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The classical S matrix theory (a theory in which all degrees of freedom are treated semiclassically) developed by Miller and Marcus is applied to the collinear H + H2 reaction on a model potential energy surface. The classical, primitive, and uniform approximations and the initial value representation integral expression for the classical S matrix are calculated for total energies in the range 15–30 kcal/mole and compared to the exact quantum mechardeal calculations of Diestler. We choose a potential energy surface for which the quasiclassical reaction probability is unity over an energy range of 6 kcal/mole in order to make the test of theory particularly meaningful. It is seen that the classical S matrix theory is in very poor agreement with the quantum mechanical results for the reaction probabilities, although the semiclassical results for reactions producing vibrationally excited products are better behaved than those for reactions producing ground state products. The collision lifetimes and the stationary trajectories show no indication at all that the semiclassical calculations include a mechanism which could account for the 75% oscillation in the quantum mechanical PR0-0

We derive a semiclassical S matrix for vibrationally inelastic collisions between two diatomic molecules, assuming a collinear geometry. Our theory incorporates a quantum mechanical superposition principle with classical dynamics and, as such, is an extension of the atom‐diatomic molecule theory of Miller. The several approximations to the S matrix differ in the complexity with which the interference between various classical trajectories is treated. We report numerical calculations for H2–D2 and D2–D2 collisions based on two different interaction potentials. The cruder approximations yield transition probabilities which agree with exact quantum mechanical results to within a factor of 2. More sophisticated approximations to the S matrix yield excellent quantitative agreement with the quantum calculations. We also examine in detail the classical dynamics of the collision process and show that the molecules pass through a number of intermediate states during the encounter.

The formalism necessary for the application of ``classical S‐matrix'' theory to collisions of an atom with a rigid asymmetric rotor is derived. This is applied to rotational excitation of formaldehyde by H2 (taken to be spherically symmetric) at energies from 10 to 15°K. Classical Monte Carlo trajectory calculations were also carried out for the same system in the energy range 10–40°K. The results support the proposal of Townes and Cheung that a collisional mechanism is responsible for the 111 →110 anomalous absorption of formaldehyde in cool interstellar dust clouds.

We utilize an extension of Miller's semi-classical S matrix theory to calculate resonant and non-resonant vibrational energy transfer probabilities. The collisions under study are collinear D 2 D 2 interactions at energies below and above the classical dynamic transition threshold and collinear H 2 D 2 interactions above the dynamic threshold. Below threshold we employ an initial angle representation. We find that the computed probabilities are in substantial agreement with exact quantum mechanical computations and represent a major improvement over quasi-classical results. At energies above threshold we apply the first order and the classical semi-classical versions of the theory. The results indicate fair agreement with quantum mechanical calculations, but no significant improvement over quasi-classical results.

Expressions for the transition probabilities in collinear collisions of an atom with a diatomic molecule in the uniform, primitive, and classical semi-classical approximations of W.H. Miller are rederived from a new integral representation of a semi-classical S matrix. This derivation treats reactive and non-reactive collisions in a unified manner and gives additional insight into the nature of the semi-classical approximation. The derivation and resulting expressions are given in terms of a general internal diatom coordinate which can be chosen for computational convenience. The new integral expression is compared with ones given previously.

A model of divergence-free S-matrix is obtained by working with Dirac's instant form of dynamics. In this form the Poincaregroup generators G/sub alpha / are given in such a way that the energy P/sub o/ and the boost operators K/sub i/ contain essential, dynamical terms, whereas the momentum P/sub i/ and the angular momentum J/sub i/consist only of trivial kinematical terms. (LBS)

The broad compound state resonance found in quantum mechanical calculations of the collinear H + H_2 reaction is shown to correspond semiclassically to multiple collisions of the atoms within the collision complex. The quantitative agreement between QM and SC results is quite reasonable.

