▼■◆■❙❚❘❨ ❖❋ ❊❉❯❈❆❚■❖◆ ❆◆❉ ❚❘❆■◆■◆● ▼■◆■❙❚❘❨ ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨ ❱■❊❚◆❆▼ ❆❚❖▼■❈ ❊◆❊❘●❨ ■◆❙❚■❚❯❚❊ ▲❨ ❉❯❨ ◆❍❆❚ ▼❆●◆❊❚■❈ ❋■❊▲❉ ❊❋❋❊❈❚ ❖◆ ❆ ❍❨❉❘❖●❊◆ ❊◆❊❘●❨ ❙❚❘❯❈❚❯❘❊ ■◆ P▲❆❙▼❆ ❆◆❉ ❊❳❈■❚❖◆ ■◆ ❚▼❉ ▼❖◆❖▲❆❨❊❘ ❙❯▼▼❆❘❨ ❖❋ ❚❍❊ P❤✳❉✳ ❚❍❊❙■❙ ❍❖ ❈❍■ ▼■◆❍ ❈■❚❨ ✕ ✷✵✷✷ ❚❤❡ ✇♦r❦ ✇❛s ❝♦♠♣❧❡t❡❞ ❛t t❤❡ ❱✐❡t♥❛♠ ❆t♦♠✐❝ ❊♥❡r❣② ■♥st✐t✉t❡✳ ❙❝✐❡♥t✐❢✐❝ s✉♣❡r✈✐s♦rs✿ Pr♦❢✳ ❉✳❙❝✳ ▲❡ ❱❛♥ ❍♦❛♥❣ ❘❡✈✐❡✇❡r ✶✿ ❘❡✈✐❡✇❡r ✷✿ ❘❡✈✐❡✇❡r ✸✿ ■♥❞❡♣❡♥❞❡♥t ❘❡✈✐❡✇❡r ✶✿ ■♥❞❡♣❡♥❞❡♥t ❘❡✈✐❡✇❡r ✷✿ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❙❝❤r☎♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❝❛♥ ♦♥❧② ❜❡ s♦❧✈❡❞ ❛❝❝✉r❛t❡❧② ❢♦r ❛ ❢❡✇ q✉❛♥t✉♠ ♣❤②s✐❝s ♣r♦❜❧❡♠s✱ ❛♥❞ ♠♦st ♦t❤❡r ♣r♦❜❧❡♠s ❛r❡ s♦❧✈❡❞ ❜② ❛♣♣r♦①✲ ✐♠❛t❡ ♠❡t❤♦❞s s✉❝❤ ❛s t❤❡ ♣❡rt✉r❜❛t✐♦♥✱ ✈❛r✐❛❜❧❡ ♠❡t❤♦❞✳ ❋♦r t❤❡ ♠❛♥②✲♣❛rt✐❝❧❡ s②st❡♠ ♣r♦❜❧❡♠✱ ❛♥♦t❤❡r ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✐♥❝♦r♣♦r❛t❡❞ ✐♥ t❤❡ ❍❛tr❡❡✲❋♦❝❦ ♠❡t❤♦❞ ❛♥❞ t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r② ✭❉❋❚✮✳ ❇❛s❡❞ ♦♥ t❤❡ ❝♦♠♣✉t❡r✱ t❤❡ s♦❧✉t✐♦♥ ♦❢ ✇❛✈❡ ❡q✉❛t✐♦♥s ✐s ❝♦♥✈❡rt❡❞ t♦ s♦❧✈✐♥❣ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ❡①♣❛♥s✐♦♥ ❝♦❡❢❢✐❝✐❡♥ts✳ ❚❤❡ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ♣❛rt✐❛❧ s♦❧✉t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ✇❛✈❡❢✉♥❝t✐♦♥s ✉s❡❞✳ ❈♦♠♠♦♥❧② ✉s❡❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✐♥❝❧✉❞❡ t❤❡ ❇✲s♣❧✐♥❡ ♠❡t❤♦❞ ❬✶❪ ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ♠❡t❤♦❞ ❬✷❪✳ ■♥ t❤❡ ❛❜♦✈❡ ♠❡t❤♦❞s✱ ✇❡ ♣❛② ❛tt❡♥t✐♦♥ t♦ t❤❡ ❋❡r❛♥❝❤✉❦✲❑♦♠❛r♦✈ ✭❋❑✮ ♦♣❡r❛t♦r ♠❡t❤♦❞✱ ✐♥ ✇❤✐❝❤ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥ s❡t ✐s ✉s❡❞ ❛s ❛ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❢♦r t❤❡ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r ✇✐t❤ ❛♥❣✉❧❛r ❢r❡q✉❡♥❝② ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ❢r❡❡ ♣❛r❛♠❡t❡r✳ ❚❤❡ ♦❜t❛✐♥❡❞ r❡s✉❧ts ❤❛✈❡ s❤♦✇♥ t❤❛t t❤✐s s❡t ♦❢ ❢✉♥❝t✐♦♥s ❝❛♥ ❞❡s❝r✐❜❡ ✇❡❧❧ ❢♦r ❛ ✈❡r② ✇✐❞❡ s♣❡❝tr✉♠ ♦❢ ❛t♦♠✐❝ ♣r♦❜❧❡♠s ❬✸❪✳ ❚❤❡r❡❢♦r❡✱ ❞❡✈❡❧♦♣✐♥❣ ❛ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ❢♦r ❛t♦♠✐❝ s②st❡♠s ❜❛s❡❞ ♦♥ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ✐s ♣r♦♠✐s✐♥❣✳ ■♥ t❤✐s t❤❡s✐s✱ ✇❡ ✇✐❧❧ ❛♣♣❧② t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ t♦ t✇♦ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❛t♦♠✐❝ ♣r♦❜❧❡♠s✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r♦❜❧❡♠ ♦❢ t✇♦ ♣❛rt✐❝❧❡s s✉❝❤ ❛s ❛ ❤②❞r♦❣❡♥ ❛t♦♠✱ t❤❡ ❢✐rst st❡♣ t♦ ❞♦ ✐s t♦ s❡♣❛r❛t❡ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ❢r♦♠ t❤❡ r❡❧❛t✐✈❡ ♠♦t✐♦♥✳ ❚❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ✐s ❢r❡❡ ❛♥❞ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ❡♥❡r❣② s♣❡❝tr✉♠ ✇❤✐❧❡ r❡❧❛t✐✈✐st✐❝ ♠♦t✐♦♥ ❤❛s ❛ ❞✐s❝♦♥t✐♥✉♦✉s ❡♥❡r❣② s♣❡❝tr✉♠✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ❛♥ ❛t♦♠ ✐s ♣❧❛❝❡❞ ✐♥ ❛♥ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✱ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ✐s ♥♦t ❢r❡❡ ❛♥❞ t❤❡ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ✐s ♥♦t ❛ tr✐✈✐❛❧ ♣r♦❜❧❡♠ ❬✹❪✳ ❆❧t❤♦✉❣❤ ✐♥ t❤❡ r❡❝❡♥t st✉❞② ♦❢ ❡①❝✐t♦♥s ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❬✺✱ ✻❪ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s ♣r❡❝✐s❡❧② s❡♣❛r❛t❡❞✱ t❤❡ ❛✉t❤♦r ♦❢ t❤❡ ✇♦r❦ ❬✼❪ ❤❛s ♣r♦♣♦s❡❞ ❛ ♠❡t❤♦❞ ❛♣♣r♦①✐♠❛t❡ ❝❡♥t❡r ♦❢ ♠❛ss s❡♣❛r❛t✐♦♥ ❜❛s❡❞ ♦♥ t✇♦ ❢❛❧s❡ ❛ss✉♠♣t✐♦♥s✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ r❡st❛t❡ t❤❡ ❝❡♥t❡r✲♦❢✲♠❛ss s❡♣❛r❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ❝❛s❡ ♦❢ t✇♦✲ ❞✐♠❡♥s✐♦♥❛❧ ❡①❝✐t♦♥s ✐♥ t❤❡ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❚❤✐s ❛❧s♦ ❧❡❛❞s t♦ ❛♥ ✐♥t❡r❡st✐♥❣ ♣❤②s✐❝❛❧ ❡❢❢❡❝t✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ r❡❧❛t❡❞ t♦ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❝❛♥ ❛❢❢❡❝t t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ❜② ❛ ♥❡✇ ♠❡❝❤❛♥✐s♠ t❤❛t ✐s ❞✐❢❢❡r❡♥t ❢r♦♠ t❤❡ ❧♦♥❣✲✐♥t❡r❡st❡❞ ♣r♦❜❧❡♠ ✇✐t❤ t❤❡ ❡①❝✐t♦♥✲♣❤♦♥♦♥ ✐♥t❡r❛❝t✐♦♥ ♠❡❝❤❛♥✐s♠ ❬✽❪✳ P❧❛s♠❛ ✐s ❛ s♣❡❝✐❛❧ ❢♦r♠ ♦❢ ♠❛tt❡r ♠❡❞✐✉♠✱ ♣r❡s❡♥t✐♥❣ ✐♥ ♠❛♥② st✉❞✐❡s ❢r♦♠ t❤❡ ♥✉❝❧❡✉s✱ ❛t♦♠s t♦ ❛str♦♣❤②s✐❝s✱ t❤❡ ✉♥✐✈❡rs ❬✾❪✳ ❖♥❡ ❞✐r❡❝t✐♦♥ ♦❢ r❡s❡❛r❝❤ t♦ ❝♦♥str✉❝t s❝r❡❡♥ ♣♦t❡♥t✐❛❧s ✐♥ ❞✐❢❢❡r❡♥t t②♣❡s ♦❢ ♣❧❛s♠❛ ✇✐t❤ t❤❡ ✉❧t✐♠❛t❡ ❛✐♠ ✐s t♦ ❢✐♥❞ ❛ ❣❡♥❡r❛❧ ♠♦❞❡❧ ❞❡s❝r✐❜✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣❧❛s♠❛ ♠❡❞✐✉♠ ♦❢ ❛♥② t❡♠♣❡r❛t✉r❡ ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ❬✶✵❪✳ ❆♥♦t❤❡r ❞✐r❡❝t✐♦♥ ✐s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❛t♦♠✐❝ ❡♥❡r❣② s♣❡❝tr✉♠ ❜❛s❡❞ ♦♥ t❤❡ ❛✈❛✐❧❛❜❧❡ ♣♦t❡♥t✐❛❧ ♠♦❞❡❧ ❬✶✶❪✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣❧❛s♠❛ t②♣❡s✱ t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣♦t❡♥t✐❛❧s ❛r❡ ♣r♦♣♦s❡❞ ✇✐t❤ ❞✐❢❢❡r❡♥t ♠♦❞❡❧s s✉❝❤ ❛s✿ ❙t❛t✐❝ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧ ✭❙❙❈✮ ❬✶✷❪✱ ❊①♣♦♥❡♥t✐❛❧ ❝♦s✐♥❡ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧ ✭❊❈❙❈✮ ❬✶✸❪✱ ●❡♥❡r❛❧✐③❡❞ ❡①♣♦♥❡♥t✐❛❧ ❝♦s✐♥❡ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧ ✶ ✭●❊❈❙❈✮ ❬✶✹❪✱ ❛♥❞ ▼♦r❡ ❣❡♥❡r❛❧✐③❡❞ ❡①♣♦♥❡♥t✐❛❧ ❝♦s✐♥❡ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧ ✭▼●❊❈❙❈✮ ❬✶✶❪✳ ■♥ r❡❝❡♥t ②❡❛rs✱ ♣❧❛s♠❛ ♣❤②s✐❝s ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❛❣❛✐♥ ❜② ♠❛♥② ❛✉t❤♦rs ❬✶✺❪ s✉❝❝❡❡❞✐♥❣ ✐♥ ♥✉♠❡r✐❝❛❧❧② s♦❧✈✐♥❣ t❤❡ ❙❝❤r☎♦❞✐♥❣❡r ❡q✉❛t✐♦♥ t♦ ❛♥ ❛❝❝✉✲ r❛❝② ♦❢ ✻ t♦ ✽ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ❬✶✶❪✳ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts s❤♦✇ t❤❛t t❤❡r❡ ❛r❡ ❡❢❢❡❝ts ❝❛✉s❡❞ ❜② t❤❡ ❡①t❡r♥❛❧ ❢✐❡❧❞ t❤❛t ❞❡s❡r✈❡ ❛tt❡♥t✐♦♥ ❛♥❞ ♥❡❡❞ t♦ ❜❡ ❢✉rt❤❡r st✉❞✐❡❞ ❜② ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ▼●❊❈❙❈ ♠♦❞❡❧✳ ❚❤❡ ❡♥❡r❣② s♣❡❝tr✉♠ ♦❢ t❤❡ ❡①❝✐t♦♥ ✐♥ ❛ ❚▼❉ ♠♦♥♦❧❛②❡r ✐s ♦❢ ❣r❡❛t ✐♥t❡r❡st ❜♦t❤ ❡①♣❡r✐♠❡♥t❛❧❧② ❛♥❞ t❤❡♦r❡t✐❝❛❧❧② ❬✻❪✳ ❚❤✐s ✇♦r❦ ❛❧s♦ ❤❛s ✐♠♣❧✐❝❛t✐♦♥s ❢♦r st✉❞②✲ ✐♥❣ ❛t♦♠✐❝ ❡♥❡r❣② s♣❡❝tr❛ ✐♥ ✇❤✐t❡ ❞✇❛r❢s ❛♥❞ ♥❡✉tr♦♥ st❛rs✳ ❚❤❡ s♣❡❝✐❛❧ ❢❡❛t✉r❡ ♦❢ t❤❡ ❡①❝✐t♦♥ ✐♥ t❤❡ ❚▼❉ ♠♦♥♦❧❛②❡r ✐s t❤❛t t❤❡ ❡❧❡❝tr♦♥✲❤♦❧❡ ✐♥t❡r❛❝t✐♦♥ ♣♦t❡♥✲ t✐❛❧ ✐s ♥♦t t❤❡ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧ ❜✉t ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ❬✶✻❪✳ ■♥ ♠♦st t❤❡♦r❡t✐❝❛❧ st✉❞✐❡s ❬✺✱ ✻❪ ♦♥❧② ❣✐✈❡s ❡♥❡r❣✐❡s ❢♦r st❛t❡s s ✭m = 0✮✳ ■♥ t❤✐s t❤❡s✐s✱ ✇❡ ❝♦♥t✐♥✉❡ t♦ ❞❡✈❡❧♦♣ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❜② ❣✐✈✐♥❣ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ t❤r♦✉❣❤ t❤❡ ✐♥✈❡rs❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❛♥❞ ♦❜t❛✐♥ t❤❡ ❡♥❡r❣② s♣❡❝tr✉♠ ✇✐t❤ ❤✐❣❤ ❛❝❝✉r❛❝② ❢♦r t❤❡ q✉❛♥t✉♠ st❛t❡s ✐♥❝❧✉❞✐♥❣ t❤❡ ✇❤♦❧❡ ❝❛s❡ ✇✐t❤ m = 0✳ ❆♥ ✐♠♣♦rt❛♥t ♣❤②s✐❝s ♣r♦❜❧❡♠ ✐s t❤❡ s❡♥s✐t✐✈✐t② ♦❢ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝✲ tr✉♠ t♦ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs✳ ❚❤❡r❡❢♦r❡✱ ✇✐t❤ ✈❡r② ♣r❡❝✐s❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ❝❛❧❝✉❧❛t❡❞ ❜② t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞✱ ✇❡ ✇✐❧❧ st✉❞② t❤❡✐r s❡♥s✐t✐✈✐t② ✇❤❡♥ ❝❤❛♥❣✐♥❣ ❛❧❧ t❤r❡❡ ♣❛r❛♠❡t❡rs ✭♠❛ss✱ s❝r❡❡♥✐♥❣ ❧❡♥❣t❤✱ ❞✐❡❧❡❝tr✐❝ ❝♦♥st❛♥t✮ ❛♥❞ ❢r♦♠ ❤❡r❡ r❡✲ tr✐❡✈❡s s✉❝❤ str✉❝t✉r❛❧ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ✇✐t❤♦✉t ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❜❛s❡❞ ♦♥ ♣✉❜❧✐s❤❡❞ ❡①♣❡r✐♠❡♥t❛❧ s♣❡❝tr✉♠ ♦❢ ♦t❤❡r r❡s❡❛r❝❤ ❣r♦✉♣s✳ ✷✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ t❤❡s✐s ✐s t♦ st✉❞② t❤❡ ❡❢❢❡❝t ♦❢ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ♦♥ ❛ ❤②❞r♦❣❡♥ ❡♥❡r❣② str✉❝t✉r❡ ✐♥ ♣❧❛s♠❛ ❛♥❞ ❡①❝✐t♦♥ ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡r ❜② ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡ s♦♠❡ r❡❧❛t❡❞ ♣❤②s✐❝❛❧ ❡❢❢❡❝ts✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s ♦❜❥❡❝t✐✈❡✱ ✇❡ ❤❛✈❡ ❝❛rr✐❡❞ ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s❡❛r❝❤ ❝♦♥t❡♥ts ✭✶✳✮ ❘❡♣r❡s❡♥t✐♥❣ t❤❡ s❡♣❛r❛t✐♦♥ ♣r♦❝❡ss ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ st✉❞② ❢♦r ❛♥ ❡①❝✐t♦♥ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞❀ ❚❤❡ st✉❞② ♦❢❢❡rs t❤❡ ♣♦ss✐❜✐❧✐t② t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝t ✐♥ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛❀ ✭✷✳✮ ❉❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤❡ t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ❛♥❞ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❢♦r ❛ ❤②✲ ❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ❣❡♥❡r❛❧ s❝r❡❡♥✐♥❣ ♣❧❛s♠❛ ▼●❊❈❙❈ ♣❧❛❝❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞❀ ✭✸✳✮ ❉❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ t♦ s♦❧✈❡ t❤❡ t✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ❢♦r t❤❡ ♥❡✉tr❛❧ ❡①❝✐t♦♥ ✐♥ ❛ ❚▼❉ ♠♦♥♦❧❛②❡r ♣❧❛❝❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞❀ ❘❡tr✐❡✈✐♥❣ ♦❢ str✉❝t✉r❛❧ ✐♥❢♦r✲ ♠❛t✐♦♥ ♦❢ ❚▼❉ ♠♦♥♦❧❛②❡rs ✭r❡❞✉❝❡❞ ♠❛ss✱ s❝r❡❡♥✐♥❣ ❧❡♥❣t❤✱ ❞✐❡❧❡❝tr✐❝ ❝♦♥st❛♥t✮ ❢r♦♠ ❡①♣❡r✐♠❡♥t❛❧ ❡♥❡r❣② s♣❡❝tr❛✳ ✷ ❈❤❛♣t❡r ✶ ❚❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ❛t♦♠✬s ❝❡♥t❡r ♦❢ ♠❛ss ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✶✳✶ ❖✈❡r✈✐❡✇ ❙❡♣❛r❛t✐♥❣ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ❢♦r ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣✲ ♥❡t✐❝ ❢✐❡❧❞ ✐s ❛ ♥♦♥✲tr✐✈✐❛❧ ♣r♦❜❧❡♠ ❤❛✈✐♥❣ ❛ s♦❧✉t✐♦♥ ❢♦r t❤❡ s♣❡❝✐❢✐❝ ❝❛s❡ ✇❤❡♥ t❤❡ s②st❡♠ ✐s ❡❧❡❝tr✐❝❛❧❧② ♥❡✉tr❛❧ ❬✹❪✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦s✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ t✇♦✲ ❞✐♠❡♥s✐♦♥❛❧ ❡①❝✐t♦♥ ♣r♦❜❧❡♠ ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❬✻✱ ✶✼❪✱ t❤❡ ❛✉t❤♦rs ✉s❡❞ t❤❡ r❡❧❛✲ t✐✈✐st✐❝ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✇✐t❤ t❤❡ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❛♥❞ ❛ss✉♠❡❞ t❤❡ ♣s❡✉❞♦✲♠♦♠❡♥t✉♠ ✈❡❝t♦r ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐s ③❡r♦ ✭K = 0✮✳ ❇✉t ❡①♣❡r✐♠❡♥t❛❧ st✉❞✐❡s ♦❢ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡rs ❛r❡ ✉s✉❛❧❧② ♠❡❛s✉r❡❞ ❛t r♦♦♠ t❡♠♣❡r❛t✉r❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❝♦♥s✐❞❡r t❤❡ ❡❢❢❡❝t ♦❢ t❡♠♣❡r❛t✉r❡ ♦♥ ❡①✲ ❝✐t♦♥ ❞❡❝❛② t✐♠❡ ✇❤✐❝❤ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❡①t❡♥s✐✈❡❧② ❬✽✱ ✶✽❪ ✇✐t❤ ❛♥ ❡①❝✐t♦♥✲♣❤♦♥♦♥ ✐♥t❡r❛❝t✐♦♥ ♠❡❝❤❛♥✐s♠✳ ❍♦✇❡✈❡r✱ ✇❡ ❞♦ ♥♦t s❡❡ ❛♥② t❤❡♦r❡t✐❝❛❧ ✇♦r❦ t❤❛t ❛♥❛❧②③❡s t❤❡ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝t t❤r♦✉❣❤ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ ♠❛❦❡s t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❛❢❢❡❝t t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ♦❢ t❤❡ ❡①❝✐t♦♥✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ✐s ♦❢ ✐♥✲ t❡r❡st t♦ ✉s ✇❤❡♥ ❛♥❛❧②③✐♥❣ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ✇♦r❦ ❬✼❪ ❜② ❉♦♥❝❦ ❡t ❛❧✳ ♣✉❜❧✐s❤❡❞ ♦♥ ❬P❤②s✳ ❘❡✈✳ ❇ 97✱ ✶✾✺✹✵✽ ✭✷✵✶✽✮❪✳ ❍❡r❡✱ t❤❡ ❛✉t❤♦rs ❞✐❞ ♥♦t ♣❛② ❛tt❡♥t✐♦♥ t♦ ❡①❛❝t❧② s❡♣❛r❛t✐♥❣ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥✱ s♦ t❤❡ r❡s✉❧ts ♦❢ t❤❡ ❛✉t❤♦r✬s t❤❡♦r❡t✲ ✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❡①❝✐t♦♥ ❡♥❡r❣② ❞❡✈✐❛t❡ ❢r♦♠ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ❢r♦♠ 5% t♦ 30%✳ P❛rt✐❝✉❧❛r❧②✱ t❤❡ ❞✐❛♠❛❣♥❡t✐❝ ❝♦❡❢❢✐❝✐❡♥t ✐s ❞✐❢❢❡r❡♥t ❢r♦♠ t❤❡ ❝♦❡❢❢✐❝✐❡♥t ♦❢ ✹ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ r❡s✉❧ts ♦❢ ♦t❤❡r ❛✉t❤♦rs✳ ✶✳✷ ❚❤❡ ♠♦t✐♦♥ ♦❢ ❛♥ ❛t♦♠✬s ❝❡♥t❡r ♦❢ ♠❛ss ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✶✳✷✳✶ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❢♦r ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❲❡ ✇r✐t❡ t❤❡ ❡①❛❝t ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❡❧❡❝tr♦♥ ✕ ❤♦❧❡ ♠♦✈✐♥❣ ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛s ❢♦❧❧♦✇s✿ ˆ ex Ψ = EΨ, H ✭✶✳✶✮ ✇❤❡r❡✿ ˆ ex = H eB ˆ eB ˆ pˆe + pˆh + lez − lhz 2me 2mh 2me 2mh e2 2 e2 2 B (xe + ye ) + B (xh + yh2 ) + V (|rh − re |) + 8me 8mh ✸ ✭✶✳✷✮ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❝♦♠♠✉t❛t♦r ♦❢ t❤❡ ♦♣❡r❛t♦r eB ˆ eB ˆ Aˆ = lez − lhz , 2me 2mh ✭✶✳✸✮ ˆ ex ❛r❡ ♥♦t ❝♦♠♠✉t❛t✐✈❡ s♦ ✇❡ ❝❛♥♥♦t ✐❣♥♦r❡ Aˆ ❜② ♣❧✉❣❣✐♥❣ ❛♥❞ ❢✐♥❛❧❧② ♦❜t❛✐♥ Aˆ ❛♥❞ H ˆ ex ✳ ③❡r♦ ✐♥ H ✶✳✷✳✷ ❍❛♠✐❧t♦♥✐❛♥ ✐♥ ❝❡♥t❡r✲♦❢✲♠❛ss ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛♥❞ r❡❧❛t✐✈❡ ♠♦✲ t✐♦♥ ❜❡t✇❡❡♥ ❡❧❡❝tr♦♥s ❛♥❞ ❤♦❧❡s ❲❡ ❝❤❛♥❣❡ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♠♦t✐♦♥ ♦❢ ❛ ♣❛rt✐❝❧❡ ❜② s❡♣❛r❛t✐♥❣ ✐t ✐♥t♦ t✇♦ ♠♦t✐♦♥s✿ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ s②st❡♠ ❛♥❞ t❤❡ r❡❧❛t✐✈❡ ♠♦t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ♣❛rt✐❝❧❡s ✇❤✐❝❤ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② t✇♦ ✈❡❝t♦rs R= me mh rh + re , mh + me mh + me r = re − rh ✭✶✳✹✮ ◆♦✇✱ ✇❡ ❢✐♥❞ t❤❡ ❡①❛❝t ❍❛♠✐❧t♦♥✐❛♥ ❜② ✇r✐t✐♥❣ ✭✶✳✹✮ ✐♥ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♦r❞✐♥❛t❡s R (X, Y, Z) ❛♥❞ r (x, y, z)✳ ˆ ex = H eB mh − me ˆ e2 B lz + (x2 + y ) + Vhe (r) pˆ + − 2µ me mh µ M 2 2 ˆ2 e B e B mh − me + (xX + yY ) P + (X + Y ) + 2M 8µ me mh e e + B × r Pˆ + B × R pˆ 2M 2µ ✭✶✳✺✮ ❲❡ s❡❡ t❤❡ ❡①❛❝t ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✺✮ ❤❛s t❤❡ ❧❛st t✇♦ ❣r♦✉♣s ♦❢ t❡r♠s t❤❛t ❤❛✈❡ ♥♦t ❜❡❡♥ s❡♣❛r❛t❡❞ ②❡t ✐♥t♦ t✇♦ ♠♦t✐♦♥s ✐♥ ❝♦♦r❞✐♥❛t❡s R ❛♥❞ r✳ ❙♦ ❜② s❡tt✐♥❣ ❛♥ ♦r❞✐♥❛r② ✈❛r✐❛❜❧❡ ❧✐❦❡ ✭✶✳✹✮✱ ✇❡ ❝❛♥♥♦t s❡♣❛r❛t❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✷✮ ✐♥t♦ t✇♦ ♠♦t✐♦♥s ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❛♥❞ t❤❡ r❡❧❛t✐✈❡ ♠♦t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡❧❡❝tr♦♥ ❛♥❞ t❤❡ ˆ ex = 0✳ ❚♦ ❞♦ t❤✐s✱ ✇❡ ♠✉st ✉s❡ t❤❡ q✉❛s✐✲♠♦♠❡♥t✉♠ ✈❡❝t♦r ❤♦❧❡ ❜❡❝❛✉s❡ ♦❢ Pˆ0 , H ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ✶✳✷✳✸ ◗✉❛s✐✲♠♦♠❡♥t✉♠ ✈❡❝t♦r ❲❡ ❢✐rst ❞❡❢✐♥❡ t❤❡ q✉❛s✐✲♠♦♠❡♥t✉♠ ✈❡❝t♦r ❬✹❪ e Pˆ0 = Pˆ − B × r ✭✶✳✻✮ ❛♥❞ t❤❡♥ ♣r♦✈❡ t❤❛t t❤❡ q✉❛♥t✐t② Pˆ0 ✐s ❝♦♠♠✉t❛t✐✈❡ ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✺✮✱ ˆ ex = 0✳ t❤❛t ✐s Pˆ0 , H ✹ ✶✳✷✳✹ ❙❡♣❛r❛t✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✈❛r✐❛❜❧❡ ✇✐t❤ ❛ q✉❛s✐✲♠♦♠❡♥t✉♠ ✈❡❝✲ t♦r ❚❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥ ♦❢ t❤❡ q✉❛s✐✲♠♦♠❡♥t✉♠ ✈❡❝t♦r Pˆ0 ✱ ❤❛✈✐♥❣ ❡✐❣❡♥✈❛❧✉❡ K ✱ ✐s Ψ R, r = U ψ (r) , ✇❤❡r❡ U = exp i e K + B×r R ˆ ex U ψ (r) = EU ψ (r) ❋r♦♠ ❤❡r❡✱ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✭✶✳✶✮ ✐s r❡✇r✐tt❡♥ ❛s H ˆ rel ❞❡s❝r✐❜✐♥❣ t❤❡ r❡❧❛t✐✈❡ ♠♦t✐♦♥ ✐s ❞❡❢✐♥❡❞ ❍❛♠✐❧t♦♥✐❛♥ H ˆ rel = U −1 H ˆ ex U H ✭✶✳✼✮ ˆ rel (r) ❢♦r t❤❡ r❡❧❛t✐✈❡ ◆♦✇✱ ✇❡ ❢✐♥❞ t❤❡ ❡①❛❝t ❡①♣r❡ss✐♦♥ ♦❢ ❍❛♠✐❧t♦♥✐❛♥ H ♠♦t✐♦♥ ♦❢ t❤❡ ❡❧❡❝tr♦♥ ✕ ❤♦❧❡ ❛s ˆ rel = H e2 B 2 (mh − me ) eB ˆ pˆ + (x + y ) + Vhe (r) + lz 2µ 8µ me mh 1 − eB × K r + K M 2M ✭✶✳✽✮ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✽✮ ❞❡s❝r✐❜❡s t❤❡ r❡❧❛t✐✈❡ ♠♦t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❡❧❡❝tr♦♥ ❛♥❞ t❤❡ ❤♦❧❡✳ Pr❡✈✐♦✉s ✇♦r❦s ♦♥❧② ❝♦♥s✐❞❡r❡❞ t❤❡ ♠♦♠❡♥t✉♠ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss K = 0✱ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✽✮ ✐s r❡✇r✐tt❡♥ e2 B 2 (mh − me ) eB ˆ Hrel = pˆ + (x + y ) + Vhe (r) + m , 2µ 8µ me mh ✭✶✳✾✮ ✇✐t❤ m = 0, ±1, · · · ± l✳ ■♥ ❝❛s❡ ♦❢ m = 0✱ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✾✮ ❜❡❝♦♠❡ ✐♥t♦ e2 B 2 ˆ Hrel = pˆ + (x + y ) + Vhe (r) 2µ 8µ ✭✶✳✶✵✮ ✶✳✸ ❉✐s❝✉ss✐♥❣ t❤❡ ❡❢❢❡❝t ♦❢ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ♦❢ t❤❡ ❡①❝✐t♦♥ ❈♦♠♠❡♥t✐♥❣ ♦♥ ❉♦♥❝❦✬s ✇♦r❦ ❬P❤②s✳ ❘❡✈✳ ❇ 97✱ ✶✾✺✹✵✽ ✭✷✵✶✽✮❪ ❬✼❪ ❚❤❡ ✇♦r❦✱ ▼✳ ❱❛♥ ❞❡r ❉♦♥❝❦ ❝❛❧❝✉❧❛t❡❞ t❤❡ ❜✐♥❞✐♥❣ ❡♥❡r❣✐❡s ❢♦r t❤❡ ❡①❝✐t♦♥s ♦❢ s♦♠❡ ❚▼❉ ♠♦♥♦❧❛②❡rs ❜② ✉s✐♥❣ ❙❱▼✳ ❚❤❡ r❡s✉❧ts ❛r❡ r❡❝♦r❞❡❞ ✐♥ ❚❛❜❧❡ ✷ ❛♥❞ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡r t❤❡♦r❡t✐❝❛❧ ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ✇♦r❦s✳ ❋r♦♠ t❤✐s ❝♦♠♣❛r✐s♦♥✱ ✐t ❤❛s ❜❡❡♥ s❤♦✇♥ t❤❛t t❤❡ t❤❡♦r❡t✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ r❡s✉❧ts ❤❛✈❡ ❞❡✈✐❛t❡❞ ❢r♦♠ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ 5%−30% ❛♥❞ t❤❡ ❛✉t❤♦r ❤❛s ❛♥❛❧②③❡❞ t❤✐s ❞✐❢❢❡r❡♥❝❡ ❜✉t ❞✐❞ ♥♦t s❤♦✇ st❛t❡ t❤❡ ❝❛✉s❡✳ ❍♦✇❡✈❡r✱ ✇❡ ❤❛✈❡ ❢♦✉♥❞ t❤❛t t❤❡ ❝❛✉s❡ ✐s ❜❡❝❛✉s❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✺ ❤❛s ♥♦t ❜❡❡♥ s❡♣❛r❛t❡❞ ❢r♦♠ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ❝♦rr❡❝t❧②✳ eB ˆ eB ˆ lez − 2m lhz ✐s ♥♦t ❝♦♠♠✉t❛t✐✈❡ ✇✐t❤ t❤❡ ❍❛♠✐❧✲ ▼♦♠❡♥t✉♠ ❝♦♠♣♦♥❡♥t Aˆ = 2m e h t♦♥✐❛♥ s♦ Aˆ ✐s r❡♠♦✈❡❞ ❢r♦♠ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ t♦ ♦❜t❛✐♥ t❤❡ s✐♠♣❧❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛✲ t✐♦♥ t❤❛t ▼✳ ❱❛♥ ❞❡r ❉♦♥❝❦✬s ✇♦r❦ ❬✼❪ ✇r✐tt❡♥ ✐♥ ❡①♣r❡ss✐♦♥ ✭✻✮ ❤❛s ♥♦ ♣❤②s✐❝❛❧ ❜❛s✐s✱ ❜✉t ✐❢ ✇❡ ❦❡❡♣ t❤✐s ❝♦♠♣♦♥❡♥t✱ ✇❡ ❝❛♥♥♦t s❡♣❛r❛t❡ t❤❡ ✈❛r✐❛❜❧❡ t❤❡ s❛♠❡ ✇❛② ❛s t❤❡ ❛✉t❤♦r✱ r❡s✉❧t✐♥❣ ✐♥ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ r❡❝♦r❞❡❞ ✐♥ ❡①♣r❡ss✐♦♥ ✭✽✮ ✐♥ ❬✼❪ ❛♥❞ ❢✐♥❛❧❧② ♦❜t❛✐♥ ❛ ❍❛♠✐❧t♦♥✐❛♥ ✇❤♦s❡ s❡❝♦♥❞ t❡r♠ ✐s r❡❧❛t❡❞ t♦ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✶✴✹ t✐♠❡s t❤❡ t❡r♠ ✐♥ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✶✳✶✵✮ ❢r♦♠ 5% → 30% ❝♦♠♣❛r❡❞ t♦ ❡①♣❡r✐♠❡♥t❛❧✳ ❚❤❡ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝t ✐♥ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ❚❤❡ ❡①❝✐t♦♥✱ tr✐♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ♦❢ ❚▼❉ ♠♦♥♦❧❛②❡rs ♠❡❛s✉r❡❞ ✐♥ t❤❡ ❧❛❜✲ ♦r❛t♦r② t♦❞❛② ❛t ❛ t❡♠♣❡r❛t✉r❡ ♦❢ − 300 ❑✳ ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡♥❡r❣② s♣❡❝tr✉♠✱ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❡①❝✐t♦♥✲♣❤♦♥♦♥ ❡❢❢❡❝t ❬✽❪✳ ■♥ t❤❡ ✇♦r❦ ❆r♦r❛ ✭✷✵✶✺✮ ❬✽❪ ❞❡♠♦♥✲ str❛t❡❞ t❤❡ ❜❛♥❞ ❣❛♣ Egap ❛♥❞ t❤❡ ❧✐❢❡ t✐♠❡ ❞❡♣❡♥❞s ♦♥ t❤❡ t❡♠♣❡r❛t✉r❡ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❡①❝✐t♦♥✲♣❤♦♥♦♥ ♠❡❝❤❛♥✐s♠✳ ❍♦✇❡✈❡r✱ t❤❡ ❛❜♦✈❡ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ✇♦r❦s ♦♥❧② ❝♦♥s✐❞❡r t❤❡ ✐♥✲ ❢❧✉❡♥❝❡ ♦❢ t❡♠♣❡r❛t✉r❡ ♦♥ t❤❡ ❡①❝✐t♦♥ ❛♥❞ tr✐♦♥ ❡♥❡r❣② s♣❡❝tr❛ ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❲❤✐❧❡✱ ✇❡ ❢♦✉♥❞ t❤❛t ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✱ t❤❡ ❡♥✲ ❡r❣② s♣❡❝tr❛ ♦❢ ❡①♣❡r✐♠❡♥t❛❧ ❚▼❉ ♠♦♥♦❧❛②❡rs ✐s ♦❢t❡♥ ❜❧✉rr❡❞✱ t❤❡ ❞❡❣r❡❡ ♦❢ ❜❧✉r ✐s ❣r❡❛t❡r ✇❤❡♥ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✐s ❤✐❣❤❡r ❛♥❞ ❝❛♥ ❜❡ ❝❧❡❛r❧② s❡❡♥ ✐♥ ❋✐❣✉r❡ ✶ ♦❢ t❤❡ r❡♣♦rt ❬✶✼❪ ❛♥❞ ❋✐❣✉r❡ ✷ ✐♥ ❬✶✾❪✳ ❚❤✐s ❡❢❢❡❝t ❝❛♥ ♦♥❧② ❜❡ ❡①♣❧❛✐♥❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ❡①❛❝t❧② ❛t ✭✶✳✽✮ ✇✐t❤ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ t❡r♠ ❣r♦✉♣ − M1 eB × K r✳ ■t ✇❛s t❤✐s ❣r♦✉♣ t❤❛t s✐❣♥✐❢✐❝❛♥t❧② ❝❤❛♥❣❡❞ t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ str❡♥❣t❤ ❛♥❞ ❧❛❜♦r❛t♦r② t❡♠♣❡r❛t✉r❡✳ ✶✳✹ ❈♦♥❝❧✉s✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ t❤❡ ❡①❛❝t s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❤②❞r♦❣❡♥ ❛t♦♠ ♣r♦❜❧❡♠ ✐♥ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ r❡❧❛t❡❞ ✐t t♦ t❤❡ ♠♦✈✐♥❣ ❡①❝✐t♦♥ ♣r♦❜❧❡♠ ✐♥ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❲❡ ❞✐s❝✉ss t❤❡ ❡❢❢❡❝t ♦❢ t❤❡ s❡♣❛r❛t✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ♦♥ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✐♥❝❧✉❞✐♥❣✿ ❝♦♠♠❡♥t✐♥❣ ♦♥ t❤❡ ✇♦r❦ ♦❢ ❉♦♥❝❦ ❬P❤②s✳ ❘❡✈✳ ❇ 97✱ ✶✾✺✹✵✽ ✭✷✵✶✽✮❪ ❛♥❞ q✉❛❧✐t❛t✐✈❡❧② ♣r❡s❡♥t✐♥❣ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✐♥✈❡st✐❣❛t✐♥❣ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝ts ✉s✐♥❣ ❛ ♥❡✇ ♠❡❝❤❛♥✐s♠✳ ❈❤❛♣t❡r ✷ ❍✐❣❤✲♣r❡❝✐s✐♦♥ ❡♥❡r❣② ❢♦r ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ♣❧❛s♠❛ ✐♥ ✉♥✐✲ ❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✷✳✶ ❖✈❡r✈✐❡✇ ❆ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ♣❧❛s♠❛ ✐s ❛♥ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ♠❛♥② ✇♦r❦s ❬✶✶✱ ✶✷✱ ✶✺❪✳ P❧❛s♠❛ ♣❤②s✐❝s ✐s ❛❧s♦ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ ❛str♦♣❤②s✐❝s✳ P❧❛s♠❛ ✐s ❢♦r♠❡❞ ❞✉r✐♥❣ t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ s❤♦rt✲✇❛✈❡❧❡♥❣t❤ ♦r ❳✲r❛② ❧❛s❡rs✱ ✐♥ s❡♠✐❝♦♥❞✉❝t♦r ❡♥❣✐✲ ♥❡❡r✐♥❣ ♣r♦❝❡ss❡s✱ ❛♥❞ ✐♥ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ t❤❡ t❤❡r♠♦♥✉❝❧❡❛r ❢✉s✐♦♥ ❡♥❡r❣② ❬✾❪✳ ✻ P❧❛s♠❛ ❤❛s ❜❡❡♥ st✉❞✐❡❞ s✐♥❝❡ t❤❡ ❧❛t❡ ✶✾✷✵s ❬✷✵❪ ✐s ♦❢ str♦♥❣ ✐♥t❡r❡st ❛❣❛✐♥ s✐♥❝❡ t❤❡ ✶✾✼✵s ❬✷✶❪✱ ❡s♣❡❝✐❛❧❧② ❛❝t✐✈❡❧② st✉❞✐❡❞ ✐♥ r❡❝❡♥t ②❡❛rs ❬✶✺❪✳ ❚❤❡♦r❡t✐❝❛❧ st✉❞✐❡s ♦❢ ♣❧❛s♠❛ ❤❛✈❡ ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t ❢♦r ❛ ❧♦♥❣ t✐♠❡ t♦ ❢✐♥❞ ❛ ❣❡♥❡r❛❧ ♠♦❞❡❧ ♦❢ t❤❡ ❛t♦♠✐❝ s②st❡♠ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ♣❧❛s♠❛✳ ❙♦♠❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠♦❞❡❧s ✐♥❝❧✉❞❡ ❙❙❈ ❬✷✷❪✱ ❊❈❙❈ ❬✶✸❪✱ ●❊❈❙❈ ❬✶✹❪ ❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧ ▼●❊❈❙❈ ❬✶✶❪✳ ❆♥ ✐♠♣♦rt❛♥t r❡s❡❛r❝❤ ❞✐r❡❝t✐♦♥ ✐s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ♦❢ t❤❡ ❤②❞r♦❣❡♥✲ ❧✐❦❡ st❛t✐♦♥❛r② ✐♦♥s ♦❢ ✐♥t❡r❡st ❜② ❞✐❢❢❡r❡♥t ❛♣♣r♦❛❝❤❡s ❬✶✶✱✶✷✱✶✹❪✳ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts s❤♦✇ t❤❛t t❤❡r❡ ❛r❡ ❡❢❢❡❝ts ❝❛✉s❡❞ ❜② t❤❡ ❡①t❡r♥❛❧ ❢✐❡❧❞ t❤❛t ❞❡s❡r✈❡ ❛tt❡♥t✐♦♥ ❛♥❞ ♥❡❡❞ ❢✉rt❤❡r r❡s❡❛r❝❤✳ ❚❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ♦❜t❛✐♥s t❤❡ ❡♥❡r❣② ✇✐t❤ ❤✐❣❤ ❛❝❝✉r❛❝② ❛♥❞ ❢❛st ❝♦♥✲ ✈❡r❣❡♥❝❡ ❬✷✸❪✳ ❚❤✐s s✉❣❣❡sts ✉s t♦ ❢✉rt❤❡r ❞❡✈❡❧♦♣ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❤②❞r♦❣❡♥ ❛t♦♠ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ♣❧❛s♠❛ ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ▼●❊❈❙❈ ♣♦t❡♥t✐❛❧ ♠♦❞❡❧✳ ✷✳✷ ❍②❞r♦❣❡♥ ❛t♦♠ ✐♥ ♣❧❛s♠❛ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✷✳✷✳✶ ❙❝r❡❡♥✐♥❣ ♣♦t❡♥t✐❛❧ ♦❢ ❝❤❛r❣❡ ✐♥ ♣❧❛s♠❛ ■♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ q✉❛♥t✉♠ ❉❡❜②❡ ♣❧❛s♠❛✱ ❙♦②❧✉ ❬✶✶❪ s✉❣❣❡sts ▼●❊❈❙❈✱ ❛♥ ❡①t❡♥❞❡❞ ❢♦r♠ ♦❢ t❤❡ ❊❈❙❈ ♣♦t❡♥t✐❛❧ ❛s ❢♦❧❧♦✇s✿ ϕ MGECSC Ze e−λr (1 + br) cos (cλr) , (r) = 4π r ✭✷✳✶✮ ✇❤❡r❡✱ t❤❡ s❝r❡❡♥❡❞ ♣❛r❛♠❡t❡r λ ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❉❡❜②❡ ❧❡♥❣t❤ ❛♥❞ ❞❡♣❡♥❞s ♦♥ t❤❡ t❡♠♣❡r❛t✉r❡ ❛♥❞ ❞❡♥s✐t② ♦❢ t❤❡ ♣❧❛s♠❛ ❬✾❪✳ ❲❡ ✉s❡ t❤✐s ❣❡♥❡r❛❧ ♣♦t❡♥t✐❛❧ ❢♦r t❤❡ ✐♥✈❡st✐❣❛t✐♦♥s ✐♥ t❤✐s ❈❤❛♣t❡r✳ ✷✳✷✳✷ ❚❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✈✐❛ t❤❡ ❑✉st❛❛♥❤❡✐♠♦✲❙t✐❡❢❡❧ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❝♦♥s✐❞❡r✐♥❣ t❤❡ s❝r❡❡♥❡❞ ♣♦t❡♥t✐❛❧ ▼●❊❈❙❈ ❤❛s t❤❡ ❢♦r♠ ✐♥ t❤❡ ❛t♦♠✐❝ ✉♥✐t s②st❡♠ t❤r♦✉❣❤ t❤❡ ❑✲❙ tr❛♥s❢♦r♠❛t✐♦♥ ✇r✐tt❡♥✿ ∂2 ∂2 ∂2 m ∂2 + + + − ε − γ u1 + v1 + u2 + v2 2 2 ∂u1 ∂v1 ∂u2 ∂v2 + γ u1 + v1 u2 + v2 u1 + v1 + u2 + v2 2 −Z + b u1 + v1 + u2 + v2 Uˆ ψ(u, v) = ✭✷✳✷✮ − ˆ ❤❛s t❤❡ ❢♦r♠ ✇❤❡r❡✱ t❤❡ ♦♣❡r❛t♦r U 2 2 Uˆ = e−(1−ic)λ(u1 +v1 +u2 +v2 ) , ✼ ✭✷✳✸✮ ♦❜t❛✐♥❡❞ ✇❤❡♥ ✇❡ t✉r♥ t❤❡ ❝♦♠♣♦♥❡♥t cos(cλr) ✐♥t♦ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢♦r♠ ❛♥❞ ❦❡❡♣ t❤❡ r❡❛❧ ♣❛rt✳ ✷✳✸ ❆❧❣❡❜r❛✐❝ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ❙❝❤r☎♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✷✳✸✳✶ ❆❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤r♦✉❣❤ ❝r❡❛t✐♦♥ ❛♥❞ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r✲ ❛t♦rs ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ❡q✉❛t✐♦♥ ✭✷✳✷✮ ❤❛s t❤❡ ❢♦r♠ ♦❢ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❢♦r t❤❡ ✹✲❞✐♠❡♥s✐♦♥❛❧ ♥♦♥✲❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r✳ ❋✐♥❛❧❧②✱ ❡q✉❛t✐♦♥ ✐s ✇r✐tt❡♥ ✐♥ ❛❧❣❡❜r❛✐❝ ❢♦r♠ ❛s ❢♦❧❧♦✇s✿ ˆ 2ˆ ωˆ K− ε˜ R + γ G + Z Sˆ (ω) |ψ = 0, 2ω 16ω ✭✷✳✹✮ ✇❤❡r❡ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ ε˜ = ε − mγ/2✳ ✷✳✸✳✷ ❇❛s✐s s❡t ❆s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ ❡q✉❛t✐♦♥ ✭✷✳✷✮ ❞❡s❝r✐❜❡s t❤❡ ♥♦♥✲❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r✱ s♦ t❤❡ ❛♣♣r♦♣r✐❛t❡ ❜❛s✐s s❡t ❢♦r s♦❧✈✐♥❣ ❡q✉❛t✐♦♥ ✐s t❤❡ ✹✲❞✐♠❡♥s✐♦♥❛❧ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r ✇❛✈❡ ❢✉♥❝t✐♦♥✳ |k1 , k2 , m (ω) = k1 !k2 !(k1 + |m|)!(k2 + |m|)! ˆ+ M k1 ˆ+ M k2 |00m (ω) ✭✷✳✺✮ ❲❡ ❝❛♥ ❢✐♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✇❤✐❝❤ ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❜❛s✐s s❡t ✭✷✳✺✮ ❛s ❢♦❧❧♦✇s✿ s |ψ (s) k (s) Ck1 ,k−k1 |k1 , k − k1 , m (ω) , = ✭✷✳✻✮ k=0 k1 =0 (s) ✇❤❡r❡ t❤❡ ❝♦❡❢❢✐❝✐❡♥ts Ck1 ,k2 ❛r❡ t❤❡ ✉♥❦♥♦✇♥s t♦ ❜❡ ❢♦✉♥❞✳ ❍❡r❡✱ t❤❡ ✐♥❞❡① s ✐s ✉s❡❞ t♦ ❧✐♠✐t t❤❡ ♥✉♠❜❡r ♦❢ ❜❛s✐s s❡ts ✐♥ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✇❡ ❝❛♥ t❤✐♥❦ ♦❢ ✐t ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ t♦ ❜❡ ❢♦✉♥❞✳ ✷✳✸✳✸ ❈❛❧❝✉❧❛t✐♥❣ ♠❛tr✐① ❡❧❡♠❡♥ts ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ♣r❡s❡♥t t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t ❝❛❧❝✉❧❛t✐♦♥s ♦❢ t❤❡ Sˆ ♦♣✲ ❡r❛t♦r✱ ✇❤✐❝❤ ♦❝❝✉rs ❞✉❡ t♦ t❤❡ ♣♦ss✐❜❧❡ s❝r❡❡♥❡❞ ♣♦t❡♥t✐❛❧✳ ❚❤❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ˆ ❛♥❞ Uˆ ✳ ❚❤❡ ♠❛tr✐① ❡❧❡♠❡♥t ♦❢ R ˆ ❛❧r❡❛❞② ❡①✐sts✱ s♦ ♦♣❡r❛t♦r Sˆ ❤❛s t✇♦ ♦♣❡r❛t♦rs R ˆ (ω)✳ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t ♦❢ t❤❡ ♦♣❡r❛t♦r U Uk1 k2 ;j1 j2 (η) = k1 k2 m| Uˆ2 Uˆ1 |j1 j2 m = Bk1 +k2 ,j1 +j2 (η) Ak1 j1 m (η) Ak2 j2 m (η) , ✭✷✳✼✮ ✽ ✇❤❡r❡ Akjm (η) = (k,j) j l l=0 j+|m| l Bn1 ,n2 (η) = k l k+|m| l+|m| η 2l (−η)n1 −n2 n1 +n2 +2|m|+2 (1 + η) ˆ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❲❡ ✉s❡ t❤❡ ❢♦r♠✉❧❛ ✭✷✳✼✮ ❢♦r t❤❡ ♠❛tr✐① ❡❧❡♠❡♥t ♦♣❡r❛t♦r U ˆ ✳ ❘❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞✿ ♠❛tr✐① ❡❧❡♠❡♥t ♦❢ S(ω) Sk1 k2 ;j1 j2 = + + b 2ω b (j1 + j2 + |m| + 1) Uk1 k2 ;j1 j2 ω j1 (j1 + |m|) Uk1 k2 ;j1 −1,j2 + j2 (j2 + |m|) Uk1 k2 ;j1 ,j2 −1 + (j1 + 1) (j1 + |m| + 1) Uk1 k2 ;j1 +1,j2 + ✭✷✳✽✮ (j2 + 1) (j2 + |m| + 1) Uk1 k2 ;j1 ,j2 +1 ✷✳✹ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❡♥❡r❣② ❛♥❞ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✷✳✹✳✶ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥s ❲❡ ✇r✐t❡ ❛ ❋♦rtr❛♥ ♣r♦❣r❛♠ ❝❛❧❧❡❞ ❍✐♥P▲❆❙✰▼❛❣ t❤❛t ✉s❡s t❤❡ s✉❜r♦✉t✐♥❡ ❞✐r❡❝t♦r② ❉❙❨●❱❳ ✐♥ t❤❡ ♣❛❝❦❛❣❡ ▲❆P❆❈❑ ❬✷✹❪ ✇✐t❤ r❡❛❧ ✈❛r✐❛❜❧❡s r❡✇r✐tt❡♥ ❛s ❘❊❆▲✯✶✻ ❡q✉✐✈❛❧❡♥t t♦ ❛♥ ❛❝❝✉r❛❝② ♦❢ ✉♣ t♦ ✸✵✲❞❡❝✐♠❛❧ ❞✐❣✐ts✳ ❲❡ ✐❧❧✉str❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ t❤r♦✉❣❤ ❚❛❜❧❡ ✷✳✶✱ r❡♣r❡s❡♥t✐♥❣ ❣r♦✉♥❞ st❛t❡ ❡♥❡r❣② 1s ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ ❜❛rr✐❡r ❧❡♥❣t❤ λ = 0.05✱ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ γ = 0.01✱ ❝♦❡❢❢✐❝✐❡♥ts b = 0.1, c = 1✳ ❙♦❧✉t✐♦♥s ✐♥ ❚❛❜❧❡ ✐s ♦❜t❛✐♥❡❞ ✇✐t❤ t✇♦ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ω = ❛♥❞ ω = 1.5✳ ❚❤❡ ♦♣t✐♠❛❧ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r ✐♥ ❋✐❣✉r❡ ✷✳✶ ❛♥❞ ❋✐❣✉r❡ ✷✳✷ ✐s ♦❜t❛✐♥❡❞ ❢♦r t❤❡ st❛t❡ 1s ✇✐t❤ ❢♦r t❤❡ s❝r❡❡♥❡❞ ♣❛r❛♠❡t❡r ❛♥❞ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✷✳✶ ✇❤✐❧❡ t❤❡ ✈❛❧✉❡ ♦❢ ♣❛r❛♠❡t❡r ω ✈❛r✐❡s ❢r♦♠ 0.5 t♦ 1.8✳ ❚❛❜❧❡ ✷✳✶✿ ✳ ■❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❣r♦✉♥❞ st❛t❡ ❡♥❡r❣② ♦✈❡r t❤❡ ❛♣♣r♦①✐♠❛t❡ ♦r❞❡r (s) ❢♦r t✇♦ ✈❛❧✉❡s ♦❢ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r✳ ❚❤❡ ❞❛t❛ ❛r❡ ❝♦❧❧❡❝t❡❞ ❢♦r t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛t♦♠✐❝ ✉♥✐ts ❛♥❞ t❤❡ s❝r❡❡♥✐♥❣ ♣❛r❛♠❡t❡rs ❚❤❡ ❛♣♣r♦①✐♠❛t❡ ♦r❞❡r c = 1✱ b = 0.1✱ ❛♥❞ γ = 0.01 λ = 0.05✳ (s) ω = ω = 1.5 ✸ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸ ✵✻✽✹ ✲✵✳✺✹✷✻✺ ✸✽✵✽ ✸✹✷✼✺✵ ✹ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✷ ✲✵✳✺✹✷✻✺✺✷ ✵✸ ✹✹✵✻✶✶ ✺ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✸ ✲✵✳✺✹✷✻✺✺✷✻ ✷✼✹✾✷✼✻ ✻ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✹ ✲✵✳✺✹✷✻✺✺✷✻✺ ✵✹✾✹✹✸ ✼ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✹ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸ ✷✹✸✶ ✽ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✹ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✷✺✽ ✾ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✹ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✶ ✶✵ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✹ ✲✵✳✺✹✷✻✺✺✷✻✺✶✸✺✸✺✸ ✾ ✐♥ Approximate order (s) 16 14 12 10 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Free parameter ❋✐❣✉r❡ ✷✳✶✿ ❉❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♦r❞❡r (s) ♦♥ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r ω ❢♦r ❣✐✈❡♥ ♣r❡❝✐s✐♦♥s ♦❢ ✶✺ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ✭r❡❞ ❧✐♥❡✮ ❛♥❞ ✶✵ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ✭❜❧❛❝❦ ❞❛s❤✮✳ ❍❡r❡✱ t❤❡ ❞❛t❛ ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❢♦r t❤❡ ❣r♦✉♥❞ st❛t❡ ✇✐t❤ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ c = 1✳ γ = 0.01 ❛♥❞ t❤❡ s❝r❡❡♥✐♥❣ ♣❛r❛♠❡t❡rs ❚❤❡ ❢✐❣✉r❡s ❝❧❡❛r❧② ❞❡♠♦♥str❛t❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ♦♣t✐♠✉♠ r❡❣✐♦♥ λ = 0.05✱ b = 0.1✱ ♦❢ ❢r❡❡ ♣❛r❛♠❡t❡r ω ✳ ❋✐❣✉r❡ ✷✳✷ s❤♦✇s t❤❡ ❡♥❡r❣✐❡s ❛t ❞✐❢❢❡r❡♥t ❛♣♣r♦①✐♠❛t✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ω ♣❛r❛♠❡t❡r✳ ❲❡ ❢✐♥❞ t❤❛t t❤❡ ❤✐❣❤❡r t❤❡ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❧♦✇❡r t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ♦♠❡❣❛ ♣❛r❛♠❡t❡r✳ ❚❤✐s ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ❢❛❝t t❤❛t t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r✳ ■♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♣r♦❣r❛♠✱ t❤❡ ♦♣t✐♠❛❧ ♣❛r❛♠❡t❡r ✈❛❧✉❡ ❤❛s ❜❡❡♥ ❝❛❧❝✉❧❛t❡❞ ✐♥ ❛❞✈❛♥❝❡ ❢♦r s♦♠❡ ❝❛s❡s ✇✐t❤ ❛ ❞✐❢❢❡r❡♥t ♠❛❣♥❡t✐❝ ❢✐❡❧❞ str❡♥❣t❤ ❢r♦♠ γ = → 10 ❛♥❞ s❝r❡❡♥❡❞ ♣❛r❛♠❡t❡r λ = → 1✳ ❚❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ✐♥t❡r♣♦❧❛t✐♦♥✳ -0.54252 -0.5426550 Engergy (a.u.) -0.54254 -0.5426551 -0.54256 -0.5426552 -0.54258 -0.5426553 -0.54260 0.9 1.0 1.1 s = -0.54262 s = s = -0.54264 -0.54266 exact 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Free parameter ❋✐❣✉r❡ ✷✳✷✿ ❉❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❡♥❡r❣② ♦♥ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r ✭❜❧❛❝❦ ❧✐♥❡✮✱ r❡❞ ❞❛s❤ s=1 ✭r❡❞ ❞❛s❤✮✱ ❛♥❞ s=3 ω ❢♦r ❣✐✈❡♥ ❛♣♣r♦①✐♠❛t❡ ♦r❞❡rs s=0 ✭❜❧✉❡ ❞♦ts✮✳ ❍❡r❡✱ t❤❡ ❞❛t❛ ❛r❡ ❝♦❧❧❡❝t❡❞ ❢♦r t❤❡ ❣r♦✉♥❞ st❛t❡ ✇✐t❤ ♣❧❛s♠❛ ♣❛r❛♠❡t❡rs ❛s ✐♥ ❋✐❣✳✷✳✶✳ ❚❤❡ ❢✐❣✉r❡s ❝❧❡❛r❧② ❝♦♥❢✐r♠ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ♦♣t✐♠✉♠ r❡❣✐♦♥ ♦❢ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r ❛s s❤♦✇♥ ✐♥ ❋✐❣✳✷✳✶ ❛♥❞✱ ♠♦r❡♦✈❡r✱ s❤♦✇ t❤❛t t❤❡ ❤✐❣❤❡r t❤❡ ❛♣♣r♦①✐♠❛t❡ ♦r❞❡r t❤❡ ❧♦✇❡r ❞❡♣❡♥❞❡♥❝❡ ♦❢ ❡♥❡r❣② ♦♥ t❤❡ ❢r❡❡ ♣❛r❛♠❡t❡r ✶✵ ω✳ 3.0 3.0 (a) 2.5 = (b) 2.5 = b = 0.1 1.0 z = 0) = 0.05 1.5 2.0 1.0 2s 1s = 0.5 = ( 2.0 ( z = 0) = 0.5 c = = 1.5 = 0.05 b = 0.1 c = 0.5 0.5 0.0 0.0 (a.u.) ❋✐❣✉r❡ ✷✳✸✿ ❚❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥s 10 (a.u.) ψ(ρ, z = 0) ❛r❡ ♣❧♦tt❡❞ ❢♦r γ = 0, 0.5✱ ❛♥❞ 2✳ t❤❡ ❣r♦✉♥❞ ❛♥❞ ❡①❝✐t❡❞ st❛t❡s 1s ✭❛✮✱ 2s ✭❜✮ ✇✐t❤ t❤r❡❡ ✈❛❧✉❡s ♦❢ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✷✳✹✳✷ ❊♥❡r❣② ❛♥❞ ✇❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠s ✐♥ ♣❧❛s♠❛ ❚❤❡ ❝❛s❡ ♦❢ ♥♦✲♠❛❣♥❡t✐❝✲❢✐❡❧❞ ♣❧❛s♠❛ ❚❤❡ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞ ❞❡s❝r✐❜❡❞ ✐♥ t❤✐s ✇♦r❦ ❛❧❧♦✇s ✉s t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡♥❡r❣② ✇✐t❤ ❛♥ ❛❝❝✉r❛❝② ♦❢ ✉♣ t♦ ✸✵ ❞❡❝✐♠❛❧ ❞✐❣✐ts✳ ❲❡ ❣✐✈❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts t❤❛t ❝♦✈❡r ❛❧❧ ♣r❡✈✐♦✉s❧② ♣✉❜❧✐s❤❡❞ r❡s✉❧ts ❢♦r ❡♥❡r❣② ❧❡✈❡❧s 1s0 ✱ 2s0 ✱ 3s0 ✱ 3d0 ✱ 4s0 ✱ 4d0 ✱ 2p0 ✱ 3p0 ❛♥❞ 4p0 ✇✐t❤ ✈❛r✐❛❜❧❡ ♣❛r❛♠❡t❡rs b, c ❛♥❞ λ✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❛❧s♦ ♦❜t❛✐♥ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ ❞✐❢❢❡r❡♥t ✈❛❧✉❡s ♦❢ λ✱ b✱ c ❛♥❞ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ r❡❢❡r❡♥❝❡ ❬✷✷❪✳ ❚❤❡ ❝❛s❡ ♦❢ ♣❧❛s♠❛ ♣❧❛❝❡❞ ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❚❤✐s s❡❝t✐♦♥ ❛❧s♦ ❣✐✈❡s ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r t❤❡ ❝❛s❡ ♦❢ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✇✐t❤ q✉❛♥t✉♠ st❛t❡s ❤❛✈✐♥❣ ♣r✐♥❝✐♣❛❧ q✉❛♥t✉♠ ♥✉♠❜❡r n ≤ 10 ❛♥❞ ♠❛❣♥❡t✐❝ q✉❛♥t✉♠ ♥✉♠❜❡r |m| ≤ 10✳ ❆❝❝✉r❛❝② ✉♣ t♦ ✷✺ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ❛♥❞ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✉♣ t♦ ✶✵ ❛✳✉✳✳ ❲❡ ♣❧♦t t❤❡ ✇❛✈❡❢✉♥❝t✐♦♥ Ψnlm (ρ, z = 0) ✐♥ ρ ❛s ❋✐❣✉r❡ ✷✳✸ ❢♦r t❤❡ ❝❛s❡ z = ❛♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ Ψnlm (ρ, z) ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s ρ, z ❛s ❋✐❣✉r❡ ✷✳✹ ❢♦r ♣❧❛s♠❛ ♣❛r❛♠❡t❡rs λ = 0.05✱ b = 0.1 ❛♥❞ c = ❛s t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✐♥❝r❡❛s❡s ❢r♦♠ 0, 0.5, ❛✳✉✳✳ ✷✳✺ ❈♦♥❝❧✉s✐♦♥ ❋✐rst✱ ✇❡ ❝♦♥✈❡rt❡❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ t❤❡ ♣❧❛s♠❛ t♦ t❤❡ ♥♦♥✲ ❤❛r♠♦♥✐❝✲♦s❝✐❧❧❛t♦r ❢♦r♠ ❛♥❞ ❝❛❧❝✉❧❛t❡❞ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ♦❢ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ s❝r❡❡♥❡❞ ♣♦t❡♥t✐❛❧ ▼●❊❈❙❈ ✐♥t♦ ❛♥ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥✱ t❤❡♥ s♦❧✈❡❞ t❤❡ s②st❡♠ ♦❢ ❡✐❣❡♥✈❛❧✉❡s✳ ◆❡①t✱ ✇❡ ✇r✐t❡ ❛ ❋❖❘❚❘❆◆ ♣r♦❣r❛♠ t❤❛t ❝♦♠♣✉t❡s t❤❡ ❣r♦✉♥❞ st❛t❡ ❡♥❡r❣② ❛s ✇❡❧❧ ❛s t❤❡ ❤✐❣❤ ❡①❝✐t❡❞ st❛t❡s ✇✐t❤ ♣r✐♥❝✐♣❛❧ q✉❛♥t✉♠ ♥✉♠❜❡rs ✉♣ t♦ n = 10✳ ❚❤❡ ♣r♦❣r❛♠ ✐s t❡st❡❞ t♦ ✇♦r❦ ✇❡❧❧ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❛r❛♠❡t❡rs✿ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✉♣ t♦ ✶✵ ❛✳✉✳ ✭2.3505 × 106 ❚❡s❧❛✮✱ t❤❡ s❝r❡❡♥❡❞ ♣❛r❛♠❡t❡r λ ✐s ✉♣ t♦ ✶ ❛✳✉✳ ❛♥❞ ♦t❤❡r ♣❛r❛♠❡t❡rs ❝♦✈❡r ♦t❤❡r ❡①♣❡r✐♠❡♥t❛❧ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❧❛s♠❛ ❞❛t❛✳ ❋✐♥❛❧❧②✱ ✇❡ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦♣t✐♠❛❧ ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ t❤❛t ❛❧❧♦✇s ❛ ❢❛st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r t❤❡ ❡♥❡r❣② ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❝❤❛♣t❡r ✶✶ (a) -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 z (a.u.) -1 -2 -3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 -1 -2 -3 (a.u.) -0.1 0.2 0.5 0.8 1.1 1.4 1.7 2.0 2.3 2.6 2.9 3.2 -1 -2 -3 (a.u.) (a.u.) (b) z (a.u.) -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -2 -4 (a.u.) ❋✐❣✉r❡ ✷✳✹✿ ❈♦♥t♦✉r ♣❧♦ts ♦❢ ✇❛✈❡ ❢✉♥❝t✐♦♥s 1s ✭❛✮✱ 2s 10 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 -5 -5 -10 (a.u.) ψ(ρ, z) ✭❜✮ ✇✐t❤ t❤r❡❡ ✈❛❧✉❡s ♦❢ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ -3.2 -2.8 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 (a.u.) ❛r❡ ♣❧♦tt❡❞ ❢♦r t❤❡ ❣r♦✉♥❞ ❛♥❞ ❡①❝✐t❡❞ st❛t❡s γ = 0, 0.5✱ ❛♥❞ 2✳ ❚❤❡ ❣r❛♣❤s ❛r❡ t❤❡ s❛♠❡ ❢♦r♠❛t✐♦♥s ❛s t❤♦s❡ ♣r❡s❡♥t❡❞ ✐♥ ❘❡❢✳ ❬✷✺❪ ❤❛✈❡ ❛ r❡❝♦r❞ ❛❝❝✉r❛❝② ♦❢ ✉♣ t♦ ✸✵ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳ ❈❤❛♣t❡r ✸ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❢♦r ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡rs ♣❧❛❝❡❞ ✐♥ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ r❡tr✐❡✈✐♥❣ ❚▼❉ ♠♦♥♦❧❛②❡rs str✉❝t✉r❡ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ✸✳✶ ❖✈❡r✈✐❡✇ ❚❤❡ ❡♥❡r❣② s♣❡❝tr❛ ♦❢ ❡①❝✐t♦♥s ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡rs ✐♥ ❛ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛r❡ ♦❢ ❣r❡❛t ✐♥t❡r❡st ❬✻✱ ✶✼✱ ✶✾❪✳ ❇❡❝❛✉s❡ ❢r♦♠ t❤❡ ❡♥❡r❣② s♣❡❝tr❛✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ q✉❛♥t✐❢② t❤❡ ❜❛s✐❝ ♠❛t❡r✐❛❧ ♣❛r❛♠❡t❡rs ♦❢ ❚▼❉ ♠♦♥♦❧❛②❡rs s✉❝❤ ❛s ❡❢❢❡❝t✐✈❡ ♠❛ss✱ ♣♦❧❛r✲ ✐③❛t✐♦♥✱ ❞✐❡❧❡❝tr✐❝ ❝❛✉s❡❞ ❜② t❤❡ s❝r❡❡♥✳ ❚❤❡s❡ ♠❛t❡r✐❛❧ ♣❛r❛♠❡t❡rs ❢♦r♠ t❤❡ ✐♥♣✉t r❡q✉✐r❡❞ ❢♦r ♠❛♥② ❞❡s✐❣♥s ✐♥ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ♦♣t♦❡❧❡❝tr♦♥✐❝s ❬✻❪✳ ■♥ ♠♦st st✉❞✐❡s✱ t❤❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❡①❝✐t♦♥ ✐♥ t❤❡ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛r❡ s♦❧✈❡❞ ❜② t❤❡ ✈❛r✐❛t✐♦♥ ❬✻✱✶✼✱✶✾❪ ✇✐t❤ ♣r❡❝✐s✐♦♥ ❡♥♦✉❣❤ t♦ ❛♥❛❧②③❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✇✐t❤ ❛♥ ❡rr♦r ♦❢ 4%✳ ❍♦✇❡✈❡r✱ ✐♥ ♦r❞❡r t♦ r❡tr✐❡✈❡ ♠♦♥♦❧❛②❡r ♣r♦♣❡rt✐❡s ❢r♦♠ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛✱ ✐♥✈❡st✐❣❛t✐♥❣ ❛♥ ❡❢❢✐❝✐❡♥t ♠❡t❤♦❞ ♦❢ s♦❧✈✐♥❣ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✐s st✐❧❧ ♥❡❝❡ss❛r②✳ ■♥ t❤✐s st✉❞②✱ ✇❡ ❞❡✈❡❧♦♣ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❢♦r ❡①❝✐t♦♥ ✇✐t❤ ♠❛♥② q✉❛♥t✉♠ st❛t❡s ❛♥❞ ♥♦♥✲③❡r♦ ❛♥❣✉❧❛r ✶✷ ♠♦♠❡♥t✉♠ ❜② ❝❛❧❝✉❧❛t✐♥❣ ♠❛tr✐① ❡❧❡♠❡♥ts ❢♦r ❑❡❧❞②s❤✲♣♦t❡♥t✐❛❧ ✉s✐♥❣ t❤❡ ✐♥t❡❣r❛❧ ❢♦r♠ ✈✐❛ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ r❡✈❡rs❡✳ ❖♥❡ ♦❢ t❤❡ ❢✐rst t♦ s✉❣❣❡st t❤❡ s❝r❡❡♥❡❞ ♣♦t❡♥t✐❛❧ ✐♥ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①❝✐t♦♥ ✇❛s ▲✳❱✳ ❑❡❧❞②s❤✱ ✶✾✼✾ ❬✶✻❪ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❛ ◆❡✉♠❛♥♥ ❢✉♥❝t✐♦♥ ✭❛❧s♦ ❦♥♦✇♥ ❛s ❛ ❇❡ss❡❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞✮ ❛♥❞ ❛ ❙tr✉✈❡ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❡❧❡❝tr♦♥ ❛♥❞ t❤❡ ❤♦❧❡ r✱ t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❢✐❧♠ d ❛♥❞ t❤❡ ❞✐❡❧❡❝tr✐❝ ❝♦♥st❛♥ts ♦❢ t❤❡ ❢✐❧♠ ε ❛♥❞ t❤❡ ❞✐❡❧❡❝tr✐❝ ❧❛②❡rs s❛♥❞✇✐❝❤❡❞ ♦♥ t❤❡ ♦✉ts✐❞❡ ♦❢ t❤❡ ❢✐❧♠ ε1 ❛♥❞ ε2 ✳ ❚❤❡ ❧❛t❡st t❤❡♦r❡t✐❝❛❧ ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ✇♦r❦s ❤❛✈❡ ♦❜t❛✐♥❡❞ s✐♠✐❧❛r r❡s✉❧ts ✇❤❡♥ ✉s✐♥❣ t❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ♠♦❞❡❧ ❢♦r ❡①❝✐t♦♥s ❛♥❞ tr✐♦♥s ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡rs ✐♥ ❝❛s❡ ♦❢ ♥♦ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❬✻❪✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❝♦♥t✐♥✉❡ t♦ ❝♦♥s✐❞❡r t❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ❢♦r t❤❡ ❡①❝✐t♦♥✳ ✸✳✷ ❙❝❤r☎ ♦❞✐♥❣❡r ❊q✉❛t✐♦♥ ❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♦❢ t❤❡ ❡①❝✐t♦♥ ✐♥ t❤❡ ❚▼❉ ♠♦♥♦❧❛②❡rs ❤❛s ❛ ❝♦♥s❡r✈❡❞ ❛♥✲ ❣✉❧❛r ♠♦♠❡♥t✉♠ ♦❢ t❤❡ ❢♦r♠ mh − me eB ˆ e2 B 2 ˆ ˆ pˆ + lz + r + Vh−e (r), Hrel = 2µ me mh 8µ ✭✸✳✶✮ ❛♥❞ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ ❛t♦♠✐❝ ✉♥✐ts ❛s ❢♦❧❧♦✇s✿ − ∂2 ∂2 + ∂x2 ∂y γ2 mαex γ + (x + y ) + Vˆh−e (r) + − E ψ(x, y) = ✭✸✳✷✮ ❚❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ❡st❛❜❧✐s❤❡❞ ❢♦r t❤❡ ❡❧❡❝tr♦♥✲❤♦❧❡ ✐♥t❡r❛❝t✐♦♥ ✐s r❡✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠ ♦❢ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ +∞ dq VK (r, α, κ) = − κ + α2 q ❡−qr , ✭✸✳✸✮ ✇❤❡r❡ ❞✐♠❡♥s✐♦♥❧❡ss ♣❛r❛♠❡t❡r α = r0 /κa∗0 ✐s ✉s❡❞ ✐♥st❡❛❞ ♦❢ s❝r❡❡♥✐♥❣ ❧❡♥❣t❤ r0 ✳ ✸✳✸ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ t♦ t❤❡ ❙❝❤r☎♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✸✳✸✳✶ ❆❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ✳ ❡q✉❛t✐♦♥ ❋✐rst✱ ❜② t❤❡ ▲❡✈✐✲❈✐✈✐t❛ tr❛♥s❢♦r♠❛t✐♦♥ ✇❡ r❡✇r✐t❡ t❤❡ ❙❝❤r☎♦❞✐♥❣❡r ✭✸✳✷✮ ❡q✉❛✲ t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ (u, v)✱ ❛♥❞ t❤❡♥ ✇❡ r❡✇r✐t❡ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ✭❄❄✮ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❛❧❣❡❜r❛✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ω2 ˆ γ2 ˆ + ˆ |ψ = 0, ˆ ˆ N − M − M + R + ω Vˆ − E R 8ω ✇❤❡r❡ E = E − mαex γ/2✳ ✶✸ ✭✸✳✹✮ ✸✳✸✳✷ ❇❛s✐s s❡t ❛♥❞ ❛♥❛❧②t✐❝ ♠❛tr✐① ❡❧❡♠❡♥ts ❲❡ ❝❤♦♦s❡ t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r ✇❛✈❡❢✉♥❝t✐♦♥s ❛s t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥ (ˆ a+ )n+m (ˆb+ )n−m |0(ω) , ✭✸✳✺✮ |n, m = (n + m)!(n − m)! ✇❤❡r❡ m ✐s t❤❡ ♠❛❣♥❡t✐❝ q✉❛♥t✉♠ ♥✉♠❜❡r ✇✐t❤ t❤❡ ✈❛❧✉❡s m = 0, ±1, ±2, ✳ ❚❤❡ ♣r✐♥❝✐♣❛❧ q✉❛♥t✉♠ ♥✉♠❜❡r n ✐s ❛♥ ✐♥t❡❣❡r ✇✐t❤ t❤❡ ✈❛❧✉❡s n ≥ |m|✳ ❚❤❡♥ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ♠❛tr✐① ❡❧❡♠❡♥ts ❢♦r t❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ✐♥ t❤❡ ❢♦r♠ ✭✸✳✸✮ ✐♥ ❛❧❣❡❜r❛✐❝ ❢♦r♠ t❤r♦✉❣❤ t❤❡ ❝r❡❛t✐♦♥ ❛♥❞ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦rs ❛s ❢♦❧❧♦✇s✿ Vjk = j, m|ω Vˆ |k, m k − m2 Uj,k−1 + = (2k + 1) Ujk + (k + 1)2 − m2 Uj,k+1 , ✭✸✳✻✮ ð ✤➙② Ujk = − κα × min(k,j) j+k−2s (−1)j+k+t s=|m| j+m s+m t=0 j−m s−m k+m s+m j + k − 2s t k−m J2s+t+1 (1/ω α2 ) s−m ■♥ ❡q✉❛t✐♦♥ ✭✸✳✼✮✱ t❤❡ ✐♥t❡❣r❛❧ ✐s ❞❡❢✐♥❡❞ ❜② Jp (x) = +∞ (1+q)p dq √ q +x2 ✭✸✳✼✮ ✇✐t❤ p ≥ ❛♥❞ x > ✐s ❡❛s✐❧② ❝❛❧❝✉❧❛t❡❞ ♥✉♠❡r✐❝❛❧❧② ✉s✐♥❣ ❛ r❡❝✉rs✐♦♥ ❢♦r♠✉❧❛✳ ✸✳✸✳✸ ❊①❛❝t ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ▲✐❦❡ ✐♥ ❈❤❛♣t❡r ✷✱ ✇❡ ❢✐♥❞ t❤❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ❡q✉❛t✐♦♥ ✭✸✳✹✮ ✐♥ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❜❛s✐s s❡t ✭✸✳✺✮ ❛s s+1 |ψ (s) (s) Cj |j − + |m|, m , = ✭✸✳✽✮ j=1 ✇❤❡r❡ (s) r❡♣r❡s❡♥ts t❤❡ ❛♣♣r♦①✐♠❛t❡ ❞❡❣r❡❡✳ ❲❡ t❤❡♥ r❡✇r✐t❡ ❛♥❞ s♦❧✈❡ t❤❡ s②♠✲ ♠❡tr✐❝ ❣❡♥❡r❛❧ ❡✐❣❡♥✈❛❧✉❡s ❛s H − E R C = 0, (s) ✭✸✳✾✮ ✐♥ ✇❤✐❝❤ ❝♦❧✉♠♥ C ❤❛s s + ✉♥❦♥♦✇♥ ❡❧❡♠❡♥t Cj ❀ R ❛♥❞ H ❛r❡ sq✉❛r❡ ♠❛tr✐❝❡s ❤❛✈✐♥❣ (s + 1) × (s + 1) ❡❧❡♠❡♥ts✳ ❋✐♥❛❧❧② ✇❡ ♦❜t❛✐♥ ❛ ♥✉♠❡r✐❝❛❧❧② ❝♦rr❡❝t ❡♥❡r❣② s♣❡❝tr❛✳ ✶✹ ✸✳✹ ❘❡s✉❧ts ❛♥❞ ❉✐s❝✉ss✐♦♥ ✸✳✹✳✶ ❙❡♥s✐t✐✈✐t② ♦❢ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ❢♦r str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ▼♦♥♦❧❛②❡r str✉❝t✉r❡ ♣❛r❛♠❡t❡rs s✉❝❤ ❛s✿ r0 ❛♥❞ µ/me ❤❛✈❡ ❛ s✐❣♥✐❢✐❝❛♥t ✐♥✲ ❢❧✉❡♥❝❡ ♦♥ t❤❡ ❡♥❡r❣② s♣❡❝tr❛ ♦❢ t❤❡ ❡①❝✐t♦♥✳ ❚❤✐s ❞✐❢❢❡r❡♥❝❡ ✐s s✐❣♥✐❢✐❝❛♥t✱ s♦ ❛♥ ❡❢❢✐❝✐❡♥t ♠❡t❤♦❞ ✐s ♥❡❡❞❡❞ t♦ r❡tr✐✈❡ ♠♦r❡ ❛❝❝✉r❛t❡❧② t❤❡ r0 ❛♥❞ µ/me ♣❛r❛♠❡t❡rs✳ ✸✳✹✳✷ ❘❡tr✐❡✈✐♥❣ t❤❡ ❡❢❢❡❝t✐✈❡ ♠❛ss ♦❢ t❤❡ ❡①❝✐t♦♥ ❛♥❞ t❤❡ ♣♦❧❛r✐③❛t✐♦♥ ❲❡ ❝❛❧❝✉❧❛t❡ t❤❡ ❛✈❡r❛❣❡ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ❢♦r t❤❡ t❤r❡❡ ❡♥❡r❣② ❧❡✈❡❧s ✶s✱ ✷s ❛♥❞ ✸s ❛♥❞ ❋✐❣✉r❡ ✸✳✶ ✐s ♦❜t❛✐♥❡❞✳ ■t ✐s s❤♦✇♥ t❤❛t t❤❡ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❛♥❞ ❬✶✼❪ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❡♥❡r❣② ❜❡t✇❡❡♥ st❛t❡s ✶s✱ ✷s ❛♥❞ ✸s ✐s δmin = 0.008% ✭8 × 10−5 ✮✳ ❋r♦♠ ❤❡r❡ ✇❡ r❡tr✐❡✈❡ t❤❡ ❜❡st str✉❝t✉r❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❢♦r ❲❙❡2 ❛s✿ κ = 4.5✱ r0 = 4.209 ± 0.003 ♥♠ ❛♥❞ µ/me = 0.2039 ± 0.0001 ❝♦rr❡s♣♦♥❞s t♦ t❤❡ tr❛♥s✐t✐♦♥ ❡♥❡r❣② ✷s ✕ ✶s ✐s 130.00 ♠❡❱ ❛♥❞ ✸s ✕ ✷s ✐s 22.00 ♠❡❱✳ (r , m/m ) 4.23 r0(nm) 4.22 4.21 4.20 4.19 4.18 4.17 4.16 0.201 0.202 0.203 /m ✐♥ ❝❛s❡ ♦❢ ♠❡❛♥ ❡❧❡❝tr✐❝❛❧ ❝♦♥st❛♥t 0.204 0.205 8.00x10 -4 5.80x10 -3 1.08x10 -3 1.58x10 -3 2.08x10 -3 2.58x10 -3 3.08x10 -3 3.58x10 -3 4.08x10 -3 4.58x10 -3 5.08x10 -3 5.58x10 -3 6.08x10 -3 6.58x10 -3 7.08x10 -3 7.58x10 -3 8.08x10 -3 8.58x10 -3 9.08x10 -3 9.58x10 -2 1.01x10 -2 1.06x10 e ❋✐❣✉r❡ ✸✳✶✿ ❉❡♣❡♥❞❡♥❝❡ ♦❢ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ µ/me e -5 4.24 δ s✐♠✉❧t❛♥❡♦✉s❧② ♦♥ s❝r❡❡♥✐♥❣ κ = 4.5 ❢r♦♠ r❡❛❧ ❞❛t❛ t❡st ❬✶✼❪✳ ❧❡♥❣t❤ r0 ❛♥❞ ♠❛ss r❛t✐♦ ❚❤✉s✱ ❢♦r t❤❡ ❣✐✈❡♥ ✈❛❧✉❡ κ = 4.5 ❛❧♦♥❣ ✇✐t❤ t❤r❡❡ ❡♥❡r❣② ❧❡✈❡❧s 1s, 2s ❛♥❞ 3s ❡①♣❡r✐♠❡♥t❛❧❧② ♠❡❛s✉r❡❞✱ ✇❡ ❤❛✈❡ r❡tr✐❡✈❡❞ t✇♦ ♣❛r❛♠❡t❡rs r0 ❛♥❞ µ/me ✳ ❚♦ r❡tr✐❡✈❡ t❤❡ t❤r❡❡ str✉❝t✉r❡ ♣❛r❛♠❡t❡rs ✇❡ ♥❡❡❞ t♦ ❤❛✈❡ ❛t ❧❡❛st ❢♦✉r ❡♥❡r❣② ❧❡✈❡❧s✳ ❲❡ ❛❧s♦ r❡tr✐❡✈❡❞ t❤r❡❡ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ❛s ❋✐❣✉r❡ ✸✳✷✳ ■♥ ✇❤✐❝❤✱ ❋✐❣✉r❡ ✸✳✷❛ s❤♦✇s t❤❛t ❢♦r t❤❡ ❝❛s❡ ♦❢ ♠❡❛♥ ❡❧❡❝tr✐❝❛❧ ❝♦♥st❛♥t κ = 4.5✱ ✇❡ ❢✐♥❞ t❤❡ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ εmin = 3.89%✳ ❲❡ ❢✉rt❤❡r ✐♥✈❡st✐❣❛t❡ t❤❡ ❞✐❢❢❡r❡♥t κ ✐♥st❛♥❝❡s ❢r♦♠ 4.0 t♦ 5.0 ❛♥❞ r❡tr✐❡✈❡ t❤r❡❡ ❚▼❉ ♠♦♥♦❧❛②❡r str✉❝t✉r❡ ♣❛r❛♠❡t❡rs κ = 4.45 ± 0.05✱ r0 = 4.249 ± 0.003 ♥♠ ❛♥❞ µ/me = 0.2102 ± 0.0002✳ ❲❡ ❛❧s♦ r❡tr✐❡✈❡❞ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ❢♦r ❚▼❉ ♠♦♥♦❧❛②❡rs ❢r♦♠ ❞✐❢❢❡r❡♥t s✉❜st❛♥❝❡s ❛♥❞ ❝♦♠♣❛r❡❞ t❤❡♠ ✇✐t❤ ♦t❤❡r ✇♦r❦s ❬✻✱ ✶✼✱ ✶✾❪ ✐♥ ❚❛❜❧❡ ✸✳✶✳ ✶✺ (r , (a) 3.89x10 -2 3.91x10 -2 3.93x10 -2 3.95x10 -2 3.97x10 -2 3.99x10 -2 4.01x10 -2 4.03x10 -2 4.05x10 -2 4.07x10 -2 4.09x10 -2 4.11x10 -2 4.13x10 -2 4.15x10 -2 4.17x10 -2 4.19x10 -2 4.21x10 -2 4.23x10 -2 4.25x10 -2 4.27x10 -2 4.29x10 -2 4.31x10 -2 4.33x10 -2 4.35x10 -2 4.37x10 -2 4.39x10 4.26 4.25 4.24 4.23 0.209 0.210 /m 0.211 0.212 (b) 5.4 5.2 5.0 4.8 4.6 4.27 r0(nm) e -2 4.28 0.208 /m ) 4.4 4.2 4.0 3.8 4.0 4.2 µ/me 4.6 4.8 5.0 e ❋✐❣✉r❡ ✸✳✷✿ ✭❛✮ ❉❡♣❡♥❞❡♥❝❡ ♦❢ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ r❛t✐♦ 4.4 ✐♥ ❝❛s❡ ♦❢ ♠❡❛♥ ❡❧❡❝tr✐❝❛❧ ❝♦♥st❛♥t ✭❜✮ ▼✐♥✐♠✉♠ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥ ❞❡♣❡♥❞❡♥❝❡ ε s✐♠✉❧t❛♥❡♦✉s❧② κ = 4.5❀ εmin ♦♥ s❝r❡❡♥✐♥❣ ❧❡♥❣t❤ ♦♥ ♠❡❛♥ ❡❧❡❝tr✐❝❛❧ ❝♦♥st❛♥t κ✳ r0 ❛♥❞ ♠❛ss ❊①♣❡r✐♠❡♥t❛❧ ❞❛t❛ t❛❦❡♥ ❢r♦♠ t❤❡ ✇♦r❦ ❬✶✾❪✱ t❤❡ ❡♥❡r❣② ❧❡✈❡❧s ♦❢ ✶s✱ ✷s✱ ✸s ❛♥❞ ✹s ❛r❡ ✶✱✼✶✷✱ ✶✳✽✹✸✱ ✶✱✽✻✹✱ ✶✱✽✼✸ ❡❱✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❛❜❧❡ ✸✳✶✿ ❙tr✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ❢♦r ❚▼❉ ♠♦♥♦❧❛②❡rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❜st❛♥❝❡s ❲❙❡2 ✱ ❲❙2 ✈➔ ▼♦❙2 ✳ ❈❤➜t µ/m0 Eb Egap ✭♠❡❱✮ ✭❡❱✮ ❲❙❡2 ✵✳✷✵✸✾ ✶✻✽✳✺✻ ✶✳✽✾✷ ✵✳✷✵ ✶✼✷ ✵✳✷✵ ✶✻✼ ✵✳✶✼✾✶ ✵✳✶✼✺ ✵✳✷✻✺✷ ✵✳✷✼✺ ❲❙2 ▼♦❙2 ✸✳✹✳✸ κ r0 r1s ✭♥♠✮ ✭♥♠✮ σ µ❡❱✴❚2 ✹✳✺ ✹✳✷✵✽✻ ✶✳✻✷✷ ✵✳✷✽✹ ❬❋❑❪ ✶✳✽✽✹ ✸✳✾✼ ✺ ✶✳✻ ✵✳✷✹ ❬✶✾❪ ✶✳✽✾ ✹✳✺ ✹✳✺ ✶✳✼ ✵✳✸✷ ❬✶✼❪ ✶✼✼✳✹✺ ✷✳✷✸✻ ✹✳✸✺ ✸✳✺✹✷✺ ✶✳✻✺✸ ✵✳✸✸✺ ❬❋❑❪ ✶✽✵ ✷✳✷✸✽ ✹✳✸✺ ✸✳✹ ✶✳✽ ✵✳✹ ❬✻❪ ✷✷✵✳✼✶ ✷✳✶✺✾ ✹✳✹✺ ✸✳✷✻✺✷ ✶✳✷✹✻ ✵✳✶✷✾ ❬❋❑❪ ✷✷✶ ✷✳✶✻ ✹✳✹✺ ✸✳✹ ✶✳✷ ✵✳✶✷ ❬✻❪ ❍✐❣❤✲♣r❡❝✐s✐♦♥ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❡♥❡r❣② t♦ ✶✷ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳ ❚❛❜❧❡s ✸✳✷ ✐♥ ❝❛s❡ ♦❢ m = 0✱ ❛♥❞ ✸✳✸ ✐♥ ❝❛s❡ ♦❢ m = −1 ❛r❡ s❤♦✇♥ ❛s ❛♥ ❡①❛♠♣❧❡ ❢♦r t❤❡ ❝❛s❡ r0 ❂ ✹✳✷✵✽✻ ♥♠✱ µ ❂ ✵✳✷✵✸✾ m0 ✱ κ = 4.5 ✇✐t❤ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ str❡♥❣t❤ ❢r♦♠ t♦ 200 ❚✳ ❚❛❜❧❡ ✸✳✷✿ ❚❤❡ ❡♥❡r❣✐❡s ♦❢ t❤❡ st❛t❡s ❢♦r ❢✐❡❧❞ B ✐s ❚ ✐♥ t❤❡ ❝❛s❡ ♦❢ r0 m = 0✿ 1s0 ✱ 2s0 ❛♥❞ 3s0 ✱ t❤❡ ❡♥❡r❣② ✉♥✐t ✐s ❡❱✱ t❤❡ ♠❛❣♥❡t✐❝ µ ❂ ✵✳✷✵✸✾ m0 ✱ κ = 4.5✳ ❂ ✹✳✷✵✽✻ ♥♠✱ B 1s0 2s0 3s0 ✵ ✲✵✳✶✻✽✺✺✻✵✶✸✻✷✷ ✲✵✳✵✸✽✺✺✹✸✷✷✸✻✵ ✲✵✳✵✶✻✺✺✶✼✸✻✷✵✵ ✶✵ ✲✵✳✶✻✽✺✷✼✻✼✵✽✽✼ ✲✵✳✵✸✽✵✻✻✻✻✶✹✵✵ ✲✵✳✵✶✹✷✵✹✷✻✼✾✷✶ ✸✵ ✲✵✳✶✻✽✸✵✶✻✺✹✺✺✵ ✲✵✳✵✸✹✺✵✼✸✻✶✻✷✸ ✲✵✳✵✵✶✷✽✷✽✸✼✽✺✹ ✺✵ ✲✵✳✶✻✼✽✺✸✸✻✻✸✾✺ ✲✵✳✵✷✽✹✷✶✻✶✽✼✵✵ ✵✳✵✶✻✷✻✻✵✵✺✵✷✼ ✶✵✵ ✲✵✳✶✻✺✽✶✶✸✻✻✽✺✶ ✲✵✳✵✵✻✼✺✵✾✵✻✶✾✺ ✵✳✵✻✽✶✷✸✹✻✵✶✽✸ ✷✵✵ ✲✵✳✶✺✽✸✺✵✺✷✺✻✷✵ ✵✳✵✺✵✵✺✻✸✷✽✻✻✸ ✵✳✶✽✺✷✽✻✵✺✶✼✽✶ ✶✻ ❚❛❜❧❡ ✸✳✸✿ ❚❤❡ ❡♥❡r❣✐❡s ♦❢ t❤❡ st❛t❡s ❢♦r ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✸✳✺ B ✐s ❚ ✐♥ t❤❡ ❝❛s❡ ♦❢ r0 ❂ m = 0✿ 2p−1 ✱ 3p−1 ✈➔ 4p−1 ✱ t❤❡ ❡♥❡r❣② ✹✳✷✵✽✻ ♥♠✱ µ ❂ ✵✳✷✵✸✾ m0 ✱ κ = 4.5✳ B 2p−1 3p−1 4p−1 ✵ ✲✵✳✵✹✾✼✽✷✵✺✸✻✻✻ ✲✵✳✵✶✾✺✺✽✼✼✼✶✻✸ ✲✵✳✵✶✵✸✷✻✻✺✹✹✸✺ ✶✵ ✲✵✳✵✺✷✸✽✹✷✾✺✾✷✷ ✲✵✳✵✷✵✼✾✼✹✸✹✵✺✽ ✲✵✳✵✵✽✸✸✵✹✼✸✽✻✺ ✸✵ ✲✵✳✵✺✻✷✼✵✸✹✷✸✶✺ ✲✵✳✵✶✼✵✷✺✷✷✼✼✸✾ ✵✳✵✵✻✾✵✸✻✽✼✽✽✾ ✺✵ ✲✵✳✵✺✽✼✸✷✹✽✸✶✽✺ ✲✵✳✵✵✾✸✼✹✷✶✺✹✻✸ ✵✳✵✷✻✺✷✼✷✹✵✼✶✾ ✶✵✵ ✲✵✳✵✻✵✽✺✶✼✵✵✺✾✶ ✵✳✵✶✻✺✵✼✽✸✷✽✺✶ ✵✳✵✽✷✶✹✶✽✾✵✻✺✻ ✷✵✵ ✲✵✳✵✺✻✵✸✺✽✻✷✻✹✽ ✵✳✵✼✾✹✸✶✾✷✶✻✷✽ ✵✳✷✵✸✼✼✽✷✶✵✹✾✼ ✉♥✐t ✐s ❡❱✱ t❤❡ ❈♦♥❝❧✉s✐♦♥s ❲❡ ❤❛✈❡ ❡①♣r❡ss❡❞ t❤❡ ❑❡❧❞②s❤ ✐♥t❡r❛❝t✐♦♥ ♣♦t❡♥t✐❛❧ ❛s ❛♥ ✐♥t❡❣r❛❧ ❢♦r♠ ✉s✲ ✐♥❣ t❤❡ ✐♥✈❡rs❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠✱ s✉❝❝❡ss❢✉❧❧② r❡tr✐❡✈❡❞ ❚▼❉ ♠♦♥♦❧❛②❡rs ❢♦r t❤❡ s✉❜st❛♥❝❡s ❲❙❡2 ✱ ❲❙2 ❛♥❞ ▼♦❙2 ❛♥❞ ✇r✐tt❡♥ ❛ ♣r♦❣r❛♠ ✐♥ ❋❖❘❚❘❆◆ ❧❛♥❣✉❛❣❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr❛ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✇✐t❤ ❛ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✶✷ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ♦✉t❧♦♦❦ ■♥ t❤✐s t❤❡s✐s✱ ✇❡ ❤❛✈❡ s♦❧✈❡❞ t❤❡ ♣r♦♣♦s❡❞ r❡s❡❛r❝❤ ♦❜❥❡❝t✐✈❡s ✇✐t❤ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ s♣❡❝✐❢✐❝ r❡s✉❧ts✿ ✶✳ ❙②st❡♠❛t✐❝❛❧❧② ♣r❡s❡♥t✐♥❣ t❤❡ ♣❤②s✐❝❛❧ ❜❛s✐s ❛♥❞ s❡♣❛r❛t✐♦♥ ♣r♦❝❡ss ♦❢ ❝❡♥t❡r ♦❢ ♠❛ss ♠♦t✐♦♥ ✐♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ t✇♦✲♣❛rt✐❝❧❡ s②st❡♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ❛♥❞ ❞✐s❝✉ss✐♥❣ t❤❡ ✐♠♣♦rt❛♥t ❡❢❢❡❝t ♦❢ t❤✐s ♦♥ t❤❡ ❡①❝✐t♦♥ ♣r♦❜❧❡♠ ✐♥ ♠❛❣♥❡t✐❝ ❢✐❡❧❞s✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✐♥✈❡st✐❣❛t✐♥❣ t❤❡ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝t ♦♥ t❤❡ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ❜② ❛ ♥❡✇ ♠❡❝❤❛♥✐s♠✳ ✷✳ ❈❛❧❝✉❧❛t✐♥❣ ✇✐t❤ ❤✐❣❤ ♣r❡❝✐s✐♦♥ t❤❡ ❡♥❡r❣② ❛♥❞ ✇❛✈❡❢✉♥❝t✐♦♥ ❢♦r ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ♣❧❛s♠❛ ✇✐t❤ ❛ ❣❡♥❡r❛❧ s❝r❡❡♥✐♥❣ ♣♦t❡♥t✐❛❧ ♣❧❛❝❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ✸✳ ❉❡✈❡❧♦♣✐♥❣ ♦❢ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❢♦r ♥❡✉tr❛❧ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ✐♥ ❚▼❉ ♠♦♥♦❧❛②❡r ♣❧❛❝❡❞ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ❋r♦♠ ❤❡r❡ ❝♦♠♠✐♥❣ t✇♦ ✐♠✲ ♣♦rt❛♥t ❛♣♣❧✐❝❛t✐♦♥s ✐♥❝❧✉❞✐♥❣✿ ✭❛✮ ❘❡tr✐❡✈✐♥❣ str✉❝t✉r❛❧ ✐♥❢♦r♠❛t✐♦♥ ♦❢ ❚▼❉ ♠♦♥♦❧❛②❡rs ✭r❡❞✉❝❡❞ ♠❛ss✱ s❝r❡❡♥✐♥❣ ❧❡♥❣t❤✱ ❞✐❡❧❡❝tr✐❝ ❝♦♥st❛♥t✮ ❢r♦♠ ❡①♣❡r✐✲ ♠❡♥t❛❧ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠❀ ✭❜✮ Pr♦✈✐❞✐♥❣ ❞❛t❛ ♦♥ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ ✇✐t❤ ❤✐❣❤ ❛❝❝✉r❛❝②✱ ❝♦♥✈❡r❣✐♥❣ t♦ ✶✷ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ❝❛❧❝✉❧❛t❡❞ ✇✐t❤ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ♦❢ ❚▼❉ ♠♦♥♦❧❛②❡rs ♦❜t❛✐♥❡❞ ✐♥ t❤❡ t❤❡s✐s✳ ❆❢t❡r t❤✐s t❤❡s✐s✱ ✇❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ❞❡✈❡❧♦♣ t❤❡ ❋❑ ♦♣❡r❛t♦r ♠❡t❤♦❞ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠s✿ ✶✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❤❡❧✐✉♠ ❛t♦♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ✷✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ♥❡❣❛t✐✈❡ ❡①❝✐t♦♥s ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✳ ✸✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥✈❡st✐❣❛t✐♥❣ ❡①❝✐t♦♥ ❡♥❡r❣② s♣❡❝tr✉♠ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ t❡♠♣❡r❛t✉r❡ ❡❢❢❡❝t✳ ✶✼ ▲✐st ♦❢ ✇♦r❦s r❡❧❛t❡❞ t♦ t❤❡ t❤❡s✐s ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ t❤❡s✐s ❛r❡ ♣✉❜❧✐s❤❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇♦r❦s✿ ✶✳ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣✱ ❉✉②✲◆❤❛t ▲②✱ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❈♦♠♠❡♥t ♦♥ ✏❊①❝✐✲ t♦♥s✱ tr✐♦♥s✱ ❛♥❞ ❜✐❡①❝✐t♦♥s ✐♥ tr❛♥s✐t✐♦♥✲♠❡t❛❧ ❞✐❝❤❛❧❝♦❣❡♥✐❞❡s✿ ▼❛❣♥❡t✐❝✲❢✐❡❧❞ ❞❡♣❡♥❞❡♥❝❡✑✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❇✱ ♣✳ ✶✷✼✹✵✶✱ ✷✵✷✵✳ ✭■❋ ✸✳✺✼✺✱ ◗✶ ❙❝✐♠❛❣♦✮ ✷✳ ỵ t ộ r ữủ ❝❤➼♥❤ ①→❝ ❝❛♦ ❝❤♦ tr↕♥❣ t❤→✐ ❝ì ❜↔♥ ❝õ❛ ♥❣✉②➯♥ tû ❤②❞r♦ ð ♠ỉ✐ tr÷í♥❣ ♣❧❛s♠❛ tr♦♥❣ tø tr÷í♥❣ ✤➲✉✱✑ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ❚r÷í♥❣ ✣❍❙P ❚P❍❈▼✱ tr✳ ✶✵✶✾✱ ✷✵✷✵✳ ✸✳ ❉✉②✲◆❤❛t ▲②✱ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❍✐❣❤❧② ❛❝❝✉r❛t❡ ❡♥❡r❣✐❡s ♦❢ ❛ ♣❧❛s♠❛✲❡♠❜❡❞❞❡❞ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ✉♥✐❢♦r♠ ♠❛❣♥❡t✐❝ ❢✐❡❧❞✱✑ P❤②s✐❝s ♦❢ P❧❛s♠❛s✱ ♣✳ ✵✻✸✸✵✶✱ ✷✵✷✶✳ ✭■❋ ✶✳✽✸✱ ◗✷ ❙❝✐♠❛❣♦✮ ❖t❤❡r ✇♦r❦s ✶✳ ❚❤❛♥❤✲❳✉❛♥ ❍✳ ❈❛♦✱ ❉✉②✲◆❤❛t ▲②✱ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣✱ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❍✐❣❤✲❛❝❝✉r❛❝② ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ t❤❡ ❜♦✉♥❞ st❛t❡s ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ❝♦♥st❛♥t ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✇✐t❤ ❛r❜✐tr❛r② str❡♥❣t❤✱✑ ❈♦♠♣✉t❡r P❤②s✐❝s ❈♦♠♠✉♥✐❝❛t✐♦♥s✱ ♣✳ ✶✸✽✱ ✷✵✶✾✳ ✭■❋ ✸✳✹✺✱ ◗✶ ❙❝✐♠❛❣♦✮ ✷✳ ❉✉②✲❆♥❤ P✳ ◆❣✉②❡♥✱ ❉✉②✲◆❤❛t ▲②✱ ❉❛✐✲◆❛♠ ▲❡✱ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣✱ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❍✐❣❤✲❛❝❝✉r❛❝② ❡♥❡r❣② s♣❡❝tr❛ ♦❢ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①❝✐t♦♥ s❝r❡❡♥❡❞ ❜② r❡❞✉❝❡❞ ❞✐♠❡♥s✐♦♥❛❧✐t② ✇✐t❤ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❝♦♥st❛♥t ♠❛❣♥❡t✐❝ ❢✐❡❧❞✱✑ P❤②s✐❝❛ ❊✿ ▲♦✇✲❞✐♠❡♥s✐♦♥❛❧ ❙②st❡♠s ❛♥❞ ◆❛♥♦str✉❝t✉r❡s✱ ♣✳ ✶✺✷✱ ✷✵✶✾✳ ✭■❋ ✸✳✶✾✱ ◗✷ ❙❝✐♠❛❣♦✮ ✶✽ ❇✐❜❧✐♦❣r❛♣❤② ❬✶❪ ❍✳ ❇❛❝❤❛✉✱ ❊✳ ❈♦r♠✐❡r✱ P✳ ❉❡❝❧❡✈❛✱ ❏✳ ❊✳ ❍❛♥s❡♥✱ ❛♥❞ ❋✳ ▼❛rt➼♥✱ ✏❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❇✲s♣❧✐♥❡s ✐♥ ❛t♦♠✐❝ ❛♥❞ ♠♦❧❡❝✉❧❛r ♣❤②s✐❝s✱✑ ❘❡♣♦rts ♦♥ Pr♦❣r❡ss ✐♥ P❤②s✐❝s✱ ✈♦❧✳ ✻✹✱ ♣✳ ✶✽✶✺✱ ✷✵✵✶✳ ❬✷❪ ❇✳ ■✳ ❙❝❤♥❡✐❞❡r ❛♥❞ ◆✳ ◆②❣❛❛r❞✱ ✏❖rt❤♦❣♦♥❛❧ ❢✉♥❝t✐♦♥s✱ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡ r❡♣r❡✲ s❡♥t❛t✐♦♥✱ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ❣❛✉ss q✉❛❞r❛t✉r❡s✱✑ ❚❤❡ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝❛❧ ❈❤❡♠✲ ✐str② ❆✱ ✈♦❧✳ ✶✵✻✱ ♣✳ ✶✵✼✼✸✱ ✷✵✵✷✳ ❬✸❪ ■✳ ❋❡r❛♥❝❤✉❦✱ ❆✳ ■✈❛♥♦✈✱ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ❛♥❞ ❆✳ ❯❧②❛♥❡♥❦♦✈✱ ◆♦♥✲♣❡rt✉r❜❛t✐✈❡ ❉❡s❝r✐♣t✐♦♥ ♦❢ ◗✉❛♥t✉♠ ❙②st❡♠s✱ ✈♦❧✳ ✽✾✹ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ P❤②s✐❝s✳ ❈❤❛♠✿ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ✷✵✶✺✳ ❬✹❪ ❍✳ ❘✉❞❡r✱ ●✳ ❲✉♥♥❡r✱ ❍✳ ❍❡r♦❧❞✱ ❛♥❞ ❋✳ ●❡②❡r✱ ❆t♦♠s ✐♥ ❙tr♦♥❣ ▼❛❣♥❡t✐❝ ❋✐❡❧❞s✳ ❆str♦♥♦♠② ❛♥❞ ❆str♦♣❤②s✐❝s ▲✐❜r❛r②✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✿ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✶✾✾✹✳ ❬✺❪ ❉✉②✲❆♥❤ P✳ ◆❣✉②❡♥✱ ❉✉②✲◆❤❛t ▲②✱ ❉❛✐✲◆❛♠ ▲❡✱ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣✱ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❍✐❣❤✲❛❝❝✉r❛❝② ❡♥❡r❣② s♣❡❝tr❛ ♦❢ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡①❝✐t♦♥ s❝r❡❡♥❡❞ ❜② r❡❞✉❝❡❞ ❞✐♠❡♥s✐♦♥❛❧✐t② ✇✐t❤ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛ ❝♦♥st❛♥t ♠❛❣✲ ♥❡t✐❝ ❢✐❡❧❞✱✑ P❤②s✐❝❛ ❊✿ ▲♦✇✲❞✐♠❡♥s✐♦♥❛❧ ❙②st❡♠s ❛♥❞ ◆❛♥♦str✉❝t✉r❡s✱ ✈♦❧✳ ✶✶✸✱ ♣✳ ✶✺✷✱ ✷✵✶✾✳ ❬✻❪ ▼✳ ●♦r②❝❛✱ ❏✳ ▲✐✱ ❆✳ ❱✳ ❙t✐❡r✱ ❚✳ ❚❛♥✐❣✉❝❤✐✱ ❑✳ ❲❛t❛♥❛❜❡✱ ❊✳ ❈♦✉rt❛❞❡✱ ❙✳ ❙❤r❡❡✱ ❈✳ ❘♦❜❡rt✱ ❇✳ ❯r❜❛s③❡❦✱ ❳✳ ▼❛r✐❡✱ ❛♥❞ ❙✳ ❆✳ ❈r♦♦❦❡r✱ ✏❘❡✈❡❛❧✐♥❣ ❡①❝✐t♦♥ ♠❛ss❡s ❛♥❞ ❞✐❡❧❡❝tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ ♠♦♥♦❧❛②❡r s❡♠✐❝♦♥❞✉❝t♦rs ✇✐t❤ ❤✐❣❤ ♠❛❣♥❡t✐❝ ❢✐❡❧❞s✱✑ ◆❛t✉r❡ ❈♦♠♠✉♥✐❝❛t✐♦♥s✱ ✈♦❧✳ ✶✵✱ ♣✳ ✹✶✼✷✱ ✷✵✶✾✳ ❬✼❪ ▼✳ ❱❛♥ ❞❡r ❉♦♥❝❦✱ ▼✳ ❩❛r❡♥✐❛✱ ❛♥❞ ❋✳ ▼✳ P❡❡t❡rs✱ ✏❊①❝✐t♦♥s✱ tr✐♦♥s✱ ❛♥❞ ❜✐❡①❝✐✲ t♦♥s ✐♥ tr❛♥s✐t✐♦♥✲♠❡t❛❧ ❞✐❝❤❛❧❝♦❣❡♥✐❞❡s✿ ▼❛❣♥❡t✐❝✲❢✐❡❧❞ ❞❡♣❡♥❞❡♥❝❡✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❇✱ ✈♦❧✳ ✾✼✱ ♣✳ ✶✾✺✹✵✽✱ ✷✵✶✽✳ ❬✽❪ ❆✳ ❆r♦r❛✱ ▼✳ ❑♦♣❡rs❦✐✱ ❑✳ ◆♦❣❛❥❡✇s❦✐✱ ❏✳ ▼❛r❝✉s✱ ❈✳ ❋❛✉❣❡r❛s✱ ❛♥❞ ▼✳ P♦t❡♠✲ s❦✐✱ ✏❊①❝✐t♦♥✐❝ r❡s♦♥❛♥❝❡s ✐♥ t❤✐♥ ❢✐❧♠s ♦❢ ❲❙❡2 ✿ ❢r♦♠ ♠♦♥♦❧❛②❡r t♦ ❜✉❧❦ ♠❛t❡✲ r✐❛❧✱✑ ◆❛♥♦s❝❛❧❡✱ ✈♦❧✳ ✼✱ ♣✳ ✶✵✹✷✶✱ ✷✵✶✺✳ ❬✾❪ ❘✳ ❏✳ ●♦❧❞st♦♥ ❛♥❞ P✳ ❍✳ ❘✉t❤❡r❢♦r❞✱ P✉❜❧✐s❤✐♥❣ ▲t❞✱ ✶✾✾✺✳ ■♥tr♦❞✉❝t✐♦♥ t♦ P❧❛s♠❛ P❤②s✐❝s✳ ■❖P ❬✶✵❪ P✳ ❑✳ ❙❤✉❦❧❛ ❛♥❞ ❇✳ ❊❧✐❛ss♦♥✱ ✏◆♦✈❡❧ ❛ttr❛❝t✐✈❡ ❢♦r❝❡ ❜❡t✇❡❡♥ ✐♦♥s ✐♥ q✉❛♥t✉♠ ♣❧❛s♠❛s✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✈♦❧✳ ✶✵✽✱ ♣✳ ✶✻✺✵✵✼✱ ✷✵✶✷✳ ❬✶✶❪ ❆✳ ❙♦②❧✉✱ ✏P❧❛s♠❛ s❝r❡❡♥✐♥❣ ❡❢❢❡❝ts ♦♥ t❤❡ ❡♥❡r❣✐❡s ♦❢ ❤②❞r♦❣❡♥ ❛t♦♠✱✑ ♦❢ P❧❛s♠❛s✱ ✈♦❧✳ ✶✾✱ ♣✳ ✵✼✷✼✵✶✱ ✷✵✶✷✳ ✶✾ P❤②s✐❝s ❬✶✷❪ ❇✳ ❙❛❤❛✱ P✳ ❑✳ ▼✉❦❤❡r❥❡❡✱ ❛♥❞ ●✳ ❍✳ ❋✳ ❉✐❡r❝❦s❡♥✱ ✏❊♥❡r❣② ❧❡✈❡❧s ❛♥❞ str✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♠♣r❡ss❡❞ ❤②❞r♦❣❡♥ ❛t♦♠ ✉♥❞❡r ❉❡❜②❡ s❝r❡❡♥✐♥❣✱✑ ❆str♦♥♦♠② ✫ ❆str♦♣❤②s✐❝s✱ ✈♦❧✳ ✸✾✻✱ ♣✳ ✸✸✼✱ ✷✵✵✷✳ ❬✶✸❪ ❖✳ ❇❛②r❛❦ ❛♥❞ ■✳ ❇♦③t♦s✉♥✱ ✏❆♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❛s②♠♣t♦t✐❝ ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ t♦ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❝♦s✐♥❡ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧✱✑ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ◗✉❛♥t✉♠ ❈❤❡♠✐str②✱ ✈♦❧✳ ✶✵✼✱ ♣✳ ✶✵✹✵✱ ✷✵✵✼✳ ❬✶✹❪ ❙✳ P❛✉❧ ❛♥❞ ❨✳ ❍♦✱ ✏❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❡①♣♦♥❡♥t✐❛❧ ❝♦s✐♥❡ s❝r❡❡♥❡❞ ❈♦✉❧♦♠❜ ♣♦t❡♥t✐❛❧✱✑ ❈♦♠♣✉t❡r P❤②s✐❝s ❈♦♠♠✉♥✐❝❛t✐♦♥s✱ ✈♦❧✳ ✶✽✷✱ ♣✳ ✶✸✵✱ ✷✵✶✶✳ ❬✶✺❪ ❩✳✲❇✳ ❈❤❡♥✱ ❨✳✲❨✳ ◗✐✱ ❍✳✲❨✳ ❙✉♥✱ ●✳✲P✳ ❩❤❛♦✱ ❛♥❞ P✳✲❋✳ ▲✐✉✱ ✏❙②st❡♠❛t✐❝ ✐♥✈❡s✲ t✐❣❛t✐♦♥s ♦❢ ❧❡✈❡❧ ❞❡❧♦❝❛❧✐③❛t✐♦♥ ❛♥❞ s♣❡❝tr♦s❝♦♣② ♦❢ ❤②❞r♦❣❡♥ ❛t♦♠ s✉❜❥❡❝t❡❞ t♦ ❛ ♣❧❛s♠❛ ❡♥✈✐r♦♥♠❡♥t ✉s✐♥❣ ✈❛r✐♦✉s st❛t✐❝❛❧❧② s❝r❡❡♥❡❞ ♣♦t❡♥t✐❛❧s✱✑ P❤②s✐❝s ♦❢ P❧❛s♠❛s✱ ✈♦❧✳ ✷✼✱ ♣✳ ✵✼✷✶✵✺✱ ✷✵✷✵✳ ❬✶✻❪ ▲✳ ❱✳ ❑❡❧❞②s❤✱ ✏❈♦✉❧♦♠❜ ✐♥t❡r❛❝t✐♦♥ ✐♥ t❤✐♥ s❡♠✐❝♦♥❞✉❝t♦r ❛♥❞ s❡♠✐♠❡t❛❧ ❢✐❧♠s✱✑ ❏❊❚P ▲❡tt❡rs✱ ✈♦❧✳ ✷✾✱ ♣✳ ✻✺✽✱ ✶✾✼✾✳ ❬✶✼❪ ❆✳ ❱✳ ❙t✐❡r✱ ◆✳ P✳ ❲✐❧s♦♥✱ ❑✳ ❆✳ ❱❡❧✐③❤❛♥✐♥✱ ❏✳ ❑♦♥♦✱ ❳✳ ❳✉✱ ❛♥❞ ❙✳ ❆✳ ❈r♦♦❦❡r✱ ✏▼❛❣♥❡t♦♦♣t✐❝s ♦❢ ❡①❝✐t♦♥ ❘②❞❜❡r❣ st❛t❡s ✐♥ ❛ ♠♦♥♦❧❛②❡r s❡♠✐❝♦♥❞✉❝t♦r✱✑ P❤②s✲ ✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✈♦❧✳ ✶✷✵✱ ♣✳ ✵✺✼✹✵✺✱ ✷✵✶✽✳ ❬✶✽❪ ❆✳ ❆r♦r❛✱ ❚✳ ❉❡✐❧♠❛♥♥✱ ❚✳ ❘❡✐❝❤❡♥❛✉❡r✱ ❏✳ ❑❡r♥✱ ❙✳ ▼✐❝❤❛❡❧✐s ❞❡ ❱❛s❝♦♥❝❡❧❧♦s✱ ▼✳ ❘♦❤❧❢✐♥❣✱ ❛♥❞ ❘✳ ❇r❛ts❝❤✐ts❝❤✱ ✏❊①❝✐t❡❞✲st❛t❡ tr✐♦♥s ✐♥ ♠♦♥♦❧❛②❡r ❲❙2 ✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✈♦❧✳ ✶✷✸✱ ♣✳ ✶✻✼✹✵✶✱ ✷✵✶✾✳ ❬✶✾❪ ❊✳ ▲✐✉✱ ❏✳ ✈❛♥ ❇❛r❡♥✱ ❚✳ ❚❛♥✐❣✉❝❤✐✱ ❑✳ ❲❛t❛♥❛❜❡✱ ❨✳✲❈✳ ❈❤❛♥❣✱ ❛♥❞ ❈✳ ❍✳ ▲✉✐✱ ✏▼❛❣♥❡t♦♣❤♦t♦❧✉♠✐♥❡s❝❡♥❝❡ ♦❢ ❡①❝✐t♦♥ ❘②❞❜❡r❣ st❛t❡s ✐♥ ♠♦♥♦❧❛②❡r WSe2 ✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❇✱ ✈♦❧✳ ✾✾✱ ♣✳ ✷✵✺✹✷✵✱ ✷✵✶✾✳ ❬✷✵❪ ▲✳ ❚♦♥❦s ❛♥❞ ■✳ ▲❛♥❣♠✉✐r✱ ✏❖s❝✐❧❧❛t✐♦♥s ✐♥ ✐♦♥✐③❡❞ ❣❛s❡s✱✑ ♥❛❧s ❆r❝❤✐✈❡✱ ✈♦❧✳ ✸✸✱ ♣✳ ✶✾✺✱ ✶✾✷✾✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❏♦✉r✲ ❬✷✶❪ ▲✳ ❙t❡♥❢❧♦ ❛♥❞ ▼✳ ❨✳ ❨✉✱ ✏P♦t❡♥t✐❛❧ ♦❢ ❛ ♠♦✈✐♥❣ t❡st ❝❤❛r❣❡ ✐♥ ❛ ❝♦❧❧✐s✐♦♥❛❧ ♣❧❛s♠❛✱✑ P❤②s✐❝❛ ❙❝r✐♣t❛✱ ✈♦❧✳ ✽✱ ♣✳ ✸✵✶✱ ✶✾✼✸✳ ❬✷✷❪ ❙✳ P❛✉❧ ❛♥❞ ❨✳ ❑✳ ❍♦✱ ✏❍②❞r♦❣❡♥ ❛t♦♠s ✐♥ ❉❡❜②❡ ♣❧❛s♠❛ ❡♥✈✐r♦♥♠❡♥ts✱✑ ♦❢ P❧❛s♠❛s✱ ✈♦❧✳ ✶✻✱ ♣✳ ✵✻✸✸✵✷✱ ✷✵✵✾✳ P❤②s✐❝s ❬✷✸❪ ❚❤❛♥❤✲❳✉❛♥ ❍✳ ❈❛♦✱ ❉✉②✲◆❤❛t ▲②✱ ◆❣♦❝✲❚r❛♠ ❉✳ ❍♦❛♥❣✱ ❛♥❞ ❱❛♥✲❍♦❛♥❣ ▲❡✱ ✏❍✐❣❤✲❛❝❝✉r❛❝② ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ t❤❡ ❜♦✉♥❞ st❛t❡s ♦❢ ❛ ❤②❞r♦❣❡♥ ❛t♦♠ ✐♥ ❛ ❝♦♥st❛♥t ♠❛❣♥❡t✐❝ ❢✐❡❧❞ ✇✐t❤ ❛r❜✐tr❛r② str❡♥❣t❤✱✑ ❈♦♠♣✉t❡r P❤②s✐❝s ❈♦♠♠✉♥✐❝❛t✐♦♥s✱ ✈♦❧✳ ✷✹✵✱ ♣✳ ✶✸✽✱ ✷✵✶✾✳ ❬✷✹❪ ◆❡t❧✐❜✳♦r❣✳ ▲❆P❆❈❑✿ ▲✐♥❡❛r ❆❧❣❡❜r❛ P❆❈❑❛❣❡✱ ✏❙✉❜r♦✉t✐♥❡ ❉❙❨●❱❳✳❢✱✑ ❤tt♣✿✴✴✇✇✇✳♥❡t❧✐❜✳♦r❣✴❧❛♣❛❝❦✴❡①♣❧♦r❡✲❤t✲♠❧✴ ❞✷✴❞✾✼✴❞s②❡✈①✲✽❢✳❤t♠❧✳ ✷✵ ... r❡✇r? ?tt? ??♥ ✐♥ ❛t♦♠✐❝ ✉♥✐ts ❛s ❢♦❧❧♦✇s✿ − ∂2 ∂2 + ∂x2 ∂y γ2 mαex γ + (x + y ) + Vˆh−e (r) + − E ψ(x, y) = ✭✸✳✷✮ ❚❤❡ ❑❡❧❞②s❤ ♣♦t❡♥t✐❛❧ ❡st❛❜❧✐s❤❡❞ ❢♦r t❤❡ ❡❧❡❝tr♦♥✲❤♦❧❡ ✐♥t❡r❛❝t✐♦♥ ✐s r❡✇r? ?tt? ??♥... P❧❛s♠❛ P❤②s✐❝s✳ ■❖P ❬✶✵❪ P✳ ❑✳ ❙❤✉❦❧❛ ❛♥❞ ❇✳ ❊❧✐❛ss♦♥✱ ✏◆♦✈❡❧ ❛ttr❛❝t✐✈❡ ❢♦r❝❡ ❜❡t✇❡❡♥ ✐♦♥s ✐♥ q✉❛♥t✉♠ ♣❧❛s♠❛s✱✑ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲? ?tt? ??rs✱ ✈♦❧✳ ✶✵✽✱ ♣✳ ✶✻✺✵✵✼✱ ✷✵✶✷✳ ❬✶✶❪ ❆✳ ❙♦②❧✉✱ ✏P❧❛s♠❛ s❝r❡❡♥✐♥❣... ❬✶✶❪✳ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts s❤♦✇ t❤❛t t❤❡r❡ ❛r❡ ❡❢❢❡❝ts ❝❛✉s❡❞ ❜② t❤❡ ❡①t❡r♥❛❧ ❢✐❡❧❞ t❤❛t ❞❡s❡r✈❡ ? ?tt? ??♥t✐♦♥ ❛♥❞ ♥❡❡❞ t♦ ❜❡ ❢✉rt❤❡r st✉❞✐❡❞ ❜② ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ▼●❊❈❙❈ ♠♦❞❡❧✳ ❚❤❡ ❡♥❡r❣② s♣❡❝tr✉♠ ♦❢