## Monday, November 3, 2014

### Quantum Numbers

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of quantum numbers of electrons, they can be defined as "The sets of numerical values which give acceptable solutions to theSchrödinger wave equation for the Hydrogen atom". Perhaps the most important aspect of quantum mechanics is thequantization of observable quantities, since quantum numbers are discrete sets of integers or half-integers, although they could approach infinity in some cases. This is distinguished from classical mechanics where the values can range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentumspin, etc. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.
There are four quantum numbers which can describe the electron completely.
The principal quantum number (n) describes the electron shell, or energy level, of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom.
The azimuthal quantum number () (also known as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum.
The magnetic quantum number (m) describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis.
The spin projection quantum number (ms) describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis. An electron has spin s = ½, consequently ms will be ±½, corresponding with "spin" and "opposite spin." Each electron in any individual orbital must have different spins because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.
NameSymbolOrbital meaningRange of valuesValue examples
principal quantum numbernshell1 ≤ nn = 1, 2, 3, …
azimuthal quantum number (angular momentum)subshell (s orbital is listed as 0, p orbital as 1 etc.)0 ≤  ≤ n − 1for n = 3:
= 0, 1, 2 (s, p, d)
magnetic quantum number, (projection ofangular momentum)menergy shift (orientation of the subshell's shape) ≤ m ≤ for  = 2:
m = −2, −1, 0, 1, 2
spin projection quantum numbermsspin of the electron (−½ = "spin down", ½ = "spin up")s ≤ ms ≤ sfor an electron s = ½,
so ms = −½, ½