For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

learn more… | top users | synonyms

1
vote
0answers
69 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
1
vote
0answers
26 views

Fourier Transform point force on a half-space [migrated]

I have to calculate the following Inverse Fourier-Transform, which describe the potential function for a point force on a half-space: ...
1
vote
0answers
29 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
4
votes
1answer
100 views

Integrating elements of a Lie group with respect to parameters of the corresponding Lie algebra

I am working with an operator $\textbf{M}$ that is represented by the Lie group SO(1,3), thus it can be written as, $$ \textbf{M} = \exp{\textbf{L}} $$ where, $$ \textbf{L} = \begin{bmatrix} ...
1
vote
2answers
71 views

Potential energy of a spherically symmetric charge density in a spherically symmetric electrostatic potential

I'm interested in calculating the potential energy of a spherically symmetric charge density in a spherically symmetric electrostatic potential. More specific, I'm currently trying to calculate the ...
1
vote
0answers
31 views

Self-energy integral at (complex) polar coordinates [migrated]

I am trying to resolve the following integral where $x,y \in \mathbb{R}$ and $a,b$ are constants in $\mathbb{R}$. $$ I = \int \frac{dxdy}{(2\pi)^2} \frac{x-ia}{\sqrt{(x-ia)^2+(y-ib)^2}} $$ This ...
3
votes
3answers
925 views

Electrostatic energy integral for point charges

The electric energy stored in a system of two point charges $Q_1$ and $Q_2$ is simply $$W = \frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$$ where $a$ is the distance between them. However, the total ...
0
votes
1answer
56 views

Energy stored in electric field [duplicate]

I know that energy stored in electric field / unit volume = $\frac{1}{2} \epsilon\,E^2$. so can I say that for any configuration calculating $\int \frac{1}{2} \epsilon\, E^2 \,d^3r$ over whole space ...
0
votes
0answers
20 views

1-loop integration for self-energy

I am trying to calculate the following integral that corresponds to 1-loop contribution to electron self-energy for a specific component of the momentum: $$ \Delta_i(q) = \int \frac{dp_1 dp_2 \ldots ...
3
votes
1answer
276 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
1
vote
0answers
10 views

How can I do a nested lightcone integral?

I want to do an integral of the form $I(x-y)=\int d^{D+1}z_1...d^{D+1}z_n f(x-z_1)...f(z_n-y)$ where $x,z_1,...z_n,y$ are all (D+1) vectors and $f$ depends only on the proper times between them. ...
7
votes
2answers
753 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
3
votes
2answers
517 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
4
votes
1answer
259 views

Imaginary Part of the Free Energy - Sohotski Plemenj theorem

I have posted this question already on Math Stack Exchange and I hope not to annoy the community if I post it here again, looking maybe for a better suited audience. I need to understand how the ...
2
votes
0answers
46 views

A box loop-integral [closed]

I am trying to evaluate the integrate $$ \int\frac{d^Dk}{(2 \pi)^D} \frac{1}{(k^2)^2(k^2-m^2)} $$ using dimensional regularisation ($D=4-2\epsilon$). From various references it appears that it should ...
12
votes
5answers
995 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 ...
2
votes
2answers
72 views

Do integrals of position make any sense? Do they have an application? [closed]

I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
1
vote
0answers
19 views

Help to verify (numerically) invariant Haar measure on unitary group

Sorry if this question is not appropriate for the forum. From the paper http://gemma.ujf.cas.cz/~brauner/files/Haar_measure.pdf I am interested to understand and verify equation (3). Can anyone please ...
1
vote
1answer
95 views

What is the meaning of $\mathrm{d}^4k$ in this integral?

From Gerardus 't Hooft's Nobel Lecture, December 8, 1999, he states the following equation (2.1): $$ \int \mathrm{d}^4k \frac{\operatorname{Pol}(k_{\mu})}{(k^2+m^2)\bigl((k+q)^2+m^2\bigr)} = \infty ...
0
votes
1answer
51 views

Deriving equipartition the (Sin Itiro Tomonaga ) way

In his book on quantum mechanics Tomonaga derives the equipartition law or energy using this integral. I am having several doubts on solving this integral! Is this solvable via this method?
0
votes
1answer
50 views

If the integral is zero when is the integrand zero? [closed]

Using Stoke's theorem we prove that the curl of the Electric field vanishes. This would be possible only if the integrand is zero when the integral is zero.
1
vote
2answers
342 views

Gauss's law… if the integral defining $\boldsymbol{E}$ diverges?

