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

Journal Article · Sun Aug 15 00:00:00 EDT 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3466802· OSTI ID:21476526

Serrano, F A ^[1]; Xiaoyan, Gu ^[2]; Dong, Shi-Hai ^[3]

---

1. Instituto Politecnico Nacional, ESIME-Culhuacan Av., Santa Ana 1000, Distrito Federal 04430 (Mexico)
2. Department of Physics, East China University of Science and Technology, Shanghai 200237 (China)
3. Escuela Superior de Fisica y Matematicas, Instituto Politecnico Nacional, Edificio 9, Unidad Profesional Adolfo Lopez Mateos, Mexico, Distrito Federal 07738 (Mexico)

We propose proper quantization rule, x{sub A}(x{sub B})k(x)dx-x{sub 0A}(x{sub 0B})k{sub 0}(x)dx=n{pi}, where k(x)={radical}(2M[E-V(x)])/({h_bar}/2{pi}). The x{sub A} and x{sub B} are two turning points determined by E=V(x), and n is the number of the nodes of wave function {psi}(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 {phi}(x) or the number of the nodes of the wave function {psi}(x) increases by 1, the momentum integral x{sub A}(x{sub B})k(x)dx will increase by {pi}. 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 Hulthen potential, the Scarf II potential, the asymmetric trigonometric Rosen-Morse potential, the Poeschl-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.

• OSTI ID: 21476526
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 8; Other Information: DOI: 10.1063/1.3466802; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMMETRY
ATOMS
ENERGY SPECTRA
EXACT SOLUTIONS
GROUND STATES
HARMONIC OSCILLATORS
HULTHEN POTENTIAL
HYDROGEN
INTEGRALS
MORSE POTENTIAL
ONE-DIMENSIONAL CALCULATIONS
QUANTIZATION
WAVE FUNCTIONS
ELEMENTS
ENERGY LEVELS
FUNCTIONS
MATHEMATICAL SOLUTIONS
NONMETALS
NUCLEAR POTENTIAL
POTENTIALS
SPECTRA