A simplified approach to the question of how ℓ is related to angular momentum—due to P. W....

A simplified approach to the question of how ℓ is related to angular momentum—due to P. W. Milonni and Richard Feynman—can be stated as follows: If Lz can take on only those values , where mℓ = 0, ±1,..., ±ℓ, then its square is allowed only the values , and the average of  should be the sum of its allowed values divided by the number of values, 2ℓ+1. Because there really is no preferred direction in space, the averages of Lx2 and Ly2 should be the same, and the sum of all three should give the average of L2. Given the sum , show that by these arguments, the average of L2 should be ℓ(ℓ+ 1 )ħ2. (Why this is not simply the average but the well-defined value of L2 requires the rigorous approach referred to in Section 7.5.)