Proper quantization rule

Abstract

We find a proper quantization rule,where n is the number of the nodes of wave function ψ(x). By this rule the energy spectra of a solvable system can be determined from its ground-state energy only. Particularly, we study three solvable quantum systems -modified Rosen-Morse potential, symmetric trigonometric Rosen-Morse potential and Manning-Rosen potential in D dimensions-with the proper quantization rule, and show that the previous complicated and tedious calculations can be greatly simplified. This proper quantization rule applies to any exactly solvable potential, and one can easily obtain its energy spectra with the rule. This work is dedicated to Professor Zhong-Qi Ma on the occasion of his 70th birthday.

Solution of Klein Gordon Equation for Some Diatomic Molecules with New Generalized Morse-like Potential Using SUSYQM

“…[1][2][3][4][5] The bound state solution of Klein Gordon equation (KGE) is of great important in nuclear and high energy physics. 6 Klein Gordon equation is a relativistic wave equation that describes spinzero particles.…”

An alternative solution of diatomic molecules

“…Additionally, the proper quantization rule [27,28], the algebraic method [29][30][31][32][33] and factorization method [34] are used to solve the corresponding systems.…”

Quantization Rule for Relativistic Klein-Gordon Equation

“…[1][2][3][4][5] The one-dimensional nonrelativistic Schrödinger equation for a bound state is (1) where m is the mass of the particle and is the Planck constant divided by 2π. n is the quantum number or a number of node in wave function Ψ n (x).…”

“…Nonetheless for exactly solvable potentials, the exact energies obtained from the quantization rule have been reported. [5][6][7][8][9][10] The quantization rule cannot be algebraically derived from the Schrödinger equation but the Schrödinger equation is utilized in the process of deducing the rule.…”