| Resumen |
We present the analytical expression of the product of the radial position and momentum uncertainties Delta r and Delta p(r) for the infinite spherical well. We find a few interesting features. First, the uncertainty Delta r 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 Delta r 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 Delta r 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 Delta r/< r > 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 Delta p(r) 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 Delta p(r) when l = 1 and n > 1. This also leads to the product Delta r Delta p(r) . Fourth, the product Delta r Delta p(r) 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 land n. We also find that the Cramer-Rao uncertainty relation is satisfied, and it increases with the quantum numbers l and n. |
