❙♣✐♥ ❞②♥❛♠✐❝s s✐♠✉❧❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ✐♥ ◆✐ 2+ ( aq ) ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ◆▼❘ ❘❡s❡❛r❝❤ ●r♦✉♣ ❉❡♣❛rt♠❡♥t ♦❢ P❤②s✐❝s ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✱ ❋✐♥❧❛♥❞ ❚❤❡ ❊①❛❝t✉s ❞♦❝t♦r❛❧ ♣r♦❣r❛♠ ♦❢ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉ ●r❛❞✉❛t❡ ❙❝❤♦♦❧ ❥②r❦✐✳r❛♥t❛❤❛r❥✉❅♦✉❧✉✳❢✐ ❏✳ ❘❛♥t❛❤❛r❥✉✱ ❏✳ ▼❛r❡➨ ❛♥❞ ❏✳ ❱❛❛r❛✱ s✉❜♠✐tt❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✶✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
■♥tr♦❞✉❝t✐♦♥ • ❚❤❡ ❛❜✐❧✐t② t♦ q✉❛♥t✐t❛t✐✈❡❧② ♣r❡❞✐❝t ❛♥❞ ❛♥❛❧②③❡ t❤❡ r❛t❡ ♦❢ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥✐❝❛❧❧② ♦♣❡♥✲s❤❡❧❧ s②st❡♠s ✐s ✐♠♣♦rt❛♥t ❢♦r ❡❧❡❝tr♦♥ ♣❛r❛♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ ❛♥❞ ♣❛r❛♠❛❣♥❡t✐❝ ♥✉❝❧❡❛r ♠❛❣♥❡t✐❝ r❡s♦♥❛♥❝❡ s♣❡❝tr♦s❝♦♣✐❡s✳ • ❚❤❡ ❇❧♦❝❤✲❘❡❞✜❡❧❞✲❲❛♥❣s♥❡ss t❤❡♦r② ✐s r❛r❡❧② ❛♣♣❧✐❝❛❜❧❡ t♦ ♣❛r❛♠❛❣♥❡t✐❝ s②st❡♠s✳ ❙♦❧✈✐♥❣ t❤❡ st♦❝❤❛st✐❝ ▲✐♦✉✈✐❧❧❡ ❡q✉❛t✐♦♥ ❤❛s ❝♦♥st✐t✉t❡❞ ❛ st❛t❡✲♦❢✲t❤❡✲❛rt ♠❡t❤♦❞ ❢♦r ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✷✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
• ■♥ ♣r✐♥❝✐♣❧❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s✐♠✉❧❛t❡ ♥✉❝❧❡❛r ❛♥❞ ❡❧❡❝tr♦♥ s♣✐♥ r❡❧❛①❛t✐♦♥ ♦❢ ❛♥② s②st❡♠ ❢♦r ✇❤✐❝❤ t❤❡ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ▼❉ ❛♥❞ ◗❈ ❡❧❡❝tr♦♥✐❝ str✉❝t✉r❡ ♠❡t❤♦❞s ❛r❡ ❛♣♣❧✐❝❛❜❧❡✳ • ❲❡ ❞❡♠♦♥str❛t❡ t❤✐s ✉s✐♥❣ ❛q✉❡♦✉s s♦❧✉t✐♦♥ ♦❢ ◆✐ 2+ ✐♦♥s ❛s ❛ ♠♦❞❡❧ s②st❡♠✳ • ❲❡ s❛♠♣❧❡ ❛ ▼❉ tr❛❥❡❝t♦r② ❜② q✉❛♥t✉♠ ❝❤❡♠✐❝❛❧ ✭◗❈✮ ❝❛❧❝✉❧❛t✐♦♥s ❛♥❞✱ ✐♥ t✉r♥✱ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ ▲✐♦✉✈✐❧❧❡✲✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s♣✐♥ ❞❡♥s✐t② ♠❛tr✐①✳ • ❲❡ ❢♦✉♥❞ ♦♥❧② ♦♥❡ ❛tt❡♠♣t ❬✶❪ ♦❢ s✉❝❤ s✐♠✉❧❛t✐♦♥ ❢r♦♠ ❧✐t❡r❛t✉r❡✳ ❆t ♣r❡s❡♥t ❞❛② ✇❡ ❤❛✈❡ ❜❡tt❡r ▼❉ ❛♥❞ ◗❈✳ ❚❤❡ r❡❧❛①❛t✐♦♥ t✐♠❡s ✇❡ ❡①tr❛❝t ❢r♦♠ ♦✉r s✐♠✉❧❛t✐♦♥ ❛r❡ ✐♥ ✈❡r② ❣♦♦❞ ❛❣r❡❡♥♠❡♥t ✇✐t❤ t❤❡ ❛✈❛✐❧❛❜❧❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳ ❬✶❪ ▼✳ ❖❞❡❧✐✉s✱ ❈✳ ❘✐❜❜✐♥❣ ❛♥❞ ❏✳ ❑♦✇❛❧❡✇s❦✐✱ ❙♣✐♥ ❞②♥❛♠✐❝s ✉♥❞❡r t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❛r②✐♥❣ ✇✐t❤ t✐♠❡ ✐♥ ❞✐s❝r❡t❡ st❡♣s✿ ▼♦❧❡❝✉❧❛r ❉②♥❛♠✐❝s ❇❛s❡❞ ❙✐♠✉❧❛t✐♦♥ ♦❢ ❊❧❡❝tr♦♥ ❛♥❞ ◆✉❝❧❡❛r ❙♣✐♥ ❘❡❧❛①❛t✐♦♥ ✐♥ ❆q✉❡♦✉s ◆✐❝❦❡❧✭■■✮✱ ❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ P❤②s✐❝s ✶✵✹ ✱ ✸✶✽✶ ✭✶✾✾✻✮✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✸✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
❍❛♠✐❧t♦♥✐❛♥ • ❚❤❡ t✇♦ ❡❧❡❝tr♦♥s ♦❢ ◆✐ 2+ ❝♦♥st✐t✉t❡ ❛♥ ❡✛❡❝t✐✈❡ s♣✐♥✲✶ s②st❡♠✳ ❚♦ t❤❡ s♣✐♥ ❍❛♠✐❧t♦✲ ♥✐❛♥ ✇❡ ✐♥❝❧✉❞❡ t❤❡ ❣ ❛♥❞ ❩❋❙ t❡♥s♦rs ˆ µ B ˆ ❙ · ❣ ( t ) · ❇ + ˆ ❙ · ❉ ( t ) · ˆ ❙ = ˆ H 0 + ˆ H ( t ) = H I ( t ) ˆ µ B gB ˆ hω 0 ˆ H 0 = S z = ¯ S z ˆ µ B ˆ ❙ · [ ❣ ( t ) − g ✶ ] · ❇ + ˆ ❙ · ❉ ( t ) · ˆ H I ( t ) = ❙ , ✭✶✮ • ◗❈ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ ▼❉ s♥❛♣s❤♦ts ♣r♦❞✉❝❡ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✱ t✐♠❡✲♦r❞❡r❡❞ s❡r✐❡s ♦❢ s♣✐♥ ❍❛♠✐❧t♦♥✐❛♥s ˆ H 1 , ˆ H 2 , ˆ H 3 , . . . , ˆ H l − 1 , ˆ H l , ✇✐t❤ t❤❡ t✐♠❡ st❡♣ τ ✳ • ■♥ t❤✐s s✐♠✉❧❛t✐♦♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉s❡❞ ▼❉ tr❛❥❡❝t♦r② ✇❛s 750 ♣s ❛♥❞ ✐t ✇❛s s❛♠♣❧❡❞ t♦ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t s❡r✐❡s ♦❢ l = 15625 s♣✐♥ ❍❛♠✐❧t♦♥✐❛♥s { ˆ H i } ✇✐t❤ t❤❡ t✐♠❡ st❡♣ ♦❢ 48 ❢s✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✹✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
❙✉♣❡r♣r♦♣❛❣❛t♦rs ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ • ❲❡ tr❛♥s❢♦r♠ t❤❡ { ˆ L 1 τ , e L 2 τ , e L 3 τ , . . . , e L l − 1 τ , e L l τ , ✇❤❡r❡ H i } t♦ s✉♣❡r♣r♦♣❛❣❛t♦rs e t❤❡ ˆ L j ❛r❡ ▲✐♦✉✈✐❧❧✐❛♥s ❞❡✜♥❡❞ ❛s ˆ ˆ L j = i [ · , ˆ ˆ H j ] • ❚❤❡ ❞❡♥s✐t② ♦♣❡r❛t♦r ✐s ❞❡✜♥❡❞ ❛s ρ ( t ) = | Ψ( t ) �� Ψ( t ) | , ˆ ✭✷✮ ✇❤❡r❡ | Ψ( t ) � ✐s t❤❡ st❛t❡ ♦❢ t❤❡ s♣✐♥ s②st❡♠✳ ˆ ρ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ▲✐♦✉✈✐❧❧❡✲✈♦♥ ◆❡✉♠❛♥♥ ❡q✉❛t✐♦♥✱ d ˆ ρ ( t ) = ˆ ˆ L ( t )ˆ ρ ( t ) , ✭✸✮ dt ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✺✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