The purpose of this paper is not to produce another new theory, but only to discuss a few common sense requirements which any decent S -matrix must satisfy. First, we require that if the initial state consists of wave packets colliding at time t ′, and the final state of wave packets colliding (when produced backwards) at time t ″, then the transition amplitude should be vanishingly small if t ″ t ′. Moreover, if there is an appreciable time delay ( t ″ > t ′), the latter should be interpretable as arising from the propagation of an unstable intermediate particle. The above requirements, which are readily generalized to multiple scattering processes, are sufficient to derive the pole structure of the S -matrix and the existence of antiparticles. However, maximal analyticity and crossing symmetry cannot be proved from these assumptions—they can only be postulated separately.

Bound and scattering states in off-mass-shell and on-mass-shell formalisms, comparing calculations by considering solutions of Bethe-Salpeter and N/D equations

The complete system of equations of the non-relativistic S -matrix theory is proposed, the system consisting of the Ekstein equations [1] and new additional equations eliminating the ambiguity in the solution of the bilinear equations known earlier. General relations between various amplitudes are found with the help of the obtained equations. The two-particle case with a separable interaction is discussed as an example.

A general method is given to calcuiate the crossing relations for arbitrary processes that is based on the transformation properties of the amplitudes and analytic continuation only. The helicity amplitudes as well as the so called R and M amplitudes are considered and some simple examples are worked out. (auth)

In the double scattering case, we show that the introduction of a pole in the S‐matrix elements, is consistent with a description in terms of successive interactions.We then show that among Olive's class of possible ``propagators'' c(k2 ‐ m2 + ie)−1 ‐ (1 ‐ c) ·(k2 ‐ m2 ‐ ie)−1, only the usual value c = 1 gives a consistent result.

S-matrix computations for quantum transitions, considering close coupling case of homonuclear diatomic molecular rotational excitation

Gupta-Bleuler condition in quantum electrodynamics, noting inadequacy for defining physical states, alternate condition, collision problem and short lived particle decay

The problem of calculating the current and admittance of a solid cylindrical antenna is of great practical interest. Yet, unlike the hollow tubular antenna, the above subject has found little attention in the literature. In the present paper, the solution to this problem is constructed in two steps. The first of these involves the derivation of an expression for the reflection coefficient at the end of a semi-infinite solid cylinder, which is obtained by solving a modified Wiener–Hopf equation using a newly-developed technique. Convenient expressions for the current and admittance of the antenna are derived next by considering multiple-reflection phenomena at the ends of the antenna. These expressions are further simplified for thin antennas, i.e. for [Formula: see text], and it is shown that the effect of the end caps, as reflected in the admittance of such an antenna, is to increase its effective length by approximately 0.26a.

The discussion on the correspondence between unstable particles and poles of the $S$ matrix, started in a previous paper, is continued with particular attention to the decay law. It is shown that the previous formalism can be modified to obtain an $S$ matrix, satisfying all the usual requirements of analyticity, unitarity, and asymptotic behavior, which exhibits an isolated resonance with exponential decay without an accompanying pole in the unphysical sheet. The existence of a local potential which yields the $S$ matrix considered is also proved.

The definition of a particle as elementary or composite and the decomposition of the scattering amplitude into one-particle reducible and irreducible parts for both types of particles are re-examined. For the composite-particle solution the decomposition is based on identifying the field-theoretic quantities for the composite particle with the corresponding quantities for the elementary particle. We then prove the cancellation mechanism between the poles of the proper vertex function and the zeros of the propagator even when the propagator satisfies a once-subtracted Kallen-Lehmann representation, so that vertex poles do not lead to poles in the scattering amplitude. The dynamical origin of the vertex poles is discussed. We also discuss the compositeness conditions used by other authors and show how these conditions follow from our construction of the various field-theoretic and S-matrix quantities. However, we find that the popular compositeness conditions Z3=0, Z1=0 are not sufficient to obtain the composite particle in field theory without a restriction on the propagator. The asymptotic behavior of both field-theoretic and S-matrix quantities is discussed briefly.

