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

We propose proper quantization rule, ∫ x A x B k ( x ) d x − ∫ x 0 A x 0 B k 0 ( x ) d x = n π , where k ( x ) = 2 M [ E − V ( x ) ] / ℏ . The x A and x B 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 o...

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Bibliographic Details
Published in:Journal of mathematical physics 2010-08, Vol.51 (8), p.082103-082103-16
Main Authors: Serrano, F. A., Gu, Xiao-Yan, Dong, Shi-Hai
Format: Article
Language:English
Subjects:
ASYMMETRY
ATOMIC AND MOLECULAR PHYSICS
ATOMS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
ELEMENTS
ENERGY LEVELS
ENERGY SPECTRA
Exact sciences and technology
EXACT SOLUTIONS
FUNCTIONS
GROUND STATES
HARMONIC OSCILLATORS
HULTHEN POTENTIAL
HYDROGEN
INTEGRALS
Mathematical functions
Mathematical methods in physics
MATHEMATICAL SOLUTIONS
Mathematics
MORSE POTENTIAL
NONMETALS
NUCLEAR POTENTIAL
ONE-DIMENSIONAL CALCULATIONS
Oscillators
Physics
POTENTIALS
QUANTIZATION
Quantum physics
Quantum theory
Sciences and techniques of general use
SPECTRA
WAVE FUNCTIONS
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Summary:We propose proper quantization rule, ∫ x A x B k ( x ) d x − ∫ x 0 A x 0 B k 0 ( x ) d x = n π , where k ( x ) = 2 M [ E − V ( x ) ] / ℏ . The x A and x B 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 ∫ x A x B k ( x ) d x 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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3466802