Proper quantization rule as a good candidate to semiclassical quantization rules

In this article, we present proper quantization rule, ∫_xA ^xBk(x) dx - ∫^x0A _x0B k₀(x) dx = nπ, where $k(x) = 2 M [E-V(x) ] h 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.