Classical picture

To nnderstand the internal molecnlar motions, we have placed great store in classical mechanics to obtain a picture of the dynamics of the molecnle and to predict associated patterns that can be observed in quantum spectra. Of course, the classical picture is at best an imprecise image, becanse the molecnlar dynamics are intrinsically quantum mechanical. Nonetheless, the classical metaphor mnst surely possess a large kernel of truth. The classical stnichire brought out by the bifiircation analysis has accounted for real patterns seen in wavefimctions and also for patterns observed in spectra, snch as the existence of local mode doublets, and the... 

In the classical picture developed above, the wavepacket is modeled by pseudo-particles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the Uajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of confignrations, X/(, t)tlt,(r, t)... 

In the classical picture of an electron orbiting round the nucleus it would not surprise us to discover that the electron and the nucleus could each spin on its own axis, just like the earth and the moon, and that each has an angular momentum associated with spinning. Unfortunately, although quantum mechanical treatment gives rise to two new angular momenta, one associated with the electron and one with the nucleus, this simple physical... 

The classical model predicts that the largest probability of finding a particle is when it is at the endpoints of the vibration. The quantum-mechanical picture is quite different. In the lowest vibrational state, the maximum probability is at the midpoint of the vibration. As the quantum number v increases, then the maximum probability approaches the classical picture. This is called the correspondence principle. Classical and quantum results have to agree with each other as the quantum numbers get large. 

During the photoelectron emission event there are electronic relaxation effects occurring, which are usually divided into intra- and inter-molecular relaxation effects. These effects can be rationalized in a classical picture as follows. An elec-... 

Figure 5.3. The classical picture the energies of dipoles varies continuously from parallel alignment with the applied magnetic field (-m-B) to antiparallel (+m-B). On the right is shown the distribution of molecules that results and the lower mean energy of the ensemble relative to the field-free environment.

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

Semi-Classical Pictures of Non-Adiabatic Induced Electron Ejection in Molecular Anions... 

The classical picture, where following the transition state the two oxygen atoms are channelled downwards to the nearest available metal atoms, is... 

As its name implies, AIM enables us to calculate such properties of atoms in a molecule as atomic charge, atomic volume, and atomic dipole. Indeed it shows us that the classical picture of a bond as an entity that is apparently independent of the atoms, like a Lewis bond line or a stick in a ball-and-stick model, is misleading. There are no bonds in molecules that are independent of the atoms. AIM identifies a bond as the line between two nuclei. 

As yet, this marks no radical departure from the classical picture of orbits, but with the 2p level (the continuation of the L shell) a difference becomes apparent. Theory now requires the existence of three 2p orbitals (quantum numbers n = 2, Z = 1, with m = +1,0, and... 

During the past 25 years, Funke and co-workers have extensively studied the emulsion polymerization of divinyl monomers alone including 1,4-DVB and ethylene glycol dimethacrylate (EDM A) under various reaction conditions. They found that the intraparticle crosslinking changes drastically the classical picture of emulsion polymerization. 

Although this classic picture evolved from "soft, mononuclear transition metal complexes suffices to explain a great deal of carbon monoxide chemistry, it is not clear that it is complete or accurate for understanding processes whereby CO is reduced, deoxygenated, and/or polymerized to form methane, long-chain hydrocarbons, alcohols, and other oxocarbons, especially in cases where heterogeneous catalysts or "hard" metals are involved (6, 7, ,9,J 0). This deficiency of information has led to the search for new modes of carbon monoxide reactivity and to attempts to understand carbon monoxide chemistry in nontraditional environments ... 

We shall now assume a classical picture to describe the transition jr — jt. ... 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

At high Reynolds numbers, we can again expect the small scales of the scalar field to be nearly isotropic. In the classical picture of turbulent mixing, one speaks of scalar eddies produced at large scales, with a distinct directional orientation, that lose their directional preference as they cascade down to small scales where they are dissipated by molecular diffusion. 

If 15N ammonium citrate was administered, and glutamate, aspartate, and glycine isolated from liver and intestinal wall protein, all showed 15N uptake. From the results of labeling studies, Schoenheimer finished his Edward K.Dunham lectures in Harvard in 1941 with the phrase— the structural materials [of the body] are in a steady state of flux. The classical picture must thus be replaced by one which takes account of the dynamic state of body structure —an idea which has become an integral part of biochemistry since that time, and which was almost totally dependent on the introduction of isotopes for its discovery. 

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