Nonrelativistic Limit of the Dirac Equation

If we partition the energy into the rest term plus a remainder. 

Using this Hamiltonian and E as the separation variable in (4.50) to obtain a time-independent equation, the only difference is that the new temporal wave function is related to the old by a phase factor. 

Henceforth, we will feel free to make use of this new definition of the Hamiltonian and of the energy when convenient, and drop the primes everywhere. We will make it clear when the rest mass term is included. 

The time-independent Dirac equation in two-component form with the rest mass subtracted becomes 

Before taking the limit c oo, this equation must be rearranged for two reasons first, because we need to change it to a form where c occurs in some form of denominator—this will provide us with terms that vanish and hopefully other terms that remain finite—and, second, because the nonrelativistic wave function is a scalar function, whereas the Dirac wave function is a four-component vector function. If we use the two-component nonrelativistic Schrodinger equation that we derived in section 4.2, we can write the nonrelativistic wave function in terms of spin-orbitals, which can be transformed to two-component spinors. Then it is only necessary to reduce the Dirac equation from four-component to two-component form. 

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It took some time until it was realized that the Dirac theory describes the spin correctly because it is a spinor-field theory, and not because it is relativistic [16]. In fact, if one takes the nonrelativistic limit of the Dirac equation, spin survives, and this is consistent with the observation that the Galilei group has spinor representations as well. So, without any doubt, spin is not a relativistic effect. 

Having recovered the potential surface from the solutions of the Born-Oppenheimer electronic problem, we can now proceed to solve the equation for nuclear motion. The Dirac-type equation for the nuclei can easily be reduced to the corresponding nonrelativistic equation by following the same reduction as we did for taking the nonrelativistic limit of the Dirac equation in section 4.6. Doing this, we abandon all pretense of Lorentz invariance for this part of the system, but we know from experiment that the nuclear relative motion in molecules takes place at rather low energies where relativistic effects may safely be neglected. 

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