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 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: 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) The vector's z-projection is given by where mj is the secondary total angular momentum quantum number, and the 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.