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arXiv:quant-ph/0306191

Multielectronic Atom in Magnetic Field Revisited

O. Chavoya-Aceves

Published Jun 27, 2003• 1 citation
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Abstract

The motion of a multi-electronic atom in an external electro-magnetic field is reconsidered. We prove that according to classical mechanics and electrodynamics, the assumption that the interaction with the magnetic field is described by means of a potential energy is no valid, and the trajectory of the center of mass can be deflected by a magnetic field, even if the internal angular momentum is zero. The characteristic equation of the corresponding hamiltonian is not separable in three degrees of freedom for the hydrogen atom.

DOI
10.48550/arxiv.quant-ph/0306191

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The motion of a multi-electronic atom in an external electro-magnetic field is reconsidered. We prove that according to classical mechanics and electrodynamics, the assumption that the interaction with the magnetic field is described by means of a potential energy is no valid, and the trajectory of the center of mass can be deflected by a magnetic field, even if the internal angular momentum is zero. The characteristic equation of the corresponding hamiltonian is not separable in three degrees of freedom for the hydrogen atom.
It is proved that, according to Classical Mechanics and Electrodynamics, the trajectory of the center of mass of a neutral system of electrical charges can be deflected by an inhomogeneous magnetic field, even if its internal angular momentum is zero. This challenges the common view about the function of the Stern-Gerlach apparatus, as resolving the eigen-states of an intrinsic angular momentum. Doubts are cast also on the supposed failure of Schrodinger's theory to explain the properties of atoms in presence of magnetic fields without introducing spin variables.
We prove that according to classical mechanics the trajectory of the center of mass of a neutral system of electrical charges can be deflected by an inhomogeneous magnetic field, even if the internal angular momentum is zero. This challenges the common interpretation of the functioning of the Stern-Gerlach apparatus, as resolving the eigen-states of a mysterious strictly quantum two valued physical observable. Even more, the main evidence we have of the failure of Schr\\"odinger's theory to explain the properties of complex atoms without introducing spin variables is not reliable.
A hydrogen atom moving across a magnetic field is considered in a wide region of magnitudes of magnetic field and atom momentum. We solve the Schr\"odinger equation of the system numerically using an imaginary time method and find wave functions of the lowest states of atom. We calculate the energy and the mean electron-nucleus separation as a function of atom momentum and magnetic field. All the results obtained could be summarized as a phase diagram on the ``atom-momentum -- magnetic-field'' plane. There are transformations of wave-function structure at critical values of atom momentum and magnetic field that result in a specific behavior of dependencies of energy and mean interparticle separation on the atom momentum P.P.P. We discuss a transition from the Zeeman regime to the high magnetic field regime. A qualitative analysis of the complicated behavior of wave functions vs PPP based on the effective potential examination is given. We analyze a sharp transition at the critical momentum from a Coulomb-type state polarized due to atom motion to a strongly decentered (Landau-type) state at low magnetic fields. A crossover occurring at intermediate magnetic fields is also studied.
An external magnetic field is applied to the classical hydrogen atom treated in classical electromagnetic theory including classical electromagnetic zero-point radiation. In an earlier article, it was shown that the average value of each of the electron's action variables appears as a discrete value, corresponding to a representation of the rotation group, because of resonance between the periodic orbit of the electron and the random classical zero-point radiation. Here it is shown that, in the presence of a magnetic field and because of resonance, the classical orbital motion of the electron is in resonance with random classical zero-point radiation only for orientations of the orbit which take integer values for the angle made with the direction of the magnetic field, but excluding the m=0 orientation where the magnetic field is parallel to the orbital plane of the electron. Classical electromagnetic explanations are given for the Stern-Gerlach result, and for the Zeeman effect.
A quantum mechanical study of a simple two-dimensional problem identifies lines in coordinate space about which the probability densities are localised. These lines in general do not correspond to extrema in the system's static potential energy surface. The subject of this study has been the hydrogen atom in externally applied parallel electric and magnetic and magnetic fields. Probability density plots of eigenstates, in which the electron experiences comparable influences from all three fields: Coulomb, electric and magnetic, are each found to display a localisation at a polar angle related to an invariant previously obtained in a classical study of the problem. The line about which localisation occurs is found to vary with energy and is in coincidence with a periodic trajectory found within a classical framework.