Abstract
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.
Original language | English |
---|---|
Pages (from-to) | 771-782 |
Number of pages | 12 |
Journal | Annalen der Physik |
Volume | 523 |
Issue number | 10 |
DOIs | |
State | Published - Oct 2011 |
Keywords
- Proper quantization rule
- bound states
- solvable potentials.