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

2024
Views:

406

Pages:

5
Author:

Jabin F.

Method of Partial wave or (Partial wave analysis) Applicable for spherically symmetric potential or Central pot? V(γ) = v(γ) SE: −ħ²/2m (∂²/∂γ² + (2/γ)∂/∂γ) + ħ²/2m * (1/sin²θ) * ∂²/∂φ² + V(γ)Ψ(γ, θ, φ) = EΨ(γ, θ, φ) ħ²/2m * (1/γ²) * ∂/∂r (γ²∂/∂γ) Ψ + ħ²/2m * (1/sinθ) * ∂/∂θ (sinθ∂/∂θ) Ψ + (ħ²/2m * (1/sin²θ) * ∂²/∂φ² + V(γ)) Ψ = EΨ (1/γ²) * ∂/∂γ (γ²∂/∂γ) Ψ + (2m/ħ²) * (E - V(γ) + (ħ²/2m * l(l+1)/γ²) Ψ = 0 radial eqn: (1/γ²) * ∂/∂r (γ²∂/∂γ) R + (2m/ħ²) * (E - V(γ) + (ħ²/2m * l(l+1)/γ²) R = 0 Spherical Bessel's function Jl(p) → p^(-1/2) * sin(p - lπ/2) Ψ(r, φ) = (1/√r) e^(-ikr - imφ) A = 1 Ψin = Ae^(ikz) = Ae^(ikr cos θ) = e^(ikr cos θ) = 1 + i(kr cos θ) + (i(kr cos θ))^2 / 2 + ... Ψin = Σ_(l=0)^(∞) i^l (2l+1) J_l(kr) P_l(cos θ) from (i): ∂^2/∂γ^2 (γR) + (2m/ħ^2) (E - V(γ)) γR = 0 ∂^2/∂r^2 (γR) + (2m/ħ^2) (E - γ(γ) + (l(l+1)ħ^2/2m r^2)) γR = 0 r → ∞ ∂^2Ψ/∂γ^2 + (2m/ħ^2) (E - γ(γ) + (l(l+1)ħ^2/2m γ^2)) Ψ = 0 Equation 1 Page - 97 V(γ) = { Vo if γ ≤ R { 0 if γ > R f(0) = (-2mV₀ / ħ²K) ∫₀ᴿ γ sin Kr V(γ) dγ = (-2mV₀ / ħ²K) ∫₀ᴿ γ sin Kγ dγ V₀ → V₀', f(0) → f'(0) f'(0) = (-2mV₀' / ħ²K) ∫₀ᴿ γ sin Kγ dγ f'(0) / f(0) = (-2mV₀' / ħ²K) ∫₀ᴿ γ sin Kγ dγ / (-2mV₀ / ħ²K) ∫₀ᴿ γ sin Kγ dγ = V₀' / V₀ V₀' / V₀ = α ⇒ f'(0) / f(0) = α ⇒ (dσ'/dΩ) / (dσ/dΩ) = |f'(0)|² / |f(0)|² = 4(e^(iphi) - α)² K = b = 2k sin(θ/2) f(0) = (-2m / ħ²b) ∫₀^∞ sin(br) V(r) dr V(γ) = (β / r) e^(-μγ) = (-2mp / ħ²b) ∫₀^∞ ((e^(ibr) - e^(-ibγ)) / 2i) e^(-μγ) dγ f(0) = (-2mp / ħ²) / (μ² + b²) (option - b) (Cue Q.11) (Page - 98) V(r) = (β / r) e^(-μγ) f(0) = (-2m / ħ²k) ∫₀^R r sin(kγ) γ(γ) dγ = (-2mβ / ħ²) / (μ² + k²) = (-2mβ / ħ²) / (μ² + (2k sin(θ/2))^2) f(0) = (-2mβ / 4k² sin²(θ/2)) M = -2mβ R² σ = |f(0)|² (A² + sin²(θ/2)) / (A² + sin²(θ/2))^2 = |f(0)|² / (A² + sin²(θ/2)) dσ / dΩ = |f(0)|² (A² + sin²(θ/2))^-2 = |f(0)|² / (A² + sin²(θ/2))^2 (option - c)