Questions tagged [integration]

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

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Motion blur integral

I was going through a biophysics paper (Berglund, Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Jul) and derived the main pieces of interest but ran into this difference of two integrals:$$R=\frac{1}{\...
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How to move a $d/dt$ inside a triple integral? [migrated]

I faced some equations while reading current density. $$\iint_S \vec J.d \vec S = - \frac{d}{dt} \iiint_V \rho.dV.$$ And then $$\iint_S \vec J.d\vec S = -\iiint_V \frac{\partial \rho}{\partial t}dV. $$...
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Integration, 2-dim isotroper harmonischer Oszillator, polar Koordinaten, Hamilton-Jacobi [closed]

I'm trying to solve the 2-dimensional isotropic harmonic oscillator using the Hamilton-Jacobi method, and come across the following integral: $$ \int \frac{dr}{r^{2}\sqrt{a-br^{2}-\frac{c}{r^{2}}}}. $$...
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4 votes
1 answer
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Far field approximation for massive Klein-Gordon equation in 3+1D

For a massless scalar, one has the familiar Green's function $$ G(t,r) = \frac{\delta(t - r)}{4\pi r}\,, $$ and one may take the far-field approximation in a rather straight-forward way: $$ \int d t d^...
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1 vote
2 answers
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Calculating average of a function of molecule's orientation (Euler angles)

In this paper, orientational average of a function of Euler angles, $f(\phi,\theta,\psi)$, is defined as: $$\langle f\rangle=\frac{1}{8 \pi^2} \int_0^\pi \int_0^{2 \pi} \int_0^{2 \pi} f(\theta, \phi, \...
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Finding the moment of inertia of a non uniform density versus uniform density rod [closed]

A thin rod 10ft long has a density which varies uniformly from 4 to 24 lb/ft. Find the moment of inertia about a perp. axis the center of mass? Here is my most recent attempt at the example problem. I ...
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2 votes
1 answer
57 views

One-loop triangle integral with equal masses

This is a question about the explicit form for an integral that is very common in QFT. $$I_3(p,p';\,m,d)\equiv \int \frac{d^d k}{(2\pi)^d} \frac{1}{\left(k^2+m^2\right)\left((k+p)^2+m^2\right)\left((...
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How should I get the expectation value of $r^2$ in the hydrogen atom? [closed]

I'm having trouble finding this expectation value: $$ \langle r^2\rangle=\frac{C_{n,l}}{\alpha^5}\int_0^{\infty}e^{-x}x^{2l+4}[L^{2l+1}_{n-l-1}(x)]^2dx $$ Where $x=\alpha r$ and $\alpha=\frac{2}{na_o}$...
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Moments of physical qualities other than density

If I have some genuine physical object $\Omega$ (that I can describe mathematically) and want to find its center of mass, I can compute its moments to get each coordinate of that center; for example, ...
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Fourier transformed quantity always representable by derivative?

Suppose we perform the following steps to go from an integral over the spatial dimension to one over the momentum: $$\int dx \,f(x) = \int d x\left(\int \frac{d k}{2\pi}\,\tilde{f}(k)\, e^{ikx} \, \...
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1 vote
1 answer
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What is the point of half-step Verlet integration

In the Wikipedia article on Verlet method, under the Velocity Verlet method, the first algorithm proposed uses half steps: Calculate $\vec{v}\left(t + \tfrac12\,\Delta t\right) = \vec{v}(t) + \...
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6 answers
90 views

Deriving Work-Kinetic Energy Theorem

I am currently reading Physics for Scientists and Engineers (Ninth Edition) by Serway and Jewett and in Chapter 7.5, a derivation of the work-kinetic energy theorem was shown. To give context, ...
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1 answer
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Question about integration limits in the special relativistic action

We can read in this article that the action of a particle in special relativity, is It seems like nitpicking maybe, but shouldn't the two coordinate time limits be changed to proper time limits, or ...
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1 answer
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Can't understand why the final result of this integration is positive

So, this excerpt is from this paper: Thermodynamics as a theory of decision-making with information-processing costs, the final equation doesn't follow the math, since the final result is positive, ...
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1 vote
0 answers
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Why are time-reversible integration algorithms in molecular dynamics simulations favorable?

I read that integration algorithms that are not time-reversible tend to be less "stable". Where stable means that the total energy stays constant (is conserved). I'd like to know what it ...
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1 answer
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Electric field at an axial location of a semicircular ring of non-uniform charge [closed]

Suppose there is a semi-circle of radius R and center O contains a non-uniform charge distribution on its perimeter. Let P be a general point at an angular position $\theta$ ( $\theta$ measured ...
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6 votes
1 answer
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Integration of Laplacian by parts

I'm trying to solve assignment (1.5) in Bellan's "Fundamentals of Plasma Physics" using Fourier transforms, but I'm stuck integrating the Laplacian. Here's the problem: Equation (1.5) is ...
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1 vote
1 answer
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Angular Integral of with Spherical Harmonics and Cross Product

