Coupled Oscillators

Two identical harmonic oscillators in one dimension each have mass m and frequency ω. Let the two oscillators be coupled by an interaction term Cx₁x₂, where C is a constant and x₁ and x₂ are the coordinates of the two oscillators. Find the exact spectrum of eigenvalues for this coupled system.

SOLUTION

The Hamiltonian of the system is

(1)

The problem is easily solved in center-of-mass coordinates. So define

(2)

These new coordinates are used to rewrite the Hamiltonian. It now decouples into separate x- and y- parts:

(3)

(4)

(5)

The x-oscillator has a frequency  and eigenvalues hΩ_x (n_x + 1/2), where n_x is an integer. The y-oscillator has a frequency of  and eigenvalues hΩ_y (n_y + 1/2), where n_y is an integer.