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{{see|Azimuthal quantum number#Addition of quantized angular momenta}} | |||
In [[quantum mechanics]], the '''total angular momentum quantum number''' parameterises the total [[angular momentum]] of a given [[Subatomic particle|particle]], by combining its [[angular momentum operator|orbital angular momentum]] and its intrinsic angular momentum (i.e., its [[spin (physics)|spin]]). | |||
If '''s''' is the particle's spin angular momentum and '''ℓ''' its orbital angular momentum vector, the total angular momentum '''j''' is | |||
= | :<math>\mathbf j = \mathbf s + \boldsymbol {\ell} </math> | ||
The. | 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: | ||
:<math>|\ell - s| \le j \le \ell + s</math> | |||
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]]) | |||
:<math> \Vert \mathbf j \Vert = \sqrt{j \, (j+1)} \, \hbar</math> | |||
the vector's ''z''-projection is given by | |||
:<math>j_z = m_j \, \hbar</math> | |||
where ''m<sub>j</sub>'' is the '''secondary total angular momentum quantum number'''. It ranges from −''j'' to +''j'' in steps of one. This generates 2''j'' + 1 different values of ''m''<sub>''j''</sub>. | |||
The total angular momentum corresponds to the [[Casimir invariant]] of the [[Lie algebra]] [[infinitesimal rotation|'''so'''(3)]] of the three-dimensional [[Rotation group SO(3)|rotation group]]. | |||
==See also== | |||
* [[Principal quantum number]] | |||
* [[Azimuthal quantum number|Orbital angular momentum quantum number]] | |||
* [[Magnetic quantum number]] | |||
* [[Spin quantum number]] | |||
* [[Angular momentum coupling]] | |||
* [[Clebsch–Gordan coefficients]] | |||
* [[Angular momentum diagrams (quantum mechanics)]] | |||
==References== | |||
*{{cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=0-13-805326-X}} | |||
==External links== | |||
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html Vector model of angular momentum] | |||
* [http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/lcoup.html#c1 LS and jj coupling] | |||
{{Electron configuration navbox}} | |||
[[Category:Atomic physics]] | |||
[[Category:Rotational symmetry]] | |||
[[Category:Rotation in three dimensions]] | |||
Latest revision as of 16:02, 7 August 2013
Template:See 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).
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:
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 mj is the secondary total angular momentum quantum number. 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
- 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)
References
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