The Energy of Spin-Orbit Coupling

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The energy stored in the interaction between a magnetic moment and external magnetic field B is:
(1.1)
We want to show that in quantum mechanics this is proportional to the dot product between the intrinsic angular momentum (spin) operator of the electron and the electron's angular momentum operator

The electron can be modeled as a charged sphere rotating about an axis, say

Say the electron has a constant charge density, radius R and rotates at angular speed
a ring of radius r carrying current i has a magnetic moment of
(1.2)
And the direction is normal to the plane of the ring
A spherical ring located at distance r from the center of the sphere contains
(1.3)

If the ring completes a revolution at time T then its angular velocity is and the current in the ring is:
(1.4)
The area of the ring is, so the contribution from a single ring is:
(1.5)
The total moment of the sphere is the sum of all the contributions of all the rings - the integral over the entire volume:
(1.6)
Note that
(1.7)
Thus:
(1.8)
And if we use where Q is the total charge of the sphere we have:
(1.9)
What is the angular momentum of such a sphere. If the mass of the sphere is then its moment of inertia is
(1.10)
And the angular momentum is therefore:
(1.11)
We see that the intrinsic angular momentum vector is proportional to the magnetic moment:
(1.12)

Now comes the external magnetic field.
Where does the external magnetic field felt by the electron in the hydrogen atom come from?
The only external source for such a thing is the proton. In the electron's rest frame it is the proton that revolves around the electron.
Since we have a charge that moves in a circle of radius r with angular velocity can be modeled as a current-carrying ring with current (same as we did with the magnetic moment):
(1.13)

 
According to Biot-Savart law the contribution of a current element is given by:
(1.14)
Where is the angle between dl and r, which in the case of magnetic field in the center of the ring is always
So the total magnetic field is given by:
(1.15)
Thus the field (which is perpendicular to plane of the ring):
(1.16)
Since the radius of the orbit is constant, we can write the field as:
(1.17)
Where is some constant.
The orbital angular momentum of the electron is given by:
(1.18)
So we see that the angular momentum vector is proportional to the magnetic field (remember that the radius is constant)
(1.19)
 
From the point of view of the electron the system looks like:

Thus, the energy stored is:
(1.20)
Or simply:
(1.21)
Where is a proportionality constant.

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