# Spin Orbit Interaction Hamiltonian

Spin Orbit Interaction Hamiltonian is defined as follows:

$H_{SO}=\frac{1}{2m_{e}c^2}\frac{1}{r} \left(\frac{\partial V}{\partial r}\right)L\cdot S$

How does one derive the above Spin Orbit Interaction Hamiltonian from relativistic treatment? Is there a good textbook that elaborates on this?

Last edited:

Dr Transport
Gold Member
Messiah, Sakaurai, Cohen-Tannouji, Bjorken & Drell, all of those texts have the spin-orbit interaction defined in them and their derivations....

Dr Transport said:
Messiah, Sakaurai, Cohen-Tannouji, Bjorken & Drell, all of those texts have the spin-orbit interaction defined in them and their derivations....
Thank you!

Thomas precession

It might be worth mentioning that the "simple derivation" presented in most or perhaps all of those textbooks obtains a final result that is in error by approximately a factor of two. To get the right answer, it is important to carry out a relativistic analysis of what is often called the "Thomas precession" effect. This is addressed at some length in Jackson's textbook on E&M, in Chapter 11 or 12, I believe.

QMfunster said:
It might be worth mentioning that the "simple derivation" presented in most or perhaps all of those textbooks obtains a final result that is in error by approximately a factor of two. To get the right answer, it is important to carry out a relativistic analysis of what is often called the "Thomas precession" effect. This is addressed at some length in Jackson's textbook on E&M, in Chapter 11 or 12, I believe.
if we began from Dirac equation, we can obtain the Hamiltonian just like the form in Jackson book. i find L. I. Schiff's book extremely well explained.

Yes, I agree. Starting from relativistic quantum theory is definitely the most straightforward way to get to the correct result.