Archive for July, 2008

Jul 20 2008

Spread of a Fermion Wave Function, Rev. A

Published by under Quantum Mechanics

For the electron in a central Coulomb potential, we have the Schrödinger equation in polar coordinates written as:
[pr2/2m + l2/2mr2 + V(r)] ψ(r, θ, φ) = Eψ(r, θ, φ)                 ([1], (IX.13), pg 348)
The term containing the square of radial linear momentum gains its prominence due to the fact that with Schrödinger wave mechanics there is no such thing as a circular electron orbit.  The trajectory followed by the electron in an attractive Coulomb field is shown in Messiah Fig. VI.2 (b), page 229.  If we can imagine the curve as being shown concave upward, instead of concave downward as in the figure, we could think of the electron as on a ski jump and rolling forward at the end of the path.  By kicking off one or more gravitons in this process, and having its magnetic dipole moment reversed in direction, both actions help place the electron on another arcing path back toward the nucleus, but in a different plane relative to the first arc.  A quantum angular step in the orientation of the nucleus, which aids in the production of a changing magnetic field, would be a third process that helps the electron start out and complete its next trajectory.
The eigenfunctions for the simplest case, of hydrogen, are:
Φnlm(r, θ, Ф) = Rnl(r) Ylm(θ, Ф)    ([2], (10.130), pg 452),
Something that can again be pondered about the stability of the electron trajectory in a quantum atomic orbit, is gravitons approaching in transverse and rear directions.  As for how some gravitons may be able to step through in these cases, there may be a spherical spread of the electron in the regions between turns, with the electron contracting as it enters a turn and expanding as it comes out of the turn in its opposite spin orientation.  This is proposed as an accordion type behavior of the sphere that contains the electron’s internal wave functions.
A spread of the particle wave function would also then be taking place for free fermions in the earth’s gravitational field, in order to allow gravitons to pass through.  An exception is when electrons are held in high voltage, such as with the GLAST instrument designed to measure gamma rays greater in energy than 100 MeV, or in the ground in the milliseconds before a lightning strike.  In these cases the electron would be concentrated enough in volume to produce Compton scattering.
[1]  Messiah, Albert, Quantum Mechanics, Two Volumes Bound as One, Dover Publications, Inc., paperback, 1999
[2] Liboff, Richard L., Introductory Quantum Mechanics, Fourth Edition, Addison Wesley, 2003


Aug 14, 2008, Rev. A: Added “radial” to second paragraph; added eigenfunction equation.

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