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 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]

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, 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.

See alsoEdit


  1. ^ Hollas, J. Michael (1996). Modern Spectroscopy (3rd ed.). John Wiley & Sons. p. 180. ISBN 0-471-96522-7.
  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
  • Albert Messiah, (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.

External linksEdit

  • Vector model of angular momentum
  • LS and jj coupling