Quantization Rule for Relativistic Klein-Gordon Equation

Abstract

Based on the exact quantization rule for the nonrelativistic Schrödinger equation, the exact quantization rule for the relativistic one-dimensional Klein-Gordon equation is suggested. Using the new quantization rule, the exact relativistic energies for exactly solvable potentials, e.g. harmonic oscillator, Morse, and Rosen-Morse II type potentials, are obtained. Consequently the new quantization rule is found to be exact for one-dimensional spinless relativistic quantum systems. Though the physical meanings of the new quantization rule have not been fully understood yet, the present formal derivation scheme may shed light on understanding relativistic quantum systems more deeply.

Perspective Chapter: Relativistic Treatment of Spinless Particles Subject to a Class of Multiparameter Exponential-Type Potentials

“…For this reason, in the quantum mechanics treatment of this kind of potentials, the method that is most often used to find the bound states solutions is the Nikiforov-Uvarov method [1], which is based on solving a hypergeometric-type differential equation (DE) by means of special orthogonal functions. Albeit, other procedures such as Asymptotic Iteration [2], Supersymmetric Quantum Mechanics [3], He's Variational iteration [4], large-N solutions [5] or Quantization-rule [6], among many other methods, have been also employed in both non-relativistic and relativistic studies; obviously, including numerical solutions [7]. In the relativistic studies of spinless particles, it is well known that the Klein-Gordon equation [8,9] can always be reduced to a Schrödinger-type equation when the Lorentz-scalar and vector potential are equal [10].…”

The Information-Theoretic Treatment of Spinless Particles with the Assorted Diatomic Molecular Potential

“…Setting Τ = 0 yields the modified hyperbolic-type potential, suggested by Schiöberg as an important diatomic molecular potential [82]. This is strongly associated with the Morse [83], Kratzer [84], Coulomb [85,86], harmonic oscillator [87,88], and other potential functions as specific cases and is widely used in various applications such as quantum statistical theory [89], conformal field theory [90], nuclear structure [91], and chemical physics [92]. Thus, given this condition, the energy eigen-spectrum may be calculated as…”