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.
The total angular momentum corresponds to the Casimir invariant of the Lie algebra so 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
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 and s is the spin quantum number.
The relation between the total angular momentum vector j and the total angular momentum quantum number j is given by the usual relation
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.
