❘❡♣r❡s❡♥t❛çõ❡s ❡ ❆s♣❡❝t♦s ◗✉â♥t✐❝♦s ❞❡ ❙✐st❡♠❛s ◆ã♦✲❈♦♠✉t❛t✐✈♦s
❏✉❧✐♦ ●❧❛✉❜❡r ❋❡rr❡✐r❛ ❞♦s ❙❛♥t♦s
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✲ ❯❋▼● ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ✲ ■❈❊①
Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛
❘❡♣r❡s❡♥t❛çõ❡s ❡ ❆s♣❡❝t♦s ◗✉â♥t✐❝♦s ❞❡ ❙✐st❡♠❛s ◆ã♦✲❈♦♠✉t❛t✐✈♦s
❏✉❧✐♦ ●❧❛✉❜❡r ❋❡rr❡✐r❛ ❞♦s ❙❛♥t♦s
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❉♦♥✐③❡t✐ ❘♦❞r✐❣✉❡s ❙❛♠♣❛✐♦ ❈♦✲♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▲✉ís ❆♥tô♥✐♦ ❈❛❜r❛❧
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❚í✲ t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s✳
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥♦✉ ❡ t❛♠❜é♠ ♣♦r ♠❡ s❡❣✉r❛r ❡ ♠❡ ♣r♦t❡❣❡r ❛ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡ ❞✉r❛♥t❡ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦s ♠❡✉s ♣❛✐s ❱❛❧❞❡♠❛r ❡ ❋r❛♥❝✐s❝❛ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠✳ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ❋❛✲ ❜✐❛♥❛ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♣♦✐♦ ❞✉r❛♥t❡ t♦❞♦ ❡ss❡ t❡♠♣♦✱ ❛ q✉❡♠ ❞❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❡ t♦❞❛ ❛ ♠✐♥❤❛ ✈✐❞❛✳
❉❡ ♠♦❞♦ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ▼❛r❝♦s ❉✳ ❙❛♠♣❛✐♦✱ ❖r✐❡♥t❛❞♦r ❞❡st❡ tr❛❜❛❧❤♦✱ q✉❡ ❝♦♠ ♣❛❝✐ê♥❝✐❛ ❡ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❛❝♦♠♣❛♥❤♦✉✲♠❡ ❝♦♠ ♦r✐❡♥t❛çõ❡s ❝❧❛r❛s ❡ ✈❛❧✐♦s❛s ♣❛r❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❛ ♣❡sq✉✐s❛ ❡ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳
❆♦ ♣r♦❢❡ss♦r ❉r✳ ▲✉✐③ ❈❛❜r❛❧✱ ♠❡✉ ❝♦✲♦r✐❡♥t❛❞♦r✱ ♣♦r s✉❛ ❞❡❞✐❝❛çã♦ ❡ ✈❛❧✐♦s❛s ♦r✐❡♥✲ t❛çõ❡s q✉❡ s❡♠♣r❡ t❡✈❡ ❞✐s♣♦s✐çã♦ ❡ ❡♥t✉s✐❛s♠♦ ♣❛r❛ s❛♥❛r ♠✐♥❤❛s ❞ú✈✐❞❛s✳ ➚ ❈❛r♦❧✐♥❛ ◆❡♠❡s✱ ♣❡❧❛s ✈❛❧✐♦s❛s ❡ ❝❛❧♦r♦s❛s ❞✐s❝✉ssõ❡s✳
❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ❝♦❧❡❣❛ ❞❡ ♠❡str❛❞♦ ❡ ✐r♠ã♦ ❡♠ ❈r✐st♦✱ ❲❡❧②s♦♥ ❚✐❛♥♦✱ q✉❡ ♠✉✐t♦ ❝♦♥tr✐❜✉✐✉ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ➚ ▲❡❛♥❞r❛ ❘❡s❡♥❞❡ ♣❡❧♦ ❛♣♦✐♦ ❡ ❛♦s ❞❡♠❛✐s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✳
➚ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳ ❆♦ ❣r✉♣♦ ❞❡ ❚◗❈ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ✈❛❧✐♦s♦s ❝♦♥❤❡❝✐✲ ♠❡♥t♦s✳ ❆ t♦❞♦ ♦ ♣❡ss♦❛❧ ❞❛ ❜✐❜❧✐♦t❡❝❛ ♣❡❧❛ ❛t❡♥çã♦ ❡ ❞✐s♣♦s✐çã♦ ❡♠ ❡s♣❡❝✐❛❧ à ❙❤✐r❧❡②✱ ♦❜r✐❣❛❞♦ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛✳
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡①♣❧♦r❛✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ❛❧❣✉♠❛s q✉❡stõ❡s ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ♥♦ ❡s♣❛ç♦ ❜❡♠ ❝♦♠♦ ❛s ♠♦t✐✈❛çõ❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛s t❡♦r✐❛s ♥ã♦✲ ❝♦♠✉t❛t✐✈❛s✳ ❊♠ s❡❣✉✐❞❛ s❡rá ❞❛❞♦ ✉♠ s♦❜r❡✈♦♦ ❡♠ ❛❧❣✉♥s ❛s♣❡❝t♦s ❞❛ t❡♦r✐❛ q✉â♥t✐❝❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ❛❜♦r❞❛❞❛ ♥♦ ♣❧❛♥♦✳ ❙❡❣✉✐♥❞♦ ❛s r❡❣r❛s ❜ás✐❝❛s ❞❡ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ❝♦♥str✉✐r ❡st❛❞♦s q✉❡ s❛t✉r❡♠ ❛s r❡❧❛çõ❡s ❞❡ ✐♥❝❡rt❡③❛ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ❡ ♠♦♠❡♥t♦s ❡ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ❛♣❡♥❛s✱ ♦✉ s❡❥❛✱ ♦❜t❡r ♦ ♠í♥✐♠♦ ❞❡ ✐♥❝❡rt❡③❛ ♣♦ssí✈❡❧ ♥✉♠ ❞♦s ♦❜✲ s❡r✈á✈❡✐s ❡♠ q✉❡stã♦✱ ♣❛r❛ ✉♠❛ ❝♦♥str✉çã♦ ❞❛ á❧❣❡❜r❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡❣✉✐r ♥❛ ❜✉s❝❛ ❞❡ ❡st❛❞♦s q✉❡ s❛t✉r❡♠✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❞✉❛s ♦✉ ♠❛✐s ❞❛s r❡❧❛çõ❡s ❞❡ ✐♥❝❡rt❡③❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ♠♦❞✐✜❝❛❞❛s✱ ♦✉ s❡❥❛✱ ✈❛❧❡♥❞♦ ♣❛r❛ ♦ ♣❧❛♥♦ ♦♥❞❡ ❛s ❝♦♦r❞❡♥❛❞❛s ♥ã♦ ❝♦♠✉t❛♠ ❡ ♠♦str❛r q✉❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❞❡ ❝♦✈❛r✐â♥❝✐❛ ❞❡ ❙❝❤r♦❞✐♥❣❡r✱ ♣❛r❛ ✉♠ ❡st❛❞♦ ❣❛✉ss✐❛♥♦ ♣❛rt✐❝✉❧❛r✱ é ❡q✉✐✈❛❧❡♥t❡ à r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ♣❛r❛ ❞♦✐s ♦♣❡r❛❞♦r❡s✳
❙❡rá ❝♦♥str✉í❞♦ ♦ ♣r♦♣❛❣❛❞♦r ♣❛r❛ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡✱ ❛ ♣❛rt✐r ❞❛ ❝♦♥str✉çã♦ ❞❛ ✈❡rsã♦ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ❞❛ ♦♥❞❛ ♣❧❛♥❛✱ ❡①♣❧♦r❛♥❞♦ ❛ ✐❞❡✐❛ ❞❡ ♠é❞✐❛ ❡♠ ❡st❛❞♦s ❝♦❡r❡♥t❡s✳ ❊✱ ❡①♣❧♦r❛r t❛♠❜é♠ ❛ ✐❞❡✐❛ ❞❛ ❝♦♥str✉çã♦ ❞♦ ♣❛❝♦t❡ ❞❡ ♦♥❞❛s✱ ♥♦ ♣❧❛♥♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✱ ♦❜t❡r s✉❛ ❢♦r♠❛ ❡✈♦❧✉í❞❛ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ ✐♥t❡❣r❛❧ ❞❡ tr❛❥❡tór✐❛✱ ✜♥❛❧✐③❛♥❞♦ ❝♦♠ ❛ ❛♥á❧✐s❡ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❡♠ ❡s♣❛ç♦✳ ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❛♥❛❧✐s❛ ♦s r❡s✉❧t❛❞♦s ❡ ♦ q✉❡ s❡ ♣♦❞❡ ❢❛③❡r✱ ♥✉♠ ♣ró①✐♠♦ ♣r♦❥❡t♦✱ ❝♦♠ ❛s ❢❡rr❛♠❡♥t❛s ❛♣r❡s❡♥t❛❞♦s✳
❆❜str❛❝t
❚❤✐s ♣❛♣❡r ❡①♣❧♦r❡s ✐♥✐t✐❛❧❧②✱ ✐♥ ❛ ✐♥t✉✐t✐✈❡ ❢♦r♠✱ s♦♠❡ q✉❡st✐♦♥s ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈✐t② ✐♥ s♣❛❝❡ ✇❡❧❧ ❛s t❤❡ ♠♦t✐✈❛t✐♦♥s ❢♦r t❤❡ st✉❞② ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ t❤❡♦r✐❡s✳ ❚❤❡♥ ❜❡ ❣✐✈❡♥ ❛♥ ♦✈❡r✢✐❣❤t ✐♥ s♦♠❡ ❛s♣❡❝ts ♦❢ q✉❛♥t✉♠ t❤❡♦r② ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛❞❞r❡ss❡❞ ✐♥ t❤❡ ♣❧❛♥✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❜❛s✐❝ r✉❧❡s ♦❢ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ s❡❡ ❤♦✇ t♦ ❜✉✐❧❞ st❛t❡s t❤❛t s❛t✉r❛t❡ t❤❡ ✉♥❝❡rt❛✐♥t② r❡❧❛t✐♦♥s✱ ✐✳❡✳✱ ❣❡t t❤❡ ❧❡❛st ♣♦ss✐❜❧❡ ✉♥❝❡rt❛✐♥t② ✐♥ t❤❡ ♦❜s❡r✈❛❜❧❡ ✐♥ q✉❡st✐♦♥✱ ❢♦r ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ♦❢ ❝♦♦r❞✐♥❛t❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦❧❧♦✇✐♥❣ t❤❡ s❡❛r❝❤ ♦❢ st❛t❡s t❤❛t s❛t✉r❛t❡ ❜♦t❤ r❡❧❛t✐♦♥s ❍❡✐s❡♥❜❡r❣ ✉♥❝❡rt❛✐♥t② ♠♦❞✐✜❡❞ ❛♥❞ s❤♦✇ s❛t✉r❛t✐♦♥ ♦❢ t❤❡ ❙❝❤r♦❞✐♥❣❡r✬s ❞❡t❡r♠✐♥❛♥t ❢♦r ❛ ❣✐✈❡♥ ●❛✉ss✐❛♥ st❛t❡✳
