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A hydrogen atom initially in its 6th excited state (principal quantum number n = 7) de-excites to the ground state through one or more electronic transitions. Given that no spectral lines are produced in either the Paschen series (n_final = 3) or the Brackett series (n_final = 4), what is the maximum number of spectral lines of distinct photon energies that can be observed?

1. 21
2. 15
3. 10
4. 7

Correct answer: 10

Solution

Excluding n = 3 and n = 4 as termination levels, the accessible quantum levels are {1, 2, 5, 6, 7} — five levels in total. The number of distinct spectral lines equals the number of ways to choose 2 levels from 5, which is C(5,2) = 10.

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