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The number of distinct (n, l, \(m_{l}\)) states of hydrogen atom with n = 3 is :

  • A.5
  • B.9
  • C.3
  • D.2

Answer: B

There are 9 distinct states.

For a given n, l runs from 0 to n - 1 and, for each l, \(m_{l}\) runs from -l to +l, giving 2l + 1 values. For n = 3:

\[ \begin{aligned} l = 0:&\ 1, \quad l = 1:\ 3, \quad l = 2:\ 5 \\ \text{total} &= 1 + 3 + 5 = 9 = n^{2} \end{aligned} \]

Including spin would double this to 18, which is not asked. The option 5 counts only the l = 2 states, 3 counts only the values of l (or the l = 1 states), and 2 has no basis.

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