Demonstration of ${\rm K^0}{\overline{\rm K}}{}^0$, ${\rm B^0}{\overline{\rm B}}{}^0$ and ${\rm D^0}{\overline{\rm D}}{}^0$ transitions with a pair of coupled pendula

Abstract

A setup of two coupled and damped pendula is used to demonstrate the main features of transitions between the neutral mesons K 0 , D 0 , B 0 and their antiparticles, including CP violation in the K 0 system. The transitions are described by two-state Schrödinger equations. Since the real parts of their solutions obey the same differential equations as the pendula coordinates, the pendulum motions can be used to represent the meson transitions. Video clips of the motions are attached as supplementary material.

Translation of time-reversal violation in the neutral K-meson system into a table-top mechanical system

“…In section 4, we show that the real parts of the Schrödinger-equation solutions for the K 0 K 0 system obey a secondorder differential equation that is formally identical to that for the coordinates of the pendulum. Therefore, as shown previously for the pair of coupled pendula [14] and by Caruso et al [12], the table-top Foucault pendulum is not only a demonstration and illustration of the K 0 K 0 system, but is also isomorphic mapping, 'translation' or equivalence in a stricter sense. The effect of the macroscopically T-violating Coriolis force is able to represent the microscopic T violation in K 0 K 0 transitions.…”

“…The real parts ℜ 1 and ℜ 2 of ψ 1 and ψ 2 , respecively, obey the second-order linear differential equation [14] R 1…”

“…with averaging-intervals T which are much larger than 1/ω 0 and much smaller than 1/Γ and |1/∆ω|. Therefore, the squares of the amplitudes x and y of the pendulum represent the K 0 and K 0 intensities, respectively, in the evolution of Ψ [14].…”

Observation of PT phase transition in a simple mechanical system

“…For example, in particle physics, it was proposed that transitions in K 0 K0 , B 0 B0 , and D 0 D0 systems could be visualized by using coupled pendula [13]. These experiments have recently been performed [14,15]. In these experiments, systems of coupled, damped pendula were used.…”

The quantum CP-violating kaon system reproduced in the electronic laboratory

“…After the proposal of Rosner [1] (see also [2,3]) of an analogy between the physics of the weak decay of neutral K-mesons (kaons) and a classical system of oscillators either electrical [1] or mechanical [2][3][4][5], it was shown [6] that this analogy is an equivalence, stricto sensu from the mathematical point of view. This equivalence is an isomorphism that connects in a univocal way 2016 The Author(s) Published by the Royal Society.…”