Showing papers on "Semiclassical physics published in 1970"
TL;DR: In this paper, a previously developed semiclassical theory of molecular collisions based on exact classical mechanics is applied to the linear atom-diatom collision (vibrational excitation).
Abstract: A previously developed semiclassical theory of molecular collisions based on exact classical mechanics is applied to the linear atom–diatom collision (vibrational excitation). Classical, semiclassical, and uniform semiclassical results for individual vibrational transition probabilities corresponding to the H2+He system are presented and compared to the exact quantum mechanical results of Secrest and Johnson. The purely classical results (the classical limit of the exact quantum mechanical transition probability) are seen to be accurate only in an average sense; the semiclassical and uniform semiclassical results, which contain interference effects omitted by the classical treatment, are in excellent agreement (within a few percent) with the exact quantum transition probabilities. An integral representation for the S‐matrix elements is also developed which, although it involves only classical quantities, appears to have a region of validity beyond that of the semiclassical or uniform semiclassical expressions themselves. The general conclusion seems to be that the dynamics of these inelastic collisions is basically classical, with all quantum mechanical structure being of a rather simple interference nature.
545 citations
TL;DR: In this paper, the theory of penning ionization and related associative ionization (AI) was examined in a classical, semiclassical, and quantum mechanical framework, the correspondence between the several descriptions being explicitly deduced; formulas for total cross sections for PI and AI, angular distributions for PI, and the distribution of energies of the ionized electron were presented.
Abstract: The theory of Penning ionization (PI) and related associative ionization (AI) is developed and examined in a classical, semiclassical, and quantum mechanical framework, the correspondence between the several descriptions being explicitly deduced; formulas for total cross sections for PI and AI, angular distributions for PI, and the distribution of energies of the ionized electron are presented. The possibility of anomalous structure is seen to appear in the energy distribution of the ejected electrons if the difference between the A*–B and A–B+ potential curves has a local extremum as a function of internuclear distance. Classically, this appears as an infinity in the distribution of electron energies, but the quantum mechanical expressions are reduced to obtain a uniformly valid approximation; the transition region through the classical infinity is characterized by an Airy function.
340 citations
TL;DR: In this article, the authors used exact solutions of the classical equations of motion (numerically obtained trajectories) to construct the corresponding classical approximation to the time-independent S-matrix elements for use in quantum mechanical expressions for cross sections; it is argued that this should accurately describe many quantum effects in heavy particle collisions.
Abstract: The aim of this work is to show how one can use exact solutions of the classical equations of motion (numerically obtained trajectories) to construct the corresponding classical approximation to the time‐independent S‐matrix elements for use in quantum mechanical expressions for cross sections; it is argued that this should accurately describe many quantum effects in heavy particle collisions. The expression for the S matrix in terms of the classical trajectory is given for systems of any number of degrees of freedom, and the matter is pursued in detail for the A + BC collision system. It is shown that within this classical limit the magnitude of an S‐matrix element is explicitly determined by its phase. Constancy of total angular momentum is used throughout to reduce the 12 first‐order differential equations of the A + BC system in its center of mass to eight equations. A practical method is also given (in Appendix B) for further reducing the number of coupled equations to six, the minimum number possible.
331 citations
IBM1
TL;DR: In this paper, it is shown that superradiance can be characterized by a specific maximum cooperation number and associated cooperation time, which are defined for the superradiant state, but their meaning and usefulness can be extended to other situations.
Abstract: Phenomena of coherent resonant propagation can be considered as resulting from the cooperative interaction of a certain number of excited two-level systems. It is shown that these phenomena can be characterized by a specific "maximum cooperation number" and by the associated "cooperation time." These are defined for the superradiant state, but their meaning and usefulness can be extended to other situations. The alternative description of of superradiance as a spontaneous or as a stimulated effect is also discussed and it is shown that with the help of the new concepts, the Dicke quantum perturbative treatment can be reconciled with the semiclassical theories.
