Moving on from Dirac's equation as an undergrad In my QM class, we're covering the basics of attempting to reconcile QM with special relativity. From what I understand, Dirac took the definition $E=\sqrt{p^2c^2+m^2c^4}$ and used it for the Schrödinger Equation. It turns out the form we want is: $$E=c\vec{\alpha}\cdot\vec{p}+\beta mc^2$$ where $\alpha$ and $\beta$ are 4x4 matrices. Is this a Hamiltonian? Does this mean I simply write: $$i\hbar\dfrac{\partial}{\partial t}|\psi\rangle = E|\psi\rangle$$It seems wrong since I don't see any potential terms. 
I tried looking around, and apparently this is a result for spin 1/2 particles? How did spin come in play here, and why 1/2? I know that the matrices involve the Pauli spin matrices, but I can't see how spin came in. In essence, I don't really know what to do now with this result and how to interpret its meaning. Can I get an undergraduate level explanation (preferably no QFT) on what Dirac's equation implies and how it does so? Wikipedia says it implies the existence of antiparticles and this just sounds so interesting. 
 A: This is what I know:
Dirac's motivation was a quantum mechanical equation for electrons that would give much more accurate treatment of atomic spectra.
Without getting into mathematical technicalities:


*

*We know we need at least 2 components wave functions to deal with the $\frac{1}{2}$ spin of an electron. 

*The main goal is to fully incorporate SR. One requirement is parity transformations. It turns out that the $(\frac{1}{2},0)$ representations transforms to the other, $(0,\frac{1}{2})$ under a parity transformation. So even if you don't want $4$ complex component wave functions, you don't have a choice.


This however makes the equation capable of expressing a multitude of representations with one equation.

Take the Klein-Gordon equation for a scalar function of a particle of mass $m$: $(\partial^2 + m^2)\psi=0$.
Dirac wanted the a linear equation. Is there a "square root" of $(\partial^2 + m^2)$?
Squaring the ansatz gives $( \frac{1}{2}\{\gamma ^ \mu , \gamma^{\nu} \} \partial_{\mu} \partial_{\nu}  + m^2)\psi=0$
To regain the original equation Dirac defined $\{\gamma ^ \mu , \gamma^{\nu} \} = 2 \eta^{\mu \nu}$
The resultants are the gamma matrices, and the linear equation is $$(i\gamma ^ \mu \partial_\mu - m)\psi=0 $$ or $$(i \partial\!\!\!/ - m)\psi=0 $$ using the convenient Feynman slash notation.

There is much much more to say and more ways to get the equation but you asked for the basics.
