Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• In Section 5.6 we treated the H atom as if the electron moves around a fixed proton. In reality, both the electron and proton orbit around their center of mass as shown in Fig. 1. Using this figure, you can repeat the analysis of Section 5.6, including the small effects of the proton’s motion, as follows:

FIGURE 1

(a) Write down the distances re and rp in terms of r, me, and mp.

---

(b) Because both e and p move, it is easiest to work with the angular velocity ω, in terms of which ve and vp are as given in Fig. 5.10. Write down the total kinetic energy K = Ke+ Kp and prove that

where μ is the reduced mass,

Notice that the expression (5.39) for K differs from its fixed proton counterpart, K =(l/2)me r2 ω2, only in the replacement of me by μ.

---

(c) Show that Newton’s law, F = ma, applied to either the electron or proton gives

(Again, this differs from the fixed proton equivalent only in that μ has replaced me.)

---

(d) Use (5.39) and (5.41) to show that K = −U/2 and E = U/2.

---

(e) Show that the total angular momentum is

(in place of L = mer2ω if the proton is fixed).

---

(f) Assuming that the allowed values of L are L = nℏ, where , use (5.41) and (5.42) to find the allowed radii r, and prove that the allowed energies are given by the usual formula E = −ER/n2, except that

This is the result quoted without proof in (5.31) and (5.32).

---

(g) Calculate the energy of the ground state of hydrogen using (5.43), and compare with the result of using the fixed-proton result ER = me(ke2)2/2ℏ2). (Give five significant figures in both answers.) The difference in your answers is small enough that we are usually justified in ignoring the proton’s motion. Nevertheless, the difference can be detected, and the result (5.43) is found to be correct.