Question

Consider two identical linear oscillators with spring constant k. The interaction potential is given by H =\(\varepsilon\)x₁x₂, where x₁ and x₂ are oscillator variables.

(a) Find the exact energy levels.

(b) Assume \(\varepsilon\) << k. Give the energy levels to first order in \(\varepsilon\)/k.

Answer 1

(a) The Hamiltonian of the system is

Hence the system can be regarded as consisting two independent harmonic oscillations of coordinates y₁, y₂. Thus, the exact total energy levels are

(b) For \(\varepsilon\) << k, the energy levels to first order in \(\varepsilon\)/k are