[Paper Review] Approximate l-state solutions of the Manning-Rosen potential by the Nikiforov-Uvarov method

Sameer M. Ikhdair, R. Sever|ArXiv.org|Jan 28, 2008

Quantum Mechanics and Non-Hermitian Physics6 references3 citations

TL;DR

This paper presents approximate analytical solutions for the bound states of the Manning-Rosen potential in the Schrödinger equation for arbitrary angular momentum quantum numbers $l$ using the Nikiforov-Uvarov (NU) method. The method yields energy eigenvalues and corresponding radial wavefunctions expressed in terms of Jacobi polynomials, with results showing good agreement with other methods for short-range potentials, small $l$, and small $\alpha$, and reducing correctly to the Hulthén and $l=0$ cases.

ABSTRACT

The Schrodinger equation for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states energies. Additionally, the corresponding wave functions are expressed by the Jacobi polynomials. The Nikiforov-Uvarov (${ m NU}$) method is used in the calculations. To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers $n$ and $l$ with two different values of the potential parameter $α.$ It is shown that the results are in good agreement with the those obtained by other methods for short potential range, small $l$ and $α.$ This solution reduces to two cases $l=0$ and Hulthen potential case.

Motivation & Objective

• To derive analytical bound state solutions for the Manning-Rosen potential with non-zero angular momentum ($l \neq 0$) in the Schrödinger equation.
• To apply the Nikiforov-Uvarov (NU) method to solve the radial Schrödinger equation with an approximate treatment of the centrifugal term.
• To validate the accuracy of the results by comparing numerical eigenvalues with existing methods for various diatomic molecules.
• To examine the limiting cases, including $l=0$ and the Hulthén potential, to confirm consistency with known solutions.

Proposed method

• The Nikiforov-Uvarov (NU) method is applied to reduce the second-order differential equation of the radial Schrödinger equation to a hypergeometric-type equation.
• A coordinate transformation is used to express the radial equation in terms of a variable $z$, transforming it into the standard form solvable by special orthogonal polynomials.
• The energy eigenvalues are derived from the quantization condition $\lambda_n = -n\tau'(z) - \frac{n(n-1)}{2}\sigma''(z)$, where $\lambda_n$ is determined by the polynomial degree.
• The radial wavefunctions are expressed as $R_{nl}(r) = N_{nl} e^{-\varepsilon \delta r} (1 - e^{-\delta r})^{l+1} P_n^{(2\varepsilon, 2l+1)}(1 - 2e^{-\delta r})$, involving Jacobi polynomials.
• The centrifugal term is approximated using a standard method, enabling analytical treatment for $l \neq 0$ states.
• The solution is verified numerically for diatomic molecules (HCl, CH, LiH, CO) and compared with other methods across different potential ranges and parameters.

Experimental results

Research questions

• RQ1 Can the Nikiforov-Uvarov method provide accurate approximate analytical solutions for the $l$-state bound states of the Manning-Rosen potential with non-zero angular momentum?
• RQ2 How does the accuracy of the NU method depend on the potential range (parameter $b$), $l$, and $\alpha$?
• RQ3 Does the derived solution correctly reduce to the known Hulthén potential and $l=0$ cases in the appropriate limits?
• RQ4 What is the form of the radial wavefunctions in terms of special orthogonal polynomials for the Manning-Rosen potential?
• RQ5 How do the calculated energy eigenvalues compare with existing results for diatomic molecules?

Key findings

• The energy eigenvalues for the Manning-Rosen potential are derived as $E_{nl} = -\frac{\left[A - (n+l+1)^2\right]^2 \hbar^2}{8\mu b^2 (n+l+1)^2}$, valid for $l \neq 0$ states.
• The radial wavefunctions are expressed in terms of Jacobi polynomials: $R_{nl}(r) = N_{nl} e^{-\varepsilon \delta r} (1 - e^{-\delta r})^{l+1} P_n^{(2\varepsilon, 2l+1)}(1 - 2e^{-\delta r})$, with normalization constant $N_{nl} = 1/\sqrt{s(n)}$.
• For short potential ranges (large $b$), the results show good agreement with other methods, particularly for small $l$ and $\alpha$.
• The solution reduces to the Hulthén potential when $\alpha = 0$, recovering known energy levels and wavefunctions.
• The $l=0$ case is recovered as a special limit, yielding $E_n = -\frac{\left[A - (n+1)^2\right]^2 \hbar^2}{8\mu b^2 (n+1)^2}$.
• Numerical results for HCl, CH, LiH, and CO show consistent energy spectra across different $n$ and $l$ quantum numbers, confirming the method's reliability.