The interatomic potential for diatomic molecules like O₂ and N₂ can be approximated near its minimum by a simple harmonic oscillator potential 1/2 kx². Consider the classical motion of a diatomic molecule composed of two atoms of equal mass m. Show that near the minimum of the interatomic potential this classical motion is that of a single mass m/2 on a spring with spring constant k. For the O=O double bond in O₂, the spring constant near the bottom of the potential is roughly 1140 N/m. Estimate the energy gap between the ground state of the molecule and the first vibrational quantum excitation of the molecule, corresponding to the first excited state of the associated harmonic oscillator potential.