Analytical solution of the class of inversely quadratic Yukawa potential with application to quantum mechanical systems

Abstract

In our study, we applied the Exact Quantization Rule approach to tackle the radial Schrödinger equation analytically, specifically addressing the class of inversely quadratic Yukawa potential. Through this method, we successfully predicted the mass spectra of heavy mesons, including charmonium and bottomonium, across a range of quantum states by leveraging the energy eigenvalues. When compared to experimental data and other researchers' findings, our model exhibited a remarkable degree of accuracy, with a maximum error of 0.0065 𝐺𝑒𝑉.We reduced our potential model to the Kratzer potential in order to further expedite our computations, and we ensured mathematical accuracy by imposing particular boundary conditions. By utilizing the acquired energy spectrum, we broadened our examination to investigate the energy spectra of homonuclear diatomic molecules, like nitrogen (N2) and hydrogen (H2). One remarkable finding was that the energy spectrum reduced as the angular momentum quantum number increased in the case where the principal quantum number stayed fixed. In a similar vein, the energy spectrum consistently decreases when the angular momentum quantum number is varied. The complex interaction between the kinetic and potential energies of the electron causes this decreasing trend in the energy spectrum as the angular momentum quantum number increases in a diatomic molecule. The energy spectrum is systematically reduced as the electron's orbit lengthens and its distance from the nucleus increases, shifting the balance between these energies.