Quantum solution of coupled harmonic oscillator systems beyond normal coordinates

Abstract

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.