The Ultimate Electron magnetic dipole moment - American History Information Guide and Reference

Orbital magnetic dipole moment

Generally, for a hydrogen atomic electron in state $Ψ n, l, m$ where $n, l$ and $m$ are the principal, azimuthal and magnetic quantum numbers respectively, the total magnetic dipole moment due to orbital angular momentum is given by

$\mu=\mu_{B}\sqrt{l(l+1)}$

Where $μ B$ is the Bohr magneton.

The z-component of the orbital magnetic dipole moment is given by

$μ z = μ B m$

The total magnetic dipole moment resulting from the electron's intrinsic $\frac{1}{2}$ spin is

$\mu=\frac{1}{2}g\mu_{B}$

where $g$ is the Lande $g$ -factor.

Since spin magnitude $\frac{1}{2}$ allows only up or down spin states,

$\mu_{z}=\pm\frac{1}{2}g\mu_{B}$

As the Lande $g$ -factor is equal to 2 to within two decimal places, it is often a sufficient approximation to take

$μ = μ B$ and $\mu_{z}=\pm\mu_{B}$ .