⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 089901 Comments on “Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential” [Chin. Phys. Lett. 34 (2017) 110301] R. C. Woods* Affiliations Department of Electrical and Computer Engineering, University of South Alabama, 2112 Shelby Hall, 150 Student Services Drive, Mobile, AL 36688, USA Received 20 June 2019; accepted 19 June 2020; published online 28 July 2020 *Corresponding author. Email: rcwoods@southalabama.edu Citation Text: Woods R C 2020 Chin. Phys. Lett. 37 089901    Abstract Some problems with the article by Oluwadare and Oyewumi [Chin. Phys. Lett. 34 (2017) 110301] are discussed. The previously proposed solution of the Schrödinger wave equation in the generalized inverse quadratic Yukawa potential is unsatisfactory for a number of reasons. DOI:10.1088/0256-307X/37/8/089901 PACS:99.10.-x, 03.65.-w, 03.65.Ca, 03.65.Ge © 2020 Chinese Physics Society Article Text In the article titled “Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential”,[1] Oluwadare and Oyewumi examined the conventional Schrödinger wave equation with a new potential energy form given as their Eq. (6) and introduced by their Ref. [37]: $$\begin{alignat}{1} V(r)&=-V_{0} (1+{e^{-\alpha r}} / r)^{2}~~ \tag {1} \end{alignat} $$ $$\begin{alignat}{1} &=-({{A}'e^{-2\alpha r}} / {r^{2}})-({{B}'e^{-\alpha r}} / r)-{C}',~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} {A}'&={C}'=V_{0},~~{B}'=2V_{0}.~~ \tag {3} \end{alignat} $$ Equations (1) and (3) cannot be valid for dimensional reasons, and the potential energy should instead be defined exclusively by Eq. (2). Equation (11) of Ref. [1], the result for parameter $R$ given in Eq. (12) of Ref. [1], and Eq. (15) of Ref. [1] are all incorrect. The trial solution $\phi_{0} =A+By$ is over-determined and so leads to a contradiction because $A$ and $B$ must simultaneously satisfy the relation $$ (A-\alpha)B=\mu Q/\hbar^{2}.~~ \tag {4} $$ Since this is inconsistent with the specified values[1] of $A$ and $B$ for the trial solution $\phi_{0}$, their Eq. (22) and subsequent analysis dependent upon it (including their Eqs. (23) and (24)) are therefore not correct. Even if the over-determined trial solution is regarded as acceptable, the final result for the energy levels (Eq. (24) in Ref. [1]) is incorrect and should be written as $$\begin{align} E={}&-{C}'-\frac{2\mu }{\hbar^{2}}\Big[ {\frac{{A}'({\alpha -\frac{{B}'}{2{A}'}})}{\frac{B}{2\alpha }+n}-\frac{\alpha \hbar^{2}}{2\mu }\Big({\frac{B}{2\alpha }+n} \Big)} \Big]^{2}\\ &+4\alpha {A}'\Big({\alpha -\frac{{B}'}{2{A}'}} \Big).~~ \tag {5} \end{align} $$ The expression for $B$ given just after Eq. (24) of Ref. [1] is also incorrect, as is the wave function given in Eq. (25) of Ref. [1] and the expressions given for parameters $\delta$ and $\varepsilon$ contained in the wave function. Equation (3) of Ref. [1] is copied inaccurately from a previous reference. Finally, the results shown in Table 1 of Ref. [1] do not justify listing numerical values to 16 significant figures. In conclusion, the analysis of Ref. [1] has been corrected as far as possible. The originally proposed simple solution of the correct equations is unsatisfactory. Full details are given in the Supplementary Material. References Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential
[1] Oluwadare O and Oyewumi K 2017 Chin. Phys. Lett. 34 110301