With the introduction of a new improved approximation scheme (Pekeris-type approximation) to deal with the centrifugal term, the energy eigenvalues and the wave functions of the Schrodinger equation of the shifted Deng-Fan molecular potential ¨ are obtained, via the asymptotic iteration method. Rotationalvibrational energy eigenvalues of some diatomic molecules are presented, these results are in good agreement with other results in the literature. For these selected diatomic molecules, energy eigenvalues obtained are in much better agreement with the results obtained from the rotating Morse potential model for moderate values of rotational and vibrational quantum numbers. Furthermore, thermodynamic properties such as the vibrational mean U, specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues are studied.

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By employing the Pekeris-type (or a new improved approximation) to deal with the (pseudo or) centrifugal term, we solve the Klein–Gordon and Dirac equations with equally mixed scalar and vector Deng–Fan molecular potentials for all values of l (orbital the angular momentum quantum number) and κ (spin–orbit coupling quantum number), respectively. Using the formalism of the Nikiforov–Uvarov method, the approximate analytical bound state energy equations and the associated two-component spinors corresponding to the two relativistic equations are obtained. Also, special cases including the non-relativistic limits of the relativistic equation are obtained.

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle’s energy is less than the potential at both negative and positive infinity. The particle’s energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions, which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues, angular momentum numbers, screening parameters, reduced mass, and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.

In recent years, an extensive survey on various wave equations of relativistic quantum mechanics with different types of potential interactions has been a line of great interest. In this regime, special attention has been given to the Dirac equation because the spin-1/2 fermions represent the most frequent building blocks of the molecules and atoms. Motivated by the considerable interest in this equation and its relativistic symmetries (spin and pseudospin) in the presence of solvable potential model, we attempt to obtain the relativistic bound states solution of the Dirac equation with double ring-shaped Kratzer and oscillator potentials under the condition of spin and pseudospin symmetries. The solutions are reported for arbitrary quantum number in a compact form. the analytic bound state energy eigenvalues and the associated upper-and lower-spinor components of two Dirac particles have been found. Several typical numerical results of the relativistic eigenenergies have also been presented. We found that the existence or absence of the ring shaped potential potential has strong effects on the eigenstates of the Kratzer and oscillator particles with a wide band spectrum except for the pseudospin-oscillator particles where there exist a narrow band gap.

A generalized Schrödinger approximation, due to Ikhdair & Sever, of the semi-relativistic two-body problem with a rectangular barrier in (1+1) dimensions is compared with exact computations. Exact and approximate transmission and reflection coefficients are obtained in terms of local wave numbers. The approximate transmission and reflection coefficients turn out to be surprisingly accurate in an energy range |ϵ − V0| < 2µc 2 , where µ is the reduced mass, ϵ the scattering energy, and V0 the barrier top energy. The approximate wave numbers are less accurate.

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