Total angular momentum quantum number

Template:Short description Template:Use American EnglishIn 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 $$\mathbf{𝐣}=\mathbf{𝐬}+\mathit{\ell }.$$

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: $$|\ell -s|\le j\le \ell +s$$ 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) $$\Vert \mathbf{𝐣}\Vert =\sqrt{j(j+1)}\hslash$$

The vector's z-projection is given by $$j_z=m_j\hslash$$ where m_j is the secondary total angular momentum quantum number, and the $\hslash$ is the reduced Planck constant. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of m_j.

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

See also

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• 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

• 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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