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 . As shown below, realQM passes this test successfully!
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
Computation with realQM in spherical coordinates with azimuthal symmetry on a mesh with 200 points in radial direction and 100 in polar angle (on my iPad), gives the value -2.904 (Hartree) for the ground state energy of Helium in good agreement the observed value and benchmark computations with stdQM:
- Pekeris (1959): -2.903724376 (best stdQM)
- Koki (2009): -2.9042 (in better agreement with observation).
Computing in cylindrical coordinates without angular variation around the cylinder axis, we obtain on a mesh (observing in particular the homogeneous Neumann condition on the plane as the free boundary again showing good agreement with observed table value: