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 first magic number for both nuclei and atoms is 2. The second is 8 for nuclei and 10 for atoms. The difference occurs because what we call the 1p and 2s levels in nuclei are widely separated, whereas the corresponding levels in atoms (called 2s and 2p by atomic physicists) are very close together. To explain this difference, compare the shape of an atomic potential energy (∝1/r) with that of the nuclear potential well. Noting that states with higher l have probability distributions that are pushed out to larger radii, qualitatively explain the difference in the positions of these two levels.
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