Schrödinger equation with Coulomb potential admits no exact solutions

Arbitrary l-solutions of the Schrödinger equation interacting with Hulthén – Hellmann potential model

“…Arbitrary -solutions play a dominant role in non-relativistic quantum mechanics since the wave function and associated eigenvalues contain all the necessary information for a full description of a quantum system [5][6][7][8]. With the experimental verification of the Schrödinger equation, researchers have devoted much interest in solving the radial Schrödinger equation to obtain bound state solutions with various methods for some potential models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

From vibrational spectroscopy and quantum tunnelling to periodic band structures – a self-supervised, all-purpose neural network approach to general quantum problems

“…All calculated transition wavenumbers are listed in Table S. 2 Even more challenging, yet highly relevant benchmark systems for the neural network are applications to double-well potentials. This way the adaptability of the FF-ANN to model wave functions of increasing complexity in terms of curvature in conjunction with small differences in values for the eigenenergy between ground and first excited state, an effect known as tunnel splitting, can be assessed.…”

“…Recently, it has even been argued that no such solutions may be formulated for comparably simple systems such as the He atom. 2 While routine applications in quantum chemistry rely on numerical frameworks such as Hartree-Fock (HF) [3][4][5][6] and post-HF 6,7 apa Theoretical Chemistry, Division, Institute of General, Inorganic and Theoretical Chemistry, Center for Chemistry and Biomedicine, University of Innsbruck, Innrain 80-82, A-6020 Innsbruck, Austria. E-mail: t.hofer@uibk.ac.at b Research Institute for Biomedical Aging Research, University of Innsbruck, Rennweg 10, A-6020 Innsbruck, Austria † Electronic Supplementary Information (ESI) available.…”

“…Recently, it has even been argued that no such solutions may be formulated for comparably simple systems such as the He atom. 2 While routine applications in quantum chemistry rely on numerical frameworks such as Hartree–Fock (HF) 3–6 and post-HF 6,7 approaches as well as various implementations of density functional theory (DFT), 8–10 the pursuit of alternative routes to provide more efficient as well as more accurate solutions to electronic structure problems is still a highly active field of research. 11–16 Typically, simplified one-dimensional prototype systems such as harmonic and anharmonic oscillators, various instances of the particle-in-a-box system and multi-well potentials act as starting points to formulate alternative strategies to solve Schrödinger's equation.…”

Exact irregular solutions to radial Schrödinger equation for the case of hydrogen-like atoms

“…The explicit representation of the independent irregular solutions at hand can help us in studying the analytic behavior of Coulomb scattered-wave amplitudes as the energy of the scattered electron is extended into the complex plane, and as the electron angular momentum quantum number takes on complex values, such as in Regge pole analysis(See Gaspard [16]). Furthermore, the closed-form expressions for the irregular solutions, rather than the usual Laurent series representation, provide explicit answers to how the irregular solutions behave for the electron waves at large distances from the nucleus, useful in presentations of the quantum Coulomb problem.Toli and Zou [17] obtained a Taylor series expansion of the regular Coulomb wave functions and concluded that the exact solutions of the SE, having the Coulomb potential for molecules consisting of more than two particles, cannot be achieved. Simos [19] developed multiderivative methods for comparing the numerical solution of the 1D SE with the existing exponentially-fitted Raptis-Allison method and Ixaru-Rizea method.…”