The Dirac equations for electron and positron

After the factorization described above Dirac obtained two operator equations. The Dirac equations (for the positron and electron) correspond to these operators acting on the wave functionThus, we obtain the equation for the negative electron energies (positron) 

If one is interested in stationary states (cf. p. 21), the wave function has the form stationary states (x, y, z, t) = (x, y, z)e, where we have kept the same symbol for the time independent factor x,y,z). After dividing by e we obtain 

The quantity q f) = V in future applications will denote the Coulomb interaction of the particle under consideration with the external potential. 

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Equation (3.125) is the required transformed Hamiltonian, and we see that in the representation in which (3 is diagonal, the Dirac equation decomposes into uncoupled equations for the upper and lower components of the wave function, i.e. for electron and positron wave functions. Setting (3 equal to +1 gives the positive energy (electron) states, whilst (3 equals -1 gives the negative energy (positron) states. 

The mathematical basis of the relativistic quantum mechanical description of many-electron atoms and molecules is much less firm than that of the nonrelativistic counterpart, which is well understood. As we do not know of a covariant quantum mechanical equation of motion for a many-particle system (nuclei plus electrons), we rely on the Dirac equation for the quantum mechanical characterization of a free electron (positron) (Darwin 1928 Dirac 1928,1929 Dolbeault etal. 2000b Thaller 1992)... 

Furthermore, starting from the Dirac equation, the assumptions of hole theory lead automatically to a formulation for an infinite number of particles occupying the Dirac sea, and the actual number of electrons and positrons cannot be deduced from hole theory. This is a most disturbing aspect of Dirac s hole theory, namely that it is actually a flni/-particle theory. So far we have considered only a single fermion in the universe and set up an equation to describe its motion. Now, we face a conceptual generalization, which in turn requires a description of the motion of infinitely many fermions. This leads to the inclusion of additional interaction operators in the Dirac equation, and it is this ineraction of electrons which makes life difficult for molecular scientists, as we shall discuss in parts III and IV of this book. 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

This process cannot be described classically, because positrons are the result of the Dirac equation, but it illustrates the fact that a vacuum current (of photons) is made up of the interaction of two Dirac currents, one for the electron, one for the positron, and these are both matter currents. Therefore, there is a transmutation of matter current to vacuum current. On the classical level, this can be described in the scalar internal gauge space as... 

Therefore the negative-energy solutions for the Dirac equation are not a mathematical fiction In principle, each fundamental particle does have its corresponding antiparticle (which has the opposite electrical charge, but the same spin and the same nonnegative mass). Equation (3.6.15) also shows the formation of a transient Coulomb-bound electron-positron pair ("positronium"), whose decay into two photons is more rapid if the total spin is S = 0 than if it is S = 1, and is dependent on the medium. 

Our discussion of the electron-positron interpretation shows that the Dirac equation can have bound states for both attractive and repulsive electric potentials. Consider for example an attractive electrostatic potential well eV(x) < 0. It may support some bound states at energies En in the gap A... 

Solutions with positive and negative energies of a one-particle Dirac equation of a molecular system are represented by states where either electronic or positronic contributions of four-component wave functions dominate. With chemical systems in mind, electronic and positronic components are also referred to as large and small components, respectively. However, small components cannot simply be neglected or projected out to arrive at a simpler two-component description because, in an intrinsic fashion they also contribute in a fully relativistic description of a chemical system. Thus, a projection step, in which positronic components are discarded, can only be applied after a suitable decoupling of electron and positron degrees of freedom. Then the effects of the small components are implicitly accounted for. 

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