1 クイズ ˆ (⃗k), N ˆ † (k⃗ ′ )] を計算せよ． 1. 実スカラー場に対して [N ˆ (⃗k)ˆ 2. N a† (⃗k)|n(⃗k)⟩ を求めよ． ˆ |0⟩ を求めよ． 3. N ˆ に対するハイゼンベルクの 4. 複素スカラー場を考えよ．電荷演算子 Q 運動方程式を考察することによって，電荷が保存されているかどうか決定せよ． Q˙ = [H, Q] この計算は式 (??) と (??) 及び，交換関係 (??) を使って行え． 2 解答問1 ˆ (⃗k), N ˆ † (⃗k)] を計算せよ．実スカラー場に対して [N ˆ (⃗k), N ˆ † (k⃗ ′ )] [N ˆ (⃗k)N ˆ † (k⃗ ′ ) − N ˆ † (k⃗ ′ )N ˆ (⃗k) =N =ˆ a† (⃗k)ˆ a(⃗k)ˆ a(k⃗ ′ )ˆ a† (k⃗ ′ ) − a ˆ(k⃗ ′ )ˆ a† (k⃗ ′ )ˆ a† (⃗k)ˆ a(⃗k) =ˆ a† (⃗k)ˆ a(⃗k)ˆ a(k⃗ ′ )ˆ a† (k⃗ ′ ) − a ˆ(k⃗ ′ )ˆ a† (⃗k)ˆ a† (k⃗ ′ )ˆ a(⃗k) =ˆ a† (⃗k)ˆ a(⃗k)ˆ a(k⃗ ′ )ˆ a† (k⃗ ′ ) − [ˆ a† (⃗k)ˆ a(k⃗ ′ ) + δ(k⃗ ′ − ⃗k)][ˆ a(⃗k)ˆ a† (k⃗ ′ ) − δ(⃗k − k⃗ ′ )] =ˆ a† (⃗k)ˆ a(⃗k)ˆ a(k⃗ ′ )ˆ a† (k⃗ ′ ) − a ˆ† (⃗k)ˆ a(k⃗ ′ )ˆ a(⃗k)ˆ a† (k⃗ ′ ) + δ(⃗k − k⃗ ′ )ˆ a† (⃗k)ˆ a(k⃗ ′ ) − δ(k⃗ ′ − ⃗k)ˆ a(⃗k)ˆ a† (k⃗ ′ ) + δ(k⃗ ′ − ⃗k)δ(⃗k − k⃗ ′ ) =δ(⃗k − k⃗ ′ )ˆ a† (⃗k)ˆ a(k⃗ ′ ) − δ(⃗k − k⃗ ′ )ˆ a(k⃗ ′ )ˆ a† (⃗k) + δ(⃗k − k⃗ ′ )2 = − δ(⃗k − k⃗ ′ )[ˆ a(k⃗ ′ ), a ˆ† (⃗k)] + δ(⃗k − k⃗ ′ )2 = − δ(⃗k − k⃗ ′ )δ(k⃗ ′ − ⃗k) + δ(⃗k − k⃗ ′ )2 =0 よって， ˆ (⃗k), N ˆ † (k⃗ ′ )] = 0 [N · · · 解答 3 問2 ˆa N ˆ† (⃗k)|n(⃗k)⟩ ∫ = d3 k ′ a ˆ† (k⃗ ′ )ˆ a(k⃗ ′ )ˆ a† (⃗k)|n(⃗k)⟩ ∫ = d3 k ′ a ˆ† (k⃗ ′ )[ˆ a† (⃗k)ˆ a(k⃗ ′ ) + δ(k⃗ ′ − ⃗k)]|n(⃗k)⟩ ∫ ∫ = d3 k ′ a ˆ† (k⃗ ′ )ˆ a† (⃗k)ˆ a(k⃗ ′ )|n(⃗k)⟩ + d3 k ′ δ(k⃗ ′ − ⃗k)ˆ a† (k⃗ ′ )|n(⃗k)⟩ ∫ = d3 k ′ a ˆ† (⃗k)ˆ a† (k⃗ ′ )ˆ a(k⃗ ′ )|n(⃗k)⟩ + a ˆ† (⃗k)|n(⃗k)⟩ ∫ =ˆ a† (⃗k) d3 k ′ a ˆ† (k⃗ ′ )ˆ a(k⃗ ′ )|n(⃗k)⟩ + a ˆ† (⃗k)|n(⃗k)⟩ ˆ |n(⃗k)⟩ + a =ˆ a† (⃗k)N ˆ† (⃗k)|n(⃗k)⟩ =ˆ a† (⃗k)n(⃗k)|n(⃗k)⟩ + a ˆ† (⃗k)|n(⃗k)⟩ =[n(⃗k) + 1]ˆ a† (⃗k)|n(⃗k)⟩ よって， ˆa N ˆ† (⃗k)|n(⃗k)⟩ = [n(⃗k) + 1]ˆ a† (⃗k)|n(⃗k)⟩ · · · 解答