A Matrix Factorization of Extended Hamiltonian Leads to $N$-Particle Pauli Equation

In this paper the Levy-Leblond procedure for linearizing the Schr\"odinger equation to obtain the Pauli equation for one particle is generalized to obtain an $N$-particle equation with spin. This is achieved by using the more universal matrix factorization, $G\tilde{G} = |G| I = (-K)^l I$. Here the square matrix $G$ is linear in the total energy E and all momenta, $\tilde G$ is the matrix adjoint of $G$, $I$ is the identity matrix, $|G|$ is the determinant of $G$, $l$ is a positive integer and $K=H-E$ is Lanczos' extended Hamiltonian where $H$ is the classical Hamiltonian of the electro-mechanical system. $K$ is identically zero for all such systems, so that matrix $G$ is singular. As a consequence there always exists a vector function $\underline\theta$ with the property $G\underline\theta=0$. This factorization to obtain the matrix $G$ and vector function $\underline\theta$ is illustrated first for a one-dimensional particle in a simple potential well. This same technique, when applied to the classical nonrelativistic Hamiltonian for $N$ interacting particles in an electromagnetic field, is shown to yield for N=1 the Pauli wave equation with spin and its generalization to $N$ particles. Finally this nonrelativistic generalization of the Pauli equation is used to treat the simple Zeeman effect of a hydrogen-like atom as a two-particle problem with spin.