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.
•• (a) Consider the helium atom to be a fixed point nucleus (charge 2e) with two spin-half fermion electrons. What is the degeneracy of its ground state? (That is, how many independent states of the whole atom have the ground state energy?) (b) Suppose instead that the electron was a spin-half boson. (It is an experimental fact that all spin-half particles are fermions, but there is nothing to stop us imagining a spin-half boson.) What then would be the degeneracy of the He atom’s ground state? (c) What would the degeneracy be if the two electrons were somehow distinguishable? (For example, they might have slightly different masses.) The different numbers that you should get in the three parts of this problem are examples of the different “statistics” of fermions, bosons, and distinguishable particles. (See Chapter 13.)
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