Problem

The Diatomic Molecule: Exercise 80 discusses the idea of reduced mass µ. Classically or qu...

The Diatomic Molecule: Exercise 80 discusses the idea of reduced mass µ. Classically or quantum mechanically, we can digest the behavior of a two-particle system into motion of the center of mass and motion relative to the center of mass. Our interest here is the relative motion, which becomes a one-particle problem if we merely use µ for the mass for that particle. Given this simplification, the quantum-mechanical results we have learned go a long way toward describing the diatomic molecule. To a good approximation, the force between the bound atoms is like an ideal spring whose potential energy is ½kx2, where x is the deviation of the atomic separation from its equilibrium value, which we designate with an a. Thus, x = r - a. Because the force is always along the line connecting the two atoms, it is a central force, so the angular parts of the Schrödinger equation are exactly as for hydrogen, (a) In the remaining radial equation (7-30), insert the potential energy ½kx2 and replace the electron mass m with p. Then, with the definition.f(r) = rR(r), show that it can be rewritten as

With the further definition g(x) =f(r), show that this becomes

(b) Assume, as is quite often the case, that the deviation of the atoms from their equilibrium separation is very small compared to that separation—that is, x«a. Show that your result from part (a) can be rearranged into a rather familial- form, from which it follows that