量子光学理論入門

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1. ྔࢠޫֶཧ࿦ʮ௒ʯೖ໳ தҪ ӻ࢘ 2018 ೥ 3 ݄ 13 ೔
2. 2 0.1 ͸͡Ίʹ ۙ೥ɺྔࢠσόΠεΛ༻͍ͨܭࢉػͷ࿩୊ΛΑࣖ͘ʹ͢ΔΑ͏ʹͳΓ·ͨ͠ɻྔࢠσό ΠεΛ༻͍࣮ͯ༻తͳܭࢉॲཧΛߦ͏ʹ͸ɺྔࢠσόΠεΛ༻͍ͨܭࢉ૷ஔͷ෺ཧతͳ࣮ ૷ํࣜʹ࢝·Γɺͦͷ૷ஔ͕΋ͨΒ͢ྔࢠޮՌͷཧ࿦తͳઆ໌ɺ͞Βʹ͸ɺྔࢠޮՌΛར ༻ͨ͠ܭࢉΞϧΰϦζϜͷ։ൃͳͲɺ͞·͟·ͳϨΠϠʔʹ͓͚Δݚڀɺ։ൃ͕ਐΉඞཁ ͕͋Γ·͢ɻͦͷΑ͏ͳதɺࡢ೥ɺ೔ຊͰ͸ޫύϧεΛྔࢠσόΠεͱͯ͠༻͍ͨܭࢉػ ͕࿩୊ͱͳΓ·ͨ͠ɻ ʕʕ ͱɺ֨ௐߴ͘ॻ͖࢝ΊͯΈͨͷͰ͕͢ɺͿͬͪΌ͚࿩Λ͢Δͱɺ౰ॳɺචऀ͸ɺ ޫύϧεΛ༻͍ͨܭࢉػͷ࢓૊Έ͕·ͬͨ͘ཧղͰ͖·ͤΜͰͨ͠ɻϝσΟΞهࣄͰ͸ɺ ʮྔࢠଌఆϑΟʔυόοΫʯ ʮ೾ଋͷऩॖʯ ʮྔࢠ૬సҠʯͳͲɺੲʑɺྔࢠྗֶͰษڧ͠ ͨΩʔϫʔυΛ໨ʹ͢Δ΋ͷͷɺ݁ہͷॴɺԿΛଌఆͯ͠ɺͲ͏͍͏ҙຯͰԿ͕૬సҠ͠ ͍ͯΔͷ͔ɺ1qbit ͨΓͱ΋ཧղ͢Δ͜ͱ͕Ͱ͖·ͤΜͰͨ͠ɻ ʮ͜Ε͸΍͹͍ɻ͜ΕΛཧ ղ͢Δʹ͸ɺ͍͍ͬͨԿΛษڧ͢Ε͹Α͍ͷͩʯͱࢥ͍ɺؔ࿈͢Δ࿦จΛಡΈ࢝Ίͨͱ͜ Ζɺੈͷதʹ͸ྔࢠޫֶʢQuantum Opticsʣͱݺ͹ΕΔ෼໺͕͋Γɺޫύϧεͷྔࢠ࿦ తৼΔ෣͍͸ɺ·͞ʹྔࢠޫֶʹΑͬͯهड़͞ΕΔ͜ͱ͕Θ͔Γ·ͨ͠ɻͬͦ͘͞ɺͦͷ ෼໺ͷ୅දతʢͱࢥΘΕΔʣڭՊॻ [1] Λߪೖͯ͠ಡΈ࢝Ίͨͱ͜Ζɺͦ͜Ͱ͸ɺͳΜͱ ΋ջ͔͍͠ௐ࿨ৼಈࢠͷੜ੒ফ໓ԋࢉࢠΛ༻͍ͨ୅਺ܭࢉ͕܁Γ޿͛ΒΕ͍ͯͨͷͰ͢ɻ ͲͲʔΜ*1ɻ ຊ࡭ࢠΛखʹͨ͠ํͷதʹ͸ɺ΋͔ͨ͠͠Βɺචऀͱಉ༷ʹɺQED ʹΑΔίϯϓτϯ ࢄཚͷஅ໘ੵʢΫϥΠϯɾਔՊͷެࣜʣΛప໷Ͱܭࢉͨ͠ࢥ͍ग़Λ࣋ͭํ΋͍Δ͔΋஌Ε ·ͤΜɻ·͞ʹ͋ͷʮϧʔϧ͸໌շͰܾͯ͠ෳࡶͰ͸ͳ͍ʹ΋͔͔ΘΒͣɺ࣮ࡍʹܭࢉΛ ͸͡ΊͯΈΔͱҟ༷ʹܭࢉ͕௕ͯ͘ɺ΋͸΍Կ͕͓͖͍ͯΔͷ͔Θ͔Βͳ͍ͷʹɺ࠷ޙͷ ܭࢉ݁Ռ͸ɺͳ͔ͥ෺ཧతʹ͖ͪΜͱղऍͰ͖ͯ͠·͏ʯੈք͕ͦ͜ʹ͸͋ͬͨͷͰ͢ɻ ͔͠΋ɺ͜ͷڭՊॻ [1] ͸ɺ͙͢ΕͨڭՊॻʹ͋Γ͕ͪͳɺ ʮ޿͍࿩୊Λద౓ͳਂ͞Ͱ໢ཏ తʹѻ͍ͬͯΔ΋ͷͷɺݸʑͷܭࢉͷલఏ৚݅΍్தͷܭࢉաఔ͕͍͍ײ͡ʹলུ͞Εͯ ͍ͯɺ͔͠΋க໋తͳޡ২͕ͨ·ʹ͋ͬͨΓͯ͠ɺࣗ෼ͷܭࢉ͕߹Θͳ͍ͷ͸ɺલఏ৚݅ ͷղऍ͕ҧ͏ͷ͔ɺܭࢉʹؒҧ͍͕͋Δͷ͔ɺͦ΋ͦ΋ڭՊॻͷهड़͕ޡ২Ͱؒҧ͍ͬͯ Δͷ͔൑அ͕͔ͭͣɺ݁ہɺଞͷॻ੶΍Β࿦จ΍ΒΛඞࢮʹࢀরͯࣗ͠෼ͷ಄Ͱߟ͑ൈ͘ Ӌ໨ʹͳͬͯɺͦΕ͕Ұ൪͍͍ษڧʹͳͬͯ͠·͏ʯͱ͍͏ྫͷύλʔϯΛ౿ऻ͍ͯͨ͠ *1 ޮՌԻ
3. 0.2 ి࣓৔ͷྔࢠԽ 3 ͷͰ͢ɻ ͱ͍͏Θ͚ͰɺຊߘͰ͸ɺ[1] ͷಋೖ෦෼Ͱઆ໌͞Ε͍ͯΔྔࢠޫֶͷجૅతͳܭࢉʹ ͍ͭͯɺܭࢉͷલఏ৚݅΍ܭࢉաఔΛͰ͖Δ͚ͩলུͤͣʹஸೡʹղઆ͢Δͱ͍͏ࢼΈΛ ߦ͍·͢ɻલड़ͷΑ͏ʹɺࣗ෼Ͱௐ΂·ͬͯ͘ߟ͑Δ͜ͱΛ༨ّͳ͘͞ΕΔ఺͕͜ͷڭՊ ॻͷΑ͍ॴͳͷͰ͕͢ɺΦϯϥΠϯॻళͷϨϏϡʔΛݟΔͱʮFrustrating to readʯ ʮToo much information without intermediate steps..ʯͳͲͷίϝϯτͱڞʹ௿͍ධՁ͕͚ͭ ΒΕ͓ͯΓɺ͜Ε͸͜ΕͰ͞Έ͍͠ؾ࣋ͪʹ΋ͳͬͯ͠·͍·͢ɻຊߘͰऔΓѻ͏಺༰ ͸ɺ͋͘·Ͱ΋ྔࢠޫֶͷʮ͞ΘΓʯͷ෦෼Ͱ͋Γɺ๯಄Ͱ৮ΕͨʮྔࢠଌఆϑΟʔυόο Ϋʯ ʮ೾ଋͷऩॖʯ ʮྔࢠ૬సҠʯͱ͍ͬͨ࿩୊·Ͱ͸ѹ౗తͳϖʔδෆ଍ʢͱචऀͷཧղ ෆ଍ʣʹΑΓɺ౸ୡ͢Δ͜ͱ͸Ͱ͖·ͤΜɻͦΕͰ΋ɺຊߘΛ͖͔͚ͬʹɺྔࢠޫֶͱͦ ͷؔ࿈෼໺ʹڵຯΛ͍͖࣋ͬͯͨͩɺϝσΟΞΛ೐Θ͢ಾͷΩʔϫʔυʹӅ͞Εͨਅ࣮Λ ʢචऀͱڞʹʂʣཧղͯ͠ΈΑ͏ͱ͍͏ಡऀ͕ݱΕΔ͜ͱΛͻ͔ͦʹظ଴͍ͯ͠·͢ɻ ຊߘͰ͸ɺௐ࿨ৼಈࢠͷྔࢠྗֶతͳऔΓѻ͍ʢੜ੒ফ໓ԋࢉࢠΛ༻͍ͨ୅਺ܭࢉʣ ɺ ͓ΑͼɺϒϥέοτදهΛ༻͍ͨྔࢠྗֶͷܭࢉख๏ʢγϡϨʔσΟϯΨʔදࣔͱϋΠθ ϯϕϧάදࣔͷҧ͍ͳͲʣʹؔ͢Δ஌ࣝΛલఏͱ͍ͯ͠·͢ɻ͜ΕΒͷ಺༰͍ͭͯ͸ɺ[2] ͳͲͷڭՊॻΛࢀߟʹ͍ͯͩ͘͠͞ɻ 0.2 ి࣓৔ͷྔࢠԽ ͸͡Ίʹɺਅۭதͷి࣓৔ΛྔࢠԽͯ͠ɺௐ࿨ৼಈࢠͷू߹ͱͯ͠هड़͢ΔྲྀΕΛઆ໌ ͠·͢ɻ͜͜͸ɺຊߘͷٞ࿦ͷग़ൃ఺ͱͳΔલఏΛ੔ཧ͢Δ͜ͱ͕໨తͰ͢ͷͰɺಋग़ͷ ྲྀΕͷΈΛ؆୯ʹ͓͓͖͑ͯ͞·͢ɻৄࡉͳಋग़աఔʹ͍ͭͯ͸ɺ[3] ͳͲΛࢀߟʹͯ͠ ͍ͩ͘͞ɻ 0.2.1 ਅۭதͷి࣓৔ ݹయి࣓ؾֶΛ༻͍Δͱɺਅۭதͷి࣓৔͸ɺޫ଎౓ c Ͱਐߦ͢Δฏ໘೾ͷॏͶ߹Θͤ ͱͯ͠දݱ͞ΕΔ͜ͱ͕Θ͔Γ·͢ɻϕΫτϧϙςϯγϟϧ A Λ༻͍ͯදݱͨ͠৔߹ɺ ೾਺ϕΫτϧΛ k ͱͯ͠ɺ࣍ͷฏ໘೾ղ͕ಘΒΕ·͢ɻ A = A0 { ae−i(ωt−k·r) + a∗ei(ωt−k·r) } (1) ͜͜Ͱɺ௕͞ L Ͱͷपظڥք৚݅Λ՝ͨ͠৔߹ɺ೾਺ϕΫτϧ͸࣍ͷ཭ࢄ஋ΛऔΓ
4. 4 ·͢ɻ kx = 2πnx L , ky = 2πny L , kz = 2πnz L (nx , ny , nz = 0, ±1, ±2, · · · ) (2) A0 ͸ภޫͷํ޲ΛදΘ͢ϕΫτϧͰɺෳૉৼ෯ a, a∗ ͕ແ࣍ݩ਺ʹͳΔΑ͏ʹௐ੔͢ Δͱɺภޫํ޲ͷ୯ҐϕΫτϧΛ ˆ e ͱͯ͠ɺ࣍ࣜͰ༩͑ΒΕ·͢ɻ A0 = √ ℏ 2ωϵ0 L3 ˆ e ·ͨɺ֯଎౓ ω ͱ೾਺ϕΫτϧ k ͷؒʹ͸ɺω = c|k| ͱ͍͏ؔ܎͕੒Γཱͪɺ͜Εʹ ΑΓɺ೾ͷਐߦ଎౓͸ޫ଎౓ c ʹҰக͠·͢ɻͳ͓ɺෳૉৼ෯ͱ͍͏ͷ͸ɺৼ෯ͱॳظҐ ૬Λ·ͱΊͯෳૉ਺ a ͰදΘͨ͠දهํ๏Ͱɺa = |a|eiδ ͱදΘ͢ͱɺ(1) ͸ɺ࣍ͷΑ͏ ʹࡾ֯ؔ਺Ͱॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ A = A0 · 2Re {ae−i(ωt−k·r)} = 2|a|A0 Re {e−i(ωt−k·r−δ)} = 2|a|A0 cos(ωt − k · r − δ) ͞Βʹɺ a = 1 2 (X1 + iX2 ), a∗ = 1 2 (X1 − iX2 ) ͱஔ͘ͱɺਤ 1 ΑΓɺ |a| = 1 2 √ X2 1 + X2 2 , cos δ = X1 √ X2 1 + X2 2 , sin δ = X2 √ X2 1 + X2 2 ͕੒ΓཱͭͷͰɺ࣍ͷΑ͏ʹల։͢Δ͜ͱ΋ՄೳͰ͢ɻ A = √ X2 1 + X2 2 A0 {cos(ωt − k · r) cos δ + sin(ωt − k · r) sin δ} = A0 {X1 cos(ωt − k · r) + X2 sin(ωt − k · r)} (3) ଓ͍ͯɺ(1) ͷϕΫτϧϙςϯγϟϧ͔Βɺి৔ E ͸࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ E = − ∂A ∂t = iωA0 { ae−i(ωt−k·r) − a∗ei(ωt−k·r) } = iE0 { ae−i(ωt−k·r) − a∗ei(ωt−k·r) } (4) ͜͜ͰɺE0 ͸ɺ࣍Ͱఆٛ͞ΕΔภޫํ޲ΛදΘ͢ϕΫτϧͰ͢ɻ E0 = ωA0 = √ ℏω 2ϵ0 L3 ˆ e
5. 0.2 ి࣓৔ͷྔࢠԽ 5 ਤ 1 ෳૉৼ෯ a ͷ࣮෦ͱڏ෦ ΋͘͠͸ɺ(3) ͷܗࣜΛ༻͍ͯܭࢉ͢Δͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ E = E0 {X1 sin(ωt − k · r) − X2 cos(ωt − k · r)} ۩ମతͳදࣜ͸লུ͠·͕͢ɺ࣓৔ B ʹ͍ͭͯ΋ɺB = ∇ × A ͷؔ܎͔Βಉ༷ʹܾ ఆ͢Δ͜ͱ͕Ͱ͖·͢ɻ ͦͯ͠ɺ͜͜·Ͱ͸ಛఆͷ೾਺ϕΫτϧ k Λ࣋ͬͨฏ໘೾Λߟ͖͑ͯ·͕ͨ͠ɺҰൠ ʹ͸ɺ(2) Λຬͨ͢͞·͟·ͳ k ʹ͍ͭͯɺ(4) Ͱද͞ΕΔฏ໘೾ͷॏͶ߹Θ͕ͤղͱͳ Γ·͢ɻ͜ͷ࣌ɺภޫͷํ޲ ˆ eɺෳૉৼ෯ aɺͦͯ͠ɺ֯଎౓ ω ͸ k ʹґଘͯ͠มԽ͢ Δ͜ͱΛߟྀ͢ΔͱɺҰൠղ͸ɺ࣍ͷΑ͏ʹද͞Ε·͢ɻ E = i ∑ k √ ℏωk 2ϵ0 L3 ˆ ek { ak e−i(ωkt−k·r) − a∗ k ei(ωkt−k·r) } (5) 0.2.2 ੜ੒ফ໓ԋࢉࢠͷಋೖ ͜͜·Ͱ͸ݹయతͳऔΓѻ͍Ͱ͕ͨ͠ɺ͜͜Ͱɺੜ੒ফ໓ԋࢉࢠΛಋೖͯ͠ɺి࣓৔ͷ ྔࢠԽΛߦ͍·͢ɻ۩ମతʹ͸ɺ(5) ͷදࣜʹ͓͍ͯɺak ͱ a∗ k Λੜ੒ফ໓ԋࢉࢠ ak , a† k ʹஔ͖׵͑·͢ɻ͜ΕΒ͸ɺ࣍ͷަ׵ؔ܎Λຬͨ͠·͢ɻ [ak , ak′ ] = 0, [a† k , a† k′ ] = 0, [ak , a† k′ ] = δkk′ (6) ͍͖ͳΓੜ੒ফ໓ԋࢉࢠ͕ొ৔ͯ͠ɺ໘৯Βͬͨಡऀ͕͍Δ͔΋஌Ε·ͤΜ͕ɺ͜Ε ͸ɺ৔ͷྔࢠԽɺ͋Δ͍͸ɺୈೋྔࢠԽͱݺ͹ΕΔྔࢠԽͷख๏ͱͳΓ·͢ɻୈೋྔࢠԽ ͷܥ౷తͳٞ࿦ʹ͍ͭͯ͸ [4] ͳͲΛࢀߟʹ͍ͯͩ͘͠͞ɻࠓͷ৔߹ɺ(5) ʹରͯ͜͠ͷ ஔ͖׵͑Λߦͬͨʮి৔ԋࢉࢠʯ͸ɺ࣍ͷΑ͏ʹͳΓ·͢ɻ
