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Angular Operators Violating the Heisenberg Uncertainty Principle

Tiago Pereira, Domingos H. U. Marchetti

Published May 9, 2008• 0 citations
OpenAlex

Abstract

The description of a quantum system in terms of angle variables may violate Heisenberg uncertainty principle. The familiar case is the azimutal angle $\phi$ and its canonical moment $L_z$. Although this problem was foreseen almost a century ago, up to the present days there are no criteria to precisely characterize the violation. In this paper, we present a theorem which provides necessary and sufficient conditions for the violation of the Heisenberg uncertainty principle. We illustrate our results with analytical examples.

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The description of a quantum system in terms of angle variables may violate Heisenberg uncertainty principle. The familiar case is the azimutal angle ϕ\phiϕ and its canonical moment LzL_zLz​. Although this problem was foreseen almost a century ago, up to the present days there are no criteria to precisely characterize the violation. In this paper, we present a theorem which provides necessary and sufficient conditions for the violation of the Heisenberg uncertainty principle. We illustrate our results with analytical examples.
A one-dimensional wave function that is defined in a bounded close interval violates the Heisenberg uncertainty principle. It was analyzed in detail. The author examined the properties of Hermitian operators and reported that an equation is the origin of the violation of the uncertainty relationship. It was also found that the boundary conditions play a crucial role in the violation of the uncertainty principle, i.e., the periodic boundary condition and the Dirichlet condition, which make a wave function zero at boundaries, yield different results. A novel elementary proof of Robertson’s uncertainty relationship, which does not use the Cauchy-Schwarz inequality, was provided in the present paper. The paper notes that if the commutation relationship between two Hermitian operators is given, there might be a case in which the fundamental properties may not be necessarily satisfied. Therefore, even if a noncommutative commutation relationship is given, the uncertainty relationship might not be valid.
We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase space (even if it violates the Robertson-Schr\"odinger uncertainty principle) will always be a quantum state of an appropriate noncommutative extension of quantum mechanics. Conversely, all canonical noncommutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle.
While there is a rigorously proven relationship about uncertainties intrinsic to any quantum system, often referred to as "Heisenberg's uncertainty principle," Heisenberg originally formulated his ideas in terms of a relationship between the precision of a measurement and the disturbance it must create. Although this latter relationship is not rigorously proven, it is commonly believed (and taught) as an aspect of the broader uncertainty principle. Here, we experimentally observe a violation of Heisenberg's "measurement-disturbance relationship", using weak measurements to characterize a quantum system before and after it interacts with a measurement apparatus. Our experiment implements a 2010 proposal of Lund and Wiseman to confirm a revised measurement-disturbance relationship derived by Ozawa in 2003. Its results have broad implications for the foundations of quantum mechanics and for practical issues in quantum measurement.
Recently, violation of Heisenberg's uncertainty relation in spin measurements is discussed [J.Erhart et al., Nature Physics 8, 185 (2012)] and [G.Sulyok et al., Phys.Rev. A 88, 022110 (2013)].We derive the optimal limitation of Heisenberg's uncertainty principle in a specific two-level system (e.g., electron spin, photon polarizations, and so on).Some physical situation is that we would measure x σ and y σ , simultaneously.The optimality is certified by the Bloch sphere.We show that a violation of Heisenberg's uncertainty principle means a violation of the Bloch sphere in the specific case.Thus, the above experiments show a violation of the Bloch sphere when we use ±1 as measurement outcome.This conclusion agrees with recent researches [K.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.