Quantum solutions of identical linearly coupled harmonic oscillators using oblique coordinates

Abstract

In this article we develop the solutions of the time-independent Schrödinger equation analytically for a system of two identical harmonic oscillators linearly coupled in the kinetic and potential energies, using oblique coordinates. These coordinates are constructed by making a non-orthogonal rotation of the original coordinates that allows us to write the matrix representation of the Hamiltonian operator in a block-diagonal form characterized by the polyadic quantum number n = n1 + n2. The expression for the oblique rotation angle is first derived as a function of the kinetic and potential coupling parameters. Then the expression for the exact energy levels of the system is obtained by making an orthogonal rotation of the oblique coordinates, and the structure of the spectrum is discussed. Finally, the eigenvectors of the Hamiltonian polyadic blocks are identified with the discrete orthogonal polynomials of Krawtchouk, which provides an analytical expression for the coefficients of the linear combinations of the wave functions in oblique coordinates.