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 j = s + ℓ   . {\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: | ℓ − s | ≤ j ≤ ℓ + s {\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) ‖ j ‖ = j ( j + 1 ) ℏ {\displaystyle \Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar }

The vector's z-projection is given by j z = m j ℏ {\displaystyle j_{z}=m_{j}\,\hbar } where mj is the secondary total angular momentum quantum number, and the ℏ {\displaystyle \hbar } is the reduced Planck 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.