total angular momentum quantum number

{{Short description|Quantum number related to rotational symmetry}}

{{further|Azimuthal quantum number#Addition of quantized angular momenta}}

{{Use American English|date=January 2019}}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

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

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:{{cite book |last1=Hollas |first1=J. Michael |title=Modern Spectroscopy |date=1996 |publisher=John Wiley & Sons |isbn=0-471-96522-7 |page=180 |edition=3rd}}

$$\backslash vert\; \backslash ell\; -\; s\backslash vert\; \backslash le\; j\; \backslash le\; \backslash ell\; +\; s$$

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)

$$\backslash Vert\; \backslash mathbf\; j\; \backslash Vert\; =\; \backslash sqrt\{j\; \backslash ,\; (j+1)\}\; \backslash ,\; \backslash hbar$$

The vector's z-projection is given by

$$j\_z\; =\; m\_j\; \backslash ,\; \backslash hbar$$

where m_j is the secondary total angular momentum quantum number, and the $\backslash hbar$ 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.