Eigensolution of the Klein–Gordon equation for modified Yukawa–Kratzer potential and its applications using parametric Nikiforov–Uvarov and SUSYQM method

Abstract

We propose here a modified Yukawa–Kratzer potential (MYKP) and determines its eigenvalue solutions. The energy spectrum in relativistic and non-relativistic regime with normalized wave functions for MYKP are obtained by parametric Nikiforov–Uvarov (pNU) method and supersymmetric quantum mechanics (SUSYQM) method using Greene–Aldrich approximation to handle 1/r and \(1/r^2\) term in the effective potential. We obtained identical eigensolutions via both methods. We recovered various potentials such as modified Kratzer potential, generalized Kratzer potential, standard Kratzer potential, modified Kratzer potential plus screened Coulomb potential (MKSCP), modified Yukawa potential (MYP), class of Yukawa potential (CYP), Hellmann potential etc from MYKP. We also obtained energy spectrum and normalized wave functions in relativistic and non-relativistic realm for all the deduced potentials from that of MYKP. The eigenvalue solution of MYKP is then applied to obtain the energy spectrum of CO and \(H_2\) molecules. Calculated numerical results of the energy spectrum for LiH and HCl molecules based on some of the deduced potentials are also presented. Further, partition function \(Z(\beta )\) for MYKP and few of the deduced potentials are computed. Other thermodynamic properties such as mean energy \(U(\beta )\), mean free energy \(F(\beta )\), entropy \(S(\beta )\) and heat capacity \(C_s(\beta )\) are computed for MYKP and CYP. Using Hellmann–Feynman theorem, the expectation values of \(1/r^2\), 1/r, kinetic energy T, and square of momentum \(p^2\) for MYKP and for other deduced potentials are presented. For validity of our results, numerical results of the energy spectrum for few recovered potentials are compared with the results available in the literature.

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Correspondence to Rajendrasinh H. Parmar.

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Parmar, R.H., Vinodkumar, P.C. Eigensolution of the Klein–Gordon equation for modified Yukawa–Kratzer potential and its applications using parametric Nikiforov–Uvarov and SUSYQM method. J Math Chem 59, 1638–1703 (2021). https://doi.org/10.1007/s10910-021-01258-y

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Keywords

  • Modified Yukawa potential
  • Modified Kratzer potential
  • Parametric Nikiforov–Uvarov method
  • Supersymmetric quantum mechanics method
  • Thermodynamic properties
  • Hellman–Feynman theorem