[Paper Review] On Potentials Integrated by the Nikiforov-Uvarov Method

Lina Ellis, Ikumi Ellis+2 more|arXiv (Cornell University)|Mar 5, 2023

Quantum Mechanics and Applications4 citations

TL;DR

This paper presents a systematic, computer-assisted integration of fundamental quantum mechanical potentials—such as harmonic oscillator, Coulomb, Pöschl-Teller, Hulthén, and Morse—using the Nikiforov-Uvarov (NU) method within the Mathematica computer algebra system. It provides unified analytical solutions for bound states, verifying and extending known results through symbolic computation, with explicit formulas for energy levels and wave functions via orthogonal polynomial structures.

ABSTRACT

We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be integrated in the Nikiforov and Uvarov paradigm with the aid of a computer algebra system. This consideration may help the readers to study analytical methods of quantum physics.

Motivation & Objective

• To provide a comprehensive, self-contained review of analytically solvable quantum potentials using the Nikiforov-Uvarov method.
• To unify the treatment of standard quantum mechanical potentials through a single computational framework enabled by symbolic computation.
• To verify and extend existing analytical solutions for bound states in quantum systems using Mathematica.
• To serve as an educational and research tool by providing a publicly available Mathematica notebook with verified calculations.
• To demonstrate the power of computer algebra systems in simplifying and automating the derivation of energy levels and wave functions for exactly solvable potentials.

Proposed method

• Adopt the Nikiforov-Uvarov method to reduce second-order differential equations to hypergeometric-type forms via a transformation $ u = \varphi(x)y(x) $.
• Apply the quantization condition $ \lambda + n\tau' + \frac{1}{2}n(n-1)\sigma'' = 0 $ to determine energy levels.
• Use the Rodrigues-type formula $ y_n(x) = \frac{B_n}{\rho(x)} \frac{d^n}{dx^n} \left( \sigma^n(x)\rho(x) \right) $ to generate orthogonal polynomial solutions.
• Employ the Mathematica computer algebra system to symbolically derive and verify all solutions, ensuring consistency across potentials.
• Utilize classical orthogonal polynomials (Jacobi, Laguerre, Hermite) as the mathematical backbone for wave function construction.
• Implement a modular notebook structure in Mathematica, with initialization cells setting up general parameters and case-specific cells solving individual potentials.

Experimental results

Research questions

• RQ1 Which standard quantum mechanical potentials can be systematically solved using the Nikiforov-Uvarov method within a computer algebra framework?
• RQ2 How can the NU method be uniformly applied to derive energy levels and wave functions for bound states across diverse potentials?
• RQ3 What is the role of symbolic computation in verifying and extending known analytical solutions in quantum mechanics?
• RQ4 How can a single computational framework handle both nonrelativistic and relativistic potentials using orthogonal polynomial structures?
• RQ5 To what extent can a Mathematica-based implementation serve as a reliable educational and research tool for quantum mechanical problems?

Key findings

• The paper successfully derives energy levels and normalized wave functions for 18 standard quantum potentials using the Nikiforov-Uvarov method.
• All results are verified and completed using the Mathematica computer algebra system, ensuring computational accuracy and consistency.
• The energy levels for each potential are obtained via the quantization rule $ \lambda + n\tau' + \frac{1}{2}n(n-1)\sigma'' = 0 $, yielding discrete spectra.
• Wave functions are expressed as classical orthogonal polynomials (Jacobi, Laguerre, Hermite) multiplied by a weight function factor, as per the Rodrigues formula.
• The method provides a unified approach that avoids the need for separate treatments per potential, streamlining the solution process.
• A publicly available Mathematica notebook is provided, enabling reproducibility and interactive exploration of all derived solutions.