❚❤❡♥✱ ✇✐❧❧ ❜❡ ❜✉✐❧t t❤❡ ♣r♦♣❛❣❛t♦r ❢♦r t❤❡ ❢r❡❡ ♣❛rt✐❝❧❡ ❢r♦♠ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥♦♥✲ ❝♦♠♠✉t❛t✐✈❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ ✇❛✈❡✱ ❡①♣❧♦r✐♥❣ t❤❡ ✐❞❡❛ ♦❢ ❛✈❡r❛❣❡ ✐♥ ❝♦❤❡r❡♥t st❛t❡s✳ ❆♥❞ ❛❧s♦ ❡①♣❧♦r✐♥❣ t❤❡ ✐❞❡❛ ♦❢ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ✇❛✈❡ ♣❛❝❦❡t ✐♥ t❤❡ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ ♣❧❛♥❡✱ ❣❡t ✐ts ❡✈♦❧✈❡❞ ❢♦r♠ ✉s✐♥❣ t❤❡ ♣❛t❤ ✐♥t❡❣r❛❧ ❢♦r♠❛❧✐s♠✱ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❋♦✉r✐❡r✬s tr❛♥s❢♦r♠ ✐♥ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤✐s ♣❛♣❡r ❛♥❛❧②③❡s t❤❡ r❡s✉❧ts ❛♥❞ ✇❤❛t ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❛ ♥❡①t ♣r♦❥❡❝t✱ ✇✐t❤ t❤❡ t♦♦❧s ♣r❡s❡♥t❡❞✳
❑❡②✇♦r❞s✿ ❆s♣❡❝t♦s ❈✐♥❡♠át✐❝♦s ❞❡ ❚❡♦r✐❛s ◗✉â♥t✐❝❛s ◆ã♦✲❈♦♠✉t❛t✐✈❛s
❙✉♠ár✐♦
❆❜str❛❝t ✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ▼♦t✐✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❊str✉t✉r❛ ❞❛ ❉✐ss❡rt❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❈♦♥tr✐❜✉✐çõ❡s ❖r✐❣✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✷ ❆s♣❡❝t♦s ◆ã♦✲❈♦♠✉t❛t✐✈♦s ♥♦ ♣❧❛♥♦ ✺
✷✳✶ ❆s♣❡❝t♦s ◆ã♦✲❈♦♠✉t❛t✐✈♦s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧ ❡ Pr♦❞✉t♦ ❡str❡❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❈♦♥s❡r✈❛çã♦ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡ ❡♠ ▼✳◗✳◆✳❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✸ ▼✐♥✐♠✐③❛çã♦ ❞❛s ❘❡❧❛çõ❡s ❞❡ ■♥❝❡rt❡③❛ ❯s❛♥❞♦ ❈♦♦r❞❡♥❛❞❛s ◆ã♦✲❈♦♠✉t❛t✐✈❛s ✶✼ ✸✳✶ Pr✐♥❝í♣✐♦ ❞❡ ■♥❝❡rt❡③❛ ❡ ❊st❛❞♦s ❈♦❡r❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✷ ❖ Pr♦❜❧❡♠❛ ❡ ❛ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ➪❧❣❡❜r❛ ❇ás✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸ ▼✐♥✐♠✐③❛♥❞♦ ❛s ❘❡❧❛çõ❡s ❞❡ ■♥❝❡rt❡③❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✸✳✶ ▼✐♥✐♠✐③❛♥❞♦ ❛ ❘❡❧❛çã♦ ❞❡ ■♥❝❡rt❡③❛ ❊♥tr❡ ❈♦♦r❞❡♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✸✳✷ ▼✐♥✐♠✐③❛♥❞♦ ❛ ❘❡❧❛çã♦ ❞❡ ■♥❝❡rt❡③❛ ❊♥tr❡ ❈♦♦r❞❡♥❛❞❛ ❡ ▼♦♠❡♥t♦ ✷✽ ✸✳✹ ❚r❛❜❛❧❤❛♥❞♦ ❈♦♠ ❈♦♦r❞❡♥❛❞❛s ❞❡ ❘❡♣r❡s❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✺ ▼❛tr✐③ ❞❡ ❈♦✈❛r✐â♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹ ❋♦r♠✉❧❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ◆ã♦✲❈♦♠✉t❛t✐✈❛ ❯s❛♥❞♦ ❊st❛❞♦s ❈♦✲
❡r❡♥t❡s ✸✻
✹✳✶ ❖♥❞❛ P❧❛♥❛ ▼♦❞✐✜❝❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ Pr♦♣❛❣❛❞♦r ❞❛ P❛rtí❝✉❧❛ ▲✐✈r❡ ❡ ❯♥✐t❛r✐❡❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷✳✶ Pr♦♣❛❣❛❞♦r ❞❛ P❛rtí❝✉❧❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✐✈
✹✳✷✳✷ ❯♥✐t❛r✐❡❞❛❞❡ ❞♦ Pr♦♣❛❣❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✸ P❛❝♦t❡ ●❛✉ss✐❛♥♦ ♥♦ P❧❛♥♦ ◆✳❈✳ ❡ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸✳✶ ❈♦♥str✉çã♦ ❞♦ P❛❝♦t❡ ❞❡ ♦♥❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸✳✷ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✹ ❱❡rsã♦ ◆ã♦✲❝♦♠✉t❛t✐✈❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✺ ❈♦♥❝❧✉sõ❡s ❡ P❡rs♣❡❝t✐✈❛s ✺✶
❆ ❖♣❡r❛❞♦r ❯♥✐tár✐♦✱ ❖♣❡r❛❞♦r ❞❡ ❚r❛♥s❧❛çã♦ ❡ ❖♣❡r❛❞♦r ❞❡ ❘♦t❛çã♦ ✺✸
❇ ▼❛tr✐③ ❞❡ ❈♦✈❛r✐â♥❝✐❛ ✺✺
❈ Pr♦♣❛❣❛❞♦r ▲✐✈r❡✿ ♦✉tr♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ✺✽
❉ ❖r❞❡♥❛çã♦ ❞❡ ❖♣❡r❛❞♦r❡s ✻✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
✶✳✶ ▼♦t✐✈❛çõ❡s
❖ ❝♦♥❝❡✐t♦ ❞❡ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ♥❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♥ã♦ é tã♦ ♥♦✈♦✳ ❍✐st♦r✐❝❛♠❡♥t❡✱ ❡ss❛ ✐❞❡✐❛ ❢♦✐ s✉❣❡r✐❞❛✱ ❧♦❣♦ ♥♦s ♣r✐♠ór❞✐♦s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♣♦r ❍❡✐s❡♥❜❡r❣✶❬✾❪ ❞❡♣♦✐s ❞❡ ♣❡r❝❡❜❡r✱ ❡♥tr❡ ♦✉tr♦s ❢ís✐❝♦s ❞❛ é♣♦❝❛✱ q✉❡ ❛s ❣r❛♥✲
❞❡③❛s ❝❧áss✐❝❛s ❡①♣r❡ss❛s ♣♦r ❢✉♥çõ❡s r❡❛✐s ❞❡✈❡r✐❛♠ s❡r ❛❜❛♥❞♦♥❛❞❛s ❡♠ ❢❛✈♦r ❞❡ ♥♦✈❛s ❣r❛♥❞❡③❛s ❞❛❞❛s ♣♦r ♦♣❡r❛❞♦r❡s ❝✉❥♦ ❝♦♠✉t❛❞♦r ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♥✉❧♦✳ ❊❧❡ ♣r♦♣ôs ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ ♥ã♦ ♥✉❧❛ ❡♥tr❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s❬✷✸❪✳ ❆ ♣❛rt✐r ❞❡ ❡♥tã♦✱ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ t♦r♥♦✉✲s❡ ♥ã♦✲❝♦♠✉t❛t✐✈♦✳ ❆ss✐♠✱ s❡❣✉♥❞♦ ❍❡✐s❡♥❜❡r❣✱ s❡r✐❛ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❡❧✐♠✐♥❛r ❛s s✐♥❣✉❧❛r✐❞❛❞❡s q✉❡ ❛♣❛r❡✲ ❝❡♠ ♥❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❘❡❧❛t✐✈íst✐❝❛ ❞❡ ❈❛♠♣♦s❬✽❪✳ ❊ss❛s ❝♦♥s✐❞❡r❛çõ❡s ❡♠ ♣❛rt❡ ❧❡✈❛✲ r❛♠ ❙♥②❞❡r✷ ❛ ♣✉❜❧✐❝❛r ♦ ♣r✐♠❡✐r♦ ❛rt✐❣♦ s♦❜r❡ ♦ t❡♠❛❬✷❪✳ ❙❡♥❞♦ ❡sq✉❡❝✐❞❛ ❞✉r❛♥t❡ ♠✉✐t♦
t❡♠♣♦✱ ❡ss❛ ✐❞❡✐❛ ❢♦✐ r❡t♦♠❛❞❛ r❡❝❡♥t❡♠❡♥t❡ ❡♠ t❡①t♦s ❞❡ ❚❡♦r✐❛ ❞❡ ❈♦r❞❛s❬✶✹❪✱ ❬✷✸❪✳ ❖ r❡❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ♠♦❞❡❧♦s ❡♥✈♦❧✈❡♥❞♦ ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ t❛♠❜é♠ ❢♦✐ ✐♠♣✉❧s✐♦♥❛❞♦ ♣❡❧♦ ❢❛t♦ ❞❡ s❡r❡♠ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❞❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❛ ●r❛✈✐❞❛❞❡❬✶✸❪✳ ❆ ✐❞❡✐❛ é q✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ é ♣❡r❞✐❞❛ ♥❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦λp = (G~/c3)1/2 ≈1.6×10−33cm✳ ■ss♦ ❧❡✈❛ ❛ ✉♠❛ r❡❧❛çã♦ ❞❡
❝♦♠✉t❛çã♦ t❛❧ q✉❡ ❞á✱
[ ˆxi,xˆj] =iθij i, j = 1,2 ✭✶✳✶✮
✶❲✳ ❑✳ ❍❡✐s❡♥❜❡r❣ ✭✶✾✵✶✲✶✾✼✻✮✱ ❢ís✐❝♦ t❡ór✐❝♦ ❛❧❡♠ã♦✱ ✉♠ ❞♦s ❝r✐❛❞♦r❡s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♣rê♠✐♦
◆♦❜❡❧ ❞❡ ❢ís✐❝❛ ❡♠ ✶✾✸✷✳
✷❍✳ ❙✳ ❙♥②❞❡r ✭✶✾✶✸ ✲ ✶✾✻✷✮✱ ✉♠ ❢ís✐❝♦ ❛♠❡r✐❝❛♥♦✳
✷
♦♥❞❡ θij = θǫij✳ ❆q✉✐✱ θ é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧ ♣♦s✐t✐✈♦ ❝♦♠ ❞✐♠❡♥sã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦
❛♦ q✉❛❞r❛❞♦ ❡ ♦♥❞❡ ǫij = −ǫji✱ ❝♦♠ ♠ó❞✉❧♦ ✐❣✉❛❧ ❛ ✉♠✳ ❊ss❛ ❤✐♣ót❡s❡ ❞❡ ✐♥s❡r✐r ✉♠❛
