Dirac equation
Schrodinger equation comes from quantizing a very simple statement:
that is, the energy is the Hamiltonian.
In particular, for a free particle
If we want to go to special relativity, we have the equation
which reflect the length of the four-momentum vector. It is the analogous to
Dirac idea: to have an equation first-order in time, analogous to Schrodinger equation, Dirac had to use matrices instead of scalars
to take the "square root" of
The Klein-Gordon equation:
can be rewritten if we consider that the coefficients are no longer complex or real numbers, but some matrices
Dirac equation is equate one of the factors to zero:
The solutions
we ca define the quantity:
satisfying:
is conserved
so it can be interpreted as a probability density. Source: this video. This does not happen with the solutions of the Klein--Gordon equation.
Keep an eye: if we take the non-relativistic limit, we should recover Schrodinger equation. But if we introduce electromagnetism by replacing