I Relativistic quantum mechanics

[PeroK]

For example, the Dirac equations should be of the form:
$$(\partial_0 - \partial_1 + i\partial_2)\psi_1 + \partial_3 \psi_2 = - im\psi_4$$
Plus three more similar equations. There's no immediate way to decouple these: they are fundamentally equations for a four-component spinor. Certainly the SDE is nowhere to be seen.

 

[vanhees71]

Sure you get the non-relativistic limit by a formal expansion with respect to powers of $1/c$, and also choosing a convenient representations for the Dirac matrices to do that. E.g., if you minimally couple the em. field you are lead to the Pauli equation with a Pauli spinor (throughing away the anti particles). All this can be found in Bjorken&Drell.

 

[PeterDonis]

the Schrodinger equation is a component of the Dirac equation

No, it isn't, it's a non-relativistic approximation to it, as has already been pointed out.

The OP question has been answered. Thread closed.