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

2024
Views:

477

Pages:

7
Author:

Jabin F.

2- Body problem or 2- Particles system. Lab frame R_cm = Σmiri / Σmi M = m1 + m2 Relative Coordinate r_12 = r1 - r2 - (i) from (i) and (ii) MR = m1r1 + m2r2 - (ii) ⇒ M R_cm = (m1/M)R + (m2/M)r2 ⇒ r1 = (m2/M)R + (1)r12 ⇒ r2 = (m1/M)R + (-1)r12 T = 1/2 m1r1² + 1/2 m2r2² = p1²/2m1 + p2²/2m2 = -ħ²/2m1 ∇1² + ħ²/2m2 ∇2² => T = 1/2 m1r1² + 1/2 m2r2² = 1/2 MR² + 1/2 μr12² = 1/2 (m1+m2) R² + 1/2 (m1m2 / (m1+m2)) r² V = √(r1²) + √(r2²) + √(r1-r2²) = V(R) + V(r1²) + V(r2²) = V(mR) + V(μr1²) + V(r1²) mass dependent # Ψ(r1, r2) = Ψm1(r1) + Ψm2(r2) If Ψ dependent on mass then include μr1², mR or Relative Coordinate Replace r1², r2² Ques.10 page-82 H = p1²/2m1 + p2²/2m2 + 1/2 mw² (x1² + x2² + (m1x1 + m2x2)² / (m1+m2)²) KE = T = p1²/2m1 + p2²/2m2 = 1/2 MR² + 1/2 μr² = 1/2 (m1+m2) X² + 1/2 (m1m2 / (m1+m2)) x² = 1/2 m1X² + 1/2 m2X² + 1/2 μx² H = 1/2 Mw_cm²x² + 1/2 Mrel² + 1/2 mx² in H.O. P²/2m + 1/2 mw²x² = 1/2 m₁x₁² + 1/2 mw²x²² w = √(mw²/m) = w => H = 1/2 MX² + 1/2 Mw²X² + 1/2 μ+x² + 1/2 μw²x² Wcm = √(M/m) * w = w ⇒ get Wrel. = √((μ+m)/μ) * w², μ = m/2 Wrel. = √3w₀ ⇒ Energy = Eₘ₁,ₘ₂ = (m₁+1/2)ħw + (m₂+3/2)ħw₀, (option-c) II-method → |V² - w²T| = 0 Ques.5 page-81 H = -ħ²/2m ∇₁² - ħ²/2m ∇₂² + 1/2 mw²r₁² + 1/2 mw²r₂² = 1/2 MR² + 1/2 μω²R² + 1/2 μω²r² - 1/4 mw²r² = 1/2 MR² + 1/2 Mw²R² + 1/4 μω²r² velocity vend. R² Toeff Wcm = √(Mw²/M) = w Wrel = √((μ+m)/μ) * w² , μ = m/2 Wrel = √3w₀ w_{rel} = \sqrt{1-\lambda} w_0 E_{min} = (n_{1} + \frac{3}{2}) \hbar w + (n_{2} + \frac{3}{2}) \sqrt{1-\lambda} \hbar w For ground state → n₁ = n₂ = 0 E_{0,0} = \frac{3}{2} \hbar w + \frac{3}{2} \sqrt{1-\lambda} \hbar w = (1 + \sqrt{1-\lambda}) \frac{3}{2} \hbar w (Option-d) Ques.11 page-95 V = 1/2 K(t) x² F = 2E = k(t) x ⇒ k(t) = 2E/x ⇒ k(t) α E If Power is Suddenly Switched off- E=0, F=0, V=0 Ques.2 page-95 V(x) = 1/2 Kx² - 2Ex in the Presence of E.F. E_n^(0) = (n+1/2)ħw - 2²E²/2mw² ≠ 0 If electric field is suddenly switched off. V'(x) = 1/2 Kx² E_n' = (n+1/2)ħw = 0