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rt t rr qts r t Pr t Ptr P

  1. ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ✽✳ ❖❦t♦❜❡r ✷✵✶✻ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  2. ❖✈❡r✈✐❡✇ ✶✳ ▼♦t✐✈❛t✐♦♥ ✷✳ ▼❡❛♥ ✜❡❧❞ ❧✐♠✐ts ✲ ❣❡♥❡r❛❧ r❡♠❛r❦s ✸✳ ❉❡r✐✈✐♥❣ t❤❡ ❍❛rtr❡❡✲▼❛①✇❡❧❧ ❡q✉❛t✐♦♥s P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  3. ❖✈❡r✈✐❡✇ ✶✳ ▼♦t✐✈❛t✐♦♥ ✷✳ ▼❡❛♥ ✜❡❧❞ ❧✐♠✐ts ✲ ❣❡♥❡r❛❧ r❡♠❛r❦s ✸✳ ❉❡r✐✈✐♥❣ t❤❡ ❍❛rtr❡❡✲▼❛①✇❡❧❧ ❡q✉❛t✐♦♥s P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  4. ❖✈❡r✈✐❡✇ ✶✳ ▼♦t✐✈❛t✐♦♥ ✷✳ ▼❡❛♥ ✜❡❧❞ ❧✐♠✐ts ✲ ❣❡♥❡r❛❧ r❡♠❛r❦s ✸✳ ❉❡r✐✈✐♥❣ t❤❡ ❍❛rtr❡❡✲▼❛①✇❡❧❧ ❡q✉❛t✐♦♥s P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  5. ❖✈❡r✈✐❡✇ ✶✳ ▼♦t✐✈❛t✐♦♥ ✷✳ ▼❡❛♥ ✜❡❧❞ ❧✐♠✐ts ✲ ❣❡♥❡r❛❧ r❡♠❛r❦s ✸✳ ❉❡r✐✈✐♥❣ t❤❡ ❍❛rtr❡❡✲▼❛①✇❡❧❧ ❡q✉❛t✐♦♥s P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  6. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  7. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  8. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  9. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  10. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

  11. ▼♦t✐✈❛t✐♦♥ ◮ ❙❡♠✐❝❧❛ss✐❝❛❧ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ id t Ψ t = (( i ∇ + A t ) ✷ + V t )Ψ t ✇✐t❤ ❝❧❛ss✐❝❛❧ A ❛♥❞ V ✳ ◮ ●♦❛❧✿ ❉❡r✐✈❡ t❤✐s ❡q✉❛t✐♦♥ ❢r♦♠ ◗❊❉✳ ◮ ❙t❛♥❞❛r❞ t❡①t❜♦♦❦ ❛r❣✉♠❡♥t✿ ❍❡✐s❡♥❜❡r❣ ❡q✉❛t✐♦♥s Pr♦❜❧❡♠✿ ❘❡s✉❧t ♦♥ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s ♦♥❧②✱ t♦ ❣❡♥❡r❛❧✳ P❡t❡r P✐❝❦❧ ▼❛t❤❡♠❛t✐s❝❤❡s ■♥st✐t✉t ▲▼❯ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❦♦❧❛✐ ▲❡♦♣♦❧❞ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ▼❛①✇❡❧❧✲❙❝❤rö❞✐♥❣❡r ❊q✉❛t✐♦♥s ❢r♦♠ t❤❡ P❛✉❧✐✲❋✐❡r③ ❍❛♠✐❧t♦♥✐❛♥

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