Azimuthal quantum number

In quantum mechanics, the azimuthal quantum numberℓ is a quantum number for an atomic orbital that determines its orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum numbern, the magnetic quantum numberm_ℓ, and the spin quantum numberm_s).

For a given value of the principal quantum number n (electron shell), the possible values of ℓ are the integers from 0 to n − 1. For instance, the n = 1 shell has only orbitals with ${\displaystyle \ell =0}$, and the n = 2 shell has only orbitals with ${\displaystyle \ell =0}$, and ${\displaystyle \ell =1}$.

For a given value of the azimuthal quantum number ℓ, the possible values of the magnetic quantum number m_ℓ are the integers from m_ℓ=−ℓ to m_ℓ=+ℓ, including 0. In addition, the spin quantum number m_s can take two distinct values. The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell.

While originally used just for isolated atoms, atomic-like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids. The quantum number ℓ plays an important role here via the connection to the angular dependence of the spherical harmonics for the different orbitals around each atom.