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Some Concepts in Reaction Dynamics

In 1929 London 1 published a very approximate solution of the Schroedinger equation for a system of chemical interest: H3. To the extent that chemistry can be regarded as existing separately from physics, this was a landmark in the history of chemistry, comparable in importance to the landmark in the history of physics marked by the appearance of the Heitler-London equation for H2. The expression for H3, was, of necessity, even less accurate than that for H2, but chemists, like the habitual poor, were accustomed to this sort of misfortune. Together with the physicists they enjoyed the sensation of living in a renaissance. The physicists still could not calculate a great deal that was of interest to them, and the chemists could calculate less, but both could now dream. It would be too easy to say that their dreams were dreams of unlimited computer time. Their dreams were a lot more productive than that. Two years after London published his equation, H. Eyring and M. Polanyi obtained the first numerical energy surface for H3. They infused the London equation with a measure of empiricism to produce an energy surface which, whether or not it was correct in its details, provided a basis for further speculations of an important sort. The existence of a tangible energy surface in 1931 stimulated speculation along two different lines. The following year Pelzer and Wigner used this London-Eyring-Polanyi (LEP) energy surface for a thermodynamic treatment of the reaction rate in H + H2. This important development reached its full flowering a few years later. In these remarks I shall be concerned with another line of development. A second more-or-less distinct category of speculation that began with (and, indeed, in) the 1931 paper has to do with the dynamics of individual reactive encounters under the influence of specified interaction potentials.

Polannyi, John C.↗

Phase-space quantum mechanics study of two identical particles in an external oscillatory potential

This simple example is used to show how the formalism of Moyal works when it is applied to systems of identical particles. The symmetric and antisymmetric Moyal propagators are evaluated for this case; from them, the correct energy levels of energy are obtained, as well as the Wigner functions for the symmetric and antisymmetric states of the two identical particle system. Finally, the solution of the Bloch equation is straightforwardly obtained from the expressions of the Moyal propagators.

Nieto, Luis M.↗

Low-energy electron scattering from CO. 2: Ab-initio study using the frame-transformation theory

The Wigner-Eisenbud R matrix method has been combined with the frame transformation theory to study electron scattering from molecular systems. The R matrix, calculated at the boundary point of the molecular core radius, has been transformed to the space frame in order to continue the solution of the scattering equations in the outer region where rotational motion of the nuclei is taken into account. This procedure has been applied to a model calculation of thermal energy electron scattering from CO.

Chandra, N.↗

Quantum Cohesion Oscillation of Electron Ground State in Low Temperature Laser Plasma

The development of radically new technological and economically efficient methods for obtaining chemical products and for producing new materials with specific properties requires the study of physical and chemical processes proceeding at temperature of 10(exp 3) to 10(exp 4) K, temperature range of low temperature plasma. In our paper, by means of Wigner matrix of quantum statistical theory, a formula is derived for the energy of quantum coherent oscillation of electron ground state in laser plasma at low temperature. The collective behavior would be important in ion and ion-molecule reactions.

Zhao, Qingxun↗

The use of the Wigner Distribution to analyze structural impulse responses

In this paper it is argued that the time-frequency analysis of structural impulse responses may be used to reveal the wave types carrying significant energy through a structure. Since each wave type is characterized by its own dispersion relation, each wave type may be associated with particular features appearing in the time-frequency domain representation of an impulse response. Here the Wigner Distribution is introduced as a means for obtaining appropriate time-frequency representations of impulse responses. Practical aspects of the calculation of the Wigner Distribution are discussed and examples of its application to the analysis of structural impulse responses are given. These examples will show that the Wigner Distribution may be conveniently used to distinguish between the contributions of various waves types to a total structural response.

