# Non-relativistic limit of the Dirac equation with electromagnetic potentials

I'm studying the Dirac equation using Walter Greiner's textbook, my question is about the following passage

Firstly I don't understand what the author means with "the rest energy being separeted by (2.78)". Secondly I don't see how (2.78) implies in (2.79).

• Apply product rule on $\tilde{X}=X e^{-i(mc^2/\hbar)t}$ with $X=(\varphi,\chi)^T$ one gets an additional term on the left side $mc^2 X e^\theta$ with $e^\theta = e^{-imc^2/\hbar t}$which brought on the RHS together with $mc^2(\varphi,-\chi)^T e^\theta$ yields $-mc^2 (\varphi,\chi)^T e^\theta+mc^2(\varphi,-\chi)^T e^\theta =-2mc^2 (0,\chi)^T e^\theta$. Of course at the end one cancels out the phase factor $e^\theta$ on both sides. Feb 15, 2023 at 16:10
• Feb 15, 2023 at 16:22

About the rest energy: That's just a definition. They define $$\varphi$$ as $$\frac{\widetilde{\varphi}}{e^{-i (m c^2/\hbar) t}}$$ and analogously $$\chi$$. Since in the case of H not being explicitely time-dependent the general solution of the Dirac equation is a linear combination of eigenstates multiplied with $$e^{-i E t/\hbar}$$, $$\varphi$$ only changes in time with E - mc^2 (as opposed to $$\widetilde{\varphi}$$ changing in time with all of E, the same is true for $$\chi$$), they have "seperated" the rest energy. About the last equation: If you use $$\varphi = \phi \cdot e^{...}$$ and the product rule, the left hand site of (2.77) gets an extra term of $$m c^2 \cdot (\begin{array}{c} \varphi \\ \chi \end{array}) \cdot e^{...}$$ (the other term is just $$i \hbar (\frac {\partial}{\partial t} (\begin{array}{c} \varphi \\ \chi \end{array})) \cdot e^{...}$$), so if you subtract that second term on both sides and after that divide everything by $$e^{...}$$ (which you are allowed to since that's never zero), you get exactly equation (2.79).