Radial position–momentum uncertainties for the infinite spherical well and the Fisher entropy

Abstract

We present the analytical expression of the product of the radial position and momentum uncertainties and for the infinite spherical well. We find a few interesting features. First, the uncertainty increases with the radius R and the quantum number n, the nth root of the spherical Bessel function, but finally arrives at a constant for a large n and decreases with the angular momentum number l. It is seen that the becomes imaginary, which arises from the fact that the moving particle is shifted to axis y suddenly from the original axis x as the quantum number n increases. Furthermore, the becomes zero when the n increases for a given l. This means that the particle is spiraling around a circle whose radius r < R changes between a varying radius and a constant but with an increasing radius r as the n increases. Finally, the particle moves around a circle with a maximum radius R. Second, the relative dispersion is independent of the radius R, and it increases with the quantum number n but decreases with the quantum number l. Third, the radial momentum uncertainty decreases with the radius R and increases with the quantum numbers l > 0 and n. We notice that there exists a turning point for the uncertainty when l = 1 and n > 1. This also leads to the product . Fourth, the product is independent of the radius R and increases with the quantum numbers l > 0 and n. Finally, we obtain the analytical expression of the Fisher entropy and notice that it decreases with the radius R but increases with the quantum numbers l and n. We also find that the Cramer–Rao uncertainty relation is satisfied, and it increases with the quantum numbers l and n.