Proper quantization rule as a good candidate to semiclassical quantization rules

• Published in: Annalen der Physik 2011-10, Vol.523 (10), p.771-782
• Main Authors: Serrano, F.A., Cruz-Irisson, M., Dong, S.-H.
• Format: Article
• Language: English
• Subjects: Asymmetry; bound states; Derivatives; Helium; Integrals; Mathematical analysis; Morse potential; Proper quantization rule; Quantization; solvable potentials; Wave functions
• ISSN: 0003-3804; 1521-3889
• DOI: 10.1002/andp.201000144

In this article, we present proper quantization rule, ∫ x A x Bk(x) dx ‐ ∫ x 0A x 0Bk0(x) dx = nπ, where $k(x) = \sqrt{2 M [E-V(x) ]}/\hbar$ 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. In this article, the authors present proper quantization rule, ∫ x A x Bk(x) dx ‐ ∫ x 0A x 0Bk0(x) dx = nπ, where $k(x) = \sqrt{2 M [E-V(x) ]}/\hbar$ and study solvable potentials. They 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. The authors 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.