Total angular momentum quantum number

Further information: Azimuthal quantum number § Addition of quantized angular momenta

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 $${\displaystyle \mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}~.}$$

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] $${\displaystyle \vert \ell -s\vert \leq j\leq \ell +s}$$ 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) $${\displaystyle \Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar }$$

The vector's z-projection is given by $${\displaystyle j_{z}=m_{j}\,\hbar }$$ where m_j is the secondary total angular momentum quantum number, and the ${\displaystyle \hbar }$ 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

• 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