Integral Calculation for Yukawa Potential Differential Cross Section

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SUMMARY

The discussion focuses on the calculation of the differential cross section for Yukawa potential, specifically addressing the conditions under which the term | e(iq - 1/a) | must be less than 1 for the value at r=∞ to vanish. Participants clarify that for q and a being real numbers, the inequality | e(1/a) | > 1 implies that a must be greater than 0. The conversation highlights the importance of understanding the behavior of the exponential term in relation to the Yukawa potential's parameters.

PREREQUISITES
  • Understanding of Yukawa potential in quantum mechanics
  • Familiarity with differential cross section calculations
  • Knowledge of complex exponentials and their properties
  • Basic grasp of limits and behavior of functions as r approaches infinity
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  • Study the derivation of Yukawa potential and its applications in particle physics
  • Learn about differential cross section calculations in quantum scattering theory
  • Explore the properties of complex numbers and their implications in physical equations
  • Investigate the significance of boundary conditions in quantum mechanics
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Students and researchers in quantum mechanics, particularly those focusing on scattering theory and potential models, will benefit from this discussion.

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Homework Statement


upload_2017-12-20_21-19-23.png


Homework Equations


This is a calculation about differential cross section of Yukawa potential.

The Attempt at a Solution


I can't understand how that highlighted part can be -1 ,
we don't know if the parenthesis term (iq-1/a) is negative or positive tho.
 

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Subtracting the value at r=0 would give you -1. For the value at r=∞ to disappear, we must have | e(iq - 1/a) | < 1. So if q and a are real, that means | e1/a | > 1, so a>0.
 
FactChecker said:
Subtracting the value at r=0 would give you -1. For the value at r=∞ to disappear, we must have | e(iq - 1/a) | < 1.
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I don't know why | iq - 1/a | should be smaller than 1 ?
 
BREAD said:
==========================================
I don't know why | iq - 1/a | should be smaller than 1 ?
Sorry. You may have looked at my post as I was correcting it. | e(iq-1/a)r | = | (e(iq-1/a))r | = | (e(iq-1/a)) |r = | (e(-1/a)) |r= 1/| (e(1/a)) |r
 
FactChecker said:
Sorry. You may have looked at my post as I was correcting it. | e(iq-1/a)r | = | (e(iq-1/a))r | = | (e(iq-1/a)) |r = | (e(-1/a)) |r= 1/| (e(1/a)) |r

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I appreciate for your quick reply
 

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