Dirac function symmetric states

Treating vibrational excitations in lattice systems of adsorbed molecules in terms of bound harmonic oscillators (as presented in Chapter III and also in Appendix 1) provides only a general notion of basic spectroscopic characteristics of an adsorbate, viz. spectral line frequencies and integral intensities. This approach, however, fails to account for line shapes and manipulates spectral lines as shapeless infinitely narrow and infinitely high images described by the Dirac -functions. In simplest cases, the shape of symmetric spectral lines can be characterized by their maximum positions and full width at half maximum (FWHM). These parameters are very sensitive to various perturbations and changes in temperature and can therefore provide additional evidence on the state of an adsorbate and its binding to a surface. 

The explicit form of (2.5.7) depends on the nature of the function q,(yjr ) in (2.4.2). One often uses the Lennard-Jones (6-12) law (2.4.4). Numerical values ofyl are given in Table 18.1.1. It should also be pointed out that the form of (2.5.8) for molecules which obey Bose-Einstein statistics (like He ) is different from that for molecules obeying Fermi-Dirac statistics (like He ). In the summation over all quantum states in (2.5.1), only symmetrical states should be retained m the former and only antis3unmetrical states in the latter. 

Systems containing symmetric wave function components are called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Dirac systems (130,131). Systems in which all components are at a single quantum state are called Maxwell-Boltzmann systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Fermi-Dirac statistics (132). 

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

Since the symmetrical quantum state of the aether, proposed by Dirac, is not normalizable it should be considered a theoretical idealization, which can never be actually realized, although it can be approached indefinitely closely. Another such a state describes a particle with specified momentum. In this case the wave function of the particle cannot be normalized because of the uncertainty principle. By analogy, an aether that conforms to both quantum mechanics and relativity has to be an unattainable idealized state, like the perfect vacuum, or void. The real vacuum is an approximation to and more complicated than that. It is of lower symmetry and structured in such a way that particle velocities can be specified relative to the aether. To make any sense, the details of electronic motion through space, demand the presence of a structured aether. 

There are therefore only two possible ways of describing a state by a wave function, viz. either by the symmetrical or the antisymmetrical wave function the second possibility corresponds to Pauli s principle, the first is another, and entirely different matter. If we count the possible states on the basis of their wave functions (i.e. of the possible wave functions which are linearly independent), two different statistics present themselves. If we confine ourselves to the symmetrical wave functions (without Pauli s principle), we get the so-called Bose-Einstein statistics if we describe the state by the antisymmetric function (with Pauli s principle), we get the Fermi-Dirac statistics (1926). Which of the two statistics we are to use in a particular case, it must be left to experience to decide. With regard to electrons, we already know that they obey Pauli s principle—we shall therefore deal with them tentatively by the Permi-Dirac statistics (see 6, p. 214) on the other hand, it turns out that we have to treat light quanta (Bose) and also gas molecules (Einstein) according to the Bose-Einstein statistics. 

For identical hydrons, the symmetry postulate of identical particles has to be fulfilled. For protons and tritons this means that the overall wave function must be antisymmetric under particle exchange and for deuterons it must be symmetric under particle exchange. Due to this correlation of spin and spatial state, the energy difference A between the lowest two spatial eigenstates can be treated as a pure spin Hamiltonian, similar to the Dirac exchange interaction of electronic spins. 

As a lowest-order approximation, we assume that each electron in an atom moves in the field of the nucleus, which is described by a potential Vaudr), and a spherically symmetric potential U(r) that accounts approximately for the remaining bound electrons. The wave functions k describing possible states of the electron satisfy the one-electron Dirac equation... 

When particles are describable only by antisymmetrical wave functions, no two may occupy identical states. The statistical distribution among energy states may be thereby profoundly affected. The particles are said to follow Fermi-Dirac statistics. When the wave functions are symmetrical, the particles are said to obey Bose-Einstein statistics. If their states were describable by symmetrical as well as by antisymmetrical wave functions, then, as shown, individual allocations to states would acquire significance and they would be obeying the Boltzmann statistics. 

Results on the investigation of atomic density functions are reviewed. First, ways for calculating the density of atoms in a well-defined state are discussed, with particular attention for the spherical symmetry. It follows that the density function of an arbitrary open shell atom is not a priori spherically symmetric. A workable definition for density functions within the multi-configuration Hartree-Fock framework is established. By evaluating the obtained definition, particular influences on the density function are illustrated. A brief overview of the calculation of density functions within the relativistic Dirac-Hartree-Fock scheme is given as well. 

Particles come in two kinds (1) bosons, with symmetric wave functions, in which case any number of them can occupy the same quantum state (i.e., having the same values of n, ri2, and 3) and (2) fermions, with antisymmetric wave functions such that there is a rigid restriction that (neglecting spin) precludes the occupation of a given state by more than one particle. Examples of bosons include photons, phonons, and entities, such as " He atoms, that are made up from an even number of fundamental particles. Examples of fermions include electrons, protons, neutrons, and entities, such as He atoms, that are made up from an odd number of fundamental particles. Bosons are described by Bose-Einstein statistics, fermions by Fermi-Dirac statistics. 

For this case of symmetric wave functions, there is no limitation on the number of particles which we can put in a given state Aside from this difference, the procedure is identical with that followed in the Fermi-Dirac statistics. The analogous table is ... 

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