Principal quantum number

Azimuthal and magnetic quantum numbers

An atomic shell consists of atomic subshells that are labeled by the azimuthal quantum number, commonly denoted by ℓ. The azimuthal quantum number is more often referred to as angular momentum quantum number, because the eigenvalues of the squared orbital angular momentum operator ${\hat {l}}^{2}$ are equal to ℓ(ℓ+1) ℏ².

For a given atomic shell of principal quantum number n, ℓ runs from 0 to n−1, as follows from the solution of the Schrödinger equation. In total, an atomic shell with quantum number n consists of n subshells and has

$\sum _{\ell =0}^{n-1}(2\ell +1)=n^{2}$

spatial (i.e., function of the position vector of the electron) atomic orbitals. An atomic subshell consists of 2ℓ+1 atomic orbitals labeled by the orbital magnetic quantum number, almost invariably denoted by m. For given ℓ, m runs over 2ℓ+1 values: m = −ℓ, −ℓ+1, …, ℓ−1, ℓ. The number m is proportional to the eigenvalues of the z-component of the orbital angular momentum operator ${\hat {l}}_{z}$ that has eigenvalues mℏ.

For historical reasons the orbitals of certain azimuthal quantum number ℓ are denoted by letters: s, p, d, f, g, for ℓ = 0, 1, 2, 3, 4, respectively. For instance an atomic orbital with n = 4 and ℓ = 2 is written as 4d. If the n = 4, ℓ = 2 subshell is occupied k times (there are k electrons in the 4d orbital), we indicate this by writing (4d)^k.

Any spatial atomic orbital can be occupied at most twice, which is because the spin magnetic quantum number m_s (proportional to the eigenvalues of the z-component of the spin angular momentum operator ${\hat {s}}_{z}\,$) can have only two values: +½ (spin up) and −½ (spin down). In addition, the Pauli exclusion principle states that no two electrons with the same four quantum numbers n, ℓ, m, and m_s can occupy the same atomic orbital. As a consequence, the spatial orbital (n,ℓ,m) can be occupied at most by two electrons with spin m_s = ±½. Hence a subshell can accommodate at most 2(2l+1) electrons. If a subshell accommodates the maximum number of electrons, it is called closed. If the atomic shell n contains the maximum of 2n² electrons, it is also called closed. For instance the noble gas neon in its lowest energy state has the electron configuration (1s)²(2s)²(2p)⁶, that is, all its shells and subshells are closed.