6. 6 E(r, t) = i ∑ k √ ℏωk 2ϵ0 L3 ˆ ek { ak e−i(ωkt−k·r) − a† k ei(ωkt−k·r) } (7) ݟ্͔͚͸ɺෳૉڞ໾ a∗ ͕Τϧϛʔτڞ໾ a† ʹஔ͖׵Θ͚ͬͨͩͰ͕͢ɺͪ͜Βͷ E(r, t) ͸ɺҐஔ r ͱ࣌ࠁ t ʹґଘͯ͠มԽ͢Δԋࢉࢠʹͳ͍ͬͯ·͢ɻ࣌ؒʹґଘ͢Δ ԋࢉࢠͰ͢ͷͰɺϋΠθϯϕϧάදࣔͷԋࢉࢠͱղऍ͍ͯͩ͘͠͞ɻ·ͨɺ͜͜Ͱ͸ɺ࣌ ؒґଘੑΛ΋ͬͨԋࢉࢠ͕͍͖ͳΓಘΒΕ·͕ͨ͠ɺ͜Ε͕ྔࢠ࿦తͳӡಈํఔࣜʢϋΠ θϯϕϧάํఔࣜʣΛຬ͍ͨͯ͠Δ͜ͱ͸ɺϋϛϧτχΞϯΛ۩ମతʹߏ੒͢Δ͜ͱͰ֬ ೝͰ͖·͢ɻ ·ͣɺମੵ L3 ͷۭؒʹ͓͚Δɺݹయతͳి࣓৔ͷΤωϧΪʔ͸ɺ࣍ࣜͰఆٛ͞Ε·͢ɻ H = 1 2 ∫ L3 ( ϵ0 E2 + µ0 H2 ) d3r ͜͜ʹɺH = 1 µ0 B Ͱ͋Γɺਅۭͷ༠ి཰ ϵ0 ͱಁ࣓཰ µ0 ͸ɺޫ଎౓ c ͱ࣍ͷؔ܎Ͱ ͭͳ͕Γ·͢ɻ c = 1 √ ϵ0 µ0 ͜ΕʹྔࢠԽ͞Εͨి৔ͷԋࢉࢠ (7)ɺ͓Αͼɺಉ༷ʹܭࢉ͞ΕΔ࣓৔ͷԋࢉࢠΛ୅ೖ ͯ͠੔ཧ͍ͯ͘͠ͱɺ్தͷܭࢉ͸ෳࡶʹͳΓ·͕͢ɺ࠷ऴతʹ࣍ͷද͕ࣜಘΒΕ·͢ɻ H = ∑ k ℏωk ( a† k ak + 1 2 ) (8) ͜Ε͸ɺ1 ͭͷ k ΛऔΓग़ͯ͠ߟ͑Δͱɺ֯଎౓ ωk ͷௐ࿨ৼಈࢠͷϋϛϧτχΞϯʹ Ұக͍ͯ͠·͢ɻͭ·Γɺ͜ͷܥ͸ɺ֯଎౓͕ҟͳΔଟ਺ͷௐ࿨ৼಈࢠΛूΊͨܥͱ౳Ձ ʹͳΓ·͢ɻ·ͨɺͦΕͧΕͷௐ࿨ৼಈࢠ͸ɺجఈঢ়ଶʹ͓͍ͯ E0 = ℏωk 2 ͱ͍͏Τω ϧΪʔΛ͍࣋ͬͯ·͢ɻ͜Ε͸ɺݹయతͳҙຯͰి࣓৔͕ଘࡏ͠ͳ͍ਅͷਅۭঢ়ଶʹ͓͍ ͯ΋ɺྔࢠ࿦తͳి࣓৔ͷΏΒ͕͗ଘࡏ͓ͯ͠ΓɺͦͷΏΒ͗ʹ൐͏ΤωϧΪʔ͕؍ଌ͞ ΕΔ΋ͷͱղऍ͞Ε·͢ɻͨͩ͠ɺ͜ͷܥʹ͸ແݶݸͷௐ࿨ৼಈࢠ͕ଘࡏ͠·͢ͷͰɺ͜ ͷܥશମͷجఈঢ়ଶͷΤωϧΪʔΛ·ͱ΋ʹܭࢉ͢Δͱɺ E = ∑ k ℏωk 2 ͱͳΓɺ஋͕ແݶେʹൃࢄͯ͠͠·͍·͢ɻҰൠʹ͸ɺωk ͷ஋͕ඇৗʹେ͖͘ͳΔྖҬ Ͱ͸ɺ·ͩ஌ΒΕ͍ͯͳ͍෺ཧతͳػߏ͕͸ͨΒ͍ͯɺൃࢄ͕͓͑͞ΒΕΔ΋ͷͱظ଴͞
7. 0.2 ి࣓৔ͷྔࢠԽ 7 Ε·͢ɻͨͩ͠ɺ͜ΕҎ߱ͷܭࢉͰ͸ɺΤωϧΪʔͷج४Λແݶେ͚ͩͣΒͨ͠΋ͷͱߟ ͑ͯɺ୯७ʹ͜ͷൃࢄ஋͸ແࢹͯ͠ߟ͑Δ͜ͱʹ͠·͢ɻ ͍ͣΕʹ͠Ζɺ͜ΕͰϋϛϧτχΞϯ͕ܾఆ͞ΕͨͷͰɺ͜ΕΛ༻͍ͯϋΠθϯϕϧά දࣔʹ͓͚Δԋࢉࢠͷ࣌ؒൃల͕ܭࢉͰ͖·͢ɻ·ͣɺϋΠθϯϕϧάදࣔͷফ໓ԋࢉࢠ ak (t) ʹର͢ΔϋΠθϯϕϧάํఔࣜ͸ɺ࣍Ͱ༩͑ΒΕ·͢ɻ dak (t) dt = i ℏ [H, ak (t)] (9) ͜͜Ͱɺak (t) = ak e−iωkt ͱԾఆ͢Δͱɺ͜Ε͸ɺ্هͷඍ෼ํఔࣜͷղͰ͋Δ͜ͱ͕ ͔֬ΊΒΕ·͢ɻ࣮ࡍɺ͜ͷ࣌ɺ(9) ͷࠨล͸ɺ dak (t) dt = −iωk ak (t) = −iωk ak e−iωkt ͱͳΓɺҰํɺ(8) ͷද͓ࣜΑͼ (6) ͷަ׵ؔ܎Λ༻͍Δͱɺӈล͸࣍ͷΑ͏ʹܭࢉ͞Ε ·͢ɻ i ℏ [H, ak (t)] = iωk [a† k ak , ak ]e−iωkt = −iωk ak e−iωkt ͜ΕΑΓɺ͔֬ʹ (9) ͕੒Γཱͪ·͢ɻಉ༷ʹͯ͠ɺa† k (t) = a†eiωkt ͕ϋΠθϯϕϧ άํఔࣜΛຬͨ͢͜ͱ΋Θ͔Γ·͢ͷͰɺ͜ΕΒΑΓɺ(7) ͷి৔ԋࢉࢠ͸ɺશମͱͯ͠ ϋΠθϯϕϧάํఔࣜΛຬͨ͢͜ͱʹͳΓ·͢ɻ 0.2.3 ϑΥοΫঢ়ଶ ి࣓৔͕ௐ࿨ৼಈࢠͷू߹ͱ౳ՁͰ͋Δ͜ͱ͔Βɺௐ࿨ৼಈࢠͷ཭ࢄతͳྭىঢ়ଶʹ Αͬͯɺి࣓৔ͷཻࢠੑɺ͢ͳΘͪɺޫࢠͷ֓೦͕ಘΒΕ·͢ɻͨͱ͑͹ɺ࿩Λ؆୯ʹ͢ ΔͨΊʹɺಛఆͷ೾਺ϕΫτϧ k ʹରԠ͢Δྭىঢ়ଶͷΈ͕ൃੜ͢ΔͱԾఆͯ͠ɺ࣍ͷ ϋϛϧτχΞϯΛߟ͑·͢ɻ H = ℏω ( a†a + 1 2 ) (10) ͜͜Ͱ͸ɺ೾਺ϕΫτϧʹର͢ΔґଘੑΛࣔ͢ఴࣈΛলུ͍ͯ͠·͢ɻ͜ͷ৔߹ɺରԠ ͢Δి৔ԋࢉࢠ (7) ͸࣍ͷΑ͏ʹͳΓ·͢ɻ E(r, t) = iE0 { ae−i(ωt−k·r) − a†ei(ωt−k·r) } (11) ͦͯ͠ɺ(10) ͸ɺ୯ମͷௐ࿨ৼಈࢠͷϋϛϧτϯͱಉ͡ܗΛ͍ͯ͠·͢ͷͰɺΑ͘஌Β Εͨɺௐ࿨ৼಈࢠΛྔࢠԽ͢Δࡍͷٞ࿦Λͦͷ··ద༻͢Δ͜ͱ͕Ͱ͖·͢ɻ݁࿦Λ·ͱ ΊΔͱ࣍ͷΑ͏ʹͳΓ·͢ɻ
8. 8 ·ͣɺ্هͷϋϛϧτχΞϯͷݻ༗஋͸ɺ E = ℏω ( n + 1 2 ) (n = 0, 1, 2, · · · ) (12) Ͱ༩͑ΒΕ·͢ɻ͜ΕΒ͸ɺޫࢠ͕ଘࡏ͠ͳ͍ਅۭঢ়ଶʢn = 0ʣ ɺ͓Αͼɺޫࢠ͕ n = 1, 2, · · · ݸͷঢ়ଶʹରԠ͢Δͱղऍ͞Ε·͢ɻn ͷ஋͕૿͑Δ͝ͱʹܥͷΤωϧΪʔ ͕ ℏω ͣͭ૿͑Δ͜ͱ͔Βɺޫࢠ 1 ݸ͋ͨΓͷΤωϧΪʔ͸ ℏω ͱ͍͏͜ͱʹͳΓ·͢ɻ (10) ͱ (12) Λݟൺ΂ΔͱΘ͔ΔΑ͏ʹɺԋࢉࢠ N = a†a ͸ޫࢠ਺Λද͓ͯ͠Γɺޫࢠ ਺͕ n ͷঢ়ଶ |n⟩ ͸ɺ N |n⟩ = a†a |n⟩ = n |n⟩ (13) ͱ͍͏ؔ܎Λຬ͓ͨͯ͠Γɺ͜Ε͸ɺ࣍ͷΤωϧΪʔݻ༗ঢ়ଶͱͳΓ·͢ɻ H |n⟩ = ℏω ( n + 1 2 ) |n⟩ ͜ͷࡍɺn = 0 ͷجఈঢ়ଶʢਅۭঢ়ଶʣʹରԠ͢Δঢ়ଶϕΫτϧ |0⟩ ͸ɺ a |0⟩ = 0 ͱ͍͏৚݅Ͱఆٛ͞Ε·͢ɻ·ͨɺੜ੒ԋࢉࢠ a† ͱফ໓ԋࢉࢠ a ͸ɺޫࢠ਺Λ 1 ͭͣͭ ૿ݮͤ͞ΔޮՌ͕͋Γɺn = 0, 1, 2, · · · ʹରͯ͠ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ a |n⟩ = √ n |n − 1⟩ (14) a† |n⟩ = √ n + 1 |n + 1⟩ (15) ͕ͨͬͯ͠ɺਅۭঢ়ଶ |0⟩ ʹੜ੒ԋࢉࢠ a† Λ n ճԋࢉͨ͠΋ͷΛେ͖͞ 1 ʹਖ਼نԽ͢ Δ͜ͱͰɺঢ়ଶ |n⟩ ΛಘΔ͜ͱ͕Ͱ͖·͢ɻ |n⟩ = 1 √ n! a†n |0⟩ (n = 1, 2, · · · ) (16) ͜ΕΒ͸ɺΤϧϛʔτԋࢉࢠͰ͋ΔϋϛϧτχΞϯͷݻ༗ঢ়ଶͰ͋Δ͜ͱ͔Βɺঢ়ଶۭ ؒͷਖ਼ن௚ަجఈΛߏ੒͓ͯ͠Γɺ࣍ͷਖ਼ن௚ަ৚݅ɺ͓Αͼɺ׬શܥͷ৚݅Λຬͨ͠ ·͢ɻ ⟨n|n′⟩ = δnn′ (n, n′ = 0, 1, 2, · · · ) (17) ∞ ∑ n=0 |n⟩ ⟨n| = 1 (18) Ұൠʹɺޫࢠ਺͕֬ఆͨ͠ঢ়ଶ |n⟩ ΛϑΥοΫঢ়ଶͱݺͼ·͢ɻ
9. 0.2 ి࣓৔ͷྔࢠԽ 9 ͜͜·ͰͰɺຊߘͷٞ࿦ͷલఏ͕ଗ͍·ͨ͠ɻҰൠʹޫ૿෯ثΛ௨ͯ͠ಘΒΕΔޫύϧ ε͸ɺෳ਺ͷϑΥοΫঢ়ଶͷॏͶ߹Θͤʹͳ͓ͬͯΓɺॏͶ߹Θͤͷํ๏ʹΑΓɺίώʔ Ϩϯτঢ়ଶɺ͋Δ͍͸ɺεΫΠʔζυঢ়ଶͱݺ͹ΕΔಛ௃తͳੑ࣭Λ࣋ͬͨঢ়ଶͱͳΓ· ͢ɻຊߘͰ͸ɺ͜ͷޙɺ͜ΕΒͷঢ়ଶΛܥ౷తʹऔΓѻ͏ख๏Λղઆ͍ͯ͘͜͠ͱʹͳΓ ·͢ɻ ͳ͓ɺ͜͜ͰɺίώʔϨϯτঢ়ଶͷઆ໌ʹਐΉલʹɺϑΥοΫঢ়ଶͷॏཁͳੑ࣭Λ࠶֬ ೝ͓͖ͯ͠·͢ɻ·ͣɺϑΥοΫঢ়ଶ͸ɺΤωϧΪʔݻ༗ঢ়ଶͰ͢ͷͰɺ࣌ؒతʹมಈ͠ ͳ͍ఆৗঢ়ଶʹ૬౰͠·͢ɻͨͱ͑͹ɺϑΥοΫঢ়ଶ |n⟩ ʹ͍ͭͯɺ(11) Ͱఆٛ͞ΕΔి ৔ԋࢉࢠ E(r, t) ͷظ଴஋Λܭࢉ͢ΔͱͲ͏ͳΔͰ͠ΐ͏͔ʁɹ͜Ε͸ɺϋΠθϯϕϧά දࣔͷԋࢉࢠͰ͢ͷͰɺ࣌ࠁ tɺ఺ r ʹ͓͚Δి৔ͷظ଴஋͕ܭࢉ͞ΕΔ͜ͱʹͳΓ·͢ ͕ɺ͜ͷޙ͙͢ʹࣔ͢Α͏ʹɺ ⟨n| E(r, t) |n⟩ = 0 (19) ͕੒Γཱͪ·͢ɻͭ·Γɺ্ۭؒͷ͢΂ͯͷ఺ʹ͓͍ͯɺి৔ͷظ଴஋͸ৗʹ 0 ʹͳΓ· ͢ɻ͜Ε͸ɺਖ਼ݭؔ਺ʹैͬͯ࣌ؒతʹৼಈ͢Δͱ͍͏ɺௐ࿨ৼಈࢠͷݹయతͳඳ૾ͱ͸ ·ͬͨ͘Ұக͠ͳ͍݁ՌͰ͢ɻ(19) ͷূ໌͸؆୯Ͱɺ·ͣɺ (14) ͓Αͼ (17) Λ༻͍Δͱɺ ⟨n| a |n⟩ = √ n ⟨n|n − 1⟩ = 0 (20) ͕੒Γཱͪ·͢ɻ·ͨɺ্ࣜͷෳૉڞ໾ΛऔΔͱɺ ⟨n| a† |n⟩ = 0 (21) ͕ಘΒΕ·͢ɻE(r, t) ͸ a ͱ a† ͷઢܗ݁߹Ͱ͢ͷͰɺ͜ΕΒΑΓ (19) ͕ಘΒΕ·͢ɻ ͳ͓ɺ্هͷܭࢉΛҰൠԽ͢Δͱɺ೚ҙͷ m = 1, 2, · · · ʹ͍ͭͯɺ ⟨n| am |n⟩ = ⟨n| a†m |n⟩ = 0 (22) ͕੒Γཱͭ͜ͱ͕Θ͔Γ·͢ɻ ͨͩ͠ɺ͜Ε͸ɺి৔ͷظ଴஋ʢෳ਺ճ؍ଌͨ͠ࡍͷฏۉ஋ʣ͕࣌ࠁʹΑΒͣʹ 0 ͱ͍ ͏͜ͱͰ͋Γɺݸผͷ؍ଌʹ͓͍ͯ͸ɺ0 Ҏ֎ͷ஋͕؍ଌ͞ΕΔ͜ͱ΋͋Γಘ·͢ɻ͜ͷ ࣄ࣮͸ɺి৔ͷ෼ࢄΛܭࢉ͢Δ͜ͱͰ֬ೝͰ͖·͢ɻࠓͷ৔߹ɺి৔ͷظ଴஋͸ 0 Ͱ͢ͷ ͰɺE2(r, t) ͷظ଴஋Λܭࢉ͢Δ͜ͱͰ෼ࢄ͕ಘΒΕ·͢ɻ·ͣɺ(11) ΑΓɺE2(r, t) Λ ܭࢉ͢Δͱɺ͕࣍ಘΒΕ·͢ɻ E2(r, t) = −E2 0 { a2e−2i(ωt−k·r) + a†2e2i(ωt−k·r) − (aa† + a†a) } = −E2 0 { a2e−2i(ωt−k·r) + a†2e2i(ωt−k·r) − (2a†a + 1) }
10. 10 2 ͭ໨ͷ౳߸Ͱ͸ɺަ׵ؔ܎ [a, a†] = 1 Λ༻͍͍ͯ·͢ɻ͜ΕΑΓɺ(22) ͓Αͼ (13) Λ༻͍ͯɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨n| E2(r, t) |n⟩ = E2 0 (2n + 1) ͜ͷ݁Ռ͔ΒɺϑΥοΫঢ়ଶʹ͓͚Δి৔͸ɺ࣌ࠁʹґଘ͠ͳ͍Ұఆͷ֬཰తͳ޿͕ ΓΛ࣋ͭ͜ͱ͕Θ͔Γ·͢ɻಛʹޫࢠ਺͕ 0 ͷجఈঢ়ଶʹ͓͍ͯ΋ E2 0 ͸ਖ਼ͷظ଴஋Λ ͓࣋ͬͯΓɺ͜Ε͕ (8) ͷ௚ޙʹ৮Εͨྔࢠ࿦తͳి࣓৔ͷΏΒ͗ʹ૬౰͢Δ΋ͷͱͳΓ ·͢ɻ 0.3 ίώʔϨϯτঢ়ଶ લষͷ࠷ޙʹ৮ΕͨΑ͏ʹɺΤωϧΪʔͷݻ༗ঢ়ଶͰ͋ΔϑΥοΫঢ়ଶ͸ɺ࣌ؒతͳม ಈΛ൐Θͳ͍ఆৗঢ়ଶͰ͋Γɺਖ਼ݭؔ਺ʹैͬͯৼಈ͢Δͱ͍͏ɺௐ࿨ৼಈࢠͷݹయతͳ ඳ૾ʹ͸Ұக͠·ͤΜɻ֎෦ͱͷ૬ޓ࡞༻͕ͳ͍ௐ࿨ৼಈࢠʹ͓͍ͯ͸ɺݹయతͳඳ૾ʹ Ұக͢Δঢ়ଶͱͯ͠ɺແݶݸͷϑΥοΫঢ়ଶͷॏͶ߹ΘͤͰදݱ͞ΕΔίώʔϨϯτঢ়ଶ ͕͋Γ·͢ɻ͜͜Ͱ͸ɺԋࢉࢠͷجຊతͳܭࢉنଇʢެࣜʣΛ੔ཧ্ͨ͠ͰɺίώʔϨϯ τঢ়ଶͷੑ࣭Λݟ͍͖ͯ·͢ɻ 0.3.1 ԋࢉࢠͷܭࢉنଇ ͸͡Ίʹɺ͜ͷޙͷܭࢉͰར༻͢ΔެࣜΛ·ͱΊ͓͖ͯ·͢ɻ·ͣɺA ͱ B Λ೚ҙͷ ԋࢉࢠͱͯ͠ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ eABe−A = ∞ ∑ n=0 1 n! (CA )nB (23) ͜͜ʹɺCA ͸ɺCA B = [A, B] Ͱఆٛ͞ΕΔԋࢉࢠͰɺ(CA )2B ͳͲ͸ɺ࣍ͷΑ͏ʹ ܭࢉ͞Ε·͢ɻ (CA )2B = [A, [A, B]] (CA )3B = [A, [A, [A, B]]] . . . ࣍ʹɺA ͱ B ͷަ׵ؔ܎ [A, B] ͕ A, B ͷͲͪΒͱ΋Մ׵ͱͳΔ৔߹ɺͭ·Γɺ [A, [A, B]] = [B, [A, B]] = 0 ͱͳΔ৔߹ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ eA+B = eAeBe− 1 2 [A, B] (24)