194 citations
TL;DR: In this article, a review of the calculations involving electromagnetic theory, where the forces are termed long-ranged van der Waals forces, or retarded dispersion forces, is presented, along with a heuristic connection between zero-point energy and semiclassical fluctuation phenomena.
111 citations
TL;DR: In this article, explicit formulas for the first and second-order semiclassical rotationally inelastic transition probability with an electric multipole potential are derived, and an expression for the interference terms between first-and secondorder perturbation theory is also derived.
Abstract: Explicit formulas are derived for the first‐ and second‐order semiclassical rotationally inelastic transition probability with an electric multipole potential. The symmetry properties of the second‐order integrals are discussed, and a method for their evaluation is presented. An expression for the interference terms between first‐ and second‐order perturbation theory is also derived.
97 citations
84 citations
TL;DR: The general problem of determining the photoelectron "counting" distribution resulting from an electromagnetic field impinging on a quantum detector is formulated and various limiting forms of this distribution are derived, including the necessary conditions for those commonly accepted.
Abstract: In this paper we formulate the general problem of determining the photoelectron "counting" distribution resulting from an electromagnetic field impinging on a quantum detector. Although the detector model used was derived quantum mechanically, our treatment is wholly classical and includes all results known to date. This combination is commonly referred to as the semiclassical approach. The emphasis, however, lies in directing the problem towards optical communication. The electromagnetic field is assumed to be the sum of a deterministic signal and a zero-mean narrow-band Gaussian random process, and is expanded in a Karhunen-Loeve series of orthogonal functions. Several examples are given. It is shown that all the results obtainable can be written explicitly in terms of the noise covariance function. Particular attention is given to the case of a signal plus white Gaussian noise, both of which are band-limited to \pm B Hz. Since the result is a fundamental one, to add some physical insight, we show four methods by which it can be obtained. Various limiting forms of this distribution are derived, including the necessary conditions for those commonly accepted. The likelihood functional is established and is shown to be the product of Laguerre polynomials. For the problem of continuous estimation, the Fisher information kernel is derived and an important limiting form is obtained. The maximum a posteriori (MAP) and maximum-likelihood (ML) estimation equations are also derived. In the latter case the results are also functions of Laguerre polynomials.
68 citations
TL;DR: In this paper, the authors used exact classical mechanics (numerically computed trajectories) to treat classically forbidden transitions in a complex collision (such as an atom plus diatom) in a manner analogous to the way one uses it to treat barrier transmission by a single particle in one dimension.
59 citations
TL;DR: In this article, a forced-common-turning-point method was proposed to overcome the difficulties of the classical trajectories associated with the various states of importance in a collision process.
Abstract: The semi-classical treatment of atom-atom collisions involving electronic transitions is discussed. As is well know n difficulties occur if the classical trajectories associated with the various states of importance in a collision process differ significantly. A method designed to overcome these is described. It will be referred to as the forced-common-turning-point method. The four coupled first-order differential equations which describe the new version of the semi-classical two-state treatment for an atom-atom collision may be reduced to a pair of generalized impact parameter equations. The first Born approximation to the cross-section obtained from the straightforward semiclassical treatment differs from the corresponding cross-section obtained from the full quantal treatment mainly in that it contains an anomalous multiplying factor equal to the ratio of the initial to the final velocity of relative motion. This anomaly does not arise with the forced-common-turning-point method. A model collision process which provides a very searching test is considered. Only two states are included. The initial interaction is zero, the final interaction is Coulombic and the transition matrix element is exponential. Curve-crossing may occur. The distorted wave approximation to the excitation cross-section may be found exactly and may also be computed using the forced-common-turning-point method. There is remarkable accord between the results. Thus in a case w here the reduced m ass of the colliding systems is 2 on the chemical scale, w here the excitation energy is 3.4 eV and where the incident kinetic energy of relative motion is only 0.85 eV above this the excitation cross-sections obtained differ by as little as 0.01 %; and, moreover, the patterns of the contributions to the cross-sections from the separate partial waves are similar.
