Total angular momentum quantum number

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

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

If s is the particle's spin angular momentum and ℓ 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]

${\displaystyle |\ell -s|\leq j\leq \ell +s}$

where ℓ 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 m_j 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 m_j.