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

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dc.creatorReed, Irving S.
dc.creatorBrun, Todd A.
dc.date2007-01-11
dc.date.accessioned2026-07-07T07:40:06Z
dc.date.available2026-07-07T07:40:06Z
dc.descriptionIn 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, $G\tilde{G} = |G| I = (-K)^l I$. 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 $\underlineθ$ 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.
dc.description32 pages, no figures
dc.identifierhttps://arxiv.org/abs/quant-ph/0701058
dc.identifierhttp://arxiv.org/abs/quant-ph/0701058
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/121681
dc.subjectQuantum Physics
dc.titleA Matrix Factorization of Extended Hamiltonian Leads to $N$-Particle Pauli Equation
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