Volume 18: Pages 321-361, 2005
Exact Classical Quantum‐Mechanical Solutions for One‐ through Twenty‐Electron Atoms
Randell L. Mills
BlackLight Power Inc., 493 Old Trenton Road, Cranbury, New Jersey 08512 U.S.A.
It is true that the Schrödinger equation can be solved exactly for the hydrogen atom, although it is not true that the result is the exact solution of the hydrogen atom. Electron spin is missed entirely, and there are many internal inconsistencies and nonphysical consequences that do not agree with experimental results. The Dirac equation does not reconcile this situation. Many additional shortcomings arise, such as instability to radiation, negative kinetic energy states, intractable infinities, virtual particles at every point in space, the Klein paradox, violation of Einstein causality, and “spooky” action at a distance. Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as a physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by Laloë [Am. J. Phys. 69, 655 (2001)]. But QM has severe limitations even as a tool. Beyond one‐electron atoms, multielectron‐atom quantum‐mechanical equations cannot be solved except by approximation methods involving adjustable‐parameter theories (perturbation theory, variational methods, self‐consistent field method, multiconfiguration Hartree‐Fock method, multiconfiguration parametric potential method, 1/Z expansion method, multiconfiguration Dirac‐Fock method, electron correlation terms, QED terms, etc.), all of which contain assumptions that cannot be physically tested and are not consistent with physical laws. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e− moving in the Coulombic field of the proton and the wave equation as modified after Schrödinger, a classical approach was explored, yielding a model that is remarkably accurate and provides insight into physics on the atomic level [R.L. Mills, Phys. Essays 16, 433 (2003); 17, 342 (2004); The Grand Unified Theory of Classical Quantum Mechanics (BlackLight Power, Inc., Cranbury, NJ, 2005)]. Physical laws and intuition are restored when dealing with the wave equation and quantum‐mechanical problems. Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrödinger boundary condition “ Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell's equations, the classical wave equation is solved with the constraint that the bound (n = 1)‐state electron cannot radiate energy. The electron must be extended rather than a point. On this basis, with the assumption that physical laws including Maxwell's equation apply to bound electrons, the hydrogen atom was solved exactly from first principles. The remarkable agreement across the spectrum of experimental results indicates that this is the correct model of the hydrogen atom. In this paper the physical approach was applied to multielectron atoms that were solved exactly, disproving the deep‐seated view that such exact solutions cannot exist according to QM. The general solutions for one‐ through twenty‐electron atoms are given. The predictions of the ionization energies are in remarkable agreement with the experimental values known for 400 atoms and ions.
Keywords: Maxwell's equations, nonradiation, quantum theory, special and general relativity, ionization energies, one‐ through twenty‐electron atom solutions
Received: April 28, 2004; Published online: December 15, 2008