Dirac picture

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

Its expectation value within the original, i.e., untransformed Dirac picture reads... 

For the DKeel and DKee2 models, this equivalence holds because the terms of both Hamiltonians related to the Hartree self-interaction are limited to the fpFW and first-order DKH transforms of the Hartree potential, Eqs. (23) and (24). Thus, the terms jointly notated by the symbols [mn k rei can consistently be used to determine the fitting coefficients of the density in the four-component Dirac picture, to build the Hartree part of the relativistic Hamiltonian at DKeel and DKee2 levels, and to evaluate the total energy. 

Figure 12.1 Hydrogen-like atoms in the limiting Schrodinger and Dirac pictures. Ground state eigenvalues of the scalar-relativistic DKH Hamiltonians up to fifth order are aiso piotted. The fourth- and fifth-order DKH ground states can hardly be distinguished from the anaiytic Dirac result on this soale of presentation.

This representation is called the Dirac picture. Alternatively, this expectation value may be reformulated within the two-component DKH framework employing the large component of the transformed spinor ip only,... 

If the property is incorporated variationally into the four-component Dirac picture (as in section 15.1), the A-dependent energy is the reference value which has to be reproduced by the DKH calculation. In order to evaluate this energy within a two-component framework, the perturbed Hamiltonian... 

This is not quite the end of the matter. What we have assumed in the QED approach is that the reference vacuum is that of the current guess. If we were to take an absolute reference, such as the free-particle vacuum, the normal ordering should take place with respect to this fixed vacuum, and then the QED approach would give the same results as the filled Dirac approach, in which rotations between the negative-energy states and the unoccupied electron states affect the energy. By this means a vacuum polarization term has been introduced into the procedure, but without the renormalization term. In atomic structure calculations in which QED effects are introduced, the many-particle states employed are usually the Dirac-Fock states (Mittleman 1981), that is, those that result from the empty Dirac picture. We will therefore take as our reference the QED approach with the floating vacuum —a vacuum defined with respect to the current set of spinors. 

Rewriting this in terms of positron and electron indices is more involved, because the two-electron part produces 16 terms. Permutational symmetry and reindexing permits the combining of six of these, and the result in the empty Dirac picture is... 

The fact that there is a relativistic correction to the property operators is a manifestation of what has been called the picture change (see for example Kello and Sadlej 1998). The Dirac operator and the corresponding Schrodinger operator do not have the same meaning. For the example given above, the position operator r does not have the same content in the Dirac picture as in the Schrodinger picture. More information on the interpretation of the picture change can be found in Moss (1973), for example. 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The classical motion corresponding to the quantum dynamics generated by the Dirac-Hamiltonian (2) can most conveniently be obtained by considering the limit h — 0 in the Heisenberg picture Consider an operator B that is a Weyl quantisation of some symbol (see (Dimassi and Sjostrand, 1999))... 

Practitioners of quantum chemistry employed both the visual imagery of nineteenth-century theoretical chemists like Kekule and Crum Brown and the abstract symbolism of twentieth-century mathematical physicists like Dirac and Schrodinger. Pauling s Nature of the Chemical Bond abounded in pictures of hexagons, tetrahedrons, spheres, and dumbbells. Mulliken s 1948 memoir on the theory of molecular orbitals included a list of 120 entries for symbols and words having exact definitions and usages in the new mathematical language of quantum chemistry. 

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

Up to this point, we have presented the fully relativistic Hamiltonian. Of course, we could set out to calculate energies of molecules employing this Hamiltonian. However, the various spin-spin interactions are easier described in terms of a perturbation picture rather than as excited states of the full-fledged Hamiltonian. Especially for the fully relativistic Dirac-Coulomb-Breit Hamiltonian, the latter calculations would be computationally very demanding. 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

A new quantum theory called wave mechanics (as formulated by Schrodinger) or quantum mechanics (as formulated by Heisenberg, Born and Dirac) was developed in 1926. This was immediately successful m accounting for a wide variety of experimental observations, and there is little doubt that, in principle, the theory is capable of describing any physical system. A strange feature of the new mechanics, however, is that nowhere does the path or velocity of the electron enter the description. In fact it is often impossible to visualize any classical motion that could be consistent with the quantum mechanical picture of the atom,... 

Relativity is expected to play an important role in several types of radiative processes in atoms. Its influence on the atomic levels fine structure has been most thoroughly investigated as its signature is easily evidenced in atomic x-ray spectra, [1], [2], [3], It manifests itself also in some delicate aspects of the chemical reactivity of the elements, [4], These effects arise from both the standard Dirac-like properties of electrons and from more sophisticated QED corrections. One of the major objective of the present paper is to show that the overall picture has dramatically changed recently, as a consequence of the considerable advances made in the design of ultra intense laser sources operated at intensities well beyond the so-called atomic unit of intensity la = 3.5 xlO16 W/cm2, [5]. 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

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