Potential Step | Lecture Note - Edubirdie

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

2024
Views:

63

Pages:

4
Author:

Jabin F.

Potential step or step potential V(x) = { 0, x < 0 { Vo, x > 0 discont. at x = 0 The pot that undergoes one discontinuous change. is Called step potential. ex. unit step fn. f(x) = { 1, x > 0 { 0, x < 0 f(m) Unit step size V(x) V = 0 V = Vo x < 0 x > 0 Case-I E > Vo V(x) = 0, x < 0 V(x) = Vo, x > 0 x = 0 d²ψ / dx² + (2mE / ħ²) (E - V(x)) ψ = 0 Region-I d²ψ₁ / dx² + (2mE / ħ²) ψ₁ = 0 K₁² = 2mE / ħ² E > 0 d²ψ₁ / dx² + K₁² ψ₁ = 0 x < 0: ψ₁(x) = Ae^(ik₁x) + Be^(-ik₁x) ψ_incident = ψ₊ = Ae^(ik₁x) → Incident wave ψ_reflected = ψ₋ = Be^(-ik₁x) → Reflected wave ψ₁(x) = ψ_incident + ψ_reflected ⇒ ψ₁(x) = ψ₊ + ψ₋ Region-II x > 0 V = Vo d²ψ₂ / dx² + (2m/ħ²) (E - Vo)ψ₂ = 0 K₂² = (2m/ħ²) (E - Vo) => d²ψ₂ / dx² + K₂²ψ₂ = 0 => ψ₂(x) = Ce^(iK₂x) + De^(-iK₂x) = Ce^(iK₂x) → Transmitted wave = De^(-iK₂x) → Not allowed (Absent) (Bcoz No Reflection) (Bcoz pot'n is const.) No Boundary in Region II => D = 0 => ψ₂(x) = Ce^(iK₂x) = ψ_transmitted. Ψ(x) = {Ψ₁(x) = Ae^(ik₁x) + Be^(-ik₁x), x < 0 {Ψ₂(x) = Ce^(ik₂x), x > 0

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