The free Dirac equation

Upon noting that u,(p)e ipt and Dirac equation, Eq. (10-213) can also be written in the form... 

It follows from (6) that every component of a solution of the free Dirac equation is a solution of the Klein-Gordon equation... 

Can we thus replace the free Dirac equation by Klein-Gordon equations, one for each component Certainly not, because the Dirac equation introduces a coupling between the components. A solution of the Dirac equation therefore contains information that is not contained in a spinor whose components are independent solutions of the Klein-Gordon equation. 

We are going to explain the procedure of forming wave packets out of plane waves for the free Dirac equation. The free stationary Dirac equation Ho tp = Erp has no square-integrable solutions at all. But it turns out that for E > me there are bounded oscillating solutions (here bounded means that the absolute value [ipix, t) of the solution remains below a certain constant M for all x and t). As for the Schrodinger equation, it is comparatively easy to find these solutions. They are similar to plane waves with a fixed momentum (wavelength), that is, they are of the form... 

The group of Poincare transformations consists of coordinate transformations (rotations, translations, proper Lorentz transformations...) linking the different inertial frames that are supposed to be equivalent for the description of nature. The free Dirac equation is invariant under these Poincare transformations. More precisely, the free Dirac equation is invariant under (the covering group of) the proper orthochronous Poincare group, which excludes the time reversal and the space-time inversion, but does include the parity transformation (space reflection). 

We are going to prove the invariance of the free Dirac equation under Poincare transformations in the form of the following statement Whenever ip ct,x) = i x) is a solution of the free Dirac equation, then (j> x) — M (A (x - a)) is also a solution of the free Dirac equation. Here it is assumed that M and A are related by (82). 

We note that the time-evolution according to the free Dirac equation is a special Poincare transformation (translation in the time-direction of Minkowski space). It is a unitary transformation generated by the free Dirac operator Hq ... 

Due to the presence of the heavy particle mass shell projector on the right hand side the wave function in (1.19) satisfies a free Dirac equation with respect to the heavy particle indices... 

We prefer to write the time-dependent free Dirac equation as a quantum-mechanical evolution equation (that is, in the familiar Schrodinger form ) in the following way... 

The Dirac equation in an external field is obtained by replacing the free Dirac operator Hq in (3) with an appropriate Dirac operator H,... 

In the following, the Fourier transformation (24) will be denoted by T, that is A Fourier transformation of the stationary free Dirac equation... 

The time-dependence of wave packets moving according to the Dirac equation usually cannot be determined explicitly. In order to get a qualitative description of the relativistic kinematics of a free particle, we investigate the temporal behavior of the standard position operator. With Ho being the free Dirac operator, we consider (assuming, for simplicity, h = l from now on)... 

Lorentz Covariance of the Field-Free Dirac Equation... 

After having derived a truly relativistic quantum mechanical equation for a freely moving electron (i.e., in the absence of external electromagnetic fields), we now derive its solutions. It is noteworthy from a conceptual point of view that the solution of the field-free Dirac equation can in principle be pursued in two ways (i) one could directly obtain the solution from the (full) Dirac equation (5.23) for the electron moving with constant velocity v or (ii) one could aim for the solution for an electron at rest — which is particularly easy to obtain — and then Lorentz transform the solution according to Eq. (5.56) to an inertial frame of reference which moves with constant velocity —v) with respect to the frame of reference that observes the electron at rest. 

Now, we have a unified substitution pattern at hand, which also comprises the time-like coordinates. Substitution of Eq. (5.116) in the field-free Dirac equation as written in Eq. (5.54) yields the covariant form of the Dirac equation with external electromagnetic fields. 

The modified Dirac equation can now be viewed from two different perspectives. The first perspective is the fact that the approximate kinetic balance condition of Eq. (5.137) has been exploited. The normalized elimination procedure then results in energy eigenvalues which deviate only in the order c from the correct Dirac eigenvalues, whereas the standard un-normalized elimination techniques are only correct up to the order c. In addition, the NESC method is free from the singularities which plague the un-normalized methods and can be simplified systematically by a sequence of approximations to reduce the computational cost [562,720,721]. From a second perspective, Eq. (14.1) defines an ansatz for the small component, which, as such, is not approximate. Hence, Eq. (14.4) can be considered an exact starting point for numerical approaches that aim at an efficient and accurate solution of the four-component SCF equations (without carrying out the elimination steps). We will discuss this second option in more detail in the next section. 

In most cases, spin-free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order. Thus a reasonable approach to the treatment of relativistic effects is to include the spin-free effects fully and treat the spin-dependent effects as a perturbation. We discuss the latter task in chapter 21. In this chapter, we will examine a modification to the Dirac equation that permits the spin-fi ee and spin-dependent terms to be separated (Kutzelnigg 1984, Dyall 1994). This separation is exact, in that no approximations have been made to obtain the separation, and therefore results obtained with the modified Dirac equation are identical to those obtained with the unmodified Dirac equation. The advantage of the separation is the identification of the genuine spin-dependent terms and the possibility of their omission in approximate calculations. This development also provides a basis for discussion and analysis of spin-free and spin-dependent operators in other approximations. 

This equation is exact just as the modified Dirac equation of chapter 15 is exact. It can also be separated into spin-free and spin-dependent terms, but now the separation must be done in both the Hamiltonian and the metric. Visscher and van Lenthe (1999) have shown that the spin separation gives different results for the two modified equations, and therefore the spin separation is not unique. This regular modified Dirac equation can be used in renormalization perturbation theory, with ZORA as the zeroth-order Hamiltonian. 

The Cowan-Griffin Hamiltonian was developed for spin-free relativistic atomic calculations (Cowan and Griffin 1976). However, it has also found some use as a starting point for developing spin-free relativistic Hamiltonians for molecular application. Here, we show the form of this operator, and the associated spin-orbit correction. For atoms, the Cowan-Griffin Hamiltonian follows directly from the radial form of the atomic Dirac equation (7.29), which may be given as... 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

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... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

The relativistic one-fermion equation in the free field presented hy Dirac in 1928 [2,3] is usually written in the contravariant form, which illustrates the relativistic equivalence of space and time in an elegant way... 

---