I always confuses on the definition, and wiki did not have any summary. so,
The Original definition is the Hamiltonian of a magnetic dipole under external magnetic field ,
where is magnetic dipole moment (MDM). It is
Here, the is the g-factor, is magneton, and is the total spin, which has a intrinsic factor inside. is gyromegnetic ratio.
We can see, the g-factor depends on the motion or geometry of the MDM. For a point particle, the g-factor is exactly equal to 2. For a charged particle orbiting, the g-factor is 1.
Put everything into the Hamiltonian,
Because energy is also equal , thus, we can see the has unit of frequency over Tesla.
Take electron as an example, the MDM is Bohr magneton . The MDM is,
The magnitude of MDM is,
The gyromagnetic ratio is,
Since using is not convenient for experiment. The gyromagnetic ratio usually divided by ,
To evaluate the magnitude of MDM of single particle state, which has orbital angular momentum and spin, the total spin . However, the g-factor for is difference from that for . Thus, the MDM is not parallel to total spin. We have to use Landé Formula,
or see wiki, sorry for my laziness.
The result is