Europhys. Lett.
Volume 69, Number 5, March 2005
Page(s) 685 - 691
Section General
Published online 04 February 2005
Europhys. Lett., 69 (5), pp. 685-691 (2005)
DOI: 10.1209/epl/i2004-10418-8

Quantum correction in exact quantization rules

Zhong-Qi Ma1 and Bo-Wei Xu2

1  Institute of High Energy Physics - P. O. Box 918(4), Beijing 100049, PRC
2  Department of Physics, Shanghai Jiaotong University - Shanghai 200030, PRC

received 22 September 2004; accepted in final form 3 January 2005
published online 4 February 2005

An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to $N\pi$, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom.

03.65.Ge - Solutions of wave equations: bound states.
03.65.Fd - Algebraic methods.

© EDP Sciences 2005