Total angular momentum quantum number

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

In quantum mechanics, the total angular quantum momentum 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

$\mathbf {j} =\mathbf {s} +{\boldsymbol {\ell }}$

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:

$|\ell -s|\leq j\leq \ell +s$

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)

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

the vector's z-projection is given by

$j_{z}=m_{j}\,\hbar$

where m_j 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 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

• principal quantum number
• orbital angular momentum quantum number
• magnetic quantum number
• spin quantum number
• angular momentum coupling
• Clebsch–Gordan coefficients

References

• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.

External links

• Vector model of angular momentum
• LS and jj coupling