Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

Abstract

In this article we extend our previous quantum-mechanical treatment of the system of identical harmonic oscillators linearly coupled in the kinetic and potential energies using oblique coordinates (Zúñiga et al 2019 J. Phys. B: At. Mol. Opt. Phys. 52 055101) to the general system of coupled non-identical harmonic oscillators. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block diagonal matrix. Accordingly, we derive the analytical formula for the oblique rotation angles, and obtain the expressions, also analytical, for the energy levels and eigenfunctions of the system in terms of the oblique parameters. We also show that oblique coordinates are in fact dependent on one of the rotational angles whose value can be freely chosen, so they form, in fact, a continuous set of coordinates that are especially flexible in dealing with more complex vibrational systems. To illustrate this, we make a numerical application to a system of kinetically coupled Barbanis oscillators, which clearly shows the advantages of using oblique coordinates to determine the energy levels and wave functions of the system variationally.

Quantum Dynamics of Oblique Vibrational States in the Hénon–Heiles System

“…In view of our recent work on the usefulness of oblique coordinates in two-dimensional systems, both in the time-independent − and time-dependent (present work) approaches, it is clear that the next step is to extend oblique coordinates to systems with a larger number of degrees of freedom such as the vibrational motions taking place in polyatomic molecules and in condensed phase systems. One of the advantages of oblique coordinates over normal coordinates is the greater number of parameters available in the former, which is useful to optimize them conveniently with greater flexibility.…”

“…The transformation from normal to local coordinates reintroduces therefore these couplings. , Generally, this transformation is linear and usually orthogonal, meaning that local coordinates are expressed as an orthogonal combination of normal coordinates. However, this linear combination is not the only one that can be used. − Our group has been working for a long time on the possibility of performing generalized nonorthogonal linear transformations to describe the vibrational stretching motions of triatomic molecules, − and we have recently focused on studying in detail this type of transformation that allows for the second-order Hamiltonian matrix of the system to be expressed in a block diagonal form, employing coupled oscillator models. − The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates …”

“…33−36 The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates. 36 For systems of two coupled harmonic oscillators, oblique coordinates are constructed by decoupling the second-order Hamiltonian matrix blocks characterized by the polyadic quantum number n = n 1 + n 2 . In a recent application 34 of these coordinates to Heńon−Heiles systems with two dissociation channels, useful for modeling the stretching vibrational modes of triatomic molecules, we have shown that oblique coordinates are much better than normal and local coordinates when calculating the energy levels and wave functions of the system using the eigenfunctions of the corresponding uncoupled Hamiltonian operator as basis functions.…”

“… 26 − 28 Our group has been working for a long time on the possibility of performing generalized nonorthogonal linear transformations to describe the vibrational stretching motions of triatomic molecules, 29 − 32 and we have recently focused on studying in detail this type of transformation that allows for the second-order Hamiltonian matrix of the system to be expressed in a block diagonal form, employing coupled oscillator models. 33 − 36 The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates. 36 …”

“… 33 − 36 The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates. 36 …”