How to obtain Dirac equation from Schrodinger equation and special relativity?

The Dirac equation is superficially similar to the Schrödinger equation for a free massive particle:

A) $-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi.$

The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energy are the space and time parts of a space-time vector, the 4-momentum, and they are related by the relativistically invariant relation

B) $\frac{E^2}{c^2} - p^2 = m^2c^2$

which says that the length of this vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,

C) $\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$

I am not sure how the equation A and B lead to equation C. It seems that it is related to substituting special relativity value into quantum mechanics operators, but I just keep failing to get a result...

First, C) isn't the Dirac Equation, it's the Klein-Gordon equation

Now, to your main question. A) comes from the classical equation for a free massive particle:

$\dfrac{p^2}{2m} = E$

by making the operator (operating on $\phi$) substitutions:

$p^2 \rightarrow - \hbar^2 \nabla^2$

$E \rightarrow i \hbar \dfrac{\partial}{\partial t}$

C) comes from B) by further recognizing that:

$E^2 \rightarrow -\hbar^2 \dfrac{\partial^2}{\partial t^2}$

protected by Qmechanic♦Aug 31 '13 at 19:06

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?