Why do Lyman transitions produce ultraviolet photons while the Balmer transitions produce visible light photons?

Answer 1

The Lyman series is the group of energies corresponding to the transitions
of an electron between the "ground state" ... the lowest energy level ... and
any other energy level that an electron can have in a hydrogen atom.

The Balmer series is the group of energies corresponding to the transitions
between the second energy level in the hydrogen atom and any other one
(except the ground state).

But the energy difference between the ground state and the second level in
the hydrogen atom is about four times the difference between the second level
and any higher one, so it's pretty clear that any line in the Lyman series ought to
have substantially more energy than any line in the Balmer series.

And that's a fact. The shortest wavelength in the Balmer series is 410 nm ...
right there in violet light ... whereas the shortest wavelength in the Lyman
series is 122 nm, almost 2 octaves above the blue end of the visible spectrum,
and well into the ultraviolet.

Similarly . . .

The Paschen series (transitions to/from the 3rd energy level),
the Brackett series (transitions to/from the 4th energy level), and
the Pfund series (transitions to/from the 5th energy level)
are groups of lines at longer and longer wavelengths, extending through the infrared
and down into the short microwave wavelengths.

The lowest-frequency/longest-wavelength transition associated with the
hydrogen atom is the "flip" transition of the electron from one spin-
orientation to the other. That's the so-called "Hydrogen-alpha" line at roughly
1420 MHz / 21 cm. Wherever Hydrogen exists, electrons are flipping, and
either absorbing or emitting "H-α" radiation.

For receivers capable of tuning 1420 MHz (no problem), the universe is alive with it
in every direction. And if you have the opportunity to examine a chart of frequency
allocations, you'll notice that this frequency (and the band about 13 MHz to either side)
is allocated for "Astronomy, Space Research, and Earth Exploration Satellites" (looking
away from space !).