Discrete Energy Levels Make Light Come in Sharp Lines
The energy levels of a hydrogen atom are discrete, like Eₙ = −13.6 eV / n². When an electron jumps from an upper rung to a lower one, it emits exactly the energy difference as a single photon. Ephoton = Ei − Ef, and a photon has a frequency f so E = hf, or as a wavelength λ = hc / E. Because the rung differences are only certain fixed values, the light comes out not as any color but as sharp lines at exact wavelengths. In the first figure we watch the level ladder and where its line lands on a spectrum bar.
Draw the hydrogen levels as a ladder. Since Eₙ = −13.6 eV / n², n = 1 sinks deep and the rungs crowd toward 0 as n grows. Pick a transition from an upper rung i to a lower rung f, and that gap Ei − Ef becomes the energy of one photon. Here f is the frequency of that photon (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 seen in the wave-particle lesson. Choose a visible transition (Hα 656, Hβ 486, Hγ 434 nm) and a correctly colored vertical line lands in its place on the 400 to 700 nm bar.
Which rung you choose for the lower level f sorts the transitions into series. Transitions falling to f = 1 form the Lyman series in the ultraviolet; falling to f = 2 gives the Balmer series in the visible; falling to f = 3 gives the Paschen series in the infrared. Pick a series with the segmented control and only that series’ transitions are highlighted on the ladder. Only the Balmer series shows up to our eyes as colored lines.
Between the same rungs, the direction gives two appearances. When the electron jumps down, it emits a photon: a bright line lit up on a dark strip, an emission spectrum. The reverse, continuous light passing through the atoms, gets exactly that energy of photon absorbed, carving a black line into a bright continuous strip: an absorption spectrum. Flip the toggle and at the same wavelength the bright line and the dark line are exactly inverted. Energy conservation alone explains it.
A single photon’s energy, frequency, wavelength, and color are all tied on one string. Raise the photon energy E with the slider and the frequency f rises with it through E = hf, while the wavelength shrinks through λ = hc / E. At the same time the on-screen color shifts from red toward violet. Higher-energy light is shorter wavelength, violet; lower-energy light is longer wavelength, red. Fix any one and the rest are set automatically.
Within one series, raising the upper rung i higher and higher makes the lines bunch toward a single place. Since Eₙ = −13.6 eV / n², the rung differences shrink as you climb, and as i grows the transition energy converges to a maximum value. That place is the series limit. Slide i up and each new line sits right beside the previous one, so the lines crowd densely toward the limit. As i goes to infinity it marks the edge where the electron just breaks free, the ionization threshold.