Quantum numbers of the state of the hydrogen atom:
- n (1, 2, 3, 4, ...) — This is the shell number / "principle quantum number" which determines to first order how much energy the electron has over the ground state: n=1 . This gives the gross structure of the hydrogen emission spectrum.
- ℓ (0, ..., n-1) — This is the "orbital quantum number" which determines the total orbital angular momentum of the electron about the proton, |L| = √(ℓ²+ℓ) h/(2π). This has a bigger effect in multielectron atoms and assigned whimsical names which were later reduced to a pattern of letters: (ℓ=0: s, ℓ=1: p, ℓ=2: d, ℓ=3: f, ℓ=4: g, ℓ=5: h, ...)
- m (or $$m_{\ell}$$) (−ℓ, 1−ℓ, ..., 0, 1, 2, ..., ℓ−1, ℓ) — because quantum mechanics does not allow us to measure every component of angular momentum, we may only know it well in one arbitrary direction. This "magnetic quantum number" which is the projection of L in our chosen direction of measurement. That is typically taken as the z-direction. $$L_z = m_{\ell} h / ( 2 \pi ) $$. This is important in figuring out how a weak magnetic field will split the emission spectrum.
- s ( −½, +½) — because the electron has its own intrinsic angular momentum this "spin quantum number" describes the component of intrinsic angular momentum in the measured direction: $$S_z = s h / ( 2 \pi ) = \pm h / ( 4 \pi )$$
- I ( −½, +½) — this describes the component of intrinsic angular momentum of the nucleus in the measured direction. For different nuclei, it will be different.
Because of magnetic interactions between the electron and its orbiting motion, the angular momentum of spin and the orbit are not conserved separately. So we need J which is L + S, with quantum numbers j and $$m_j$$ which replace ℓ, m, and s. Likewise F = I + J, which is important in hyperfine splitting which gives us the 21-cm hydrogen microwave signature used to map the galaxy and the cesium line responsible for the definition of the second.
- j (1/2, ..., n−1/2) — total angular momentum quantum number: $$| J | = \sqrt{j (j+1)} \frac{h}{2 \pi}$$ .
- $$m_j$$ ( −j, −j+1, ..., −1/2, 1/2, ..., j−1, j) — analogue of magnetic quantum number. $$J_z = j \frac{h}{2 \pi}$$ .
It might look like you can add s to ℓ and $$m_{\ell}$$ to get j and $$m_j$$ but that's a deceit because these are really labels for eigenstates of the physical systems and the nℓm states of the Schrödinger solution don't correspond simply to the njm states of the Dirac solution. The Dirac solution does not include effects from quantum electrodynamics like the Lamb shift.
https://en.wikipedia.org/wiki/Hydrogen-like_atom has a summary which pretty much requires going to quantum mechanics textbooks to cover better. Because of the variety of names, notation, and pedagogical motivations behind introducing various approximation schemes, it's difficult to find a nice summary.
http://hyperphysics.phy-astr.gsu.edu/hbase/qunoh.html#c2 has an overview of how n, l and m go into solutions of the Schrödinger equation.
So in all cases, the measured component of orbital angular momentum is an integer multiple of $$\hbar = \frac{h}{2 \pi}$$ (this is hard-coded into the math of quantum mechanics) while the measured component of intrinsic angular momentum of any electron, proton or electron is $$\pm \frac{1}{2} \hbar $$. Thus we say that these particles have spin-1/2 since $$\hbar$$ is the scale of quantum angular momentum measurements. The total angular momentum is the square-root of the sum of the squares of angular momentum component on three perpendicular axes which is for an integer, ℓ, (or possible half-integral, j) given as $$\sqrt{\ell (\ell + 1) } \hbar$$. Because of quantum physics we can't know $$L_x$$ or $$L_y$$ if we know $$L_z$$ but we can know both one component, $$L_z$$ and the total, $$|L| = \sqrt{L_x^2 + L_y^2 + L_z^2}$$.
