Rigid Rotator | Lecture Note - Edubirdie

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

2024
Views:

354

Pages:

5
Author:

Jabin F.

⇒ Rigid Rotator:- A Particle that undergoing purely Rotation motion with fix radius is Called Rigid Rotator. Rigid Rotator with fixed axis of Rotation (or fixed plane) -> R= const, θ= const, φ= variable, V(γ) = 0γ Const. Case 2. Rigid Rotator with variable plane of rotation -> θ= Variable, φ= variable, γ= R= Const., V(γ)= 0γ Const. Case (1) θ= const = 90°, r= Const., φ= variable H = Lz^2 / 2I, I = mR^2 (moment of inertia) ψ(γ,0,φ)= Ф(φ) HΦ(φ) = EΦ(φ) L_z^2 / 2I Φ(φ) = E Φ(φ) Φ(φ) = 1/sqrt(2π) e^(imφ), m = 0, ±1, ±2, ±3... L_z^2 / 2I Φm(φ) = E Φm(φ) (mh)^2 / 2I Φm(φ) = E Φm(φ) Em = m^2 h^2 / 2I Φm(φ) = 1/sqrt(2π) e^(imφ), m = 0, ±1, ±2, ±3... m = 0 E = 0 Φ_0(φ) = 1/sqrt(2π) non-degenerate. G.S. E_m = m^2 h^2 / 2I Doubly Degenerate except ground state. l=2 γ= R = Const, V=0 θ and φ are variable Y_lm(θ,φ) = Θ_l(θ) Φ_m(φ) Φ_m(φ) = 1/sqrt(2π) e^(imφ) I = mR^2 H = L^2 / 2I, Y_lm(θ,φ) = Y_lm(θ,φ) - spherical harmonics... H Y_lm(θ,φ) = E Y_lm(θ,φ) L^2 / 2I Y_lm(θ,φ) = E Y_lm(θ,φ) l(l+1)ħ^2 / 2I Y_lm(θ,φ) = E Y_lm(θ,φ) => E_l = l(l+1)ħ^2 / 2I Y_lm(θ,φ) = Θ_l(θ) Φ_m(φ), Φ_m(φ) = 1/sqrt(2π) e^(imφ), m = 0, ±1, ±2... Ques. 4 Page - 62 Unitary matrix U = e^(iTH/2) (where H^2 = 1) ... e^(iθH) = I cos θ + i H sin θ e^(iθA) If A^2 = I then e^(iθA) = I cos θ + i A sin θ H = [1/√2, 0, 1/√2] [0, 1, 0] [1/√2, 0, -1/√2] H^2 = [1, 0, 0] [0, 1, 0] [0, 0, 1] = I => U = e^(iπH/4) = I cos(π/4) + iH sin(π/4) = iH/√2 => U = [i/√2, 0, i/√2] [0, i, 0] [i/√2, 0, -i/√2] (option - b)

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