In Schroedinger equation, which is second order differential equation, one normally, equates both $\psi(x)$ and $\psi'(x)$ across the boundary, as boundary conditions. However, the dirac equation is a first order differential equation. So, if one were to have a dirac equation like $$\gamma^{\mu}\partial_{\mu}\psi -m\psi + V(x)\psi = 0,$$ where lets say $V(x)$ is a potential barrier, would one require to equate both $\psi$ and $\psi'$ as boundary conditions? Since Dirac equation is a 1st order differential equation, mathematically, is seems only $\psi(x)$ needs to be equated across the boundary. Also, since Dirac equation is a matrix equation, equating both $\psi(x)$ and $\psi'(x)$ would lead to over constraining the system; and rendering it unsolvable.

Another point of confusion on this is that a particle satisfying Dirac equation would satisfy klein gordon equation which is a second order differential equation.



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