Nonrelativistic Limit and Pauli Equation

In this section, we derive the two-component Pauli equation from the Dirac equation in external electromagnetic fields. It is also desirable to recover the Schrodinger equation in order to see the connection between relativistic theory and nonrelativistic quantum mechanics. For this purpose, we rewrite Eq. 

We now focus on the small component and write the lower part of Eq. (5.135) 

For nonrelativistic energies, the energy ihd/dt — E and the potential V are small compared to the rest energy nteC so that we may approximate Y by 

This relation between the components of the spinor ensures that states below -2meC are omitted (otherwise ihd/dt E would not be small compared to the rest energy). This approximation will turn out to be very important in the relativistic many-electron theory so that a few side remarks might be useful already at this early stage. Eq. (5.137) will become important in chapter 10 as the so-called kinetic-balance condition (in the explicit presence of external vector potentials also called magnetic balance). It shows that the lower component of the spinor Y is by a factor of 1/c smaller than Y (for small linear momenta), which is the reason why Y is also called the large component and Y the small component. In the limit c oo, the small component vanishes. 

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