Proper quantization rule

Abstract

We find a proper quantization rule, ∫xAxBk(x)dx- ∫x0Ax0Bk0(x)dx=nπ, 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.