In relativistic quantum mechanics, the solution of the free Dirac equation is assumed to be $$\Psi(\textbf{r},t)=u(\textbf{p})e^{i(\textbf{p}\cdot \textbf{r}-Et)}$$ How do I know that this is the most general separable solution?
I was trying to derive this explicitly by the method of separation of variables as $$\Psi(\textbf{r},t)=u(\textbf{p})\Phi(\textbf{r})T(t)$$ Substituting, this into the free Dirac equation I trivially obtained $$T(t)\sim e^{-iEt}$$ but I cannot solve the space-part of the equation (in a representation, I used the Dirac Pauli representation). The equation I'm stuck with is: $$-i\frac{\alpha\cdot\nabla\Phi}{\Phi}u+\beta mu=Eu$$ where $E$ is the separation constant (which is dimensionally the energy). How to solve this part to show that $\Phi\sim e^{i\textbf{p}\cdot \textbf{r}}$. Any suggestion in this regard will be helpful.
$\bullet$ I have written the equation in natural units i.e., $c=\hbar=1$.