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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Uncertainty relation for N fermions

David M. Goodmanson, John R. Taylor, Enrique Campesino-Romeo

Published Jan 15, 1979• 1 citation
Publisher

Abstract

We define uncertainties in position and momentum for states of $N$ identical fermions of spin $s$. We prove that in a one-dimensional world the product of these uncertainties is greater than or equal to $[\frac{N}{(2s+1)}]\frac{\ensuremath{\hbar}}{2}$. We generalize this to arbitrary numbers of dimensions; in particular, the corresponding result in three dimensions has the factor $\frac{\ensuremath{\hbar}}{2}$ replaced by $(\frac{3}{4}){(\frac{\ensuremath{\hbar}}{2})}^{3}$.

DOI
10.1103/physrevd.19.485

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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.
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Abstract For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by N linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an N -dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle.