The hydrogen atom according to wave mechanics, parts I, II, III, IV, V

TL;DR

This paper provides a comprehensive analysis of the hydrogen atom's wave functions across five coordinate systems, offering accurate plots and insights into the implications of multiple coordinate representations in wave mechanics.

Contribution

It introduces previously unreported amplitude surface plots in ellipsoidal and spheroconical coordinates, enhancing understanding of atomic orbitals in various coordinate systems.

Findings

Accurate surface plots of amplitude functions in multiple coordinate systems.

Comparison of amplitude surfaces across spherical, paraboloidal, ellipsoidal, and spheroconical coordinates.

Identification of previously unreported amplitude surfaces in ellipsoidal and spheroconical coordinates.

Abstract

Based on an exposition of the underlying physics and applied mathematics in arXiv:1603.00899, this paper in five separate parts presents a description of the properties of the amplitude functions of the hydrogen atom according to wave mechanics in the coordinate representation in all four systems of coordinates in which Schroedinger's equations are separable, explicitly I spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal coordinates and IV spheroconical coordinates; part V discusses the implications of these multiple systems in which these atomic orbitals can be directly calculated. Most published plots of orbitals are inaccurate. In particular, parts I - IV contain many accurate plots of the surfaces of amplitude functions in the various systems with a common criterion for the amplitude at which the surface is constructed. Although the plots are generated from amplitude functions in the specified systems of coordinates, the plots are translated to cartesian coordinates that the eye can recognise. The surfaces of the amplitude functions in ellipsoidal and spheroconical functions are previously unreported, with the accurate surfaces in spherical polar and paraboloidal coordinates for a direct comparison.