Author: Dr. Jonathan Kenigson, FRSA*
Abstract. In this brief lecture, we provide a convergent series representing the expected overlap of n circles of fixed radius r lying entirely within a bounding circle of unit radius. We furthermore relate this development to quantum theory. The Cauchy probability distribution, also known as the Lorentzian or Breit-Wigner distribution, is a continuous probability distribution with applications to particle scattering phenomena. In astrophysics, Cauchy distributions are used to model the shape of spectral lines produced by the emission or absorption of light by atoms and molecules in plasmas and particle accelerators. The Breit-Wigner distribution (a type of Cauchy distribution) is used to model the resonance behavior of particles in scattering experiments. It is characterized by a resonant peak at the particle’s mass and a width that represents the particle’s decay rate or inverse lifetime. Relativistic Breit-Wigner distributions appear in mathematical physics, particularly in the context of particle physics and scattering experiments. They are used to model the behavior of resonances (unstable particles) in high-energy reactions and decays, such as those observed in particle accelerators like the Large Hadron Collider (LHC).
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