Scattering amplitude in scattering from a delta function

In summary, the problem is to show that for a potential of type V(r) = gδ(r), where g is a constant and δ(r) is the delta function of Dirac 3D, the scattering amplitude can be determined using the Born approximation. The equation f(θ) = C ∫ dr rV(r)sin(qr) is relevant to this problem. The person has attempted to calculate the Fourier transform to determine the amplitude, but has encountered some difficulties and is seeking help. However, as per the rules of the PF, the person needs to show their work first before receiving assistance.
  • #1
joaosantossilva
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Homework Statement
Show that for a potential of type 𝑉 (𝑟⃗) = 𝑔𝛿 (𝑟⃗), where 𝑔 is a constant and 𝛿 (𝑟⃗) is the delta function of Dirac 3D. Using the Born approximation, determine the scattering amplitude and derive an expression for 𝑔 so that the scattering length is b.
Relevant Equations
f(θ) = C ∫ dr rV(r)sin(qr)
I tried to calculate the Fourier transform to get the amplitude, but I got lost
 
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  • #2
Welcome to the PF. :smile:
joaosantossilva said:
Homework Statement:: Show that for a potential of type 𝑉 (𝑟⃗) = 𝑔𝛿 (𝑟⃗), where 𝑔 is a constant and 𝛿 (𝑟⃗) is the delta function of Dirac 3D. Using the Born approximation, determine the scattering amplitude and derive an expression for 𝑔 so that the scattering length is b.
Homework Equations:: f(θ) = C ∫ dr rV(r)sin(qr)

I tried to calculate the Fourier transform to get the amplitude, but I got lost
Fun problem. But per the PF rules, we can only help you after you show your work. Please show what you have done so far. Thanks.
 
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1. What is a scattering amplitude?

A scattering amplitude is a mathematical quantity used to describe the probability of a particle scattering off of a potential or scatterer. It is typically denoted by the symbol "f" and is related to the differential cross section of the scattering process.

2. How is a scattering amplitude calculated?

A scattering amplitude can be calculated using quantum mechanical calculations or through experimental measurements. In quantum mechanics, it is calculated by solving the Schrödinger equation for the relevant potential or scatterer. In experiments, it is calculated by analyzing the scattering data and fitting it to a theoretical model.

3. What is a delta function in scattering?

A delta function, also known as a Dirac delta function, is a mathematical function that is used to describe the concentration of mass or charge at a single point in space. In the context of scattering, it is used to represent the point-like nature of the scattering potential or scatterer.

4. How does a delta function affect the scattering amplitude?

A delta function in the scattering potential results in a sharp peak in the scattering amplitude at the corresponding scattering angle. This is because the delta function represents a localized potential, causing the scattered particles to interact with it only at a single point in space.

5. What information can be extracted from the scattering amplitude in scattering from a delta function?

The scattering amplitude from a delta function contains information about the strength and location of the scattering potential or scatterer. It also provides information about the angular distribution of the scattered particles, which can be used to determine the size and shape of the scatterer.

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