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.

In the Bohr-Sommerfeld ("old") quantum theory, the electron in a hydrogen-like (one-electron) atom moves in an elliptic orbit. The principal quantum number appears in this theory at two places: in the energy E of the electron and in the length a of the major axis of the ellipse.

$E=-{\frac {m_{e}}{2}}\left({\frac {Z\,e^{2}}{4\pi \epsilon _{0}\hbar \;n}}\right)^{2}\quad {\hbox{and}}\quad a={\frac {4\pi \epsilon _{0}\;n^{2}\hbar ^{2}}{m_{e}\,Z\,e^{2}}}$

Here m_e is the mass of the electron, e is the charge of the electron, Ze is the charge of the nucleus, ε₀ is the electric constant, and $\hbar$ is Planck's reduced constant. In the "new" quantum mechanics the energy E of a hydrogen-like atom is exactly the same, but an electron orbit is replaced by an electron orbital that has no radius. However, in the new quantum theory a appears as the expectation value of r (the length of the position vector of the electron).

Strictly speaking, the principal quantum number is not defined for many-electron atoms. However, in a fairly good approximate description (central field and independent particles) of the many-electron atom, the principal quantum number does appear and hence n is a label that is often applied to many-electron atoms as well.

Azimuthal and magnetic quantum numbers

An atomic shell consists of atomic subshells that 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.