Exact solution of the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators

TL;DR

This paper provides an exact algebraic solution for the energy spectrum and eigenfunctions of two coupled harmonic oscillators, covering both isotropic and anisotropic cases using group theory methods.

Contribution

It introduces an algebraic approach employing $SU(1,1)$ and $SU(2)$ groups to exactly solve the coupled harmonic oscillator Hamiltonian.

Findings

Exact energy spectrum and eigenfunctions derived

Solutions applicable to both isotropic and anisotropic cases

Results recover known solutions when coupling is neglected

Abstract

We study the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators from an algebraic approach of the $SU(1,1)$ and $SU(2)$ groups. In order to obtain the energy spectrum and eigenfunctions of this problem, we write its Hamiltonian in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. We use the one boson and two boson realizations of the $su(1,1)$ Lie algebra, and the one boson realization of the $su(2)$ Lie algebra to apply three tilting transformations to diagonalize the original Hamiltonian. These transformations let us to obtain the exact solutions of the isotropic and the anisotropic cases, from which the particular expected results are obtained for the cases where the coupling is neglected.