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Chinese Physics C

ON THE ENERGY LEVELS AND THE STATIONARY STATES OF THE EXOTIC ATOMS CONTAINING MONOPOLES

Xiaohu Dai

Published Jan 1, 1978• 0 citations
OpenAlex

Abstract

In this short paper, which is based on an analysis of some contradictions of the principles for the determination of the solution in usual quantum mechanics, and we suggest an adjusted framework of quantum mechanics, which naturally contains essentailly singular states. By using the monopole harmonics, which were studied by Yang and Wu, we solve the stationary state problems of the exotic atoms, which contain the magnetic monopole and different elementary particles.

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•May 1, 2013•0 citations
In this short paper, which is based on an analysis of some contradictions of the principles for the determination of the solution in usual quantum mechanics, and we suggest an adjusted framework of quantum mechanics, which naturally contains essentailly singular states. By using the monopole harmonics, which were studied by Yang and Wu, we solve the stationary state problems of the exotic atoms, which contain the magnetic monopole and different elementary particles.
The standard quantum mechanical description of the motion of a charged particle in the field of a stationary magnetic monopole is notorious for the presence of unnatural singularities in the Hamiltonian operator originating in the vector potential A(r) used to describe the magnetic field of the monopole. In this paper, an elementary quantum mechanical formulation of the problem which involves only the physically observable field B(r) is presented. This is achieved by treating as a fundamental observable of the charged particle its kinetic momentum instead of the linear momentum p . An irreducible representation of the fundamental commutation relations involving the operators and is explicitly constructed. It is shown that the existence of an irreducible representation requires that Dirac’s charge quantization condition is satisfied. Also, it is demonstrated that, from the quantum mechanical perspective, the singularities (appearing when the vector potential is introduced) are in fact properties of coordinate representations of the fundamental commutation relations.
We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).
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.
The familiar concepts of state vectors and operators in quantum mechanics rely on associative products of observables. However, these notions do not apply to some exotic systems such as magnetic monopoles, which have long been known to lead to nonassociative algebras. Their quantum physics has remained obscure. This Letter presents the first derivation of potentially testable physical results in nonassociative quantum mechanics, based on effective potentials. They imply new effects which cannot be mimicked in usual quantum mechanics with standard magnetic fields.