Connection Between the Dirac and Schrodinger Equations

In the presence of electric and magnetic fields the Dirac equation is modified to 

In the time-independent case the Dirac equation may be written as 

Here T l and T s arc (large and small) two-component wave functions which include the a and /3 spin functions. The latter equation can be solved for T s. 

The term involving jr ir = is the usual kinetic energy operator. The vector product (ji X ji) gives 

The first and last terms are zero (since a x a = 0). With p = i the other two yield 

The two last terms cancel (since a x b = —b x a), and the curl of the vector potential is the magnetic field, eq. (8.9). The final result is 

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Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

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. 

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