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  1. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❙✐♥❦✴s♦✉r❝❡ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ✌♦✇ P♦t❡♥t✐❛❧ ❙❖❊✷✶✺✻✿ ❋❧✉✐❞s ▲❡❝t✉r❡ ✾ ✈♦rt❡① ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  2. ❯② ✰ ❢ ✭ ① ✮ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ U ❙t❡❛❞② ✉♥✐❢♦r♠ ✌♦✇ ✐♥ t❤❡ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ① ✲❞✐r❡❝t✐♦♥✱ ✐✳❡✳ ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ ✉ ① ❂ ❯ P♦t❡♥t✐❛❧ y ✈♦rt❡① ✉ ② ❂ ✵ ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s x ❲❤❛t ✐s t❤❡ str❡❛♠ ❢✉♥❝t✐♦♥❄ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❙✐♥❝❡ ✉ ① ❂ ❅✥ ❘❛♥❦✐♥❡ ❜♦❞② ❅ ② ❂ ❯ ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s t❤✐s ♠❡❛♥s ♣❛r❛❞♦① ❩ ✥ ❂ ❯ ❞② ❂

  3. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ U ❙t❡❛❞② ✉♥✐❢♦r♠ ✌♦✇ ✐♥ t❤❡ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ① ✲❞✐r❡❝t✐♦♥✱ ✐✳❡✳ ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ ✉ ① ❂ ❯ P♦t❡♥t✐❛❧ y ✈♦rt❡① ✉ ② ❂ ✵ ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s x ❲❤❛t ✐s t❤❡ str❡❛♠ ❢✉♥❝t✐♦♥❄ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❙✐♥❝❡ ✉ ① ❂ ❅✥ ❘❛♥❦✐♥❡ ❜♦❞② ❅ ② ❂ ❯ ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s t❤✐s ♠❡❛♥s ♣❛r❛❞♦① ❩ ✥ ❂ ❯ ❞② ❂ ❯② ✰ ❢ ✭ ① ✮

  4. ✒ ✓ ✵ ✰ ❞❢ ❂ ✵ � ❞① ❞❢ ❞① ❂ ✵ ❢ ✭ ① ✮ ❂ ❈ ✮ ❙tr❡❛♠ ❢✉♥❝t✐♦♥ ✥ ❂ ❯② ✰ ❈ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❈❤❡❝❦ ✿ ✉ ② ❂ � ❅✥ ❯♥✐❢♦r♠ ❅ ① ❂ str❡❛♠ ✌♦✇ ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ s♦ ♦✉r ❢✉♥❝t✐♦♥ ♠✉st ❜❡ P♦t❡♥t✐❛❧ ✈♦rt❡① ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  5. ❞❢ ❞① ❂ ✵ ❢ ✭ ① ✮ ❂ ❈ ✮ ❙tr❡❛♠ ❢✉♥❝t✐♦♥ ✥ ❂ ❯② ✰ ❈ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❈❤❡❝❦ ✿ ✒ ✓ ✉ ② ❂ � ❅✥ ✵ ✰ ❞❢ ❯♥✐❢♦r♠ ❅ ① ❂ � ❂ ✵ str❡❛♠ ✌♦✇ ❞① ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ s♦ ♦✉r ❢✉♥❝t✐♦♥ ♠✉st ❜❡ P♦t❡♥t✐❛❧ ✈♦rt❡① ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  6. ❢ ✭ ① ✮ ❂ ❈ ✮ ❙tr❡❛♠ ❢✉♥❝t✐♦♥ ✥ ❂ ❯② ✰ ❈ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❈❤❡❝❦ ✿ ✒ ✓ ✉ ② ❂ � ❅✥ ✵ ✰ ❞❢ ❯♥✐❢♦r♠ ❅ ① ❂ � ❂ ✵ str❡❛♠ ✌♦✇ ❞① ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ s♦ ♦✉r ❢✉♥❝t✐♦♥ ♠✉st ❜❡ P♦t❡♥t✐❛❧ ✈♦rt❡① ❞❢ ❞① ❂ ✵ ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  7. ❙tr❡❛♠ ❢✉♥❝t✐♦♥ ✥ ❂ ❯② ✰ ❈ ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❈❤❡❝❦ ✿ ✒ ✓ ✉ ② ❂ � ❅✥ ✵ ✰ ❞❢ ❯♥✐❢♦r♠ ❅ ① ❂ � ❂ ✵ str❡❛♠ ✌♦✇ ❞① ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ s♦ ♦✉r ❢✉♥❝t✐♦♥ ♠✉st ❜❡ P♦t❡♥t✐❛❧ ✈♦rt❡① ❞❢ ❞① ❂ ✵ ❢ ✭ ① ✮ ❂ ❈ ❈♦♠❜✐♥✐♥❣ ✮ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  8. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❈❤❡❝❦ ✿ ✒ ✓ ✉ ② ❂ � ❅✥ ✵ ✰ ❞❢ ❯♥✐❢♦r♠ ❅ ① ❂ � ❂ ✵ str❡❛♠ ✌♦✇ ❞① ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ s♦ ♦✉r ❢✉♥❝t✐♦♥ ♠✉st ❜❡ P♦t❡♥t✐❛❧ ✈♦rt❡① ❞❢ ❞① ❂ ✵ ❢ ✭ ① ✮ ❂ ❈ ❈♦♠❜✐♥✐♥❣ ✮ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❙tr❡❛♠ ❢✉♥❝t✐♦♥ ❉r❛❣ ❢♦r❝❡s ✥ ❂ ❯② ✰ ❈ ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  9. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ❲❡ ❝♦✉❧❞ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ❢♦r t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ t♦ ☞♥❞ ■■ ✣ ❂ ❯① ✰ ❈ ✶ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ ❉r❛✇ t❤❡ ✌♦✇ ♥❡t ✿ str❡❛♠ ✌♦✇ ❙✐♥❦✴s♦✉r❝❡ y ✌♦✇ P♦t❡♥t✐❛❧ ✈♦rt❡① ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s x ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦①

