p^ | p₁p₂ >= (p₁ + p₂)| p₁p₂ > (3-3) Ĥ| p₁p₂ > | 𝑴𝒂𝒙 𝑷𝒍𝒂𝒏𝒄𝒌 𝒏𝒆𝒘𝒔𝒑𝒂𝒑𝒆𝒓

p^ | p₁p₂ >= (p₁ + p₂)| p₁p₂ > (3-3)

Ĥ| p₁p₂ >=(Eₚ₁ + Eₚ₂)| p₁p₂ > (3-4)

Ĥ| n₁n₂n₃...>=Σₘ nₚₘ Eₚₘ| n₁n₂n₃...> (3-5)

Quanta in oscillators → Particles in momentum states. (3-6)
2-
kᵗʰ oscillator → mᵗʰ momentum mode pm

3-

E = Σₖ₌₁ᴺ nₖℏωₖ → E=Σₚₘ nₚₘ Eₚₘ

| n₁n₂...) =Πₖ(1/nₖ!)½(âₖ†)ⁿᵏ | 0 > (3-7)

| 21000...) =(1/2!)½(â₁†)² (1/1!)½ â₂† | 0 > (3-8)

âₚ₁† | 0 > = | 10 > ، âₚ₂† | 0 >= | 01 > (3-9)

âₚ₂†âₚ₁† | 0 > α | 11 > ، âₚ₁†âₚ₂† | 0 > α | 11 > (3-10)

|âₚ₁†âₚ₂†=λâₚ₂†âₚ₁† (3-11)

[âᵢ†، âⱼ†]= âᵢ†âⱼ†- âⱼ†âᵢ†=0 (3-13)

[âᵢ ، âⱼ†]=δᵢⱼ (3-14)

| n₁n₂...>=Πₚₘ[1/√nₚₘ](âₚₘ†)ⁿᵖᵐ | 0 > (3-15)

âₚ₁†âₚ₂†| 0 >= âₚ₂†âₚ₁†| 0 >= | 1ₚ₁1ₚ₂ > (3-16)

âᵢ†| n₁...nᵢ...> =√(nᵢ +1)| n₁...nᵢ +1...> (3-17)

âᵢ | n₁...nᵢ...> =√nᵢ | n₁...nᵢ -1...> (3-18)


{ĉᵢ†، ĉⱼ†}= ĉᵢ†ĉⱼ†+ ĉⱼ†ĉᵢ†=0 (3-19)

ĉᵢ†ĉᵢ†+ ĉᵢ†ĉᵢ†=0 ⇒ ĉᵢ†ĉᵢ†=0 (3-20)

{ĉᵢ†، ĉⱼ†}=0 (3-21)

{ĉᵢ ، ĉⱼ†}=δᵢⱼ (3-22)

n₁^ |11> = | 11 > (3-23)

ĉᵢ†| n₁...nᵢ...> =(-1)ᵞ√(1- nᵢ)| n₁...nᵢ +1...> (3-24)

ĉᵢ | n₁...nᵢ...> =(-1)ᵞ √nᵢ | n₁...nᵢ -1...> (3-25)
حيث
γ=Σᵢ=n₁ - n₂...+ nᵢ₋₁ (3-26)

|110 > a state with particles in states 1 and 2 (3-27)
و هي الحالة بها جسيمان في الحالتين 1 و 2

→|101> move particle from state 2 to 3

تحريك الجسيم من الحالة ٢ إلى ٣

→|011> move particle from state 1 to 2

تحريك الجسيم من الحالة ١ إلى ٢

→|110 > move particle from state 3 to 1

â₁† â₃ â₂† â₁ â₃† â₂ |110 >=±|110 > (3-28)

â₁† â₃ â₂† â₁ â₃† â₂|110 > =â₃ â₃† â₁† â₁ â₂† â₂|110 > (3-29)

â₁† â₃ â₂† â₁ â₃† â₂|110 > =(|110 >+ n^₃|110 >) (3-30)

â₃ â₃†|110 >=1+ â₃† â₃|110 >=1+ n₃|110 > (3-31)

ĉ₁† ĉ₃ ĉ₂† ĉ₁ ĉ₃† ĉ₂|110 > =- ĉ₃ ĉ₃† ĉ₁† ĉ₁ ĉ₂† ĉ₂|110 > (3-32)

âₚ , âₕ†] =δ⁽³⁾(p - h) (3-33)

وللطاقات

Ĥ=∫d³p Eₚ âₚ†âₚ (3-34)

< p|p' >=< 0 |âₚâₚ,†| 0 > (3-35)

< p|p' >=< 0 |[δ⁽³⁾(p - p') ± âₚ†âₚ,]| 0 > (3-36)

< x | p > ==∫d³h Φₕ(χ)< h | p >=Φₚ(χ) (3-37)

< p'h' | hp > =< 0 |âₚ,âₕ,âₕ†âₚ†| 0 > (3-38)

< p'h' | hp > =δ⁽³⁾(p' - p)δ⁽³⁾(h' - h) ± δ⁽³⁾(p' - h)δ⁽³⁾(h' - p) (3-39)