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Pendulum Represents Binary Quantum State of Oscillations

Masataka Ohta

Published May 1, 2018• 0 citations
OpenAlex

Abstract

There is a straightforward correspondence between superposition of polarization modes of photons and that of classical radio waves. While classical particles cannot be superpositioned, classical waves can be, which is within classical intuition. Even more intuitively, two dimensional oscillations of a pendulum represent oscillating binary quantum state such as polarization state of photons.

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There is a straightforward correspondence between superposition of polarization modes of photons and that of classical radio waves. While classical particles cannot be superpositioned, classical waves can be, which is within classical intuition. Even more intuitively, two dimensional oscillations of a pendulum represent oscillating binary quantum state such as polarization state of photons.
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Abstract The wave equation of electromagnetism, the Helmholtz equation, has the same form as the Schrödinger equation, and so optical waves can be used to study quantum mechanical problems. The electromagnetic wave solutions for non-diffracting beams lead to the two-dimensional Helmholtz equation. When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum. The resulting optical eigenmodes, Mathieu modes, have an optical Fourier transform with a spatial intensity distribution that is proportional to the quantum mechanical probability for the pendulum. Comparison of Fourier intensities of eigenmodes are in excellent agreement with calculated quantum mechanical probabilities of pendulum stationary states. We further investigate wave-packet superpositions of a few modes and show that they mimic the libration and the nonlinear rotation of the classical pendulum, including revivals due to the quantized nature of superpositions. The ability to ‘dial a wavefunction’ with the optical modes allows the exploration of important aspects of quantum wave-mechanics and the pendulum that may not be possible with other physical systems.
Abstract Although the classical examples of harmonic oscillators such as springs or the simple pendulum are of limited use in the submicroscopic world of quantum mechanics, there are plenty of other applications that make the oscillator worth studying. Atoms in a crystal, for instance, oscillate about their mean positions and it is their vibrations that carry sound and heat through the material. Indeed, we know from classical theory that by using normal coordinates we may regard any system performing small oscillations about a point of stable equilibrium as a collection of harmonic oscillators. This extends even to the normal modes of vibration of a string or of the electromagnetic field. The harmonic oscillator thus provides the key to understanding how wave phenomena such as light can be quantized, and lies at the heart of Einstein’s idea of the photon and of Planck’s quantum.