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

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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: ...
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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 ...
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67 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 ...
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1answer
99 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} ...
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1answer
29 views

Optimal angle of projectile with air drag [closed]

I am trying to find the maximum distance of a projectile and the optimal angle to achieve this distance. I've been given the projection has mass $1kg$ and initial velocity $150m/s$ with air drag ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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1answer
49 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.
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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, ...
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31 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 $$S = 1 + \frac{1 + R_j^2}{R_i^2}$$ ...
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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' \, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
46 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})}} ...
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2answers
69 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 ...
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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 ...
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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, \, \, ...
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83 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
50 views

How do you interpret definite-integrating of both sides of an equation?

Consider a particle is moving at an acceleration of $a=f(s)$ [$s$ stands for particle's location) if we have the initial velocity of a particle then what is the final velocity? So... we know that ...
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1answer
46 views

Avoiding a singularity in the simulation of a spherica pendulum

I didn't know whether to put this here or in StackOverflow - so I open to answers just telling me to go there! I am looking to simulate the motion of a spherical pendulum. The Lagrangian is $$ ...
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27 views

Calculate the parabollic trajectory

In order to calculate the parabollic trajectory (or ballistic) I need intrinsic equation. Now I need to integrate this equation but I don't know how. I would appreciate so much if you help me!
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32 views

Why does the angular average of the divergence result in a factor 1/3?

In going from eq.(6.13) to (6.14) in http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf it is assumed that ...
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2answers
183 views

Can you take the integral of $ d^2x\over dt^2$? [closed]

I am messing around with physics problems, and as silly as this maybe how do you take the integral of $$\int_0^\infty xd^2x$$ For example taking Newton's Second law $F=ma$ $$ F=m{d^2x\over dt^2} ...
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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 ...
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41 views

Why is the angular average of the direction of the momentum squared a third instead of one?

In the article http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf (in going from eq. (6.13) to eq. (6.14)) it is stated that ...
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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: ...
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1answer
47 views

Calculating force repulsive between a line segment and a point [closed]

As shown in the diagram below, let there be a line segment $L$ and a point $p$ such that they both repel each other electrically. Let $\vec{v} = \vec{p} - \vec{q}$ be the vector from $p$ to the point ...
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1answer
130 views

Book on gamma functions with applications in physics

I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions. Does anybody know ...
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0answers
73 views

Angular frequency integral to wavelength integral

I have this integral $$ \bar{\omega}^2 = \int\limits_{-\infty}^{\infty}\omega^2|\tilde{F}(j\omega)|^2\frac{d\omega}{2\pi} $$ and I want to convert it to wavelength domain $\lambda $. I know that the ...