How to prove the width of a resonance using the cross section of a S wave?

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SUMMARY

The discussion focuses on proving that the parameter 'b' represents the width of a resonance in the context of the total cross section of an S wave, defined by the equation Sigma=(pi/k)*(b^2)/((E-En)^2+(b/2)^2). The user successfully calculated the maximum cross section at E=En and derived two solutions, E1 and E2, corresponding to half the maximum, confirming that the difference E1-E2 equals 'b'. The conversation emphasizes the relationship between mathematical calculations and physical interpretations in resonance phenomena.

PREREQUISITES
  • Understanding of S wave resonance and cross section analysis
  • Familiarity with the equation Sigma=(pi/k)*(b^2)/((E-En)^2+(b/2)^2)
  • Basic knowledge of resonance width and its physical significance
  • Proficiency in solving quadratic equations
NEXT STEPS
  • Explore the concept of resonance in quantum mechanics
  • Study the implications of cross section in particle physics
  • Learn about the physical significance of parameters in wave equations
  • Investigate alternative methods for analyzing resonance phenomena
USEFUL FOR

Students and researchers in physics, particularly those studying quantum mechanics and particle physics, will benefit from this discussion, especially in understanding resonance and cross section analysis.

ziad1985
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Homework Statement


The total cross section of a S wave is given by the following equation
Sigma=(pi/k)*(b^2)/((E-En)^2+(b/2)^2))
prove that b is the width of the resonance.


Homework Equations


Nothing that is the only thing


The Attempt at a Solution


I already solved the question using basic calculation.
For E=En i get the maximum of Sigma , for half of that maximum , i get a second degree equation of E, when solving it i get 2 solution representing that half maximum(E1 & E2) , the difference between E1 and E2 , E1-E2=b
But I'm not really comfortable with the way i did it , I know i got the solution , but I'm looking for another way , something more to do with physics.
idea anyone?
 
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