måndag 19 augusti 2013

Quantum Contradictions 12: Electron Pair of Helium vs Spin

Let us return with an elaboration of post 6 in this series about the electron configuration of the ground state of Helium. Let us make an Ansatz with the distribution of the two electrons of Helium given by the following two (squared) wave functions (in standard notation):
• $\psi_1(r,\theta)^2 = (1 + \beta\cos(\theta))exp(-2\alpha r)\times\alpha^3/\pi$
• $\psi_2(r,\theta)^2 = (1 - \beta\cos(\theta))exp(-2\alpha r)\times\alpha^3/\pi$,
where $\alpha$ and $\beta$ are positive parameters to determine. This corresponds to a configuration with the two electrons being separated, with electron 1 shifted towards the upper part of a spherical atom and electron 2 towards the lower part.

Standard text book computation gives the following total energy $E$ as the sum of potential energy, radial kinetic energy and interelectronic repulsion energy in the setting of Schrödinger's equation:
• $E = - 4 \alpha + \alpha^2 + 5 \alpha/8 - \alpha\beta^2/24$
with the last term resulting from the separation weights $(1 + - \beta\cos(\theta ))$. Optimization in $\alpha$ with $\beta = 1$ gives
• $\alpha = 2 - (5/16 - 1/48)$
• $E = (41/24)^2 = - 2.918$
to be compared with the observed $E = - 2.903$. For a configuration without separation ($\beta = 0$),
$E = - 2.85$.  We thus find theoretical support of an idea of electronic Zweideutigkeit in the ground state of Helium represented by an electron pair in opposition.

In the above computation we neglected azimuthal kinetic energy based on the idea that the electron pair should be averaged over azimuthal angle to eliminate azimuthal variation.  The considered electron pair is thus to be viewed as "typical" combined with averaging over angle reducing azimuthal kinetic energy to zero.

The total (multi-dimensional) wave function $\psi(r_1, \theta_1, r_1, \theta_2)$ in the standard setting thus has a Hartree product form
• $\psi (r_1, \theta_1, r_2, \theta_2) = \psi_1(r_1, \theta_1)\times \psi_2(r_2,\theta_2)$
for which only radial kinetic energy is taken into account.

PS The angular momentum of $\psi_1$ equals $- \sin(\theta ) (\sin(\phi ), - \cos(\phi ), 0)$ and that of $\psi_2$ equals $+ \sin(\theta ) (\sin(\phi ), - \cos(\phi ), 0)$ up to the common factor $\beta\exp(-2\alpha r)\times \alpha^3/\pi$, which can be associated with a two-valued spin up/down.

The mysterious apparent Zweideutigkeit of electrons thus may simply be an effect of electron separation by repulsion.