Finite Potential Well | Lecture Note - Edubirdie

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

2024
Views:

68

Pages:

4
Author:

Jabin F.

Finite potential well: (Symmetric) V(x) = { 0, |x| < L/2 { Vo, |x| > L/2 { 0, otherwise V(x) = V(-x) Symmetric V = 0 V = Vo V = Vo x = -L/2 x = 0 x = L/2 Consider a particle of mass m inside finite potential well K₁² = (2m(Vo - E)) / ħ² = K_II² (V = 0) K_III² = (2mE) / ħ² Region-I (x < -L/2) d²ψ₁ / dx² + (2m/ħ²) (E - Vo) ψ₁ = 0 Region-II (-L/2 < x < L/2) => d²ψ_II / dx² + (2mE / ħ²) ψ_II = 0 Region-III (x > L/2) => d²ψ_III / dx² + (2mE / ħ²) ψ_III = 0 ψ_I = Ae^(k_Ix) + Be^(-k_Ix), x < -L/2 ψ_II = C cos(k_II x) + D sin(k_II x), -L/2 < x < L/2 ψ_III = Ee^(k_Ix) + Fe^(-k_Ix), x > L/2 [k_II = k_I] Region-I x < -L/2, as x -> -∞ Be^(-k_Ix) -> ∞, Ae^(k_Ix) -> 0 => B = 0 ψ_I(x) = Ae^(k_Ix) Region-II (No change) Ψ_II = C cos K_II x + D sin K_II x Region-III x > L/2, as x -> +∞ Ee^(K_Ix) -> ∞, Fe^(-K_Ix) -> 0 So E = 0 then Ψ_III(x) = Fe^(-K_Ix) Ψ(x) = { Ae^(K_Ix), x < -L/2 { C cos K_II x + D sin K_II x, -L/2 < x < L/2 { Fe^(-K_Ix), x > L/2 Boundary Con'n Ψ_I | x = -L/2 = Ψ_II | x = -L/2 Ψ_I' | x = -L/2 = Ψ_II' | x = -L/2 Ψ_II | x = L/2 = Ψ_III | x = L/2 Ψ_II' | x = L/2 = Ψ_III' | x = L/2 dΨ I ∣ x=− 2 L = dx dΨ II ∣ x=− 2 L dx dΨ II ∣ x=+ 2 L = dx dΨ III ∣ x=+ 2 L Continuity cond^n of wave fn Ψ I ∣ x=− 2 L =Ψ II ∣ x=− 2 L Even fn odd function ψ I (x)=Ae K I x ψ I (x)=Ae K I x ψ II (x)=CcosK II x ψ II (x)=DsinK II x ψ III (x)=Fe −K I x ψ III (x)=Fe −K I x

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