In quantum mechanics
, the total angular quantum momentum numbers
parameterize 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 values:
where 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. 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.
- Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.