Spectral Lines From Energy Levels

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A sample of excited hydrogen atoms has its electrons populating the fourth energy level. As these electrons cascade down to the ground state through all possible routes, how many distinct spectral lines can at most appear?

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When electrons de-excite from a level with principal quantum number $n$, the maximum number of distinct spectral lines produced by all possible downward transitions is $\dfrac{n(n-1)}{2}$, because each line corresponds to a unique pair of levels between which a transition can occur. For $n = 4$ this gives $\dfrac{4 	imes 3}{2} = 6$ lines. The value 4 wrongly counts only transitions ending at the ground state ($4	o1, 3	o1, 2	o1$ plus one extra) and misses intermediate jumps such as $4	o3$, $4	o2$, and $3	o2$. The value 3 counts only the direct drops from $n = 4$ to each lower level. The value 10 corresponds to $n = 5$ using the same formula and overcounts. This is the standard NCERT combinatorial result for hydrogen emission. A plausibility check by listing the pairs ($4	o3, 4	o2, 4	o1, 3	o2, 3	o1, 2	o1$) confirms exactly six distinct transitions, matching the formula.

Spectral Lines From Energy Levels

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