Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. It only takes a minute to sign up.
Sign up to join this community
Recently, I found that when electrons in an atomic sample de-excite from a higher energy level ($n_2$) to a lower energy level ($n_1$), the number of spectral lines observed in the spectrum is
$$\frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}$$
Can anyone please tell me how to derive this?
When you move from level $n_1$ to level $n_2$, the total number of energy levels are $n_2-n_1+1$ (including $n_1$ and $n_2$).
Now, you want to find the total number of possible spectral lines in the atomic sample.
Note that if you chose any two energy levels (say $n_i$ and $n_j$), you will get a unique spectral line corresponding to those energy levels.
So, the total number of spectral lines possible are $$\binom{n_2-n_1+1}{2}=\dfrac{(n_2-n_1+1)(n_2-n_1)}{2}$$
