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 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.
See also
Principal quantum number Orbital angular momentum quantum number Magnetic quantum number Spin quantum number Angular momentum coupling Clebsch–Gordan coefficients Angular momentum diagrams (quantum mechanics) Rotational spectroscopy
References
Albert Messiah, (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
External links Vector model of angular momentum LS and jj coupling