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Journal of Modern Optics

Quasi-Hermitian Hamiltonians, minimum-uncertainty (MU) angular momentum states and interferometers

Y. Ben-Aryeh

Published Dec 6, 2007• 3 citations
Publisher

Abstract

Minimum-uncertainty (MU) angular momentum states which are eigenstates of a nonunitary angular momentum displacement operator are found also to be eigenstates of a quasi-Hermitian Hamiltonian which has a certain metric with real eigenvalues. Various properties of these states are discussed. The relations between electromagnetic (EM) MU angular momentum states which have been proposed for improving the resolution in interferometers and the present theory are discussed.

DOI
10.1080/09500340701403152

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Minimum-uncertainty (MU) angular momentum states which are eigenstates of a nonunitary angular momentum displacement operator are found also to be eigenstates of a quasi-Hermitian Hamiltonian which has a certain metric with real eigenvalues. Various properties of these states are discussed. The relations between electromagnetic (EM) MU angular momentum states which have been proposed for improving the resolution in interferometers and the present theory are discussed.
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.
Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum. \textcopyright{} 1996 The American Physical Society.
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.
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.