Particle number and interactions in the entropic uncertainty relations

Abstract

Abstract The dependence of the entropic uncertainty relations on particle number and on attractive and repulsive interparticle potential strengths, is examined in the ground state of coupled harmonic oscillators. The reduced entropy sums depend on the normalization constants of the position and momentum densities, which in turn depend on particle number and interaction strength. These sums exhibit maxima as the particle number is varied, when the particles are in the presence of attractive potentials. On the other hand, no maxima are observed for systems with a repulsive potential. All reduced entropy sums increase and move away from the bound with greater interactions. Consistent behaviours are observed for both reduced one- and two-variable entropy sums. Further reductions of the N-particle densities induce that the respective entropy sums are now farther away from the corresponding bounds. The interpretation of these results is that reduction and increased interaction translates into a weakening of the strength of the uncertainty principle statement. The one- and two-variable entropy sums of these systems are shown to approximate the a + b ln N relation, when the intensity of the one-body harmonic potential is set at the number of particles.