Physics:Total angular momentum quantum number
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 [math]\displaystyle{ \mathbf j = \mathbf s + \boldsymbol {\ell} ~. }[/math]
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] [math]\displaystyle{ \vert \ell - s\vert \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]\displaystyle{ \Vert \mathbf j \Vert = \sqrt{j \, (j+1)} \, \hbar }[/math]
The vector's z-projection is given by [math]\displaystyle{ j_z = m_j \, \hbar }[/math] where mj is the secondary total angular momentum quantum number, and the [math]\displaystyle{ \hbar }[/math] 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
- Canonical commutation relation § Uncertainty relation for angular momentum operators
- 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
- ↑ 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. https://archive.org/details/introductiontoel00grif_0.
- Albert Messiah, (1966). Quantum Mechanics (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
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