Total angular momentum quantum number

The total angular quantum momentum number parameterizes the total angular momentum of a given electron, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).

If s is the electron spin angular momentum and l its orbital angular momentum vector, the total angular momentum j is

$\mathbf {j} =\mathbf {s} +\mathbf {l}$

The associated quantum number j can take the following values:

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

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 of angular momentum quantum numberss

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

the vector's z-projection is given by

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

where m_j ranges from −j to +j in steps of one. This generates 2j + 1 different values of m_j.

LS coupling

In monoelectronic atoms or light atoms in relatively weak magnetic fields, electron spins interact among themselves so they combine to form a single spin angular momentum S and so do orbital angular momenta, forming a single orbital angular momentum L. This is called Russell-Saunders coupling or LS coupling. Then S and L add together and form a total angular momentum J:

$\mathbf {J} =\mathbf {L} +\mathbf {S}$ where $\mathbf {L} =\sum _{i}\mathbf {l} _{i}$ and $\mathbf {S} =\sum _{i}\mathbf {s} _{i}$

This situation is valid as long as extern magnetic fields are weak, so the coupling between orbital and spin angular momenta is stronger than with the external magnetic field. Strong magnetic fields cause these two momenta to decouple (Paschen-Back effect) which gives rise to a different splitting pattern in the energy levels.

jj coupling

In heavier atoms the situation is different. Having bigger nuclear charges, interactions spin-orbit become so prominent as spin-spin interactions or orbit-orbit interactions. In this situation, each orbital angular momentum tends to combine with each individual spin angular momentum, originating individual total angular momenta. These then add up to form the total angular momentum J

$\mathbf {J} =\sum _{i}\mathbf {j} _{i}=\sum _{i}(\mathbf {l} _{i}+\mathbf {s} _{i})$

This situation is known as jj coupling.

See also

• Principal_quantum_number
• orbital angular momentum quantum number
• magnetic quantum number
• spin angular momentum quantum number

External links

• Vector model of angular momentum
• LS and jj coupling