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量子光学理論入門

 量子光学理論入門

Etsuji Nakai

April 12, 2023
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  1. ྔࢠޫֶཧ࿦ʮ௒ʯೖ໳
    தҪ ӻ࢘
    2018 ೥ 3 ݄ 13 ೔

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  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

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  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

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  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
    =


    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

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  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

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  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଴͞

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  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

    [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

    [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

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  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

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  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)
    }

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  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)

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  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)

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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 ≥

    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

    View full-size slide

  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

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  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

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  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

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  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

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  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)

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  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

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  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

    ɻ

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  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

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  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

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  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

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  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

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  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)]

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  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)

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  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

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  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)

    View full-size slide

  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)]

    View full-size slide

  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
    )

    View full-size slide

  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

    View full-size slide

  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

    [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

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  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)

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  38. 38
    ਤ 10 ૬ޓ࡞༻ʹΑΔεΫΠʔζͷൃੜ
    ͜͜Ͱɺ2 ͭ໨ͷ౳߸͸ɺ(70) ͔ΒಘΒΕ·͢ɻ͜ͷ࣌ɺe− i

    VI t ͱ͍͏ԋࢉࢠͷத਎
    Λߟ͑Δͱɺ(69) ΑΓɺ
    e− i

    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

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  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

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  40. 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

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