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).

Further information: Azimuthal quantum number § Addition of quantized angular momenta

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: Canonical commutation relation § Uncertainty relation for angular momentum operators

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