In this paper, we demonstrated that the multiple turning point problems within the framework of the Wentzel-Kramers-Brillouin (WKB) approximation method can be reduced to two turning point one for a nonsymmetric potential function by using an appropriate Pekeris-type approximation scheme. We solved the Schrödinger equation with the Killingbeck potential plus an inversely quadratic potential (KPIQP) function. The special cases of the modeled potential are discussed. We obtained the energy eigenvalues and the mass spectra of the heavy QQ¯ and heavy-light Qq¯ mesons systems. The results in this present work are in good agreement with the results obtained by other analytical methods and available experimental data in the literature.

The energy levels of the Klein–Gordon equation in hyper-radial space under the Deng-Fan potential energy function are studied by the SWKB and WKB approximation methods. We obtained the analytic solution of the energy spectra and the ground state wave function in closed form. Furthermore, we obtained the energy equation corresponding to the Schrodinger equation by invoking the non-relativistic limit. The variations of the non-relativistic N -dimensional energy spectra with the potential parameters and radial quantum number are investigated. The energy levels are degenerate for N = 2 , N = 4 and increase with the dimensionality number. The ground state wave function and its gradient are continuous at the boundary r = 0 , r = ∞ . Our results for the energy spectra are in excellent agreement with the ones obtained by other analytical methods where similar centrifugal approximations were applied.

We present reports for the structural, electronic, mechanical (elastic), and thermodynamic properties of NiHfX (X = Si, Sn, Ge) half Heusler compounds. We performed the calculation based on the density functional theory as implemented in the quantum espresso computational suite. We also predict the structural stability of the alloys, and the mechanical and thermodynamic properties using the linear response density functional perturbation theory. The compounds are non-magnetic semiconductors with narrow bandgaps. Results for the equilibrium lattice parameter for NiHfSn is in reasonable agreement with reports in the existing literature. The elastic parameters obey Born’s stability criteria, and there are no negative phonon dispersions, hence, establishing the compounds’ mechanical stability. NiHfGe is a brittle and hard material with directional covalent bonding, while NiHfSi and NiHfSn alloys are ductile and malleable. The alloys obey the Dulong-Petit law at a heat capacity of 72 and 74 J/Kmol and temperatures of 720 K and 800 K. The phonon frequencies shows that the wavelength of the materials is the far-infrared region. NiHfSi has the highest ${\mathrm{\Theta }}_D$ of 411.68 K at zero pressure and will have the highest thermal conductivity implying less viability as a thermoelectric material than NiHfSn with a ${\mathrm{\Theta }}_D$ of 340.19 K. GRAPHICAL ABSTRACT

In this paper, we obtained the analytical N-dimensional bound state solutions of exponential-type potential functions using the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation method with an improved approximation to the orbital centrifugal barrier. Furthermore, we derived the partition function with the associated thermodynamic properties. We calculated the expectation values (〈1/r²〉) via the Hellmann–Feynman theorem. Our results for the energy levels are in excellent agreement with the best available results obtained by other methods in the existing literature. The variations of the thermodynamic functions with the temperature parameter (β) agree very well with the other results obtained under different potential energy functions. The expectation values conformed to the ones obtained by another approach in previous work. This work demonstrates the exactness of the leading-order WKB approximation.