Chaotic dynamics of an electron

Abstract

First, we construct the image of the torus on the two-layer shell of the sphere and note that the isometries of the image of the torus on the sphere generate the unitary group $U(2)$, and then we establish that, as a result of the action of the modular group on the sphere, it is factorized in such a way that the minimal (one-element) equivalence classes are given by the set of primes. Next, we form a representation of the sphere winding for the Jacobi theta function and the Riemann zeta function, and then, considering the chaotic dynamics on the sphere, we notice that in the problem of random walk along the broken lines of the winding of the sphere, the concept of a complex probability amplitude arises quite naturally, and the dynamics of the probability amplitude of a wandering particle obeys a differential equation generalizing the Schrödinger equation.