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arXiv:1407.0083 · Physical Review A

Heisenberg uncertainty relation for three canonical observables

Spiros Kechrimparis, Stefan Weigert

Published Dec 15, 2014• 51 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.90.062118

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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.
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 .
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An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.
Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.
We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables [Formula: see text] and [Formula: see text] and such vectors that the lower bound for the product of standard deviations [Formula: see text] and [Formula: see text] calculated for these vectors is zero: [Formula: see text]. We also show that for some pairs of non-commuting observables the sets of vectors for which [Formula: see text] can be complete (total). The Heisenberg, [Formula: see text], and Mandelstam–Tamm (MT), [Formula: see text], time–energy uncertainty relations ([Formula: see text] is the characteristic time for the observable [Formula: see text]) are analyzed too. We show that the interpretation [Formula: see text] for eigenvectors of a Hamiltonian [Formula: see text] does not follow from the rigorous analysis of MT relation. We show also that contrary to the position–momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of [Formula: see text] and [Formula: see text].
Uncertainty relations play a fundamental role in quantum mechanics by quantifying the intrinsic limitations on the simultaneous sharpness of incompatible observables. Beyond the standard two-observable product form, additive uncertainty relations for triples of observables provide a natural framework for describing collective constraints among three noncommuting components. In this work, we study an additive uncertainty relation for three Hermitian observables from the viewpoint of rotational symmetry and covariance geometry. We give a short rotational derivation by rotating the observable triple and applying the Robertson uncertainty relation to the two transverse observables. This derivation makes the saturation mechanism transparent and leads to a necessary and sufficient condition for equality for general density operators. In the nontrivial equality case, the covariance ellipsoid of the observable triple degenerates into a disk perpendicular to the expectation value of the commutator vector. We also discuss an inverse construction based on finite-dimensional representations of the Lie algebra su(2)\mathfrak{su}(2)su(2), which provides a systematic way to construct observable triples with prescribed saturating states. These results clarify the geometric and representation-theoretic structure underlying the tightness of additive three-observable uncertainty relations.