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 Problem 1 it was shown that the speed of an electron in a Bohr orbit is of order v ≈ αc, where α is the fine-structure constant (9.40). Using this value, you can estimate the importance of relativistic corrections to the hydrogen energies, as follows: (a) Write down the correct relativistic expression for the electron’s kinetic energy, and use the binomial series (Appendix B) to show that

provided that v ≪ c (which is certainly true if v ≈ αc). Thus the relativistic correction ΔErel to the energy is about 3mv4/8c2. (b) Substitute v ≈ αc and show that this gives

Comparing this with (9.39), we see that relativistic corrections to the hydrogen energy are of the same order as the spin-orbit correction. Therefore, a correct treatment of fine structure must include both effects.

Problem 1

•• (a) Derive an expression for the electron’s speed in the nth Bohr orbit, (b) Prove that the orbit with highest speed is the n = 1 orbit, with v1 = ke2/ℏ. Compare this with the speed of light, and comment on the validity of ignoring relativity (as we did) in discussing the hydrogen atom, (c) The ratio

is called the “fine-structure constant” and is generally quoted as α ≈ 1/137. Verify this value.