Journal of Mathematical Chemistry

, Volume 55, Issue 10, pp 1964–1984 | Cite as

Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

  • José ZúñigaEmail author
  • Adolfo Bastida
  • Alberto Requena
Original Paper


Normal coordinates can be defined as orthogonal linear combinations of coordinates that remove the second order couplings in coupled harmonic oscillator systems. In this paper we go further and explore the possibility of using linear although non-orthogonal coordinate transformations to get the quantum solution of coupled systems. The idea is to use as non-orthogonal linear coordinates those which allow us to express the second-order Hamiltonian matrix in a block diagonal form. To illustrate the viability of this treatment, we first apply it to a system of two bilinearly coupled harmonic oscillators which admits analytical exact solutions. The method provides in this case, as an extra mathematical result, the analytical expressions for the eigenvalues of a certain type of symmetrical tridiagonal matrices. Second, we carry out a numerical application to the Barbanis coupled oscillators system, which contains a third order coupling term and cannot be solved in closed form. We demonstrate that the non-orthogonal coordinates used, named oblique coordinates, are much more efficient than normal coordinates to determine the energy levels and eigenfunctions of this system variationally.


Coupled harmonic oscillators Normal coordinates Non-orthogonal linear coordinates Barbanis oscillator system Quantum treatment 



We thank Domingo Jimenez for useful comments. This work was partially supported by the Spanish Agencia Estatal de Investigación (AET) and Fondo Europeo de Desarrollo Reginal (FEDER, UE) under Project CTQ2016-79345-P, and by the Fundación Séneca de la Agencia de Ciencia y Tecnología de la Región de Murcia under Project 19419/PI/14-2.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departamento de Química FísicaUniversidad de MurciaMurciaSpain

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