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Schrödinger solution for the Morse oscillator

Lucjan Piela

Published Jul 1, 2008• 0 citations

Abstract

Piela replies: The Morse oscillator is a single point mass subject to the Morse original potential cited in Ilya Kaplan’s book, equation 5.22. Contrary to what Donald Truhlar writes, the Morse oscillator does not represent two point masses with a spring, not to mention a diatomic molecule. Therefore, Kaplan’s equation 5.23 is an exact solution of the Schrödinger equation for the Morse oscillator. The same solution is, of course, an approximate one for the Schrödinger equation for two point masses with a Morse-like spring or any real diatomic molecule.Truhlar could literally repeat his arguments for the harmonic oscillator, instead of the Morse one. His conclusion in such a case would mean that the widely known solution to the Schrödinger equation for the harmonic oscillator is not exact. 1 1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. Google Scholar© 2008 American Institute of Physics.

DOI: 10.1063/1.4796905

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Piela replies: The Morse oscillator is a single point mass subject to the Morse original potential cited in Ilya Kaplan’s book, equation 5.22. Contrary to what Donald Truhlar writes, the Morse oscillator does not represent two point masses with a spring, not to mention a diatomic molecule. Therefore, Kaplan’s equation 5.23 is an exact solution of the Schrödinger equation for the Morse oscillator. The same solution is, of course, an approximate one for the Schrödinger equation for two point masses with a Morse-like spring or any real diatomic molecule.Truhlar could literally repeat his arguments for the harmonic oscillator, instead of the Morse one. His conclusion in such a case would mean that the widely known solution to the Schrödinger equation for the harmonic oscillator is not exact. 1 1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. REFERENCESSection:ChooseTop of pageREFERENCES <<1. L. Piela, Ideas of Quantum Chemistry, Elsevier, Amsterdam (2007), p. 239. Google Scholar© 2008 American Institute of Physics.

The position dependent mass Schrödinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrödinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrödinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.

A method for calculating the analytical solutions of the one-dimensional Schr\"odinger equation is suggested. A general discussion of the possible forms of the potentials and wave functions that are necessary to get the analytical solution is presented. In general, the analytical solutions appear in multiplets corresponding to the quantum number n of the harmonic oscillator. As an application, known solutions for the anharmonic oscillators are critically recalculated and a few additional results are found. Analytical solutions are also found for the generalized Morse oscillators. \textcopyright{} 1996 The American Physical Society.

Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.

This study has described the approximate solution of Schrodinger equation to diatomic molecule for harmonic oscillator. The solution procedure is developed by the Power series method and Newton’s second law. It consider an approximate solution of harmonic oscillator using Schrödinger equation in one dimension only because other analytical approaches are limited to the widely known method and consider two to five dimensions with various iteration method to obtain their results but here the solutions to be obtained and their efficiency will help other research to comprehend how the solution of this harmonic oscillator has been done over the years and also to use the most efficient approximate solution.