The Quantum Metronome · AI Generator #GEN-C67E7527

Physics · quantum-mechanics

Problem

A novel quantum device consists of an electron confined to a one-dimensional box of length L=2.0×10−9  mL = 2.0 \times 10^{-9}\;\text{m}. The electron can tunnel between this box and an identical adjacent box through a thin barrier. The tunneling occurs with a characteristic frequency ω0=1.5×1015  rad/s\omega_0 = 1.5 \times 10^{15}\;\text{rad/s}, creating a "quantum metronome" that oscillates between the two boxes. At t=0t = 0, the electron is prepared in a superposition state: ∣ψ(0)⟩=13∣1L⟩+23∣1R⟩|\psi(0)\rangle = \frac{1}{\sqrt{3}}|1_L\rangle + \sqrt{\frac{2}{3}}|1_R\rangle, where ∣1L⟩|1_L\rangle and ∣1R⟩|1_R\rangle are the ground states of the left and right boxes respectively. The Hamiltonian of this two-level system can be written as: H=E1(∣1L⟩⟨1L∣+∣1R⟩⟨1R∣)−ℏω02(∣1L⟩⟨1R∣+∣1R⟩⟨1R∣)H = E_1(|1_L\rangle\langle 1_L| + |1_R\rangle\langle 1_R|) - \frac{\hbar\omega_0}{2}(|1_L\rangle\langle 1_R| + |1_R\rangle\langle 1_R|) where E1E_1 is the ground state energy of each individual box. (a) Find the probability of detecting the electron in the left box at time t=π3ω0t = \frac{\pi}{3\omega_0}. (b) At what time t>0t > 0 will the electron first have equal probability of being found in either box? (c) Calculate the uncertainty in energy ΔE\Delta E for the initial state ∣ψ(0)⟩|\psi(0)\rangle.

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