Deriving the Pauli-Schrödinger equation from the Dirac equation

Since the Schrödinger Pauli equation describes a non-relativistic spin ½ particle. This equation must be an approximation of the Dirac equation in an electromagnetic field. I was trying to derive this but I got stuck at a point. The free particle Dirac can be reduced to the equations \begin{align} σ^{i}(p_{i}+eA_{i}) u_B & = (E-m+eA_0)u_A. \\ \sigma^{i}(p_{i}+eA_{i})u_A & = (E+m+A_{0})u_B \end{align}

I multiplied both sides of the first equation by $$(E+m+eA_0)$$ to get the Schrödinger Pauli equation. I was not able to eliminate $$u_B$$ completely from the equation. Can someone help me with the derivation?

• Hi user215742, I've started MathJax for you, but please edit the post to ensure it's done correctly. For reference on the math fonts, search "notation" in help center. – Kyle Kanos Mar 25 at 14:34
• Which components of $A_\mu$ are non-zero here? – GuestGuestGuest1 Mar 25 at 14:40
• @GuestGuestGuest1 none of the components are zero. – Manvendra Somvanshi Mar 25 at 15:12
• Solve second for $u_B$ and plug it back in first equation. – Sunyam Mar 25 at 20:20
• You can find more information about this by searching for Foldy-Wouthuysen transformation. – Chiral Anomaly Mar 26 at 0:54