Statistical approach to quantum mechanics II: Nonrelativistic spin

TL;DR

This paper develops a classical statistical model for nonrelativistic quantum spin using spinning top analogies, reproducing key quantum properties and deriving a modified Pauli-Schrödinger equation.

Contribution

It introduces a classical spinning top framework to represent quantum spin, including half-integer spins, and derives a modified Schrödinger equation consistent with quantum spin properties.

Findings

Models half-integer spin using classical tops.

Predicts supraluminal speeds only for relativistic particles.

Derives a modified Pauli-Schrödinger equation with additional terms.

Abstract

In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schr\"odinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.