Hamiltonian classical relativistic

The functions h (x, p) are the classical relativistic Hamiltonians of spinless particles with positive and negative energies, respectively. 

We now turn to the quantization of the classical relativistic Hamiltonian, and in particular to its representation in the Dirac equation. Quantization of the classical relativistic Hamiltonian has been treated in detail in many texts, such as Rose (1961),... 

While this provides us with an equation for the relativistic electron, the u and matrices arose from the mathematical treatment, and only indirectly from the physics. It would be nice if we could also give these quantities a physical interpretation. In order to find some classical operator or quantity corresponding to a and p, we compare the Dirac Hamiltonian with the classical relativistic Hamiltonian. For this purpose we use the classical relativistic expression for the energy from (2.60) for a field-free system. 

The first of the relativistic correction terms is called the mass-velocity operator. If we expand the square root operator in the classical relativistic Hamiltonian for a free particle, we find... 

A system of N non-relativistic spinless particles is described, using standard terminology, by the classical Hamiltonian... 

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 Hamiltonian of the relativistic hydrogen-like atom in a uniform magnetic field is given as (Landau, 1980)... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

A. Relativistic Semi-Classical Many-Electron Hamiltonian 180... 

F. Halbwachs, F. Piperno, and J. P. Vigier, Relativistic Hamiltonian description of the classical photon behaviour A basis to interpret aspect s experiments, Lett. Nuovo Cimento 33(11) (1982). 

Although this spin-orbit interaction is essentially a relativistic effect it may be approached classically. For hydrogen-like atoms the spin-orbit hamiltonian is... 

Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spin-orbit interaction is already included in the one-electron Dirac Hamiltonian but the two-electron interaction should also include interactions classically known as spin-other-orbit, spin-spin etc... Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 emd we just concentrate on some practical considerations. 

The refinement is based on classical electrodjmamics and the usual quantum mechanical rules for forming operators (Chapter 1) or, alternatively, on the relativistic Breit Hamiltonian (p. 156). This is how we get the Hamiltonian equation(12.67), which contains the usual non-relativistic Hamiltonian plus the perturbation equation [Eq. (12.69)] with a number of terms (p. 766). 

Confirming the total eneigy expressions as the conventional sum between kinetic and potential eneigy components. Having the Hamiltonian-Lagrangian expression proven as reliable in classical side its relativistic extension naturally follows in the same way with relativistic Lagrangian as... 

The second frontier is to improve the MCP operator. The model core potential hamiltonian employs the classical kinetic energy operator (see Eq. 8.6), however, a genuine DK hamiltonian employs the relativistic kinetic operator ... 

Therefore, we may here start from the Breit interaction to derive the pseudo-relativistic Hamiltonians instead of following the somewhat meandering historical path from 1926 to about 1932, which was mainly based on classical considerations. As usual, a quantum-electrodynamical derivation is also possible and has been presented by Itoh [678], but the sound basis of our semi-classical theory, which we pursue throughout this book, is necessarily the Breit equation. Needless to say, the rigorous transformation approach to the Dirac-Coulomb-Breit Hamiltonian yields results identical to those from the QED-based derivation. 

The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to the one-electron Dirac equation, which does contain spin operators. From the one-electron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the non-relativistic case, because the electron-electron interaction is not unambiguously defined. The non-relativistic Coulomb interaction is often used as a reasonable first approximation. The relativistic treatment of atoms and molecules based on the many-electron Dirac equation leads to so-called four-component methods. The name stems from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable to the electron-electron interactions this is the preferred way to explain the electronic structure of the lowest states. 

This equation is obviously not Lorentz invariant— it has x, y, and z appearing quadrat-ically but t appearing linearly, which violates the relativistic principle of equivalence of spatial and temporal variables. Since we know that the nonrelativistic classical Hamiltonian is not Lorentz invariant, it is no surprise that neither is the nonrelativistic... 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

In this book the interaction between fields and molecules is treated in a semi-classical fashion. Quantum mechanics is used for the description of the molecule, whereas the treatment of the electromagnetic fields is based on classical electromagnetism. A complete quantmn mechanical description using quantmn electrodynamics is beyond the scope of this presentation, although we will make use of the correct value of the electronic g-factor as given by quantum electrodynamics. Furthermore, only ab initio methods derived from the non-relativistic Schrodinger equation are discussed. Nevertheless, the Dirac equation is briefly discussed in order to introduce the electronic spin via the Pauli Hamiltonian. 

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