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arXiv:0809.4843

Magnetic Charge of the Stark States of Hydrogen Atoms

T. Pradhan

Published Sep 28, 2008• 0 citations
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Abstract

It is conjectured that Stark states of excited hydrogen atom posses magnetic charge for which the quantum mechanical operator is $${\cal G}_{op} = {e\over \hbar} (\vecσ\cdot\vec A)$$ where $\vec A$ is the Runge-Lenz vector. The expectation value $g$ of this operator for Stark states is found to be $$ g = e(n_1-n_2)$$ which obeys a Dirac-Saha type quantization formula $${eg\over c} = (n_1-n_2)α$$ where $α$ is the fine structure constant and $n_1$ and $n_2$ are parabolic quantum numbers. An experimental arrangement is outlined to test this conjecture.

DOI
10.48550/arxiv.0809.4843

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It is conjectured that Stark states of excited hydrogen atom posses magnetic charge for which the quantum mechanical operator is Gop=eℏ(σ⃗⋅A⃗){\cal G}_{op} = {e\over \hbar} (\vecσ\cdot\vec A)Gop​=ℏe​(σ⋅A) where A⃗\vec AA is the Runge-Lenz vector. The expectation value ggg of this operator for Stark states is found to be g=e(n1−n2) g = e(n_1-n_2)g=e(n1​−n2​) which obeys a Dirac-Saha type quantization formula egc=(n1−n2)α{eg\over c} = (n_1-n_2)αceg​=(n1​−n2​)α where ααα is the fine structure constant and n1n_1n1​ and n2n_2n2​ are parabolic quantum numbers. An experimental arrangement is outlined to test this conjecture.
It is shown that the excited states of hydrogen atom in a uniform electric field (Stark States) posess magnetic charge whose magnitude is given by a Dirac-Saha type relation: egℏc=3n {eg\over \hbar c} = \sqrt 3 n ℏceg​=3​n An experiment is proposed to fabricate such states and to detect their magnetic charge.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
Formulations of magnetic monopoles in a Hilbert-space formulation of quantum mechanics require Dirac's quantization condition of magnetic charge, which implies a large value that can easily be ruled out for elementary particles by standard atomic spectroscopy. However, an algebraic formulation of nonassociative quantum mechanics is mathematically consistent with fractional magnetic charges of small values. Here, spectral properties in nonassociative quantum mechanics are derived and applied to the ground state of hydrogen with a magnetically charged nucleus. The resulting energy leads to new strong upper bounds for the magnetic charge of various elementary particles that can appear as the nucleus of hydrogenlike atoms, such as the muon or the antiproton.
Consider the problem: why does the bound state spectrum E(n)<0E(n) < 0E(n)<0, of hydrogen atom Hamiltonian HHH have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, A⃗\vec AA, of HHH, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of HHH and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector A⃗\vec AA.
Weak magnetic monopoles with a continuum of charges less than the minimum implied by Dirac's quantization condition may be possible in nonassociative quantum mechanics. If a weakly magnetically charged proton in a hydrogen atom perturbs the standard energy spectrum only slightly, magnetic charges could have escaped detection. Testing this hypothesis requires entirely new methods to compute energy spectra in nonassociative quantum mechanics. Such methods are presented here, and evaluated for upper bounds on the magnetic charge of elementary particles.