Proper quantization rule as a good candidate to semiclassical quantization rules

Abstract

In this article, we present proper quantization rule, ∫k(x) dx - ∫k0(x) dx = nπ, where and study solvable potentials. We find that the energy spectra of solvable systems can be calculated only from its ground state obtained by the Sturm-Liouville theorem. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of proper quantization rule come from its meaning – whenever the number of the nodes of the logarithmic derivative ϕ(x) = ψ(x)-1dψ(x) /dx or the number of the nodes of the wave function ψ(x) increases by one, the momentum integral will increase by π. We apply two different quantization rules to carry out a few typically solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization as well as the asymmetrical trigonometric Scarf potential and show a great advantage of the proper quantization rule over the original exact quantization rule.