Classical and quantum potentials

The archetype of quantum objects is the photon. It is massless, has unit spin, carries no charge, and responds to the quantum potential field. By comparison, an electron is a massive fermion with half-spin and unit negative charge. It responds to both classical and quantum potentials. The only property that these two entities have in common is their wave nature,... 

The interacting waves from myriads of charge centres constitute the electromagnetic radiation field. In particle physics the field connection between balanced charge centres is called a virtual photon. This equilibrium is equivalent to the postulated balance between classical and quantum potentials in Bohmian mechanics, which extends holistically over all space. 

There is no way to understand the energy difference between the S and A states (figure 5) in terms of pre-assigned interaction of the particles constituting the molecule. As the wave function is real, the momenta of all particles are zero and the energy is just the sum of classical and quantum potential energies... 

Classical and quantum potentials of mean force for the proton and deuteron abstraction of Ala-PLP by Tyr265, and the re-protonation of Ala-PLP carbanion intermediate by Lys39 in the active site of alanine... 

We briefiy review results we have obtained on model potentials with the VQRS reference system. The results obtained with the diagonal approximation to the propagator are superior to any previous such approximations that we are aware of. In table 3 classical and quantum results are presented for various moments of the quartic oscillator ... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

The analytic potential energy surfaces, used for the Cl + CH3Clb and Cl + CHjBr trajectory studies described here, should be viewed as initial models. Future classical and quantum dynamical calculations of SN2 nucleophilic substitution should be performed on quantitative potential energy functions, derived from high-level ab initio calculations. By necessity, the quantum dynamical calculations will require reduced dimensionality models. However, by comparing the results of these reduced dimensionality classical and quantum dynamical calculations, the accuracy of the classical dynamics can be appraised. It will also be important to compare the classical and quantum reduced dimensionality and classical complete dimensionality dynamical calculations with experiment. 

We now use a trick to partition this exact expression for the chemical potential into classical and quantum correction parts [29]. To do this we multiply and divide inside the logarithm of the excess term by the classical average... 

An important difference between classical and quantum particles is the way they interact with potential barriers. It is a principle of classical mechanics that the only way to overcome a potential barrier is with sufficient energy. Quantum-mechanically this is not always the case. The effect is illustrated by a beam of particles (e.g. electrons) approaching a potential barrier. 

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

So far we have illustrated the classic and quantum mechanical treatment of the harmonic oscillator. The potential energy of a vibrator changes periodically as the distance between the masses fluctuates. In terms of qualitative considerations, however, this description of molecular vibration appears imperfect. For example, as two atoms approach one another, Coulombic repulsion between the two nuclei adds to the bond force thus, potential energy can be expected to increase more rapidly than predicted by harmonic approximation. At the other extreme of oscillation, a decrease in restoring force, and thus potential energy, occurs as interatomic distance approaches that at which the bonds dissociate. 

Cross section and potential. Collision cross sections are related to the intermolecular potential by well-known classical and quantum expressions (Hirschfelder et al, 1965 Maitland et al, 1981). Based on Newton s equation of motion the classical theory derives the expression for the scattering angle,... 

Line shapes of a great many collision-induced absorption spectra have been obtained in recent years, using classical and quantum formalisms. These will be discussed in Chapters 5 and 6. Line shape calculations require the same input as the moment calculations, namely the dipole moment and interaction potential. They offer the advantage of generating certain parts of the spectra with remarkable precision, for example the... 

Results. In Table 5.1 we compare a few results of classical, semi-classical and quantum moment calculations. An accurate ab initio dipole surface of He-Ar is employed (from Table 4.3 [278]), along with a refined model of the interaction potential [12]. A temperature of 295 K is assumed. The second line, Table 5.1, gives the lowest three quantum moments, computed from Eqs. 5.37, 5.38, 5.39 the numerical precision is believed to be at the 1% level. For comparison, the third line shows the same three moments, obtained from semi-classical formulae, Eqs. 5.47 along with 5.37 with the semi-classical pair distribution function inserted. We find satisfactory agreement. We note that at much lower temperatures, and also for less massive systems, the semi-classical and quantal results have often been found to differ significantly. The agreement seen in Table 5.1 is good because He-Ar at 295 K is a near-classical system. 

