Dirac equation

Schrodinger equation comes from quantizing a very simple statement:

(1)E=H=12mp2+V(q),

that is, the energy is the Hamiltonian.
In particular, for a free particle E=12m(px2+py2+pz2) gives rise to

itψ=12((ix)2+(iy)2+(iz)2)itψ=22(x2+y2+z2)ψ

If we want to go to special relativity, we have the equation

(2)E2px2py2pz2=m2c4,

which reflect the length of the four-momentum vector. It is the analogous to (1) in the relativistic case. If you try to quantize this equation you arrive at the Klein--Gordon equation. This is an equation of second-order in time, so it is not a good replacement for Schrodinger equation in the relativistic case.

Dirac idea: to have an equation first-order in time, analogous to Schrodinger equation, Dirac had to use matrices instead of scalars

αx=(0001001001001000),αy=(000i00i00i00i000),αz=(0010000110000100),β=(1000010000100001)

to take the "square root" of (2)

E=αxpx+αypy+αzpz+βm.

The Klein-Gordon equation:

(μμ+(mc)2)Ψ=0

can be rewritten if we consider that the coefficients are no longer complex or real numbers, but some matrices γν and γμ:

(γννimc)(γμμ+imc)Ψ=0

Dirac equation is equate one of the factors to zero:

γννimc=0

The solutions ψ are no longer complex-valued functions, but spinor-valued functions. The good news is that
we ca define the quantity:

ρc=Ψ¯γtΨ

satisfying:

Keep an eye: if we take the non-relativistic limit, we should recover Schrodinger equation. But if we introduce electromagnetism by replacing p with something that take into account the vector potential A (see electromagnetic field), we obtain Pauli equation.