Proper quantization rule as a good candidate to semiclassical quantization rules |
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Authors: | FA Serrano M Cruz‐Irisson S‐H Dong |
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Institution: | 1. ESIME‐Culhuacan, Instituto Politécnico Nacional, Av. Santa Ana 1000, Mexico, D. F. 04430, Mexico;2. Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edificio 9, Unidad Profesional Adolfo Lopez Mateos, Mexico D. F. 07738, MexicoPhone: +52 55 57296000 ext. 55255, Fax: +52 55 57296000 ext. 55015 |
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Abstract: | In this article, we present proper quantization rule, ∫![urn:x-wiley:00033804:media:ANDP201000144:tex2gif-inf-1](/cms/asset/9296b8c0-cfad-4cf3-b6a2-08fc4dd5c5a4/tex2gif-inf-1.gif) k(x) dx ‐ ∫![urn:x-wiley:00033804:media:ANDP201000144:tex2gif-inf-4](/cms/asset/9811e87d-1f1a-48fc-bc1e-6eeaf3a7361b/tex2gif-inf-4.gif) k0(x) dx = nπ, where and study solvable potentials. We find that the energy spectra of solvable systems can be calculated only from its ground state obtained by the Sturm‐Liouville theorem. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of proper quantization rule come from its meaning – whenever the number of the nodes of the logarithmic derivative ?(x) = ψ(x)‐1dψ(x) /dx or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral will increase by π. We apply two different quantization rules to carry out a few typically solvable quantum systems such as the one‐dimensional harmonic oscillator, the Morse potential and its generalization as well as the asymmetrical trigonometric Scarf potential and show a great advantage of the proper quantization rule over the original exact quantization rule. |
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Keywords: | Proper quantization rule bound states solvable potentials |
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