University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

244

Pages:

6
Author:

Jabin F.

Time Dependent Perturbation Theory H = H0 + Hp Hp = V(t) = depends on time probability of transition Without Perturb H = 0 Time Indepen Hp = V(t) Time depend Per Hp = V(t) H₀ |φᵢ⟩ = Eᵢ⁽⁰⁾ |φᵢ⟩ initial state H₀ |φf⟩ = Ef⁽⁰⁾ |φf⟩ final state The probability of transition from an initial unperturbed state |φᵢ⟩ to another state |φf⟩ under time dependent perturbation V(t) = { V₀(r,t), t₁ < t < t₂ { 0, otherwise Pᵢ→f = |(-i/ħ) ∫(t₁ to t₂) e^(iωfᵢt) <φf|V₀|φᵢ> dt|² ωfᵢ = (Ef - Eᵢ)/ħ future (+ve) t=0 ← past (-ve) -∞ < t < +∞ The rate of transition Wᵢ→f = dPᵢ→f/dt For constant perturbation: V(t) = { V₀, 0 < t' < t { 0, otherwise Constant perturbation: V₀ is independent of time Pᵢ→f = |(-i/ħ) ∫(0 to t) e^(iωfᵢt') <φf|V₀|φᵢ> dt'|² = |(-i/ħ) <φf|V₀|φᵢ> ∫(0 to t) e^(iωfᵢt') dt'|² = (1/ħ²) |<φf|V₀|φᵢ>|² |∫(0 to t) e^(iωfᵢt') dt'|² ∫(0 to t) e^(iωfᵢt') dt' = (1/iωfᵢ) (e^(iωfᵢt) - 1) Pᵢ→f = (1/ħ²ωfᵢ²) |<φf|V₀|φᵢ>|² (e^(-iωfᵢt) - 1) (e^(iωfᵢt) - 1) = (1/ħ²ωfᵢ²) |<φf|V₀|φᵢ>|² (1 + 1 - e^(iωfᵢt) - e^(-iωfᵢt)) = (1/ħ²ωfᵢ²) |<φf|V₀|φᵢ>|² (2 - 2 cos ωfᵢt) P_i -> f = (1/ħ²ω_fi²) |<φ_f|v_0|φ_i>|² (2 - 2 cos ω_fi t) = (|\<φ_f|v_0|φ_i>|^2 * 2(1 - 1 + 2 sin²(ω_fi t / 2))) / (ħ²ω_fi²) P_i -> f = (4 |\<φ_f|v_0|φ_i>|^2 sin²(ω_fi t / 2)) / (ħ²ω_fi²) if t -> 0, lim sin²(ω_fi t / 2) / ((ω_fi t / 2)²) P_i -> f = (|\<φ_f|v_0|φ_i>|^2) / (ħ²) * t² P_i -> f α t² if t -> ∞ sin²(ω_fi t / 2) / (πt(ω_fi / 2)²) = πt δ(ω_fi / 2) πt lim sin²(ω_fi t / 2) / (πt(ω_fi / 2)²) = πt δ(ω_fi / 2) πtδ(Ef - Ei) 2ħ 2πħδ(Ef - Ei) Pi -> f = (2πt |<φf|V₀|φᵢ>|²) / ħ * δ(Ef - Ei) {Nonzero, if Ef = E₀ {zero, if Ef ≠ E₀ ⇒ Pi -> f α t The rate of transition Wi -> f = dPi -> f / dt = (2π / ħ) |<φf|V₀|φᵢ>|² * δ(Ef - Ei) Ei = E₀ p ≠ 0 Wi -> f = 0, 2π/t, 4π/t, 6π/t ρ = (ufct / exo_j) * (2/2) Time Indep. ΔE ≠ 0 ρ = 0 sin²(ωfift / 2) / (ωfift / 2)²