The first
real clue to the nature of atoms was the discovery of
characteristic atomic spectra, which suggested that atoms
have discrete energy levels, and can create and emit real
photons that carry off the energy difference between two
levels.

Quantum description of
hydrogen, 1 proton and 1 electron, in terms of standing waves
of probability for the one electron! The two quantum numbers
are the principal quantum number n and the “angular
momentum” quantum number ℓ, represented
by letters s (0), p (1), d (2), etc.

The space quantum numbers are
n, ℓ and m_{ℓ}. It's important to
understand that the “angular momentum” quantum number ℓ has
nothing to do with classical angular momentum. Rather, it is
related to the form of the three-dimensional standing
probability wave for that state. The values of ℓ and m_{ℓ}
determine the specific form of a ubiquitous function known as Y_{ℓmℓ}(θ,
φ), aka “spherical harmonics,” in spherical polar coordinates.

To understand more, we need
to learn about spin, an intrinsic or internal quantum number for
all fundamental particles, and the Pauli Principle!