måndag 9 september 2013

Quantum Contradictions 20: Averaged Hartree Model between Scylla and Charybdis

The present sequence of posts Quantum Contradictions 1 - 20 with the key posts 6,  9 and 12  focussing on Helium (and two-electron ions) lead to a modified Hartree model for the ground state as a system of single electron wave functions defined by minimization of the total energy as the sum of (i) kernel potential energy, (i) inter-electron potential energy and (iii) kinetic energy, where the electrons keep individuality defined by individual presence in space as expressed by the single electron wave functions as concerns (i) and (ii), while the kinetic energy is computed after angular averaging as an expression of lack of individuality.

This model gives according to 12 a ground state energy of Helium of - 2.918 for spherical wave functions with polar decentration, to be compared with the observed - 2.903. Not bad.

In this model electrons thus keep individuality as concerns potential energies but lack individuality as concerns kinetic energy as a result of polar averaging.

Helium is thus described by two electronic wave functions, defined on 3-dimensional space, of the form:
  • $\psi_1(r,\theta )^2 = (1 + \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$,
  • $\psi_2(r,\theta )^2 = (1 -  \beta\cos(\theta ))\exp(-2\alpha r)\times\frac{\alpha^3}{\pi}$,
where $\alpha$ and $\beta$ are positive parameters determined by total energy minimization. This corresponds to a configuration with the two electrons being (more or less) separated, with electron 1 shifted towards the North pole of a spherical atom and electron 2 towards the South pole. The kinetic energy is computed after polar averaging or summation of $\psi_1$ and $\psi_2$.

We compare with the full wave function $\psi (r_1,\theta_1,\phi_1, r_2,\theta_2, \phi_2)$ satisfying Schrödinger's linear wave equation in 6 spatial dimension, where the electrons have lost all individuality as being indistinguishable and the wave function is given a statistical meaning. We have understood that this model is unphysical and should not be used.

The averaged Hartree model is a system in 3 dimensions and as such can be given a physical meaning without statistics, while the angular averaging removes the observed unphysical nature of the original Hartree model as a classical electron cloud (or Bohr) model of the atom.

The averaged Hartree model thus can be seen as a semi-classical physical model obtained by angular averaging in a classical model, instead of the full statistics of the full quantum model necessarily introducing non-physical aspects.

The averaged Hartree model steers between the Scylla of a classical model with full electronic individuality, which does not seem to describe the atomistic world, and the Charybdis of a full multidimensional quantum model with no electronic individuality, which is an unphysical model loaded with contradictions.

PS For the hydrogen ion H- with two electrons surrounding a +1 kernel, we obtain similarly a ground state energy of - 0.531 to be compared with observed - 0.528, and with - 0.500 for Hydrogen, thus indicating that H- is a stable configuration.

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