Quantum Contradictions 16: An Inconvenient Truth (about Schrödinger's Equation)

Roothaan and Weiss states in Correlated Orbitals for the Ground State of Heliumlike Systems 

(Rev. Mod. Phys. 32, 194–205 (1960)):

• For the ground, state of atoms, molecules, and solids, the explicit solution of the Schrodinger equation to obtain an accurate wave function is virtually a hopeless task. Instead, one usually chooses a convenient function with adjustable parameters, and determines the best wave function within this class by minimizing the expectation value of the energy, computed from that wave function, with respect to these parameters. The form of the function chosen represents a certain restriction, and it is essential to use physical arguments in choosing such a form; in doing so, we essentially choose a model. The success of such a wave function can, to a certain extent, be judged by comparing the calculated energy with the experimental total energy.

This statement delivers an inconvenient truth about Schrödinger's equation as the basis of quantum mechanics: The equation can be solved exactly only for Hydrogen with one electron and computational solution for many electrons is beyond the capacity of any thinkable computer. Already Helium with two electrons presents mysteries since the spherically symmetric 1s2 state presented as the ground state in text books, does not have the experimentally observed energy and the true ground state is hidden to the reader. 

Physicists thus take for granted that solutions of Schrödinger's equation describe the atomistic world, but this hypothesis is impossible to test against observation since solutions cannot be determined, except in a few elementary cases. The second pillar of modern physics, Einstein's equations of general relativity, shares the same quality. 

This is an inconvenient truth about modern physics which is not told in text books. No wonder that modern physicists no longer speak about quantum mechanics and general relativity but instead about string theory and multiverse for which no equations at all are presented.