<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>W</italic><sub>0</sub> 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.</p></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.

Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

Institute of Computer Science Pedagogical

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

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,...

Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

Published Online: Dec 05, 2019

Page range: 157 - 165

Received: Apr 23, 2019

Accepted: Jul 10, 2019

DOI: https://doi.org/10.2478/aupcsm-2019-0012

Keywords
Schrodinger equation, Ricatti equation, unit circle, quantization condition

© 2019 Kazimierz Rajchel, published by Sciendo

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

Published Online: Dec 05, 2019

Page range: 157 - 165

Received: Apr 23, 2019

Accepted: Jul 10, 2019

DOI: https://doi.org/10.2478/aupcsm-2019-0012

KeywordsSchrodinger equation, Ricatti equation, unit circle, quantization condition

© 2019 Kazimierz Rajchel, published by Sciendo

This work is licensed under a Creative Commons Attribution Share-Alike 4.0 License.

Keywords
Schrodinger equation, Ricatti equation, unit circle, quantization condition