The Quantum Pendulum Clock · AI Generator #GEN-95147883

Physics · modern-physics

Problem

An unusual timepiece uses a microscopic diamond crystal of mass m=2.0×10−15  kgm = 2.0 \times 10^{-15}\;\text{kg} suspended as a pendulum bob. The crystal oscillates in simple harmonic motion with period T=1.0  sT = 1.0\;\text{s} and amplitude A=5.0×10−6  mA = 5.0 \times 10^{-6}\;\text{m}. Due to quantum uncertainty, the position of the crystal cannot be determined more precisely than its de Broglie wavelength. At the moment when the crystal passes through the equilibrium position (maximum speed), calculate: (a) The de Broglie wavelength of the crystal at this instant. (b) The ratio of the de Broglie wavelength to the amplitude of oscillation. (c) If we require the de Broglie wavelength to be at least 10%10\% of the amplitude for quantum effects to become observable, what would be the maximum mass of the crystal bob? Use h=6.626×10−34  J⋅sh = 6.626 \times 10^{-34}\;\text{J·s}.

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