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American Journal of Physics

Uncertainty relations for angular momentum eigenstates in two and three spatial dimensions

Christian Bracher

Published Feb 16, 2011• 13 citations

Abstract

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.

DOI: 10.1119/1.3534840

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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.

The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.

The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.

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.

Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.

The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.