Showing papers on "Semiclassical physics published in 1973"
TL;DR: In this article, the frequency shifts in emission and absorption arising from resonant many-body interactions in a system of two-level atoms are discussed from several points of view: (1) in the language of superradiance, Dicke states, quantum electrodynamics and perturbation theory, with emphasis on the impact approximation; (2) by means of diagrams related to the temperature-Green's function formalism.
284 citations
TL;DR: In this article, the authors developed a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators.
Abstract: We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.
164 citations
TL;DR: In this paper, an angle-dependent intermolecular potential was obtained by a least squares fit to the available experimental data, and suggested which other experiments will provide useful information about the potential.
Abstract: Extensive calculations on the rotationally inelastic collisions between HCl and argon have been carried out using the semiclassical method described in the preceding paper. By a detailed study of a single system it is hoped to clarify the behavior of an atom‐diatomic system during a collision, and the nature of the semiclassical model. This study also elucidates the relations between the different parts of the intermolecular potential and various experiments. We obtain an angle‐dependent intermolecular potential by a least squares fit to the available experimental data, and suggest which other experiments will provide useful information about the potential.
121 citations
TL;DR: In this article, the binary-encounter approximation (BEA) is transformed from momentum space to configuration space, and the impact-parameter representation allows one to calculate a variety of quantities pertinent to the general problem of ionization.
Abstract: The binary-encounter approximation (BEA) is transformed from momentum space to configuration space. In this frame the impact-parameter representation allows one to calculate a variety of quantities pertinent to the general problem of ionization. Among these are cross sections for proton ionization of hydrogen and helium; in the latter case, croas sections for ejection of both electrons are also given. A number of tables and formulas are given, enabling one to correct the simple hydrogenlike-model predictions of the BEA for effects which arise in multielectron atoms. Multiple-ionization probabilities (K + L shell) are calculated and compared to experimental results and to the predictions of the semiclassical approximation.
115 citations
TL;DR: Complex-valued classical trajectories for three-dimensional reactive collisions of H+H2 have been calculated at collision energies below the classical threshold for reaction, and from such trajectories classical S-matrix elements for the 0 → 1 rotational transition have been constructed as discussed by the authors.
Abstract: Complex‐valued classical trajectories for three‐dimensional reactive collisions of H+H2 have been calculated at collision energies below the classical threshold for reaction, and from such trajectories classical S‐matrix elements for the 0 → 1 rotational transition have been constructed. Comparison with available quantum mechanical results for the same system are encouraging and suggest that this semiclassical theory is capable of accurately describing reactive tunneling in a physically realistic model of a chemical reaction. Ways of simplifying the practical aspects of applying classical S‐matrix theory to three‐dimensional reactive systems are also described.
107 citations
Abstract: A method has been developed for calculating rotationally inelastic cross sections of atom‐diatom collisions in a semiclassical approximation (classical path, quantum internal states). An analysis of the relations among the intermolecular potential, the cross sections, and the experimental quantities indicates that many experiments related to inelastic scattering give significant information about the intermolecular potential. A computer program embodying this method of calculation has been developed and tested. Quantities related to experimental measurements have been calculated to within experimental accuracy in a reasonable computing time, which suggests that the information about the intermolecular potential contained in various experimental results can be extracted by a semiclassical calculation.
107 citations
TL;DR: In this article, the averaging procedure in Strutinsky's method of shell corrections is formulated for a general type of averaging function, and the method is proved analytically to give the same results as semiclassical methods.
90 citations
88 citations
TL;DR: In this paper, the authors used the classical (CSC), primitive (PSC), and uniform (USC) expressions for transition probabilities given by Miller and co-workers to calculate the reactive and nonreactive 0 --> 0 and 0 --> 1 transition probabilities for the collinear H + H2 exchange reaction.
