From bohr-sommerfeld semiclassical quantization rule to Qiang-Dong proper quantization rule

We give a brief review of the fundamental development of the quantization rule from the Bohr-Sommerfeld semiclassical quantization rule to proper quantization rule. The latter was based on recently exact quantization rule proposed by Ma-Ou in 2004. This method has become one of several important formalisms to deal with solvable quantum systems, but the integrals, in particular the calculation of the quantum correction term involved in it become rather complicated. To overcome this difficulty, we have improved it and found a proper quantization rule(Formula presented), where (Formula presented) and x_B two turning points determined by E = V (x), and n is the number of the nodes of wave function Ψ (x) . By this rule the energy spectra of all solvable systems can be determined from its ground state energy only. The beauty and simplicity of the rule come from its meaning-whenever the number of the nodes of φ (x) or the number of the nodes of the wave function Ψ (x) increases by one, the momentum integral (formula presented) dx will increase by π. As illustrations, we first carry out the asymmetric trigonometric Rosen-Morse potential by exact quantization rule method and find that the integral calculations become rather complicated, and then we solve the modified Rosen-Morse potential, the Coulombic ring-shaped noncentral Hartmann system and the Manning- Rosen effective potential by the proper quantization rule. The evaluation of the Langer modification and Maslov index in D dimensions via the proper quantization rule are also studied.