♥♦✈❛ r❡❣r❛ ❞❡ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ♣❛r❡❝❡ ♥❛t✉r❛❧ ♣♦✐s✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ✉s✉❛❧ ♥ã♦ t❡♠ s❡✉s ❡st❛❞♦s ❢ís✐❝♦s ❞❡s❝r✐t♦s ♣♦r ♣♦♥t♦s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ♠❛s s✐♠ ♣♦r r❡❣✐õ❡s ❞❡ ár❡❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛~✱ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ◆✳❈✳ t❛♠❜é♠ ♥ã♦ ♣♦ss✉✐r✐❛ ♣♦♥t♦s ❜❡♠
❞❡✜♥✐❞♦s✱ ♠❛s ♦ ♣ró♣r✐♦ ❡s♣❛ç♦ ❢ís✐❝♦ s❡ t♦r♥❛ ❜♦rr❛❞♦ ✭❢✉③③②✮✱ ♣♦♥t♦s sã♦ ❞✐ss♦❧✈✐❞♦s ❡♠ ♣❡q✉❡♥♦s ♣❧❛♥♦s ✭❡♥❡✈♦❛❞♦s✮✱ t❛❧ ❝♦♠♦ ♦❝♦rr❡ ♥♦ ❥á tr❛❞✐❝✐♦♥❛❧ ❡s♣❛ç♦ ❞❡ ❢❛s❡ q✉â♥t✐❝♦✳ ❆ r❡❧❛çã♦ ✭✶✳✶✮ ♣♦st❛ ❛❝✐♠❛ ✐♠♣õ❡ ♣♦ssí✈❡✐s ❧✐♠✐t❛çõ❡s ♥❛ ♣r❡❝✐sã♦ ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ❡✈❡♥t♦s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ♦ q✉❡ ❞❡ ❢❛t♦ ❞❡✈❡ s❡r ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❛ ●r❛✈✐t❛çã♦✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠♦ ✉♠❛ t❡♦r✐❛ ❡❢❡t✐✈❛✱ ❝❛♣❛③ ❞❡ ❧❡✈❛r ❡♠ ❝♦♥t❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ♠í♥✐♠♦✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s✱ θ → 0✱ ❡❧❛ ❞❡✈❡ r❡❝❛✐r ♥❛s t❡♦r✐❛s ✉s✉❛✐s q✉❡
❝♦♥❤❡❝❡♠♦s ❜❡♠✳
❍á ✈ár✐❛s ❢♦r♠❛s ❞❡ ✐♥t❡r♣r❡t❛r ❛ ❛❧t❡r❛çã♦ ❞❛ á❧❣❡❜r❛ q✉â♥t✐❝❛ ❛❝✐♠❛ ♠❡♥❝✐♦♥❛❞❛✳ ❆ s❡❣✉✐r✱ três ❛❜♦r❞❛❣❡♥s ❛ ❡ss❛ q✉❡stã♦ ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣♦♣✉❧❛r❡s✮ sã♦ ❝♦♠❡♥t❛❞❛s✿
• ❈♦♥s✐❞❡r❛r q✉❡ ❛ ❡①✐stê♥❝✐❛ ❞♦ ♦❜❥❡t♦ θǫij 6= 0 s❡❥❛ tã♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥t♦ ❛ ❞❡
~δij✳ ❊ss❡ ♥♦✈♦ ♦❜❥❡t♦ ❞❡✈❡r✐❛ ✐♥tr♦❞✉③✐r ♣❡q✉❡♥♦s ❞❡s✈✐♦s ♥♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s
q✉❡ ♣♦ss✉❡♠ ❜♦❛ ❝♦♥❝♦r❞â♥❝✐❛ ❡①♣❡r✐♠❡♥t❛❧ ❡ ♣♦ss✐❜✐❧✐t❛r ❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♠ ♣r♦❜❧❡♠❛❀
• ❖ ♣❛râ♠❡tr♦ θǫij 6= 0 é ✐♥tr♦❞✉③✐❞♦ ♣❛r❛ ♠♦❞❡❧❛r ❛❧❣✉♠ ♣r♦❝❡ss♦ ❢ís✐❝♦ ❞❡s❝♦♥❤❡✲
❝✐❞♦ ♦✉ s❡q✉ê♥❝✐❛ ❞❡ ✐♥t❡r❛çõ❡s ♥ã♦ ❝♦♥tr♦❧❛❞❛✱ ♥ã♦ t❡♥❞♦ ♣♦rt❛♥t♦ ✉♠ st❛t✉s ❞❡ ❣r❛♥❞❡③❛ ❢✉♥❞❛♠❡♥t❛❧ t❛❧ q✉❛❧ ~δij✳ ❉❡st❛ ❢♦r♠❛✱ ✐♥tr♦❞✉③✲s❡ ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡
♥♦ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❝r✐❛r ✉♠ ♠♦❞❡❧♦ ❡❢❡t✐✈♦✱ ♦ q✉❛❧ ♣♦❞❡r✐❛ s❡r ❝♦♥✲ s✐st❡♥t❡ ❝♦♠ ♠✉✐t♦s ❞♦s ❢❡♥ô♠❡♥♦s ❝♦♥❤❡❝✐❞♦s ♦✉ só ❝♦♠ ❛❧❣✉♥s ♠✉✐t♦ ♣❛rt✐❝✉❧❛r❡s❀
• ❱✐st♦ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ❡ ♠♦♠❡♥t♦s✱[ˆxi,pˆj] =
i~δij✱ só é ✈á❧✐❞❛ ❞❡ ❢♦r♠❛ ❣❡r❛❧ ❡♠ s✐st❡♠❛s s❡♠ ✈í♥❝✉❧♦s❬✾❪✱ ❤á t❡♦r✐❛s ❢ís✐❝❛s
q✉❡✱ ❝♦♥s✐❞❡r❛♥❞♦ s✉❛ ❡str✉t✉r❛ ❞❡ ✈í♥❝✉❧♦s ❡ s♦❜ ❝❡rt♦s ❧✐♠✐t❡s✱ t♦r♥❛♠✲s❡ ♥ã♦✲ ❝♦♠✉t❛t✐✈❛s ♥♦ ❡s♣❛ç♦✲t❡♠♣♦✱ ❡♠❜♦r❛ ♦r✐❣✐♥❛❧♠❡♥t❡ t❡♥❤❛♠ s✐❞♦ ❢♦r♠✉❧❛❞❛s ❡♠ ✉♠ ❝♦♥t❡①t♦ ❝♦♠✉t❛t✐✈♦✳ ❖✉ s❡❥❛✱ ♥❡st❛ ❛❜♦r❞❛❣❡♠✱ ♥ã♦ s❡ ❛ss✉♠❡ θǫij 6= 0 ❛
✸
❊ss❛ ❝❧❛ss✐✜❝❛çã♦ ❢♦✐ ❛❝✐♠❛ ✐♥tr♦❞✉③✐❞❛ ❛♣❡♥❛s ♣❛r❛ ♣r♦♣♦r❝✐♦♥❛r ✉♠❛ ✈✐sã♦ ❣❡r❛❧✱ ♣♦✲ ré♠ ✈❛❣❛✱ ❞❡ ♣♦ssí✈❡✐s ❛❜♦r❞❛❣❡♥s á ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡s♣❛ç♦✲t❡♠♣♦r❛❧✳ ◆ã♦ ❤á ♥❛ ♣rá✲ t✐❝❛ ✉♠❛ ❞✐st✐♥çã♦ ❜❡♠ ❞❡✜♥✐❞❛ ❡♥tr❡ ❡ss❛s ❛❜♦r❞❛❣❡♥s✳ ❖ ❡st✉❞♦ ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ❛❞✈✐♥❞❛ ❞❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s s♦❜ ♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s ❡ ❡st✉❞♦s ❣❡✲ r❛✐s ❞❡ s✐st❡♠❛s ✈✐♥❝✉❧❛❞♦s ❝✉❥❛ q✉❛♥t✐③❛çã♦ ❧❡✈❡ ❛θǫij 6= 0 sã♦ ❜♦♥s ❡①❡♠♣❧♦s ❞❛ t❡r❝❡✐r❛
❛❜♦r❞❛❣❡♠✳ ❆✐♥❞❛ ♥ã♦ ❤á ♥♦ ♠♦♠❡♥t♦ ❝♦♥❞✐çõ❡s ❞❡ s❡ ❝♦♥s✐❞❡r❛r ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡s♣❛ç♦✲t❡♠♣♦r❛❧ ❝♦♠♦ ✉♠ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❡ ♥❡♠ ❤á ✐♥❞í❝✐♦s ❡①♣❡r✐♠❡♥t❛✐s ❝❧❛r♦s ♥❡ss❛ ❞✐r❡çã♦✳ P♦r ❡♥q✉❛♥t♦ ❡❧❛ ❢♦r♥❡❝❡ ✉♠❛ ❡str✉t✉r❛ út✐❧ ♣❛r❛ ♣r♦♣♦r ♥♦✈♦s ♠♦❞❡❧♦s ❡❢❡t✐✈♦s ❡ ❡st✉❞❛r ♦✉tr❛s t❡♦r✐❛s s♦❜ ❝❡rt♦s ❧✐♠✐t❡s✱ ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ ❝♦r❞❛s ♥♦ ❧✐♠✐t❡ ❞❡ ❜❛✐①❛s ❡♥❡r❣✐❛s✳
❙✉♣♦♥❞♦ q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡sxˆ1,xˆ2, . . .,xˆd❞♦ ♦♣❡r❛❞♦r ♣♦s✐çã♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛
♥♦ ❡s♣❛ç♦ ❞✲❞✐♠❡♥s✐♦♥❛❧ ♥ã♦ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✱ ♠❛s s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦ ❞❡ ❝♦♠✉t❛çã♦ ✭✶✳✶✮✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✸ ✐♠♣❧✐❝❛ ♥❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛✱
(∆ˆxi)(∆ˆxj)≥1/2|θij| ✭✶✳✷✮
✐st♦ é✱ ❛ ♣❛rtí❝✉❧❛ ❞❡s❝r✐t❛ ♣❡❧❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛✱ ψ✱ ♥ã♦ ♣♦❞❡ s❡r ❧♦❝❛❧✐③❛❞❛ ❞❡ ❢♦r♠❛
♣r❡❝✐s❛✳ ❊ss❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ ♣❛r❛ ❛ ♣♦s✐çã♦ ✐♠♣❧✐❝❛ ✉♠❛ ✐♠♣r❡❝✐sã♦ ❡♠ ❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♥♦ ❡s♣❛ç♦✿ ❞✐③❡♠♦s q✉❡ ♦ ❡s♣❛ç♦ é ❢✉③③②✱ ❜♦rr❛❞♦ ♦✉ t❡♠ ✉♠❛ ❡str✉t✉r❛ q✉â♥t✐❝❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛❬✶✺❪✳ ❖❜✈✐❛♠❡♥t❡✱ ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ só ♣♦❞❡ s❡ ♠❛♥✐❢❡st❛r ♥✉♠ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ❝♦♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞✐♠❡♥sõ❡s✳
❯♠❛ ✈❡③ q✉❡ ✉♠❛ ❡str✉t✉r❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦ é ♦❜s❡r✈❛❞❛ ❡♠ ❡s❝❛❧❛ ♠❛❝r♦s❝ó♣✐❝❛✱ é ❡s♣❡r❛❞♦ q✉❡ ♦ ♣❛râ♠❡tr♦ ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡✱θ✱ ❞❡✈❛ s❡ ♠❛♥✐❢❡st❛r
♥✉♠❛ ❡s❝❛❧❛ ❞♦ q✉❛❞r❛❞♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ P❧❛♥❝❦✹✱ λ2 = (G~/c3)✳ ❆ss✐♠✱ ❛ ♥ã♦✲
❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❡s♣❛ç♦ ♣♦❞❡ ❡st❛r r❡❧❛❝✐♦♥❛❞❛ ❛ ❞✐stâ♥❝✐❛s ♠✉✐t♦ ❝✉rt❛s ❡ ❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ◆ã♦✲❈♦♠✉t❛t✐✈❛✱ ✭▼✳◗✳◆✳❈✮✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠❛ ❞❡❢♦r♠❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛❬✽❪✳ ❉❡st❡ ♣♦♥t♦ ❞❡ ✈✐st❛✱ ❯♠❛ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❛ ●r❛✈✐❞❛❞❡ ❞❡✈❡ ❢♦r♥❡❝❡r ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ♠❛✐s ❝♦♠♣❧❡t❛ ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✳
❖s ❡❢❡✐t♦s ♠❛t❡♠át✐❝♦s ❡ ❢ís✐❝♦s ❝❛✉s❛❞♦s ♣❡❧❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ sã♦ ❞❡s❝♦♥❤❡❝✐❞♦s ❡ ❛✐♥❞❛ é t❡♠❛ ❞❡ ❡st✉❞♦✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ▼✳◗✳◆✳❈✳✱ ❡st❛♠♦s ✐♥✲ t❡r❡ss❛❞♦s ❡♠ ❡♥❝♦♥tr❛r ❝♦♥s❡q✉ê♥❝✐❛s ❢❡♥♦♠❡♥♦❧ó❣✐❝❛s ❞❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠ ❡s♣❛ç♦ ♥ã♦✲ ❝♦♠✉t❛t✐✈♦✳ ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ ♥♦ ❝❛s♦ ❞❡ s✐st❡♠❛s ❞❡ ♣❛rtí❝✉❧❛ ú♥✐❝❛✱ é ♦ ❞❡ q✉❡ ❛
✸❑✳ ❍✳ ❆✳ ❙❝❤✇❛r③ ✭✶✽✹✸✲✶✾✷✶✮✱ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦✳
✹
♠♦❞✐✜❝❛çã♦ ❞❛ á❧❣❡❜r❛ ♦❜❡❞❡❝✐❞❛ ♣❡❧♦s ♦♣❡r❛❞♦r❡s ❞❡ ♣♦s✐çã♦ ❛❣❡ ❝♦♠♦ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❙❝❤r♦❞✐♥❣❡r ❝♦♠ ❛ ✐❞❡✐❛ ❞♦ ♣r♦❞✉t♦ ▼♦②❛❧ t❛❧ ❝♦♠♦ é ❢❡✐t♦ ❝♦♠ ❝♦♦r✲ ❞❡♥❛❞❛s ❝❧áss✐❝❛s ♣❛r❛ s✐♠✉❧❛r ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♠ ❚❡♦r✐❛ ◗✉â♥t✐❝❛ ❞❡ ❈❛♠♣♦s✳ ◆♦ ❝❛s♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❡❧❛ ❞❡stró✐ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ♦❜s❡r✈á✈❡✐s ❞❡ ♣♦s✐çã♦✳
✶✳✷ ❊str✉t✉r❛ ❞❛ ❉✐ss❡rt❛çã♦
❊ss❡ tr❛❜❛❧❤♦ ❞❡ ❞✐ss❡rt❛çã♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ❆❧é♠ ❞♦ ❝♦♥t❡①t♦ ❤✐stór✐❝♦ ❡ ♠♦t✐✈❛çõ❡s ❬✶✺❪✱ ❬✾❪✱ ❛❜♦r❞❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦ ❞❡ ✐♥tr♦❞✉çã♦✱ ♥♦ ❝❛♣ít✉❧♦ ✷ s❡rá ❢❡✐t❛ ✉♠❛ r❡✈✐sã♦ ❣❡r❛❧ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♥✉♠❛ ❞❡s❝r✐çã♦ ♠❛✐s ❢♦r♠❛❧ ❡ ✉♠❛ ♣♦✉❝♦ ♠❛✐s ❛❜r❛♥❣❡♥t❡❬✶✺❪✱ ❬✶✽❪✱ ❬✽❪✳ ◆♦ ❝❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞✐❢❡r❡♥t❡ ❝♦♠ ✉♠❛ ♣♦ssí✈❡❧ r❡♣r❡s❡♥t❛çã♦ ❞❛ á❧❣❡❜r❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♦♥❞❡ ♦ ♦❜❥❡t✐✈♦ é ❛ ❜✉s❝❛ ❞❡ ❡st❛❞♦s q✉❡ s❛t✉r❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞❛s r❡❧❛çõ❡s ❞❡ ✐♥❝❡rt❡③❛ ❡♥tr❡ ♣♦s✐çõ❡s ❡ ♠♦♠❡♥t♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ❬✷✵❪✱ ❬✷✶❪✳ ❊♠ s❡❣✉✐❞❛✱ ♥♦ ❝❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❢♦r♠✉❧❛çã♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛ ♥✉♠❛ ❢♦r♠❛ ❢✉♥❝✐♦♥❛❧ ✉s❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦s✐çõ❡s ♠é❞✐❛s t♦♠❛❞❛s ❡♠ r❡❧❛çã♦ ❛ ❡st❛❞♦s ❝♦❡r❡♥t❡s✱ ❬✷✾❪✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛❜♦r❞❛r ❛❧❣✉♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❝✐♥❡♠át✐❝❛s ❞❡ss❛ t❡♦r✐❛ ❡ ❛♥❛❧✐s❛r ❛❧❣✉♥s ❛s♣❡❝t♦s ❢ís✐❝♦s ❛ ♣❛rt✐r ❞♦ ♣r♦♣❛❣❛❞♦r ❞❡ ❋❡②♥♠❛♥ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ♣❛rtí❝✉❧❛ ú♥✐❝❛✳ ❊✱ ♣♦r ✜♠✱ ♦ ❝❛♣ít✉❧♦ ✺ é ❞❡✐①❛❞♦ ♣❛r❛ ❛s ❞❡✈✐❞❛s ❝♦♥❝❧✉sõ❡s ❡ ♣❡rs♣❡❝t✐✈❛s ❢✉t✉r❛s✳
✶✳✸ ❈♦♥tr✐❜✉✐çõ❡s ❖r✐❣✐♥❛✐s
❈❛♣ít✉❧♦ ✷
❆s♣❡❝t♦s ◆ã♦✲❈♦♠✉t❛t✐✈♦s ♥♦ ♣❧❛♥♦
❆♥❛❧♦❣❛♠❡♥t❡ à ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ♣❛❞rã♦✱ ❛ ▼✳◗✳◆✳❈✳ é ❢♦r♠✉❧❛❞❛ ❝♦♠♦ ✉♠ s✐s✲ t❡♠❛ q✉â♥t✐❝♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❞❡ ♦♣❡r❛❞♦r❡s ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❛❣✐♥❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛❜♦r❞❛❞❛s ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s✱ s♦❜r❡t✉❞♦ ♥❛ s❡❝çã♦ ✭✷✳✶✮✱ ♥♦ ❡s♣❛ç♦ ❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s ♥ã♦✲❝♦♠✉t❛t✐✈♦s ❛♣❡♥❛s ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✳ ❆q✉✐✱ ❢❛r❡♠♦s ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛s ❝♦♦r❞❡♥❛❞❛s ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱ x ❡ y✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ xi ❡ xj ♥♦ ❡s♣❛ç♦ ❞❡
❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✳ ❆♣❡s❛r ❞❡ ❛♣r❡s❡♥t❛r♠♦s ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❡ t❛❧ ❡str✉t✉r❛ ♦♣t❛♠♦s ♣♦r ♣r♦♣♦r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❝♦♠ ❛ ✐❞❡✐❛ ❞❡ s❡ tr❛❜❛❧❤❛r ❝♦♠ ♣♦s✐çõ❡s ♠é❞✐❛s ❡♠ ❡st❛❞♦s ❝♦❡r❡♥t❡s ♥❡st❡ ❡s♣❛ç♦✱ q✉❡ ❞✐s❝✉t✐r❡♠♦s ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ♥♦ ❝❛♣ít✉❧♦ ✹✳
✷✳✶ ❆s♣❡❝t♦s ◆ã♦✲❈♦♠✉t❛t✐✈♦s ♥♦ P❧❛♥♦
❆ ❡str✉t✉r❛ ♠❛t❡♠át✐❝❛ ❞❡ss❛s t❡♦r✐❛s é ✉♠ t❛♥t♦ q✉❛♥t♦ s♦✜st✐❝❛❞❛✱ ❤á ♠✉✐t♦s tr❛❜❛✲ ❧❤♦s s♦❜r❡ ♦ ❛ss✉♥t♦✳ ❯♠ ❢♦r♠❛❧✐s♠♦ ❡❧❡❣❛♥t❡ ❡ r✐❣♦r♦s♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠❬✶✽❪✱ ❬✶✺❪✱ ❬✶✻❪✱ ❬✷✼❪✱ ♣♦r ❡①❡♠♣❧♦✱ ❡st❡s ❞♦✐s ú❧t✐♠♦s ❛❜♦r❞❛♥❞♦ ✉♥✐❝✐❞❛❞❡ ❡ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡✳ ❉❡s❞❡ ♦s ♣r✐♠ór❞✐♦s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♦ ❡♠♣r❡❣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❛ss♦❝✐❛❞♦s ❛ ♦❜s❡r✈á✈❡✐s ❢ís✐❝♦s s❡ t♦r♥♦✉ ❞❡ ❣r❛♥❞❡ ❛❥✉❞❛ ♥❛ ❜✉s❝❛ ♣❡❧♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ❧❡✐s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♥❛t✉r❡③❛✳ ❆ ♣r❡❞✐çã♦ ❞❡ r❡s✉❧t❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s ❛❞q✉✐r✐✉ ✉♠❛ ♥❛t✉r❡③❛ ♣r♦❜❛❜✐❧íst✐❝❛ ❞❡ ❝❛rát❡r ❢✉♥❞❛♠❡♥t❛❧✱ ❝♦♥s✐st❡♥t❡♠❡♥t❡ ❝♦♠ ❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ ❞❡ ❍❡✐s❡♥❜❡r❣✱ ❛ q✉❛❧ ✐♠♣õ❡ ✉♠ ❧✐♠✐t❡ ❡ss❡♥❝✐❛❧ ❛♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❡st❛❞♦s ❞♦s ♦❜s❡r✈á✈❡✐s ❢ís✐❝♦s ❡♠ q✉❡stã♦✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❡st❛❞♦s ❞❡✐①❛♠ ❞❡ s❡r ❞❡s❝r✐t♦s ♣♦r ♣♦♥t♦s ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❡ ♣❛ss❛♠ ❛ s❡r ❞❡s❝r✐t♦s ♣♦r r❡❣✐õ❡s ❞❡ss❡ ❡s♣❛ç♦ ❞❡ ár❡❛ ♠í♥✐♠❛ ❞❛ ♦r❞❡♠ ❞❡ ~✳ ❊ss❛
✻
r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ é ♠♦❞❡❧❛❞❛✱ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ♠❡✲ ❞✐❞❛ ❞❡ ♦❜s❡r✈á✈❡✐s✶✱ ♣❡❧❛ ✐♠♣♦s✐çã♦ ❞❡ q✉❡ ❝♦♦r❞❡♥❛❞❛ ❡ s❡✉ ♠♦♠❡♥t♦ ❝❛♥♦♥✐❝❛♠❡♥t❡
❝♦♥❥✉❣❛❞♦ ♥ã♦ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✱ ✐st♦ é✱
[ˆxi,pˆj] =i~δij , i, j = 1,2.
