The ionisation energy of hydrogen is 13.6 eV. This represents an electron excited from the ground state becoming free of the atom.

There exist other energy states above the ground state that the atom can occuoy by exciting the electron to these states abosrbing the right amount of energy.

But the atom can not exist with energy between these states. The energy of the atom is quantised.

This is a fundamental aspect of sub-atomic physics. It bears no relation to anything described by Newton in classical mechanics.

The hydrogen atom can become excited by absorbing a photon of exactly the correct energy to jump to another energy level.

$$ E_2 - E_1 = hf $$

So to jump from the ground state to n=2

$$ hf = -3.4 \text{ eV} - ( - 13.6\text{ eV}) $$

$$ hf = 10.2 \text{ eV} $$

$$ hf = 10.2 \times 1.6 \times 10^{-19} \text{ J} $$

$$ f = \frac{1.63 \times 10^{-18}}{6.6 \times 10^{-34}} $$

$$ f = 2.5 \times 10^{15} \text{ Hz}$$

$$ \lambda = \frac{3 \times 10^8}{2.5 \times 10^{15}} $$

$$ \lambda = 122 \text{ nm} $$

Once an atom is excited to a higher energy level it can de-excite by dropping to a lower energy emitting a photon equal to the energy difference.

This can happen via any allowed energy level.

So in the example shown above there are three possible photons emitted as the atom de-excites down to the ground state.