58 citations
TL;DR: In this paper, the classical scattering observable is obtained by exp(iΔ) where Δ is the interaction, and elements of the scattering matrix in the semiclassical limit are then defined, using the correspondence procedure, as the Fourier components of the classical observable.
TL;DR: In this paper, the classical Stokes polarization parameters are extended to treat quantum systems by using the semiclassical radiation theory and considering only radiation by the dipole mechanism, and the identification of the Stokes parameters with the density matrix elements is noted.
Abstract: The classical Stokes polarization parameters are extended to treat quantum systems. This is accomplished by using the semiclassical radiation theory and considering only radiation by the dipole mechanism. The identification of the Stokes parameters with the density matrix elements is noted. The Stokes parameters for a number of familiar quantum systems which undergo dipole radiation are then calculated.
TL;DR: In this paper, the theory of rotational excitation in collisions of diatomic molecules is transformed to obtain equations for a set of generalized phase shifts, in a form which may be interpreted in terms of trajectories and interference effects.
Abstract: The previous development of the theory of rotational excitation in collisions of diatomic molecules is transformed to obtain equations for a set of generalized phase shifts. The resulting equations are in a form which may be interpreted in terms of trajectories and interference effects. Approximate equations valid in the semiclassical limit are obtained. A further approximation leads to the previously developed sudden approximation.
TL;DR: In this article, the numerical calculation of semiclassical rotational transition probabilities up to second order is presented and the results are used to calculate rotationally inelastic collision cross sections.
Abstract: The numerical calculation of semiclassical rotational transition probabilities up to second order is presented. Various electric multipole potentials are considered. The results are used to calculate rotationally inelastic collision cross sections. This is applied to energy and angular momentum transfer in HCN–HCN and ICN–ICN collisions, respectively. Applications to rotational linewidth calculations are also discussed.
TL;DR: In this article, an improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom, and the resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions.
Abstract: An improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom. This formalism is applied to the problem of a single "two-level atom" interacting with a monochromatic field. The resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions. Not only stimulated emission and absorption, but also spontaneous emission with the proper Einstein $A$ coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.
TL;DR: Semiclassical scattering theory is applied to the optical model for two different complex potentials in this paper, and the results for the imaginary part of the phase shift are in excellent agreement with the exact calculation of Marriott and Micha.
Abstract: Semiclassical scattering theory is applied to the optical model for two different complex potentials. For the case of a Lennard‐Jones interaction, the results for the imaginary part of the phase shift are in excellent agreement with the exact calculation of Marriott and Micha. The use of a modified wavenumber approximation yields useful estimates. For a complex exponential potential, the semiclassical and quantum expressions for the opacity function at zero impact parameter are identical.
TL;DR: In this paper, the authors compared the results of the perturbed stationary states (PSS) approximation with those of the more usual (static) approximation and with the exact results, showing that the PSS results were good at low energies at all mass ratios studied, unlike the static results.
Abstract: Vibrational–translational energy transfer is examined in the near‐adiabatic (or perturbed stationary states) approximation. The results are classical, and the method used is related to that of Marcus [J. Chem. Phys. 49, 2617 (1968)]. The results are compared with those of the more usual (“static”) approximation and with the exact results. The PSS results were good at low energies at all mass ratios studied, unlike the static results. For certain mass ratios the static approximation fails badly, even at very low transition probabilities. For other mass ratios, the results are of comparable accuracy except at high energies where the static one is somewhat better. Reasons for the above behavior are discussed, and implications regarding existing infinite‐order distorted wave and semiclassical calculations are noted. The relation to a recent correction of the Jackson–Mott calculation is described.
TL;DR: In this paper, the N-state semiclassical approach is used in the calculation of transition probabilities for atom-diatomic molecule collinear collisions, where the diatomic molecule is allowed to oscillate during the generation of the classical trajectories.
