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I was studying a very simplified intro to Spectroscopy. The following diagram shows the emission spectra of Hydrogen Gas:

enter image description here

Credit: NASA, ESA, and L. Hustak (STScI)

My Question: In the spectra, the brightness of the red light is shown as maximum. Doesn't the brightness correspond to the energy emitted by the electron when it jumps back to the ground state?

In that case, shouldn't the amplitude be the highest in the case of purple light (corr. $e^-$ jumps down from level 6 to level 2)?

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Brightness does not correspond to the energy, it corresponds to the transition that happens most often (producing the most photons). Which will often be the ones at lowest energy levels. In fact as your graph shows, the brightness (I think a better word is Intensity) of each peak is typically in inverse proportion to the energy level. Higher energy transitions occur less often.

Energy gap in the transition corresponds to the color (frequency) of the photon emitted.

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