11. 0.3 ίώʔϨϯτঢ়ଶ 11 ଓ͍ͯɺA, B, C Λ೚ҙͷԋࢉࢠͱͯ͠ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ [A, BC] = [A, B]C + B[A, C] [AB, C] = A[B, C] + [A, C]B ੜ੒ফ໓ԋࢉࢠͷ 2 ݸҎ্ͷੵʹର͢Δަ׵ؔ܎Λܭࢉ͢Δ࣌͸ɺ͜ͷؔ܎Λ܁Γฦ͠ ద༻͢Δ͜ͱͰɺ(6) ͷؔ܎ʹؼண͢Δ͜ͱ͕Ͱ͖·͢ɻͨͱ͑͹ɺ࣍ͷΑ͏ͳܭࢉྫ͕ ߟ͑ΒΕ·͢ɻ [a, a†2] = [a, a†]a† + a†[a, a†] = 2a† [a, a†3] = [a, a†2]a† + a†2[a, a†] = 2a†2 + a†2 = 3a†2 ਺ֶతؼೲ๏Λ༻͍Δͱɺ্هͷܭࢉྫ͸ɺ࣍ͷΑ͏ʹҰൠԽ͢Δ͜ͱ͕Ͱ͖·͢ɻ [a, a†n] = na†(n−1) ಉ༷ʹͯ͠ɺ࣍ͷؔ܎Λࣔ͢͜ͱ΋Ͱ͖·͢ɻ [an, a†] = nan−1 0.3.2 ίώʔϨϯτঢ়ଶͷఆٛ ϑΥοΫঢ়ଶʹ͓͍ͯి৔ͷظ଴஋͕ 0 ʹͳΔͷ͸ɺ(20)(21) ͰݟͨΑ͏ʹɺੜ੒ফ໓ ԋࢉࢠ a, a† ͷظ଴஋͕ 0 ʹͳΔ఺ʹ༝དྷ͠·͢ɻͦΕͰ͸ɺԾʹɺα Λ೚ҙͷෳૉ਺ͱ ͯ͠ɺ ⟨α| a |α⟩ = α, ⟨α| a† |α⟩ = α∗ (25) Λຬͨ͢ঢ়ଶ |α⟩ ͕ଘࡏͨ͠ͱ͢Ε͹ɺͲ͏ͳΔͰ͠ΐ͏͔ʁɹ͜ͷ৔߹ɺి৔ͷظ଴஋ ͸ (11) ΑΓɺ ⟨α| E(r, t) |α⟩ = iE0 { αe−i(ωt−k·r) − α∗ei(ωt−k·r) } = −2E0 Im { αe−i(ωt−k·r) } ͱͳΓ·͢ɻ͜Ε͸ɺෳૉৼ෯Λ α ͱ͢Δฏ໘೾ͷํఔࣜʹଞͳΒͣɺݹయతͳ೾ಈͷ ඳ૾͕࠶ݱ͞ΕΔ͜ͱʹͳΓ·͢ɻͦͯ͠ɺ࣮ࡍʹ͜ͷΑ͏ͳ৚݅Λຬͨ͢ঢ়ଶͷ 1 ͭ ͕ɺ͜͜Ͱઆ໌͢ΔίώʔϨϯτঢ়ଶͱͳΓ·͢ɻ ͸͡Ίʹɺ΍΍ఱԼΓతͰ͕͢ɺα Λ೚ҙͷෳૉ਺ͱͯ͠ɺ࣍ͷฒਐԋࢉࢠΛఆٛ͠ ·͢ɻ D(α) = exp ( αa† − α∗a ) (26)
12. 12 D(α) ͸ɺ൓Τϧϛʔτԋࢉࢠ A = −(αa† − α∗a) Λ༻͍ͯɺD(α) = e−A ͱॻ͚Δ ͜ͱʹ஫ҙ͢ΔͱɺD†(α) = e−A† = eA ͱͳΔ͜ͱ͔Βɺ D(α)†D(α) = e−AeA = 1 ͕੒Γཱͪ·͢ɻͭ·ΓɺD(α) ͸ɺD†(α) = D−1(α) Λຬͨ͢ϢχλϦʕԋࢉࢠͰ͋Δ ͜ͱ͕Θ͔Γ·͢ɻ·ͨɺઌͷఆٛΑΓɺࣗ໌ʹ D(−α) = D†(α) ͕੒ΓཱͭͷͰɺ͜Ε ΒΛ·ͱΊͯɺ͕࣍ಘΒΕ·͢ɻ D†(α) = D−1(α) = D(−α) (27) ࣍ʹɺ͜ͷฒਐԋࢉࢠʹΑΔফ໓ԋࢉࢠͷม׵ D†(α)aD(α) Λܭࢉ͠·͢ɻ·ͣɺઌ ΄Ͳͷ A = −(αa† − α∗a) ͱ a ͷަ׵ؔ܎͸ɺ CA a = [A, a] = α ͱͳΔͷͰɺ n = 2, 3, · · · ʹରͯ͠ɺ (CA )na = 0 ͕ಘΒΕ·͢ɻ͕ͨͬͯ͠ɺ D(α) = e−A ʹରͯ͠ɺ(23) Λద༻͢Δͱ͕࣍ಘΒΕ·͢ɻ D†(α)aD(α) = a + α (28) ͜Ε͸ɺϋΠθϯϕϧάදࣔͰݟͨ࣌ʹɺϢχλϦʕԋࢉࢠ D(α) ʹΑΔঢ়ଶͷม׵ ͸ɺෳૉৼ෯ͷظ଴஋Λ α ͚ͩʮฏߦҠಈʯ͢ΔޮՌ͕͋Δ͜ͱΛҙຯ͠·͢ɻ͜ͷࣄ ࣮͸ɺγϡϨʔσΟϯΨʔදࣔͰ࣍ͷΑ͏ʹදΘ͢͜ͱ΋Ͱ͖·͢ɻ·ͣɺίώʔϨϯτ ঢ়ଶ |α⟩ Λ࣍ࣜͰఆٛ͠·͢ɻ |α⟩ = D(α) |0⟩ D(α) ͕ϢχλϦʔม׵Ͱ͋Δ͜ͱ͔Βɺ͜Ε͸ɺਖ਼نԽ͞Εͨঢ়ଶʹͳΓ·͢ɻ ⟨α|α⟩ = ⟨0| D†(α)D(α) |0⟩ = ⟨0|0⟩ = 1 ͜ͷ࣌ɺ(28) ͷ྆ลʹࠨ͔Β D(α) Λԋࢉ͢Δͱɺ(27) Λ༻͍ͯɺ aD(α) = D(α)a + αD(α) ͕ಘΒΕ·͢ɻ͜ΕΛར༻͢Δͱɺ࣍ͷΑ͏ʹɺίώʔϨϯτঢ়ଶ |α⟩ ͸ফ໓ԋࢉࢠ a ͷ ݻ༗ঢ়ଶʢݻ༗஋ αʣͰ͋Δ͜ͱ͕Θ͔Γ·͢ɻ a |α⟩ = aD(α) |0⟩ = {D(α)a + αD(α)} |0⟩ = αD(α) |0⟩ = α |α⟩ ͜ΕΑΓɺίώʔϨϯτঢ়ଶ |α⟩ ͸๯಄ͷ (25) ͷ৚݅Λຬ͓ͨͯ͠Γɺݹయతͳ೾ಈ ͷඳ૾ʹରԠ͢Δঢ়ଶͱݴ͑·͢ɻ
13. 0.3 ίώʔϨϯτঢ়ଶ 13 0.3.3 ίώʔϨϯτঢ়ଶͷੑ࣭ ͜͜Ͱ͸·ͣɺίώʔϨϯτঢ়ଶͱϑΥοΫঢ়ଶͷؔ܎Λௐ΂·͢ɻ·ͣɺฒਐԋࢉࢠ ͷఆٛ (26) ʹ͓͍ͯɺA = αa†, B = −α∗a ͱͯ͠ɺެࣜ (24) Λద༻͠·͢ɻࠓͷ৔ ߹ɺ[A, B] = |α|2 Ͱ͋Γɺ(24) ͷલఏ৚݅͸͔֬ʹຬͨ͞Ε͓ͯΓɺ࣍ͷ݁Ռ͕ಘΒΕ ·͢ɻ D(α) = e− 1 2 |α|2 eαa† e−α∗a ͜͜Ͱɺa |0⟩ = 0 ΑΓɺe−α∗a |0⟩ = 1 ͱͳΔ͜ͱʹ஫ҙ͢Δͱɺ(16) Λ༻͍ͯɺί ώʔϨϯτঢ়ଶΛϑΥοΫঢ়ଶͷઢܗ݁߹ʹల։͢Δ͜ͱ͕Ͱ͖·͢ɻ |α⟩ = D(α) |0⟩ = e− 1 2 |α|2 eαa† |0⟩ = e− 1 2 |α|2 ∞ ∑ n=0 αn n! a†n |0⟩ = e− 1 2 |α|2 ∞ ∑ n=0 αn √ n! |n⟩ ͜ΕΑΓɺίώʔϨϯτঢ়ଶ |α⟩ ͸͞·͟·ͳޫࢠ਺Λ࣋ͬͨঢ়ଶͷॏͶ߹Θͤʹͳͬ ͍ͯΔ͜ͱ͕Θ͔Γ·͢ɻͦͯ͠ɺޫࢠ਺Λ؍ଌͨ͠ࡍʹɺn ݸͷޫࢠ͕؍ଌ͞ΕΔ֬཰ ͸ɺ࣍ࣜͰ༩͑ΒΕ·͢ɻ P(n) = | ⟨n|α⟩ |2 = e−|α|2 |α|2n n! ͜Ε͸ɺظ଴஋ͱ෼ࢄ͕ |α|2 ͷϙΞιϯ෼෍ʹҰக͍ͯ͠·͢*2ɻ·ͨɺ্هͷ݁Ռ Λ༻͍Δͱɺ2 छྨͷίώʔϨϯτঢ়ଶ |α⟩ , |β⟩ ͷ಺ੵ͸࣍ࣜͰ༩͑ΒΕ·͢ɻ ⟨α|β⟩ = e− 1 2 (|α|2+|β|2) ∞ ∑ n=0 α∗nβn n! = e− 1 2 (|α|2+|β|2−2α∗β) ͞Βʹɺ͜ͷେ͖͞ͷ 2 ৐Λܭࢉ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ | ⟨α|β⟩ |2 = ⟨α|β⟩ ⟨β|α⟩ = e− 1 2 (|α|2+|β|2−2α∗β)e− 1 2 (|α|2+|β|2−2αβ∗) = e−(|α|2+|β|2−α∗β−αβ∗) = e−|α−β|2 ͜Ε͸ɺίώʔϨϯτঢ়ଶͷू߹͸௚ަܥʹ͸ͳΒͳ͍͜ͱΛ͍ࣔͯ͠·͢ɻͨͩ͠ɺ |α − β| ≫ 1 ͷ৔߹͸ɺۙࣅతʹɺ| ⟨α|β⟩ |2 ∼ 0 ͕੒Γཱͪ·͢ɻ *2 ҰൠʹϙΞιϯ෼෍Ͱ͸ɺظ଴஋ͱ෼ࢄͷ஋͕Ұக͠·͢ɻ
14. 14 ଓ͍ͯɺίώʔϨϯτঢ়ଶʹ͓͚Δి৔ͷ෼ࢄΛܭࢉ͠·͢ɻίώʔϨϯτঢ়ଶʹ͸࣌ ؒมಈ͕ൃੜ͢ΔͷͰɺ͜ͷ఺ʹ஫ҙ͠ͳ͕Βܭࢉ͢Δඞཁ͕͋Γ·͢ɻ͜͜Ͱ͸·ͣɺ ϋΠθϯϕϧάදࣔΛ༻͍ͯɺܥͷঢ়ଶʹґଘ͠ͳ͍Ұൠతͳ෼ࢄͷܭࢉެࣜΛಋ͖· ͢ɻ͸͡Ίʹɺ(11) ͷి৔ԋࢉࢠʹ͓͍ͯɺภޫํ޲ͷมҟΛදΘ͢εΧϥʔԋࢉࢠΛ࣍ Ͱఆٛ͠·͢ɻ E(r, t) = iE0 { ae−i(ωt−k·r) − a†ei(ωt−k·r) } (29) ͜͜ʹɺE0 = |E0 | ͱ͠·͢ɻ͜ͷ࣌ɺ a = 1 2 (X1 + iX2 ), a† = 1 2 (X1 − iX2 ) ͢ͳΘͪɺ X1 = a + a†, X2 = −i(a − a†) (30) ͰΤϧϛʔτԋࢉࢠ X1 , X2 Λఆٛ͢Δͱɺ(3) Λಋ͍ͨࡍͱಉ༷ͷܭࢉʹΑΓɺ࣍ͷؔ ܎͕ಘΒΕ·͢ɻ E(r, t) = E0 {X1 sin(ωt − k · r) − X2 cos(ωt − k · r)} (31) ͕ͨͬͯ͠ɺ೚ҙͷঢ়ଶ |Ψ⟩ ʹରͯ͠ɺԋࢉࢠ A ͷظ଴஋ΛҰൠʹ ⟨A⟩ = ⟨Ψ| A |Ψ⟩ ͱදه͢Δͱɺి৔ͷظ଴஋ʹ͍ͭͯɺ͕࣍੒Γཱͪ·͢ɻ ⟨E(r, t)⟩ = E0 {⟨X1 ⟩ sin(ωt − k · r) − ⟨X2 ⟩ cos(ωt − k · r)} (32) ͞Βʹɺ(31) ΑΓɺ E(r, t)2 = E2 0 {X2 1 sin2(ωt − k · r) + X2 2 cos2(ωt − k · r) − {X1 , X2 } sin(ωt − k · r) cos(ωt − k · r)} (33) ͕੒ΓཱͭͷͰɺ͜ͷظ଴஋ΛऔΔͱɺ ⟨E(r, t)2⟩ = E2 0 {⟨X2 1 ⟩ sin2(ωt − k · r) + ⟨X2 2 ⟩ cos2(ωt − k · r) − ⟨{X1 , X2 }⟩ sin(ωt − k · r) cos(ωt − k · r)} (34) ͕ಘΒΕ·͢ɻ͜͜ʹɺ{X1 , X2 } ͸ɺ࣍ͷ൓ަ׵ؔ܎Λද͠·͢ɻ {X1 , X2 } = X1 X2 + X2 X1
