Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Physical significance of minimum uncertainty states of an angular momentum system

H. Bacry

Published Aug 1, 1978• 18 citations
Publisher

Abstract

The minimization of the uncertainty relation $2\ensuremath{\Delta}x\ensuremath{\Delta}p\ensuremath{\ge}\ensuremath{\hbar}$ characterizes the coherent states. The spincoherent states of Radcliffe minimize the uncertainty relation $2\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}})\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})\ensuremath{\ge}|〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}〉|$ (where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}$ form an arbitrary orthonormal basis), but this property does not characterize them. Here, we analyze that situation and suggest an alternative to restore the parallelism.

DOI
10.1103/physreva.18.617

Related papers

Minimum uncertainty states of angular momentum and angular position

Roberta Zambrini, D.T. Pegg, S.M. Barnett +2 more

In this paper, the much less known uncertainty relation for angular position and momentum to find the system at the edge of the phase interval is considered. States that minimize the uncertainty product with a constraint given either by the angular variance or by the angular momentum variance is examined.

European Quantum Electronics Conference (EQEC)•Jan 18, 2006•3 citations

Angular Momentum Uncertainty Relation and the Three-Dimensional Oscillator in the Coherent States

Michael Martin Nieto

An angular momentum uncertainty relation is obtained in terms of sine and cosine operators that have a meaning even in a second-quantization formalism. For a threedimensional oscillator in coherent states the new uncertainty product is a minimum for large $\ensuremath{\Delta}{L}_{z}$. Even for small $\ensuremath{\Delta}{L}_{z}$ the uncertainty product is very small.

Physical Review Letters•Jan 30, 1967•16 citations

Entropic uncertainty minimum for angle and angular momentum

Alison M. Yao, Thomas Brougham, Electra Eleftheriadou +2 more

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.

Journal of Optics•Sep 23, 2014•1 citation

Angular minimum uncertainty states with large uncertainties

Jörg B. Götte, PAUL M. RADMORE, Roberta Zambrini +1 more

The uncertainty relation for angle and angular momentum has a lower bound which depends on the form of the state. Surprisingly, this lower bound can be very large. We derive the states which have the lowest possible uncertainty product for a given uncertainty in the angle or in the angular momentum. We show that, if the given angle uncertainty is close to its maximum value, the lowest possible uncertainty product tends to infinity.

arXiv:quant-ph/0512077•Journal of Physics B Atomic Molecular and Optical Physics•May 30, 2006•5 citations

On the uncertainty relation in the coherent spin-state representation

L Kolodziejczyk, A Ryter

The product ( delta s x ) 2 ( delta s y ) 2 is computed where ( delta s x,y ) 2 =(s x,y 2 )-(s x,y ) 2 ; averaging is performed in the coherent spin-state representation given by Radcliffe (1971). After applying the Holstein- Primakoff transformation s - =(2s) 1 2/a Dagger , s + =(2s) 1 2/a and mu = alpha /(2s) 1 2/, and putting s to infinity one proceeds from Radcliffe space into the Glauber space. After this procedure the product ( delta s x ) 2 ( delta s y ) 2 becomes ( delta x) 2 ( delta p) 2 . Using the Jackiw equation it is shown that the function mod mu =0> is the only one which minimizes the uncertainty product, for every s.

Journal of Physics A Mathematical Nuclear and General•Jan 21, 1974•6 citations

Minimum uncertainty states of angular momentum and angular position

David Thomas Pegg, Stephen Mark Barnett, Roberta Zambrini +2 more

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.

New Journal of Physics•Feb 17, 2005•41 citations
The minimization of the uncertainty relation 2\ensuremathΔx\ensuremathΔp\ensuremath≥\ensuremathℏ2\ensuremath{\Delta}x\ensuremath{\Delta}p\ensuremath{\ge}\ensuremath{\hbar}2\ensuremathΔx\ensuremathΔp\ensuremath≥\ensuremathℏ characterizes the coherent states. The spincoherent states of Radcliffe minimize the uncertainty relation 2\ensuremathΔ(J\ensuremath→\ifmmode⋅\else\textperiodcentered\fiu\ensuremath→)\ensuremathΔ(J\ensuremath→\ifmmode⋅\else\textperiodcentered\fiv\ensuremath→)\ensuremath≥∣〈J\ensuremath→\ifmmode⋅\else\textperiodcentered\fiw\ensuremath→〉∣2\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}})\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})\ensuremath{\ge}|〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}〉|2\ensuremathΔ(J\ensuremath→​\ifmmode⋅\else\textperiodcentered\fiu\ensuremath→)\ensuremathΔ(J\ensuremath→​\ifmmode⋅\else\textperiodcentered\fiv\ensuremath→)\ensuremath≥∣〈J\ensuremath→​\ifmmode⋅\else\textperiodcentered\fiw\ensuremath→〉∣ (where u\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}}u\ensuremath→, v\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}v\ensuremath→, w\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}w\ensuremath→ form an arbitrary orthonormal basis), but this property does not characterize them. Here, we analyze that situation and suggest an alternative to restore the parallelism.
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.
The product ( delta s x ) 2 ( delta s y ) 2 is computed where ( delta s x,y ) 2 =(s x,y 2 )-(s x,y ) 2 ; averaging is performed in the coherent spin-state representation given by Radcliffe (1971). After applying the Holstein- Primakoff transformation s - =(2s) 1 2/a Dagger , s + =(2s) 1 2/a and mu = alpha /(2s) 1 2/, and putting s to infinity one proceeds from Radcliffe space into the Glauber space. After this procedure the product ( delta s x ) 2 ( delta s y ) 2 becomes ( delta x) 2 ( delta p) 2 . Using the Jackiw equation it is shown that the function mod mu =0> is the only one which minimizes the uncertainty product, for every s.
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.