atomic spectra energy levels
The energy levels of the hydrogen atom are separated as Eₙ = −13.6 eV / n². When an electron jumps from a higher rung to a lower one, it sends out exactly that difference as one photon. Ephoton = Ei − Ef, and the photon carries a frequency so E = hf, and as a wavelength λ = hc / E. The differences are fixed values, so the light comes out only as lines of fixed wavelength. The first picture shows the level ladder and where those lines land on the spectrum.
Draw the hydrogen levels as a ladder. Eₙ = −13.6 eV / n², so n = 1 sits deep, and as n grows the rungs gather toward 0. Pick a transition from an upper rung to a lower one and the difference Ei − Ef is the energy of one photon. f is the frequency of that photon, so E = hf, and λ = hc / E is the wavelength of the light. This λ is the light's own wavelength, different from the matter wave λ = h/p. Pick the visible transitions 656, 486, 434 nm and a colored line lands in its place on the bar from 400 to 700 nm.
Which lower rung you choose splits the series. Transitions that fall to f = 1 are the Lyman series, ultraviolet, and those that fall to f = 2 are the Balmer series, visible light. Those that fall to f = 3 are the Paschen series, infrared. Choose a series and only the transitions of that series are marked on the ladder. What the eye sees as colored lines is only the Balmer series.
Even across the same rungs, the direction makes two pictures. When the electron jumps down it sends out a photon, a bright line lighting on a dark band, emission. When continuous light passes an atom, only photons of that exact difference are absorbed and a dark line is cut into a bright band, absorption. Change the direction and a bright line and a dark line swap places at the same wavelength. Energy conservation explains it.
The energy, the frequency, the wavelength, and the color of one photon are tied on one string. Raise the photon energy E and the frequency f rises with it as E = hf, and the wavelength shrinks as λ = hc / E. At the same time the color moves from red toward violet. Light of large energy is violet of short wavelength, and light of small energy is red of long wavelength. Fix one and the rest follow.
Inside one series, raise the upper rung and the lines gather toward one place. Eₙ = −13.6 eV / n², so the gaps shrink as you climb, and as the upper rung grows the transition energy converges on a maximum. That place is the series limit. Raise the upper rung and a new line sticks right beside the previous one, crowding toward the limit. When the upper rung grows without end, that is the edge where the electron is just freed, the ionization threshold.