How to Calculate Scattering Amplitude for a Particle in a 1/r^2 Potential?

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Homework Help Overview

The discussion revolves around calculating the scattering amplitude for a quantum particle interacting with a potential of the form V(r) = a/r^2 using the partial waves method. Participants are exploring the implications of the potential on the solutions to the radial Schrödinger equation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to calculate phase shifts for different angular momentum values but express difficulty in solving the Schrödinger equation for the given potential. There is a discussion about the nature of solutions for integer versus non-integer angular momentum values and the relevance of Bessel functions.

Discussion Status

Some participants have provided insights regarding the properties of Bessel functions and their applicability regardless of whether the angular momentum index is an integer. However, there is still uncertainty about how to derive the phase factors needed for the scattering amplitude calculation.

Contextual Notes

Participants are grappling with the implications of the centrifugal term in the radial Schrödinger equation and the challenges posed by the specific form of the potential. There is an acknowledgment of the need for hints or guidance to navigate the complexities of the problem.

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Homework Statement



Use partial waves method to calculate scattering amplitude for a quantum particle scattering off the potential V(r) = a/r^2.

Homework Equations





The Attempt at a Solution



To calculate phase shifts \delta_{l} for each angular momentum's value l, it's necessary to solve Schrodnger's equation for the specified potential - but I'm unable to do it. I know that solution for a free particle can be expressed in terms of Bessel spherical functions, but in radial Schrödinger's equation we have the centrifugal term of the form l(l+1)/r^2, where l is an integer, not for arbitary potential a/r^2. Any hints?
 
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The radial equation doesn't care whether or not l is an integer ...
 
Ok, but in the noninteger case the solutions are special functions, from which I don\t know how to calculate the phase factors in question.
 
The asymptotic properties of Bessel functions hold whether or not the index is an integer.
 

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