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 (i.e., its spin).

The total angular momentum corresponds to the Casimir invariant of the Lie algebra **so**(3) 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:^{[1]}

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 *m _{j}* 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 2

*j*+ 1 different values of

*m*

_{j}.

## 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

**^**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.