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
![{\displaystyle \mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a082e9ca912d5becaedb4e461672914076046d)
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 \vert \ell -s\vert \leq j\leq \ell +s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b48c97a66521161958457e8e6d0dbb843e3b4bc3)
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac043be2745472d5fb33a19a7da2371b7db7f3b)
The vector's z-projection is given by
![{\displaystyle j_{z}=m_{j}\,\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00d438233276e29fa293a275a6d9fc4c86b943c)
where
mj is the
secondary total angular momentum quantum number, and the
![{\displaystyle \hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41)
is the
reduced Planck constant. It ranges from −
j to +
j in steps of one. This generates 2
j + 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
- ^ Hollas, J. Michael (1996). Modern Spectroscopy (3rd ed.). John Wiley & Sons. p. 180. ISBN 0-471-96522-7.
External links