Abstract: The N‐state semiclassical approach is used in the calculation of transition probabilities for atom–diatomic molecule collinear collisions. The diatomic molecule is allowed to oscillate during the generation of the classical trajectories. Comparisons are made with techniques in which the diatomic molecule is constrained to be nonoscillatory as the classical trajectories are traced out. The semiclassical method is extended to include diatomic–diatomic and atom–triatomic molecule collisions with use of the energy conservation equation.
TL;DR: In this paper, the transition probabilities for collision induced dissociation reaction are calculated for a harmonic oscillator model using a semiclassical approximation, where the number of oscillations is correlated to the number bound states of the BC molecule.
TL;DR: In this article, it was shown how classical trajectories, determined by a scalar interaction, can be used to expand the scattering operator in the sum of operators characteristic of the radiator's internal states.
Abstract: The classical-path approximation reduces the problem of pressure broadening of spectral lines to the evaluation of matrix elements of the scattering operator. If the intermolecular potential is long range and the interaction volume is large, the broadening is caused by distant or weak collisions. In this case, the scattering operator can be approximated by a second-order expansion, and the perturber trajectories can be taken to be straight paths. For neutral atoms or molecules, the intermolecular potential is short range and broadening arises from close or "strong" collisions. In this paper it is shown how classical trajectories, determined by a "scalar interaction" (i.e., one that does not depend upon the state of the radiator), can be used to expand the scattering operator in the sum of operators characteristic of the radiator's internal states.
TL;DR: In this article, a semi-classical approximation for two assumed interaction potentials is presented for proton-helium scattering, and the cross sections for these two potentials are shown for protons incident at energies of 7, 19, 58, and 116 eV in the laboratory frame and for scattering angles, at each energy, out to the rainbow angle.
Abstract: Elastic differential cross sections for proton-helium scattering are calculated in the semi-classical approximation for two assumed interaction potentials. Both potentials are of the form $V(r;A,B,C)=(\frac{2}{r}){e}^{\frac{\ensuremath{-}r}{A}}[1+\frac{r}{A}+\frac{1}{2}{r}^{2}(\frac{1}{{A}^{2}}\ensuremath{-}\frac{U}{B})]\ensuremath{-}U{[1+\frac{r}{B}+{(\frac{r}{C})}^{2}+\frac{2U{r}^{4}}{\ensuremath{\alpha}}]}^{\ensuremath{-}1},$ where $U$ is the difference between the ground-state energies of He and ${\mathrm{Li}}^{+}$ (4.373 11 hartree) and $\ensuremath{\alpha}$ is the polarizability of He (1.3835 ${\mathrm{bohr}}^{3}$). The first potential ${V}_{M}\ensuremath{\equiv}V(r;0.423,0.483,0.441)$ fits the He-${\mathrm{H}}^{+}$ ground-state energies which Michels has calculated. The second, ${V}_{W}\ensuremath{\equiv}V(r;0.442,0.505,0.451)$, is similar to ${V}_{M}$ except that its minimum is decreased by 10% to agree with the value obtained by Wolniewicz. The cross sections for these two potentials are shown for protons incident at energies $T$ of 7, 19, 58, and 116 eV in the laboratory frame and for scattering angles, at each energy, out to the rainbow angle ${\ensuremath{\theta}}_{R}$. ${\ensuremath{\theta}}_{R}$ is given in center-of-mass coordinates by the expression ${\ensuremath{\theta}}_{R}T\ensuremath{\cong}0.1$ rad hartree. As the collision energy decreases, the cross sections develop oscillatory structure not present in the classical cross sections. This structure and the rainbow angle are sensitive to the choice of potential, which suggests that measurements of ${\mathrm{H}}^{+}$-He cross sections may be used to test the suitability of, e.g., the Born-Oppenheimer potential for scattering phenomena. It is also suggested that many-body calculations of these cross sections would allow, by comparison with the present results, an evaluation of the potential scattering model.
