[Paper Review] Stationary Solutions of the Klein-Gordon Equation in a Potential Field

Guang-Qing Bi, Yuekai Bi|arXiv (Cornell University)|Aug 25, 2010

Quantum chaos and dynamical systems11 references3 citations

TL;DR

This paper presents a relativistic formulation of the Klein-Gordon equation in a potential field by introducing the concept of 'system mass' to explicitly account for mass defect and binding energy. By defining stationary solutions as Ψ(r,t) = ψ(r)e^(-iEt/ℏ) and equating m = E/c², the authors derive a form of the Klein-Gordon equation that mirrors the Schrödinger equation under equal scalar and vector potentials, proving these solutions possess probability significance and enabling exact solutions via the same methods used for the Schrödinger equation.

ABSTRACT

We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.

Motivation & Objective

• To establish a rigorous relativistic wave equation framework that explicitly incorporates mass defect and binding energy via the concept of system mass.
• To demonstrate that stationary solutions of the Klein-Gordon equation in a potential field retain probability significance, analogous to the Schrödinger equation.
• To prove that when scalar and vector potentials are equal, exact bound state solutions of the Klein-Gordon equation can be obtained using the same method as for the Schrödinger equation.
• To extend the applicability of exact solutions from the Schrödinger equation to the Klein-Gordon equation under equal scalar and vector potentials, including scattering states.
• To provide a unified mathematical approach for solving relativistic bound and scattering states by leveraging known non-relativistic solution techniques.

Proposed method

• Defining the system mass as m = m₀ + E′/c², where E′ is the sum of kinetic and potential energy, to incorporate mass defect explicitly in relativistic systems.
• Using the relativistic energy-momentum relation E² = c²p² + m₀²c⁴ as the characteristic equation for the base functions of quantum mechanics.
• Deriving the Klein-Gordon equation in a potential field by applying Lorentz transformations to wave functions and momentum operators between inertial frames.
• Introducing the stationary state ansatz Ψ(r,t) = ψ(r)e^(-iEt/ℏ) to reduce the relativistic wave equation to a time-independent form.
• Establishing a formal similarity between the resulting equation and the Schrödinger equation under the condition S(r) = U(r), where S and U are scalar and vector potentials.
• Applying the same mathematical techniques used to solve the Schrödinger equation to the transformed Klein-Gordon equation, enabling exact solutions for potentials like Hulthén and Manning-Rosen.

Experimental results

Research questions

• RQ1 Can the Klein-Gordon equation in a potential field be reformulated in a way that explicitly accounts for mass defect and binding energy?
• RQ2 Do stationary solutions of the Klein-Gordon equation in a potential field retain the same probability interpretation as in the Schrödinger equation?
• RQ3 Under what conditions can exact bound state solutions of the Klein-Gordon equation be obtained using the same method as for the Schrödinger equation?
• RQ4 Can the exact solution technique for the Schrödinger equation with the Hulthén potential be extended to the Klein-Gordon equation under equal scalar and vector potentials?
• RQ5 Is there a formal correspondence between the radial Klein-Gordon equation and the radial Schrödinger equation when scalar and vector potentials are equal?

Key findings

• The Klein-Gordon equation in a potential field can be reformulated using the concept of system mass m = m₀ + E′/c², which explicitly accounts for mass defect and binding energy.
• Stationary solutions of the form Ψ(r,t) = ψ(r)e^(-iEt/ℏ) satisfy the Klein-Gordon equation and possess probability significance, analogous to the Schrödinger equation.
• When scalar and vector potentials are equal (S(r) = U(r)), the Klein-Gordon equation reduces to a form formally identical to the Schrödinger equation: E′ψ = −(ℏ²/(m₀ + m))∇²ψ + 2Uψ.
• Under equal scalar and vector potentials, exact bound state solutions of the Klein-Gordon equation can be obtained using the same method as for the Schrödinger equation, as demonstrated for Hulthén and Manning-Rosen potentials.
• The radial Klein-Gordon equation with the Hulthén potential takes the form d²u/dr² + (m₀ + m)/ℏ² [E′ + 2Ze²λ e^(-λr)/(1−e^(-λr)) − l(l+1)ℏ²/((m₀ + m)r²)] u(r) = 0, which is solvable by the same method as the Schrödinger equation.
• The exact solution technique for scattering states of the Schrödinger equation with the Hulthén potential can be extended to the Klein-Gordon equation under equal scalar and vector potentials, preserving the solvability.