dorsal/arxiv
View SchemaA Matrix Factorization of Extended Hamiltonian Leads to $N$-Particle Pauli Equation
| Authors | Irving S. Reed, Todd A. Brun |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701058 |
| URL | https://arxiv.org/abs/quant-ph/0701058 |
Abstract
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.
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"abstract": "In this paper the Levy-Leblond procedure for linearizing the Schr\\\"odinger\nequation to obtain the Pauli equation for one particle is generalized to obtain\nan $N$-particle equation with spin. This is achieved by using the more\nuniversal matrix factorization, $G\\tilde{G} = |G| I = (-K)^l I$. Here the\nsquare matrix $G$ is linear in the total energy E and all momenta, $\\tilde G$\nis the matrix adjoint of $G$, $I$ is the identity matrix, $|G|$ is the\ndeterminant of $G$, $l$ is a positive integer and $K=H-E$ is Lanczos\u0027 extended\nHamiltonian where $H$ is the classical Hamiltonian of the electro-mechanical\nsystem. $K$ is identically zero for all such systems, so that matrix $G$ is\nsingular. As a consequence there always exists a vector function\n$\\underline\\theta$ with the property $G\\underline\\theta=0$. This factorization\nto obtain the matrix $G$ and vector function $\\underline\\theta$ is illustrated\nfirst for a one-dimensional particle in a simple potential well. This same\ntechnique, when applied to the classical nonrelativistic Hamiltonian for $N$\ninteracting particles in an electromagnetic field, is shown to yield for N=1\nthe Pauli wave equation with spin and its generalization to $N$ particles.\nFinally this nonrelativistic generalization of the Pauli equation is used to\ntreat the simple Zeeman effect of a hydrogen-like atom as a two-particle\nproblem with spin.",
"arxiv_id": "quant-ph/0701058",
"authors": [
"Irving S. Reed",
"Todd A. Brun"
],
"categories": [
"quant-ph"
],
"title": "A Matrix Factorization of Extended Hamiltonian Leads to $N$-Particle Pauli Equation",
"url": "https://arxiv.org/abs/quant-ph/0701058"
},
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