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 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]

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

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.

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