Analysis of Transverse Vibration of Antenna Structure Resonator Using Bernoulli–Euler Beam Theory and Quantum Mechanical Examination of Its Quantized Displacement

Abstract

We analytically investigated the vibration of a macroscopic antenna structure resonator of 10.7 µm length showing a quantized displacement at 110 mK using Bernoulli–Euler beam theory. According to our calculations on the vibrational displacement and mode of the resonator, the vibration mode increased step-by-step from the 1st to 10th mode and then repeated again from the 1st to 10th mode. We found the 10th mode of the resonator at 1.592 GHz when using every paddle length of 490 nm, which is 10 nm shorter than that of a usual antenna structure resonator, existed near a frequency of 1.49 GHz at which the quantized displacement was observed. In the 10th mode of the antenna structure resonator at 1.592 GHz, the displacement of the central beam was so extensively damped that its mechanical energy may be considered to be zero. Therefore, the mechanical energy of the antenna structure resonator could be approximately calculated from the displacement of forty paddles regarded as cantilever beams. We could explain the quantized displacement of the antenna structure resonator using the equivalent mass of a harmonic oscillator, to which the antenna structure resonator was approximately equivalent in mechanical energy, and the quantization of a harmonic oscillator in a textbook on quantum mechanics if the wave function of quantum mechanics can be applied to this harmonic oscillator.