Solutions of the time-independent Schrödinger equation by uniformization on the unit circle
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica (2019)
- Volume: 18, page 157-165
- ISSN: 2300-133X
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topKazimierz Rajchel. "Solutions of the time-independent Schrödinger equation by uniformization on the unit circle." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 18 (2019): 157-165. <http://eudml.org/doc/296813>.
@article{KazimierzRajchel2019,
abstract = {The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.},
author = {Kazimierz Rajchel},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {Schrödinger equation; Ricatti equation; unit circle; quantization condition},
language = {eng},
pages = {157-165},
title = {Solutions of the time-independent Schrödinger equation by uniformization on the unit circle},
url = {http://eudml.org/doc/296813},
volume = {18},
year = {2019},
}
TY - JOUR
AU - Kazimierz Rajchel
TI - Solutions of the time-independent Schrödinger equation by uniformization on the unit circle
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2019
VL - 18
SP - 157
EP - 165
AB - The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.
LA - eng
KW - Schrödinger equation; Ricatti equation; unit circle; quantization condition
UR - http://eudml.org/doc/296813
ER -
References
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- Dubrowin, B.A., A.T. Fomienko and S.P. Novikov. Modern Geometry - Methods and Applications. Part II. The Geometry and Topology of Manifolds. Vol. 104 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1985.
- Konishi, Kenichi and Giampiero Paffuti. Quantum Mechanics. A New Introduction. New York: Oxford University Press. 2009.
- Rajchel, Kazimierz and Jerzy Szczesny. "New method to solve certain differential equations." Ann. Univ. Paedagog. Crac. Stud. Math. 15 (2016): 107-111.
- Rajchel, Kazimierz. "New solvable potentials with bound state spectrum." Acta Phys. Polon. B 48, no. 4 (2017): 757-764.
- Reid, William T. Riccati differential equations. Vol. 86 of Mathematics in Science and Engineering. New York-London: Academic Press, 1972.
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