The Time-Independent Dirac Equation

Now that we have a valid equation, we need to manipulate it to obtain a workable form. It is almost invariably the case that we are looking for stationary states, and most considerations of time-dependent phenomena proceed from a set of states that are time-independent. 

A stationary state is the eigenfunction of a Hamiltonian that is independent of the time variable and has the energy as its eigenvalue. If we have such a Hamiltonian, we can write the wave function as a product of a spatial and a temporal part. 

If ii is time-independent, the left side is a function of r only and the right side a function of t only. For the two to be equal, they must both be equal to a constant, which is the total energy E. The solution of the temporal part is easily obtained as a simple exponential  

The spatial part is the time-independent Dirac equation = cu (—IW + eA) r r - -which we may now proceed to solve. 

In the light of the chapter on special relativity (chapter 2), it is apparent that there is a possible problem in performing this separation of the space and time variables, because the Lorentz transformation mixes them. The separation would have to be performed in a particular frame of reference, and only be valid in this frame of reference. If we want results in another frame of reference, we must perform a Lorentz transformation to that frame, and there is no guarantee that we will still have a stationary state. However, if our Hamiltonian is Lorentz invariant, the choice of the frame of reference is arbitrary, and, as we saw above, the probability density is independent of time and of the frame of reference. We may therefore choose the frame that is most convenient. In molecules (and in atoms) the Born-Oppenheimer frame is the most convenient frame of reference for electronic stmcture calculations because the nuclear potential is then simply the static Coulomb potential. Regardless of whether the Hamiltonian is Lorentz invariant or not, it is this frame that we work in from here on. 

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It is also common in the literature to write the time-independent Dirac equation in terms of Pauli-spin matrices... 

The experimental success in spectroscopy and the mentioned characteristics of few-electron high-Z systems lead to a challenge for theoreticians to predict the spectroscopic data including all known corrections most precisely. In the following we will examine all corrections to the electron binding energy of the lowest lying states of heavy one-electron ions. The time independent Dirac equation (units chosen so that h = c = l,a = e )... 

In this section we shall discuss the interactions that arise upon the introduction of electromagnetic fields in the relativistic electronic Hamiltonian, and we shall also consider the form of electromagnetic interactions in the non-relativistic limit. To simplify matters, we shall first limit our attention to one-electron systems. Consider the time-independent Dirac equation for a free particle... 

The (time-independent) Dirac equation (DE), which describes a single electron or positron in an external electric field V... 

In the presence of an electric potential V (e.g. from nuclei), the time-independent Dirac equation may be written as in eq. (8.8), where we have again explicitly indicated the electron mass. 

With the generalized momentum operator n replacing p, the time-independent Dirac equation may be separated analogously to the procedure in Section 8.2.1 to give the equivalent of eq. (8.13). 

It is interesting to compare the time-independent limit of the Dirac equation with the Schrddinger equation. The time-independent Dirac equation is given as... 

Based on Eq. (6.61), the time-independent Dirac equation is obtained as... 

From section 6.6 we understand that the nonrelativistic energy is not the first term of the series expansion, but the rest energy mgC is. Therefore, the energy eigenvalues will differ by meC. We will later see that this result transfers to the many-electron case. Since a constant W can be added to the time-independent Dirac equation, Eq. (6.7),... 

The time-independent Dirac equation for a free particle follows easily from (4.52) as... 

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

For the relativistic case of the time-independent Dirac equation for an electron subjected only to time-independent electrostatic fields, we have... 

As in the nonrelativistic case, most of the salient features of the atomic systems are exposed in the treatment of the simplest of these, the hydrogen-like one-electron atoms. In Hartree atomic units the time-independent Dirac equation yields the coupled equations... 

Most studies of molecular properties are founded on the response of the molecule to electromagnetic fields. A relativistic electronic system is described by the time-independent Dirac equation... 

For a time-independent scalar potential, the electron-positron field operator, (a ), is expanded in a complete basis of four-component solutions of the time-dependent Dirac equation [19],... 

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

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

The wave function < (r, r) will have four components, as the operators arc 4 x 4 matrices. Separation of the time leads to the time-independent version of the Dirac equation for a free election (positron)... 

Here we do not follow the original derivation of Levy-Leblond [16], but we derive the LLE as the non-relativistic limit (nrl) of the Dirac equation (DE) [12], and do this at the time-independent level. We hence start by... 

The functions in Fig. 7.5 approximate the Dirac delta function. Draw graphs of the corresponding functions that approximate the Heaviside step function with successively increasing accuracy. Quantum mechanics postulates that the present state of an undisturbed system determines its future state. Consider the special case of a system with a time-independent Hamiltonian H. Suppose it is known that at time to the state function is fo). Derive Eq. (7.101) by substituting the expansion (7.66) with g = i/i into the time-dependent Schrodinger equation (7.97) multiply the result by i/i, integrate over all space, and solve for c . 

To obtain further information on the nature of the Dirac wave function, we can solve the equation for a simple model system. The simplest case is the time-independent equation for a free particle. In the nonrelativistic case the Schrodinger equation for a free particle moving along the x axis is... 

The free-particle Dirac equation provides a nice demonstration of some of the properties of the 4-spinor solutions, but quantum chemistry is mainly concerned with electrons bound in molecules by electromagnetic forces. In a static potential V, such as that provided by the nuclei in the Bom-Oppenheimer frame, where the vector potential A is zero, the time-independent electronic Dirac equation is... 

We now have the necessary tools to discuss the transformation of the Dirac Hamiltonian. We consider the time-independent equation with a static scalar potential, (4.74)... 

The Dirac Equation in a Central Field.—The previous sections have indicated that at times it is useful to have an explicit representation of the matrix element <0 (a ) n> where tfi(x) is the Heisenberg operator satisfying Eq. (10-1). Of particular interest is the case when the external field A (x) is time-independent, Ae = Ae(x), so that the states > can be assumed to be eigenstates of the then... 

The theory discussed until now is based on the Kramers-Heisenberg-Dirac dispersion relation for the transition polarizability tensor as given in Eq. (6.1-1). The expression shown in this equation describes a steady state scattering process and contains no explicit reference to time. Therefore, the resonance Raman theory which is based on the KHD dispersion relation is sometimes also termed as time-independent theory (Ganz et al., 1990). 

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