Comment on " Approximate Analytical Versus Numerical Solutions of Schr\"odinger Equation Under Molecular Hua Potential "

TL;DR

This paper critiques a previous study on the Schrödinger equation with molecular Hua potential, correcting the energy spectrum calculations and addressing errors in the original analysis.

Contribution

It identifies errors in prior work and provides the correct energy spectrum for the D-dimensional Schrödinger equation with molecular Hua potential.

Findings

Previous results contained inconsistencies in parameter q

Corrected energy spectrum derived using supersymmetry method

Clarified the valid parameter range for the model

Abstract

We present arguments proving that the results obtained by Hassanabadi and coworkers in the study of the D-dimensional Schr\"odinger equation with molecular Hua potential through the supersymmetry method in quantum mechanics are incorrect. We identified the inconsistencies in their reasoning on the allowed values of the parameter q and we constructed the correct energy spectrum.

Equations

$$V(r)=V_0{\left[\frac{1-e^{-b_h(r-r_e)}}{1-q{e}^{-b_h(r-r_e)}}\right]}^2$$

$$\frac{1}{r^2}\approx \frac{1}{r_e^2}\left[D_0+D_1\frac{e^{-b_h(r-r_e)}}{1-q{e}^{-b_h(r-r_e)}}+D_2\frac{e^{-2b_h(r-r_e)}}{{\left(1-q{e}^{-b_h(r-r_e)}\right)}^2}\right]$$

$$\left\{-\frac{d^2}{d{r}^2}+\frac{2\mu }{{\hslash }^2}V(r)+A_l\left[D_0+\frac{D_1{e}^{-b_h(r-r_e)}}{1-q{e}^{-b_h(r-r_e)}}+\frac{D_2{e}^{-2b_h(r-r_e)}}{{\left(1-q{e}^{-b_h(r-r_e)}\right)}^2}\right]\right\}{R}_{n_r,l}(r)$$

$$\varphi (r)=\frac{A}{1-q{e}^{-\alpha x}}+B;x=\frac{r-r_e}{r_e}\text{ and }\alpha =b_h{r}_e,$$

$${\varphi }^2(r)-{\varphi }^{\prime }(r)=V_{eff}(r)-{\tilde{E}}_{0,l},$$

$$A=-\frac{\alpha }{2}\left[1\pm \sqrt{1-\frac{4V_1}{{\alpha }^2}}\right],$$

$$B=\frac{\alpha }{2}\left[1-\frac{V_2}{\alpha A}\right],$$

$${\tilde{E}}_{0,l}=-B^2.$$

$$R_{0,l}(r)=N{e}^{-\int \varphi \left(x\right)dx}=\mathcal{N}{e}^{-\left(A+B\right)x}{\left(1-q{e}^{-\alpha x}\right)}^{-\frac{A}{\alpha }},$$

$$R_{0,l}(r)\underset{r\to \infty }{\to }0,$$

$$R_{0,l}(r)\underset{r\to r_0}{\to }0.$$

$$A=-\frac{\alpha }{2}\left[1+\sqrt{1-\frac{4V_1}{{\alpha }^2}}\right],$$

$$B=-\frac{1}{2}\left(A+\frac{V_1+V_2}{A}\right).$$

$$V_{eff+}\left(x,a_0\right)=V_{eff-}\left(x,a_1\right)+R\left(a_1\right),$$

$$R\left(a_1\right)=\frac{1}{4}\left[{\left(a_0+\frac{V_1+V_2}{a_0}\right)}^2-{\left(a_1+\frac{V_1+V_2}{a_1}\right)}^2\right],$$

$$a_1=a_0-\alpha .$$

$${\tilde{E}}_{n_r,l}^{\left(-\right)}=\underset{k=1}{\sum ^{n_r}}R\left(a_k\right)=\frac{1}{4}\left[{\left(a_0+\frac{V_1+V_2}{a_0}\right)}^2-{\left(a_0-n_r\alpha +\frac{V_1+V_2}{a_0-n_r\alpha }\right)}^2\right].$$

$${\tilde{E}}_{n_r,l}={\tilde{E}}_{n_r,l}^{\left(-\right)}+{\tilde{E}}_{0,l}=-\frac{1}{4}{\left(a_0-n_r\alpha +\frac{V_1+V_2}{a_0-n_r\alpha }\right)}^2,$$

$$E_{n_r,l}$$

$$N_r=n_r+{\delta }_l+\frac{1}{2},{\lambda }_l=\frac{2\mu V_0}{{\hslash }^2{b}_h^2}\left(\frac{1}{q^2}-1\right)+\frac{\left(D+2l-1\right)\left(D+2l-3\right)}{4b_h^2{r}_e^2}\left(\frac{D_2}{q^2}-\frac{D_1}{q}\right),$$

$${\delta }_l=\sqrt{\frac{1}{4}+\frac{2\mu V_0}{{\hslash }^2{b}_h^2}{\left(1-\frac{1}{q}\right)}^2+\frac{\left(D+2l-1\right)\left(D+2l-3\right)}{4b_h^2{r}_e^2}\frac{D_2}{q^2}}.$$