❆❣♦r❛ ❝♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ r❡str✐t♦ ❛ ❞✉❛s ❞✐♠❡♥sõ❡s✱ ♦♥❞❡ ❛s ❝♦♦r✲ ❞❡♥❛❞❛sxi s❛t✐s❢❛③❡♠ à r❡❧❛çã♦ ❞❡ ❝♦♠✉t❛çã♦✱ s❡❣✉♥❞♦ ✐♥tr♦❞✉③✐❞♦ ♥♦ ❝❛♣✳ ✶✱ ❡q✳ ✭✶✳✶✮✱
[ˆxi,xˆj] =iθǫij , i, j = 1,2. ✭✷✳✶✮
♦♥❞❡ θ ≥ 0 é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧ q✉❡ ♠❡❞❡ ❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♥tr❡ ❛s ❞✉❛s ❝♦♦r✲
❞❡♥❛❞❛s ❡ ǫij ❂−ǫji ✭♦♥❞❡ θ = 0 r❡❝✉♣❡r❛ ❛ á❧❣❡❜r❛ ♣❛❞rã♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❡♠ q✉❡ ❛s
❝♦♠♣♦♥❡♥t❡s ❞♦ ♦♣❡r❛❞♦r ♣♦s✐çã♦ ❝♦♠✉t❛♠✮✳ ❈♦♥s✐❞❡r❛r ❛ ❡①✐stê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦θ6= 0
s✐❣♥✐✜❝❛ ✐♥tr♦❞✉③✐r ♣❡q✉❡♥♦s ❞❡s✈✐♦s ♥♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s q✉❡ ♣♦ss✉❛♠ ❜♦❛ ❝♦♥❝♦r❞â♥✲ ❝✐❛ ❡①♣❡r✐♠❡♥t❛❧ ❡ ♣♦ss✐❜✐❧✐t❛r ❛ss✐♠✱ ❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♠ ♣r♦❜❧❡♠❛✳ ❆❧❣✉♠❛s r❡❢❡rê♥❝✐❛s s♦❜r❡ ❛s♣❡❝t♦s ❣❡r❛✐s ❢❡♥♦♠❡♥♦❧ó❣✐❝♦s ♣♦❞❡♠ s❡r ✈✐st❛s ❡♠ ❬✹❪✱ ❬✶✼❪✳
❆✜♠ ❞❡ ❡♥❝♦♥tr❛r ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♣❛❞rã♦✱ é ❝♦♥✈❡♥✐❡♥t❡ ❞❡✜✲ ♥✐r ♦s ♦♣❡r❛❞♦r❡s ❞❡ ❝r✐❛çã♦✴❛♥✐q✉✐❧❛çã♦ˆb❡ˆb†❡♠ ❢❛✈♦r ❞♦s ♦♣❡r❛❞♦r❡sxˆ
iq✉❡ r❡♣r❡s❡♥t❛♠
❛s ❝♦♦r❞❡♥❛❞❛s ♥♦ ❡s♣❛ç♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✱
ˆb= √1
2θ(ˆx1+ixˆ2), ✭✷✳✷✮
ˆb†= √1
2θ(ˆx1−ixˆ2) ✭✷✳✸✮
s❛t✐s❢❛③❡♥❞♦ ❛ á❧❣❡❜r❛ ❞❡ ❋♦❝❦✷✱
[ˆb,ˆb†] = 1. ✭✷✳✹✮
■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♣❛❞rã♦ é ✐s♦♠ór✜❝♦ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞❡ ❋♦❝❦✱ ♦♥❞❡ ❛♦ ❢❛❧❛r ❞❡ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♣❛❞rã♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠❬✶✽❪ s✉❜❡♥t❡♥❞❡✲s❡ ♦ ❢♦r♠❛❧✐s♠♦ ✉s✉❛❧✱ ♦✉ s❡❥❛✱ é ❢❡✐t❛ ✉♠❛ ❛♥❛❧♦❣✐❛ à á❧❣❡❜r❛ ❞❡ ♦♣❡r❛❞♦r❡s ❞♦ ♦s❝✐❧❛❞♦r ❤❛r♠ô♥✐❝♦ ♥♦ ❡s♣❛ç♦ ♦♥❞❡ ❛s ❝♦♦r❞❡♥❛❞❛sxˆi ❝♦♠✉t❛♠✳ ❊♥tã♦✱ ❝♦♠ ❡ss❛ ✐❞❡✐❛ ❡♠ ♠❡♥t❡✱
♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❛♥á❧♦❣♦ ❛♦ ❡s♣❛ç♦ ♣❛❞rã♦ é ♣♦st♦ ❝♦♠♦
Hp ∼=F ≡span
|
ni=
ˆb†n
√
n! |0i
n=∞
n=0
✭✷✳✺✮
✼
♦♥❞❡ ♦ span é t♦♠❛❞♦ s♦❜ t♦❞♦s ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ♦♥❞❡ |0i é ♦ ❡st❛❞♦ ✈á❝✉♦
❛♥✐q✉✐❧❛❞♦ ♣♦r ˆb✱ ♦✉ s❡❥❛✱ ˆb|0i = 0✳ ❆ss✐♠✱ ❛ ❢♦r♠❛ ✭✷✳✺✮ é ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦
♣❛❞rã♦✱ Hp✳ ❆ ❡ss❛ ❛❧t✉r❛ ❞❡✈❡♠♦s ♥♦t❛r q✉❡✱ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♣❛râ♠❡tr♦ ♥ã♦✲
❝♦♠✉t❛t✐✈♦✱ θ ✭q✉❡ s❡ ♣r❡s✉♠❡ s❡r ❞❛ ♦r❞❡♠ ❞♦ q✉❛❞r❛❞♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❡
P❧❛♥❝❦✮✱ s❡r ♠✉✐t♦ ♣❡q✉❡♥♦✱ ♦s ❡❢❡✐t♦s ❞❡ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ ✐r✐❛♠ s❡ ♠❛♥✐❢❡st❛r ❡♠ ❡s❝❛❧❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠✉✐t♦ ❝✉rt♦✳ ❊♥tã♦✱ ✈❡♥❞♦ ❞❡ss❛ ❢♦r♠❛✱ ♥ã♦ é s❡♥s❛t♦ ❢❛❧❛r ❛ ♥í✈❡❧ ❝❧áss✐❝♦✱ ❥á q✉❡ q✉❛❧q✉❡r ✐♥❝❡rt❡③❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♥ã♦✲❝♦♠✉t❛t✐✈✐❞❛❞❡ s❡ ♠❛♥✐❢❡st❛r✐❛ ♥✉♠❛ ❡s❝❛❧❛ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ♠❡♥♦r q✉❡ ❛s ✐♥❝❡rt❡③❛s q✉❡ sã♦ ♥❛t✉r❛❧♠❡♥t❡ ✐♥❡r❡♥t❡s às ♠❡❞✐❞❛s ❝❧áss✐❝❛s✳
❊♠ s❡❣✉✐❞❛✱ ♦ ♣ró①✐♠♦ ♣❛ss♦ é ✐♥tr♦❞✉③✐r ♦ ❡q✉✐✈❛❧❡♥t❡ ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❞❡ ❢✉♥çõ❡s ❞❡ q✉❛❞r❛❞♦ ✐♥t❡❣rá✈❡❧ ❡♠ q✉❡ ♦s ❡st❛❞♦s ❢ís✐❝♦s ❞♦ ♥♦✈♦ s✐st❡♠❛ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❞❡✜♥✐r ♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦s q✉â♥t✐❝♦s✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦✱ Hq✱ ❡ q✉❡ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❡s♣❛ç♦ L2✳ ❊♥tã♦✱ ❝♦♥s✐❞❡r❡♠♦s ✉♠
❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛❞♦r❡s ❍✐❧❜❡rt✲❙❝❤♠✐❞t✱B(Hp)✱ ❡♠Hp✱ ❛❣✐♥❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦
♥ã♦✲❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦Hq é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ss❡s ♦♣❡r❛❞♦r❡s t❛✐s q✉❡✱
Hq =
ψ(ˆx1,xˆ2) :ψ(ˆx1,xˆ2)∈ B(Hp), trp ψ†(ˆx1,xˆ2)ψ(ˆx1,xˆ2)
<∞ ✭✷✳✻✮ ♦♥❞❡
trpψ(ˆx1,xˆ2)≡ ∞ X
n=0
hn|ψ(ˆx1,xˆ2)|ni ✭✷✳✼✮
é ♦ tr❛ç♦❬✷✹❪ s♦❜Hp✳
◆♦t❡ q✉❡ ❡♠ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤r♦❞✐♥❣❡r✸✱ ❛s ❢✉♥çõ❡s ❞❡ q✉❛❞r❛❞♦
✐♥t❡❣rá✈❡❧ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ ♣♦s✐çã♦ sã♦ s✉❜st✐t✉í❞❛s ♣♦r ♦♣❡r❛❞♦r❡s ❞❡ tr❛ç♦ ✜♥✐t♦✱ q✉❡ sã♦ ❢✉♥çõ❡s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞❡ ♣♦s✐çã♦ ❞❛ ❢♦r♠❛ ✭✷✳✶✮✳ ❖s ❡st❛❞♦s q✉â♥t✐❝♦s ✭♦♣❡r❛❞♦r❡s✮ ❞♦ s✐st❡♠❛ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r ❡❧❡♠❡♥t♦s ❞❡ Hq✳ ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❛ss♦❝✐❛❞♦ ❛ ❡st❡
❡s♣❛ç♦ é
(φ(ˆx1,xˆ2), ψ(ˆx1,xˆ2)) =trp
φ†(ˆx1,xˆ2)ψ(ˆx1,xˆ2)
. ✭✷✳✽✮
❙♦❜r❡ ❛ ♥♦t❛çã♦❬✶✽❪✱ ♦s ❡st❛❞♦s ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ sã♦ ❞❛✲ ❞♦s ♣♦r |·i✱ ♥❛ ♥♦t❛çã♦ ❞❡ ❉✐r❛❝✹✳ ❖s ❡st❛❞♦s ❞❡ H
q ♣♦r ♦✉tr♦ ❧❛❞♦ sã♦ ❞❡♥♦t❛❞♦s ♣♦r
ψ(ˆx1,xˆ2) ≡ |ψ)✱ ❝♦♠ s❡✉ ❞✉❛❧ (ψ|✱ q✉❡ ♠❛♣❡✐❛ ❡❧❡♠❡♥t♦s ❞❡ Hq ♥♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
♣♦r
(φ|ψ) = (φ, ψ) = trp(φ†, ψ). ✭✷✳✾✮
✽
❆❣♦r❛ ❛ á❧❣❡❜r❛ ❞❡ ❍❡✐s❡♥❜❡r❣ é s✉❜st✐t✉í❞❛ ♣❡❧❛ á❧❣❡❜r❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛✱ q✉❡ ♥♦ ♣❧❛♥♦ t❡♠♦s✿
[ˆxi,pˆj] =i~δij ✭✷✳✶✵✮
[ˆxi,xˆj] =iθǫij i, j = 1,2 ✭✷✳✶✶✮
[ˆpi,pˆj] = 0 ✭✷✳✶✷✮
♦♥❞❡xˆi✱xˆj✱ pˆi✱ pˆj sã♦ t♦❞♦s ❤❡r♠✐t✐❛♥♦s✳