  10. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❲❡ ❝♦✉❧❞ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ❢♦r t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ t♦ ☞♥❞ ❯♥✐❢♦r♠ ✣ ❂ ❯① ✰ ❈ ✶ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❉r❛✇ t❤❡ ✌♦✇ ♥❡t ✿ ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ y P♦t❡♥t✐❛❧ ✈♦rt❡① ψ= const ❈♦♠❜✐♥✐♥❣ (different C) s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s x ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦① Q= ∆ψ

  11. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❲❡ ❝♦✉❧❞ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ❢♦r t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✱ t♦ ☞♥❞ ❯♥✐❢♦r♠ ✣ ❂ ❯① ✰ ❈ ✶ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❉r❛✇ t❤❡ ✌♦✇ ♥❡t ✿ ❙✐♥❦✴s♦✉r❝❡ y ✌♦✇ P♦t❡♥t✐❛❧ ✈♦rt❡① ψ= const (different C) ❈♦♠❜✐♥✐♥❣ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❘❛♥❦✐♥❡ ❜♦❞② x ❉r❛❣ ❢♦r❝❡s ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦① Q= ∆ψ φ= const (different C1)

  12. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ■■ ❙✐♥❦✴s♦✉r❝❡ ✌♦✇ ❯♥✐❢♦r♠ y str❡❛♠ ✌♦✇ ❋❧♦✇ r❛❞✐❛❧❧② ♦✉t✇❛r❞ ✭✐♥✇❛r❞ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❢♦r s✐♥❦✮✳ ❙✐♥❦✴s♦✉r❝❡ ❋❧♦✇ ♦❜❡②s ❝♦♥t✐♥✉✐t② ④ t❤✉s ✌♦✇ x P♦t❡♥t✐❛❧ ✈♦rt❡① ✉ ✴ ✶ ❈♦♠❜✐♥✐♥❣ r s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ u ❊❛s✐❡st t♦ ✇♦r❦ ✐♥ t❡r♠s ♦❢ u ❘❛♥❦✐♥❡ ❜♦❞② ♣♦❧❛r ❝♦♦r❞✐♥❛t❡s ✱ s♦ ❉r❛❣ ❢♦r❝❡s u θ u r ❞✬❆❧❡♠❜❡rt✬s ✣ ✭ r ❀ ✒ ✮ ❀ ✥ ✭ r ❀ ✒ ✮ ♣❛r❛❞♦① r θ ❢♦r ✇❤✐❝❤ x ✉ r ❂ ♠ r ❀ ✉ ✒ ❂ ✵

  13. ✷✲❞ ■❞❡❛❧ ❋❧♦✇ ❊✈❛❧✉❛t✐♥❣ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ✿ ■■ ✉ r ❂ ❅✣ ✉ ✒ ❂ ✶ ❅✣ ❅ r ❀ ❯♥✐❢♦r♠ r ❅✒ str❡❛♠ ✌♦✇ ❯♥✐❢♦r♠ str❡❛♠ ✌♦✇ ❚❤✉s✱ ❩ ♠ ❙✐♥❦✴s♦✉r❝❡ ❅ r ❂ ♠ ❅✣ ✌♦✇ r ❀ ✐✳❡✳ ✣ ❂ r ❞r P♦t❡♥t✐❛❧ ✈♦rt❡① s♦ ❈♦♠❜✐♥✐♥❣ ✣ ❂ ♠ ❧♦❣ r ✰ ❈ s♦❧✉t✐♦♥s ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ❚❤✐s ✐s ❛ ❝♦♥st❛♥t ❢♦r r ❂ ❝♦♥st✱ ✐✳❡✳ ❛ ❝✐r❝❧❡✳ ❘❛♥❦✐♥❡ ❜♦❞② ❉r❛❣ ❢♦r❝❡s ❙✐♠✐❧❛r❧②✱ ❡✈❛❧✉❛t❡ t❤❡ str❡❛♠ y ❞✬❆❧❡♠❜❡rt✬s ♣❛r❛❞♦① ❢✉♥❝t✐♦♥✱ t♦ ☞♥❞ ✥ ❂ ♠ ✒ ✰ ❈ ✶ x ④ r❛❞✐❛❧ ❧✐♥❡s ❢r♦♠ t❤❡ ❝❡♥tr❡

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