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European Journal of Physics

Momentum measurements in external fields

D E Dugdale

Published Mar 1, 1992• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/13/2/004

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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.
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.
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 shown in two examples that to measure the momentum of a particle with great accuracy, allowing for large uncertainties in the position, it is still necessary to consider additional quantum effects. For the case of measurements by electromagnetic means, the uncertainties in the measurability of field strengths and of charge-current densities should be considered. Within the framework of the Bohr-Rosenfeld analysis, uncertainties in the field measurements do not limit the accuracy of momentum measurements. What happens when the Bohr-Rosenfeld approximations are not made is an open question.
The Hamiltonian and the spin operators for a spin 3/2 are represented in the basis formed by the Kronecker productions of the [Formula: see text] Pauli matrices. This reformulation allows us to represent a spin 3/2 as a system of two coupled fictitious spins 1/2. Correlations between these fictitious spins are studied using well-developed methods. We investigate the temperature and field dependences of correlations, such as mutual information, classical correlations, entanglement, and geometric and quantum discords in the fictitious spin-1/2 system describing a nuclear spin 3/2 which is placed in magnetic and inhomogeneous electric fields. It is shown that the correlations between the fictitious spins demonstrate properties which differ from those of real two-spin systems. In contrast to real systems all the correlations between the fictitious spins do not vanish with increasing external magnetic field; at a high magnetic field the correlations tend to their limiting values. Classical correlations, quantum and geometric discords reveal a pronounced asymmetry relative to the measurements on subsystems (fictitious spins) even in a uniform magnetic field and at symmetrical EFG, [Formula: see text]. The correlations depend also on the distribution of external charges, on the parameter of symmetry [Formula: see text]. At [Formula: see text] quantum and geometric discords have finite values in a zero magnetic field. The proposed approach may be useful in analysis of properties of particles with larger angular momentum, can provide the way to discover new physical phenomenon of quantum correlations, and can be a useful tool for similar definitions of other physical quantities of complex systems.
This chapter discusses quantum measurements of relativistic systems, such as quantum fields. The quantum theory of fields can be constructed with the help of a functional integral with integration over field configurations. It is natural that a description of measurements of quantum fields can be achieved by the restriction of such integrals. The main point is that different definitions of electromagnetic field strength lead to different conclusions about the existence or nonexistence of an absolute quantum restriction on the measurability of the field strength. To analyse the measurement process in quantum mechanics, one can use a simple model of the sensitive element of the instrument, sometimes called a 'meter'. The purely classical part of the instrument is not then explicitly considered. The coupling of the apparatus to the measured system is described by an interaction Lagrangian that is proportional to the product of the measured quantity and the coordinate of the meter.
Studying the localisation-delocalisation property of confined quantum systems is crucial for advancements in quantum technology and information science. Here, we investigate the localisation-delocalisation property for a particle in a quantum well under the influence of an electric field by calculating position-momentum space standard deviations (Δx, Δp) and the Shannon entropies (Sx, Sp) as a function of the quantum well width (L) or the electric field strength (F). Our results suggest that when the width L or the strength F changes, both the standard deviations and the Shannon entropies can lead to consistent interpretations in measuring the localisation-delocalisation property of this system. Specifically, for a fixed magnitude of F, Δx and Sx increases with the width L, suggesting an extensive delocalisation, while they diminishes with increasing F for a given width of L, revealing the localisation property of this system. By comparing the behaviours of position-momentum uncertainty relation (ΔxΔp) and the Shannon entropy uncertainty relation (St = Sx + Sp), we find that the behaviour of ΔxΔp with L or F varies for different quantum state. In contrast, the variation of St with L or F is consistent across different quantum states. The underlying physical reasons for the effects of quantum well width and electric field strength on the uncertainty relations of this system are analysed from the differentiation perspective. This study provides deep insights into localisation-delocalisation property of a particle in a quantum well under external perturbations, and has significant applications in the field of quantum physics, condensed physics, semiconductor physics and material science.