8.3 Electron spin  (Page 3/9)

Even in the ground state ( $l=0$ ), a hydrogen atom has magnetic properties due the intrinsic (internal) electron spin. The magnetic moment of an electron is proportional to its spin.

For all electrons, $s=\text{½}$ and $m_s=\pm \text{½}.$ As we will see, not all particles have the same spin quantum number. For example, a photon as a spin 1 ( $s=1$ ), and a Higgs boson has spin 0 ( $s=0$ ).

An electron has a magnetic moment associated with its intrinsic (internal) spin. Spin-orbit coupling occurs when this interacts with the magnetic field produced by the orbital angular momentum of the electron.

Spin up (relative to positive z -axis):
$\theta =55\text{°}.$
Spin down (relative to positive z -axis):
$\theta ={\text{cos}}^{\mathrm{-1}}\left(\frac{S_z}{\text{S}}\right)={\text{cos}}^{\mathrm{-1}}\left(\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)={\text{cos}}^{\mathrm{-1}}\left(\frac{\mathrm{-1}}{\sqrt{3}}\right)=125\text{°}.$

The spin projection quantum number is $m_s=\pm \text{½}$ , so the z- component of the magnetic moment is
${\mu }_z=\pm {\mu }_{\text{B}}.$
The potential energy associated with the interaction between the electron and the external magnetic field is
$U=\mp {\mu }_{\text{B}}B.$
The energy difference between these states is $\text{Δ}E=2{\mu }_{\text{B}}B$ , so the wavelength of light produced is
$\lambda =8.38\times {10}^{\mathrm{-5}}\text{m}\approx 84\mu \text{m}$

It is increased by a factor of 2.