Journal of Mathematical Chemistry

, Volume 53, Issue 9, pp 2018–2027 | Cite as

Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

  • Damian Mikulski
  • Jerzy Konarski
  • Krzysztof Eder
  • Marcin Molski
  • Stanisław Kabaciński
Original Paper


The study involves finding exact eigenvalues of the radial Schrödinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.


Factorization method Riccati equation Supersymmetric quantum mechanics Isospectral Hamiltonians Anharmonic potentials 


Compliance with Ethical Standards

Conflict of interest

We declare that all authors of this article have no financial disclosures and conflict of interest.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Damian Mikulski
    • 1
  • Jerzy Konarski
    • 3
  • Krzysztof Eder
    • 1
  • Marcin Molski
    • 3
  • Stanisław Kabaciński
    • 2
  1. 1.Gen. Zamoyska and Helena Modrzejewska High SchoolPoznańPoland
  2. 2.Faculty of ComputingPoznan University of TechnologyPoznańPoland
  3. 3.Department of Theoretical Chemistry, Faculty of ChemistryA. Mickiewicz UniversityPoznańPoland

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