I have been told (here) that, under particular conditions, the electric field produced by a charge present in space $D$, defined by ...
0
votes
3answers
42 views

Why does work in the Work Energy Therom equal the Sum of the integral F*dr?

I am studying the work energy therom and I understand that 1/2(mv^2)=Wnet, however, I saw this picture below online. I understand the summation of the force times the change in distance. However, ...
-1
votes
1answer
78 views

Commutation relations in Quantum Field Theory [closed]

\begin{align} [a, a^\dagger] =& \left[\int d^3 x e^{-ikx} (\omega \phi(x) + i \Pi^\dagger(x)), \int d^3 x' e^{ikx'} (\omega \phi^\dagger(x') - i \Pi(x')) \right] \\ =& \int d^3x \, d^3x' \, ...
1
vote
0answers
32 views

How to integrate to find the view factor of two parallel disks of different radii? [closed]

You have two parallel coaxial disks of different radii. I have tables that give me the value as $$F_{ij} = \tfrac{1}{2} [S - \sqrt{S^2 - 4(r_j/r_i)^2}]$$ where $$S = 1 + \frac{1 + R_j^2}{R_i^2}$$ ...
1
vote
1answer
47 views

Slow-roll approximation for potential $V=\frac{1}{2}m^{2}\varphi^{2}$

I'm attempting to derive a solution to the slow-roll approximation for a scalar potential of the form $V=\frac{1}{2}m^{2}\varphi^{2}$. For the solution for $a(t)$ I will start by taking the slow-roll ...
1
vote
0answers
29 views

Divergence theorem for cylindrical coordinates [closed]

I have a Vector field in a cylinder where x^2+y^2=4 and goes from z=0 to z=3 and a vector field A=(4x)i-(2y^2)j+(z^2)k and I'm trying to verify the divergence theorem for the vector field i set set ...
1
vote
0answers
26 views

Relationship between Lippmann-Schwinger integrals of different dimensions

Define $G_n (\mathbf{x},\mathbf{x}')$ as $$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i ...
0
votes
1answer
60 views

Electric field integration? [closed]

If we have a rod uniformly charged with $Q$ stretching from $-a$ to $a$ on the $x$-axis as shown in the picture And we want to calculate electric field in point $2a$ on the $x$-axis, we know that ...
0
votes
1answer
14 views

Motivation for integrals over scalar field

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've seen: If you want to know the final temperature of an object that travels through a ...
1
vote
1answer
47 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})}} ...
0
votes
0answers
23 views

Showing $\frac{d^3\underline{k}}{2\omega_{k}(2\pi)^3}$ is Lorentz invariant [duplicate]

Show $\frac{d^3\underline{k}}{2\omega_{k}(2\pi)^3}$ is Lorentz invariant. Hint: try to evaluate $\int dk_0\delta(k_0^2 - M^2)\theta(k_0)$ where $M^2 = \underline{k} + m^2$ My attempt is as ...
1
vote
1answer
42 views

Asymptotic behaviour of the propagator for a scalar field

When discussing causality in Chapter 2 of Peskin & Schroeder a couple of equations giving the asymptotic behaviour of the propagator for a scalar field appear: $$ \text{If} \,\, x^0-y^0=t, \, \, ...
5
votes
4answers
846 views

How to compute the expectation value $\langle x^2 \rangle$ in quantum mechanics?

$$\langle x^2 \rangle = \int_{-\infty}^\infty x^2 |\psi(x)|^2 \text d x$$ What is the meaning of $|\psi(x)|^2$? Does that just mean one has to multiply the wave function with itself?
3
votes
3answers
128 views

Why doesn't $\vec{E} =\frac{1}{4\pi\epsilon_0} \int\frac{\rho \hat{r}\;dxdydz}{r^2}$ blow up at $r=0$, when $\rho$ is finite?