15. 0.3 ίώʔϨϯτঢ়ଶ 15 ͕ͨͬͯ͠ɺE(r, t) ͷ෼ࢄ V [E(r, t)] ͸ɺ࣍ͷΑ͏ʹܾ·Γ·͢ɻ V [E(r, t)] = ⟨E(r, t)2⟩ − ⟨E(r, t)⟩2 = E2 0 {⟨X2 1 ⟩ sin2(ωt − k · r) + ⟨X2 2 ⟩ cos2(ωt − k · r) − ⟨{X1 , X2 }⟩ sin(ωt − k · r) cos(ωt − k · r)} − [E0 {⟨X1 ⟩ sin(ωt − k · r) − ⟨X2 ⟩ cos(ωt − k · r)}]2 = E2 0 { (⟨X2 1 ⟩ − ⟨X1 ⟩2) sin2(ωt − k · r) + (⟨X2 2 ⟩ − ⟨X2 ⟩2) cos2(ωt − k · r) − (⟨{X1 , X2 }⟩ − 2⟨X1 ⟩⟨X2 ⟩) sin(ωt − k · r) cos(ωt − k · r) } = E2 0 { V (X1 ) sin2(ωt − k · r) + V (X2 ) cos2(ωt − k · r) −2V (X1 , X2 ) sin(ωt − k · r) cos(ωt − k · r) } (35) ͜͜ʹɺV (X1 ), V (X2 ), V (X1 , X2 ) ͸ɺͦΕͧΕɺX1 , X2 ͷ෼ࢄɺ͓Αͼɺڞ෼ࢄ Λද͠·͢ɻ V (X1 ) = ⟨X2 1 ⟩ − ⟨X1 ⟩2, V (X2 ) = ⟨X2 2 ⟩ − ⟨X2 ⟩2 V (X1 , X2 ) = 1 2 ⟨{X1 , X2 }⟩ − ⟨X1 ⟩⟨X2 ⟩ ࣍ʹɺίώʔϨϯτঢ়ଶʹ͍ͭͯ͜ΕΒͷ஋Λܭࢉ͠·͕͢ɺͦͷ४උͱͯ͠ɺ(28) Λ ֦ுͨ࣍͠ͷެࣜΛ͓͖ࣔͯ͠·͢ɻ D†(α)a†manD(α) = (a† + α∗)m(a + α)n (m, n = 0, 1, 2, · · · ) (36) ͜Ε͸ɺD(α) ͕ϢχλϦʔԋࢉࢠͰ͋ΓɺD(α)D†(α) = 1 Λຬͨ͢͜ͱ͔Β੒Γཱ ͪ·͢ɻ·ͣɺ(28) ͷΤϧϛʔτڞ໾ΛऔΔͱɺ D†(α)a†D(α) = a† + α∗ ͕ಘΒΕ·͢ɻ͕ͨͬͯ͠ɺͨͱ͑͹ m = n = 2 ͷ৔߹ɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ D†(α)a†2a2D(α) = D†(α)a†D(α)D†(α)a†D(α)D†(α)aD(α)D†(α)aD(α) = (a† + α∗)2(a + α)2 Ұൠͷ m, n ʹ͍ͭͯ΋ɺಉ༷ͷܭࢉ͕੒Γཱͭ͜ͱ͸͙͢ʹΘ͔ΔͰ͠ΐ͏ɻ͜ΕΑ ΓɺίώʔϨϯτঢ়ଶ |α⟩ = D(α) |0⟩ ʹରͯ͠ɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ ⟨α| a†nam |α⟩ = ⟨0| D†(α)a†namD(α) |0⟩ = ⟨0| (a† + α∗)m(a + α)n |0⟩ = α∗mαn
16. 16 ࠷ޙͷ౳߸ʹ͍ͭͯ͸ɺa |0⟩ = 0ɺ͓ΑͼɺͦͷΤϧϛʔτڞ໾ ⟨0| a† = 0 ͔Β੒Γཱ ͪ·͢ɻ ͜ͷ݁ՌΛ༻͍ΔͱɺX1 = a + a† ͷ෼ࢄ V (X1 ) ʹ͍ͭͯ͸ɺ X2 1 = a2 + a†2 + aa† + a†a = a2 + a†2 + 2a†a + 1 (37) ʹ஫ҙͯ͠ɺ࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ V (X1 ) = ⟨α| X2 1 |α⟩ − ⟨α| X1 |α⟩2 = (α2 + α∗2 + 2α∗α + 1) − (α + α∗)2 = 1 X2 = −i(a − a†) ʹ͍ͭͯ΋ಉ༷ͷܭࢉʹΑΓɺ V (X2 ) = ⟨α| X2 2 |α⟩ − ⟨α| X2 |α⟩2 = 1 ͕ಘΒΕ·͢ɻ ଓ͍ͯɺڞ෼ࢄ V (X1 , X2 ) ʹ͍ͭͯ͸ɺ 1 2 {X1 , X2 } = 1 2 (X1 X2 + X2 X1 ) = −i 2 { (a2 − aa† + a†a − a†2) + (a2 − a†a + aa† − a†2) } = −i(a2 − a†2) ͱͳΔ͜ͱ͔Βɺ V (X1 , X2 ) = 1 2 ⟨{X1 , X2 }⟩ − ⟨X1 ⟩⟨X2 ⟩ = −i(α2 − α∗2) + i(α + α∗)(α − α∗) = 0 ͕ಘΒΕ·͢ʢίϥϜʮ࠷খෆ֬ఆঢ়ଶͱڞ෼ࢄͷؔ܎ʯ΋ࢀরʣ ɻ Ҏ্ͷ݁ՌΛ (35) ʹ୅ೖ͢ΔͱɺίώʔϨϯτঢ়ଶʹ͓͚Δి৔ E(r, t) ෼ࢄ͸ɺ V [E(r, t)] = E2 0 ͱܾ·Γ·͢ɻͭ·ΓɺίώʔϨϯτঢ়ଶʹ͓͚Δి৔͸ɺҰఆͷ෼ࢄΛอͬͨ··ৼಈ Λଓ͚Δ͜ͱ͕Θ͔Γ·͢ɻ
17. 0.4 εΫΠʔζυঢ়ଶ 17 ίϥϜɿ࠷খෆ֬ఆঢ়ଶͱڞ෼ࢄͷؔ܎   ɹҰൠʹɺඇՄ׵ͳ 2 ͭͷԋࢉࢠ X1 , X2 ʹ͍ͭͯɺ࣍ͷෆ౳͕ࣜ੒Γཱͭ͜ͱ͕஌ ΒΕ͍ͯ·͢ɻ √ V (X1 )V (X2 ) ≥ 1 2 |⟨[X1 , X2 ]⟩| ɹ͜Ε͸ϩόʔτιϯͷෆ౳ࣜͱݺ͹ΕΔ΋ͷͰɺϋΠθϯϕϧάͷෆ֬ఆੑݪ ཧΛ਺ֶతʹදݱͨ͠΋ͷͱߟ͑ΒΕ·͢ɻͨͱ͑͹ɺ࠲ඪͱӡಈྔͷަ׵ؔ܎ [x, p] = iℏ ʹ͜ΕΛద༻͢Δͱɺ∆x = √ V (x), ∆p = √ V (p) ͱͯ͠ɺ ∆x∆p ≥ ℏ 2 ͱͳΓɺΑ͘஌ΒΕͨ ∆x∆p ∼ ℏ ͱ͍͏ؔ܎͕࠶ݱ͞Ε·͢ɻ ɹຊจͰఆٛͨ͠ X1 = a + a† ͓Αͼ X2 = −i(a − a†) ͷ৔߹ɺ͜ΕΒͷަ׵ؔ ܎͸ɺ [X1 , X2 ] = −i[a + a†, a − a†] = i[a, a†] − i[a†, a] = 2i ͱͳΔ͜ͱ͔Βɺϩόʔτιϯͷෆ౳ࣜΑΓɺ √ V (X1 )V (X2 ) ≥ 1 ͕੒Γཱͪ·͢ɻ͕ͨͬͯ͠ɺV (X1 ) = V (X2 ) = 1 ͱͳΔίώʔϨϯτঢ়ଶ͸ɺ͜ ͷҙຯͰͷෆ֬ఆੑ͕࠷খݶʹ͓͑͞ΒΕͨঢ়ଶͱݴ͑·͢ɻ ɹͦͯ͠ɺϩόʔτιϯͷෆ౳ࣜʹ͓͚Δ౳߸͕੒ཱ͢Δঢ়ଶʹ͍ͭͯ͸ɺඞͣɺڞ ෼ࢄ V (X1 , X2 ) ͕ 0 ʹͳΓ·͢ɻ͜Ε͸ɺϩόʔτιϯͷෆ౳ࣜΑΓڧ੍͍໿Λද Θ͢ɺ࣍ͷγϡϨʔσΟϯΨʔͷෆ֬ఆੑؔ܎͔Β֬ೝ͢Δ͜ͱ͕Ͱ͖·͢aɻ V (X1 )V (X2 ) ≥ |V (X1 , X2 )|2 + 1 4 |⟨[X1 , X2 ]⟩|2 a γϡϨʔσΟϯΨʔͷෆ֬ఆੑؔ܎ͷূ໌͸ɺ[5] Λࢀরɻ   0.4 εΫΠʔζυঢ়ଶ ͜͜Ͱ͸ɺ͍Α͍ΑɺຊߘͷϝΠϯςʔϚͰ΋͋ΔεΫΠʔζυঢ়ଶʹ͍ͭͯௐ΂͍ͯ ͖·͢ɻ͸͡ΊʹεΫΠʔζԋࢉࢠΛఆٛͯ͠ɺਅۭঢ়ଶʹର͢Δ͜ͷԋࢉࢠͷޮՌΛ֬
18. 18 ೝ͠·͢ɻͦͷޙɺਅۭΛεΫΠʔζͨ͠ঢ়ଶΛฒਐԋࢉࢠͰฏߦҠಈ͢Δ͜ͱͰɺҰൠ ͷεΫΠʔζυঢ়ଶΛߏ੒͠·͢*3ɻ 0.4.1 εΫΠʔζυԋࢉࢠͷఆٛ ͸͡Ίʹɺ΍΍ఱԼΓతͰ͕͢ɺϵ Λ೚ҙͷෳૉ਺ͱͯ͠ɺεΫΠʔζԋࢉࢠΛ࣍ࣜͰ ఆٛ͠·͢ɻ S(ϵ) = exp { 1 2 (ϵ∗a2 − ϵa†2) } (38) ͜ͷ࣌ɺఆٛΑΓࣗ໌ʹ S(−ϵ) = S†(ϵ) ͕੒Γཱͪ·͢ɻ·ͨɺ൓Τϧϛʔτԋࢉࢠ A = − 1 2 (ϵ∗a2 − ϵa†2) Λ༻͍ͯɺS(ϵ) = e−A ͱॻ͚Δ͜ͱ͔Βɺฒਐԋࢉࢠʹ͍ͭͯ (27) Λಋ͍ͨࡍͱಉٞ͡࿦ʹΑΓɺS(ϵ) ͸ϢχλϦʔԋࢉࢠͰ͋Γɺ࣍ͷؔ܎Λຬͨ͢ ͜ͱ͕Θ͔Γ·͢ɻ S†(ϵ) = S−1(ϵ) = S(−ϵ) ͦͯ͠ɺ͜ͷεΫΠʔζԋࢉࢠ͸ɺa ͱ a† Λ࣍ͷΑ͏ʹม׵͢ΔޮՌ͕͋Γ·͢ɻ S†(ϵ)aS(ϵ) = a cosh r − a†e2iϕ sinh r (39) S†(ϵ)a†S(ϵ) = a† cosh r − ae−2iϕ sinh r (40) ͜͜ʹɺr ͱ ϕ ͸ɺϵ = re2iϕ ͱۃදࣔͨ͠ࡍͷ ϵ ͷେ͖͞ͱҐ૬෦෼Λද͠·͢ɻ͜ ͷؔ܎͸ɺA = − 1 2 (ϵ∗a2 − ϵa†2), B = a ͱͯ͠ɺ(23) ͷެࣜΛద༻͢Δ͜ͱͰࣔ͞Ε· ͢ɻ·ͣɺA ͱ a ͷަ׵ؔ܎Λ 2 ճଓ͚ͯܭࢉ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ CA a = − 1 2 [ϵ∗a2 − ϵa†2, a] = 1 2 ϵ[a†2, a] = −ϵa† (CA )2a = − 1 2 [ϵ∗a2 − ϵa†2, −ϵa†] = 1 2 |ϵ|2[a2, a†] = |ϵ|2a = r2a ͜ΕΑΓɺҰൠͷ n = 1, 2, · · · ʹ͍ͭͯɺ͕࣍੒Γཱͭ͜ͱ͕਺ֶతؼೲ๏Ͱࣔ͞Ε ·͢ɻ (CA )na = { −ϵrn−1a† = −e2iϕrna† (n = 1, 3, 5, · · · ) rna (n = 0, 2, 4, · · · ) ͕ͨͬͯ͠ɺsinh r ͱ cosh r ͷϚΫϩʔϦϯల։ɺ sinh r = ∞ ∑ n=0 r2k+1 (2k + 1)! , cosh r = ∞ ∑ n=0 r2k (2k)! *3 εΫΠʔζʢSqueezeʣ͸ɺӳޠͰʮԡͭ͠Ϳ͢ʯͱ͍͏ҙຯʹͳΓ·͢ɻ
19. 0.4 εΫΠʔζυঢ়ଶ 19 Λ༻͍ͯɺ(23) ΑΓɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ S†(ϵ)aS(ϵ) = ∞ ∑ n=0 1 n! (CA )na = −e2iϕa† ∞ ∑ n=0 r2k+1 (2k + 1)! + a ∞ ∑ n=0 r2k (2k)! = a cosh r − a†e2iϕ sinh r ͜ΕͰ (39) ͕ࣔ͞Ε·ͨ͠ɻ(40) ͸ɺ͜ͷΤϧϛʔτڞ໾ͱͯ͠ಘΒΕ·͢ɻͦ͠ ͯɺS(ϵ) ͕ϢχλϦʔԋࢉࢠͰ͋Δ͜ͱ͔Βɺ(36) ͱಉ༷ʹͯ͠ɺm, n = 0, 1, 2, · · · ʹ ͍ͭͯɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ S†(ϵ)a†manS(ϵ) = (a† cosh r − ae−2iϕ sinh r)m(a cosh r − a†e2iϕ sinh r)n (41) ͜͜Ͱಛʹɺϕ = 0ɺ͢ͳΘͪɺϵ = r ͷ৔߹Λߟ͑Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ S†(r)aS(r) = a cosh r − a† sinh r (42) S†(r)a†S(r) = a† cosh r − a sinh r (43) S†(r)a†manS(r) = (a† cosh r − a sinh r)m(a cosh r − a† sinh r)n (44) 0.4.2 εΫΠʔζԋࢉࢠͷੑ࣭ εΫΠʔζԋࢉࢠ S(ϵ) ͷಇ͖Λௐ΂ΔͨΊʹɺҰྫͱͯ͠ɺਅۭঢ়ଶ |0⟩ ʹ S(r) Λ ԋࢉͨ࣍͠ͷঢ়ଶΛ༻ҙ͠·͢ɻ |0, r⟩ = S(r) |0⟩ ͜ͷ࣌ɺి৔ԋࢉࢠ (31) ͷظ଴஋ɺ͓Αͼɺ෼ࢄ͸ͲͷΑ͏ʹ࣌ؒมԽ͢ΔͰ͠ΐ͏ ͔ʁɹ·ͣɺ(42) ΑΓɺ ⟨0, r| a |0, r⟩ = ⟨0| S†(r)aS(r) |0⟩ = cosh r ⟨0| a |0⟩ − sinh r ⟨0| a† |0⟩ = 0 ͱͳΓɺ͞Βʹ͜ͷෳૉڞ໾ΛऔΔ͜ͱͰɺ ⟨0, r| a† |0, r⟩ = 0 ͱͳΓ·͢ɻ͕ͨͬͯ͠ɺ(30) ΑΓɺ ⟨0, r| X1 |0, r⟩ = 0, ⟨0, r| X2 |0, r⟩ = 0