The interatomic V(r) found by potential-model calculations is compared to an ab initio calculation of the intermolecular potential for NeH+ due to Peyerimhoff (1965). The reason for a rather significant difference in the two results is discussed, and a new method due to Remler (1971) involving Regge poles is applied. The method can be used to calculate the differential cross section, where the starting point for such calculations is based on intuition in light of the classical deflection function.

Three particle production in the high angular momentum limit is related to the residue at the pole of the elastic amplitude. If there is no three particle production there can be no pole. Unitarity forces the coupling constant to be real. The results depend only on unitarity, crossing, and Mandelstam analyticity.

Attenuation cross section and refractive index using collision theory, calculating resonance profiles of autoionizing lines based on scattering theory

On the basis of unitarity and the Mandelstam representation, the analyticity and threshold behavior of the positions and residues of the poles of the scattering amplitude in angular momentum as a function of energy are discussed. The right and left cuts of the partial wave amplitude for complex l are separated, and, in a model theory in which elastic unitarity holds, the domain of analyticity in the l plane is extended to the region Rel > 1. (auth)

A unified formal treatment of the two-state potential-curve-crossing problem in atomic collision theory is presented, and the case of close crossings analyzed in detail. A complete solution for this case, including necessary computations, is given using a suitable generalization of the linear model originally suggested by Landau, Zener, and Stueckelberg. Our solution is based upon a hierarchy of approximations concerned with (i) choice of a discrete basis set for electronic coordinates, (ii) semiclassical treatment of the nuclear motion, (iii) an appropriate model for the two-state electronic Hamiltonian, and (iv) a complete solution to that model.

BS>S-matrix theory is reviewed. The postulates underlying the S-matrix theory of strong interactions are outlined, and the results of Froissart and Regge are given. The properties of Regge poles and the experimental status of the principle of maximum strength are discussed. Finally, the Smatrix theory is applied to high-energy scattering. (L.T.W.)

Just as the derivative of the argument of the S matrix with respect to energy gives a time interval for events, it is shown that the corresponding derivative with respect to momentum transfer gives a space interval. This space interval corresponds to the classical impact parameter in the classical limit. More generally, it is suggested that these two derivatives may provide a basis for introducing space-time intervals into physical theory.

A mathematically more precise derivation of the Pugh equations for the S matrix is presented. These equations are applied to the calculation of the third-order vertex term. It is found that λ v is finite. Also, a formal relationship is established between the Pugh vertex term and the renormalized Feynman vertex term.

Scattering matrices accurately computed via close‐coupling techniques are analyzed and discussed as a function of critical parameters of an atom–diatomic molecule scattering system. The usefulness of the t test for delineating the region of strong coupling and ascertaining the range over which statistical approximations may be appropriate is demonstrated.

The Lamb shift in hydrogen is calculated by means of dispersion theory. Relativistic invariance is maintained throughout the calculation, and electron spin is included, although the spin of the proton is neglected. Divergence problems which have plagued other calculations are eliminated by the exploitation of the Regge behavior of the amplitudes which appear in the dispersion integrals. The preliminary results are in excellent agreement with the lowest-order values given by Yennie and Erickson.

The Gunson‐Olive technique for exploiting physical region poles and the unitarity equations to derive the analytic properties of S‐matrix elements is applied to particles which belong to isospin multiplets and to particles with spin. In the former case it is shown that, in the framework of S‐matrix theory, crossing symmetry leads to the requirement that particles and antiparticles transform according to complex conjugate representations; and it is possible to clarify the origin of the familiar isospin crossing matrices. In the case of particles with spin the construction of appropriate ``spinor'' basic states is discussed and the analytic properties of the corresponding spinor S‐matrix elements or ``M‐functions'' are derived.

The correspondence (unstable particle) \ensuremath{\rightarrow} (pole of the $S$ matrix) is discussed. In particular, it is explicitly shown that $S$ matrices, satisfying all requirements of analyticity, unitarity, and threshold and asymptotic behavior in energy, can be constructed in such a way that a sharp isolated resonance is produced without an accompanying pole in the unphysical sheet. As a matter of fact, in two of the examples which we consider, no singularities, except kinematical and dynamical branch lines, are found in any sheet of the energy Riemann surface.

