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arXiv:1411.0823 · Nanosystems Physics Chemistry Mathematics

Uncertainty relation between angle and orbital angular momentum: interference effect in electron vortex beams

Sh. Tanimura

Published Apr 16, 2015• 3 citations
arXivHTMLPDF

Abstract

The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator.Instead of the angle variable operator, we introduce the complex position operator Ẑ = x + iŷ and interpret the order parameter µ = Ẑ / Ẑ † Ẑ as a measure of certainty of the angle distribution.We prove the relation between the uncertainty of angular momentum and the angle order parameter.We also prove its generalizations and discuss experimental methods for testing these relations.

DOI
10.17586/2220-8054-2015-6-2-205-212

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The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator.Instead of the angle variable operator, we introduce the complex position operator Ẑ = x + iŷ and interpret the order parameter µ = Ẑ / Ẑ † Ẑ as a measure of certainty of the angle distribution.We prove the relation between the uncertainty of angular momentum and the angle order parameter.We also prove its generalizations and discuss experimental methods for testing these relations.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employing the exponential of the angle instead of the angle itself, which leads to circular variance 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 also discuss a number of physically feasible approximations to these optimal states. The theory is applied to the orbital angular momentum of a beam of photons and verified in an experiment that employs computer-controlled spatial light modulators at both the state preparation and analyzing stages.
The violation of local uncertainty relations is a valuable tool for detecting entanglement, especially in multi-dimensional systems. The orbital angular momentum of light provides such a multi-dimensional system. We study quantum correlations for the conjugate variables of orbital angular momentum and angular position. We determine an experimentally testable criterion for the demonstration of an angular version of the EPR paradox. For the interpretation of future experimental results from our proposed setup, we include a model for the indeterminacies inherent to the angular position measurement. For this measurement angular apertures are used to determine the probability density of the angle. We show that for a class of aperture functions a demonstration of an angular EPR paradox, according to our criterion, is to be expected.
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.