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Journal of the Korean Physical Society

Relativistic spinless particles with position dependent mass: Bound states and scattering phase shifts

Ahmet Taş, Oktay Aydogdu A, Mustafa Saltı

Published May 1, 2017• 9 citations
Publisher

Abstract

The scattering and bound states of relativistic spinless particles with position-dependent mass are investigated in the framework of the improved Frost-Musulin potential, which is used to explain some properties of diatomic molecules. After we obtain approximate analytical solution of the Klein-Gordon equation, we find the normalized wave functions in terms of the hypergeometric functions in order to deal with the bound and the scattering states. We also analyze the scattering phase shifts and the bound states numerically.

DOI
10.3938/jkps.70.896

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The scattering and bound states of relativistic spinless particles with position-dependent mass are investigated in the framework of the improved Frost-Musulin potential, which is used to explain some properties of diatomic molecules. After we obtain approximate analytical solution of the Klein-Gordon equation, we find the normalized wave functions in terms of the hypergeometric functions in order to deal with the bound and the scattering states. We also analyze the scattering phase shifts and the bound states numerically.
We have obtained the approximate solutions of the Klein–Gordon equation with the Yukawa ring-shaped potential, by using the Nikiforov–Uvarov method for a special case of equal scalar and vector potentials. The energy eigenvalues for bound states and the corresponding wave functions are also obtained in a proper approximation. We have also shown that the results can be used to evaluate the energy eigenvalues of the Yukawa, angle-dependent, and Coulomb potentials. The numerical results are discussed and presented in the table and in the figure, which suggest their applicability to other systems. With the adjusted potential parameters given in the table, it is shown that the interaction of spinless (Klein–Gordon) particles with the Yukawa ring-shaped potential gives positive energy eigenvalues for the various quantum states.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.