Quantum mechanics is supposed to explain the electronic shell structure of the periodic table based on solutions of the Schrödinger equation named wave functions. Eric Scerri questions this narration on the ground that the Schrödinger equation of an atom with more than one electron cannot be solved exactly and for several electrons not even computationally because of the high dimensionality of the Schrödinger equation with 3N spatial dimensions for an atom with N electrons, which is difficult already for Helium with N = 2.
Various drastic reductions of the dimensionality of the Schrödinger equation are thus being made in practice such as Hartree-Fock methods based on combinations of hydrogenic electron orbitals as exact solutions to the solvable Schrödinger equation for Hydrogen, and density functional methods, but the relation between these brute ad hoc approximate solutions and solutions to the full Schrödinger equation is unknown and so the claim that the periodic table comes out the Schrödinger equation is not well founded.
In any case computation of approximate solutions of the Schrödinger takes most of the capacity in scientific computing, and so electron orbitals and electron densities are produced in massive numbers. Eric Scerri asks Have Electron Orbitals Really Been Observed? and answers: No: only electron densities have been (can be) experimentally observed.
Direct experimental support of the validity of Schrödinger equation as a model of atomistic physics thus appears (is) impossible. Even worse, it seems (is) impossible to compare predictions by the Schrödinger equation to experimental observation because solutions cannot be computed. Questions are inevitable:
- What is the scientific reason to believe that the Schrödinger equation describes the atomistic world?
- What is the scientific reason to believe that electrons interact precisely through the Coulomb potential of the Schrödinger equation?