# Physics:Total angular momentum quantum number

In quantum mechanics, the total angular momentum quantum number parametrises 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 its orbital angular momentum vector, the total angular momentum j is $\displaystyle{ \mathbf j = \mathbf s + \boldsymbol {\ell} ~. }$

The associated quantum number is the main total angular momentum quantum number j. It can take the following range of values, jumping only in integer steps:[1] $\displaystyle{ |\ell - s| \le j \le \ell + s }$ where 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) $\displaystyle{ \Vert \mathbf j \Vert = \sqrt{j \, (j+1)} \, \hbar }$

The vector's z-projection is given by $\displaystyle{ j_z = m_j \, \hbar }$ where mj is the secondary total angular momentum quantum number, and the $\displaystyle{ \hbar }$ is the reduced Planck's constant. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of mj.

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

## References

1. Hollas, J. Michael (1996). Modern Spectroscopy (3rd ed.). John Wiley & Sons. p. 180. ISBN 0 471 96522 7.