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, &hellip;,  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&minus;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 = &minus;l, &minus;l+1, &hellip;, l&minus;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.