The Large and Small Components of the Dirac Wavefunction

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

{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\\

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

\begin{displaymath}\bgroup\color{black} j_0=\bar{\psi}\gamma_4\psi=\psi^\dagger\gamma_4\gamma_4\psi=\psi^\dagger\psi \egroup\end{displaymath}

Jim Branson 2013-04-22