20. 20 ͱͳΔ͜ͱ͔Βɺి৔ԋࢉࢠͷظ଴஋͸ 0 ͱͳΓ·͢ɻҰํɺ(37) ͰݟͨΑ͏ʹɺ X2 1 = a2 + a†2 + 2a†a + 1 Ͱ͋Δ͜ͱ͔Βɺ⟨0, r| X2 1 |0, r⟩ Λܭࢉ͢Δʹ͸ɺ ⟨0, r| a2 |0, r⟩ , ⟨0, r| a†2 |0, r⟩ , ⟨0, r| a†a |0, r⟩ ͷ஋ΛٻΊΔඞཁ͕͋Γ·͢ɻ͜ΕΒ͸ɺ(44) Λ༻͍ͯܭࢉ͢Δ͜ͱ͕Ͱ͖·͢ɻͨͱ ͑͹ɺ(44) Ͱ m = 0, n = 2 ͷ৔߹Λߟ͑Δͱɺ ⟨0, r| a2 |0, r⟩ = ⟨0| S†(r)a2S(r) |0⟩ = ⟨0| (a cosh r − a† sinh r)2 |0⟩ ͱͳΓ·͕͢ɺ(a cosh r − a† sinh r)2 Λల։ͨ͠ࡍʹਅۭظ଴஋͕ 0 ʹͳΒͳ͍ͷ͸ɺੵ aa† ΛؚΉ߲ͷΈͰɺ ⟨0| aa† |0⟩ = ⟨0| (a†a + 1) |0⟩ = 1 ͱ͍͏ؔ܎ʹ஫ҙ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨0, r| a2 |0, r⟩ = − cosh r sinh r ⟨0| aa† |0⟩ = − cosh r sinh r ͜ͷෳૉڞ໾ΛऔΔͱɺ ⟨0, r| a†2 |0, r⟩ = − cosh r sinh r ͕ಘΒΕ·͢ɻಉ༷ʹͯ͠ɺ ⟨0, r| a†a |0, r⟩ = ⟨0| S†(r)a†aS(r) |0⟩ = ⟨0| (a† cosh r − a sinh r)(a cosh r − a† sinh r) |0⟩ = sinh2 r ⟨0| aa† |0⟩ = sinh2 r ͕ಘΒΕΔͷͰɺ͜ΕΒΛ·ͱΊΔͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨0, r| X2 1 |0, r⟩ = −2 cosh r sinh r + 2 sinh2 r + 1 = −2 ( er + e−r 2 ) ( er − e−r 2 ) + 2 ( er − e−r 2 )2 + 1 = e−2r (45) ͕ͨͬͯ͠ɺX1 ͷ෼ࢄ͸ɺ V (X1 ) = ⟨0, r| X2 1 |0, r⟩ − ⟨0, r| X1 |0, r⟩2 = e−2r
21. 0.4 εΫΠʔζυঢ়ଶ 21 ͱܾ·Γ·͢ɻಉ༷ʹͯ͠ɺX2 ͷ෼ࢄ΋ܭࢉ͢Δ͜ͱ͕Ͱ͖ͯɺ݁Ռ͸࣍ͷΑ͏ʹͳΓ ·͢ɻ V (X2 ) = ⟨0, r| X2 2 |0, r⟩ − ⟨0, r| X2 |0, r⟩2 = e2r ͜ͷ݁ՌΛݟΔͱɺ|0, r⟩ ͸ɺX1 ͱ X2 ʹ͍ͭͯɺલষͷίϥϜʮ࠷খෆ֬ఆঢ়ଶͱ ڞ෼ࢄͷؔ܎ʯͰઆ໌ͨ͠ɺෆ֬ఆੑ͕࠷খݶͷঢ়ଶɺ͢ͳΘͪɺ V (X1 )V (X2 ) = 1 Λຬͨ͢ঢ়ଶʹͳ͓ͬͯΓɺ͜ΕΑΓɺڞ෼ࢄʹ͍ͭͯɺ V (X1 , X2 ) = 0 ͕੒Γཱͭ͜ͱ͕Θ͔Γ·͢ɻ Ҏ্ͷ݁ՌΛ (35) ʹ୅ೖ͢Δͱɺ݁ہͷॴɺి৔ԋࢉࢠͷ෼ࢄ͸࣍ͷΑ͏ʹܾ·Γ ·͢ɻ V [E(r, t)] = E2 0 { e−2r sin2(ωt − k · r) + e2r cos2(ωt − k · r) } (46) ͨͱ͑͹ɺ࠲ඪݪ఺ r = 0 ͷҐஔΛߟ͑Δͱɺ࣌ࠁ t = 0 ʹ͓͚Δ෼ࢄ͸ɺE2 0 e2r Ͱ ͋ΓɺͦͷޙɺE2 0 e−2rʙE2 0 e2r ͷൣғΛपظతʹมಈ͢Δ͜ͱ͕Θ͔Γ·͢ɻ ͜ͷ݁Ռ͸ɺ࣍ͷΑ͏ʹਤܗతʹղऍ͢Δ͜ͱ͕Ͱ͖·͢ɻ·ͣɺ(29) Ͱ r = 0 ͱ͢ Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ E(0, t) = iE0 (ae−iωt − a†eiωt) = −E0 X2 (t) ࠷ޙͷ౳߸͸ɺX2 = −i(a − a†) ʹɺa, a† ͷ࣌ؒൃల a(t) = ae−iωt, a†(r) = a†eiωt Λద༻ͨ͠ɺ࣍ͷ݁Ռ͔ΒಘΒΕ·͢ɻ X2 (t) = −i(ae−iωt − a†eiωt) ͜Ε͸ෳૉฏ໘্Λ֯଎౓ −ω Ͱճస͢Δԋࢉࢠɺ a(t) = ae−iωt = 1 2 {X1 (t) + iX2 (t)} ͷڏ෦ʹΑͬͯి৔ E(0, t) ͕༩͑ΒΕΔ͜ͱΛҙຯ͠·͢*4ɻ͜͜Ͱ͸ɺԋࢉࢠ a(t) ͕෺ཧత࣮ମͰ͋Γɺి৔ E(0, t) ͸ͦͷڏ෦ͷΈΛऔΓग़ͨ͠ԋࢉࢠͩͱղऍͯͩ͘͠ *4 ͜ͷٞ࿦Ͱ͸ɺઆ໌Λ؆໌ʹ͢ΔͨΊʹɺఆ਺ഒͷҧ͍Λແࢹͯ͠ߟ͍͑ͯ·͢ɻ·ͨɺҰൠͷ఺ r ʹ͓ ͍ͯ͸ɺa(t) ͷॳظҐ૬͕Ұఆ஋ k · r มԽ͢Δ͚ͩͰຊ࣭తͳҧ͍͸͋Γ·ͤΜɻ
22. 22 ਤ 2 ԋࢉࢠ a(t) ͕ճస͢Δ༷ࢠ ͍͞ɻ͜ͷΑ͏ͳཧղͷ΋ͱͰ͸ɺઌʹܭࢉͨ͠ V (X1 ) ͱ V (X2 ) ͸ɺԋࢉࢠ a(t) ͷ࣌ ࠁ t = 0 ʹ͓͚Δ࣮࣠ํ޲ͱڏ࣠ํ޲ͷ෼ࢄΛදΘ͢͜ͱʹͳΓ·͢ɻ ͦͯ͠ɺҰൠͷ࣌ࠁ t ʹ͓͍ͯ͸ɺਤ 2 ͷΑ͏ʹɺԋࢉࢠ a(t) ͸֯ −ωt ͚ͩճస͢Δ ͨΊɺV (X1 ) ͱ V (X2 ) ͸ɺ−ωt ͚ͩճసͨ͠ํ޲ͷ෼ࢄͱͳΓ·͢ɻ ͕ͨͬͯ͠ɺͨͱ͑͹ɺωt = π 2 ͱͳΔ࣌ࠁʹ͓͍ͯ͸ɺ࣮෦ͱڏ෦͕ͪΐ͏ͲೖΕସ ΘͬͯɺE(0, t)ɺ͢ͳΘͪɺa(t) ͷڏ෦ͷ෼ࢄ͸ V (X1 ) ʹҰக͠·͢ɻͦͷଞͷҰൠͷ ࣌ࠁʹ͓͍ͯ͸ɺE(0, t) ͷ෼ࢄ͸ɺV (X2 ) ͱ V (X1 ) ͷؒΛৼಈ͢Δ͜ͱ͕ਤܗతʹཧ ղͰ͖ΔͰ͠ΐ͏ɻͳ͓ɺঢ়ଶ |0, r⟩ ͷ৔߹ɺa ͷظ଴஋͸ 0 Ͱ͢ͷͰɺΑΓਖ਼֬ʹ͸ɺ ਤ 3 ͷঢ়گʹ૬౰͠·͢ɻͭ·Γɺݪ఺Λத৺ͱ͢ΔɺV (X1 ) = e−2r, V (X2 ) = e2r ͷ ପԁܗͷ֬཰෼෍͕֯଎౓ −ωt Ͱճస͢ΔܗʹͳΓ·͢ɻ ͪͳΈʹɺr = 0ɺ͢ͳΘͪɺϵ = 0 ͷ৔߹ɺεΫΠʔζԋࢉࢠ͸߃౳ԋࢉࢠͱ ͳΔͷͰɺ|r, 0⟩ ͸ਅۭঢ়ଶ |0⟩ ʹҰக͠·͢ɻͦͯ͜͠ͷ࣌ɺX1 , X2 ͷ෼ࢄ͸ɺ V (X1 ) = 1, V (X2 ) = 1 ͱͳΓɺਤ 3 ͷପԁ͸ਅԁͱͳΓ·͢ɻٯʹݴ͏ͱɺr > 0 ͷ৔ ߹ɺঢ়ଶ |0, r⟩ ͸ɺ࣮࣠ํ޲ͷ෼ࢄ͕ԡͭ͠Ϳ͞Εͯɺͦͷ୅ΘΓʹɺڏ࣠ํ޲ͷ෼ࢄ͕ ޿͕Δ͜ͱʹͳΓ·͢ɻ͜ͷΑ͏ʹɺෳૉৼ෯ a ͷ෼ࢄΛҰఆͷํ޲ʹԡͭ͠Ϳ͢ޮՌ ͕͋Δ͜ͱ͔ΒɺS(ϵ) ͸εΫΠʔζԋࢉࢠͱݺ͹Ε·͢ɻ ͳ͓ɺ͜͜·ͰɺεΫΠʔζԋࢉࢠͷύϥϝʔλʔ ϵ = re2iϕ ʹ͍ͭͯɺϕ = 0 ͷ৔߹ ͷΈΛߟ͖͑ͯ·͕ͨ͠ɺҰൠʹ ϕ ̸= 0 ͷ৔߹͸ɺ෼ࢄΛԡͭ͠Ϳ͢ํ޲͕֯ ϕ ͚ͩม Խ͠·͢ɻ͜Ε͸ɺ࣍ͷܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻ͸͡Ίʹɺ2a = (X1 + iX2 ) Λ −ϕ ͚ͩճసͨ͠ԋࢉࢠͷ࣮෦ͱڏ෦Λ Y1 , Y2 ͱ͠·͢ɻ Y1 + iY2 = 2ae−iϕ = (X1 + iX2 )e−iϕ (47)
23. 0.4 εΫΠʔζυঢ়ଶ 23 ਤ 3 ঢ়ଶ |0, r⟩ ͷෳૉৼ෯͕ճస͢Δ༷ࢠ ͜ͷ࣌ɺ(39) ΑΓɺ ⟨0, ϵ| a |0, ϵ⟩ = ⟨0| S†(ϵ)aS(ϵ) |0⟩ = ⟨0| (a cosh r − a†e2iϕ sinh r) |0⟩ = 0 ͱͳΔ͜ͱ͔ΒɺY1 ͱ Y2 ͷظ଴஋͸ͲͪΒ΋ 0 ʹͳΓ·͢ɻ ⟨0, ϵ| Y1 |0, ϵ⟩ = 0, ⟨0, ϵ| Y2 |0, ϵ⟩ = 0 ࣍ʹɺY1 = Re (2ae−iϕ) = ae−iϕ + a†eiϕ ΑΓɺ Y 2 1 = e−2iϕa2 + e2iϕa†2 + aa† + a†a = e−2iϕa2 + e2iϕa†2 + 2a†a + 1 (48) ͱͳΔͷͰɺ⟨0, ϵ| Y 2 1 |0, ϵ⟩ Λܭࢉ͢Δʹ͸ɺ ⟨0, ϵ| a2 |0, ϵ⟩ , ⟨0, ϵ| a†2 |0, ϵ⟩ , ⟨0, ϵ| a†a |0, ϵ⟩ ΛٻΊΔඞཁ͕͋Γ·͢ɻ͜Ε͸ɺ(41) Λ༻͍ͯܭࢉ͠·͢ɻͨͱ͑͹ɺm = 0, n = 2 ͷ৔߹Λߟ͑Δͱɺ(45) ΛٻΊͨ࣌ͱಉ༷ʹͯ͠ɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ ⟨0, ϵ| a2 |0, ϵ⟩ = ⟨0| S†(ϵ)a2S(ϵ) |0⟩ = ⟨0| (a cosh r − a†e2iϕ sinh r)2 |0⟩ = −e2iϕ cosh r sinh r ⟨0| aa† |0⟩ = −e2iϕ cosh r sinh r ্ࣜͷෳૉڞ໾ΛऔΔͱɺ͕࣍ಘΒΕ·͢ɻ ⟨0, ϵ| a†2 |0, ϵ⟩ = −e−2iϕ cosh r sinh r
24. 24 ਤ 4 ԋࢉࢠ 2a ͷճసલޙͷ෼ࢄ ࣍ʹɺ(41) Ͱ m = 1, n = 1 ͷ৔߹Λߟ͑Δͱɺ͕࣍ಘΒΕ·͢ɻ ⟨0, ϵ| a†a |0, ϵ⟩ = ⟨0| S†(ϵ)a†aS(ϵ) |0⟩ = ⟨0| (a† cosh r − ae−2iϕ sinh r)(a cosh r − a†e2iϕ sinh r) |0⟩ = sinh2 r ⟨0| aa† |0⟩ = sinh2 r Ҏ্ͷ݁ՌΛ·ͱΊΔͱɺ(48) ΑΓɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨0, ϵ| Y 2 1 |0, ϵ⟩ = −2 cosh r sinh r + 2 sinh2 r + 1 = e−2r ͕ͨͬͯ͠ɺY1 ͷ෼ࢄ͸ɺ V (Y1 ) = ⟨0, ϵ| Y 2 1 |0, ϵ⟩ − ⟨0, ϵ| Y1 |0, ϵ⟩2 = e−2r ͱܾ·Γ·͢ɻಉ༷ʹͯ͠ɺY 2 2 ͷ෼ࢄ΋ܭࢉ͢Δ͜ͱ͕Ͱ͖ͯɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ V (Y2 ) = ⟨0, ϵ| Y 2 2 |0, ϵ⟩ − ⟨0, ϵ| Y2 |0, ϵ⟩2 = e2r ͭ·Γɺঢ়ଶ |0, ϵ⟩ ʹ͓͍ͯ͸ɺԋࢉࢠ 2a Λ֯ −ϕ ͚ͩճస͢Δͱɺ࣮෦ Y1 ͱڏ෦ Y2 ͷ෼ࢄ͕ɺͪΐ͏Ͳ e−2r ͓Αͼ e2r ͱͳΓ·͢ɻ͜Ε͸ɺճస͢ΔલͰ͍͏ͱɺ࣮ ࣠ͱڏ࣠Λ֯ ϕ ͚ͩճసͨ͠ํ޲ͷ෼ࢄ͕ e−2r ͓Αͼ e2r Ͱ͋Δ͜ͱΛҙຯ͠·͢ʢਤ 4ʣ ɻ
