All Questions
Tagged with scattering integration
15
questions
0
votes
0
answers
35
views
How to compute an integral for double scattering process? [migrated]
I need help with the following integral:
\begin{equation}
\int_{0}^{\pi}\int_{0}^{2\pi} \frac{(sm_2^2+\ s^2)C(A+B)+s\left(m_2^2+s\right)C^2 }{AB} \sin\theta_1 \ d\phi_1d\theta_1
\end{equation}
where $...
- 41
2
votes
1
answer
136
views
Transition rate derivation in non-relativistic quantum scattering
I am reading Principles of Quantum Mechanics by Shankar, here's a derivation I am puzzled.
To evaluate probability of particle entering detector in some solid angle, using $S$-matrix and Fermi's ...
- 567
0
votes
1
answer
42
views
Particular case of Green's theorem
Suppose we have $u(r)=\sum_{\lambda=1}^{\infty} a_{\lambda} u_{\lambda}(r), \, 0 \leq r \leq a$ in this article Introduction to R-matrix theory in atomic
physics
they say that
$$\int_{0}^{a}\left[u_{\...
- 445
0
votes
1
answer
236
views
Integrating amplitude for electron scattering from Coulomb potential
I am following Zee's QFT book in Section II.6. I have found the amplitude for an electron to scatter from a static Coulomb potential as
\begin{align*}
\mathcal{M}&=ie\!\int\!d^4x\,\big\langle ...
- 713
0
votes
1
answer
216
views
About Rutherford scattering
I am trying to derive Rutherford's scattering formula, with the coordinate system and polar coordinates chosen as in the picture below.
Angular momentum conservation yields $mvb = mr^2 \dot{\varphi}$....
- 123
1
vote
0
answers
70
views
Singular Integrals in Scattering Theory [closed]
This might seem a bit off-topic question but it comes into different physical theories. While learning quantum scattering I came across certain singular integrals but could not compute them. I am ...
- 387
4
votes
2
answers
231
views
Contour for integration in 1D scattering problem
A plane wave scattered by a 1D potential can be described by,
$$\psi(x) = \begin{cases}
e^{ikx} + R e^{-ikx}, & x<0\\
T e^{ikx}, & x>0
\end{cases}$$
where $R$ is the reflection ...
- 964
5
votes
2
answers
210
views
Integral involving two energy Green's functions
The problem
I am attempting to evaluate this integral:
\begin{equation}
I(\vec{k}) = \lim_{\epsilon\to 0} \int d^3q \,
\frac{1}{E-E_{\vec{q}}+i\epsilon}
\frac{1}{E-E_{\vec{k}+\vec{q}}+i\epsilon}
\...
- 153
0
votes
1
answer
275
views
Volume element in radial integration
I am reading the quantum scattering book by Taylor. He pointed out that since we are not observing the magnitude of incoming momentum we integrate over it. Then he replaces the volume element $d^3p$ ...
2
votes
1
answer
361
views
Integration boundaries in formula of central-force at classical scattering development
In "Classical Mechanics" book by Goldstein, Poole & Safko they develop the formulas of scattering to due to central-force problem.
In page 88, after receiving the expression for the differential ...
- 185
1
vote
1
answer
591
views
Scattering amplitude Green's function integral
On page 208 of Weinberg's QM book, he calculates the following integral
\begin{align}
G_k (\vec{x}-\vec{y})
=& \int \frac{d^3 q}{(2\pi \hbar)^3} \frac{e^{i\vec{q} \cdot (\vec{x}-\vec{y})}} {E(k)-...
- 129
8
votes
1
answer
1k
views
Strange use of complex analysis in Weinberg QFT 1?
In the beginning of chapter 3 on scattering theory in Weinberg's QFT book there is a use of the Cauchy residual theorem that I just cannot get.
First some notation, we are looking at states that are ...
- 390
6
votes
1
answer
2k
views
Integral in Peskin and Schroeder
I'm having a bit of a slow day, and can't see how to do the following integral in Peskin and Schroeder (page 107 for anyone with the book). We've derived in the centre of mass frame the integral over ...
- 7,936
3
votes
1
answer
1k
views
Meaning of $\mathrm{d}\Omega$ in basic scattering theory?
In basic scattering theory, $\mathrm{d}\Omega$ is supposed to be an element of solid angle in the direction $\Omega$. Therefore, I assume that $\Omega$ is an angle, but what is this angle measured ...
- 2,251
4
votes
1
answer
777
views
Spinor integration
I am learning on-shell methods for one loop integrals from this paper: Loop amplitudes in gauge theory: modern analytic approaches by Britto. Starting with formula (18) spinor integration is explained....
- 719