Relativistic Dynamics

So far we have only considered kinematic effects of special relativity, but no forces or energies have been discussed yet. We will close this gap in this section with the discussion of dynamical aspects of special relativity. This will reveal some of the most remarkable and famous peculiarities of this theory, which have a striking influence on relativistic quantum chemistry. 

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Historical Background.—Relativistic quantum mechanics had its beginning in 1900 with Planck s formulation of the law of black body radiation. Perhaps its inception should be attributed more accurately to Einstein (1905) who ascribed to electromagnetic radiation a corpuscular character the photons. He endowed the photons with an energy and momentum hv and hv/c, respectively, if the frequency of the radiation is v. These assignments of energy and momentum for these zero rest mass particles were consistent with the postulates of relativity. It is to be noted that zero rest mass particles can only be understood within the framework of relativistic dynamics. 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

However, it is possible to probe such a relativistic dynamics with a second field with a lower frequency, so that multiphoton absorption, leading to ATI, can take place. In order to ionize such a stabilized atom with a significant, probability, the second field must force the electron wave function to explore again the vicinity of the nucleus to be able to absorb energy (i.e. photons). This can he achieved by chosing parallel polarizations and field strength intensities and frequencies such that the characteristic excursion lengths [Pg.114]

As an introduction to relativistic dynamics, it is of interest to treat time as a dynamical variable rather than as a special system parameter distinct from particle coordinates. Introducing a generic global parameter r that increases along any generalized system trajectory, the function t(r) becomes a dynamical variable. In special relativity, this immediately generalizes to A (r) for each independent particle, associated with spatial coordinates x (r). Hamilton s action integral becomes... 

Again anticipating relativistic dynamics, energy is related to momenta as time is related to spatial coordinates. 

With the Lagrangian form known one can unfold the relativistic dynamics according with the analytical mechanics principles actually, for the momentum one uses the form of the conjugated canonical momentum, widely checked in the classical framework with the analytical Lagrangian formulation... 

The form of the Lorentz transformation for a boost in x-direction as given by Eqs. (3.63) and (3.67) suggests that the relative velocity v between IS and IS is always smaller than the speed of light c. This is indeed true and will be shown to follow from the relativistic equation of motion, i.e., the relativistic dynamics, to be discussed in section 3.3. 

One of the most interesting and distinctive features of the narrow resonance spectroscopy as compared with the spectroscopy of the older particles, where only light u, d, s quarks come into play, i.e. up to masses below 3 GeV/c, is the highly satisfactory predictive power of the phenomenological models used to describe the particle spectrum in this new sector. This is mainly due to the fact that the large mass of the quarks involved allows one to make use of non-relativistic dynamics (i.e. Schrodinger equation) which would not have been a sensible approximation in the old sector with light quarks. 

The application of non-relativistic dynamics to the description of the bottomonium system is quite successful and is discussed in Chapter 12. 

Efforts to use relativistic dynamics to describe nuclear phenomena began in the 1950s with application to infinite nuclear matter. Johnson and Teller [Jo 55] developed a nonrelativistic field theory for interacting nucleons and neutral, scalar mesons which served as a catalyst for Duerr, who, in a landmark paper [Du 56], developed a relativistic invariant version of the Johnson and Teller model which included both scalar and vector meson fields. He showed that nuclear saturation and the strong spin-orbit potential of the shell model could be readily understood. He also predicted a single particle potential which qualitatively reproduced the real part of the central optical potential well depth and its energy dependence for incident kinetic energies up to 200 MeV. 

The scalar densities depend critically on the assumed relativistic dynamics. One-baryon-loop vacuum polarization corrections to the relativistic Hartree solution of QHD for finite nuclei reduce the scalar... 

We have now discussed how the description of coordinates and velocities is affected by the postulates of special relativity. To develop a relativistic dynamics, we also need to account for any relativistic effects on the mass of the particles involved. We imagine a simple collision experiment in the frame S A particle with mass m moves along the y direction, that is, perpendicular to the direction of motion of S relative to S. The particle undergoes a totally elastic collision with a wall in the x z plane and rebounds in the -y direction. If the speed along the y axis before the collision was u y, the total change of momentum is... 

The results obtained for the spinor eigenvalues (which relate to mean spinor radii through the virial theorem) are presented in table 22.1. For the s shell, the screening effects have little influence on the energy. The major influence is the change from nonrelativistic to relativistic dynamics. For the p shell, both the dynamics and the potential are important. The contributions are almost the same for the pi/2 subshell, but for the p3/2 subshell the effect of the dynamics is smaller by about a factor of three. For the d shell the dominant effect is the change in potential. 

Whitney, C. K. (2007). Relativistic dynamics in basic chemistry. Found. Phys. 37, 788-812. 

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