Wahl, T. J.↗

The application of the Wigner Distribution to the identification of structure-borne noise components

In this paper it is shown that time-frequency analysis of a transient structural response may be used to identify the wave types carrying significant energy through a structure. The identification of various wave types is possible since each is characterized by its own dispersion relation, with the result that each wave type may be associated with characteristic features in the time-frequency domain representation of a structural response. In the work described here, the Wigner Distribution has been used to obtain time-frequency representations of structural transient responses. Practical aspects of the calculation of the Wigner Distribution are discussed briefly, and examples of its application to the analysis of transient structural responses are given. Additionally, time-frequency filtering techniques are described that can be used both to detect the presence of and to quantify the energy carried by both non-dispersive and dispersive waves.

Wahl, T. J.↗

Ground state energy of electrons in a static point-ion lattice

The ground state energy of a neutral collection of protons and electrons was investigated under the assumption that in the ground state configuration, static protons occupy the sites of a rigid Bravais lattice. The Wigner-Seitz method was used in conjunction with three postulated potentials: bare Coulomb, Thomas-Fermi screening, and screening by a uniform bare background charge. Within these approximations, the exact band-minimum energy and wave functions are derived. For each of the three potentials, the approximate minimum ground state energy per proton (relative to isolated electrons and protons) is, respectively, -1.078 Ry, -1.038 Ry, and -1.052 Ry. These three minima all fall at a density of about 0.60 gm/cu cm, which is thus an approximate lower bound on the density of metallic hydrogen at its transition pressure.

Styer, D. F.↗

Identification of structural-borne noise components by the use of time-frequency filtering

In this paper it is shown that a time-frequency analysis of a structural impulse response may be used to identify the wave types carrying significant energy through a structure. The identification of various wave types is possible since each is characterized by its own dispersion relation, with the result that each wave type may be associated with characteristic features in the time-frequency domain representation of an impulse response. In the work described here, the Wigner Distribution has been used to obtain appropriate time-frequency representations of impulse responses. Practical aspects of the calculation of the Wigner Distribution are discussed briefly and examples of its application to the analysis of structural impulse responses are given. Additionally, filtering techniques are described that can be used both to detect the presence of and to quantify the energy carried by both nondispersive and dispersive waves.

Wahl, T. J.↗

Analytic computation of energy derivatives - Relationships among partial derivatives of a variationally determined function

Formulas are presented relating Taylor series expansion coefficients of three functions of several variables, the energy of the trial wave function (W), the energy computed using the optimized variational wave function (E), and the response function (lambda), under certain conditions. Partial derivatives of lambda are obtained through solution of a recursive system of linear equations, and solution through order n yields derivatives of E through order 2n + 1, extending Puley's application of Wigner's 2n + 1 rule to partial derivatives in couple perturbation theory. An examination of numerical accuracy shows that the usual two-term second derivative formula is less stable than an alternative four-term formula, and that previous claims that energy derivatives are stationary properties of the wave function are fallacious. The results have application to quantum theoretical methods for the computation of derivative properties such as infrared frequencies and intensities.

King, H. F.↗

Metallic bond effects on mean excitation energies for stopping powers

Mean excitation energies for first row metals are evaluated by means of the local plasma approximation. Particle corrections based on Pines' (1953) procedure and the Wigner Seitz (1934) model of the metallic state are included. The agreement with experimental values is remarkably good. In contrast to previous work, the calculations given here estimate shifts in the plasma frequency according to the theory for plane wave states in an extended plasma as calculated by Pines. It is demonstrated that the effects of the metallic bond in lithium and beryllium are quite large and that they appear mainly as a result of collective oscillations in the 'free' electron gas formed from the valence electrons. The usefulness of the plasma frequency shift derived for a degenerate electron gas in predicting the plasma frequency shift within the ion core is considered surprising.

Wilson, J. W.↗

Evolution of wave function in a dissipative system

For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert space. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner line width theory follows easily.