We discuss the system of two overlapping resonances along two parallel lines: (A) the general restrictions imposed on the $S$ matrix by unitarity, and (B) the results of the Weisskopf-Wigner approximation. The equivalence between the two is used to get a physical understanding of the formalism applicable to both scattering and decay processes. We show that the connection between the overlap in channel space, ${\ensuremath{\chi}}_{c}$, and the overlap of the two resonances, $\ensuremath{\chi}$, is the unitarity sum rule.

A general discussion is carried out here of resonance poles in multi-two-body channel S-matrix theory. After Regge, the poles are defined by surfaces in energy-angular-momentum space. Using the inverse of the Regge trajectory α(ν), a study is made of the poles in the energy plane arising from a given Regge trajectory and how these poles move near threshold with angular momentum. These pole motions have important similarities with conclusions recently derived by others on the basis of a picture in which the coupling strength is varied. One of our results is a unified understanding of the behavior of poles near threshold for the s-wave case and the corresponding behavior for poles of higher angular momentum. The function α(ν) is found to possess all information needed to describe a resonance including a specification of associated energy poles on a number of Riemann sheets. A number of related questions are also considered here: the general question of defining a resonance pole in the many-channel problem; the precise formulation of those conditions under which a resonance pole is “near” enough to the physical region to produce an observable peak; the meaning of a resonance near threshold; the possibility of diagonalizing the S-matrix for nonintegral angular momenta and unphysical energies; properties of α(ν) for the many-channel problem including the positive definiteness of Im α; a discussion of the possibility of Regge trajectories decoupling from certain channels and the relation of this possibility to conservation laws and their derivation. The entire discussion is kept on very general grounds in which only unitarity and plausible assumptions about the analyticity of scattering amplitudes in angular momentum and energy are employed.

Threshold effects on nuclear reaction cross sections are re-examined in the frame of S -matrix theory. The Thomas-Ehrmann shift is rederived. The behaviour of poles of the S -matrix near thresholds, their distribution on the sheets of the Riemann surface and their influence on nuclear cross sections are investigated. The possibility of the occurrence of “threshold states” and of “virtual states’ is discussed. The effect of the large number of thresholds encountered in the investigation of statistical fluctuations of nuclear cross sections on the average width Γ of the compound nucleus is found to be negligible.

A definition of a relativistic generalized potential is given, suitable at arbitrary energies for a pair of particles whose elastic scattering amplitude satisfies the Mandelstam representation. It is shown that the generalized potential plays a role in the dynamics analogous to that of the ordinary nonrelativistic potential in a Schrodinger equation and determines the scattering to the same extent. Below the threshold for inelastic processes the generalized potential is real and its energy dependence in the elastic region is expected for certain particle combinations (such as the nucleon-nucleon) to be weak. In such cases one may uniquely define, for use in the Schrodinger equation, an energy-independent ordinary potential that coincides with the potential of Charap and Fubini (Abstr. 1960A01252). In general, when the potential is complex and energy-dependent the dynamical problem involves iteration of an integral equation deduced by Mandelstam (Abstr. 1959A04941). The generalized potential may be decomposed according to range and it is shown that keeping only the long-and medium-range parts, corresponding to transfer of one or two particles, is almost equivalent to the "strip approximation." Finally, a general definition is given of "pure potential scattering" as opposed to scattering associated with "independent" particles, either stable or unstable, and a variety of experimental situations are discussed with respect to this distinction, which is shown to be susceptible to experimental test.

Some recent equations for final-state interactions based on the explicit insertion of unitarity singularities in an on-mass-shell analyticS-matrix theory are considered. Numerical solutions for the model case of spinless particles are given. This forms the framework for a discussion of the expected magnitudes of multiple-rescattering effects, which are of topical interest. The equations are straightforward numerically, and comparison of the results shows the features which are independent of the details of the individual theories.