Volume integral to spherical coords to contour integral

In summary, the homework statement is trying to solve for the potential in momentum space of two fermions scattering off of each other. The problem is that the term \frac{exp^{iqr}-exp^{-iqr}}{iqr} doesn't seem to make sense and the limits of integration are changed. I think this problem might be related to the transition matrix element for two fermions scattering off of each other.
  • #1
sbh77
3
0

Homework Statement


V(x) = \int \frac{d^3q}{(2\pi)^3} \frac{-g^2}{|\vec{q}|^2 + m^2} \exp^{i \vec{q} \cdot \vec{x}}

= -\frac{g^2}{4\pi^2} \int_0^{\infty} dq q^2 \frac{exp^{iqr}-exp^{-iqr}}{iqr} \frac{1}{q^2+m^2}

= \frac{-g^2}{4\pi^2 i r} \int_{-\infty}^{\infty} dq \frac{q exp^{iqr}}{q^2+m^2}


Homework Equations



none

The Attempt at a Solution



I understand how the measure changes to a single integral with a factor of 4\pi (integrating out the angles leaving the radial one) and I also know that a vector in spherical coordinates is represented just by a quantity in the radial direction. The problem I am having is I don't know where the term \frac{exp^{iqr}-exp^{-iqr}}{iqr} came from. It almost looks like sin but the "i" is in the denominator.

My last question is why does the second line become the third? I see the limits of integration are changed but why did the second exponential term disappear? Is it a symmetry thing?

Thanks!
 
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  • #2
here it is in tex, just need the tags either side
[tex] V(x) = \int \frac{d^3q}{(2\pi)^3} \frac{-g^2}{|\vec{q}|^2 + m^2} e^{i \vec{q} \cdot \vec{x}} [/tex]

[tex]= -\frac{g^2}{4\pi^2} \int_0^{\infty} dq q^2 \frac{e^{iqr}-e^{-iqr}}{iqr} \frac{1}{q^2+m^2}[/tex]

[tex]= \frac{-g^2}{4\pi^2 i r} \int_{-\infty}^{\infty} dq \frac{q e^{iqr}}{q^2+m^2}[/tex]
 
  • #3
note 1/i = -i

can you decribe the problem a bit more, looks like an integral over momentum space?
what are x & r this problem?
 
  • #4
also the 3rd line probably comes about by splitting the second line into a sum of 2 intergals and changing variables in one, q -> -q then recombining...
 
  • #5
ah, my mistake, x and r are the same thing. I am just comparing the first Born approximation with that of the transition matrix element for two fermions scattering off of each other. By making this comparison it can be seen that the potential (in momentum space) is

V(q) = \frac{-g^2}{q^2+m^2} (sorry, I don't know what these tags are)

where "g" is the coupling constant in the Yukawa theory and m is the mass of the fermion. So I am just inverting this back to coordinate space. "q" is the momentum difference between the inbound and outbound fermion - along the propagator in a Feynman diagram.

Fourier transforms have never been my strong point! :)

Thanks again!
 

Related to Volume integral to spherical coords to contour integral

1. What is a volume integral in spherical coordinates?

A volume integral in spherical coordinates is a mathematical tool used to calculate the volume of a three-dimensional region in space. It is typically denoted as ∫∫∫ f(x,y,z) dV and involves integrating a function f(x,y,z) over the entire region in spherical coordinates.

2. How do you convert a volume integral from Cartesian coordinates to spherical coordinates?

To convert a volume integral from Cartesian coordinates to spherical coordinates, you need to use the Jacobian determinant of the transformation. This is given by r^2 sinθ, where r is the distance from the origin and θ is the angle from the positive z-axis. The integral in spherical coordinates then becomes ∫∫∫ f(r,θ,φ) r^2 sinθ dr dθ dφ.

3. What is the relationship between a volume integral and a contour integral?

A volume integral and a contour integral are two different types of integrals used in different coordinate systems. A volume integral is used in three-dimensional space, while a contour integral is used in two-dimensional space. The relationship between them is that a volume integral can be converted to a contour integral by using the divergence theorem.

4. How is a contour integral related to Green's theorem?

A contour integral is related to Green's theorem in that Green's theorem provides a way to evaluate a contour integral by transforming it into a double integral. This is useful for calculating the area enclosed by a curve or the work done by a vector field along a closed path.

5. What are some real-life applications of volume integrals in spherical coordinates?

Volume integrals in spherical coordinates have various real-life applications, such as calculating the mass of a planet, finding the electric field within a charged sphere, and determining the flow of a fluid around a spherical object. They are also used in physics and engineering to calculate moments of inertia and solve problems involving spherical symmetry.

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