Qiang–Dong proper quantization rule and its applications to exactly solvable quantum systems

Abstract

We propose proper quantization rule, ∫xAxBk(x)dx−∫x0Ax0Bk0(x)dx=nπ, where k(x)=2M[E−V(x)]/ℏ. The xA and xB are two turning points determined by E=V(x), and n is the number of the nodes of wave function ψ(x). We carry out the exact solutions of solvable quantum systems by this rule and find that the energy spectra of solvable systems can be determined only from its ground state energy. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. 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 1, the momentum integral ∫xAxBk(x)dx will increase by π. We apply this proper quantization rule to carry out solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization, the Hulthén potential, the Scarf II potential, the asymmetric trigonometric Rosen–Morse potential, the Pöschl–Teller type potentials, the Rosen–Morse potential, the Eckart potential, the harmonic oscillator in three dimensions, the hydrogen atom, and the Manning–Rosen potential in D dimensions.