# Why does the scattering length it take the form it does?

Physics Asked by Quantum spaghettification on October 20, 2020

In partial-wave analysis, and particularly $s$-wave scattering, the scattering length can be defined as
[lim_{krightarrow 0} kcot(delta_0)=-frac{1}{a_0}.]
My question is why is it defined this way? i.e. what is the physical importance of the quantity $kcot(delta_0)$ that it should be used in this expression?

A brief section on S-wave scattering is found here - http://www.int.washington.edu/users/mjs5/Class_560/lec560_2/node2.html it discusses the physical significance of the cot term. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I’ll summarize and try not to plagiarize at the same time. The cot term, as I’m sure you know, just comes from the mathematics. Many years ago others also questioned the presence of the cot term. They went on to show mathematically that using a power series expansion, it represents correctly the limits of the scattering amplitude. Also, it can be used to calculate the bound states as well.

Answered by jmh on October 20, 2020

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