Quantum solution of coupled harmonic oscillator systems beyond normal coordinates
 作者： José Zúñiga,  Adolfo Bastida,  Alberto Requena 作者单位： 1Universidad de Murcia 刊名： Journal of Mathematical Chemistry, 2017, Vol.55 (10), pp.1964-1984 来源数据库： Springer Nature Journal DOI： 10.1007/s10910-017-0777-1 关键词： Coupled harmonic oscillators;  Normal coordinates;  Non-orthogonal linear coordinates;  Barbanis oscillator system;  Quantum treatment; 英文摘要： 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. 原始语种摘要： 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.

• coordinates　坐标
• coupled　耦合
• oscillator　振荡器
• harmonic　谐波
• diagonal　对角线
• demonstrate　说明
• efficient　有用的
• mathematical　数学的
• linear　线形的
• orthogonal　直交的