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 \vec{B},

H = -\vec{\mu}\cdot \vec{B},

where \vec{\mu} is magnetic dipole moment (MDM). It is

\vec{\mu} = g \frac{q}{2 m} \vec{J} = g \frac{\mu}{\hbar} \vec{J} = \gamma \vec{J}.

Here, the g is the g-factor, \mu is magneton, and \vec{J} is the total spin, which has a intrinsic factor m\hbar / 2 inside. \gamma 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,

H = -\gamma \vec{J}\cdot \vec{B} = -\gamma J_z B = -\gamma \hbar \frac{m}{2} B [J],

Because energy is also equal E = \hbar f , thus, we can see the \gamma has unit of frequency over Tesla.


Take electron as an example, the MDM is Bohr magneton \mu_{e} = e\hbar/(2m_e). The MDM is,

\vec{\mu_e} = g_e \frac{e}{2 m_e} \vec{S} = g_e \frac{\mu_e}{\hbar}\vec{S} = \gamma_e \vec{S}.

The magnitude of MDM is,

|\vec{\mu_e}|= g_e \frac{e}{2 m_e} \frac{\hbar}{2} = \gamma_e \frac{\hbar}{2} [JT^{-1}],

The gyromagnetic ratio is,

\gamma_e = g_e \frac{\mu_e}{\hbar} [rad s^{-1} T^{-1}].

Since using rad s^{-1} is not convenient for experiment. The gyromagnetic ratio usually divided by 2\pi,

\gamma_e = g_e \frac{\mu_e}{2\pi\hbar} [Hz T^{-1}].


To evaluate the magnitude of  MDM of  single particle state, which has orbital angular momentum and spin, the total spin \vec{J} = \vec{L} + \vec{S}. However, the g-factor for \vec{L} is difference from that for \vec{S}. Thus, the MDM is not parallel to total spin. We have to use Landé Formula,

\left< JM|\vec{V}|JM'\right> = \frac{1}{J(J+1)} \left< JM|(\vec{J}\cdot\vec{V})|JM\right> \left<JM|\vec{J}|JM'\right>

or see wiki, sorry for my laziness.

The result is


For J = L \pm 1/2,

g = J(g_L \pm \frac{g_S-g_L}{2L+1})