Quantum Numbers
By solving the Schrödinger equation (HΨ = EΨ), we obtain a set of mathematical equations, called wave functions (Ψ). Each wave function is associated with an energy level E, eigenvalue of the Hamilonian operator H.
The wave functions describe the spatial probability distribution of the
electrons: The probability of finding an electron of a certain energy
at some position in space.
A wave function for an electron in an atom is called an atomic orbital;
this atomic orbital describes a region of space in which there is a
high probability of finding the electron. Energy changes within an atom
are the result of an electron changing from a wave pattern with a given
energy to a wave pattern with a different energy (usually accompanied
by the absorption or emission of a photon of light).
Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) is related to the spin.

n = main quantum number (1, 2, 3…). This number specifies the energy of
an electron and the orbital size, it is related to the radial part of
Schrödingers equation
 l = azimuthal quantum number
(0,..., n1) (orbital angular momentum) which defines the shape of an
orbital with a particular principal quantum number
 l = 0 → Sorbital; S for Sharp
 l = 1 → Porbital; P for Principle
 l = 2 → Dorbital; D for Difuse
 l = 3 → Forbital, F for Fundamental
 s = spin quantum number (1/2). This one specifies the orientation of the spin axis of an electron.
j = inner quantum number (l ± s) = (l ± 1/2) > 0) (total angular Momentum
m = magnetic quantum number number indicates the orbital orientation in space of a given energy (n) and shape (l)
m_{l}: +l → l
m_{s}: +s → s
m_{j}: +j → j
Element 
Be 
Single Configuration 
1s^{2 }2s^{2} 
Configuration Interaction 
a(1s^{2 }2s^{2})+b(1s^{2 }2p^{2}) 
Each electron in the atom is given a unique set of 4 quantum numbers.
This is called the Single Configuration Approximation. Normally
only the first two quantum numbers are shown as these determine the
electron energy. For example, the ground state (ie the state with
the lowest energy) of Helium, which has two electrons, is 1s^{2},
where 1 is the value of n, s the value of l and the superscript ^{2} is the number of equivalent electrons.
A more complete description of the electron states involves Configuration
Interaction,
ie a correction is included for the distortions of the wave functions
caused by the interactions between electrons. An exemple is given in
the adcacent table.The ground state of Berylium, having for electrons,
is not given by a single configuration, but by a mixture of (1s^{2 }2s^{2}) and (1s^{2 }2p^{2}).This
configuration interaction is due to the fact that the 4 electrons do
not behave exactly like four independent electrons such but the do
interact. As a result, the one electron states, derived for Hydrogen,
do not exactly fit the multi electron system.
In spectroscopy, an energy level is represented as ^{2S+1}L_{J } (for example ^{2}P_{3/2}), where 2S+1 is the multiplicity and J the total angual momentum.
Example: LS or Russel  Sounderscoupling
In this case angular momentums (li) of the individual electrons are
coupled to form the total orbital angular momentum (L) and the spins
(si) of the electrons for the total spin (S). These two angular momenta
then couple by spinorbit interaction to form the total angular
momentum (J) of the atom.Depding on the relative orientation of L and S
the total angular momentum J may take different values :
J= LS ; LS +1;....; L+S
An
other coupling mechanism is the jj – coupling which can be seen at
heavy atoms.In this case the orbital angular momentum and the spin of a
single electron in the atom will couple to form the total angular
momentum of this electron. All the total angular momenta of the
electrons in the atom will then couple to form the total angular
momentum of the atom.
The intensity of an emission line depends on the number of emitting
atoms and the likelihood of the transition, described by its Atomic
Transition Probability (ATP). An ATP depends on the angular and radial
components of the wave functions involved. No exact solutions exist for
multielectron atoms. The angular components can be solved for many
transitions using Racah coefficients with various schemes that approximate
the coupling of the angular momenta of the electrons. The most important
schemes are LS or RussellSaunders Coupling, jj or JjCoupling,
and JlCoupling. The radial components
can be obtained using various approximations such as the HartreeFock
approach or the hydrogenic approach of Bates and Damgaard.
References:
[1] http://en.wikipedia.org/wiki/Image:Nalamp3.jpg
[2] Skriptum Experimentalphysik III, Institut für Experimentalphysik, Auflage WS 2001/2002
[3] http://de.wikipedia.org/wiki/Bild:NaDterms.png
First published on the web: 15 November 1999 by Richard Payling
Authors of the latest version (March 2008): Lara Lobo Revilla and Karl Preiss . The text is based on a lecture given by Prof E.B. Steers at the first Gladnet training course in Antwerp Sept. 2007