Yu, Li-Hua↗

Cosmic ray anisotropies at high energies

The directional anisotropies of the energetic cosmic ray gas due to the relative motion between the observers frame and the one where the relativistic gas can be assumed isotropic is analyzed. The radiation fluxes formula in the former frame must follow as the Lorentz invariance of dp/E, where p, E are the 4-vector momentum-energy components; dp is the 3-volume element in the momentum space. The anisotropic flux shows in such a case an amplitude, in a rotating earth, smaller than the experimental measurements from say, EAS-arrays for primary particle energies larger than 1.E(14) eV. Further, it is shown that two consecutive Lorentz transformations among three inertial frames exhibit the violation of dp/E invariance between the first and the third systems of reference, due to the Wigner rotation. A discussion of this result in the context of the experimental anisotropic fluxes and its current interpretation is given.

Martinic, N. J.↗

Strong-field cyclotron scattering. I - Scattering amplitudes and natural line width

The introduction of resonance line width into the QED cyclotron scattering amplitudes is considered. It is shown that the width arises from loop corrections to the electron propagator, which also bring about shifts in the Landau energy levels. A formalism is developed that allows the dressed electron propagator to be derived. It is shown that the states of Herold et al. (1982) and of Sokolov and Ternov (1968), which diagonalize the component of the magnetic moment operator parallel to the external magnetic field, are appropriate for calculation of the scattering amplitudes, whereas the states of Johnson and Lippmann (1949) are not. In addition, it is shown that the Breit-Wigner broadening approximation E tends to E - i(Gamma)/2 is consistent with the perturbation-theoretic order of the calculation, if the former basis states are chosen, but not the latter.

Graziani, Carlo↗

Kinematics of the CS method for the treatment of molecular collisions

The problem of the quantum treatment of nonreactive collisions of two simple molecules or of an atom and a molecule is considered mathematically. Kinematical structure theorems analogous to the Wigner-Eckart theorem for scalar operators are derived for the transition operator (T-operator) of a two-molecule system, where, in addition to the usual conservation laws, certain additional conservation laws are presumed satisfied in a collision. The additional conservation laws were examined in order to develop a testable approximation scheme (CS method) for the specification of the kinematics of a system of two rigid diatomic molecules. The rand p-helicity basis states for a two-rigid rotor system are defined and unitary transformations established between these and conventional bases. It is found that only questionable criteria exist for testing the on-the-energy-shell matrix of a T(energy) operator for the presence of properties of two of the additional conservation laws.

Hahne, G. E.↗

Boson localization and universality in YBa2Cu(3-x)M(x)O(7-delta)

We consider a two component mixture of charged fermions on neutralizing background with all sign combinations and arbitrarily small mass ratios. In the two impurity limit for the heavier component we show that the pair forms a bound state for all charge combinations. In the lowest order approximation we derive a closed form expression Veff(r) for the binding potential which has short-range repulsion followed by attraction. In the classical limit, when the mass of embedded particles is large m2 much greater than m, we can calculate from Veff(r) also the cohesive energy E and the bond length R of a metallic crystal such as lithium. The lowest order result is R = 3.1 A, E = -0.9 eV, not entirely different from the experimental result for lithium metal. The same interaction for two holes on a parabolic band with m2 greater than m gives the quantum mechanical bound state which one may interpret as a boson or local pair in the case of high-Te and heavy fermion superconductors. We also show that for compounds of the type YBa2Cu(3 - x)M(x)O(7 - delta) one can understand most of the experimental results for the superconducting and normal states with a single temperature dependent boson breaking function f(T) for each impurity content x governing the decay of bosons into pairing fermions. In the normal state f(T) turns out to be a linear, universal function, independent of the impurity content I and the oxygen content delta. We predict with universality a depression in Tc(x) with slight down bending in agreement with experiment. As a natural consequence of the model the bosons become localized slightly above Tc due to the Wigner crystallization, enhanced with lattice local field minima. The holes remain delocalized with a linearly increasing concentration in the normal state, thus explaining the rising Hall density. The boson localization temperature T(sub BL) shows up as a minimum in the Hall density R(sub ab)(exp -1). We also give explanation for very recently observed scaling of temperature dependent Hall effect in La(2 - x)Sr(x)CuO4.

Kallio, A.↗