Total angular momentum quantum number

In quantum mechanics, the total angular quantum momentum numbers parameterize the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).

If s is the particle's spin angular momentum and l its orbital angular momentum vector, the total angular momentum j is

:$mathbf\; j\; =\; mathbf\; s\; +\; mathbf\; l$

The associated quantum number is the main total angular momentum quantum number "j". It can take the following values:

:$|ell\; -\; s|\; le\; j\; le\; ell\; +\; s$

where $scriptstyleell$ is the azimuthal quantum number (parameterizing the orbital angular momentum) and "s" is the spin quantum number (parameterizing the spin).

The relation between the total angular momentum vector j and the total angular momentum quantum number "j" is given by the usual relation (see angular momentum quantum number)

:$Vert\; mathbf\; j\; Vert\; =\; sqrt\{j\; ,\; (j+1)\}\; ,\; hbar$

the vector's "z"-projection is given by

:$j\_z\; =\; m\_j\; ,\; hbar$

where "m_j" is the secondary total angular momentum quantum number. It ranges from −"j" to +"j" in steps of one. This generates 2"j" + 1 different values of "m"_"j".

The total angular momentum corresponds to the Casimir invariant of the Lie algebra SO(3) of the three-dimensional rotation group.

ee also

* principal quantum number
* orbital angular momentum quantum number
* magnetic quantum number
* spin quantum number
* angular momentum coupling
* Clebsch-Gordan coefficients

References

*cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |id=ISBN 0-13-805326-X

External links

* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html Vector model of angular momentum]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/lcoup.html#c1 LS and jj coupling]