I will give here some comments and clarifications about the derivation and developing of the Dirac equation from a historical point of view.
The Dirac equation for a free particle is given by the following definitions and expressions:
In the above ψ is the wave function for the free particle , p is the momentum operator with three components , c is the speed of light , ħ is the reduced Planck constant , i=√(-1) is of course the imaginary number unit , H is the Hamiltonian , and t is time.Dirac first elaborated and published his equation in 1928. The spatial coordinates have been written in different books and textbooks using different notations:
With the superscript notation , the Dirac equation takes the following condensed form:
The α symbols in the equation are the original Dirac matrices :
These matrices take the more detailed form:
And the wave function ψ is expressed by:
If we multiply , develop and expand the terms of the Dirac equation , we get the following set of four equations:
Rearranging and reordering these equations we obtain:
In his book about The Theory of Spinors (originally published in 1938) , Elie Cartan obtained the following four Dirac equations:
The letter V represents the magnetic potential , which is mostly expressed by the letter A in modern notation. But since we’re considering the equation for a free particle we can neglect V and make it equal to zero. Cartan noted that we pass from his notation to that of Dirac by making the following replacements or substitutions:
In fact after replacing we get the following set of equations:
Now if we combine ,add or subtract these equations two by two we get the Dirac equations.
Adding equations (2.1) and (2.4) and simplifying gives equation (1.2) above.
Similarly: Adding equations (2.2) and (2.3) and simplifying gives equation (1.3) above.
(2.1) – (2.4) give (1.4) , and (2.3) – (2.3) give (1.1) .
The Dirac equation presented a relativistic version of the Schrodinger equation for spin 1/2 particles and predicted the existence of antimatter. It is of importance in many domains such as relativistic quantum mechanics and Quantum Field Theory.