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.

•• The fine structure of an atomic spectrum results from the magnetic field “seen” by an orbiting electron. In this question you will make a semiclassical estimate of the B field seen by a 2p electron in hydrogen. The B field at the center of a circular current loop, i, of radius r is known to be B = μ0i/2r. (a) Treating the electron and proton as classical particles in circular orbits (each as seen by the other), show that the B field seen by the electron is

where L is the electron’s orbital angular momentum (L − mevr for a circular orbit). Remember that the current produced by the orbiting proton is i = ev/2πr, where v is the speed of the proton as seen by the electron (or vice versa). (b) For a rough estimate, you can give L and r their values for the n = 2 orbit of the Bohr model, L = 2ℏ and r = 4aB. Show that this gives B ≈ 0.39 T and hence that the separation, 2μBB, of the two 2p levels is about 4.5 × 10−5 eV.

It should be clear that this semiclassical calculation is only a rough estimate. You have used the Bohr values for L and r. If, for example, you had used the quantum value , this would have changed your answer by a factor of . There is another very important reason that the argument used here is only roughly correct: The electron’s rest frame is noninertial (since it is accelerated) and a careful analysis by the British physicist L. H. Thomas showed that the energy separation calculated here should include an additional factor of . That our answer, 4.5 × 10−5 eV, is correct to two significant figures is just a lucky accident.