Relativistic interpretation

J. P. Vigier, Relativistic interpretation (with non-zero photon mass) of the small ether drift velocity detected by Michelson, Morley and Miller, Apeiron 4(2-3) (Special Issue The Field Beyond Maxwell) (April-July 1997). 

An Extended (Sufficiency) Criterion for the Vanishing of the Tensorial Field Observability of Molecular States in a Hamiltonian Formalism An Interpretation Lagrangeans in Phase-Modulus Formalism A. Background to the Nonrelativistic and Relativistic Cases Nonreladvistic Electron... 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

Actually Schrddinger s original paper on quantum mechanics already contained a relativistic wave equation, which, however, gave the wrong answer for the spectrum of the hydrogen atom. Due to this fact, and because of problems connected with the physical interpretation of this equation, which is of second order in the spaoe and time variables, it was temporarily discarded. Dirac took seriously the notion of first... 

It is a characteristic feature of all these relativistic equations that in addition to positive energy solutions, they admit of negative energy solutions. The clarification of the problems connected with the interpretation of these negative energy solutions led to the realization that in the presence of interaction, a one particle interpretation of these equations is difficult and that in a consistent quantum mechanical formulation of the dynamics of relativistic systems it is convenient to deal from the start with an indefinite number of particles. In technical language this is the statement that one is to deal with quantized fields. 

We have not as yet however treated the charge-transfer data available for complexes of the 5 d series. For these latter species though the effective spin-orbit coupling constants are often of the order of 3 kK. or more, as compared with only about 1 kK. for Ad systems, and smaller values still for the 3d elements. Consequently, as for the d—d transitions it is often necessary explicitly to consider relativistic effects in the interpretation of charge-transfer spectra, and in particular to make allowance for the changes in spin-orbit contributions which may accompany a given di transition. In fact one of us has shown (18) that these changes are... 

The fundamental quantum-mechanical interpretation by Ramsey of the NMR parameters of chemical shift [123, 124] and spin-spin J coupling [125, 126] still forms the basis of our understanding of these terms, as has been recently noted [127]. However, Ramsey neglected relativistic effects, which are very significant for the heavier nuclei such as 199Hg, 207Pb, and 203 205xi [127]. Derivations and discussion of Ramsey s theoretical results are found in the textbook by Slichter [26]... 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

From the physical point of view the condition a < 3/2 and b < 3 can be interpreted as a criteria for a relativistic ideal gas to be degenerate. In other words, this means that if these conditions are satisfied, there exists a non-zero most probable velocity. 

EXO 0748-676, Cottam et al. (2002) have found absorption spectral line features, which they identify as signatures of Fe XXVI (25-time ionized hydrogenlike Fe) and Fe XXV from the n = 2 —> 3 atomic transition, and of O VIII (n = 1 —> 2 transition). All of these lines are redshifted, with a unique value of the redshift z = 0.35. Interpreting the measured redshift as due to the strong gravitational field at the surface of the compact star (thus neglecting general relativistic effects due to stellar rotation on the spectral lines (Oezel Psaltis 2003)), one obtains a relation for the stellar mass-to-radius ratio ... 

The spin (angular momentum) quantum number ms. In their interpretation of many features of atomic spectra Uhlenbeck and Goudsmit (1925) proposed for the electron a new property called spin angular momentum (or simply spin) and assumed that only two states of spin were possible. In relativistic (four-dimensional) quantum mechanics this quantum number is related to the symmetry properties of the wave function and may have one of the two values designated as A. 

The first topic has an important role in the interpretation and calculation of atomic and molecular structures and properties. It is needless to stress the importance of electronic correlation effects, a central topic of research in quantum chemistry. The relativistic formulations are of great importance not only from a formal viewpoint, but also for the increasing number of studies on atoms with high Z values in molecules and materials. Valence theory deserves special attention since it improves the electronic description of molecular systems and reactions with the point of view used by most laboratory chemists. Nuclear motion constitutes a broad research field of great importance to account for the internal molecular dynamics and spectroscopic properties. 

Methods for treating relativistic effects in molecular quantum mechanics have always seemed to me, if I may say so without appearing too impertinent to those who work in the field, a complete dog s breakfast. The difficulty is to know to what question they are supposed to be the answer, in the circumstances in which we find ourselves. We do not know what a relativistically invariant theory applicable to molecular behaviour might look like. As was pointed out to us at the last meeting, the Dirac equation certainly will not do to describe interacting electrons and even at the single particle level, where it seems to work, there is an inconsistency in interpreting its solutions in terms... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

A crucial feature of PNC experiments in atoms, molecules, liquids or solids is that for interpretation of measured data in terms of fundamental constants of the P,T-odd interactions, one must calculate those properties of the systems, which establish a connection between the measured data and studied fundamental constants (see section 4). These properties are described by operators heavily concentrated near or on heavy nuclei they cannot be measured and their theoretical study is not a trivial task. During the last several years the significance of (and requirement for) ab initio calculation of electronic structure providing a high level of reliability and accuracy in accounting for both relativistic and correlation effects has only increased (see sections 3 and 10). 

The electronic structure parameters describing the P,T-odd interactions of electrons (sections 7, 8, and 10) and nucleons (section 9) including the interactions with their EDMs should be reliably calculated for interpretation of the experimental data. Moreover, ab initio calculations of some molecular properties are usually required even for the stage of preparation of the experimental setup. Thus, electronic structure calculations suppose a high level of accounting for both correlations and relativistic effects (see below). Modern methods of relativistic ab initio calculations (including very... 

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