The Large and Small Components of the Dirac Wavefunction

Returning to the pair of equations in $\psi_A$ and $\psi_B$. Note that for $E\approx mc^2$, that is non-relativistic electrons, $\psi_A$ is much bigger than $\psi_B$.

$$\begin{eqnarray*} {1\over c}(E+eA_0+mc^2)\psi_B = \vec{\sigma}\cdot\left(\vec{p}... ...p}+{e\over c}\vec{A}\right)\psi_A\approx{pc\over 2mc^2}\psi_A\\ \end{eqnarray*}$$

In the Hydrogen atom, $\psi_B$ would be of order ${\alpha\over 2}$ times smaller, so we call $\psi_A$ the large component and $\psi_B$ the small component. When we include relativistic corrections for the fine structure of Hydrogen, we must consider the effect $\psi_B$ has on the normalization. Remember that the conserved current indicates that the normalization condition for the four component Dirac spinor is.

$$\bgroup\color{black} j_0=\bar{\psi}\gamma_4\psi=\psi^\dagger\gamma_4\gamma_4\psi=\psi^\dagger\psi \egroup$$