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

<math display="block">\mathbf j = \mathbf s + \boldsymbol {\ell} ~.</math>

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:

<math display="block">\vert \ell - s\vert \le j \le \ell + s</math>

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)

<math display="block"> \Vert \mathbf j \Vert = \sqrt{j \, (j+1)} \, \hbar</math>

The vector's z-projection is given by

<math display="block">j_z = m_j \, \hbar</math>

where m<sub>j</sub> is the secondary total angular momentum quantum number, and the <math> \hbar</math> is the reduced Planck constant. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of m<sub>j</sub>.

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.
  • Vector model of angular momentum
  • LS and jj coupling