Solved Problems on Differential Equations - Homework 5 | PHYS 6124, Assignments of Physics

Download Solved Problems on Differential Equations - Homework 5 | PHYS 6124 and more Assignments Physics in PDF only on Docsity! PHYS 6124 Solution Set # 5 Fall 2006 Ψ(x) =    Aeikx + B e−ikx ; for x < 0 C eikx ; for x > 0 Continuity of Ψ at x = 0 implies A + B = C (1) Integrate the differential equation over a small neighborhood x ∈ (−², ²): − h̄ 2 2m ∫ ² −² Ψ ′′(x) dx + α ∫ ² −² δ(x) Ψ(x) = E ∫ ² −² Ψ(x) dx − h̄ 2 2m [Ψ′(²)−Ψ′(−²)] + α Ψ(0) = E ∫ ² −² Ψ(x) dx In the limit ² → 0, the integral on the righthand side goes to zero (since Ψ is bounded), so that − h̄ 2 2m [ Ψ′(0+)−Ψ′(0−)] + α Ψ(0) = 0 Evaluate Ψ′(0±) and Ψ(0) from the expression at the top of the page: Ψ′(0+) = ikC ; Ψ′(0−) = ik(A−B) ; Ψ(0) = C so that − h̄ 2 2m ik (C − A + B) + α C = 0 (2) Equations (1) and (2) can be solved for the two ratios B/A and C/A. After some algebra, I find B A = −ıβ 1 + ıβ and C A = 1 1 + ıβ where β = (mα)/(kh̄2) and so R = |B|2 |A|2 = β2 1 + β2 and T = |C|2 |A|2 = 1 1 + β2