25. 0.4 εΫΠʔζυঢ়ଶ 25 ਤ 5 εΫΠʔζυঢ়ଶʹ͓͚Δ a ͷ֬཰෼෍ 0.4.3 εΫΠʔζͱฒਐͷ૊Έ߹Θͤ ਅۭঢ়ଶ |0⟩ ʹεΫΠʔζԋࢉࢠ S(ϵ) Λԋࢉ͢Δ͜ͱͰɺਤ 4ʢӈʣͷΑ͏ʹɺҰఆ ํ޲ʹ෼ࢄΛԡͭ͠Ϳͨ͠ঢ়ଶ͕ಘΒΕΔ͜ͱ͕Θ͔Γ·ͨ͠ɻ͜ΕΛ͞Βʹฒਐԋࢉࢠ D(α) ͰฏߦҠಈͨ͠ɺ࣍ͷঢ়ଶΛҰൠͷεΫΠʔζυঢ়ଶͱͯ͠ఆٛ͠·͢ɻ |α, ϵ⟩ = D(α)S(ϵ) |0⟩ ͜Ε͸ɺਤ 5 ͷΑ͏ʹɺෳૉৼ෯ʢফ໓ԋࢉࢠʣ a ͷظ଴஋͕ α Ͱɺ(47) Ͱఆٛͨ͠ɺ ֯ ϕ ͚ͩճసͨ͠ํ޲ͷ෼ࢄ V (Y1 ), V (Y2 ) ͕ͦΕͧΕ e−2rɺ͓Αͼɺe2r ʹҰக͢Δ΋ ͷͱظ଴͞Ε·͢*5ɻ͜͜Ͱ͸·ͣɺ͜ΕΒͷࣄ࣮Λ۩ମతͳܭࢉͰ֬ೝ͓͖ͯ͠·͢ɻ εΫΠʔζυঢ়ଶʹؔ͢ΔܭࢉΛߦ͏ࡍ͸ɺ(36)ɺ͓Αͼɺ(41) Ͱࣔͨ͠ɺ࣍ͷؔ܎ࣜ Λར༻͍͖ͯ͠·͢ɻ D†(α)a†manD(α) = (a† + α∗)m(a + α)n S†(ϵ)a†manS(ϵ) = (a† cosh r − ae−2iϕ sinh r)m(a cosh r − a†e2iϕ sinh r)n ͨͱ͑͹ɺফ໓ԋࢉࢠ a ʹ͍ͭͯɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ S†(ϵ)D†(α)aD(α)S(ϵ) = S†(ϵ)(a + α)S(ϵ) = a cosh r − a†e2iϕ sinh r + α *5 ݫີʹ͸ V (Y1), V (Y2) ͸ԋࢉࢠ 2a ʹର͢Δ෼ࢄͰ͕͢ɺ͜͜Ͱ͸ఆ਺ഒ͸ແࢹͯ͠ߟ͍͑ͯ·͢ɻ
26. 26 ͕ͨͬͯ͠ɺεΫΠʔζυঢ়ଶʹ͓͚Δ a ͷظ଴஋͸࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ ⟨α, ϵ| a |α, ϵ⟩ = ⟨0| S†(ϵ)D†(α)aD(α)S(ϵ) |0⟩ = ⟨0| (a cosh r − a†e2iϕ sinh r + α) |0⟩ = α ͜ΕͰ·ͣ͸ɺa ͷظ଴஋͕ α ʹͳΔ͜ͱ͕Θ͔Γ·ͨ͠ɻෳૉڞ໾ΛऔΔͱ࣍ͷؔ ܎͕ಘΒΕ·͢ɻ ⟨α, ϵ| a† |α, ϵ⟩ = α∗ ͜ΕΒΛ༻͍Δͱɺ Y1 = Re (2ae−iϕ) = ae−iϕ + a†eiϕ (49) Y2 = Im (2ae−iϕ) = −i(ae−iϕ − a†eiϕ) (50) ͷظ଴஋͸ɺ࣍ͷΑ͏ʹܾ·Γ·͢ɻ ⟨α, ϵ| Y1 |α, ϵ⟩ = αe−iϕ + α∗eiϕ (51) ⟨α, ϵ| Y2 |α, ϵ⟩ = −i(αe−iϕ − α∗eiϕ) (52) ଓ͍ͯɺ(49) ΑΓɺ Y 2 1 = e−2iϕa2 + e2iϕa†2 + aa† + a†a = e−2iϕa2 + e2iϕa†2 + 2a†a + 1 (53) ͱͳΔ͜ͱ͔ΒɺY 2 1 ͷظ଴஋ΛٻΊΔʹ͸ɺ⟨α, ϵ| a2 |α, ϵ⟩ , ⟨α, ϵ| a†2 |α, ϵ⟩ɺͦͯ͠ɺ ⟨α, ϵ| a†a |α, ϵ⟩ Λܭࢉ͢Δඞཁ͕͋Γ·͢ɻ͸͡Ίʹɺa2 ʹ͍ͭͯߟ͑Δͱɺ࣍ͷΑ͏ ʹͳΓ·͢ɻ·ͣɺ S†(ϵ)D†(α)a2D(α)S(ϵ) = S†(ϵ)(a + α)2S(ϵ) = S†(ϵ)(a2 + 2αa + α2)S(ϵ) = (a cosh r − a†e2iϕ sinh r)2 + 2α(a cosh r − a†e2iϕ sinh r) + α2 ͱͳΔͷͰɺ͜Εʹରͯ͠ਅۭظ଴஋ΛऔΔͱɺaa† ΛؚΉ߲ɺ͓Αͼɺఆ਺߲ͷΈ͕࢒ Δ͜ͱʹ஫ҙͯ͠ɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨α, ϵ| a2 |α, ϵ⟩ = ⟨0| S†(ϵ)D†(α)a2D(α)S(ϵ) |0⟩ = ⟨0| { (a cosh r − a†e2iϕ sinh r)2 + 2α(a cosh r − a†e2iϕ sinh r) + α2 } |0⟩ = −e2iϕ cosh r sinh r + α2
27. 0.4 εΫΠʔζυঢ়ଶ 27 ্هͷෳૉڞ໾ΛऔΔͱɺ͕࣍ಘΒΕ·͢ɻ ⟨α, ϵ| a†2 |α, ϵ⟩ = −e−2iϕ cosh r sinh r + α∗2 a†a ʹ͍ͭͯ΋ಉ༷ͷܭࢉΛߦ͍·͢ɻ·ͣɺ S†(ϵ)D†(α)a†aD(α)S(ϵ) = S†(ϵ)(a† + α∗)(a + α)S(ϵ) = S†(ϵ)(a†a + αa† + α∗a + |α|2)S(ϵ) = (a† cosh r − ae−2iϕ sinh r)(a cosh r − a†e2iϕ sinh r) + α(a† cosh r − ae−2iϕ sinh r) + α∗(a cosh r − a†e2iϕ sinh r) + |α|2 ͱͳΔͷͰɺ͜ͷਅۭظ଴஋ΛऔΔͱɺઌͱಉ༷ʹ aa† ΛؚΉ߲ͱఆ਺߲ͷΈ͕࢒Γɺ࣍ ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨α, ϵ| a†a |α, ϵ⟩ = ⟨0| S†(ϵ)D†(α)a†aD(α)S(ϵ) |0⟩ = ⟨0| { (a† cosh r − ae−2iϕ sinh r)(a cosh r − a†e2iϕ sinh r) + α(a† cosh r − ae−2iϕ sinh r) + α∗(a cosh r − a†e2iϕ sinh r) + |α|2 } |0⟩ = sinh2 r + |α|2 Ҏ্ͷ݁ՌΛ (53) ʹద༻͢ΔͱɺY 2 1 ͷظ଴஋͸࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ ⟨α, ϵ| Y 2 1 |α, ϵ⟩ = e−2iϕ(−e2iϕ cosh r sinh r + α2) + e2iϕ(−e−2iϕ cosh r sinh r + α∗2) + 2(sinh2 r + |α|2) + 1 = −2 cosh r sinh r + e−2iϕα2 + e2iϕα∗2 + 2 sinh2 r + 2|α|2 + 1 ͕ͨͬͯ͠ɺ(51) ͱ͋ΘͤͯɺY1 ͷ෼ࢄ V (Y1 ) ͸࣍ͷΑ͏ʹܾ·Γ·͢ɻ V (Y1 ) = ⟨α, ϵ| Y 2 1 |α, ϵ⟩ − ⟨α, ϵ| Y1 |α, ϵ⟩2 = (−2 cosh r sinh r + e−2iϕα2 + e2iϕα∗2 + 2 sinh2 r + 2|α|2 + 1) − (e−2iϕα2 + e2iϕα∗2 + 2|α|2) = −2 cosh r sinh r + 2 sinh2 r + 1 = e−2r ͜ΕͰɺY1 ͷ෼ࢄ͕͔֬ʹ e−2r ͱͳΔ͜ͱ͕֬ೝͰ͖·ͨ͠ɻಉ༷ͷܭࢉʹΑͬͯɺ Y2 ͷ෼ࢄʹ͍ͭͯ΋ɺ V (Y2 ) = ⟨α, ϵ| Y 2 2 |α, ϵ⟩ − ⟨α, ϵ| Y2 |α, ϵ⟩2 = e2r ͱͳΔ͜ͱ͕֬ೝͰ͖·͢ɻ͜ΕͰઌ΄Ͳͷਤ 5 ͷඳ૾͕ɺ࣮ࡍͷܭࢉͰ΋֬ೝ͞Ε· ͨ͠ɻ
28. 28 ਤ 6 α ͕࣮਺Ͱ r > 0 ͷ৔߹ 0.4.4 εΫΠʔζυঢ়ଶͷ࣌ؒൃల લઅͷਤ 5 ͸ɺεΫΠʔζυঢ়ଶ |α, ϵ⟩ ͷ࣌ࠁ t = 0 ʹ͓͚Δෳૉৼ෯ a ͷ֬཰෼෍ Λ໛ࣜతʹදͨ͠΋ͷͰ͢ɻ͜ͷޙɺԋࢉࢠ a ͸ɺa(t) = ae−iωt ʹ͕ͨͬͯ࣌ؒ͠ൃల ͢ΔͷͰɺ͜ͷ֬཰෼෍͸ɺ֯଎౓ −ω Ͱෳૉฏ໘্Λճస͍ͯ͘͜͠ͱʹͳΓ·͢ɻͦ ͯ͠ɺෳૉৼ෯ͷڏ਺෦෼͕࠲ඪݪ఺ r = 0 ʹ͓͚Δి৔ E(0, t) ʹରԠ͢Δ͜ͱ͔Βɺ ి৔ͷ෼ࢄ V [E(0, t)] ͸ɺa ͷڏ࣠ํ޲ͷ෼ࢄͱͯ࣌ؒ͠มಈ͢Δ͜ͱʹͳΓ·͢ɻ ͜͜ͰɺಛʹɺฒਐԋࢉࢠʹΑΔฏߦҠಈ α Λ࣮਺ͱͯ͠ɺ͞Βʹɺϵ = re2iϕ ʹ͓͍ ͯɺϕ = 0 ͱͯ͠ɺr > 0 ͷ৔߹ͱ r < 0 ͷ৔߹Λൺֱͯ͠Έ·͢*6ɻ·ͣɺr > 0 ͷ ৔߹ɺ֬཰෼෍ͷ࣌ؒมԽ͸ਤ 6 ͷΑ͏ʹදΘ͢͜ͱ͕Ͱ͖·͢ɻෳૉৼ෯ a ͷڏ෦͕ ి৔ʹରԠ͢Δ͜ͱΛࢥ͍ग़͢ͱɺి৔ͷظ଴஋͕ 0 ͷࡍʹ෼ࢄ V [E(0, t)] ͕࠷େͱͳ Γɺͦͷޙɺి৔ͷظ଴஋͕େ͖͘ͳΔͱɺͦΕʹ͋Θͤͯ෼ࢄ͕খ͘͞ͳΔͱ͍͏ಈ͖ Λ͠·͢ɻٯʹ r < 1 ͷ৔߹͸ɺਤ 7 ͷΑ͏ʹɺి৔ͷظ଴஋͕ 0 ͷࡍʹ෼ࢄ͸࠷খͱ ͳΓɺి৔ͷظ଴஋͕େ͖͘ͳΔͱ෼ࢄ΋େ͖͘ͳΔͱ͍͏ಈ͖Λ͠·͢ɻ ͜ͷঢ়گ͸ɺ(35) Λ༻͍ͯɺ࣮ࡍʹܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻ·ͣɺϕ = 0 Ͱ͋ *6 r ͸ ϵ ͷେ͖͞ͳͷͰɺݫີʹ͸ r < 0 ͷ৔߹ͱ͍͏ͷ͸ଘࡏͤͣɺ࣮ࡍʹ͸ɺr > 0 Ͱ ϕ = π 2 ͷ৔ ߹ʹ૬౰͠·͢ɻͨͩ͠ɺܭࢉ্͸ܗࣜతʹ r < 0 ͱߟ͑ͯ΋݁Ռ͸ಉ͡ʹͳΓ·͢ɻ
29. 0.4 εΫΠʔζυঢ়ଶ 29 ਤ 7 α ͕࣮਺Ͱ r < 0 ͷ৔߹ Δ͜ͱ͔ΒɺY1 , Y2 ͸ X1 , X2 ʹҰகͯ͠ɺ V (X1 ) = V (Y1 ) = e−2r, V (X2 ) = V (Y2 ) = e2r ͕੒Γཱͪ·͢ɻ͕ͨͬͯ͠ɺ ʮ0.4.2 εΫΠʔζԋࢉࢠͷੑ࣭ʯͰɺঢ়ଶ |0, r⟩ ʹ͓͚Δ ෼ࢄ V [E(r, t)] Λܭࢉͨ͠ࡍͱಉ͡ܭࢉ͕੒Γཱͪɺ(46) ͱಉ݁͡Ռ͕ಘΒΕ·͢ɻಛ ʹ r = 0 ͷ৔߹Λߟ͑Δͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ V [E(0, t)] = E2 0 ( e−2r sin2 ωt + e2r cos2 ωt ) Ұํɺి৔ͷظ଴஋ͷ࣌ؒมಈ͸ɺ(32) Ͱܭࢉ͢Δ͜ͱ͕Ͱ͖·͢ɻ(51)(52) Λ༻͍ Δͱɺࠓͷ৔߹͸ɺϕ = 0 Ͱ α ͕࣮਺Ͱ͋Δ͜ͱʹ஫ҙͯ͠ɺ ⟨X1 ⟩ = ⟨α, r| Y1 |α, r⟩ = α + α∗ = 2α ⟨X2 ⟩ = ⟨α, r| Y2 |α, r⟩ = −i(α − α∗) = 0 ͕ಘΒΕ·͢ɻ͜ΕΒΛ (32) ʹ୅ೖͯ͠ɺr = 0 ͷ৔߹Λߟ͑Δͱ࣍ͷ݁Ռ͕ಘΒΕ ·͢ɻ ⟨E(0, t)⟩ = 2E0 α sin(ωt) ͜ΕΒͷ݁ՌΛ༻͍ͯɺr > 0, r = 0, r < 0 ͷͦΕͧΕͷ৔߹ʹ͍ͭͯɺඪ४ภࠩͷ෯ ΛՃ͑ͨৼಈͷ༷ࢠɺ͢ͳΘͪɺ ⟨E(0, t)⟩ ± √ V [E(0, t)]
