The Quantum Pendulum Clock · AI Generator #GEN-1FF611A3

Physics · quantum-mechanics

Problem

A physicist designs a conceptual "quantum pendulum clock" where a quantum particle of mass m=2.0×10−26  kgm = 2.0 \times 10^{-26}\;\text{kg} is confined in a harmonic potential well that mimics a classical pendulum. The potential energy is given by V(x)=12mω2x2V(x) = \frac{1}{2}m\omega^2 x^2 where ω=1.5×1014  rad/s\omega = 1.5 \times 10^{14}\;\text{rad/s}. The particle is initially prepared in a superposition state: ∣ψ⟩=12∣0⟩+12∣2⟩|\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|2\rangle, where ∣n⟩|n\rangle are the energy eigenstates of the quantum harmonic oscillator. The physicist wants to use the oscillating probability density of finding the particle at position x=0x = 0 as a timing mechanism. (a) Find the period TT of oscillation of the probability density ∣ψ(0,t)∣2|\psi(0,t)|^2 at the center of the well. (b) Calculate the minimum uncertainty in position Δxmin\Delta x_{\text{min}} for this quantum pendulum, and compare it to the classical amplitude A=2Emω2A = \sqrt{\frac{2E}{m\omega^2}} where EE is the average energy of the superposition state. Use ℏ=1.055×10−34  J⋅s\hbar = 1.055 \times 10^{-34}\;\text{J·s}.

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