2007/01/11 by I.S. Reed, Todd A. Brun, Reed, Irving S. +1
Chemistry · Computer Science · Mathematics · #Advanced NMR Techniques and Applications #Algebraic and Geometric Analysis #FOS: Physical sciences #Matrix Theory and Algorithms #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.quant-ph/0701058
openalex publication_date 2007/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, GG = |G| I = (-K)l I. Here the square matrix G is linear in the total energy E and all momenta, 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 \underlineθ with the property G\underlineθ=0. This factorization to obtain the matrix G and vector function \underlineθ 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.