Approximate analytical solutions and mean energies of stationary Schrödinger equation for general molecular potential

Abstract

The Schrödinger equation is solved with general molecular potential via the improved quantization rule. Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann–Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods–Saxon potential, Morse potential, Möbius squared and Tietz–Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H2, CO, HF, and O2. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.