Exactly Sduable 2-State Problem

University:
Massachusetts Institute of Technology
Course:
8.322 | Quantum Theory II
Academic year:

2021
Views:

51

Pages:

6
Author:

Baylee Callahan

6.2 Exactly solvable 2-state problem Consider a two-state system with [H_{0}=(\begin{matrix}E_{1}&0\ 0&E_{2}\end{matrix})] [V(t)=(\begin{matrix}0&Be^{-iat}\ Be^{iat}&0\end{matrix})] In interaction picture, ih ∂C₁/∂t = ∫e^(i(E₁-E₂)/ħ)t C₂(t) ih ∂C₂/∂t = ∫e^(i(E₂-E₁)/ħ)t C₁(t) → dc/dt = (-i/ħ) ( ∫e^(i(ω-ω₀)t) 0 0 ∫e^(-i(ω-ω₀)t) ) C(t) where C(t) = (C₁(t), C₂(t))ᵀ ω₀ = (E₂-E₁)/ħ Can find exact solution of (*) With initial conditions C₁(0) = 1, C₂(0) = 0, |C₁(t)|² = (Ω² + ħ²(ω-ω₀)²/4) Sin² Ωt / Ω² Ω = √(Ω² + ħ²(ω-ω₀)²/4) |C₁(t)|² = 1 - |C₂(t)|² 6.8 |C₁(t)|² frequency = 2Ω/π = (1/π)e^sx |C₁(t)|²min |C₁(t)|² absorb emit absorb emit absorb emit t |C₁(t)|²min = (ω-ω₂1)² + 4Ω²/ħ² At resonance, ω = ω₂1 Ω = Ω/ħ |C₁(t)|²min = 0. Amplitude as a function of ω: k/ħ Amplitude peaked @ resonance with α ∝ Ω (strength of perturbation) Time-dep. potentials H_0 = H_0 + V(t) I. picture M(t) >_I = e^(iH_0t/ħ) φ(t) = Σ G_m M_m EOM ih ∂C_m(t)/∂t = Σ U_mn(t) e^(iω_mn t) C_n(t) V_mn(t) = ω_mn = (E_m - E_n)/ħ 2-state system: [H_{0}=(\begin{matrix}E_{1}&0\ 0&E_{2}\end{matrix})] [V(t)=(\begin{matrix}0&\delta e^{i\omega t}\ \delta e^{-i\omega t}&0\end{matrix})] |C₁(t)|² = (Ω² + ħ²(ω-ω₂1)²/4) Sin² Ωt / Ω² ω₂1 = (E₂-E₁)/ħ Ω = √(Ω² + ħ²(ω-ω₂1)²/4) |C₁(t)|² = 1 - |C₂(t)|² |C₁(t)|² t Periodically forced 2-state system is a basic problem - demonstrates fundamental features of absorption & emission. Analogous to absorption & emission of radiation by particles in EM fields Simplify atom to 2-level system Couple to background rad field @ frequency ω (V ~ (e^(iwt) 0)) When ω near ω₂₁ = (E₂-E₁)/ħ, system can absorb a quantum of radiation from the field. Same with stimulated emission when in E₂. Again, we are doing semiclassical approximation; complete picture of spontaneous emission requires quantizing b.g. field. Examples of 2-state systems a) Spin magnetic resonance Consider spin ½ particle (|↑>, |↓>) in magnetic field B = Boẑ + B₁ (x cos ωt + y sin ωt) [Diagram of a rotating magnetic field vector]

Exactly Sduable 2-State Problem

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