Total angular momentum quantum number

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

${\displaystyle \mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}~.}$

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]

${\displaystyle |\ell -s|\leq j\leq \ell +s}$

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)

${\displaystyle \Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar }$

The vector's z-projection is given by

${\displaystyle j_{z}=m_{j}\,\hbar }$

where m_j is the secondary total angular momentum quantum number, and the ${\displaystyle \hbar }$ is the reduced Planck's constant. It ranges from −j to +j in steps of one. This generates 2j + 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

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 links

• Vector model of angular momentum
• LS and jj coupling