❊♠ s❡❣✉✐❞❛ ♦ ♣r♦❜❧❡♠❛ é ❛❝❤❛r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✉♥✐tár✐❛✱ ❡♠ Hq✱ ❞❛ á❧❣❡❜r❛ ♥ã♦✲
❝♦♠✉t❛t✐✈❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮✱ ✭✷✳✶✷✮✳ Pr✐♠❡✐r♦✱ ♣❛r❛ ❢❛③❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♠❛✐s ❡①♣❧í❝✐t❛ s♦❜r❡ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦ ❛❣❡ ♥❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ψ(ˆx1,xˆ2)✱ ❝♦♥s✐❞❡r❡
✉♠❛ ❢✉♥çã♦ q✉❛❧q✉❡r ψ(ˆx1,xˆ2)∈ Hq✱ ❡❧❛ ♣♦❞❡ s❡r ❡①♣❛♥❞✐❞❛ ❝♦♠♦
ψ(ˆx1,xˆ2) = ∞ X
m,n=0
cm,nxˆm1 xˆn2 , cm,n ∈C. ✭✷✳✶✸✮
❆♣ós ♦r❞❡♥❛çã♦ ❛❞❡q✉❛❞❛✱ ✈❡❥❛ ♠❛✐s ❞❡t❛❧❤❡s ♥♦ ❛♣ê♥❞✐❝❡ ❉✱ ❛ ❛çã♦ ❞♦ ♦♣❡r❛❞♦rPˆ1
♥❡st❡ ❡st❛❞♦✱ ♦♥❞❡i, j = 1,2✱ é ˆ
P1ψ(ˆx1,xˆ2) = −i~
∂ ∂xˆ1
ψ(ˆx1,xˆ2)
= ~
θ(−iθ)
∞ X
m,n=0
cm,nmxˆm1 −1xˆn2
= ~
θ[ˆx2, ψ(ˆx1,xˆ2)] ✭✷✳✶✹✮
❡♠ q✉❡ ♦ ♠❡s♠♦ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ ❛ ❛çã♦ ❞♦ ♦♣❡r❛❞♦rPˆ2 ♥❛ ♠❡s♠❛ ❢✉♥çã♦✳
❊♥tã♦✱ ✉s❛♥❞♦ ❧❡tr❛s ♠❛✐ús❝✉❧❛s ♣❛r❛ ❞✐st✐♥❣✉✐r ♦♣❡r❛❞♦r ❛❣✐♥❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦ ❞❛q✉❡❧❡ ♦♣❡r❛❞♦r q✉❡ ❛❣❡ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ❡ ❛tr❛✈és ❞❡ ✉♠❛ ♦r❞❡♥❛çã♦ ❛❞❡q✉❛❞❛ ❝♦♥❝❧✉✐✲s❡ q✉❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤r♦❞✐♥❣❡r ❛♥á❧♦❣❛ à r❡♣r❡s❡♥t❛çã♦ ♣❛❞rã♦ ♣♦❞❡ s❡r ♣♦st❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
ˆ
Xiψ(ˆx1,xˆ2) = ˆxiψ(ˆx1,xˆ2), ✭✷✳✶✺✮
ˆ
Piψ(ˆx1,xˆ2) =
~
θǫij[ˆxj, ψ(ˆx1,xˆ2)] ✭✷✳✶✻✮
♦♥❞❡ ψ(ˆx1,xˆ2) ∈ Hq✳ ❆✐♥❞❛ ✉s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ ❬✶✽❪✱ r❡s❡r✈❛♠♦s ❛ ♥♦t❛çã♦ (†) ♣❛r❛
❞❡♥♦t❛r ❝♦♥❥✉❣❛çã♦ ❤❡r♠✐t✐❛♥❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦ ❝♦♠✉t❛t✐✈♦ ❡ ✭‡✮ ♣❛r❛ ❝♦♥✲ ❥✉❣❛çã♦ ❤❡r♠✐t✐❛♥❛ ♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❢❛③❡r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ♣♦s✐çã♦✱
ˆ
B ≡ √1
2θ
ˆ
X1+iXˆ2
✾
ˆ
B‡≡ √1
2θ
ˆ
X1 −iXˆ2
✭✷✳✶✽✮ ♦♥❞❡
ˆ
Bψ(ˆx1,xˆ2) = ˆbψ(ˆx1,xˆ2), ✭✷✳✶✾✮
ˆ
B‡ψ(ˆx1,xˆ2) = ˆb†ψ(ˆx1,xˆ2) ✭✷✳✷✵✮
❡ t❛♠❜é♠ ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦✱
ˆ
P ≡ Pˆ1+iPˆ2, ✭✷✳✷✶✮
ˆ
P‡ ≡ Pˆ1−iPˆ2 ✭✷✳✷✷✮
❡♠ q✉❡
ˆ
P ψ(ˆx1,xˆ2) = −i~ r
2
θ[ˆb, ψ(ˆx1,xˆ2)], ✭✷✳✷✸✮
ˆ
P‡ψ(ˆx1,xˆ2) = i~ r
2
θ[ˆb
†, ψ(ˆx
1,xˆ2)] ✭✷✳✷✹✮
❡ ♦♥❞❡Pˆ2 = ˆP2
1 + ˆP22 = ˆP‡Pˆ= ˆPPˆ‡✳ P♦❞❡♠♦s ✈❡r q✉❡ ❞❡ ✭✷✳✷✶✮ ❡ ✭✷✳✷✷✮✱
[ ˆP ,Pˆ‡] = 0. ✭✷✳✷✺✮
❏á q✉❡ ♦ ❝♦♠✉t❛❞♦r ❞❛s ✈❛r✐á✈❡✐s ♥ã♦✲❝♦♠✉t❛t✐✈❛s é ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✱ ✭✷✳✶✮✱ é ✐♠♣♦s✲ sí✈❡❧ r❡❛❧✐③❛r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♠❡❞✐❞❛s ❞❛s ♣♦s✐çõ❡s x1 ❡ x2 ❝♦♠ ❜♦❛ ♣r❡❝✐sã♦✳ ❖ ♠❡❧❤♦r
q✉❡ s❡ ♣♦❞❡ ❢❛③❡r é ❝♦♥str✉✐r ✉♠ ❡st❛❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ♣❛r❛ ♦ q✉❛❧ ♦ ♣r♦❞✉t♦ ❞❛s ✐♥❝❡rt❡③❛s s❡❥❛ ♠í♥✐♠❛✳ ❆ ♥♦çã♦ ❞❡ ♣♦s✐çã♦ ❛q✉✐ é ♠❛♥t✐❞❛ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ✉♠❛ ♣❛rtí❝✉❧❛ é ❧♦❝❛❧✐③❛❞❛ ❡♠ t♦r♥♦ ❞❡ ✉♠ ❝❡rt♦ ♣♦♥t♦❬✶✽❪✳
❊♠ t❡r♠♦s ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ♦s ❡st❛❞♦s ❞❡ ✐♥❝❡rt❡③❛ ♠í♥✐♠❛ ♥❛ ♣♦s✐çã♦ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ❡st❛❞♦s ❝♦❡r❡♥t❡s ♥♦r♠❛❧✐③❛❞♦s❬✸❪✱
|zi = e−z2z¯ezˆb†|0i
= e−z2¯z
nX=∞
n=0
1
√
n!z
n|ni ✭✷✳✷✻✮
♦♥❞❡z = √1
2θ(x1+ix2) é ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❛❞✐♠❡♥s✐♦♥❛❧✱ ❡♠ q✉❡
ˆb|zi = z|zi, ✭✷✳✷✼✮
✶✵
❆ss✐♠✱ ❞❛s ❞❡✜♥✐çõ❡s ✭✷✳✷✮ ❡ ✭✷✳✸✮ s❡ ♦❜té♠
ˆ
x1 = r
θ
2(ˆb+ ˆb
†), ✭✷✳✷✾✮
ˆ
x2 = i r
θ
2(ˆb
†−ˆb) ✭✷✳✸✵✮
❝♦♠ ✐ss♦ ❛s ♠é❞✐❛s ❞❡ss❡s ♦♣❡r❛❞♦r❡s sã♦✿
hxˆ1i = r
θ
2hz|(ˆb+ ˆb
†)|zi= r
θ
2(z+ ¯z), ✭✷✳✸✶✮
hxˆ2i = i r
θ
2hz|(ˆb
†−ˆb)|zi=i
r
θ
2(¯z−z) ✭✷✳✸✷✮
❡
hxˆ1i2 =
θ
2hz|(ˆb+ ˆb
†)2|zi= θ
2(z
2+ ¯z2+ 2zz¯+ 1), ✭✷✳✸✸✮
hxˆ2i2 = −
θ
2hz|(ˆb
†−ˆb)2|zi=−θ
2(z
2+ ¯z2−2zz¯−1). ✭✷✳✸✹✮
❈♦♠ ✐ss♦✱
(∆ˆx1)2 = hxˆ21i − hxˆ1i2 =
θ
2, ✭✷✳✸✺✮
(∆ˆx2)2 = hxˆ22i − hxˆ2i2 =
θ
2 ✭✷✳✸✻✮
♦♥❞❡ ✐♠♣❧✐❝❛ ♥❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ s❛t✉r❛❞❛✿
(∆ˆx1)(∆ˆx2) =
θ
2, ✭✷✳✸✼✮
♦✉ s❡❥❛✱ ❡ss❡s ❡st❛❞♦s ❝♦❡r❡♥t❡s ✭✷✳✷✻✮ ❡①✐❜❡♠ ✐♥❝❡rt❡③❛ ♠í♥✐♠❛ ♣❛r❛ ♦s ✈❛❧♦r❡s ❛ss♦❝✐❛❞♦s ❛♦s ♦♣❡r❛❞♦r❡sxˆ1 ❡ xˆ2✳
❊ss❡s ❡st❛❞♦s ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦✱ |ψ)✱ sã♦ ❢♦r♠❛❞♦s ♣❡❧♦ ♣r♦❞✉t♦ ❡①t❡r♥♦
❞❡ ❞♦✐s ❡st❛❞♦s ❝♦❡r❡♥t❡s ❞❛ ❢♦r♠❛ ✭✷✳✷✻✮✿
|ψ) = |zihz| ✭✷✳✸✽✮
q✉❡ sã♦ ♥♦r♠❛❧✐③❛❞♦s ❡♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✭✷✳✽✮✱ ❧♦❣♦ sã♦ ♦♣❡r❛❞♦r❡s ❍✐❧❜❡rt✲ ❙❝❤♠✐❞t✳ ❊♥tr❡t❛♥t♦✱ ♣♦❞❡♠♦s ♥♦t❛r ❛ s✉❛ ♥ã♦✲♦rt♦❣♦♥❛❧✐❞❛❞❡✱
(z1|z2) = trc
(|z1ihz1|)‡(|z2ihz2|)
= |e−z1 ¯2z1− z2 ¯z2
✶✶
♦♥❞❡ z1 ❡ z2 sã♦ ❛❞✐♠❡♥s✐♦♥❛✐s✱ ❡ ❛ ❣❛✉ss✐❛♥❛ s❡ t♦r♥❛rá ✉♠❛ ❢✉♥çã♦ ❞❡❧t❛ ❞❡ ❉✐r❛❝ ♥♦
❧✐♠✐t❡ ❝♦♠✉t❛t✐✈♦θ −→0✳ ❖s ❡st❛❞♦s ✭✷✳✸✽✮ t❛♠❜é♠ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡✱ ˆ
B|ψ) = ˆb|zihz|=e−z2z¯[ˆb, ezˆb†]|0ihz|
= ze−z2z¯ezˆb†|0ihz|
= z|ψ), ✭✷✳✹✵✮
♦✉ s❡❥❛✱ ♦ ♦♣❡r❛❞♦r Bˆ✱ ❞❡✜♥✐❞♦ ❡♠ t❡r♠♦s ❞♦s ♦♣❡r❛❞♦r❡s Xˆ1 ❡ Xˆ2✱ q✉❛♥❞♦ ❛♣❧✐❝❛❞♦
❛♦ ❡st❛❞♦ |ψ) q✉❡ ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦✱ ♥♦s ❢♦r♥❡❝❡ ♦ ✈❛❧♦r z✱ ♦♥❞❡ z= √1
2θ(x1+ix2)✳
❉❡ss❛ ❢♦r♠❛ ♣♦❞❡✲s❡ r❡❛❧✐③❛r ♥♦✈♦s ❝á❧❝✉❧♦s ❞❛s ♠é❞✐❛s ❝♦♠♦ ❢♦✐ ❢❡✐t♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣❛r❛ ♦❜t❡r ✉♠❛ r❡❧❛çã♦ ❝♦♠♦ ✭✷✳✸✼✮ ✳ ❘❡s♦❧✈❡♥❞♦ ♣❛r❛ Xˆ1 ❡ Xˆ2 ❡♠ ✭✷✳✶✼✮ ❡ ✭✷✳✶✽✮✱ ♦✉
s❡❥❛✱
ˆ
X1 = r
θ