Results. Figure 5.6 compares the classical and quantum profiles of the spectral function, VG (o), of He-Ar pairs at 295 K (light and heavy solid curves, respectively), over a wide frequency band. Whereas at the lowest frequencies the profiles are quite similar, at the higher frequencies we observe increasing differences which amount up to an order of magnitude. The induced dipole [278] and the potential function [12] are the same for both computations. Bound state contributions have been suppressed we have seen above that for He-Ar at 295 K, the spectroscopic effects involving van der Waals molecules amount to only 2% at the lowest frequencies, and to much less than that at higher frequencies. 

For any given potential and dipole function, at a fixed temperature, the classical and quantum profiles (and their spectral moments) are uniquely defined. If a desymmetrization procedure applied to the classical profile is to be meaningful, it must result in a close approximation of the quantum profile over the required frequency band, or the procedure is a dangerous one to use. On the other hand, if a procedure can be identified which will approximate the quantum profile closely, one may be able to use classical line shapes (which are inexpensive to compute), even in the far wings of induced spectral lines a computation of quantum line shapes may then be unnecessary. 

In the first place, we have two atoms fiee to move, not just the single particle treated previously. This is not in fact a serious problem. Provided that we are only interested in the relative motion of the nuclei, with the potential energy depending only on the distance between them, we can apply directly the classical and quantum results obtained above, x now represents a coordinate giving the displacements of the atoms relative to the centre of mass of the molecule. The mass must be altered to take account of the fact that both atoms move. We replace m by the reduced mass given in terms of the two atomic masses by... 

In this introductory chapter the concepts of linear and nonlinear polarization are discussed. Both classical and quantum mechanical descriptions of polarizability based on potential surfaces and the "sum over states" formalism are outlined. In addition, it is shown how nonlinear polarization of electrons gives rise to a variety of useful nonlinear optical effects. 

Both the classical and quantum approaches ultimately lead to a model in which the polarizability is related to the ease with which the electrons can be displaced within a potential well. The quantum mechanical picture presents a more quantitative description of the potential well surface, but because of the number of electrons involved in nonlinear optical materials, theoreticians often use semi-empirical calculations with approximations so that quantitative agreement with experiment is not easily achieved. 

The only difference between the classical and quantum formulations resides in the additional potential-energy term h2V2A/2mA, known as the quantum potential, Vq. In the classical case Vq 0. A quantum-mechanically stationary state occurs when Vq = k, a constant independent of x, i.e. 

We are especially interested in calculating accurate intermolecular potentials to be used with classical and quantum dynamics programs. Particularly for systems with larger numbers of valence electrons, we need to be able to obtain very accurate SCVB wavefunction by means of even smaller numbers of structures. To accomplish this we have devised a new approach in order to generate one or more optimal virtual orbitals for each occupied SC orbital [10]. We concentrate here on the case of a single properly optimized virtual orbital for each SC occupied orbital < r... 

The H3 potential surface which has been most widely used in transition state,2X4 classical dynamical,213 215 and quantum mechanical dynamical216-218 calculations has been the semi-empirical surface of Porter and Karplus.22 Because of the thin potential barrier of the PK surface, one would expect a larger amount of quantum mechanical tunnelling to be predicted at room temperature (this has been found to be the case in calculations performed by Johnston219 on barriers in the very similar LEPS surfaces). However, Karplus et a/.213-216 compared classical and quantum mechanical calculations on the PK surface and found that reaction cross-sections for both are very similar, and therefore that the tunnelling effect in the Ha system is small. 

Kupperman et a/.223-228 compared transition-state, classical, and quantum mechanical thermal rate constants using the SSMK surface.112 In ab initio potential energy surface calculations, the potential energy is known only as a list of values at selected geometries of the system. It becomes necessary, then, if trajectory calculations are to be made, to fit the calculated points to a smooth and continuous map . In their calculations, Kupperman et al. fit the collinear SSMK surface by the rotating Morse function procedure of Wall and Porter.227... 

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