torsdag 11 augusti 2016

New Quantum Mechanics 16: Relation to Hartree and Hartree-Fock

The standard computational form of the quantum mechanics of an atom with N electrons (Hartree or Hartree-Fock) seeks solutions to the standard multi-dimensional Schrödinger equation as linear combinations of wave functions $\psi (x1,x2,...,xN)$ depending on $N$ 3d space coordinates $x1$,...,$xN$ as a product:
  • $\psi (x1,x2,...,xN)=\psi_1(x1)\times\psi_2(x2)\times ....\times\psi_N(x_N)$ 
where the $\psi_j$ are globally defined electronic wave functions depending on a single space coordinate $xj$.

The new model takes the form of a non-standard free boundary Schrödinger equation in a wave function $\psi (x)$ as a sum:
  • $\psi (x)=\psi_1(x)+\psi_2(x)+....+\psi_N(x)$,
where the $\psi_j(x)$ are electronic wave functions with local support on a common partition of 3d space with common space coordinate $x$.

The difference between the new model and Hartree/Hartree-Fock is evident and profound.  A big trouble with electronic wave functions having global support is that they overlap and demand an exclusion principle and new physics of exchange energy.  The wave functions of the new model do not overlap and there is no need of any exclusion principle or exchange energy.

PS Standard quantum mechanics comes with new forms of energy such as exchange energy and correlation energy. Here correlation energy is simply the difference between experimental total energy and total energy computed with Hartree-Fock and thus is not a physical form of energy as suggested by the name, simply a computational /modeling error.

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