Understanding Scattering Length in Atom Interaction: A Comprehensive Guide

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The discussion focuses on the concept of scattering length in atomic interactions, describing it as the minimum separation at which significant scattering occurs. The user expresses confusion over the presence of two different scattering lengths, labeled as a_{1,-1} and a_{2,2}, both measured in units of a_0, which is the Bohr radius. It is clarified that the scattering length is derived from the S-matrix and is crucial for understanding low-energy scattering in short-ranged potentials. The conversation emphasizes the importance of these parameters in scattering theory. Overall, the scattering length serves as a key factor in characterizing atomic interactions.
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Hi all,
I am reading some materials on the scattering. There is simple model to consider atom as a rigid body but with radius replaced by the so-called scattering length. I didn't learn the scattering theory systematically but my understanding on the scattering length is the minimum separation between two atoms when the scatter or interaction becomes significant. I do some research on some typical atoms like sodium. But what confusing me is the article gives 2 scattering length labeled as ##a_{1,-1}## and ##a_{2,2}##, which both are in some unit of ##a_0##. So why there are two different scattering length? Is ##a_0## the bohr radius?
 
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As any physically relevant quantity on scattering, the scattering length is a parameter derived from the S-matrix. It characterizes low-energy (long-wavelength) scattering on a short-ranged potential. For first details, see the pretty good Wikipedia article:
http://en.wikipedia.org/wiki/Scattering_length
 
Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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