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arXiv:0705.2641 · Physics of Particles and Nuclei

Wave packets in the quantum theory of collisions

M. I. Shirokov

Published Sep 1, 2008• 0 citations
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Abstract

Two methodological troubles 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 $a+b\to c+d$. In order to calculate the cross section one uses the matrix element $$ of the $S$-matrix. The element is proportional to $δ$-function expressing the energy conservation. The corresponding probability $|< cd|S|ab>|^2$ contains $δ^2$ which is mathematically senseless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the troubles are resolved using wave packets of incident particles.

DOI
10.1134/s1063779608050067

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Two methodological troubles 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+da+b\to c+da+b→c+d. In order to calculate the cross section one uses the matrix element $$ of the SSS-matrix. The element is proportional to δδδ-function expressing the energy conservation. The corresponding probability ∣<cd∣S∣ab>∣2|< cd|S|ab>|^2∣<cd∣S∣ab>∣2 contains δ2δ^2δ2 which is mathematically senseless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the troubles are resolved using wave packets of incident particles.
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.
Quantum elastic potential scattering of a particle is re-examined taking into account exact solutions of the corresponding Schrödinger equation. In addition to the scattering of stationary plane waves and stationary finite-width wave beams, nonstationary wave packets having finite duration times are studied and some corresponding examples are presented. The role of interference between the scattered wave and the advancing incident beam is studied. Several two-dimensional scattering problems, involving axially symmetric, generic examples of nonuniform attractive and repulsive potentials, are discussed in more detail. This discussion concentrates on finding proper conditions when the solutions of the Schrödinger equation may resemble the corresponding solutions of the classical Newton equation. Examples are shown where such similarities occur.
THE classical theory of a point charge and the quantum theory of wave packets contain in their usual form well-known divergences. Dirae1 has elaborated a classical particle theory which avoids these difficulties, but his results cannot as yet be applied to quantum theory. On the other hand, Heisenberg1 has recently proposed a new formalism which permits the calculation of collision cross-sections in quantum theory without being disturbed by diverging terms. However, no connecting link such as the correspondence principle has been given in order to apply this quantum formalism to a given problem such as Rutherford scattering, Compton effect or radiation damping.
Potential scattering of free-electron wave packets is considered in the framework of the nonstationary quantum-mechanical theory. To characterize the efficiency of this process, the average angle of scattering is calculated in the lowest (first and second) orders of perturbation theory in the potential. In the approximation of a well-localized, narrow, and nonspreading wave packet, the first-order term is shown to agree with the classical angle of scattering, linear in the potential. This term is shown to vanish at a very large size of the initial wave packet (much larger than the size of a target). An analogy between the potential scattering of wave packets and electron-light ponderomotive scattering is discussed.
Scattering theory is extended to the description of processes during collision. Indeed, for large wave packets we may consider situations where time t, while large with respect to characteristic frequencies, is still smaller than the duration of the collision tc{\mathit{t}}_{\mathit{c}}tc​. This leads to an asymptotic theory which is different from the usual S-matrix approach. There appear intermediate states with scattering cross sections which result from secular effects for ttc{\mathit{t}}_{\mathit{c}}tc​, but differ from the values obtained for t\ensuremath{\gtrsim}tc{\mathit{t}}_{\mathit{c}}tc​ in the S-matrix theory. The consideration of such intermediate time scales is the normal procedure in many-body situations as realized, for example, in chemical reactions. Our method is closely connected to our work on the extension of quantum theory beyond the Hilbert space for large Poincar\'e systems. This theory applies to persistent scattering using delocalized density matrices (an example is plane waves). For this case, the S-matrix theory is not valid, while the results of our asymptotic approach remain valid for all times. Our theoretical predictions have been validated by numerical simulations. \textcopyright{} 1996 The American Physical Society.