Abstract
In this work we have applied ansatz method to solve for the approximate ℓ-state solution of time independent Schrödinger wave equation with modified Möbius squared potential plus Hulthén potential to obtain closed form expressions for the energy eigenvalues and normalized radial wave-functions. In dealing with the spin-orbit coupling potential of the effective potential energy function, we have employed the Pekeris type approximation scheme, using our expressions for the bound state energy eigenvalues, we have deduced closed form expressions for the bound states energy eigenvalues and normalized radial wave-functions for Hulthén potential, modified Möbius square potential and Deng-Fan potential. Using the value 0.976865485225 for the parameter ω, we have computed bound state energy eigenvalues for various quantum states (in atomic units). We have also computed bound state energy eigenvalues for six diatomic molecules: HCl, LiH, TiH, NiC, TiC and ScF. The results we obtained are in near perfect agreement with numerical results in the literature and a clear demonstration of the superiority of the Pekeris-type approximation scheme over the Greene and Aldrich approximation scheme for the modified Möbius squares potential plus Hulthén potential.
Highlights
The need for exact solution of Schrödinger wave equation in quantum mechanics cannot be over emphasized, this is due to the vital information derivable from them (Miranda et al, 2010; Qiang, et al, 2009), information such as energy, momentum, wavelength and frequency of the system can only be obtained with the knowledge of the wave function (Eyube et al, 2019)
The Schrödinger equation was studied by an improved approximation scheme for the Hulthén potential (Ikhdair,2009), Okorie et al (2018) have studied the solution of the Schrödinger equation with modified Möbius square potential, they used their results to explore the thermodynamic properties of the potential
For small values of the screening parameters, our computed bound state energy eigenvalues values are almost indistinguishable from literature results
Summary
The need for exact solution of Schrödinger wave equation in quantum mechanics cannot be over emphasized, this is due to the vital information derivable from them (Miranda et al, 2010; Qiang, et al, 2009), information such as energy, momentum, wavelength and frequency of the system can only be obtained with the knowledge of the wave function (Eyube et al, 2019). Exact solution of the Schrödinger equation is restricted to only few potential models such as the Coulombic potential and harmonic oscillator potential (Hitler et al, 2017; Tsaur and Wang, 2014) for all quantum states nl where n is the principal quantum number and l is the angular momentum quantum number. The Hulthén, Morse, and Eckart potentials are among the few potential energy functions which give exact solution for zero angular momentum quantum number (l = 0), these solutions are often referred to as s-wave solutions (Hitler et al, 2017; Tsaur and Wang, 2014). We are encouraged to solve for the approximate l-state solution of time independent Schrödinger wave equation with modified Möbius squared potential plus Hulthén potential, which, to the best of our knowledge has never been solved in the literature. The modified Mobius Squared Potential (Okorie et al, 2018) plus the Hulthén potential (MMSPHP) (Jia et al, 2008) is given by: V r
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