A two-level system has energies and

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem

Problem

A two-level system has energies E1E_1 and E2E_2, with ℏω21=E2−E1\hbar\omega_{21}=E_2-E_1, and starts at t=0t=0 in ∣ψ(0)⟩=cos⁡α ∣1⟩+sin⁡α ∣2⟩|\psi(0)\rangle=\cos\alpha\,|1\rangle+\sin\alpha\,|2\rangle, with 0<α<π/20<\alpha<\pi/2. We watch A^=a(∣1⟩⟨2∣+∣2⟩⟨1∣)\hat A=a(|1\rangle\langle2|+|2\rangle\langle1|), with aa real and positive. ((a)) Show that, up to an overall phase, ∣ψ(t)⟩=cos⁡α ∣1⟩+sin⁡α e−iω21t∣2⟩|\psi(t)\rangle=\cos\alpha\,|1\rangle+\sin\alpha\,e^{-i\omega_{21}t}|2\rangle, and find ⟨A^⟩(t)\langle\hat A\rangle(t) and ΔA(t)\Delta A(t). ((b)) Find ΔE\Delta E, and say why it does not change with time. ((c)) For α=π/6\alpha=\pi/6, evaluate ΔE τA\Delta E\,\tau_A, where τA=ΔA/∣d⟨A^⟩/dt∣\tau_A=\Delta A/|d\langle\hat A\rangle/dt|, in units of ℏ/2\hbar/2 at ω21t=π/3\omega_{21}t=\pi/3 and at ω21t=π/2\omega_{21}t=\pi/2. ((d)) Show that ΔE τA≥ℏ/2\Delta E\,\tau_A\ge\hbar/2 at every instant where τA\tau_A is defined, and find exactly when equality holds. Say in one sentence why tt enters this calculation only as a parameter.

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