•• Consider two classical atoms of mass m and m′, separated by a distance R, with their ce...

•• Consider two classical atoms of mass m and m′, separated by a distance R, with their center of mass fixed at the origin. (a) Write down expressions for the atoms’ distances, r and r′, from the origin, in terms of m, m′, and R. (b) Suppose that the two atoms vibrate in and out along the internuclear axis. Show that their total kinetic energy can be expressed as , where μ is the reduced mass defined in Eq. (12.41) (Problem 1). This result proves for a classical molecule (what is true in quantum mechanics as well) that the vibrational motion of both atoms can be treated as if only one atom were moving, provided that we take its mass to be μ.

Problem 1

• Consider two atoms of masses m and m′, bound a distance R0 apart and rotating about their center of mass (as in Fig. 12.26). (a) Calculate their moment of inertia, I, and prove that it can be written as I = μR02, where μ is the reduced mass

This shows that one can treat the rotational motion of any diatomic molecule as if only one of the atoms were moving, provided that one takes its mass to be μ. (b) Prove that if m ≪ m′ then μ ≈ m.