TL;DR: A theoretical study of field-induced predissociations is presented in this paper, where it is found that the rate of predissociation in general depends on the rotational quantum number.
Abstract: A theoretical study of field‐induced predissociations is presented. It is found that the rate of predissociation in general depends on the rotational quantum number. However, for several limiting cases the rotational dependence cancels out. The vibrational effect on predissociation is treated in the semiclassical approximation. Comparison is made with the recent experimental results on the magnetic quenching of iodine fluorescence by Degenkolb, Steinfeld, Wasserman, and Klemperer.
TL;DR: In this paper, a numerical method is described for calculating the first order response of an electron distribution to changes of field or scattering rate or for deriving the time development of any small distortion of the distribution.
Abstract: A numerical method is described for calculating the first order response of an electron distribution to changes of field or scattering rate or for deriving the time development of any small distortion of the distribution. The initial state may be in thermal equilibrium or may be modified by biassing fields. The technique simplifies the analysis of transport problems in which low symmetry perturbations distort a basic system of higher symmetry. The application of the method to the calculation of the galvanomagnetic properties of materials for small electric fields is set out as a detailed example and is illustrated by reference to n-type GaAs.
TL;DR: In this article, a simple method is described for the evaluation of classical trajectory integrals which play an important role in semiclassical collision theory, and the results for a Lennard-Jones potential are compared with the exact results of Bernstein and Kramer and then are applied to an optical analysis of glory quenching for low energy Li-HBr collisions.
Abstract: A simple method is described for the evaluation of classical trajectory integrals which play an important role in semiclassical collision theory. The results for a Lennard‐Jones potential are compared with the exact results of Bernstein and Kramer and then are applied to an optical analysis of glory quenching for low energy Li–HBr collisions. Calculations and comparison with exact results are also presented for an exponential interaction.
TL;DR: In this article, a semiclassical theory for inelastic collisions is used to calculate direct and multiple transition probabilities for the collinear collision of an atom with a harmonic oscillator.
Abstract: A semiclassical theory for inelastic collisions is used to calculate direct and multiple transition probabilities for the collinear collision of an atom with a harmonic oscillator. The method is based upon a single optimal choice of the effective interaction potential for the entrance and exit scattering channels. In light of the simplicity of this approach as well as the agreement with exact quantum‐mechanical results for this model problem, extensions of the theory to more complex inelastic processes are possible.
TL;DR: In this article, a special case of the equations for Lamb's semiclassical laser theory is solved, and the results are used to discuss the asymptotic behavior of the solution for large electromagnetic field and the power-series expansion in the field amplitude of the population inversion.
Abstract: This paper solves a special case of the equations for Lamb's semiclassical laser theory. The results are used to discuss the asymptotic behavior of the solution for large electromagnetic field and the power-series expansion in the field amplitude of the population inversion. A comparison with the continued-fraction method used earlier is made.
TL;DR: A review of graphical angular momentum methods is presented in this article, where the graphical methods are applied to the coupling problem in the semiclassical first and second Born approximations with an electric multipole potential.
Abstract: A review of graphical angular momentum methods is presented. As an example, the graphical methods are applied to the coupling problem in the semiclassical first and second Born approximations with an electric multipole potential. The general nth‐order Born term is then considered. A simple expression is derived for the matrix elements of the anisotropic terms in the potential. The evaluation of the remaining integrals over time is discussed.
TL;DR: In this paper, the authors examined the implication in the fact that the initial state of an atom plays a critical role in determining its average lifetime in an excited state, in a semiclassical theory of atomic structure proposed by Jaynes and Crisp.
Abstract: We examine the implication in the fact that, in a semiclassical theory of atomic structure proposed by Jaynes and Crisp, the initial state of an atom plays a critical role in determining its average lifetime in an excited state.