Abstract: Using the classical (CSC), primitive (PSC), and uniform (USC) semiclassical expressions for transition probabilities given by Miller and co-workers, we have calculated the reactive and nonreactive 0 --> 0 and 0 --> 1 transition probabilities for the collinear H + H2 exchange reaction. Comparison with previously calculated exact quantum and quasiclassical results for the reactive and nonreactive 0 --> 0 transitions reveals that the semiclassical approximations are not very good, especially the CSC and PSC ones. All three semiclassical probabilities for the reactive 0 --> 0 transition exceed unity in the collision energy range from 0.0 to 0.2 eV above the quasiclassical reaction threshold. This feature coupled with the failure of any of the semiclassical approximations to produce the marked quantum effects present in this transition causes these results to be less accurate than the corresponding quasiclassical ones. For the reactive and nonreactive 0 --> 1 transitions the USC results are in qualitative agreement with the exact quantum ones and are better than the standard quasiclassical results. However, the reverse quasiclassical results are almost as good as the USC ones for these transitions. A probable reason for the inability of the USC expression to produce the strong oscillations observed in the exact quantum results is that the latter are due to interference between direct and resonant (i.e., compound state) processes whereas the present formulation of the semiclassical method does not encompass such phenomena. A comparison of the total reaction probabilities obtained by the USC and quasiclassical methods with the exact quantum one indicates that the USC result is more accurate than the quasiclassical one, except at collision energies less than 0.50 eV. This improved accuracy is due to a partial cancellation of errors in the contributing 0 --> 0 and 0 --> 1 USC reactive transition probabilities.
76 citations
TL;DR: In this article, the authors proposed a uniform approximation involving Bessel functions of the first kind, which approaches unity for the elastic collision, and also gives good agreement with exact quantum results, even if probabilities are large.
Abstract: It has been observed in the past that the usual Airy uniform approximation gives probabilities greater than one, especially for near elastic collisions. By mapping the phase onto −ζ cos y + ky + A rather than (1∕3)y^3 − ζy + A one obtains a uniform approximation involving Bessel functions of the first kind, which approaches unity for the elastic collision. This Bessel uniform approximation is no more complicated than the Airy and also gives good agreement with exact quantum results, even if probabilities are large.
68 citations
01 Jul 1973
TL;DR: In this article, the authors measured various coincidence rates between four photomultiplier tubes viewing cascade photons on opposite sides of dielectric beam splitters and showed that the experimental configuration is sensitive to differences between the classical and quantum field-theoretic predictions for the photoelectric effect.
Abstract: We have measured various coincidence rates between four photomultiplier tubes viewing cascade photons on opposite sides of dielectric beam splitters. This experimental configuration, we show, is sensitive to differences between the classical and quantum field-theoretic predictions for the photoelectric effect. The results, to a high degree of statistical accuracy, contradict the predictions by any classical or semiclassical theory in which the probability of photoemission is proportional to the classical intensity.
TL;DR: In this article, a simple canonical transformation is proposed to remove the singularities of the wave function at the turning points of the trajectory, which yields an integral expression for the S matrix by having produced wave functions which can be integrated over all space.
Abstract: Sometimes, as in reactive systems, action‐angle variables are not conveniently defined at all points of the trajectory and recourse must be made to conventional coordinates. A simple canonical transformation converts the latter to coordinates of which one is time and the remainder are constant along the trajectory. The transformation serves to remove the singularities of the semiclassical wavefunction at the turning points of the trajectory. It yields, thereby, an integral expression for the S matrix by having produced wavefunctions which can be integrated over all space. The result supplements that of Paper III [R. A. Marcus, J. Chem. Phys. 56, 311 (1972)], which was derived for systems for which action‐angle variables could be defined throughout the collision.
TL;DR: In this article, an approximation to the Wigner distribution function for a one-dimensional dense Fermi gas in a potential well is presented, where quantum oscillations occurring near the turning line are expressed in terms of a simple universal function, suitable for incorporation into a Thomas-Fermi self-consistent scheme.
TL;DR: In this article, a method for calculating the eigenvalues for systems not permitting separation of variables is described, where trajectories are supplemented by interpolation to connect open ends of quasi-periodic trajectories.
Abstract: Semiclassical theory for bound states is discussed and a method is described for calculating the eigenvalues for systems not permitting separation of variables. Trajectory data are supplemented by interpolation to connect open ends of quasi-periodic trajectories. The method is also applied to quasi-bound states.Previously, semiclassical S-matrix theory has focused on “direct” reactions. Processes involving complexes (compound state resonances) are treated in the present paper and an expression is derived for the S-matrix. Use is made of the above analysis of quasi-bound states and of trajectories connecting those states with open channels. The result deduced for the S-matrix has the expected factorization property, and expressions are given for computing the quantities involved. Some extensions and applications will be described in later papers. An implication for classical trajectory calculations of complexes is noted.
