•• (Section 12.2) If you haven’t already done Problem 1, do so now, and then make a plot o...

•• (Section 12.2) If you haven’t already done Problem 1, do so now, and then make a plot of the potential energy U(s) as a function of s. (For the purposes of the plot, you may as well choose units such that ke2 and R0 are equal to 1. Choose the ranges of U and s to show the interesting behavior.) Does your graph confirm the results of Problem 1?

Problem 1

••As a simple classical model of the covalent bond, suppose that an H2 molecule is arranged symmetrically as shown in Fig. 1. Write down the total potential energy U of the four charges and, treating the protons as fixed, find the value of the electrons’ separation s for which U is a minimum. Show that the minimum value is Umin ≈ −4.2ke2/R0. (Note that you have here treated the two protons as fixed at separation R0.Classically, they could not remain fixed in this way; it is one of the triumphs of quantum mechanics that it explains how the protons have a stable equilibrium separation, as we describe in Section 12.4. Note also that you should not take the number 4.2 in our answer too seriously — especially in estimating the dissociation energy. First, the electrons have appreciable kinetic energy, which we have not considered, second, even after the molecule has been dissociated into two atoms, each atom still has an energy −ER. Both of these effects lower the dissociation energy, and a more realistic estimate would be B ~ ke2/R0.)

FIGURE 1