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

Kazimierz Rajchel

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.

Keywords


Schrödinger equation; Ricatti equation; unit circle; quantization condition

Mathematics Subject Classification (2010)


34-XX; 35A24; 65Lxx; 97I40

References


Cao, Zhuangqi and Cheng Yin. Advanced in One Dimensional Wave Mechanics. Towards A Unified Classical View. Berlin Heidelberg: Springer-Verlag, 2014.

Derezinski, Jan and Michał Wrochna. "Exactly solvable Schrödinger operators." Ann. Henri Poincaré 12, no. 2 (2011): 397-418.

Derezinski, Jan and Michał Wrochna. "Exactly solvable Schrödinger operators." Arxiv (2018): arxiv.org/pdf/1009.0541.pdf

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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