dorsal/arxiv
View SchemaAnalytic calculation of energies and wave functions of the quartic and pure quartic oscillators
| Authors | E. Z. Liverts, V. B. Mandelzweig, F. Tabakin |
|---|---|
| Categories | |
| ArXiv ID | physics/0603165 |
| URL | https://arxiv.org/abs/physics/0603165 |
| DOI | 10.1063/1.2209769 |
Abstract
Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schr\"{o}dinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.
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"abstract": "Ground state energies and wave functions of quartic and pure quartic\noscillators are calculated by first casting the Schr\\\"{o}dinger equation into a\nnonlinear Riccati form and then solving that nonlinear equation analytically in\nthe first iteration of the quasilinearization method (QLM). In the QLM the\nnonlinear differential equation is solved by approximating the nonlinear terms\nby a sequence of linear expressions. The QLM is iterative but not perturbative\nand gives stable solutions to nonlinear problems without depending on the\nexistence of a smallness parameter. Our explicit analytic results are then\ncompared with exact numerical and also with WKB solutions and it is found that\nour ground state wave functions, using a range of small to large coupling\nconstants, yield a precision of between 0.1 and 1 percent and are more accurate\nthan WKB solutions by two to three orders of magnitude. In addition, our QLM\nwave functions are devoid of unphysical turning point singularities and thus\nallow one to make analytical estimates of how variation of the oscillator\nparameters affects physical systems that can be described by the quartic and\npure quartic oscillators.",
"arxiv_id": "physics/0603165",
"authors": [
"E. Z. Liverts",
"V. B. Mandelzweig",
"F. Tabakin"
],
"categories": [
"physics.atom-ph"
],
"doi": "10.1063/1.2209769",
"title": "Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators",
"url": "https://arxiv.org/abs/physics/0603165"
},
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