TL;DR: In this paper, a uniform approximation for a one-dimensional integral with many coalescing saddle-points is derived by applying the asymptotic techniques of Chester et al and Ursell.
Abstract: The uniform asymptotic evaluation of integral representations for S-matrix elements and scattering amplitudes in semiclassical collision theory is considered A uniform approximation for a one-dimensional integral with many coalescing saddle-points is derived by applying the asymptotic techniques of Chester et al and Ursell Physically, each saddle-point is associated with a real or complex-valued classical trajectory A concrete example of an integral with four coalescing classical trajectories whose positions depend on two parameters is discussed qualitatively to provide motivation for the calculations The uniform approximation is expressed in terms of a canonical integral and its derivatives The topological structure of the classical trajectories determines the canonical integral A series representation is derived for the canonical integral The series repesentation can be used to evaluate the canonical integral for small to moderate values of its arguments The extension of the one-dimensional uni
TL;DR: In this article, the asymptotic evaluation of multidimensional integrals for the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered, where the n-dimensional integral is assumed to possess two saddle points, whose position depends on a parameter and which may coalesce for a certain value of the parameter.
Abstract: The asymptotic evaluation of multidimensional integrals for the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered The n-dimensional integral is assumed to possess two saddle points, whose position depends on a parameter and which may coalesce for a certain value of the parameter A uniform asymptotic approximation is obtained by evaluating n-1 of the integrals by the ordinary non-uniform method of Chester et al The resulting uniform asymptotic formulae are generalizations of the results for one dimensional integrals given in a previous paper
TL;DR: In this paper, the authors describe a quantum-mechanical theory of the inelastic scattering of low-energy electrons by multiphonon processes, from the surface of a semi-infinite crystal.
Abstract: We describe a quantum-mechanical theory of the inelastic scattering of low-energy electrons by multiphonon processes, from the surface of a semi-infinite crystal. A model introduced in an earlier paper is also employed in this work. The model describes the interaction of an incident low-energy electron with surface optical phonons by means of the macroscopic electric field set up outside the crystal by the ion motion. The model may be used to describe scattering either from ionic crystals, such as ZnO, or from nonionic crystals. In this paper, we find an explicit expression for the wave function of the outgoing electron, and we obtain an expression for the probability that $n$ phonons are created or absorbed in the scattering process. Two cases are considered. First we examine the cross section for scattering off thermal phonons, and second from a coherent surface wave excited by external means. For the first case, our result agrees with the earlier semiclassical theory of Lucas and Sunjic. However, the model here is more general than theirs, since it is fully quantum mechanical. We show explicitly that the energy-loss cross section is proportional to the intensity of the specular beam, for scattering off both ionic and covalent crystals. For the second case (scattering from surface optical phonons generated coherently by an external source), we obtain a closed expression for the cross section. The physical origin of differences between the expressions is discussed.
TL;DR: In this paper, the radial equation for scattering from a cylindrically symmetrical potential is examined, where two-dimensional scattering arises in high-energy electron diffraction from crystals.
Abstract: The radial equation for scattering from a cylindrically symmetrical potential is examined, because two-dimensional scattering arises in high-energy electron diffraction from crystals. Particular attention is paid to the case of s waves, where there is a centripetal attractive potential for free particles. After showing that the Langer transformation, which leads to correct semiclassical wavefunctions for all other cases in two and three dimensions, fails for s waves, the method of comparison equations is applied, which enables the phase shifts and bound state conditions to be expressed in a simple form valid for all angular momenta. The theory is tested for s waves by comparison with exactly-calculated energy levels.
TL;DR: In this article, a semiclassical model for inner Coulomb corrections to pion-nucleus scattering in the Δ-resonance region is suggested, and its main consequence is a change of the nuclear radius R, giving R(π − ) R (π + ) = 1+2Z αE k 2 R.
