The Non-relativistic Limit of the Dirac Equation

One important requirement for the Dirac equation is that it reproduces what we know from non-relativistic quantum mechanics. Note that we have derived this equation from something that did give the right answers so we expect the Dirac equation to pass this test. Perhaps we will learn something new though.

We know that our non-relativistic Quantum Mechanics only needed a two component spinor. We can show that, in the non-relativistic limit, two components of the Dirac spinor are large and two are quite small. To do this, we go back to the equations written in terms of $\psi_A$ and $\psi_B$, just prior to the introduction of the $\gamma$ matrices. We make the substitution to put the couplings to the electromagnetic field into the Hamiltonian.