Derive dirac's equation in quantum mechanics?

Answer 1

I was considering writing out the answer, and maybe that would be "correct". However, I'm wondering if it may be better, especially as a beginning here, to instead, point the way for the questioner, and others to answer this for themselves.

To this end I propose that anyone looking for the answer to how to derive Dirac's Equation (for spin 1/2 particles) may simply type "Dirac equation" in the appropriate Search/Research field, here. Near the top of the references will probably be one to Wikipedia. (So you could have gone straight to Wikipedia and searched for "Dirac equation".)

Under the heading "Dirac's coup" you will find the motivating principle, and an outline of Dirac's own approach to deriving his equation. Above that, is additional motivation for the relationship with the Klein-Gordon equation.

Now, it may be of interest to note that in a very real sense the Dirac equation can be seen to be related to an operator that is a "square root" of the Kein-Gordon operator (similarly related to the Klein-Gordon equation). The thing is, that within non-commutative algebraic systems (like matrices) there are often multiple "square roots" (if there exist any at all). (Consider, for instance, the many different "square roots" of positive definite real matrices.) So Dirac's operator is only one such "square root" of the Klein-Gordon operator.

(My Ph.D. dissertation took a more general approach, and finds, for instance, that there are two component "square roots". So left- and right- handed particles are not required to be symmetric, as they are within Dirac's equation. Which is why neutrinos "require" a "projection" operator to be placed within Dirac's equation in order to account for their lack of such symmetry.)