Quantum description of linearly coupled harmonic oscillator systems using oblique coordinates

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 (J. Phys. B. At. Mol. Opt. Phys., \textbf{52}, 055101 (2019)) 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.