Full text

About a decade ago Hassanabadi and his coworkers [1] claimed to have approximately solved the D-dimensional Schrödinger equation with the Hua potential in the framework of the supersymmetric quantum mechanics approach (SUSY QM) by employing a Pekeris-type approximation to replace the centrifugal potential term. We point out however that there are several inconsistencies in the application of the SUSY QM and in the derivation of the energy spectrum.

First, the Hua potential [2] is given by the expression

[TABLE]

with the deformation parameter $q$ contained in the interval $-1<q<1$. For $q>0$, it is obvious that the potential (1) has a strong singularity at the point $r=r_{0}=r_{e}+\frac{1}{b_{h}}\ln q$, and on the other hand the Pekeris approximation

[TABLE]

is valid only for $qe^{b_{h}r_{e}}\geq 1$ (see Refs. [3, 4]). The D-dimensional Schrödinger equation (4) in Ref. [1] should be written in the range $r_{0}<r<\infty$ and for $e^{-b_{h}r_{e}}\leq q<1$, as

[TABLE]

where $A_{l}=\frac{\left(D+2l-1\right)\left(D+2l-3\right)}{4r_{e}^{2}}.$

Second, with the superpotential $\phi(r)$ defined as

[TABLE]

the authors of Ref. [1] obtained the Riccati equation

[TABLE]

from which the quantities $A,B$ and $\widetilde{E}_{0,l}$ are found to be

[TABLE]

[TABLE]

[TABLE]

Without correctly specifying the signs of $A$ and $B$, they then used the shape invariance approach to obtain the energy spectrum. Therefore, the result given by Eq. (21) in Ref. [1] is not correct. In this case, the signs of $A$ and $B$ can be fixed by considering the ground state wave function $R_{0,l}(r)$ defined by

[TABLE]

where $\mathcal{N}$ is the normalisation constant. For $R_{0,l}(r)$ to be a physically acceptable solution, it has to satisfy the boundary conditions

[TABLE]

and

[TABLE]

From this we see that $A<0$ and $A+B>0$ or $B>\left|A\right|$. The solution of the problem should be re-examined starting from the resolution of equations (15a) and (15b) in Ref. [1] . As a result, $A$ and $B$ can be expressed as

[TABLE]

[TABLE]

Then, by putting $a_{0}=A$ and using the shape invariance condition

[TABLE]

we find after some simple calculation that

[TABLE]

and

[TABLE]

The energy eigenvalues of Hamiltonian $H_{-}=-\frac{d^{2}}{dx^{2}}+V_{eff\text{ }-}\left(x\right)$ are then given by

[TABLE]

From Eqs. (8) and (17) it follows immediately that

[TABLE]

By using Eq. (12) together with Eqs. (10) in the Ref. [1] and since $\widetilde{E}_{n_{r},l}=V_{3}$ (see Eq. (12) in Ref. [1]) we arrive at the following expression for the energy levels:

[TABLE]

where we have set

[TABLE]

and

[TABLE]

This result can be verified in three-dimensional space. Indeed, if one substitutes $D_{0}=C_{0}r_{e}^{2},D_{1}=B_{0}r_{e}^{2},D_{2}=A_{0}r_{e}^{2}$, and $q=c_{h},$ one recovers the discrete energy spectrum derived by path integration [5].

Third, the numerical results obtained from Eq. (21) in the Ref. [1] for $r_{e}=1,q=0.170066$ and $b_{h}=1.61890$ in table $1$, are wrong. In this case, the correct numerical values must be calculated from the expression of our Eq. (LABEL:E.16) which is valid for $q\geq 0.198116507$ when $r_{e}=1$ and $b_{h}=1.61890$. We can also point out that the variation of $E_{n_{r},0}$ in terms of the parameter $q$ is valid only for $e^{-b_{h}r_{e}}\leq q<1$ (see Fig. (6) in Ref. [1] ). In addition, when $q=1$, the potential (1) becomes a step potential for which there are no bound states. This makes it possible to affirm that the curves plotted by the authors of Ref. [1] in Fig. (6) are incorrect.

In conclusion, the approximate analytical and numerical results obtained by the authors of Ref. [1] are unsatisfactory because  the SUSY QM method is used without taking into account the conditions for its application. The radial Schrödinger equation (3) can only be approximately solved by this method when $e^{-b_{h}r_{e}}\leqslant c_{h}<1$ and $r_{0}<r<+\infty$.