TL;DR: In this article, a semiclassical theory of the width and shift of molecular spectral lines is developed for gases, and overlapping and nonoverlapping lines are considered, within the framework of the impact approximation.
Abstract: A semiclassical theory of the width and shift of molecular spectral lines is developed for gases. Overlapping
and nonoverlapping lines are considered, within the framework of the impact approximation. Use is made of
"exact" semiclassical theory of molecular collisions, recently developed by Miller and by Marcus, and of
developments in the quantum mechanical theory of spectral line shapes, by introducing the former into the
latter. Comparison is made with a classical-like approach.
TL;DR: In this paper, a general semiclassical theory that combines exact classical dynamics and quantum superposition is presented, where the quantum-like degrees of freedom are quantized semiclassically via use of double-ended boundary conditions, while the unquantized classical-like degree of freedom enter only through a phase space average over their initial coordinates and momenta.
Abstract: Within the framework of a general semiclassical theory that combines exact classical dynamics and quantum superposition it is shown how a certain averaging procedure allows one to treat some degrees of freedom in a strictly classical sense while others are quantized semiclassically. This enormously simplifies the application of the theory to three-dimensional collision systems and also leads to an interesting formal structure in the theory: the quantum-like degrees of freedom are quantized semiclassically via use of double-ended boundary conditions, while the unquantized classical-like degrees of freedom enter only through a phase space average over their initial coordinates and momenta. Preliminary results for vibrational excitation of H2 by He are presented and compared with available quantum mechanical calculations.
TL;DR: In this paper, the force on a moving ion in an electron gas to which an external electric field may be applied, is investigated in a semiclassical approximation, using a simple collision-time model for the mechanism which keeps the electron gas in equilibrium.
Abstract: The force on a moving ion in an electron gas to which an external electric field may be applied, is investigated in a semiclassical approximation, using a simple collision-time model for the mechanism which keeps the electron gas in equilibrium. It is found that in this simple model the screening of the ion has no effect on the force, so that the only correction to the force acting is that due to the 'wind effect', the momentum transfer due to the scattering of the electrons which are streaming past the ion. A complete solution of the equations is given for a weakly charged ion, to first order in the charge.
TL;DR: In this article, a semiclassical scattering matrix for the two-state problem with one non-adiabaticity region is discussed and the Stueckelberg phases for an exponential model are obtained and some limiting cases are considered.
TL;DR: In this paper, the distortion constants are calculated from diatomic potential curves using the phase-integral relationships which occur in the derivation of the potential curve's phase integral relationship.
TL;DR: There are a number of areas in chemical kinetics where generalizations have been helpful in interpreting and correlating a large body of experimental data in gas phase or solution reactions.
Abstract: There are a number of areas in chemical kinetics where generalizations have been helpful in interpreting and correlating a large body of experimental data in gas phase or solution reactions I am reminded here of Bronsted’s relation between rate constants and equilibrium constants, Eyring’s and Evans and Polanyi’s work on transition state theory, Rice, Ramsperger and Kassel’s work on unimolecular reaction, later augmented to RRKM, treatments of the curve crossing problems, Hammett’s σρ relation and acidity function, and the subsequent equations they stimulated, theories of three-body recombination of atoms and of electron transfers in solution and at electrodes, simple BEBO calculations on activation energies, the Woodward-Hoffman rules and their implications for activation energies, Breit-Wigner and later treatments of resonances, models for ion-molecule reactions, to name a few In the case of inelastic non-reactive collisions one would include the SSH theory, distorted wave theory for some systems, the Anderson theory of spectral line broadening and its later extensions
The interested observer, as well as the seasoned practitioner, might well ask which of these generalizations of analytical thought apply to current problems of molecular dynamics, what new ones have been developed, or what experimental generalizations are there, if any, which literally cry out for a theoretical answer He might ask, too, whether the present field is sufficiently different from the previous ones that the approximate analytical theory will be literally swept under by a Spartan-like phalanx of exact classical trajectories and their semiclassical and quantum mechanical counterparts, with much analytical thought going into this army
We shall not attempt to answer all of these questions here, but shall summarize instead some of the trends which appear to be developing in the field Calculations in the area are diverse, and some classification would be useful A possible scheme for dynamical calculations is proposed in this introductory paper
TL;DR: In this article, a three-dimensional theory for the resonant interaction of electromagnetic waves with a gas of two-level atoms is formulated in terms of macroscopic variables, and the resulting attenuation and reflection coefficients are expressed as velocity integrals of continued fractions.
