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American Journal of Physics

Triple pendulum as an analog to three coupled stationary states

Joseph L. McKibben

Published Nov 1, 1977• 10 citations
Publisher

Abstract

The contest entry was a spring-coupled triple pendulum. It was used to demonstrate the character of normal modes and in particular a mode that has high Q even with the center pendulum highly damped. The coupling of one of the springs to a pair of pendulums could be varied while they were swinging and with this adiabaticity in the effect upon a normal mode was observed. In the discussion the mathematical simularity to the equations of three coupled quantum mechanical levels is pointed out. The eigenstate in metastable hydrogen atoms that survives passage through the spin filter of a polarized ion source is compared to the high-Q mode observed in the triple pendulum.

DOI
10.1119/1.10965

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Acta Physica Sinica•Dec 22, 2021•0 citations
The contest entry was a spring-coupled triple pendulum. It was used to demonstrate the character of normal modes and in particular a mode that has high Q even with the center pendulum highly damped. The coupling of one of the springs to a pair of pendulums could be varied while they were swinging and with this adiabaticity in the effect upon a normal mode was observed. In the discussion the mathematical simularity to the equations of three coupled quantum mechanical levels is pointed out. The eigenstate in metastable hydrogen atoms that survives passage through the spin filter of a polarized ion source is compared to the high-Q mode observed in the triple pendulum.
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.
We discuss two mechanical systems in which the approximate equations of motion are the same as the rapid adiabatic passage of a two-state quantum-mechanical system, such as that which occurs with quasistatic interactions and from the quasiperiodic action of a chirped-frequency laser pulse. We discuss a pair of masses on springs, weakly coupled by another spring, one of whose spring constants varies slowly, and a pair of coupled pendula, where the length of one pendulum slowly changes. In each example an approximation analogous to the rotating-wave approximation used in the corresponding quantum system brings the second-order macroscopic equations of motion into first-order form, and a slow variation in a system parameter characteristic leads to adiabatic change and rapid adiabatic passage.
Since the birth of quantum mechanics, its classical correspondence (or analogy) has been a hot topic for physicists. In this paper, we first discuss whether there is a classical correspondence of quantum entanglement. We give a positive answer through the following examples: in the framework of quantization of mesoscopic circuits, two mesoscopic capacitance inductance (LC) circuits with mutual inductance are proved to be the source of quantum entanglement by using the integration within an ordered product, and then the formula of their characteristic frequency is obtained, It is found that it is similar to the expression of the small oscillating frequency of a classical system described below. The classical system is shown in Fig. 1 . Two walls are connected with the same spring. And between the two springs a sliding trolley can move on a smooth table. The trolley is hung with a simple pendulum, The small oscillating frequency of the system is calculated by analytical mechanics. It is found that the swing of the simple pendulum will cause the trolley to oscillate back and forth. The mutual restraint effect of the pendulum, the trolley and the spring reflects the “entanglement” between them.