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

2024
Views:

371

Pages:

5
Author:

Jabin F.

V(γ) = V(γ') = V(γ) ex V(x,y,z) = Axy = Axrsinθ cosφ sinφ ≠ V(γ) This is not central poth. V(x,y,z) = A(x² + y² + z²) = Ar² = V(γ) → central poth 3-D (-ħ²/2m ∇² + V(γ)) Ψ(γ) = EΨ(γ) (-ħ²/2m ∇² + V(γ,θ,φ)) Ψ(γ,θ,φ) = EΨ(γ,θ,φ) ∇² = (1/γ²) ∂/∂γ (r² ∂/∂γ) + (1/rγsinθ) ∂/∂θ (sinθ ∂/∂θ) + (1/γ²sin²θ) ∂²/∂φ² L^2 = -ħ^2 (1/sinθ * d/dθ (sinθ * d/dθ) + 1/sin^2θ * d^2/dφ^2) L_z = -iħ * d/dφ Compare ∇^2 = 1/γ^2 * d/dr (γ^2 * d/dr) - L^2 / ħ^2 Now. S.E (-ħ^2 / 2m) * 1/γ^2 * d/dγ (r^2 * d/dr) - L^2 / ħ^2 + V(γ,θ,φ) * ψ(r,θ,φ) = Eψ(γ,θ,φ) H = -ħ^2 / 2m * 1/γ^2 * d/dr (γ^2 * d/dγ) + L^2 / 2mħ^2 + V(γ,θ,φ) => [L^2, H] = [L^2, V(γ,θ,φ)] ≠ 0 => [L_z, H] = [L_z, V(γ,θ,φ)] ≠ 0 But ↓ For central or spherically potential V(γ) = V(γ') = V(γ) H = -ħ^2 / 2m * 1/γ^2 * d/dr (γ^2 * d/dγ) + L^2 / 2mħ^2 + V(γ) [L^2, H] = 0 = [L_z, H] H, L^2, L_z can have simultaneous eigenfunctions. For central poth -> Ψ(γ,θ,φ) = R(γ)Θ(θ)Φ(φ) Non central poth -> (-ħ^2/2m ∇^2 + V(γ,θ,φ))Ψ(r,θ,φ) = EΨ(γ,θ,φ) Ψ(γ,θ,φ) ≠ R(γ)Θ(θ)Φ(φ) for normalization ∫(from 0 to infinity) ∫(from 0 to π) ∫(from 0 to 2π) |Ψ(γ,θ,φ)|^2 r^2 sinθ drdθdφ = 1 Central Pot'l -> (-ħ^2/2m ∇^2 + V(r)) Ψ(r,θ,φ) = EΨ(γ,θ,φ) Ψ(r,θ,φ) = R(γ)Θ(θ)Φ(φ) = R(γ)Y_lm(θ,φ) ∫(from 0 to infinity) ∫(from 0 to π) ∫(from 0 to 2π) |Ψ(γ,θ,φ)|^2 γ^2 sinθ drdθdφ = 1 => ∫(from 0 to infinity) |R(γ)|^2 r^2 dr ∫(from 0 to π) |Θ(θ)|^2 sinθ dθ ∫(from 0 to 2π) |Φ(φ)|^2 dφ = 1 => ∫(from 0 to infinity) |R(γ)|^2 r^2 dr = 1 => ∫(from 0 to π) |Θ(θ)|^2 sinθ dθ = 1 => ∫(from 0 to 2π) |Φ(φ)|^2 dφ = 1 => Radial Prob. density -> P(γ) = γ^2 |R(γ)|^2 ⇒ ∫(from 0 to π) ∫(from 0 to 2π) Y_lm*(θ,φ) Y_lm(θ,φ) sinθ dθ dφ = 1 The prob. of finding the particle lies in small interval (γ→γ+dγ), θ→θ+dθ φ→φ+dφ = |ψ(γ,θ,φ)|^2 r^2 dr sinθ dθ dφ = |ψ(γ,θ,φ)|^2 dτ Total prob. density = |ψ(γ,θ,φ)|^2 The prob. of finding the particle lies b/w r to r+dr = |R(γ)|^2 γ^2 dγ