Total angular momentum quantum number

Quantum number related to rotational symmetry

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

j = s + ℓ   . {\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]

| ℓ − s | ≤ j ≤ ℓ + s {\displaystyle \vert \ell -s\vert \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)

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

The vector's z-projection is given by

j z = m j ℏ {\displaystyle j_{z}=m_{j}\,\hbar }
where mj is the secondary total angular momentum quantum number, and the ℏ {\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 mj.

The total angular momentum corresponds to the Casimir invariant of the Lie algebra so(3) of the three-dimensional rotation group.

See also

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
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Quantum numbers
  • Principal quantum number (n)
  • Azimuthal quantum number (ℓ)
  • Magnetic quantum number (m)
  • Spin quantum number (s)
Ground-state configurations
  • Periodic table (electron configurations)
  • Electron configurations of the elements (data page)
Electron fillingElectron pairingBonding participationElectron counting rules


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