30. 30 ਤ 8 εΫΠʔζυঢ়ଶʹ͓͚Δి৔ͷมҐʹ൐͏෼ࢄͷมԽ Λάϥϑʹ͋ΒΘ͢ͱɺਤ 8 ͷ݁Ռ͕ಘΒΕ·͢ɻ͜ΕΛݟΔͱɺલड़ͨ͠ɺి৔ͷมҐ ʹ൐͏෼ࢄͷมԽ͕͔֬ʹݱΕ͍ͯΔ͜ͱ͕Θ͔Γ·͢ɻಛʹ r > 0 ͷ৔߹͸ɺి৔ͷৼ ෯ʢE(0, t) ͷ࠷େ஋ʣʹର͢ΔΏΒ͕͗খ͘͞ͳ͍ͬͯ·͢ɻ͜Ε͸ɺr = 0 ͷҰൠత ͳίώʔϨϯτঢ়ଶʹରͯ͠ɺৼ෯ͷྔࢠ࿦తͳΏΒ͗ΛΑΓ͓͑ͨ͞ঢ়ଶ͕ඞཁͳࡍʹ ׆༻Ͱ͖Δٕज़ͱͳΓ·͢ɻ 0.4.5 εΫΠʔζԋࢉࢠͱฒਐԋࢉࢠͷೖΕସ͑ ຊষͰ͸ɺਅۭঢ়ଶ |0⟩ ΛεΫΠʔζԋࢉࢠͰԡͭ͠Ϳͨ͠ޙʹɺ͜ΕΛฒਐԋࢉࢠͰ ฏߦҠಈͨ͠΋ͷͱͯ͠ɺεΫΠʔζυঢ়ଶΛఆٛ͠·ͨ͠ɻҰํɺਅۭঢ়ଶΛઌʹฏߦ Ҡಈ͓͖ͯ͠ɺޙ͔ΒεΫΠʔζԋࢉࢠͰԡͭ͠Ϳ͢ͱ͍͏ૢ࡞΋ՄೳͰ͢ɻͦͯ͠ɺ͜ ΕΒͷૢ࡞͸ɺ࣍ͷؔ܎Ͱ݁ͼ͖ͭ·͢ɻ D(α)S(ϵ) = S(ϵ)D(β) (54)
31. 0.4 εΫΠʔζυঢ়ଶ 31 ͜͜ʹɺα ͱ β ͸࣍ͷؔ܎Ͱ݁ͼ͍͍ͭͯ·͢ɻ α = β cosh r − β∗e2iϕ sinh r (55) ͭ·ΓɺฏߦҠಈͷํ޲Λ α ͔Β β ʹม͓͚͑ͯ͹ɺઌʹฏߦҠಈͯ͠΋ಘΒΕΔঢ় ଶ͸ಉ͡ͱ͍͏͜ͱʹͳΓ·͢ɻ ͜ͷূ໌͸ɺͦΕ΄Ͳ೉͘͠͸͋Γ·ͤΜɻ͸͡Ίʹɺফ໓ԋࢉࢠ a Λ S(−ϵ) ͰϢχ λϦʔม׵ͨ͠ԋࢉࢠ b Λఆٛ͠·͢ɻ b = S†(−ϵ)aS(−ϵ) = a cosh r + a†e2iϕ sinh r 2 ͭ໨ͷ౳߸͸ɺ(39) Ͱ r ͷූ߸Λม͑ͨ΋ͷͱͯ͠ಘΒΕ·͢ɻ͜ͷΤϧϛʔτڞ ໾ΛऔΔͱɺ͕࣍ಘΒΕ·͢ɻ b† = a† cosh r + ae−2iϕ sinh r ·ͨɺS(−ϵ) = S−1(ϵ) = S†(ϵ) Ͱ͋Δ͜ͱ͔Βɺ b = S(ϵ)aS†(ϵ) (56) ͱॻ͘͜ͱ΋Ͱ͖·͢ɻ͞Βʹɺ͜ͷΤϧϛʔτڞ໾ΛऔΔ͜ͱͰɺ͕࣍ಘΒΕ·͢ɻ b† = S(ϵ)a†S†(ϵ) (57) ͜͜Ͱɺb Λ༻͍ͨฒਐԋࢉࢠ Dg (β) Λ࣍ͷΑ͏ʹఆٛ͠·͢ɻ Dg (β) = exp ( βb† − β∗b ) ͜ͷ࣌ɺ࣍ͷܭࢉʹΑΓɺ͜Ε͸ɺ(55) Ͱఆٛ͞ΕΔ α Λ༻͍ͨɺ௨ৗͷฒਐԋࢉࢠ D(α) ʹҰக͢Δ͜ͱ͕Θ͔Γ·͢ɻ Dg (β) = exp ( βb† − β∗b ) = exp { β(a† cosh r + ae−2iϕ sinh r) − β∗(a cosh r + a†e2iϕ sinh r) } = exp { (β cosh r − β∗e2iϕ sinh r)a† − (β∗ cosh r − βe−2iϕ sinh r)a } = D(α) (58) Ұํɺ A = −(βa† − β∗a), B = −(βb† − β∗b) ͱ͢Δ࣌ɺ(56)(57) ΑΓɺ S(ϵ)AS†(ϵ) = B
32. 32 ͕੒Γཱͪ·͢ɻ͕ͨͬͯ͠ɺ Dg (β) = exp(−B) = exp { −S(ϵ)AS†(ϵ) } ͕੒Γཱͪɺ͜ΕΛϚΫϩʔϦϯల։͢ΔͱɺS†(ϵ)S(ϵ) = 1 Λ༻͍ͯɺ࣍ͷ݁Ռ͕ಘΒ Ε·͢ɻ Dg (β) = ∞ ∑ n=0 1 n! { −S(ϵ)AS†(ϵ) }n = S(ϵ) { ∞ ∑ n=0 1 n! (−A)n } S†(ϵ) = S(ϵ)e−AS†(ϵ) = S(ϵ)D(β)S†(ϵ) (59) ࠷ޙʹ (58) ͱ (59) Λ౳஋ͨ͠΋ͷʹɺӈ͔Β S(ϵ) Λԋࢉ͢Δͱ (54) ͕ಘΒΕ·͢ɻ 0.5 ޫύϥϝτϦοΫ૿෯ث લষͰઆ໌ͨ͠εΫΠʔζυঢ়ଶΛ࣮ࡍʹ࡞Γग़࣮͢ݧ૷ஔͱͯ͠ɺޫύϥϝτϦοΫ ૿෯ث͕஌ΒΕ͍ͯ·͢ɻ͜Ε͸ɺޫʹର͢Δඇઢܗ૬ޓ࡞༻ΛҾ͖ى݁͜͢থମʹ޲͚ ͯɺڧ͍ϙϯϓޫΛૹΓࠐΉ͜ͱʹΑΓɺεΫΠʔζυঢ়ଶͷ৴߸ޫΛಘΔͱ͍͏΋ͷͰ ͢ɻ͜͜Ͱ͸ɺͦͷΑ͏ͳඇઢܗ૬ޓ࡞༻ͷ࠷΋جຊతͳܗࣜΛ༩͑ɺ࣮ࡍʹεΫΠʔζ υঢ়ଶ͕ಘΒΕΔ͜ͱΛཧ࿦తʹࣔ͠·͢ɻ 0.5.1 ૬ޓ࡞༻දࣔ ֎෦͔ΒϙϯϓޫΛૹΓࠐΜͰڧ੍తʹ৴߸ޫΛൃੜ͢Δͱ͍͏ॲཧ͸ɺϋϛϧτχΞ ϯʹରͯ͠ɺ࣌ؒʹґଘ͢Δ૬ޓ࡞༻߲Λ෇͚Ճ͑Δ͜ͱͰهड़͞Ε·͢ɻ͜͜Ͱ͸ɺͦ ͷΑ͏ͳ૬ޓ࡞༻ʹΑΔ࣌ؒൃలΛ؆໌ʹܭࢉ͢ΔखஈΛ༩͑Δɺ૬ޓ࡞༻දࣔΛಋೖ͠ ·͢ɻ ·ͣɺҰൠʹɺ࣌ؒʹґଘ͠ͳ͍ࣗ༝ܥͷϋϛϧτχΞϯ H0 ʹɺ࣌ؒʹґଘ͢Δ૬ޓ ࡞༻߲ V (t) Λ෇͚Ճ͑ͨܥΛߟ͑·͢ɻ H(t) = H0 + V (t) ͜͜ͰɺγϡϨʔσΟϯΨʔදࣔͷঢ়ଶϕΫτϧ |Ψ(t)⟩ ʹରͯ͠ɺ૬ޓ࡞༻දࣔͷঢ় ଶϕΫτϧ |ΨI (t)⟩ Λ࣍ࣜͰఆٛ͠·͢ɻ |ΨI (t)⟩ = e i h H0t |Ψ(t)⟩ (60)
33. 0.5 ޫύϥϝτϦοΫ૿෯ث 33 ͜Ε͸ɺ௚ײతʹ͸ɺϋϛϧτχΞϯશମ H0 + V (t) ʹΑΔ࣌ؒൃల͔ΒɺH0 ʹΑΔ د༩Λʮר͖໭ͨ͠ʯ΋ͷͱߟ͑ΒΕ·͢ɻΑΓਖ਼֬ʹݴ͏ͱɺ|ΨI (t)⟩ ͸ɺ࣍ʹࣔ͢ɺ૬ ޓ࡞༻߲ͷΈʹΑΔγϡϨʔσΟϯΨʔํఔࣜΛຬͨ͠·͢ɻ d dt |ΨI (t)⟩ = − i h VI (t) |ΨI (t)⟩ (61) ͜͜ʹɺVI (t) ͸ɺγϡϨʔσΟϯΨʔදࣔͷ૬ޓ࡞༻߲ V (t) ʹରͯ͠ɺ࣍Ͱఆٛ͞ ΕΔɺ૬ޓ࡞༻දࣔͷ૬ޓ࡞༻߲Ͱ͢ɻ VI (t) = e i h H0tV (t)e− i h H0t (62) (61) ͕੒Γཱͭ͜ͱ͸ɺ࣍ͷΑ͏ʹɺ௚઀ͷܭࢉͰ֬ೝ͢Δ͜ͱ͕Ͱ͖·͢ɻ·ͣɺ (60) Λ࣌ؒඍ෼͢ΔͱɺγϡϨʔσΟϯΨʔํఔࣜɺ d dt |Ψ(t)⟩ = − i h {H0 + V (t)} |Ψ(t)⟩ Λ༻͍ͯɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ d dt |ΨI (t)⟩ = d dt { e i h H0t |Ψ(t)⟩ } = i h H0 e i h H0t |Ψ(t)⟩ + e i h H0t d dt |Ψ(t)⟩ = i h H0 e i h H0t |Ψ(t)⟩ − i h e i h H0t {H0 + V (t)} |Ψ(t)⟩ = − i h e i h H0tV (t) |Ψ(t)⟩ ࠷ޙͷ౳߸͸ɺH0 ͱ e i h H0t ͕Մ׵Ͱ͋Δ͜ͱ͔Β੒Γཱͪ·͢ɻ͜ͷ݁ՌΛ࣍ͷΑ͏ ʹมܗ͢Δͱɺ(61) ͕ಘΒΕ·͢ɻ d dt |ΨI (t)⟩ = − i h e i h H0tV (t)e− i h H0te i h H0t |Ψ(t)⟩ = − i h VI (t) |ΨI (t)⟩ ଓ͍ͯɺγϡϨʔσΟϯΨʔදࣔͰද͞ΕͨҰൠͷԋࢉࢠ O(t) ʹରͯ͠ɺͦͷ૬ޓ࡞ ༻දࣔΛ࣍ࣜͰఆٛ͠·͢ɻ OI (t) = e i h H0tO(t)e− i h H0t (63) ͜Ε͸ɺϋΠθϯϕϧάදࣔʹ͓͚Δɺ࣌ؒʹґଘ͠ͳ͍ϋϛϧτχΞϯ H0 ʹΑΔ࣌ ؒൃలͱಉ౳Ͱ͢ͷͰɺOI (t) ͸ɺH0 Λ༻͍ͨϋΠθϯϕϧάํఔࣜΛຬͨ͠·͢ɻ d dt OI (t) = i h [H0 , OI (t)]
34. 34 ͨͩ͠ɺγϡϨʔσΟϯΨʔදࣔʹ͓͍ͯɺO ࣗ਎͕࣌ؒґଘੑΛ࣋ͭ৔߹͸ɺ࣍ͷΑ ͏ʹ໌ࣔతͳ࣌ؒґଘ߲Λ௥Ճ͢Δඞཁ͕͋Γ·͢ɻ d dt OI (t) = i h [H0 , OI (t)] + e i h H0t ∂O(t) ∂t e− i h H0t ࠷ޙʹ (60) ͱ (63) ͷఆٛΑΓɺ࣌ࠁ t ʹ͓͚Δԋࢉࢠ O ͷظ଴஋͸ɺ૬ޓ࡞༻දࣔ ͷԋࢉࢠͱঢ়ଶϕΫτϧͷ૊Έ߹ΘͤͰܭࢉͰ͖Δ͜ͱ͕Θ͔Γ·͢ɻ ⟨O(t)⟩ = ⟨ΨI (t)| OI (t) |ΨI (t)⟩ (64) ͜͜·Ͱ͸૬ޓ࡞༻දࣔͷҰൠ࿦Ͱ͕͢ɺಛʹɺ૬ޓ࡞༻දࣔʹ͓͍ͯɺ૬ޓ࡞༻߲ VI (t) ͕࣌ؒґଘੑΛ࣋ͨͳ͘ͳΔͱ͍͏ಛผͳ৔߹͸ɺ(61) Λੵ෼͢Δ͜ͱ͕Ͱ͖ͯɺ ૬ޓ࡞༻දࣔʹ͓͚Δঢ়ଶͷ࣌ؒൃల͕ɺ࣍ͷΑ͏ʹ໌ࣔతʹܾ·Γ·͢ɻ |ΨI (t)⟩ = e− i h VI t |Ψ(0)⟩ ͜͜Ͱɺ࣌ࠁ t = 0 Ͱ͸ɺ૬ޓ࡞༻දࣔͱγϡϨʔσΟϯΨʔද͕ࣔҰக͢Δ͜ͱΛ༻ ͍͍ͯ·͢ɻ͜ͷ݁ՌΛ (64) ʹ୅ೖ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ ⟨O(t)⟩ = ⟨Ψ(0)| e i h VI tOI (t)e− i h VI t |Ψ(0)⟩ (65) ͜Ε͕೚ҙͷঢ়ଶ |Ψ(0)⟩ ʹରͯ͠੒Γཱͭͱ͍͏͜ͱ͸ɺ্هͷظ଴஋ΛऔΔԋࢉࢠ e i h VI tOI (t)e− i h VI t ͸ɺϋΠθϯϕϧάදࣔʹΑΔ O(t) ͷදݱͰ͋Δ͜ͱΛҙຯ͠·͢ɻ OH (t) = e i h VI tOI (t)e− i h VI t (66) ͜͜Ͱ͞ΒʹɺOI (t) ͕࣌ؒґଘੑΛ࣋ͨͳ͍৔߹Λߟ͑Δͱɺ্هΛ࣌ؒඍ෼ͯ͠ɺ ࣍ͷؔ܎͕ࣜಘΒΕ·͢ɻ d dt OH (t) = i h [VI , OH (t)] (67) ͜ͷ࠷ޙͷ݁Ռ͸ɺϋΠθϯϕϧάදࣔͱ૬ޓ࡞༻දࣔΛϛοΫεͨ͠ಛผͳϋΠθϯ ϕϧάํఔࣜͱݟΔ͜ͱ͕Ͱ͖ΔͰ͠ΐ͏ɻ 0.5.2 ॖୀޫύϥϝτϦοΫ૿෯ ͜͜Ͱ͸ɺॖୀޫύϥϝτϦοΫ૿෯ͱݺ͹ΕΔ૬ޓ࡞༻ܥʹ͍ͭͯɺ࠷΋୯७Խ͞Ε ͨϞσϧΛ঺հ͠·͢ɻ·ͣɺࣗ༝ܥͷϋϛϧτχΞϯ͸ɺ֯଎౓ ω ͷϞʔυͷΈΛ࣋ ͭి࣓৔ͱ͠·͢ɻ H0 = ℏω ( a†a + 1 2 )
