Radial position-momentum uncertainties for the infinite circular well and Fisher entropy

Abstract

We show how the product of the radial position and momentum uncertainties can be obtained analytically for the infinite circular well potential. Some interesting features are found. First, the uncertainty Δr increases with the radius R and the quantum number n, the n-th root of the Bessel function. The variation of the Δr is almost independent of the quantum number n for n>4 and it will arrive to a constant for a large n, say n>4. Second, we find that the relative dispersion Δr/〈r〉 is independent of the radius R. Moreover, the relative dispersion increases with the quantum number n but decreases with the azimuthal quantum number m. Third, the momentum uncertainty Δp decreases with the radius R and increases with the quantum numbers m>1 and n. Fourth, the product ΔrΔpr of the position-momentum uncertainty relations is independent of the radius R and increases with the quantum numbers m and n. Finally, we present the analytical expression for the Fisher entropy. Notice that the Fisher entropy decreases with the radius R and it increases with the quantum numbers m>0 and n. Also, we find that the Cramer–Rao uncertainty relation is satisfied and it increases with the quantum numbers m>0 and n, too.