(a) Show that the number of electron states in a sub-shell is 4 + 2. (b) By...

(a) Show that the number of electron states in a sub-shell is 4ℓ + 2.

(b) By summing the number of states in each of subshells, show that the number of states in a shell is 2n². Use the fact that the sum of the first n odd integers, from 1 to 2n − 1, is n². That comes from regrouping the sum in pairs, starting by adding the largest to the smallest:

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1+ 3 + 5 + + (2n-5) + (2n - 3) + (2n-1) = [1 + (2n-1)] + [3+ (2n-3)] + [5+ (2n-5)] + .. = 2n + 2n + 2n + = nx²2/² = n² ... X