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The Journal of Chemical Physics

Wave packet study of a linear collision problem

Andreas Bringer, John F. Harris

Published Dec 15, 1989• 15 citations
Publisher

Abstract

Using real-time wave packet propagation we consider the scattering properties of a linear collision system consisting of a harmonic oscillator coupled by truncated Morse potentials to two particles. This system is viewed as a model for the collision of a projectile D with a three particle target ABC with interactions such that the possible final states comprise backscattering of the projectile with the target intact, the ‘‘exchange’’ reaction ABC+D→A+BCD and the ‘‘fragmentation’’ reaction ABC+D→A+BC+D. For the types of interaction considered, the exchange reaction was found to proceed primarily via inelastic resonant processes and to be the dominant event for incident energies below the oscillator energy. Fragmentation occurred via nonresonant processes and dominated at higher energy. Other features of the scattering are discussed and compared with simple quantum theory and the description of the system given by classical mechanics.

DOI
10.1063/1.457238

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Using real-time wave packet propagation we consider the scattering properties of a linear collision system consisting of a harmonic oscillator coupled by truncated Morse potentials to two particles. This system is viewed as a model for the collision of a projectile D with a three particle target ABC with interactions such that the possible final states comprise backscattering of the projectile with the target intact, the ‘‘exchange’’ reaction ABC+D→A+BCD and the ‘‘fragmentation’’ reaction ABC+D→A+BC+D. For the types of interaction considered, the exchange reaction was found to proceed primarily via inelastic resonant processes and to be the dominant event for incident energies below the oscillator energy. Fragmentation occurred via nonresonant processes and dominated at higher energy. Other features of the scattering are discussed and compared with simple quantum theory and the description of the system given by classical mechanics.
The scattering of a particle by a target with periodic time dependence is usually formulated in Floquet extended space. This leads to time-independent transition probabilities which correspond to the scattering of wave packets that are close to being monoenergetic. Because of the wide space extension of such packets, many oscillations take place during the time particle and target interact. The purpose of this study consists in showing that the scattering amplitudes can also be used to describe the way the particle feels the instantaneous configuration of the target at the time of the collision. The application is made to an oscillating barrier in one dimension. Several tests of the formalism validate the analysis, by changing either the initial position of the wave packet, or the phase of the particle–target interaction. Two equivalent necessary conditions can be formulated to recover the results of standard scattering theory. One is that no energies differing by at least one quantum are present in the energy range effectively covered by the initial wave packet. The other is that there is appreciable self-overlapping of the wave packet after one period of oscillation.
The projectile beam coherence effects occurring in ion-atom collisions are analyzed on the basis of the recent theory of Karlovets et al. [Phys. Rev. A 92, 052703 (2015)] developed for the elastic scattering of wave packets of particles off a potential field. This theory is generalized to estimate the loss of coherence for inelastically scattered projectiles in ionizing collisions. The results obtained by the suggested model are compared with experimental data for the ionization of hydrogen atoms and molecules by 75-keV proton impact. Significantly improved agreement is observed between the theory and experiment.
The one-dimensional scattering of a two body interacting system by an infinite wall is studied in a quantum-mechanical framework. This problem contains some of the dynamical features present in the collision of atomic, molecular and nuclear systems. The scattering problem is solved exactly, for the case of a harmonic interaction between the fragments. The exact result is used to assess the validity of two different approximations to the scattering process. The adiabatic approximation, which considers that the relative co-ordinate is frozen during the scattering process, is found to be inadequate for this problem. The uncorrelated scattering approximation, which neglects the correlation between the fragments, gives results in accordance with the exact calculations when the scattering energy is high compared to the oscillator parameter.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
We begin by considering the following scattering experiment. A projectile, e.g., an atom A in a given electronic state, collides with a target which we will take as a diatomic molecule BC in its ground electronic, vibrational and rotational state. For a short period of time an activated complex [ ABC ] # is generated. As time passes the activated complex can break into different products. For instance, in our example these products will be A + BC , B + AC, C + AB and A + B + C . Each one of the possible products can be in different electronic, vibrational and rotational quantum states. The total energy which is originally the sum of the electronic and translational energies of the projectile A and the electronic, vibrational, rotational and translational energies of target BC is conserved during the scattering process. Time-independent scattering theory enables one to calculate the probability of obtaining the specific products in given quantum states and the kinetic energy distribution of the products as a function of the total energy of the system without the need to solve the time-dependent Schrödinger equation. The time-independent formulation of scattering theory is based on the ability to propagate analytically an initial given wavepacket, Φ(0), to infinite times. That is, we need to get a closed form expression for lim t →±∞ e −iĤ t /ħ |Φ( t = 0)〉. To quote from the introduction of the excellent book on scattering theory written by Taylor: “The most important experimental technique in quantum physics is the scattering experiment. That this is so is clear from even the briefest review of modern physics”.
Two methodological difficulties of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type a + b → c + d. In order to calculate the cross section one uses the matrix element 〈cd|S|ab〉 of the S-matrix. The element is proportional to δ-function expressing the energy conservation. The corresponding probability |〈cd|S|ab〉|2 contains δ2, which is mathematically meaningless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the problems are resolved using wave packets of incident particles.