Principal quantum number

The principal quantum number, usually designated by n, is a quantum number appearing in the description of the electronic structure of atoms. The number arises naturally in the solution of the Schrödinger equation for hydrogen-like atoms. It is a positive integral number, n = 1, 2, 3, …, labeling atomic shells. An atomic shell consists of atomic subshells that in turn are labeled by the azimuthal quantum number, commonly denoted by l. 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

$\sum _{l=0}^{n-1}2l+1=n^{2}$

atomic orbitals.

An atomic subshell consists of 2l+1 atomic orbitals labeled by the magnetic quantum number, almost invariably denoted by m. For given l, the magnetic quantum number runs over 2l+1 values: m = −l, −l+1, …, l−1, l.

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 = 3 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 atomic orbital can be occupied at most twice, once with electron spin up and once with spin down. Hence a subshell can accommodate at most 2(2l+1) electrons. If it does, the subshell is called closed. If the atomic shell n contains 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.