Quantum Mechanics · Lesson 08

Chapter practice

Bring the chapter together: two-level Hamiltonians, avoided crossings, Rabi oscillations, the rotating frame, detuning, the Landau–Zener formula and adiabatic passage. No worked example sits above the answers.

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H^= \frac{1}{2} (ϵ σ_{z} + Δ σ_{x})

E_{\pm} = \pm \frac{1}{2} ϵ^{2} + Δ^{2}

P_{e} (t) = sin^{2} \frac{Ω t}{2}

Ω_{R} = Ω^{2} + δ^{2}

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Intuition

Bring the chapter together: two-level Hamiltonians, avoided crossings, Rabi oscillations, the rotating frame, detuning, the Landau–Zener formula and adiabatic passage. No worked example sits above the answers.

A conductor brings an orchestra in, holds it, and lets it fall away. Driving a two-level system is conducting with one instrument: timing, pitch and pace decide what it plays.

Before you practise

$E_{\pm}=\pm\tfrac12\sqrt{\epsilon^{2}+\Delta^{2}}$ ; $P_{e}=\frac{\Omega^{2}}{\Omega'^{2}}\sin^{2}\frac{\Omega't}{2}$ with $\Omega'=\sqrt{\Omega^{2}+\delta^{2}}$ ; $P_{\text{jump}}=e^{-\pi\Delta^{2}/2\hbar v}$ .

\hat{H}_{R}=\frac{\hbar}{2}\bigl(-\delta\,\sigma_{z}+\Omega\,\sigma_{x}\bigr)

Reminders

A $\pi$ pulse inverts; a $\pi/2$ pulse makes an equal superposition.
Coupled levels keep a gap of at least $\Delta$ .
Slow sweeps follow; fast sweeps jump.

Applications