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

For ,

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