dorsal/arxiv
View SchemaQuasilinearization approach to quantum mechanics
| Authors | R. Krivec, V. B. Mandelzweig |
|---|---|
| Categories | |
| ArXiv ID | physics/0112024 |
| URL | https://arxiv.org/abs/physics/0112024 |
Abstract
The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schr\"{o}dinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precison of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.
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"abstract": "The quasilinearization method (QLM) of solving nonlinear differential\nequations is applied to the quantum mechanics by casting the Schr\\\"{o}dinger\nequation in the nonlinear Riccati form. The method, whose mathematical basis in\nphysics was discussed recently by one of the present authors (VBM), approaches\nthe solution of a nonlinear differential equation by approximating the\nnonlinear terms by a sequence of the linear ones, and is not based on the\nexistence of some kind of a small parameter. It is shown that the\nquasilinearization method gives excellent results when applied to computation\nof ground and excited bound state energies and wave functions for a variety of\nthe potentials in quantum mechanics most of which are not treatable with the\nhelp of the perturbation theory or the 1/N expansion scheme. The convergence of\nthe QLM expansion of both energies and wave functions for all states is very\nfast and already the first few iterations yield extremely precise results. The\nprecison of the wave function is typically only one digit inferior to that of\nthe energy. In addition it is verified that the QLM approximations, unlike the\nasymptotic series in the perturbation theory and the 1/N expansions are not\ndivergent at higher orders.",
"arxiv_id": "physics/0112024",
"authors": [
"R. Krivec",
"V. B. Mandelzweig"
],
"categories": [
"physics.comp-ph"
],
"title": "Quasilinearization approach to quantum mechanics",
"url": "https://arxiv.org/abs/physics/0112024"
},
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