Wave Mechanics VII: Pauli’s Spinor Theory

H. M. Schwartz

Philip L. Bowers

Michel Gondran, Alexandre Gondran

The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.

Ralph Schiller

We derive a modified Hamilton-Jacobi theory for a classical spinning dipole and show that this classical theory can be put in a form almost identical with the Pauli spin theory. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single-valued, and that it satisfy the usual energy eigenvalue equation. In this manner we deduce the correct energy levels for a hydrogen atom in an external magnetic field. We derive the integral kernel for the Pauli equation from this classical model and discuss the properties of our spinor solutions under canonical transformation. We exhibit the charge-conjugate solution to the classical spin equation.

Jaroslav Zamastil, Jakub Benda