Abstract: A three-dimensional theory for the resonant interaction of electromagnetic waves with a gas of two-level atoms is formulated in terms of macroscopic variables. The theory is utilized to find the steady-state attenuation of a plane wave in the presence of another plane wave running in the opposite direction with different amplitude. Contributions are included from the reflection of the oppositely running wave by an induced standing-wave inhomogeneity in the population inversion of the medium. The resulting attenuation and reflection coefficients are expressed as velocity integrals of continued fractions. Correspondence is made with existing gas-laser theories, yielding the formulation of a high-intensity ring-laser theory. Analytic approximations for the coefficients are presented for the Doppler-limit cases of both waves weak, one wave weak, and negligible reflection (rate-equation approximation). More-general cases have been calculated numerically. The attenuation coefficients exhibit a Lamb-dip feature. The relative depth of the dip increases rapidly with power at low saturation levels, slowly at high saturation, and is greater in the attenuation of the weaker wave. The width of the dip is nonlinearly power broadened. The shape of the dip is very nearly Lorentzian, except for one special case at high power in which the line splits. The propagation equations for the two waves are integrated over long absorption paths. A large resulting attenuation increases the relative size of the dip while decreasing the power broadening.
TL;DR: In this article, an exact series expansion is obtained with the help of convergence factors for the non-separable two-dimensional canonical integral considered earlier, and compared with the approach adopted in the present paper.
Abstract: The evaluation of the multidimensional canonical integrals that occur in the uniform asymptotic representations of the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered. For the non-separable two-dimensional canonical integral considered earlier, an exact series expansion is obtained with the help of convergence factors. This method avoids the complex variable techniques used previously. The uniform asymptotic formulae derived by Miller and Marcus are discussed, and compared with the approach adopted in the present paper.
TL;DR: In this article, the concepts of "semiclassical mechanics" are applied to the propagation of an electron beam through a crystal lattice, leading to simple physical descriptions and numerically accurate formulae for phenomena which have previously only emerged after lengthy computations based on Fourier analysis.
Abstract: The concepts of “semiclassical mechanics” are applied to the propagation of an electron beam through a crystal lattice. We obtain a variety of approximate descriptions which bridge the gap between the full quantum treatment necessary at lower voltages and the purely classical analysis which becomes valid at higher voltages. These methods, which are all based on “real space”, lead to simple physical descriptions and numerically accurate formulae for phenomena which have previously only emerged after lengthy computations based on Fourier analysis: examples are the existence of “critical voltages” and “critical angles”. It is shown how the purely classical concept of “caustic” can explain features on micrographs which are usually considered as interference fringes; this bypasses the need to construct “dispersion surfaces”. Absorption from the diffracted beams, which is caused by inelastic scattering, is incorporated into the semiclassical analysis.
TL;DR: A semiclassical treatment of electronic transitions in the collinear rearrangement H+ + D2 → HD+ (ν = 0,1) + D is presented in this article.
TL;DR: In this paper, the uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered and a non-separable two-dimensional canonical integral with four coalescing saddle points is presented.
Abstract: The uniform asymptotic evaluation of multidimensional integrals for the S-matrix in semiclassical collision theory is considered. A concrete example of a non-separable two-dimensional integral with four coalescing saddle points is chosen since it exhibits many of the features of more general cases. It is shown how a uniform asymptotic approximation can be obtained in terms of a non-separable two-dimensional canonical integral and its derivatives. This non-separable two-dimensional canonical integral plays a similar role to the Airy integral in one-dimensional integrals with two coalescing saddle points. The uniform approximation is obtained by applying to the two-dimensional case the asymptotic techniques introduced by Chester et al. for one-dimensional integrals. An exact series representation is obtained for the canonical integral by means of complex variable techniques. The series representation can be used to evaluate the canonical integral for small to moderate values of its arguments, whilst for large values of its arguments existing asymptotic techniques may be used.