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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,..., n-1) (orbital angular momentum) which defines the shape of an orbital with a particular principal quantum number
  • l = 0 → S-orbital; S for Sharp
  • l = 1 → P-orbital; P for Principle
  • l = 2 → D-orbital; D for Difuse
  • l = 3 → F-orbital, 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)

                        ml: +l → -l
                        ms: +s → -s
                        mj: +j → -j




Single Configuration

 1s2 2s2

Configuration Interaction

 a(1s2 2s2)+b(1s2 2p2)

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 1s2, 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 (1s2 2s2) and (1s2 2p2).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+1LJ (for example 2P3/2), where 2S+1 is the multiplicity and J the total angual momentum.

Example: LS or Russel - Sounders-coupling
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 spin-orbit 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= ‌ L-S ‌ ; ‌ L-S ‌ +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 multi-electron 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 Russell-Saunders Coupling, jj- or Jj-Coupling, and Jl-Coupling. The radial components can be obtained using various approximations such as the Hartree-Fock approach or the hydrogenic approach of Bates and Damgaard.


[2] Skriptum Experimentalphysik III, Institut für Experimentalphysik, Auflage WS 2001/2002

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