Kimball Tutorial 1                                            Tutorial 2

Content: Using the exact solution of the Schrödinger equation for the ground state of the H atom, derive the Ansatz of G.E.Kimball (für deutsche Leser: die erstern 10 Seiten der Einführung enthalten etwa das gleiche)                                                                                         ES 07 February 2017//2002//1982

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Properties of the H1s wavefunction

H atom, 1s ground state wavefunction: Proton at origin, r distance of electron from proton
Exact H1s wavefunction in atomic units! e is the base of the natural logarithms.

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Graphics:H atom, 1s wavefunction

H atom ground state electron probability density

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Graphics:H atom, 1s 'point' density

H atom ground state probability density summed over all space. Must be 1, meaning: We will find the electron of the atom with certainty somewhere in the universe

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The same two functions in 3D:

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Graphics:H atom, 1s wavefunction

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Graphics:H atom, 1s 'point' density

This shows the electron density cusp at the location of the proton

Plot of the radial density of the H atom ground state: This is the point density at r multiplied with the volume of an infinitesimal spherical shell K_tutorial1a_13.pngdr, plotted as function of r. Dimensionwise, it is the charge in that spherical shell at r. Its maximum is at r = 1 a0, the radius of the 1s orbit of the old atom model of Niels Bohr.

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Graphics:H atom, 1s 'radial' density plotted

The next image is the surface of revolution of the above curve around the z-axis
with proton at the origin

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Graphics:H atom, 1s 'radial' density

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H atom, average distance of electron from proton R, bohr units

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Inside this average distance we find about 58% of the total charge of 1 electron:

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Energy components:

H atom, potential energy

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H atom, total potential energy

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H atom, kinetic energy

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H atom, total kinetic energy

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H atom, total Energy

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H atom, ground state equilibrium energy terms expressed with R, the average distance of the electron from the proton

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Kimball’s “Ansatz”: Total energy Etot = T+V
T has dimension K_tutorial1a_36.png, V K_tutorial1a_37.png, hence:

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and should be a minimum! Set its derivative to zero to find the optimal R=x:

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Substitute x for R:

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We have already determined, that Etot = -1/2 Eh, and R=3/2 a0.
Solve for the constants a,b using these results:

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Substitute a,b into the “Ansatz” above and obtain Kimball’s equation for the H atom in its stationary ground state:
This is exact! It is a significant mathematical transformation of Schrödinger’s equation for the H-atom ground state, contains the same information and has been misunderstood by all practitioners of a qualitative Kimball model:

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Kinetic energy Eh:  Full dimension  K_tutorial1a_47.png

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Potential energy Eh:

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Virial Theorem

Check the ratio

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This should be -2.00... by the Virial Theorem, which is valid in classical and quantum physics. The value -2 of this ratio signifies, that the force maintaining the observed system in a state of equilibrium varies proportional to K_tutorial1a_54.png. It is the electrostatic force between every charged particle pair of the system! If computations of molecules and solid lattices are reasonably complete, they yield Vir of 2.000±0.002. Most texts on quantum chemistry give a proof of this theorem.  

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Note: In this tutorial I have used the qualifier "exact" for the H1s wavefunction and the "Kimball equation". Those two expressions are equivalent. They are "exact" in the usual context of quantum chemical computations, but not "absolutely true". We have neglected the finite kinetic energy of the proton (see Tutorial2 and 6), which is ~1/1836 of the kinetic energy of the electron, and have assumed that the proton is a point charge with a size neglectable compared to the "size" of the H-atom. Furthermore, all movements within the atom leave the barycenter fixed. We have neglected relativistic effects, quantum electrodynamic corrections, magnetic interactions between electron and proton spins, and others. They are all of an absolute size 10-5 or less of the values shown.

Created with the Wolfram Language