35. 0.5 ޫύϥϝτϦοΫ૿෯ث 35 ͜͜Ͱɺภޫํ޲͸Ұํ޲ͷΈͱԾఆ͍ͯ͠·͢ɻ͜ͷܥʹରͯ͠ɺ֯଎౓ 2ω Ͱมಈ ͢Δɺ࣍ͷ૬ޓ࡞༻߲Λ༩͑·͢ɻ V (t) = − iℏ 2 χ(a2e2iωt − a†2e−2iωt) ͜Ε͸ɺΤωϧΪʔ͕ 2ℏω ͷޫࢠ܈͔ΒͳΔϙϯϓޫΛૹΓࠐΜͰɺϙϯϓޫͷ 1 ͭ ͷޫࢠ͔ΒɺΤωϧΪʔ͕ ℏω ͷ৴߸ޫͷޫࢠΛ 2 ݸੜ੒͢Δͱ͍͏૬ޓ࡞༻Λهड़ͯ͠ ͍·͢ɻຊདྷ͸ϙϯϓޫͷޫࢠͷੜ੒ফ໓΋ߟ͑Δඞཁ͕͋Γ·͕͢ɺ͜͜Ͱ͸ɺϙϯϓ ޫ͸ݹయతͳ֎৔ͱͯ͠औΓѻ͍ͬͯ·͢ɻॖୀޫύϥϝτϦοΫ૿෯ͱ͍͏໊લͷʮॖ ୀʯͱ͍͏෦෼͸ɺϙϯϓޫ͔Βੜ੒͞ΕΔ৴߸ޫͷ 2 ݸͷޫࢠ͕ɺಉҰͷΤωϧΪʔ ℏω Λ࣋ͭ͜ͱʹ༝དྷ͠·͢ɻ࣮ࡍͷ࣮ݧͰ͸ɺ͜ͷΑ͏ͳ૬ޓ࡞༻ΛҾ͖ى͜͢ޮՌΛ ࣋ͬͨ݁থମʹϙϯϓޫΛরࣹ͢Δͱɺ݁থମ͔Β͸ɺϙϯϓޫͱڞʹɺ৽͘͠ൃੜͨ͠ ৴߸ޫ͕ग़ྗ͞ΕΔ͜ͱʹͳΓ·͢ɻ ͦΕͰ͸ɺ͜ͷܥͷ࣌ؒൃలΛ૬ޓ࡞༻දࣔΛ༻͍ͯܭࢉ͍͖ͯ͠·͢ɻ·ͣɺ૬ޓ࡞ ༻߲Λ૬ޓ࡞༻දࣔʹม׵͢Δͱɺ(62) ͷఆٛΑΓɺ͕࣍ಘΒΕ·͢ɻ VI (t) = e i h H0tV (t)e− i h H0t = − iℏ 2 χ ( e i h H0ta2e− i h H0te2iωt − e i h H0ta†2e− i h H0te−2iωt ) ͜͜Ͱɺࣗ༝ܥ H0 ʹ͓͚Δԋࢉࢠ a, a† ͷ࣌ؒൃల͸ɺ e i h H0tae− i h H0t = e−iωt, e i h H0ta†e− i h H0t = eiωt (68) Ͱ༩͑ΒΕΔ͜ͱΛࢥ͍ग़͢ͱɺ্ࣜ͸࣍ͷΑ͏ʹॻ͖׵͑Δ͜ͱ͕Ͱ͖·͢ɻ VI (t) = − iℏ 2 χ ( e i h H0tae− i h H0te i h H0tae− i h H0te2iωt − e i h H0ta†e− i h H0te i h H0ta†e− i h H0te−2iωt ) = − iℏ 2 χ(a2 − a†2) (69) ͜ΕΑΓɺ૬ޓ࡞༻දࣔͷ૬ޓ࡞༻߲ VI ͸࣌ؒґଘੑΛ࣋ͨͳ͍͜ͱ͕Θ͔Γ·͢ɻ ࣍ʹɺԋࢉࢠ a, a† Λ (63) Ͱ૬ޓ࡞༻දࣔʹม׵͠·͢ɻ͜Ε͸ࣗ༝ܥ H0 ʹ͓͚Δ ࣌ؒൃలͱಉ౳Ͱ͢ͷͰɺࠓͷ৔߹͸ɺ໌ࣔతʹղΛॻ͖Լ͢͜ͱ͕Ͱ͖·͢ɻ aI (t) = ae−iωt, a† I (t) = a†eiωt, ͜͜ͰɺγϡϨʔσΟϯΨʔදࣔʹ͓͍ͯɺ࣌ࠁ t ʹґଘ͢Δԋࢉࢠ Y1 (t), Y2 (t) Λ࣍ ࣜͰఆٛ͠·͢ɻ Y1 (t) + iY2 (t) = 2aeiωt
36. 36 ઌʹ (47) Ͱఆٛͨ͠ Y1 , Y2 ͸ɺෳૉฏ໘Λ֯ ϕ ͚ͩճసͨ͠ɺ৽͍͠࠲ඪܥʹ͓͚ Δɺෳૉৼ෯ 2a ͷ࣮෦ͱڏ෦ΛදΘ͢΋ͷͰͨ͠ɻ͜Εͱಉ༷ʹɺ্هͷ Y1 (t), Y2 (t) ͸ɺෳૉฏ໘Λ֯ −ωt ͚ͩճసͨ͠࠲ඪܥʹ͓͚Δ࣮෦ͱڏ෦Λද͠·͢ɻͦͯ͠ɺ͜ ͷ Y1 (t), Y2 (t) Λ૬ޓ࡞༻දࣔʹม׵͢Δͱɺ (68) Λ༻͍ͯɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ Y1I (t) + iY2I (t) = e i h H0t2ae− i h H0teiωt = 2a ͕ͨͬͯ͠ɺY1 (t), Y2 (t) ͸૬ޓ࡞༻දࣔʹ͓͍ͯ࣌ؒґଘੑΛ࣋ͨͳ͍ԋࢉࢠͰ͋Γɺ X1 + iX2 = 2a ͱ͍͏ؔ܎Λࢥ͍ग़͢ͱɺ݁ہɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ Y1I = X1 = a + a† (70) Y2I = X2 = −i(a − a†) (71) ͦͯ͠ɺVI , Y1I , Y2I ͕͢΂ͯ࣌ؒґଘੑΛ࣋ͨͳ͍͜ͱ͔Βɺ͜ΕΒʹ͍ͭͯɺઌʹ ࣔͨؔ͠܎ࣜ (67) Λద༻͢Δ͜ͱ͕Ͱ͖·͢ɻͨͱ͑͹ɺY1 (t) ͷϋΠθϯϕϧάදࣔ ʹ͓͚Δ࣌ؒൃల͸ɺ࣍ࣜͰܾ·Γ·͢ɻ d dt Y1H (t) = i ℏ [VI , Y1H (t)] ্ࣜͷӈลʹ (66) Λ୅ೖͯ͠ɺVI ͱ e± i h VI t ͕Մ׵Ͱ͋Δ͜ͱΛ༻͍Δͱ͕࣍ಘΒΕ ·͢ɻ d dt Y1H (t) = i h e i h VI t[VI , Y1I ]e− i h VI t ͞Βʹɺ(69)ɺ͓Αͼɺ(70) Λ༻͍Δͱɺ্هͷަ׵ؔ܎͸࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ [VI , Y1I ] = − iℏ 2 χ[a2 − a†2, a + a†] = −iℏχ(a† + a) = −iℏχY1I ͜ͷ݁ՌΛ୅ೖͯ͠ɺ࠶౓ɺ(66) Λ༻͍ΔͱɺY1H (t) ʹର͢ΔϋΠθϯϕϧάํఔࣜ ͕࣍ͷΑ͏ʹܾ·Γ·͢ɻ d dt Y1H (t) = χe i h VI tY1I e− i h VI t = χY1H (t) ࣌ࠁ t = 0 ʹ͓͍ͯ Y1H (0) = Y1I (0) = X1 ͱͳΔ͜ͱΛߟྀ͢Δͱɺ͜ͷղ͸࣍ͷ Α͏ʹܾ·Γ·͢ɻ Y1H (t) = eχtX1 ·ͨɺY2 (t) ʹ͍ͭͯಉ༷ͷٞ࿦Λ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ Y2H (t) = e−χtX2
37. 0.5 ޫύϥϝτϦοΫ૿෯ث 37 ਤ 9 ૬ޓ࡞༻ͷͳ͍ίώʔϨϯτঢ়ଶͷ࣌ؒൃల ͜ΕΒͷ݁Ռ͸ɺͲͷΑ͏ͳ෺ཧݱ৅ʹରԠ͢Δ͔ཧղͰ͖ΔͰ͠ΐ͏͔ʁɹͨͱ͑ ͹ɺχ = 0 Ͱ͋Ε͹ɺ࣌ࠁʹΑΒͣʹɺY1H (t) = X1 , Y2H (t) = X2 ͕੒ΓཱͪɺY1 , Y2 ͸࣌ؒมԽ͠ͳ͍͜ͱʹͳΓ·͢ɻY1 ͱ Y2 ͸ɺ֯଎౓ −ω Ͱճస͢Δ࠲ඪܥʢճసܥʣ ʹ͓͚Δෳૉৼ෯ͷ࣮෦ͱڏ෦Ͱͨ͠ͷͰɺ͜Ε͸ɺճసܥ͔ΒݟΔͱෳૉৼ෯͕࣌ؒత ʹมԽ͠ͳ͍ɺ͢ͳΘͪɺܥશମ͸ɺҰఆͷ֬཰෼෍Λอͬͯ֯଎౓ −ω Ͱճస͢Δ͜ͱ ʹͳΓ·͢ɻt = 0 ͷॳظঢ়ଶ͕ίώʔϨϯτঢ়ଶͰ͋Ε͹ɺਤ 9 ͷΑ͏ʹɺਅԁܗͷ֬ ཰෼෍Λ΋ͬͨෳૉৼ෯͕֯଎౓ −ω ͰճసΛଓ͚Δͱ͍͏Θ͚Ͱ͢ɻ Ұํɺχ ̸= 0 ͷ৔߹͸ɺճసܥ͔Βݟͨ࣌ʹɺ࣮࣠ํ޲ͱڏ࣠ํ޲ͷ֬཰෼෍͕ɺࢦ਺ తʹ֦େʗॖখ͞ΕΔ͜ͱʹͳΓ·͢ɻͨͱ͑͹ɺਤ 9 ͱಉ͡ίώʔϨϯτঢ়ଶΛॳظঢ় ଶͱ͢Δͱɺͦͷޙͷ࣌ؒൃల͸ɺਤ 10 ͷΑ͏ʹͳΓ·͢ɻͭ·Γɺ͕࣌ؒܦա͢Δʹ ͕ͨͬͯ͠ɺ֬཰෼෍ʹεΫΠʔζ͕ൃੜ͢Δͱͱ΋ʹɺৼ෯͕খ͘͞ͳ͍͖ͬͯ·͢ɻ ࣮ࡍͷ࣮ݧ؀ڥͰ͸ɺϙϯϓޫͱ৴߸ޫͷ૬ޓ࡞༻͕ൃੜ͢Δͷ͸ɺ݁থମͷ಺෦ʹݶఆ ͞ΕΔͷͰɺ݁থମ͔Β͸ɺ༗ݶ࣌ؒޙʹ͋Δஈ֊·ͰεΫΠʔζ͞Εͨ৴߸ޫ͕ग़ྗ͞ ΕΔ͜ͱʹͳΓ·͢ɻ ·ͨɺઌ΄Ͳͷܭࢉ݁Ռ͸ɺ࣍ͷΑ͏ʹղऍ͢Δ͜ͱ΋Ͱ͖·͢ɻࠓɺVI ͸࣌ؒґଘੑ Λ΋ͨͳ͍͜ͱ͔ΒɺY1 (t), Y2 (t) ͷظ଴஋͸ɺ(65) Λ༻͍ͯܭࢉ͢Δ͜ͱ͕Ͱ͖·͢ɻ ͨͱ͑͹ɺY1 (t) ʹ͍ͭͯॻ͖Լ͢ͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ ⟨Y1 (t)⟩ = ⟨Ψ(0)| e i h VI tY1I (t)e− i h VI t |Ψ(0)⟩ = ⟨Ψ(0)| e i h VI tX1 e− i h VI t |Ψ(0)⟩ (72)
38. 38 ਤ 10 ૬ޓ࡞༻ʹΑΔεΫΠʔζͷൃੜ ͜͜Ͱɺ2 ͭ໨ͷ౳߸͸ɺ(70) ͔ΒಘΒΕ·͢ɻ͜ͷ࣌ɺe− i ℏ VI t ͱ͍͏ԋࢉࢠͷத਎ Λߟ͑Δͱɺ(69) ΑΓɺ e− i ℏ VI t = exp { − 1 2 (χa2 − χa†2) } ͕ಘΒΕ·͢ɻ࣮͸ɺ͜Ε͸ɺ(38) Ͱఆٛͨ͠εΫΠʔζԋࢉࢠͱಉ͡΋ͷͰ͋Γɺ(72) ͸࣍ͷΑ͏ʹॻ͖௚͢͜ͱ͕Ͱ͖·͢ɻ ⟨Y1 (t)⟩ = ⟨Ψ(0)| S†(−χt)X1 S(−χt) |Ψ(0)⟩ ͜ΕΑΓɺ͜ͷܥͷ૬ޓ࡞༻͸ɺॳظঢ়ଶ |Ψ(0)⟩ ʹରͯ͠ɺεΫΠʔζԋࢉࢠ S(−χt) ͱಉ͡࡞༻Λ΋ͨΒ͢͜ͱ͕Θ͔Γ·͢ɻ 0.6 ࠓޙͷൃల ຊߘͰ͸ɺྔࢠޫֶͷཧ࿦తͳجૅͱͳΔίώʔϨϯτঢ়ଶɺ͓ΑͼɺεΫΠʔζυঢ় ଶʹ͍ͭͯɺͦͷجຊతͳੑ࣭Λઆ໌͠·ͨ͠ɻ·ͨɺ࣮ࡍʹεΫΠʔζυঢ়ଶΛൃੜ͞ ͤΔඇઢܗ૬ޓ࡞༻ͷҰྫͱͯ͠ɺॖୀޫύϥϝτϦοΫ૿෯ثͷ࠷΋୯७Խ͞ΕͨϞσ ϧΛ঺հ͠·ͨ͠ɻ ͜ͷޙɺΑΓݱ࣮ͷ݁থମʹ͍ۙϞσϧΛߏங͢Δʹ͸ɺલষͷ๯಄Ͱ৮ΕͨΑ͏ʹɺ ϙϯϓޫࣗ਎ͷྔࢠ࿦తͳऔΓѻ͍ɺ͋Δ͍͸ɺ݁থ֎෦ͱͷΤωϧΪʔͷ΍ΓͱΓʢ֎ ք΁ͷΤωϧΪʔͷྲྀग़ʣͳͲΛߟྀͨ͠मਖ਼͕ඞཁͱͳΓ·͢ɻ͜ͷΑ͏ͳɺΑΓҰൠ
39. 0.6 ࠓޙͷൃల 39 తͳϞσϧͷ৔߹ɺඇઢܗ૬ޓ࡞༻Λݫີʹܭࢉ͢Δ͜ͱ͸ࠔ೉ͳͨΊɺܥͷঢ়ଶΛٖࣅ తͳ֬཰෼෍Ͱهड़͢Δ΢Οάφʔؔ਺Λಋೖͯ͠ɺ΢Οάφʔؔ਺ʹର͢Δۙࣅతʢ൒ ݹయతʣͳӡಈํఔࣜΛಋ͘ͱ͍ͬͨૢ࡞͕ߦΘΕ·͢ɻ ͋Δ͍͸·ͨɺεΫΠʔζυঢ়ଶ |α, ϵ⟩ ʹ͓͍ͯɺ࣌ࠁ t = 0 ʹ͓͚ΔॳظҐ૬ɺ͢ͳ Θͪɺα ͷҐ૬ δ Λ 0ɺ΋͘͠͸ɺπ ʹઃఆ͢Δ͜ͱʹΑΓɺූ߸ΛؚΊͨৼ෯͕ූ߸ҧ ͍ͱͳΔ 2 छྨͷޫΛಘΔ͜ͱ͕Ͱ͖·͢ɻޫύϧεΛྔࢠσόΠεͱ͢ΔܭࢉػͰ͸ɺ ͜ͷ 2 छྨͷޫͷॏͶ߹ΘͤʹΑΓɺྔࢠϏοτΛදݱ͢Δͱ͍͏ख๏͕༻͍ΒΕ·͢ɻ ͦͷͨΊɺޫύϧεΛ༻͍ͨܭࢉػΛ࣮ݱ͢Δ্Ͱ͸ɺৼ෯ʢҐ૬ʣͷҟͳΔޫʹରͯ͠ɺ બ୒తͳ૬ޓ࡞༻Λ࣮ݱ͢Δػߏ͕ॏཁͳ໾ׂΛՌͨ͢͜ͱͱͳΓ·͢ɻ ຊߘͷ࠷ॳʹ঺հͨ͠ [1] Ͱ͸ɺΑΓຊ֨తͳޫύϥϝτϦοΫ૿෯ثͷཧ࿦ܭࢉɺ͋ Δ͍͸ɺྔࢠίϯϐϡʔςΟϯάͷ࿩୊ʹ΋৮ΕΒΕ͍ͯ·͢ͷͰɺڵຯͷ͋Δಡऀͷํ ͸ɺҰ౓ɺ໨Λ௨ͯ͠ΈΔͱΑ͍Ͱ͠ΐ͏ɻ
41. 41 ࢀߟจݙ [1]ʮQuantum Optics (2nd Edition)ʯ D. F. Walls, Gerard J. Milburn ʢஶʣ Springer [2]ʮݱ୅ͷྔࢠྗֶʢୈ̎൛ʣ ʯ J.J. αΫϥΠʢஶʣ, J. φϙϦλʔϊʢஶʣ٢Ԭॻళ [3]ʮి࣓৔ͷྔࢠԽʯhttp://eman-physics.net/elementary/em expand.html [4]ʮ৔ͷྔࢠ࿦ʯ http://kscalar.kj.yamagata-u.ac.jp/˜endo/kougi/QFT/QFT2013.pdf [5]ʮHeisenberg’s uncertainty principleʯ https://en.wikipedia.org/wiki/Heisenberg’s uncertainty principle