( ˆH0+gΨˆ
†Ψ) ˆˆ Ψ
=
−¯h
2
2m∇
2+V
ext+gΨˆ†Ψˆ
ˆ
Ψ. ✭✷✳✶✶✮
❙❡❥❛♠ ♦s ♦♣❡r❛❞♦r❡s ❞❡ ❝❛♠♣♦ Ψˆ ❡s❝r✐t♦s ♥❛ ❜❛s❡ ❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❞❡ ♣❛rtí❝✉❧❛
ú♥✐❝❛✳ ❉❡ss❡ ♠♦❞♦
ˆ
Ψ(r, t) =X k
ψk(r)ak(t), ✭✷✳✶✷✮
❡
ˆ
Ψ†(r, t) =X k
ψk∗(r)a†k(t), ✭✷✳✶✸✮ ♦♥❞❡ ψk(r) sã♦ ❢✉♥çõ❡s ❞❡ ♦♥❞❛ ❞❡ ♣❛rtí❝✉❧❛ ú♥✐❝❛ ❞♦ k✲és✐♠♦ ♦r❜✐t❛❧✳ ak†(t) ❡ ak(t) sã♦✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ♦♣❡r❛❞♦r❡s ❞❡ ❝r✐❛çã♦ ❡ ❛♥✐q✉✐❧❛çã♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♥♦ ❡st❛❞♦ ❝♦r✲ r❡s♣♦♥❞❡♥t❡ ❛♦ ♦r❜✐t❛❧ψk(r)✱ s❡♥❞♦ q✉❡
a†k(t) =e−iHt/ˆ ¯ha†k(0)eiHt/ˆ ¯h ❡ ak(t) =eiHt/ˆ ¯hak(0)e−iHt/ˆ ¯h. ✭✷✳✶✹✮
P❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s {|n0, n1, . . . , n∞i} ♥♦ ❡s♣❛ç♦ ❞♦s ♥ú♠❡r♦s ❞❡ ♦❝✉♣❛çã♦
q✉❡ ♦❜❡❞❡❝❡♠ ❛s r❡❧❛çõ❡s ❞❡ ♦rt♦♥♦r♠❛❧✐❞❛❞❡
hn′0, n′1, . . . , n∞′ |n0, n1, . . . , n∞i=δn0,n′0δn1,n′1. . . δn∞,n′∞, ✭✷✳✶✺✮
❝♦♠♣❧❡t❡③❛
X
n0,n1,...,n∞
|n0, n1, . . . , n∞ihn0, n1, . . . , n∞|= 1, ✭✷✳✶✻✮
❡ ♣r♦❞✉t♦ ❞✐r❡t♦
|n0, n1, . . . , n∞i=|n0i|n1i. . .|n∞i, ✭✷✳✶✼✮
s❡❣✉❡✱ ❥✉♥t♦ às r❡❧❛çõ❡s ❞❡ ❝♦♠✉t❛çã♦ ✭♣♦r s✐♠♣❧✐❝✐❞❛❞❡✱a†k≡a†k(0)❡ ak ≡ak(0)✮✱
[ak, a†k′] =δk,k′, ✭✷✳✶✽✮
❡
[ak, ak′] = [a†k, a†k′] = 0, ✭✷✳✶✾✮
♦♥❞❡δk,k′ é ♦ ❞❡❧t❛ ❞❡ ❑r♦♥❡❝❦❡r✱ q✉❡
a†kak|nki=nk|nki, ✭✷✳✷✵✮
ak|nki= (nk)1/2|nk−1i, ✭✷✳✷✶✮
❡
a†k|nki= (nk+ 1)1/2|nk+ 1i, ✭✷✳✷✷✮
♦♥❞❡nk= 0,1,2, . . . ,∞ ✐♥❞✐❝❛ ♦ ♥ú♠❡r♦ ♣❛rtí❝✉❧❛s ♥♦ ❡st❛❞♦k✳
❆ ❡①♣r❡ssã♦ ✭✷✳✶✷✮ ♣♦❞❡ s❡r ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ ❡s❝r✐t❛ ♥❛ ❢♦r♠❛
ˆ
Ψ(r, t) =ψ0(r)a0(t) +
X
k6=0
ψk(r)ak(t), ✭✷✳✷✸✮
♦♥❞❡ ♦s t❡r♠♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ Ψ0(r, t) ❡ ❛♦s ❡st❛❞♦s ❡①❝✐t❛❞♦s Ψk(r, t) ❢♦r❛♠ s❡♣❛r❛❞♦s✳
❆ ♠é❞✐❛ s♦❜r❡ ♦ ♦♣❡r❛❞♦rΨ(ˆ r, t) ❝♦♥s✐st❡ ❡♠
