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arXiv:2210.13018 · Physica Scripta

Angular time delay in quantum mechanical scattering

Jochen Zahn

Published Jul 12, 2023• 2 citations
arXivHTMLPDF

Abstract

Abstract We apply Brunetti and Fredenhagen’s concept of the time of occurrence of an event in quantum mechanics Brunetti and Fredenhagen (2002 Phys. Rev. A 66 044101) to the example of scattering off a spherical potential. Thereby, we re-derive the expression of Froissart, Goldberger, and Watson for the angular time delay Froissart et al (1963 Phys. Rev. 131 2820), clarifying some conceptual issues with their derivation. We also present an elementary re-derivation of the ‘space shift’ (essentially the impact parameter) defined in the quantum mechanical context by the same authors. We clarify the relation of both quantities to their classical counterparts in the context of the WKB approximation. As an example, we apply the concepts to scattering at a hard sphere. We find pronounced peaks in the both the time delay and the space shift at the minima of intensity in the forward diffraction region for short wavelength scattering and discuss whether these could in principle be observable.

DOI
10.1088/1402-4896/ace703

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The classical deflection function for scattering by a radially symmetric potential corresponds to twice the angular momentum derivative of the scattering phase shifts calculated in the conventional WKB approximation. A ``quantum-mechanical deflection function'' defined as twice the angular momentum derivative of the quantum-mechanical phase shifts (without WKB approximation) is able to incorporate into the simple trajectory picture certain quantum aspects of the scattering process, e.g., the influence of the Goos-H\"anchen shift. We demonstrate the performance of this quantum-mechanical deflection function for sharp-edged and smooth-edged two-dimensional scattering potentials.
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