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When dealing with electrons and the electromagnetic spectrum, why are there only four visible transitions in the hydrogen atom?

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This all falls back to the hydrogen absorption spectra. There are multiple "series" discovered by different scientists. The series in the visible region is the Balmer series.

Since the electronic energy levels of the hydrogen atom are quantized, there can only be certain discrete transitions.

As it turns out, if you solve the Balmer series (n=2), you get:

$$\frac{1}{\lambda} = \frac{4}{B}\left(\frac{1}{2^2} - \frac{1}{n^2}\right) = R_\mathrm{H}\left(\frac{1}{2^2} - \frac{1}{n^2}\right) \quad \mathrm{for~} n=3,4,5,...$$

You will then get lines at $\lambda$ = 656.3 nm, 486.1 nm, 434.1 nm, 410.2 nm...

Only a few of those fall in the visible. The Lyman series (n=1) occurs in the UV spectra, and the Paschen series (n=3) occurs in the IR.

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