How to calculate scattering aplitude?

[Boris]

Homework Statement

Hello, everyone . My teacher has given me an assignment to show that the exact Coulomb amplitude (2) satisfies the unitary condition for an elastic Scattering theory (1).

(1) ## Imf(k',k'')=( \frac {κ} {4\pi} ) ∫ f^*(k',k'')f(k',k'')dΩ ##

(2) ## f(θ)= \frac {-α} {2mϑ^2} \frac{\Gamma(1+in)}{\Gamma(1-in)} \frac{\exp(-2in\ln\sin(\frac{\theta}{2}))}{\sin^2\frac{\theta}{2}} ##

Homework Equations

As I understood the goal is to show that this condition is suitable for scattering in every potential including weak Coulomb potentional, right?

The Attempt at a Solution

That is why I took the formula (1) which combines I am part of amplitude with the whole amplitude and subtract Coulomb amplitude (2) . The problem is that we haven't studied Quantum mechanics yet, so it seems kinda difficult. I'm not asking you to show me every step of this calculation , but can you just give me any sort of advice how to start it ? Thank you in advance.

 

[mark726]

Dear student,

Thank you for sharing your assignment with us. It seems like you have a good understanding of the problem and the goal is indeed to show that the Coulomb amplitude (2) satisfies the unitary condition for elastic scattering (1) in any potential.

Since you haven't studied quantum mechanics yet, it might be helpful to start by understanding the concepts of scattering and unitarity in classical mechanics. In classical mechanics, scattering refers to the process of particles interacting with each other and changing their direction or momentum. Unitarity, on the other hand, refers to the conservation of probability in a physical system.

Next, you can try to understand the mathematical expressions (1) and (2) and how they relate to the classical concepts of scattering and unitarity. You can also look into the properties of the Gamma function, which appears in the Coulomb amplitude (2), and how it relates to the scattering process.

Once you have a good understanding of the classical concepts and the mathematical expressions, you can start by substituting the Coulomb amplitude (2) into the unitary condition (1) and see if it satisfies the condition. You might also need to use some properties of the Gamma function to simplify the expression.

I hope this helps you get started on your assignment. Good luck!