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New Journal of Physics

Minimum uncertainty states of angular momentum and angular position

David Thomas Pegg, Stephen Mark Barnett, Roberta Zambrini, Sonja Franke‐Arnold, Miles J. Padgett

Published Feb 17, 2005• 41 citations
Publisher

Abstract

The states of linear momentum that satisfy the equality in the Heisenberg uncertainty principle for position and momentum, that is the intelligent states, are also the states that minimize the uncertainty product for position and momentum. The corresponding uncertainty relation for angular momentum and angular position, however, is more complicated and the intelligent states need not be the constrained minimum uncertainty product states. In this paper, we investigate the differences between the intelligent and the constrained minimum uncertainty product states for the angular case by means of instructive approximations, a numerical iterative search and the exact solution. We find that these differences can be quite significant for particular values of angular position uncertainty and indeed may be amenable to experimental measurement with the present technology.

DOI
10.1088/1367-2630/7/1/062

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The states of linear momentum that satisfy the equality in the Heisenberg uncertainty principle for position and momentum, that is the intelligent states, are also the states that minimize the uncertainty product for position and momentum. The corresponding uncertainty relation for angular momentum and angular position, however, is more complicated and the intelligent states need not be the constrained minimum uncertainty product states. In this paper, we investigate the differences between the intelligent and the constrained minimum uncertainty product states for the angular case by means of instructive approximations, a numerical iterative search and the exact solution. We find that these differences can be quite significant for particular values of angular position uncertainty and indeed may be amenable to experimental measurement with the present technology.
The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty product. In this paper, we highlight the difference between intelligent states and minimum uncertainty states by investigating a class of intelligent states which obey the equality in the angular uncertainty relation while having an arbitrarily large uncertainty product. To develop an understanding for the uncertainties of angle and angular momentum for the large-uncertainty intelligent states we compare exact solutions with analytical approximations in two limiting cases.
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.
Uncertainty relations are key components in the understanding of the nature of quantum mechanics. In particular, entropic relations are preferred in the study of angular position and angular momentum states. We propose a new form of angle–angular momentum state that provides, for all practical purposes, a lower bound on the entropic uncertainty relation, , for any given angular uncertainty, thus improving upon previous bounds. We establish this by comparing this sum with the absolute minimum value determined by a global numerical search. These states are convenient to work with both analytically and experimentally, which suggests that they may be of use for quantum information purposes.
We present an accurate description of the conjugate pair angle-angular momentum in terms of the exponential of the angle instead of the angle itself, which leads to dispersion as a natural measure of resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum turn out to be given by Mathieu wave functions. We discuss Gaussian approximations to these optimal states in terms of von Mises distributions. The theory is successfully applied to the spatial degrees of freedom of a photon and verified in an experiment that employs computer-controlled spatial light modulators at both the state preparation and the analyzing stages.
I reexamine Heisenberg’s uncertainty relation for two- and three-dimensional wave packets with fixed angular momentum quantum numbers m or ℓ. A simple proof shows that the product of the average extent Δr and Δp of a two-dimensional wave packet in position and momentum space is bounded from below by ΔrΔp≥ℏ(|m|+1). The minimum uncertainty is attained by modified Gaussian wave packets that are special eigenstates of the two-dimensional isotropic harmonic oscillator, which include the ground states of electrons in a uniform magnetic field. Similarly, the inequality ΔrΔp≥ℏ(ℓ+3/2) holds for three-dimensional wave packets with fixed total angular momentum ℓ and the equality holds for a Gaussian radial profile. I also discuss some applications of these uncertainty relations.