Ja n 20 07 A Matrix Factorization of Extended Hamiltonian Leads to N - Particle Pauli Equation

In this paper the Levy-Leblond procedure for linearizing the Schrödinger 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, GG̃ = |G|I = (−K)lI. Here the square matrix G is linear in the total energy E and all momenta, G̃ 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 θ with the property Gθ = 0. This factorization to obtain the matrix G and vector function θ is illustrated first for a one-dimensional particle in a simple potential well. The quantization of the matrix relation, Gθ = 0, is performed, using the Planck-de Broglie laws and a Fourier transform to obtain the generalized wave equation, G(p̂, q)ψ(q) = 0, where q and p̂ are, respectively, the generalized position and momentum-operator vectors of (n+1)-dimensions. This quantization procedure, when applied to the above one-dimensional example, yields immediately a pair of coupled operator equations which become, after eliminating one component of ψ, the classical second-order partial-diffferential Schrödinger equation of a single one-dimensional particle in a force field. 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. This analysis of the Zeeman effect in a weak magnetic field exhibits the usual two-fold splitting of the energy levels, obtained first by Pauli using the Pauli equation for N = 1, except for a slight change in the particle mass needed in the Larmor frequency. This “increased” mass, given by mL = m1m2/(m2 −m1), agrees with the g-factor correction for nuclear motion obtained in 1952 by W.H. Lamb using the relativistic, one-particle, Dirac equation with potentials. Also it is consistent with the known fact that positronium has no weak-field Zeeman effect.