Variational Method Note | Santa Fe College - Edubirdie

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

2024
Views:

460

Pages:

2
Author:

Jabin F.

Variational method ψ(η) = { A(a^2-η^2), -a ≤ η ≤ a { 0, otherwise = √(15/16a^5) * (a^2-η^2), -a ≤ η ≤ a = 0, otherwise = 5ħ^2 / 4ma^2 V(η) = 1/2 mω^2 η^2 = ∫(0 to 1) (15/16a^5) * (mω^2/2) * ∫(-a to a) (a^2-η^2)^2 * η^2 dη = (15mω^2 / 16a^5) * ∫(0 to a) (a^4η^2 + η^6 - 2a^2η^4) dη = (15mω^2 / 16) * a^2 * (35+15-42) / 105 = (15mω^2 / 16a^5) * ∫(0 to 1) (a^4π^2 + π^6 - 2a^2π^4) dη 2P.E = mw^2a^2/14 47 ⇒ −∂E>/∂a=0 =5ℏ^2/4m a^2+mw^2a^2/14 => 5ℏ^2/2ma^3=mω^2a/7 5ℏ^2/4m[−2/a^3]+mω^2a/7=0 a^4=35ℏ^2/2m^2ω^2 ⇒ a^2=±(35ℏ^2/2m^2ω^2)^1/2 ⇒ a^2=+(35ℏ^2/2m^2m^2)^1/2 ⇒ = 4m/35/2/5+ 2/14mw^2 √7×√5=√35552 mw²/35)25 ⇒ (=(\begin{matrix}\frac{1}{2\sqrt[4]{14}}+\frac{1}{2}\sqrt{\frac{5}{14}})\hbar~\omega) 14 ħw (option-b)

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