Electric field at $(x,y,z)$ produced by a continuous distribution of charges is given by:$$\mathbf{E}(x,y,z) =\dfrac{1}{4\pi\epsilon_0} \int\dfrac{\rho(x',y',z') \mathbf{\hat{r}} ...
6
votes
3answers
3k views

What does an integral symbol with a circle mean?

I have frequently seen this symbol used in advanced books in physics: $$\oint$$ What does the circle over the integral symbol mean? What kind of integral does it denote?
3
votes
0answers
85 views

Electric field due to charged disc, on the plane of the disc [closed]

A standard problem in finding the field is that of a uniformly charged disc, on its axis, but for this problem I'm supposed to find the potential and the field on the edge of the disc, i.e. in the ...
2
votes
1answer
174 views

Recovering QM from QFT

Reading through David Tong lecture notes on QFT. On pages 43-44, he recovers QM from QFT. See below link: QFT notes by Tong First the momentum and position operators are defined in terms of ...
13
votes
2answers
829 views

Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
0
votes
4answers
85 views

Proving that the Center of Mass of a solid sphere is at the origin [closed]

For my own knowledge and to understand why. I am trying to convince myself that the center of mass for a rigid solid sphere is at the origin (0,0,0). I begin with the basic definition of CM ...
1
vote
1answer
72 views

How does Galitskii's integral converge?

In V M Galitskii's 1958 paper "$\textit{Energy spectrum of a non-ideal Fermi Gas}$," he builds the following integral as part of a longer expression for the real part of the self-energy (eqn 26'). It ...
0
votes
1answer
47 views

Integration to find general solution of free particle [closed]

I was attempting problems in Griffiths Intro to QM when I came across the following: A free particle has the initial wavefunction: $$\Psi(x, 0) = Ae^{-ax^2} \, .$$ Find $\Psi(x, t)$. I ...
2
votes
1answer
1k views

Momentum variance in momentum space for particle in a box

My assignment asks me to compute the momentum space wavefunction of the nth energy eigenstate of the particle in a one-dimensional infinite square well, then "show that your result is in agreement ...
3
votes
1answer
86 views

Do wave functions really belong to $L^2$ space, or do we need to restrict our physical Hilbert space even further?

I am beginning to study quantum mechanics and I got stuck right at the beginning. I am trying to prove that the time derivative of the expected value of momentum of a particle is the (negative) ...
0
votes
0answers
12 views

Calculate distance with variable acceleration [duplicate]

I know the time and I know the force that acts horizontally on a particle. The problem is that the force is F=k(D-V), where k and D are constants. So at the beginning where the particle has zero ...
2
votes
1answer
74 views

Cancelling the partial of a coordinate, $\partial q$, with the element of a coordinate, $dq$ in Physics [closed]

I've seen in many books, things like this ( I will be simple ): $$\int \frac{\partial f}{\partial q} dq=\int df$$ where $f$ is a function of $q$ and other coordinates. I just axiomatically assumed ...
0
votes
0answers
26 views

How to intergate the cross section over the surface of a detector?

My beam moves along the $X$ axis. I know the cross section $\frac{d \sigma}{d \Omega}$. My rectangular detector is perpendicular to the $XY$ plane and its surface is perpendicular to the line ...
1
vote
2answers
130 views

Mathematical confusion in quantum mechanics

During a class about Ehrenfest theorem, my teacher use an equation to proceed its derivation (to prove $\frac{d<r>}{dt}=\frac{<p>}{m}$ ) and that is: ...
1
vote
0answers
53 views

Fraunhofer diffraction at circular aperture - integration of bessel function

I'm trying to understand the calculation here: http://en.wikipedia.org/wiki/Fraunhofer_diffraction_(mathematics)#Circular_aperture for the solution by integration, but I plain and simple fail to see ...
1
vote
0answers
48 views

Finding the dipole moment of a continous charge distribution [closed]

I am looking for the dipole moment given by the charge density \begin{equation}\rho = -e_0\frac{1}{\pi a^3}e^{\frac{-2r}{a}}+e_0\delta(\vec{r})\end{equation} where $e_0$ and $a$ are positive ...