# Why do we need to account for relativity to describe spin-orbit coupling

I am struggling to understand why one has to rely on relativity to explain spin-orbit coupling.

In fact, I always thought that the very nature of spin arises when one includes relativity into the Schrödinger equation. While it is true that spin naturally arises from Dirac's equation, I also read that alternative non-relativistic quantum formalism such as the Lévy-Leblond equation are able to describe the "emergence" of spin. Consequently, the spin is a purely quantum property and one does not need to rely on relativity to explain its nature.

However, the spin-orbit coupling is always refereed to as a relativistic effect. What I dont understand is that, since spin and (orbital) angular motion are non-relativistic and spin-orbit operator can be expressed based on classical argument, we still call spin-orbit a relativistic effects.

Although I understand that accounting for relativity is needed to FULLY describe the SO coupling, why is it so difficult to find a clear explanation that separates the classical part from the relativistic one.

I have to emphasis that I am chemist, working in the field of quantum chemistry. As such I have, for example, only little knowledge about representation theory. Hence, I am more seeking for an "intuitive" (if that is possible when mixing quantum theory and relativity ...) explanation. However I have some good understanding of quantum mechanics in general and some basic knowledge of (special) relativity.

• The electron doesn't interact with itself (unless you get into quantum field theory). The moving electron does indeed produce a $\mu e v/2\pi r$ field but that only effects other electrons and the nucleus and is a smaller effect. It's the transformed $Ze/4 \pi \epsilon r^2$ that gives the spin-orbit energy shift. – RogerJBarlow Jul 4 '18 at 16:05