EIGENVALUES OF KLEIN-GORDON EQUATION ON EQUAL VECTOR AND SCALAR COULOMB POTENTIAL

ABSTRACT The Klein-Gordon equation, a fundamental relativistic wave equation in quantum mechanics, describes the dynamics of spinless bosonic particles. This thesis presents an analytical investigation of the eigenvalues of the Klein-Gordon equation in the presence of equal vector and scalar Coulomb potentials, a physically significant configuration commonly encountered in nuclear, particle, and atomic physics. The condition of equal scalar and scalar potential, S(r) = V(r) = -Zα/r, simplifies the Klein-Gordon equation by eliminating the scalar potential from the rest mass term, leading to a Schrödinger-like second-order differential equation that is amenable to exact analytic treatment. Using the Nikiforov-Uvarov (NU) method as the primary analytical framework, exact bound-state energy eigenvalues and the corresponding normalised wave functions are derived for the s-wave (l = 0) and arbitrary angular momentum quantum number l. The energy spectrum is analysed across different quantum numbers n and l, revealing a structure that reduces to the well-known hydrogen-like energy levels in the non-relativistic limit. Special attention is given to the role of the fine-structure constant α, the coupling strength Z, and the rest mass energy in shaping the relativistic energy spectrum. Numerical results are computed for selected meson-like systems and pionic atoms where the Coulomb interaction dominates. The results demonstrate good agreement with prior analytical and computational studies reported in the literature. The thesis confirms that the equal scalar-vector Coulomb potential condition leads to a discrete energy spectrum bounded by |E| < mc², with eigenvalues that depend strongly on the coupling constant Zα. The analysis provides important physical insight into the relativistic behaviour of spinless charged particles and contributes to the broader understanding of relativistic quantum mechanical systems.