The Hydrogen Atom and Its Series

The formula for hydrogen's levels

For the hydrogen atom, the simplest one (a proton, an electron), the energy levels follow a remarkably simple formula, where n is an integer called the principal quantum number:

   E_n = -13,6 / n^2   (en eV),   avec n = 1, 2, 3, ...

Let's compute the first few:

   n=1 :  E = -13,6 / 1   = -13,60 eV   (fondamental)
   n=2 :  E = -13,6 / 4   =  -3,40 eV
   n=3 :  E = -13,6 / 9   =  -1,51 eV
   n=4 :  E = -13,6 / 16  =  -0,85 eV
   n=∞ :  E = 0                          (ionise)

We recover exactly the diagram seen in the previous chapter, and we see why the levels crowd together toward the top: as 1/n^2, they accumulate near E = 0.

The line series

All the transitions that end at the same low level form a series of lines. Each series falls in a different range of wavelengths:

   E=0  ------------------------------  (ionisation)
        |   |   |
   n=4  ---+---+---+----------
        |  |   |   |
   n=3  -+-+---+---+-----  ---.        transitions VERS n=3
        | |    |   |          |----> serie de PASCHEN (infrarouge)
   n=2  -+-+---+----  ---.
        | |    |        |----------->  serie de BALMER (visible !)
        | |    |        |
   n=1  -+-+----  ---.
                    |------------------> serie de LYMAN (ultraviolet)

• Lyman series: all the transitions toward n=1. Large energy gaps -> very energetic photons -> ultraviolet (invisible).
• Balmer series: transitions toward n=2. Medium gaps -> visible light. It is the only series visible to the eye: these are the famous red, blue, and violet lines of hydrogen.
• Paschen series: transitions toward n=3. Small gaps -> infrared (invisible).

A complete calculation: the red line of hydrogen

Let's compute the wavelength of the photon emitted during the transition n=3 -> n=2 (the red Balmer line).

   ΔE = E3 - E2 = (-1,51) - (-3,40) = 1,89 eV

Converting to joules: ΔE = 1,89 × 1,6 × 10^-19 ≈ 3,0 × 10^-19 J. Then, from ΔE = hc/λ, we get:

   λ = h c / ΔE
     = (6,6 × 10^-34 × 3,0 × 10^8) / (3,0 × 10^-19)
     ≈ 6,6 × 10^-7 m
     = 660 nm

660 nm is red — exactly the color of hydrogen's most famous line. The level diagram therefore predicts, within calculation accuracy, the colors observed in the spectrum.

In summary

Hydrogen's levels are E_n = -13,6/n^2 eV, which explains their crowding toward the top. Transitions toward the same low level form series: Lyman (toward n=1, UV), Balmer (toward n=2, visible), Paschen (toward n=3, IR). Computing ΔE and then λ = hc/ΔE reproduces precisely the colors of the observed lines — theory and experiment meet.