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Journal of Particle Physics

The One-Dimensional, Non-Relativistic, Quanitum Morse Oscillator at the Classical Limit

G. A., M. Grado-Caffaro, www.sapienzastudies.com, Grado-Caffaro M., Permanent address: M.A. Grado-Caffaro and M. Grado-Caffaro- Scientific Consultants, C/Julio Palacios 11, 9-B, 28029-Madrid (Spain)

Published Jan 17, 2018• 0 citations
Publisher

Abstract

The one-dimensional, non-relativistic, quantum Morse oscillator is studied at the classical limit.In fact, near the classical limit, the energy eigenvalues relative to the eigenstates of the nonrelativistic, time-independent, Schrödinger equation with Morse potential are negative and approximately proportional to the square of the corresponding vibrational quantum number.Within this framework, the mass of the oscillator in question is found to be negative.This can take place in certain particle phenomena and, in fact, occurs, for instance, in semiconductor superlattices.These cases are outlined very briefly in the present paper.

DOI
10.22606/jpp.2018.21001

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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.

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The one-dimensional, non-relativistic, quantum Morse oscillator is studied at the classical limit.In fact, near the classical limit, the energy eigenvalues relative to the eigenstates of the nonrelativistic, time-independent, Schrödinger equation with Morse potential are negative and approximately proportional to the square of the corresponding vibrational quantum number.Within this framework, the mass of the oscillator in question is found to be negative.This can take place in certain particle phenomena and, in fact, occurs, for instance, in semiconductor superlattices.These cases are outlined very briefly in the present paper.
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.
The Schrodinger equation is considered for a one-dimensional potential function composed of the Morse oscillator and an infinite barrier at long range. The boundary condition introduced by the barrier implies discretization of the unbound states. The eigenstates and eigenvalues are determined along with the discretization condition and the proper normalization for positive and negative energies. An analysis is presented for a particle in this potential driven by an external field coupled in through a dipole. Analytical formulae are derived for the matrix elements of an exponential and linear dipole function with respect to the eigenstates. Numerical time-dependent solutions to the Schrodinger equation are obtained for a periodic external field.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.
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.