PX262 - G6b - spin-orbit coupling

in the reference frame of an electron orbiting a nucleus, the nucleus is 'seen' to be orbiting it
the electron is at the centre of a current carrying loop , so it is subject to magnetic field given by the biot-savart law:

B = \frac{μ_{0} Z e ω}{4 π r} = \frac{μ_{0} Z e}{4 π m r^{3}} l

where, $\frac{Z e ω}{2 π}$ corresponds to the current

the energy of the interaction is:

W = - \frac{1}{2} {\vec{μ}}_{s} \cdot \vec{B} = \frac{μ_{0} Z e^{2}}{8 π m^{2} r^{3}} \vec{l} \cdot \vec{s}

where, the factor of $1 / 2$ comes from a relativistic effect known as thomas precession

$\vec{l} \cdot \vec{s}$ can be expressed through the total angular momentum:

\begin{aligned} \vec{j} & = \vec{l} + \vec{s} \\ ⟹ j^{2} & = l^{2} + s^{2} + 2 \vec{l} \cdot \vec{s} \\ ∴ \vec{l} \cdot \vec{s} & = \frac{1}{2} (j^{2} - l^{2} - s^{2}) \end{aligned}

therefore, the interaction energy is:

W = \frac{μ_{0} Z e^{2}}{16 π m^{2} r^{3}} (j^{2} - l^{2} - s^{2})

using postulate 3, the interaction energy operator can be written as:

{\hat{H}}_{L S} = f (r) ({\hat{J}}^{2} - {\hat{L}}^{2} - {\hat{S}}^{2})

where, $f (r) = \frac{μ_{0} Z e^{2}}{4 π m^{2} r^{3}}$

this is called spin-orbit (LS) coupling
additional note:
- if $s = \frac{1}{2 :} j = l \pm \frac{1}{2} \to$ 2 states, splits into $2$
- but, if $s = 1 / 2$ and $l = 0 : j = \frac{1}{2} \to 1$ state, no splits
- therefore, each state with a given value of $j$ consists of $(2 j + 1)$ degenerate components
- these can be separated by an applied magnetic field