New Horizons in Mathematical Physics

Download PDF (397.2 KB) PP. 105 - 117 Pub. Date: December 31, 2019

DOI: 10.22606/nhmp.2019.34001

• Spiros Konstantogiannis^*
  4 Antigonis Street, Nikaia 18454, Athens, Greece

Considering the stationary Schrödinger equation for a general pseudo-Coulomb potential as the normal form of the associated Laguerre equation, we review, in one and three dimensions, the bound-state solutions for the potential, when the inverse-square-term coupling is not less than a negative critical value. We show that, as a consequence of the inverse-square-term coupling being a two-to-one mapping for all but one of the allowed negative values of its parameter, an additional sequence of bound-state energies emerges for each of the respective potentials. In this framework, the slightest relaxation of the boundary condition for the radial wave function at the origin results in minus-infinity ground-state energy for the Coulomb potential, rendering the hydrogen atom unstable.

associated Laguerre equation; pseudo-Coulomb potential; inverse square potential; Kratzer potential; valence electron model; additional energies; instability.

[1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics. Pergamon Press Ltd., Second Edition, 1965, pp. 113-115.

[2] A. Kratzer, “Die ultraroten Rotationsspektren der Halogenwasserstoffe”, Z. Phys. 3, 289 (1920); E. Fues, “Zur Intensität der Bandenlinien und des Affinitätsspektrums zweiatomiger Moleküle”, Ann. Phys. 80, 281 (1926).

[3] R. J. Le Roy and R. B. Bernstein, “Dissociation Energy and Long-Range Potential of Diatomic Molecules from Vibrational Spacings of Higher Levels”, J. Chem. Phys. 52, 3869 (1970).

[4] S. Frish, Optical Spectra of Atoms. Fizmatgiz, Moscow, 1963.

[5] Teimuraz Nadareishvili and Anzor Khelashvili, “Pragmatic SAE procedure in the Schrodinger equation for the inverse-square-like potentials”, Available: https://arxiv.org/abs/1209.2864

[6] Mary L. Boas, Mathematical Methods in the Physical Sciences. John Wiley & Sons, Inc., Third Edition, 2006, pp. 605-606.

[7] Alexander V. Turbiner, “One-dimensional quasi-exactly solvable Schrödinger equations”, Phys. Rep. 642, 1-71 (2016).

[8] Arfken, Weber, and Harris, Mathematical Methods for Physicists. Elsevier Inc., Seventh Edition, 2013, p. 892.

[9] Michael Martin Nieto, “Hydrogen atom and relativistic pi-mesic atom in N-space dimensions”, Am. J. Phys. 47, 1067 (1979).

[10] David J. Griffiths, Introduction to Quantum Mechanics. Pearson Education, Inc., Second Edition, 2005, pp. 131- 141.

[11] Anzor A. Khelashvili and Teimuraz P. Nadareishvili, “What is the boundary condition for the radial wave function of the Schrödinger equation?”, Am. J. Phys. 79 (6), 668-671 (2011).

[12] Djamil Bouaziz, “Kratzer’s molecular potential in quantum mechanics with a generalized uncertainty principle”, Ann. Phys. 355, 269-281 (2015).

[13] O. Bayrak, I. Boztosun, and H. Ciftci, “Exact Analytical Solutions to the Kratzer Potential by the Asymptotic Iteration Method”, Int. J. Quantum Chem. 107 (3), 540-544 (2007).

[14] Sameer M. Ikhdair and Ramazan Sever, “Exact quantization rule to the Kratzer-type potentials: an application to the diatomic molecules”, J. Math. Chem. 45, 1137 (2009).

[15] Mohammad R Setare and Ebrahim Karimi, “Algebraic approach to the Kratzer potential”, Phys. Scr. 75, 90-93 (2007).

[16] Mohammad Reza Pahlavani, Theoretical Concepts of Quantum Mechanics. InTech, 2012, p. 242.

[17] Spiros Konstantogiannis, “Polynomial and Non-Polynomial Terminating Series Solutions to the Associated Laguerre Differential Equation”, ResearchGate, DOI: 10.13140/RG.2.2.24904.98569.