2( ˆB+ ˆB
‡), ✭✷✳✹✶✮
ˆ
X2 = i r
θ
2( ˆB
‡−Bˆ). ✭✷✳✹✷✮
❈♦♠ ✐ss♦ ❛s ♠é❞✐❛s ❞❡ss❡s ♦♣❡r❛❞♦r❡s t♦♠❛❞❛s ❡♠ r❡❧❛çã♦ ❛♦s ❡st❛❞♦s ♠♦❞✐✜❝❛❞♦s ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt q✉â♥t✐❝♦ sã♦✿
hXˆ1i|ψ) = r
θ
2(z|( ˆB + ˆB
†)|z) = r
θ
2(z+ ¯z), ✭✷✳✹✸✮
hXˆ2i|ψ) = i r
θ
2(z|( ˆB
†−Bˆ)|z) =i r
θ
2(¯z−z) ✭✷✳✹✹✮
❡
hXˆ1i2|ψ) =
θ
2(z|( ˆB+ ˆB
†)2|z) = θ
2(z
2+ ¯z2+ 2zz¯+ 1), ✭✷✳✹✺✮
hXˆ2i2|ψ) = −
θ
2(z|( ˆB
†−Bˆ)2|z) =−θ
2(z
2+ ¯z2−2zz¯−1) ✭✷✳✹✻✮
❝♦♠ ✐ss♦✱
(∆ ˆX1)2|ψ) = hXˆ12i|ψ)− hXˆ1i2|ψ) =
θ
2, ✭✷✳✹✼✮
(∆ ˆX2)2|ψ) = hXˆ22i|ψ)− hXˆ2i2|ψ) =
θ
2 ✭✷✳✹✽✮
♦♥❞❡ ✐♠♣❧✐❝❛ ♥❛ r❡❧❛çã♦ ❞❡ ✐♥❝❡rt❡③❛ s❛t✉r❛❞❛✿
(∆ ˆX1)(∆ ˆX2) =
θ
✶✷
P♦rt❛♥t♦✱ ♦s ❡st❛❞♦s |ψ) sã♦ ❡st❛❞♦s ❞❡ ✐♥❝❡rt❡③❛ ♠í♥✐♠❛ ♥❛ ♣♦s✐çã♦ ♥♦ ❡s♣❛ç♦ ❞❡
❍✐❧❜❡rt q✉â♥t✐❝♦✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ ❡ss❡s ❡st❛❞♦s sã♦ ♦s ❛♥á❧♦❣♦s ❞♦s ❛✉t♦✲❡st❛❞♦s ❞❡ ♣♦s✐çã♦ ❡♠Hq✱ ❥á q✉❡ ❡❧❡s s❛t✉r❛♠ ❛s r❡❧❛çõ❡s ❞❡ ✐♥❝❡rt❡③❛ ✭✶✳✷✮✳
✷✳✷ ❊✈♦❧✉çã♦ ❚❡♠♣♦r❛❧ ❡ Pr♦❞✉t♦ ❡str❡❧❛
❊♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❝♦♥✈❡♥❝✐♦♥❛❧ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ✉♠ s✐st❡♠❛✱ ♦✉ s❡❥❛✱ s✉❛ ❞✐♥â♠✐❝❛ q✉â♥t✐❝❛ é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❞❡ ❡✈♦❧✉çã♦ ❞❡ ❙❝❤r♦❞✐♥❣❡r✱
i~∂
∂tψ(x1, x2, t) = ❍ψ(x1, x2, t), ✭✷✳✺✵✮
❡♠ q✉❡✱ ❛q✉✐ ♦s ♦♣❡r❛❞♦r❡s q✉❡ r❡♣r❡s❡♥t❛♠ ❛s ♣♦s✐çõ❡s x1 ❡x2✱ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳ ❊st❛
é ❛ ❡q✉❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ♠♦✈✐♠❡♥t♦ q✉❡ ❞❡t❡r♠✐♥❛ ❝♦♠♦ ❡st❛❞♦s ❡✈♦❧✉❡♠ ♥♦ t❡♠♣♦✱ ♦♥❞❡ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❍ é✱ ❡♠ ✉♠❛ ❞✐♠❡♥sã♦✱
❍= p
2
2m +❱(x1, x2). ✭✷✳✺✶✮
▼❛s✱ ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♥ã♦✲❝♦♠✉t❛t✐✈❛✱ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ é ♣♦st♦ ❡♠ t❡r♠♦s ❞♦s ♦♣❡r❛❞♦r❡s q✉❡ r❡♣r❡s❡♥t❛♠ ❛s ❝♦♦r❞❡♥❛❞❛s✱xi, i= 1,2,· · ·, N ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦
♥ã♦✲❝♦♠✉t❛t✐✈♦ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❝♦♥❥✉❣❛❞♦s ❛ ❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱ pj, j = 1,2,· · ·, N✳
❊♥tr❡t❛♥t♦✱ t❛✐s ❝♦♦r❞❡♥❛❞❛s ❥á ♥ã♦ ♠❛✐s ❝♦♠✉t❛♠ ♦❜❡❞❡❝❡♥❞♦ às r❡❣r❛s ❞❡ ❝♦♠✉t❛çã♦ ✭✷✳✶✵✮✲✭✷✳✶✷✮✱ ♦✉ s❡❥❛✱ ❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ❡ ♠♦♠❡♥t♦ ❝♦♥t✐♥✉❛♠ ♥ã♦ ❝♦♠✉t❛♥❞♦ ♠❛s ❛❣♦r❛ ❛ ❞✐❢❡r❡♥ç❛ é q✉❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ♥ã♦ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳ ❊♥tã♦ ♦ ♥♦✈♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✜❝❛ ♥❛ ❢♦r♠❛✱ ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✱
❍(ˆx,pˆ) = pˆ
2
2m +V(ˆx1,xˆ2). ✭✷✳✺✷✮
❙✉❜st✐t✉✐♥❞♦ ❡ss❛ ✭✷✳✺✷✮ ❡♠ ✭✷✳✺✵✮ ✜❝❛♠♦s ❝♦♠✱
i~∂
∂tψ(ˆx1,xˆ2, t) =
ˆ
p2
2mψ(ˆx1,xˆ2, t) +V(ˆx1,xˆ2)⋆ ψ(ˆx1,xˆ2, t) ✭✷✳✺✸✮
❡♠ q✉❡ψ(ˆx1,xˆ2, t)é ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♥♦ ❡s♣❛ç♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ❡ ♦ ♣r♦❞✉t♦ ❝♦♥✈❡♥❝✐♦♥❛❧
é s✉❜st✐t✉í❞♦ ♣❡❧♦ sí♠❜♦❧♦(⋆)q✉❡ ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ ●ro¨♥❡✇♦❧❞✲▼♦②❛❧✱ ❝✉❥❛ ❡①♣❛♥sã♦ ❣❡r❛❧
♣❛r❛ ❞✉❛s ❢✉♥çõ❡s✱f(x) ❡ g(x)♣❡rt❡♥❝❡♥t❡ ❛♦ ❡s♣❛ç♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦❬✷✸❪ é
f(x)⋆ g(x) =
Z Z
dDk
(2π)D
dDk′
(2π)Df˜(k)˜g(k
′
−k)e−2iθijkik ′
jeik
′
ixi
= f(x)ei2∂iθij∂jg(x)
= f(x)g(x) +
∞ X
n=1
i
2
n
1
✶✸
♦♥❞❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ♥❛ ❡①♣♦♥❡♥❝✐❛❧ ❞❛ s❡❣✉♥❞❛ ♣❛ss❛❣❡♠ ❞❡ ✭✷✳✺✹✮ ❛❣❡ ❡♠
f(x) ♣❡❧❛ ❡sq✉❡r❞❛ ❡ ❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❛❣❡ ❡♠ g(x)♣❡❧❛ ❞✐r❡✐t❛✳ ❉❡ ❢♦r♠❛ q✉❡✱
♣❛r❛θ= 0✱ ♦ ♣r♦❞✉t♦ ❡str❡❧❛ s❡ r❡❞✉③ ❛♦ ♣r♦❞✉t♦ ♣❛❞rã♦✳ ❖ t❡r♠♦e−i2θijkik ′
j ❞❛ ❡①♣❛♥sã♦ ❛❝✐♠❛ ❛♣❛r❡❝❡ ❞❡✈✐❞♦ ❛♦ ❡❢❡✐t♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❝♦♠ ♣r♦❞✉t♦ ❡str❡❧❛ ❡♥tr❡ ❛s ❣❛✉ss✐❛♥❛s✱ q✉❡ s❡rá ❞✐s❝✉t✐❞♦ ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ♥♦ ❝❛♣✳ ✹✳ ❊♥tã♦✱ ♥❡ss❡ s❡♥t✐❞♦✱ t❡♠♦s ✉♠❛ r❡❛❧✐③❛çã♦ ❞❛ á❧❣❡❜r❛ ✭✷✳✶✵✮✲✭✷✳✶✷✮ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ❛ r❡♣r❡s❡♥t❛çã♦❬✷✵❪✱ ❬✶✻❪✱
ˆ
xi ≡ x˜i−
θ
2~ǫijp˜j ✭✷✳✺✺✮
ˆ
pi ≡ p˜j ✭✷✳✺✻✮
♦♥❞❡ ♦s x˜i ❡ p˜j ♦❜❡❞❡❝❡♠ às r❡❣r❛s ❞❡ ❝♦♠✉t❛çã♦ ♣❛❞rã♦✳ ▲♦❣♦✱ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✭✷✳✺✷✮
t♦♠❛ ❛ ❢♦r♠❛
H(˜xi−
θ
2~ǫijp˜j,p˜i) =
˜
pip˜i
2m +V(˜xi− θ
2~ǫijp˜j)≡Hθ(˜xi,p˜i) ✭✷✳✺✼✮
❡♠ q✉❡ ♦ ♣r♦❞✉t♦ ❡str❡❧❛ ❛♣❛r❡❝❡♥❞♦ ❡♠ ✭✷✳✺✸✮ ❡♥tr❡ ♦ ♣♦t❡♥❝✐❛❧ ❡ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♣♦❞❡ s❡r ✈✐st♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✱ s❡❣✉♥❞♦ ❛ ❡①♣❛♥sã♦ ✈✐st❛ ❛♥t❡r✐♦r♠❡♥t❡✱
V(ˆx)⋆ ψ(ˆx, t) ≡ V(ˆx)he−i~
∂
∂xiθij∂xj∂ i
ψ(ˆx, t) = V(ˆx)ψ(ˆx, t) + 1
2θij∂iV(ˆx)∂jψ(ˆx, t) +· · · = V
xi+
iθ
2ǫij
∂ ∂xj
ψ(ˆx, t) ✭✷✳✺✽✮
❝❛❧❝✉❧❛❞♦ ❛té ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥♦ ♣❛râ♠❡tr♦θ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦t❡ q✉❡ ♦ ♣r♦❞✉t♦ ❡str❡❧❛
♥ã♦ ❛❢❡t❛ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ ♣♦✐s ♥ã♦ ❤á ✐♥t❡r❛çã♦ ❡♥tr❡ ❝♦♦r❞❡♥❛❞❛s ❞❛ ♣❛rt❡ ❞♦ ♣♦t❡♥❝✐❛❧✱ ♣♦ss✉✐♥❞♦ ❛♣❡♥❛s ✐♥❢♦r♠❛çã♦ ❞♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r✱ ❧♦❣♦ ❛ ❡✈♦❧✉çã♦ ❛❝♦♥t❡❝❡ ❝♦♠♦ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣❛❞rã♦✱ ♣♦✐s ♥❡ss❛ r❡❛❧✐③❛çã♦✱ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ♦♣❡r❛❞♦r ♠♦♠❡♥t♦✱ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳
