magnetic quantum number (m)
The magnetic quantum number, written m (or m_ℓ), tells you not how much angular momentum a particle has but which way it points — specifically, the size of its component along a chosen axis, usually called the z-axis. For a state with azimuthal number ℓ, m can take the integer values from −ℓ up to +ℓ, giving 2ℓ + 1 allowed orientations in all. Each one corresponds to a measured z-component of exactly m·ℏ.
The name comes from magnetism. With no external field, all these orientations have the same energy, so they are hidden from view. Switch on a magnetic field and the different values of m acquire slightly different energies, because angular momentum carries a magnetic moment that prefers to line up with the field. The once-overlapping energy levels fan out, and a single spectral line splits into a small family — the Zeeman effect.
It is important to be honest about what m describes. It is not a literal compass needle frozen at some angle; the orientation is itself quantum-mechanical, and the components perpendicular to the chosen axis remain fundamentally undefined. Choosing m fixes one component sharply, at the price of leaving the others fuzzy — a direct expression of how angular momentum obeys the uncertainty principle.
For each ℓ there are 2ℓ+1 orientations, each with a z-component of m·ℏ.
Do not confuse this m with the spin magnetic number m_s. The orbital m runs in integer steps from −ℓ to +ℓ; the spin version takes only the two half-integer values ±1/2 for an electron.