Non-relativistic and relativistic energy of molecules in external fields with time-dependent moving boundaries

We studied the solution of the Schr & ouml;dinger equation in the presence of external magnetic and Aharonov-Bohm (AB) fields, using the Deng-Fan potential and the Nikiforov-Uvarov method. The energy spectra were computed and used to investigate the energy levels of cesium diatomic molecules. The Hellmann-Feynman theorem was applied to compute the expectation values of certain physical observables. Additionally, we solved the time-dependent Schr & ouml;dinger equation, studying the time-dependent density distribution function, average energy, and disequilibrium. The Klein-Gordon equation was also solved, and the approximate bound state energy equations as well as the corresponding radial wave function were obtained in closed form. Additionally, the expression for the scattering phase shift was obtained in D-dimensions. Ultimately, we investigated the bound-state solutions of the Dirac equation under spin and pseudospin symmetries, considering Coulomb-like tensor interactions. We derived the energy eigenvalue equations and the associated upper and lower spinor wavefunctions for spin and pseudospin limits.