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

2024
Views:

83

Pages:

5
Author:

Jabin F.

Sudden Approximation → If perturbation is suddenly applied to the system V(t') = { V₀(t'), 0 < t' < t { 0 , otherwise e^(iωt) ≈ Const from② Pif = 1/(ħ²ωf²) | ∫₀ᵗ e^(-iωft') <φf|V(t')|φi> dt' |² Pif = 1/(ħ²ωf²) | <φf|V₀|φi> |² Prob. Consider a particle of mass 'm' in ground state of the 1-D infinite poth well; defined as- V(x) = { 0, 0 < x < L { ∞, otherwise time dep. perturbation (i) V(t) = { (Ax)e^(-t/τ), 0 < t < ∞ { 0, otherwise τ and A Const. (ii) V(t) = { An te^(-t/τ), 0 < t < ∞ { 0, otherwise τ and A Const. n << 1 then Calculate the prob. of transition to n-th state. Sol'n Φ(lm) = √(2/L) sin(lπx/L) , 0 < x < L Φ(lm) = { √(2/L) sin(lπx/L), 0 < x < L { 0, other wise (i) V(t) = { Axe^(-t/τ), 0 < t < ∞ { 0, other wise P(i→n) = |(-i/ħ) ∫₀∞ e^(iωft) <φf|Axe^(-t/τ)|φi> dt|² Pif = |A|^2 |<Φn|x|Φi>|^2 / ħ² * 1 / (1/τ² + ωf²) Pif+ = |A|^2 |<Φn|x|Φi>|^2 / ħ² * 1 / (1/τ² + ωf²) finther <Φn|x|Φi> = 2/L ∫₀ᴸ sin(nπx/L) x sin(iπx/L) dx = ? (ii) Similarly Pif→f = |A|^2 |<Φn|x²|Φi>|^2 / ħ² * 1 / (1/τ² + ωf²) prob. → Consider a particle of mass 'm' in ground state of 1-D, H.O Rot'n V(x) = (1/2) mw²x². Calculate the prob. of finding the particle in n-th state on applying a time dep.- pot Perturbation~ (i) V(t) = { An e^(-t/τ), 0 < t < ∞ { 0, otherwise (ii) V(t) = { An² e^(-t/τ), 0 < t < ∞ { 0, otherwise Soln P₀ₙ = | -iħ ∫₀ᵗ A <φₙ|p|φ₀> e^(iωft') e^(-t'/τ) dt' |² = |A|²/ħ² |<φₙ|p|φ₀>|² | ∫₀ᵗ e^(iωft' - t'/τ) dt' |² = |A|²/ħ² |<φₙ|p|φ₀>|² (τ² / (1 + ωf²τ²)) <φₙ|p|φ₀> = = { 0, n=even non-zero, n=odd. Hence μ = √ħ/2mω (a+a†) Similarly→ (ii) Posm = |A|^2 <ϕn|x²|ϕ₀> τ² / ħ² * 1/(1+w₀²τ²) <ϕn|x²|ϕ₀> = = { nonzero, n = even zero, n = odd @web14 page-91 E(t) = Ae^(-(t/τ)²), (H-O) ⇒ V(t) = H'(t) = 2AE^(-(t/τ)²)/τ x, ωfi = Ef - Eo = ħω₀/2 Po→f = |(-i/ħ) ∫₀ᵗ e^(iωft') <φf|V(t')|φi> dt'|² = |(-i/ħ) 2A ∫₀ᵗ e^(iωft') <φf|x|φi> e^(-t'^2/τ²) dt'|² = q²/ħ² |<φ_1|φ_0>|² | ∫_-∞^∞ e^(-t'^2/τ² + iωft') dt' |² = q²/ħ² |<φf|φi>|² √πτ² | e^(-ωf²τ²/4) |² = √πτ² 4A²/ħ² |<φf|φi>|² e^(-ωf²τ²/2) P₀∝ e^(-ωf²τ²/2) (option-a) @yes.15 page-91 Constant perturo v(t) = { V₀, 0 < t < t' ; t' = t { 0, otherwise Pif = 4 ||^2 sin²(ωft'/2) / ħ²ωf²t² Pif ∝ (Δt)² , Δt → ∞ Pif ∝ (Δt) , Δt → ∞ ∴ H'Δt << 1 ⇒ H' << 1/Δt Δt → ∞ ⇒ (Pif)² / (Pif₁) = (Δt₂)² / (Δt₁)² = (2)² = 4 (option-c) Δt → ∞ ⇒ (Pif₂)² / (Pif₁) = Δt₂ / Δt₁ = 2 → (option-b)