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Journal of Physics B Atomic and Molecular Physics

Quantum-mechanical streamlines and classical trajectories in elastic scattering

H. J. Korsch, R Möhlenkamp

Published Jun 14, 1978• 6 citations
Publisher

Abstract

The hydrodynamic formulation of quantum mechanics is considered in the semiclassical limit and the relationship between quantum streamlines and classical trajectories is discussed. The flow pattern of the quantum streamlines can be described in terms of the interference of classical trajectories. The flow inside the classically forbidden regions is related to complex-valued classical trajectories. The analytically solvable cases of Coulomb and hard-sphere scattering are investigated in detail.

DOI
10.1088/0022-3700/11/11/011

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The hydrodynamic formulation of quantum mechanics is considered in the semiclassical limit and the relationship between quantum streamlines and classical trajectories is discussed. The flow pattern of the quantum streamlines can be described in terms of the interference of classical trajectories. The flow inside the classically forbidden regions is related to complex-valued classical trajectories. The analytically solvable cases of Coulomb and hard-sphere scattering are investigated in detail.
We present a detailed discussion of the semiclassical approximations to path integrals when there exist caustics in the relevant family of classical trajectories. We show carefully how the Airy functions arise from the post-semiclassical approximation of the path integral and interpret the coefficients in the Airy functions in terms of the flow of classical trajectories. Finally, we present a configuration-space path integral for the scattering matrix and extend our discussion of caustics to the case of rainbow scattering.
In this paper, we apply the one dimensional quantum law of motion, that we recently formulated in the context of the trajectory representation of quantum mechanics, to the constant potential, the linear potential and the harmonic oscillator. In the classically allowed regions, we show that to each classical trajectory there is a family of quantum trajectories which all pass through some points constituting nodes and belonging to the classical trajectory. We also discuss the generalization to any potential and give a new definition for de Broglie's wavelength in such a way as to link it with the length separating adjacent nodes. In particular, we show how quantum trajectories have as a limit when ℏ → 0 the classical ones. In the classically forbidden regions, the nodal structure of the trajectories is lost and the particle velocity rapidly diverges.
The formulation of quantum mechanics in terms of the picture of trajectory ensemble or of the hydrodynamical picture, developed in a previous article, is further supplemented for some points. Individual trajectories for some typical non-stationary state are explicitly obtained, illustrating that they generally show complicated fluctuations due to the action of quantum potential. Next, emphasis is placed on the fact that the Schrödinger field admits a mechanical model. Also the Klein-Gordon field is represented with a quasi-irrotational flow under some additional scalar field. This representation of Klein-Gordon field suggests its possible generalization to include the vortical motion. Such treatments of interacting Klein-Gordon and electromagnetic fields involve the formalism of Dirac's recent classical electrodynamics as a limiting case. Some remarks are given on this Dirac's theory. Finally, the picture of trajectory ensemble is generalized so as to represent wave equations with sources. In an appendix criticism is given on the attempt to interpret quantum mechanics in terms of Markoff process.
We present experimental studies of geometry-specific quantum scattering in microwave billiards of a given shape. We perform full quantum-mechanical scattering calculations and find excellent agreement with experimental results. We also carry out semiclassical calculations where the conductance is given as a sum over all classical trajectories between the leads, each of the trajectories carrying a quantum-mechanical phase. We unambiguously demonstrate that the characteristic frequencies of the oscillations in the transmission and reflection amplitudes t and r are related to the length distribution of the classical trajectories between the leads, whereas the frequencies of the probabilities T=/t/(2) and R=/r/(2) can be understood in terms of the length difference distribution in the pairs of classical trajectories. We also discuss the effect of nonclassical "ghost" trajectories, i.e., trajectories that include classically forbidden reflection off the lead mouths.