• ✇❤✐❝❤ ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ ❡①♣r❡ss ˆ ρ ( t ) ✱ ✐♥ r♦t❛t✐♥❣ ❢r❛♠❡✱ ❛t t❤❡ t✐♠❡ ✐♥st❛♥t t = nτ ❛s ρ r ( nτ ) = e − ˆ ˆ ˆ ˆ L n − 1 τ · · · e L 1 τ ˆ ˆ ( n )ˆ ˆ ˆ ˆ ˆ ˆ L 0 nτ e L n τ e ˆ ρ (0) ≡ L ρ (0) , ✭✹✮ ✇❤❡r❡ ˆ L 0 ✐s t❤❡ ▲✐♦✉✈✐❧❧✐❛♥ ❢♦r♠ ♦❢ ˆ ˆ H 0 ✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✻✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
❊♥s❡♠❜❧❡ • ❈♦♥s✐❞❡r✐♥❣ ❛♥ ❡♥s❡♠❜❧❡ ♦❢ s♣✐♥ s②st❡♠s✱ ✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t ❛❧❧ ✐ts ♠❡♠❜❡rs ❛r❡ ✐♥✐t✐❛❧❧② ✐♥ t❤❡ ˆ ( n ) � ˆ ρ r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❛s � ˆ ˆ ρ r ( nτ ) � = �L ρ (0) . s❛♠❡ st❛t❡✱ t❤❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡ ♦❢ ˆ ˆ ( n ) ✐s ˆ • ❆ss✉♠✐♥❣ ❡r❣♦❞✐❝✐t② ❛♥❞ t❤❛t t❤❡ s❡r✐❡s { ˆ H i } ✐s ❧♦♥❣ ❡♥♦✉❣❤✱ t❤❡ ❡♥s❡♠❜❧❡ ❛✈❡r❛❣❡ ♦❢ L ♦❜t❛✐♥❡❞ ❛s m ( n ) 1 m ( n ) e − ˆ ˆ ˆ ˆ ˆ ( n ) � = L i + n − 1 τ · · · e ˆ ˆ ˆ ˆ ˆ � L 0 nτ L i + n τ e L i τ , �L e ✭✺✮ i =1 ✇❤❡r❡ m ( n ) ✐s t❤❡ ♥✉♠❜❡r ♦❢ t❤❡ s✉❜✲s❡r✐❡s ˆ H i , . . . , ˆ H i + n t❤❛t ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ { ˆ H i } ✳ ˆ m ( n ) s❤♦✉❧❞ ❜❡ ❧❛r❣❡ ❡♥♦✉❣❤ t♦ r❡♥❞❡r (1 /m ( n )) � m ( n ) ˆ H i ( nτ ) ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ i =1 ˆ ˆ H 0 ✳ ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✼✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
▼❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ρ r ❧✐✈❡s ✐♥ ❛ ♥✐♥❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ • ˆ ❢♦❧❧♦✇✐♥❣ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ s❤✐❢t ♦♣❡r❛t♦rs ❛♥❞ t❤❡ z ❝♦♠♣♦♥❡♥t ♦❢ ˆ ❙ ✿ S − ˆ ˆ S − ˆ ˆ S z + ˆ S z ˆ ˆ ˆ 2 , 3 ˆ S z ˆ S z − 2ˆ S − S − S − S z 1 { ˆ B i } = , , 2 , √ √ , 2 2 6 ˆ S + ˆ ˆ S z + ˆ S z ˆ ˆ S + ˆ ˆ 1 S + S + S + √ 3 , , 2 , . ✭✻✮ 2 2 ❏②r❦✐ ❘❛♥t❛❤❛r❥✉ ✭✽✮ ♣◆▼❘ ♠❡❡t✐♥❣✱ ▼❛r✐❛♣❢❛rr✱ ❋❡❜r✉❛r② ✷✸✱ ✷✵✶✹
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