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We find a proper quantization rule,where n is the number of the nodes of wave function ψ(x). By this rule the energy spectra of a solvable system can be determined from its ground-state energy only. Particularly, we study three solvable quantum systems -modified Rosen-Morse potential, symmetric trigonometric Rosen-Morse potential and Manning-Rosen potential in D dimensions-with the proper quantization rule, and show that the previous complicated and tedious calculations can be greatly simplified. This proper quantization rule applies to any exactly solvable potential, and one can easily obtain its energy spectra with the rule. This work is dedicated to Professor Zhong-Qi Ma on the occasion of his 70th birthday.

The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position S x and momentum S p Shannon information entropies for the low-lying states = n 0, 1 are calculated. Some interesting features of the information entropy densities ρ x ( )and ρ p ( ) s as well as the probability densities ρ x ( ) and ρ p ( ) are demonstrated. We find that the choices of the values for those parameters have to satisfy the condition on n max . We also notice that the ρ p ( ) and ρ p ( ) s are symmetric to the momentum p and the ρ x ( ) or ρ p ( ) is equal or greater than 1 at some positions r or momentum p. In addition, the Bialynicki-Birula-Mycielski inequality is tested from different cases and found to hold for these cases.

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