The stationary Dirac equation

As a first step in solving the Dirac equation one may split off the time factor exp —iEt/%). That is, one inserts the expression 

The solution ij of this equation of course depends on the eigenvalue parameter 

One has to be aware of the fact that the time-dependent equation is not only solved by ilj E,x,t), but also by arbitrary superpositions of solutions with different values of E. 

For a given value of E the solution tl) E,x) of equation (28) may or may not be square-integrable. If ip E,x) is square-integrable (that is, if E is an eigenvalue of H), then the corresponding solution 4 E,x,t) of the time-dependent Dirac equation is a bound state with stationary position and momentum densities (according to our tentative interpretation). Bound states occur in the presence of an external force that attracts the particle to some region of space. 

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 

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With the above ansatz for the spinor we can rewrite the stationary Dirac equation for hydrogen-like atoms with the Hamiltonian of the form of Eq. (6.55)... 

This is the set of all energies for which the free stationary Dirac equation has plane-wave like solutions (out of which square-integrable wave packets can be formed). The spectrum a Ho) is the continuum of all real numbers except the numbers in the spectral gap, the open interval (—mc, mc ). 

It is occasionally argued that the missing nodes in the Dirac radial density explain how electrons get across the nodes [129] as there are no longer any nodes. However, one must keep in mind that this is a somewhat artificial question as time has been eliminated from the stationary Dirac and Schrodinger equations. The question remains why the density dramatically... 

So far only the position-space formulation of the (stationary) Dirac Eq. (6.7) has been discussed, where the momentum operator p acts as a derivative operator on the 4-spinor Y. However, for later convenience in the context of elimination and transformation techniques (chapters 11-12), the Dirac equation is now given in momentum-space representation. Of course, a momentum-space representation is the most suitable choice for the description of extended systems under periodic boundary conditions, but we will later see that it gains importance for unitarily transformed Dirac Hamiltonians in chapters 11 and 12. We have already encountered such a situation, namely when we discussed the square-root energy operator in Eq. (5.4), which cannot be evaluated if p takes the form of a differential operator. 

For an energy-independent X-operator it is necessary to employ the Dirac equation in a different form. Multiplication of the upper of the two Dirac equations in split notation, Eq. (5.80), by X from the left produces a right-hand side that reads in stationary form XEip. From Eq. (11.2) we understand that this is identical to XEtp = Etp, and hence the two left-hand sides of the split Dirac equation become equal. 

In order to resolve the two terms on the right-hand side, we rearrange the eigenvalue equation of the one-electron stationary Dirac equation so that we obtain an expression for the 4-spinor tp,... 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

In the standard (Dirac-Pauli) representation, the Dirac equation for an electron in the field of a stationary potential V reads... 

Band theory is a one-electron, independent particle theory, which assumes that the electrons are distributed amongst a set of available stationary states following the Fermi-Dirac statistics. The states are given by solutions of the Schrodinger equation... 

We choose a reference frame in which the nucleus is stationary and the vector potential A is zero, and consider a scalar potential V r). The Dirac equation becomes... 

In the 0-Hartree method [19] the Dirac equation is also used as the starting point, but the Lagrangian of quantum field theory is made stationary by altering the balance of direct and exchange terms in a very specific way. Like Hartree s original theory without exchange, this method is consistent with the fundamental principles of quantum field theory (the Hartree-Fock method is not), and allows the central field to be further... 

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

We shall use the notation of [76, 22.5]. The central field Dirac equation has stationary states Fg(x) of energy E such that (using Hartree units)... 

For the quantization of the electron-positron field the expansion of the arbitrary solution of the Dirac equation (x) with the fidl set of stationary... 

Therefore the Dirac equation for stationary states represents in fact a system of coupled equations... 

Using these expressions, the upper-component Dirac equation for stationary states can be written as... 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

From Eqs. (5.136) and (5.137) in chapter 5 we know that the kinetic balance condition relates the small (lower) and large (upper) 2-spinors. In the stationary case, in which the time-dependence of the Dirac equation drops out and the total energy E if a one-electron state comes in as a separation constant according to (ih) d/df — e (cf. Eqs. (6.5)-(6.7)), we may write Eq. (5.136) as... 

When properly interpreted, the Klein-Gordon equation gives quite satisfactory results for bosonic particles. However, there are reasons for rejecting it for the description of an electron. For instance, it does not accommodate the spin i nature of the electron. Furthermore, the occurrence of a second derivative with respect to time makes it difficult to introduce the notion of stationary states. To derive an alternative equation, Dirac" tried to find a Lorentz invariant equation of the form... 

The application of this model in physics and chemistry has had a long history. We shall give some examples of the early works. The work of Einstein S2) on the theory of Brownian motion is based on a random walk process. Dirac S3) used the model to discuss the time behavior of a quantum mechanical ensemble under the influence of perturbations this development enables one to discuss the probability of transition of a system from one unperturbed stationary state to another. Pauli 34) [also see Tolman 35)], in his treatment of the quantum mechanical H-theorem, is concerned with the approach to equilibrium of an assembly of quantum states. His equations are identical with those of a general monomolecular... 

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