The generalized pseudospectral approach to the bound states of Hulthen and Yukawa potentials

Abstract: The generalized pseudospectral method is employed to calculate the bound states of Hulth\'en and Yukawa potentials in quantum mechanics, with special emphases on higher excited states and stronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a nonuniform and optimal spatial discretization of the radial Schr\"odinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials with $n\leq$10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high as up to $n=17$ have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. The $n>6$ states of Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulth\'en potential are reported here for the first time. Excepting the $1s$ and $2s$ states of Yukawa potential, the present method surpasses in accuracy all other existing results in the stronger coupling region for all other states of both these systems. This offers a simple and efficient scheme for the accurate calculation of these and other screened Coulomb potentials.