Volume 89, Number 1, January 2010
Article Number 10003
Number of page(s) 4
Section General
Published online 01 January 2010
EPL, 89 (2010) 10003
DOI: 10.1209/0295-5075/89/10003

Proper quantization rule

Wen-Chao Qiang1 and Shi-Hai Dong2

1   Faculty of Science, Xi'an University of Architecture and Technology - Xi'an, 710055, China
2   Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edificio 9, Unidad Profesional Adolfo López Mateos - Mexico D. F. 07738, Mexico

received 28 September 2009; accepted in final form 7 December 2009; published January 2010
published online 4 January 2010

We find a proper quantization rule, $\int _{x_{A}}^{x_{B}}k(x){\rm d}x-
\int _{x_{0A}}^{x_{0B}}k_{0}(x){\rm d}x=n\pi $, where n is the number of the nodes of wave function $\psi (x)$. 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.

03.65.Fd - Algebraic methods.
34.20.Cf - Interatomic potentials and forces.
03.65.Ta - Foundations of quantum mechanics; measurement theory.

© EPLA 2010