a†2 + 2a†a + 1 Ͱ͋Δ͜ͱ͔Βɺ⟨0, r| X2 1 |0, r⟩ Λܭࢉ͢Δʹɺ ⟨0, r| a2 |0, r⟩ , ⟨0, r| a†2 |0, r⟩ , ⟨0, r| a†a |0, r⟩ ͷΛٻΊΔඞཁ͕͋Γ·͢ɻ͜ΕΒɺ(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։ͨ͠ࡍʹਅۭظ͕ 0 ʹͳΒͳ͍ͷɺੵ aa† ΛؚΉ߲ͷΈͰɺ ⟨0| aa† |0⟩ = ⟨0| (a†a + 1) |0⟩ = 1 ͱ͍͏ؔʹҙ͢Δͱɺ࣍ͷ݁Ռ͕ಘΒΕ·͢ɻ ⟨0, r| a2 |0, r⟩ = − cosh r sinh r ⟨0| aa† |0⟩ = − cosh r sinh r ͜ͷෳૉڞΛऔΔͱɺ ⟨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 ͷࢄɺ V (X1 ) = ⟨0, r| X2 1 |0, r⟩ − ⟨0, r| X1 |0, r⟩2 = e−2r