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

2024
Views:

401

Pages:

5
Author:

Jabin F.

Amarch Harmonic Oscillator Linear Harmonic oscillator (L.H.U) Simple Harmonic Oscillator (S.H.0) Harmonic Oscillator in 1- D A particle that executes simple harmonic motion, is called S. H.0. Cond" of S.H. motion (i) acc^2 d - displacement (ii) motion must be in straight lin F = -kn m d^2u / dt^2 = -ku => d^2u / dt^2 = -k/m u acc^2 x - displacement ẍ + k/m x = 0 V(x) = ∫ F.dr = ∫ -kndr V(n) = 1/2 Kn^2 K/m = w^2 ⇒ K = mw^2 V(x) = 1/2 mw^2 x^2 S.E for L.H.O ⇒ S.E for L.H.O ⇒ d²ψ/dx² + 2m/ħ² (E - 1/2 kx²) ψ = 0 Operator Method for S.H.O -> Ĥ = ṕ²/2m + 1/2 mw²x̂² Ĥ/ħw = mw²/2ħw x̂² + ṕ²/2ħmw Ĥ/ħw = (√(mw/2ħ) x̂)² + (ṕ/√(2ħmw))² z = x + iy z* = x - iy zz* = |z|² = x² + y² â = √(mw/2ħ) x̂ + i/√(2ħmw) p̂ â⁺ = √(mw/2ħ) x̂ - i/√(2ħmw) p̂ ââ† = mw/2ħ x̂² + ṕ²/2ħmw - i/2ħ (x̂ṕ - ṕx̂) = Ĥ/ħw - i/2ħ [x̂, ṕ] ââ† = Ĥ/ħw - i/2ħ (iħ) ââ† = Ĥ/ħw + 1/2 ⇒ Ĥ = (ââ† - 1/2) ħw => â†â = H/ħw + iħ [x̂, ṕ] / 2ħ â†â = H/ħw - 1/2 H = (â†â + 1/2) ħw [â, â†] = ââ† - â†â = H/ħw + 1/2 - H/ħw - 1/2 [â, â†] = 1 = ââ† - â†â ⇒ [â†â, â] = ? ⇒ [â†â, â] = -â = [ââ†, â] ⇒ [â†â, â†] = â† = [ââ†, â†] ⇒ [H, â] = ? = -âħw ⇒ [H, â†] = ? = â†ħw ⇒ [H, â†â] = ? = 0 ⇒ [H, ââ†] = ? = 0 [â†a, â] = â†[a, â] + [â†, â]a [â†a, â] = -â ⇒ [H, â] = [(â†a + 1/2)ħw, â] = (â†a + 1/2)ħw(â†a) - (â†a)(â†a + 1/2)ħw = ħw[(â†a)(â†a) + 1/2(â†a) - (â†a)(â†a) - (â†a)/2] ⇒ [H, â†, â] = 0 ⇒ [â†â, â†] = â†[â, â†] + [â†, â†]â [â†â, â†] = -â ⇒ [â†â, â] = â†[â, â] + [â†, â]â [â†â, â] = â† ⇒ [â†â, â†] = â†[â, â†] + [â†, â†]â [â†â, â†] = â† ⇒ [H, â] = [(â†â + 1/2)ħw, â] = ħw[â†â, â] + [1/2, â]â ⇒ [H, â†] = [(â†â + 1/2)ħw, â†] = ħw[â†â, â†] + [1/2, â†]â†