Half Harmonic Oscillator in 1 Dimension | Lecture Note - Edubirdie

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

2024
Views:

186

Pages:

5
Author:

Jabin F.

Half Harmonic oscillator in 1-D V(x) = 1/2 * m * w^2 * x^2, x > 0 infinity, x <= 0 V(x) = 0, x > 0 1/2 * m * w^2 * x^2, x <= 0 for (i) for (ii) V = infinity V = infinity psi = 0 psi = 0 x = 0 x = 0 psi(x)|x=0 = 0 psi(x)|x=0 = 0 or psi(x)|x=0 = 0, x >= 0 or psi(x)|x=0 = 0, x <= 0 psi_n(x) = zero at x=0, for n=odd psi_n(0) ≠ 0 at x=0, for n=even [not allowed in Half H.O. but this is in L-H.] E_n = (n + 1/2)ħω, n=1,3,5... odd = 3ħω/2, 7ħω/2, 11ħω/2... When Given n = 0,1,2,3... n' = 1,3,5... = (2n+1) E_n' = (n' + 1/2)ħω = (2n+1+1/2)ħω E_n = (2n+3/2)ħω = 3ħω/2, 7ħω/2, 11ħω/2... When Given n = 0,2,4... n' = n+1 = 1,3,5,7... => E_n = (n+3/2)ħω, n=0,2,4,6... = 3ħω/2, 7ħω/2, 11ħω/2... Graph for "half" Harmonic oscillator: V(x) = 1/2 * m * w^2 * x^2, x > 0 V(x) = infinity, x <= 0 n = odd allowed We Can Compare with full Harmonic oscillator *Ques.2 at, t=0 page-30 ψ(η,0) = (1/2)u₀(η) + (i√3/2)u₁(η) |ψ> = (1/2)|u₀> + (i√3/2)|u₁> = 1, = -1 = 0, = 0 for ground state probability p₀ = |<ψ₀|ψ>|² / <ψ|ψ> = |C₀|² / Σₙ|cₙ|² = 1/4 p₁ = |(i√3/2)|² = 3/4 = ΣₙPₙEₙ = (1/4)(1/2)ħω + (3/4)(3/2)ħω = (10/8)ħω = (5/4)ħω (Option - a) Ques. 21 $22 501721 Page-33 ψ₁ = Φ₀ - 2Φ₁ + 3Φ₂ ψ₂ = Φ₀ - Φ₁ + αΦ₂ ψ₁ |ψ₁⟩ = |φ₀⟩ - 2|φ₁⟩ + 3|φ₂⟩ |ψ₂⟩ = |φ₀⟩ - |φ₁⟩ + α|φ₂⟩ <ψ₁|ψ₂> = 0 1 + 2 + 3α = 0 α = -1 (option - c) Soln (22) |ψ₂⟩ = |φ₀⟩ - |φ₁⟩ - |φ₂⟩ same for all 3 Pn = |cn|² / Σn|cn|² P₀ = 1/3, P₁ = 1/3, P₂ = 1/3 = 1/3 * 1/2 ħω + 1/3 * 3/2 ħω + 1/3 * 5/2 ħω = (1 + 3 + 5)ħω / 6 = 3/2 ħω (option - b)

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