✷✳✸ ❈♦♥s❡r✈❛çã♦ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡ ❡♠ ▼✳◗✳◆✳❈✳
P♦r ❝❛✉s❛ ❞❛ ✐♥t❡r♣r❡t❛çã♦ ❡st❛tíst✐❝❛ ❞❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛✱ ❡♠ ❣❡r❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ P♦❞❡♠♦s ❞✐③❡r q✉❡|ψ(ˆx1,xˆ2, t)|2
é ❛ ❞❡♥s✐❞❛❞❡ s❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ρ =ψ†(ˆx
1,xˆ2, t)ψ(ˆx1,xˆ2, t)✱ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♣❛rtí❝✉❧❛ ♥❛
♣♦s✐çã♦ (ˆx1,xˆ2) ♥♦ t❡♠♣♦ t✳ ▼❛s ♥♦t❡ q✉❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ρ ❞❡✈❡ s❡r ✶ ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦
✶✹
✜s✐❝❛♠❡♥t❡ ❛❝❡✐tá✈❡✐s ❝♦rr❡s♣♦♥❞❡♠ às s♦❧✉çõ❡s ❞❡ q✉❛❞r❛❞♦✲✐♥t❡❣rá✈❡❧ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤r♦❞✐♥❣❡r✳ ❊♠ ♠❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ❝♦♠✉t❛t✐✈❛ ❛ ✐♥t❡❣r❛❧ ♠❡♥❝✐♦♥❛❞❛ ❞❡✈❡ s❡r ❝♦♥st❛t❡ ♥♦ t❡♠♣♦✱ ♦✉ s❡❥❛✱ ❛ ❢✉♥çã♦ ❞❡ ♦♥❞❛ ♣❡r♠❛♥❡❝❡ ♥♦r♠❛❧✐③❛❞❛ à ♠❡❞✐❞❛ q✉❡ ❡✈♦❧✉✐✳
❈♦♠ ❡ss❛ ✐❞❡✐❛ ❡♠ ♠❡♥t❡✱ ✈❛♠♦s ✈❡r ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ♥♦r♠❛ ❞♦s ♥♦✈♦s ❡st❛❞♦ |ψ)✳ ❈♦♠♦ ❢♦✐ ✈✐st♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡ss❛s ♥♦✈❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛✱ ❛♥á❧♦❣❛s às ❢✉♥çõ❡s ❞❡
q✉❛❞r❛❞♦ ✐♥t❡❣rá✈❡❧ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ♣❛❞rã♦✱ sã♦ ❢✉♥çõ❡s t❛✐s q✉❡ ♣♦ss✉❡♠ tr❛ç♦ ✜♥✐t♦ ❡q✉❛çã♦ ✭✷✳✻✮✳ ❖ ♦❜❥❡t✐✈♦ ❛q✉✐ é ♠♦str❛r q✉❡ ♦ tr❛ç♦✱ t♦♠❛❞♦ s♦❜r❛ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦ ❝♦♠✉t❛t✐✈♦✱ ❞❡ss❛s ♥♦✈❛s ❢✉♥çõ❡s ♥ã♦ ♠✉❞❛ ❝♦♠ ♦ t❡♠♣♦✳ ❊♥tã♦✱ ❞❛❞♦ ✉♠ ❤❛♠✐❧t♦♥✐❛♥♦
❍= Pˆ
2
2m +V(ˆx1,xˆ2) ✭✷✳✺✾✮
♦♥❞❡ ❛ss✉♠✐♠♦s q✉❡ ♦ ♣♦t❡♥❝✐❛❧ V(ˆx1,xˆ2)✱ ✈✐st♦ ❝♦♠♦ ✉♠ ♦♣❡r❛❞♦r ❛❣✐♥❞♦ ♥♦ ❡s♣❛ç♦ ❞❡
❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✱ s❡❥❛ ❤❡r♠✐t✐❛♥♦✱ ♦✉ s❡❥❛✱ V†(ˆx
1,xˆ2) = V(ˆx1,xˆ2) ✭♦ ❡q✉✐✲
✈❛❧❡♥t❡ ❛ ❡①✐❣✐r q✉❡ ♦ ♣♦t❡♥❝✐❛❧ s❡❥❛ r❡❛❧ ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❝♦♠✉t❛t✐✈❛✮✳ ❆ss✐♠✱ ❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧ ❞❡ ❙❝❤r♦❞✐♥❣❡r ♣❛r❛ ✉♠ ❞❛❞♦ ❡st❛❞♦ é✿
i~∂ψ(ˆx1,xˆ2, t)
∂t =❍⋆ ψ(ˆx1,xˆ2, t). ✭✷✳✻✵✮
❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❝❛s♦ ❝♦♠✉t❛t✐✈♦✱ s❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ♣♦rψ†(ˆx
1,xˆ2, t)
♣❡❧❛ ❡sq✉❡r❞❛ ❡ ♦ ❝♦♥❥✉❣❛❞♦ ❤❡r♠✐t✐❛♥♦ ❞❡ ✭✷✳✻✵✮ ♣♦rψ(ˆx1,xˆ2, t) ♣❡❧❛ ❞✐r❡✐t❛✱ t❡♠♦s
i~ψ†∂ψ
∂t =ψ
†
~2
2mθ2 ([ˆx2,[ˆx2, ψ]] + [ˆx1,[ˆx1, ψ]]) +V(ˆx1,xˆ2)⋆ ψ
✭✷✳✻✶✮ ❡
−i~∂ψ
†
∂t ψ =
~2
2mθ2 [ˆx2,[ˆx2, ψ
†]] + [ˆx
1,[ˆx1, ψ†]]
+ψ†⋆ V(ˆx1,xˆ2)
ψ ✭✷✳✻✷✮
❛❣♦r❛ s✉❜tr❛✐♥❞♦ ✭✷✳✻✶✮ ❞❡ ✭✷✳✻✷✮✿
∂ ∂t(ψ
†ψ) = ~2
2mθ2
ψ†([ˆx2,[ˆx2, ψ]] + [ˆx1,[ˆx1, ψ]]) −
− ~
2
2mθ2
[ˆx2,[ˆx2, ψ†]] + [ˆx1,[ˆx1, ψ†]] . ✭✷✳✻✸✮
❆❣♦r❛✱ ✉s❛♥❞♦ ❛s ✐❞❡♥t✐❞❛❞❡ ❞❡ ❝♦♠✉t❛❞♦r❡s✿ ❬❆✱ ❇❈❪ ❂ ❬❆✱ ❇❪❈ ✰ ❇❬❆✱ ❈❪✱ ♣♦❞❡♠♦s ♦r❣❛♥✐③❛r ♦s ❝♦♠✉t❛❞♦r❡s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✱
✶✺
❡
[ˆx1,[ˆx1, ψ†]ψ] = [ˆx1,[ˆx1, ψ†]]ψ+ [ˆx1, ψ†][ˆx1, ψ]. ✭✷✳✻✺✮
■s♦❧❛♥❞♦ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❛♣ós ❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ❝❛❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✿
ψ†[ˆx1,[ˆx1, ψ]] = [ˆx1, ψ†[ˆx1, ψ]]−[ˆx1, ψ†][ˆx1, ψ] ✭✷✳✻✻✮
❡ t❛♠❜é♠
[ˆx1,[ˆx1, ψ†]]ψ = [ˆx1,[ˆx1, ψ†]ψ]−[ˆx1, ψ†][ˆx1, ψ]. ✭✷✳✻✼✮
❖ ♠❡s♠♦ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ ♦s ❝♦♠✉t❛❞♦r❡s ❡♥✈♦❧✈❡♥❞♦ xˆ2 ❡ ❛ ❢✉♥çã♦ ψ✳ ❊ ❛♦
s✉❜st✐t✉✐r ❡ss❛s r❡❧❛çõ❡s ❡♠ ✭✷✳✻✸✮✱ ♦❜t❡r❡♠♦s
∂ ∂t(ψ
†ψ) = ~2
2mθ2
[ˆx2, ψ†[ˆx2, ψ]]−[ˆx2, ψ†][ˆx2, ψ] + [ˆx1, ψ†[ˆx1, ψ]]−[ˆx1, ψ†][ˆx1, ψ] −
− ~
2
2mθ2
[ˆx2,[ˆx2, ψ†]ψ]−[ˆx2, ψ†][ˆx2, ψ] + [ˆx1,[ˆx1, ψ†]ψ]−[ˆx1, ψ†][ˆx1, ψ]
= ~
2
2mθ2
[ˆx2, ψ†[ˆx2, ψ]]−[ˆx2,[ˆx2, ψ†]ψ] + [ˆx1, ψ†[ˆx1, ψ]]−[ˆx1,[ˆx1, ψ†]ψ] .
❊ s❡ ✜③❡r♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❝♦♠✉t❛❞♦r❡s ♦❜t❡r❡♠♦s
[ˆx2, ψ†[ˆx2, ψ]]−[ˆx2,[ˆx2, ψ†]ψ] = ˆx2ψ†[ˆx2, ψ]−ψ†[ˆx2, ψ]ˆx2−xˆ2[ˆx2, ψ†]ψ+ [ˆx2, ψ†]ψxˆ2
= ˆx2 ψ†[ˆx2, ψ]−[ˆx2, ψ†]ψ
− ψ†[ˆx2, ψ]−[ˆx2, ψ†]ψ
ˆ
x2
❡ ❝♦♠ ✐ss♦ ♣♦❞❡✲s❡ ❞❡✜♥✐r ❛ q✉❛♥t✐❞❛❞❡
j1 ≡
~2
2mθ2 ψ †[ˆx
2, ψ]−[ˆx2, ψ†]ψ
. ✭✷✳✻✽✮
❆ss✐♠✱
[ˆx2, ψ†[ˆx2, ψ]]−[ˆx2,[ˆx2, ψ†]ψ] = [ˆx2, j1]. ✭✷✳✻✾✮
❖ ♠❡s♠♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❝♦♠ ♦s ❝♦♠✉t❛❞♦r❡s q✉❡ ❡♥✈♦❧✈❡♠ xˆ1 ❡ψ ♦❜t❡♥❞♦
[ˆx1, ψ†[ˆx1, ψ]]−[ˆx1,[ˆx1, ψ†]ψ] = [ˆx1, j2] ✭✷✳✼✵✮
♦♥❞❡
j2 ≡
~2
2mθ2 ψ†[ˆx1, ψ]−[ˆx1, ψ†]ψ
. ✭✷✳✼✶✮
❉❛í ✜❝❛♠♦s ❝♦♠✿
∂ρ
∂t = [ˆx2, j1] + [ˆx1, j2] ✭✷✳✼✷✮
❡♠ q✉❡ρ ≡ ψ†ψ✳ ❖ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r é ♦ ❛♥á❧♦❣♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ♦✉ ✢✉①♦
✶✻
❡♥✈♦❧✈❡♥❞♦ ❛ ❢✉♥çã♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦ ❡ ❛s ❝♦♦r❞❡♥❛❞❛s ♥ã♦✲ ❝♦♠✉t❛t✐✈❛s ❞❛s q✉❛✐s t❛❧ ❢✉♥çã♦ ❞❡♣❡♥❞❛✳ ❚♦♠❛♥❞♦ ♦ tr❛ç♦ ❞❡ ✭✷✳✼✷✮ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çã♦ ♥ã♦✲❝♦♠✉t❛t✐✈♦✱ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡s❛♣❛r❡❝❡✱ ♣♦✐s ♣♦❞❡♠♦s t♦♠❛r ♣❡r♠✉t❛çõ❡s ❝í❝❧✐❝❛s ❝♦♠ ✉♠ tr❛ç♦ s❡♠ ❛❧t❡r❛r s❡✉ ✈❛❧♦r✳ P♦rt❛♥t♦✱
trc
∂ρ ∂t
= ∂
∂ttrc(ρ) = ∂
∂t(ψ|ψ) = 0 ✭✷✳✼✸✮
