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Physica Scripta

Uncertainty relations for the triple components of canonical momentum

Zhi-Jie Liu, Jie Zhou, Hao-Nan Qiang, Mi Xie, Jing‐Ling Chen

Published Mar 18, 2025• 2 citations
Publisher

Abstract

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 .

DOI
10.1088/1402-4896/adc215

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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 .
The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of noncommuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially for spin-1/2. We derive uncertainty relations encompassing the triple components of angular momentum and show that, compared with the relations involving only two components, a triple constant 2/sqrt[3] often arises. Intriguingly, this constant is the same for the position and momentum case. Experimental verification is carried out on a single spin in diamond, and the results confirm the triple constant in a wide range of experimental parameters.
Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.
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​.
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 focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.