Particles interpretation

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. 

The commutation rules (11-465) and (11-466), together with the existence of the no-particle states, allow us to formulate an asymptotic particle interpretation for photons and electrons in the remote past, albeit in terms of a formalism involving an indefinite metric. 

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. 

Consideration of the symmetry of the Poincare group also shows that the cyclic theorem is independent of Lorentz boosts in any direction, and also reveals the physical meaning of the E(2) little group of Wigner. This group is unphysical for a photon without mass, but is physical for a photon with mass. This proves that Poincare symmetry leads to a photon with identically nonzero mass. The proof is as follows. Consider in the particle interpretation the PL vector... 

It follows that the same analysis can be applied to the particle interpretation, giving... 

In the particle interpretation, these are part of the Lie algebra of rotation and boost generators of the Poincare group ... 

From this analysis, it is concluded that the B<3> component is identically nonzero, otherwise all the field components vanish in the B cyclic theorem (812) and Lie algebra (809). If we assume Eq. (803) and at the same time assume that B(3) is zero, then the Pauli-Lubanski pseudo-4-vector vanishes for all A0. Similarly, in the particle interpretation, if we assume the equivalent of Eq. (803) and assume that J(3) is zero, the Pauli-Lubanski vector vanishes. This is contrary to the definition of the helicity of the photon. Therefore, for finite field helicity, we need a finite B(3). 

This is also the relation obtained in the hypothetical rest frame. Therefore, the B cyclic theorem is Lorentz-invariant in the sense that it is the same in the rest frame and in the light-like condition. This result can be checked by applying the Lorentz transformation rules for magnetic fields term by term [44], The equivalent of the B cyclic theorem in the particle interpretation is a Lorentz-invariant construct for spin angular momentum ... 

In a simple two-particle interpretation, the spin of a given particle depends on that of the second spin (entangled state) or is independent of the second particle s spin (nonentangled or separable state). In the Schmidt decomposition, the two-particle wave function can be written as... 

In QM without particle interpretation, the statement interference fringes disappear once we have which-path information has truth content. The statement disappearance originates in correlations between the measuring apparatus and the system being observed, is not fully granted. From the present stand point, it is false. 

It is apparent why the separation of translation is problematic for the identification of electrons and nuclei. In the translation-free Hamiltonian the inverse effective mass matrix and the form of the potential functions fij depend intimately on the choice of V and the choice of this is essentially arbitrary. In particular it should be observed that because there are only N—1 translation-free variables they cannot, except in the most conventional of senses, be thought of as particle coordinates and that the non-diagonal nature of fi 1 and the peculiar form of the fij also militate against any simple particle interpretation of the translation-free Hamiltonian. It is thus not an entirely straightforward matter to identify electrons and nuclei once this separation has been made. 

Problem The particle model of matter may be introduced through the solution of two substances (see E4.6). Students will not come up independently with the particle interpretation - they need help by the verbal particle explanation or better by demonstrating model experiments. One can use one big sphere as a model for the iodine particle and one little sphere as a model for the ethanol particle. The model experiment can be done without many comments - one could then ask the students to associate certain aspects of the model with the original (see E4.6). In the model discussion it is important to expose the model traits, but also the irrelevant items. One can compare the model experiment with the model drawing (see Fig. 4.6). 

So, where is the electron and what is it doing We do not really know, and modern physics tells us that we never will know with 100% certainty. However, with information derived from wave mathematics for an electron, we can predict where the electron probably is. According to the particle interpretation of the wave character of the electron, the surface that surrounds 90% of an electron s charge is the surface within which we have a 90% probability of finding the electron. We could say that the electron spends 90% of its time within the space enclosed by that surface. Thus an orbital can also be defined as the volume within which an electron has a high probability of being found. 

The price for this relative simplicity is the loss of straightforward one-particle interpretation of one-particle eigenvalues which requires additional attention and will be discussed in more detail in following paragraphs. Only brief rudimentaries of the Density Functional Theory itself and its previous applications will be given here as extensive reviews on these subjects have already appeared [6-9] or are included in other chapters of this book. 

Given a solution V (t) of the Dirac equation, the interpretation is a consistent one-particle interpretation in the following sense If ip describes a one-particle state initially, then there is one and only one particle for all times. Or, in a more mathematical language If the initial wave packet is normalized t/ (0) = 1, then this normalization remains constant in time, that is,... 

CT and Hole-Particle Interpretation Other Structural Indices... 

The Hartree-Fock approximation is free of experimentally determined or adjustable parameters, allows a simple, single-particle interpretation, is often satisfactory for ground-state geometries, and can be improved systematically... 

The surprising implication is that Dirac s equation does not allow of a self-consistent single-particle interpretation, although it has been used to calculate approximate relativistic corrections to the Schrodinger energy spectrum of hydrogen. The obvious reason is that a 4D point particle is without duration and hence undefined. An alternative description of elementary units of matter becomes unavoidable. Prompted by such observation, Dirac [3] re-examined the classical point model of the electron only to find that it has three-dimensional size, with an interior that allows superluminal signals. It all points at a wave structure with phase velocity > c. 

These observations, especially the dependency on frequency, could not be explained by classical wave theory. However, Albert Einstein showed that they are exactly what would be expected with a particle interpretation of radiation. In 1905, Einstein proposed that electromagnetic radiation has particle-like... 

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