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American Journal of Physics

The measurability of the kinetic momentum of a charged particle in a magnetic field

W P Healy

Published Dec 1, 1988• 0 citations
Publisher

Abstract

Uncertainty relations are derived for the components of the kinetic momentum (mass times velocity) of a charged particle in a magnetic field. In constrast to the components of the canonical momentum, the components of the kinetic momentum do not all commute with each other when the field is not zero. The uncertainty relations are illustrated by two Gedanken experiments in which the kinetic momentum is measured by using (a) the Doppler shift of emitted radiation and (b) the angle of deflection of the particle by the field.

DOI
10.1119/1.15733

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Uncertainty relations are derived for the components of the kinetic momentum (mass times velocity) of a charged particle in a magnetic field. In constrast to the components of the canonical momentum, the components of the kinetic momentum do not all commute with each other when the field is not zero. The uncertainty relations are illustrated by two Gedanken experiments in which the kinetic momentum is measured by using (a) the Doppler shift of emitted radiation and (b) the angle of deflection of the particle by the field.
Abstract Uncertainty relations are one of the fundamental components of quantum theory, typically used to describe the limitations on the range of measurement results for two noncommuting observables. Recently, there has been growing interest in uncertainty relations concerning multiple observables. In this work, we study the uncertainty relations of the three components of the canonical momentum of an electron moving in a magnetic field. We derive the uncertainty relations for both the addition and multiplication of these components. Our results indicate that, compared to uncertainty relations involving only two components, the uncertainty relations for the three components are associated with a constant τ = 2 / 3 .
We discuss the uncertainty relation for the azimuthal angle ϕϕϕ and the zzz-component of the angular momentum LzL_zLz​. To this end we derive the uncertainty relation for an arbitrary pair of observables and discuss the conditions for its validity. By means of a simple parameter-dependent state we illustrate the well known fact that the standard uncertainty relation for the coordinate and its conjugate momentum does not apply the pair of observables ϕ−Lzϕ-L_zϕ−Lz​. This analysis is motivated by a discussion of the motion of an electron in a constant magnetic field appeared recently in this journal (Eur. J. Phys. \textbf{33} (2012) 1147) where the author assumed the validity of the standard uncertainty relation for the pair ϕ−Lzϕ-L_zϕ−Lz​.
Canonical and kinetic momentum have representations in quantum mechanics that imply differences in the measurability of these related quantities in external fields. The measurability of kinetic momentum in external electric and magnetic fields and in free space is examined for several different measurement procedures. It is shown that kinetic momentum is a measurable physical observable, at least in uniform external fields. The quantum limitations of measurement are found to be greater for measurement of kinetic momentum in magnetic fields than in either free space or electric fields, in a way that is consistent with the standard operator representation of kinetic momentum.
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.
It is demonstrated that the collapse of the wavefunction in the quantum measurement process is reflected in the two-particle correlation function for the electron spin, which can in principle be studied by a triple Stern-Gerlach experiment. The change in momentum of a Stern-Gerlach magnet is itself sufficient to allow the determination of the electron spin, subject to the uncertainty principle, if the coherence of the two components of the spin is destroyed by the magnetic field. Similar considerations apply to the two-particle correlation function for the photon polarisation, and two-photon atomic cascades can be used in practice to investigate the measurement process and determine the extension of a photon wavepacket. The irreversibility of quantum mechanical measurements and its relation to the uncertainty principle is briefly considered.