One reasonable and memorable approximation excluding any <fine structure> is:
$$
E_n = -\frac{13.6eV}{n^2}
$$
{title=Hydrogen spectral series mnemonic}
see for example example: <hydrogen 1-2 spectral line>.


Gross hydrogen emission spectrum

{tag=Hydrogen emission spectrum}

= 21 centimeter line
{synonym}
{title2}

21 cm is very long and very low energy, because he energy split is very small!

Compare it e.g. with the <hydrogen 1-2 spectral line> which is 121.6 nm!


Hydrogen line

{title2=121.6 nm}
{title2=10.2e V}

<equation Hydrogen spectral series mnemonic> gives for example from <principal quantum number> 1 to 2 a difference:
$$
E_n = -13.6eV\left[\frac{1}{2^2} - \frac{1}{1^2}\right] = 10.2 eV
$$
which with <Planck-Einstein relation> gives about 121.6 <nm> ($2.47 \times 10^15$ <Hz>), which is a reasonable match with the value of 121.567... from the <NIST Atomic Spectra Database>.


Ciro Santilli @cirosantilli 37

 Incoming links: Hydrogen 1-2 spectral line