The Helium atom as two electron charge densities restricted to half-spaces meeting at plane through the kernel as free boundary under a Bernoulli condition = homogeneous Neumann + continuity. The basic “two-valuedness” in stdQM expressed by Pauli’s Exclusion Principle is here replaced by a the two-valuedness of the half-space subdivision, not asking for any ad hoc introduction of the notion of spin.
As first crucial test case beyond Hydrogen, we now consider realQM for Helium with seeking a minimiser among wave functions arguing that by symmetry we may assume that is supported by the half-space and by with the plane as free boundary. More precisely, we assume by symmetry that for , which allows restriction of attention to .
The ground state then appears as the minimiser over of the total energy
under the normalisation
where the electron potential for is given by
We observe that is the solution of the equation in with for and else, which gives an alternative way of computing other then direct integration as above.
The condition of stationarity with respect to variations of takes the form of the eigenvalue problem
for the Hamiltonian
We compute stationary points by a gradient method, and the total energy as . Computing in cylindrical coordinates without angular variation around the cylinder axis and a far-field condition , we obtain on a mesh (observing in particular the homogeneous Neumann condition on the plane as the free boundary showing good agreement with observed table value):