Derivation of A Conserved Linear Energy and Klein Gordon Equations to Describe Spin and Magic Numbers Using Generalized Special Relativity

Abstract

The concept of energy in special relativity does not satisfy Newton limit. It does not also conforms to observations. The aim of this work is to use generalized special relativity to formulate a new expression for the linear energy. One also needs to construct a new quantum relativistic Klein Gordon equation, which can describe spin phenomena instead of using two different quantum equations to describe particles having integral and half-integral spin quantum numbers, one is the Klein- Gordon equation, while the other is the Dirac equation. The energy conservation of the potential dependent special relativity has been developed. The energy expression can be simplified and the conservation can be secured when using vector four-dimensional representation with the fourth time component is imaginary and related to the momentum. Treating particles as strings an imaginary energy has been found to be quantized and proportional to the harmonic energy .This resembles the imaginary wave number reflecting the energy liberated by electromagnetic waves that interact with matter. Potential dependent energy- momentum relativistic relation was utilized to derive new quantum Klein- Gordon equation. This equation reduces to the ordinary Klein- Gordon equation in the absence of potential. Treating nucleons as strings, a new energy-quantized expression was obtained. This energy resembles that of Schrödinger harmonic oscillator with additional term representing the rest mass. This model also predicts the magic numbers. The conservation of energy in potential dependent special relativity has been found using 4- dimensional representation. In this version, the square of the momentum multiplied by the square of the free speed of light subtracted from the curved space energy is invariant and constant everywhere. The quantum equation derived from this equation in a weak field limit shows that the momentum is quantized and the energy is reduced to that of Schrodinger harmonic oscillator when neglecting the rest mass term. In addition, the proposed model shows the possibility of using potential dependent Klein-Gordon equation to find the magic numbers. This shows that, this equation can describe fermions as well as bosons.