Phase-space representations atoms

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

In this chapter we present Fock s construction of a representation of the group SO (4) on the phase space of the hydrogen atom. This representation... 

While in some analogous works, it was possible to devise surfaces of section or even full representations of phase space this is hardly thinkable here. Let us recall that an on-shell (or constant energy H = E = 0.001 atomic units) Poincare section would be of dimension = D(phasespace) — 1 — 1=6. Instead we... 

Phase space is a 6N- 12 dimensional representation of the atomic (3n - 6) coordinates and their associated (3N- 6) momenta. Reactive trajectories in phase space move from reactant to product. The TS is the hyperplane such that all trajectories that cross this plane do so only once. In other words, trajectories that cross this plane from the reactant side will go on to products without ever turning back and recrossing the plane toward reactant. Given this definition, the rate of reaction is the number of tfajectories that cross the plane per unit time. ... 

Applications of exact quantum mechanics to systems with four atoms involved is still a major numerical burden, not to mention application to even bigger systems, and so approximate methods based on the solution of the classical equations of motion are nevertheless highly desired. In this context the semiclassical initial value representation (IVR), a precursor of which had already been derived in Miller s 1970 paper [2], may turn out to become a significant tool in molecular dynamics (see Ref. [16] for a comprehensive list of recent references). In the IVR approach the quantum mechanical 5-matrix elements are approximated by a (multi-dimensional) integral over the initial classical phase space and the integrand contains only ingredients from clas-... 

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

The concepts of coherence and incoherence are related to the way in which the neutron, both as a wave and as a particle, interacts with the scattering sample. Wave-like representations of the neutron view its interaction with solids as occurring simultaneously at several atomic centres these atoms become the sources of new wavefronts. Since the scattering occurs simultaneously from all of these atoms the new wavefronts will spread out spherically from each new source and remain in phase. Provided the lattice is ordered, the coherence of the incident wave has been conserved. Constmctive interference between the new wavefronts leads to the generation of distinctive diffraction patterns with well-defined beams, or reflections, appearing only in certain directions in space and no intensity in other directions. 

Here, we discuss an alternative scheme where the superposition state <1>) can be generated in two identical atoms driven in free space by a coherent laser field. This can happen when the atoms are in nonequivalent positions in the driving field, where the atoms experience different intensities and phases of the driving field. The populations of the collective states of the system can be found from the master equation (31). We use the set of the collective states (35) as an appropriate representation for the density operator... 

One of the major trends of current research is the study of transmission of information between the atom and photons in the process of emission and absorption. In particular, the conservation of angular momentum provides the transmission of the quantum phase information in the atom-held system. The atomic quantum phase can be constructed as the 57/(2) phase of the angular momentum of the excited atomic state (Section III). It is shown that this phase has very close connection with the EPR paradox and entangled states in general. Via the integrals of motion, it is mapped into the Hilbert space of multipole photons (Section IV.A). This mapping is adequately described by the dual representation of multipole photons, constructed in another study [46] (see also Section IV.B, below). Instead of the quantum number m, corresponding to the... 

The boundary surfaces in Figure 3.11 are for the electron density, the probability of an electron being at any point in space. The electron density is given by the square of the wavefunction. Points with the same electron density will have the same numerical value for the wavefunction, but the wavefunction may be positive or negative. For example, if the probability of finding an electron at a particular point was one-quarter, 0.25, then the wavefunction at that point would have the value plus one half, + 0.5, or minus one half, -0.5, since both (0.5)2 and (-0.5)2 are equal to 0.25. The sign of the wavefunction gives its phase. To represent the wavefunction itself we can use the same contours as for electron density, but we also need to indicate the phase of the wavefunction. In atomic and molecular orbital representations, we shall use colour to show differences in phase. Is orbitals are all one phase and so are shown in one colour. 2p orbitals have two lobes, which are out... 

The term structure in chemistry usually refers to a chemical structure, meaning the spatial arrangement of atoms or its representation in a structural formula (see Skill 6.1c). This is a chemical property, not a physical property, so the CSET standard may be referring to physical structure in a broader sense. All matter has mass and takes up space with an associated size. Matter experiencing gravity has a weight. Most matter we encounter exists in one of three phases (see Skill 12.1d). 

Like the sine function, the more complicated wave functions for atomic orbitals can also have phases. Consider, for example, the representations of the Is orbital in T Figure 9.38. Note that here we plot this orbital a bit differently from what is shown in Section 6.6. The origin is the point where the nucleus resides, and the wave function for the Is orbital extends from the origin out into space. The plot shows... 

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