# Shift in reference frame argument in Spin-Orbit coupling

Assume a hydrogen atom with an electron $e^{-}$of spin and orbital about a proton $e^{+}$ of no spin.

The spin of the electron produces a spin magnetic moment and its orbital motion produces a magnetic field since the electron can be model as a current proportional to charge $e^{-}$. The key idea- to my current understanding- is to model the system such that there is an interaction between the proton and the electron.

The standard argument to this is to switch the reference frame from the lab frame to the electron's frame. In this electron frame, the electron is $\mathbf{stationary}$ while the proton orbits about the electron is a counter direction to the direction of the electron as if the system were viewed from the lab frame. The proton behaves like a current loop and generates a magnetic field.

This interaction produces a potential energy generated by the spin magnetic moment of the electron and magnetic field due to the orbit of the proton.

Mathematically: $\vec{\mu_{s}}\cdot \vec{B}_{orbit}=V_{spin-orbit}$ where $\vec{\mu_{s}}$ is the spin magnetic moment of the electron and $\vec{B}_{orbit}$ is the magnetic field due to the orbiting proton.

Here's the confusion:

If the electron is stationary, in its reference frame it would experience 'no' spin and 'no' orbital motion. Instead, it would see the proton possessing a spin and an orbit.

This would then render the shift in reference moot.

How then would $\vec{\mu_{s}}\cdot \vec{B}_{orbit}=V_{spin-orbit}$ hold? The electron is stationary so 'no' spin, right?

• Stationary electrons have spin. Every electron has spin. – Jahan Claes Jul 28 '17 at 2:46
• You really shouldn't think of electrons as actually spinning. But if you really want to, you should think that we've gone into the electron's COM frame, but we haven't allowed our frame to rotate along with the electron. Think of a ball that is both travelling through the air and spinning. We start running beside it, so it looks stationary now. But we DON'T start spinning along with it, so it still looks like it's spinning. We've gone into the ball's reference frame, and the ball is still spinning. – Jahan Claes Jul 28 '17 at 2:48
• I think the COM frame was helpful. A physical analogy I tried to came up with was a sphere with x,y and z axis. The sphere spins about the z-axis so the only 'spin' is the x and y axis just rotating about the z-axis. The z-axis remains stationary. If I were 'in' that z-axis I would see the collocation of the electron's mass 'spinning' and the proton orbiting about the electron's z-axis.@JahanClaes – Physkid Jul 29 '17 at 1:04