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arXiv:math-ph/0207042 · Journal of Mathematical Physics

Scattering into cones and flux across surfaces in quantum mechanics: A pathwise probabilistic approach

Andrea Posilicano, Stefania Ugolini

Published Oct 24, 2002• 4 citations
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Abstract

We show how the scattering-into-cones and flux-across-surfaces theorems in quantum mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behavior of paths of Nelson’s diffusions. The quantum mechanical results can then be recovered by taking expectations in our pathwise statements.

DOI
10.1063/1.1504884

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We show how the scattering-into-cones and flux-across-surfaces theorems in quantum mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behavior of paths of Nelson’s diffusions. The quantum mechanical results can then be recovered by taking expectations in our pathwise statements.
The relation between the physical meaning of a nonrelativistic NNN-particle flux and its mathematical representation in quantum mechanics is discussed. We prove a theorem that equates the probability of finding an NNN-fragment system in the distant future in a given cone with the total probability that the fragments cross a distant surface subtended by the cone. Together with the scattering-into-cones theorem this result proves that the usually calculated number of fragments whose momenta in the distant future lie in a given cone is equal to the total counted number of fragments that, at any time, cross a subtended distant surface. It thus adds to both physically and mathematically cleaner underpinnings of scattering theory.
The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the flux-across-surfaces theorem, which was conjectured by Combes, Newton, and Shtokhamer [Phys. Rev. D 11, 366–372 (1975)]. We prove the flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. The proof is based on the free flux-across-surfaces theorem (Daumer et al.) [Lett. Math. Phys. 38, 103–116 (1996)], and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V(x) decays like |x|−γ at infinity with γ>n∈N then the generalized eigenfunctions of the corresponding Hamiltonian −1/2Δ+V are n−2 times continuously differentiable with respect to the momentum variable.
We review the foundations of the scattering formalism for one-particle potential scattering and discuss the generalisation to the simplest case of many non-interacting particles. We point out that the “straight path motion” of the particles, which is achieved in the scattering regime, is at the heart of the crossing statistics of surfaces, which should be thought of as detector surfaces. We prove the relevant version of the many-particle flux across surfaces theorem and discuss what needs to be proven for the foundations of scattering theory in this context.
We review the foundations of the scattering formalism for one particle potential scattering and discuss the generalization to the simplest case of many non interacting particles. We point out that the "straight path motion" of the particles, which is achieved in the scattering regime, is at the heart of the crossing statistics of surfaces, which should be thought of as detector surfaces. We sketch a proof of the relevant version of the many particle flux across surfaces theorem and discuss what needs to be proven for the foundations of scattering theory in this context.
In this paper we study the sample path trajectories of the diffusions which are to describe potential scattering experiments in stochastic mechanics. Our main result asserts that the behavior of the trajectories is physically correct in that they settle down to straight line motion at constant speed with probability one. This limiting velocity determines a final momentum random variable, and we show that it has the same distribution as does the final momentum for the corresponding quantum state. Finally, we discuss the possible role of such a pathwise description of quantum scattering in Mathematical Physics On etudie les trajectoires des processus de diffusion qui doivent decrire les experiences de diffusion par un potentiel en mecanique stochastique. Le resultat principal est que le comportement des trajectoires est physiquement correct en ce sens qu'elles tendent vers des mouvements rectilignes uniformes avec probabilite un, la vitesse limite determine une grandeur aleatoire representant l'impulsion finale et on montre qu'elle a la meme distribution que l'impulsion finale dans l'etat quantique correspondant. On discute le role possible d'une telle description de la diffusion quantique en termes de chemins en physique mathematique