Answer to Question #88103 in Atomic and Nuclear Physics for Sridhar

Question #88103
If the elements with principal quantum number n>4 where not allowed in the nature the number of possible elements would be 1)60 2)32 3)2 4) 64
Expert's answer

In order to answer the question, one has to calculate the number of different electronic states with the top value n = 4 for the principal quantum number (because electrons occupy these states one by one for every subsequent atom). Let us recall that every atom can be characterised by a set of electron occupation numbers written in the form


Taking into account that ranges for these numbers vary between

@$m_s = \pm \frac{1}{2}\\ m_l = -l, -l+1, .., 0, .., l-1,l\\ l = 0,..,n-1@$

we obtain

@$N = \sum_{n=1}^{4} \sum_{l=0}^{n-1} \sum_{m_l = -l}^{l} \sum_{m_s = \pm \frac{1}{2}} 1 = \sum_{n=1}^{4} \sum_{l=0}^{n-1} 2(2l+1) = \sum_{n=1}^{4} 2n^2 = 60 @$

Answer: 1) 60.

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