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
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]
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)
The vector's z-projection is given by
The total angular momentum corresponds to the Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.