I have an integral involving spherical harmonics and a cross product. It reads $$ \int d^3k d^3Q \phi^{*}_{L'}(k+Q)Y^{*}_{L',M'_{L'}}(\widehat{k+Q})C^{J'M'_{J'}}_{L'S'}\mathbf{S}\cdot \mathbf{(k\times ...
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5 votes
1 answer
108 views

Table of integrals for dimensional regularization

Is there any reference (book or paper) that contains a list of integrals useful for dimensional regularization? I would need it to approach integrals like these $$ \int d^dx \frac{x^\mu}{|x|^{2d-4}}, \...
0 votes
1 answer
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Extra second-order term in Dyson Formula Expansion in David Tong's Notes

Right at the bottom of page 52 of David Tong's QFT notes we have just defined the time ordered Dyson formula, David Tong then shows the expansion of $(3.20)$ however an extra second-order term has ...
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Direct calculation of the gravitational potential inside a hollow sphere

I calculated the gravitational potential inside a massive sphere with constant density and got the result: $$\Phi = -2\pi G\rho R^2 + \frac{2}{3}\pi G\rho R_p^2$$ Where $R$ is the radius of the sphere ...
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How can I calc the density from 2D density gradient field; Schlieren exp; integrate total differential; scalar potential from gradient vector field;

I did Schlieren experiments with vertical and horizontal cutoff direction. Now I have the density gradient in both horizontal and vertical direction stored in matrices (I have a value for every pixel ...
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0 answers
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Logarithmic behavior in pair annihilation cross section?

I am calculating a tree-level annihilation cross section ($\chi\chi\rightarrow \gamma \gamma$) for a certain model and am getting logarithmic behavior in the cross section, which I've never seen so it ...
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0 votes
2 answers
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Motion of free fall [duplicate]

We know that according to law of free falls object, all bodies fall with the same constant acceleration. But in distance formula ($s = \frac12 gt^2$), why the acceleration is just half?
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Maxwell Ampere law derivation

I have seen a proof on the internet regarding the derivation of the maxwell ampere law in this link: Deriving the Ampère-Maxwell law and even though I am pretty much satisfied in the way he derives ...
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0 votes
1 answer
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Multiple questions on equation 1.6 after deriving Planck's law on page 5 (Schwartz QFT)

After deriving Planck's law for the expectation energy of a single mode, Schwartz takes the limit $L \rightarrow \infty$ and turns the sums into integrals. The average total energy of the blackbody ...
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0 votes
2 answers
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How to calculate integrals in Ideal Fermi Gas theory? [closed]

I'm having troubles solving integrals in the Ideal Fermi Gas theory. In particular the ones of the type: $$ \int\frac{d\vec{k}}{(2π)^3}θ(k_F − k)( \vec{k} \cdot \vec{q})^n$$ but I actually don't ...
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2 votes
2 answers
41 views

Trajectory of a free particle in a sinusoidal force field

I'm struggling to derive the analytical form for the initial (transient) motion of a free particle in a sinusoidal force field. For the sake of simplification, I'm operating in a one-dimensional space....
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31 votes
3 answers
5k views

Why aren't Runge-Kutta methods used for molecular dynamics simulations?

One of the most used schemes for solving ordinary differential equations numerically is the fourth-order Runge-Kutta method. Why isn't it used to integrate the equation of motion of particles in ...
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0 answers
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How is the pressure derived from the Gibbs-Duhem equation, within the Local Density approximation? (Problem in calculating an integral)

Following this reference https://www.nature.com/articles/nphys1477.pdf (starting from Eq 3), I am trying to understand the details in the derivation of the pressure along the x axis p(x,y=0,z=0) from ...
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In the calculation of precession of Mercury or in the Kepler problem, how can we evaluate a definite integral by using a contour integral

In the calculation of Mercury using perturbation, we have to evaluate the integral: $$ \delta \phi =-\frac{GMm}{2a^3}\sqrt{\frac{m}{2E}}\frac{\partial}{\partial J}\int^{r_{max}}_{r_{min}}\frac{r^3 dr}{...
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0 votes
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Spherical harmonics integral

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi $$ I've tried to use the definition of the ...
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Integrate continuity equation in QM

From Shankar's QM book pg. 166: The continuity equation for probability density in QM is $$\frac{\partial P(\vec{r},t)}{\partial t}=-\nabla \cdot \vec{j}(\vec{r},t),$$ where $P=\psi^*\psi$ is the ...
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1 vote
1 answer
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Comoving distance in $\Lambda$CDM - Understanding an approximation

I am trying to find the comoving distance, $$\chi = c\int_0^z \frac{dz}{H(z)}$$ for the $\Lambda$CDM model (spatially flat universe, containing only matter and $\Lambda$). $$H^2 = H_0^2[\Omega_{m,0}(1+...
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How can we ignore the diverging term $e^\infty$ in the integral?

In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution. In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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0 votes
1 answer
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Time evolution of Gaussian packet

From Shankar's QM book pg. 154: Consider the Gaussian wave packet at time $t=0:$ $$\psi(x',0)=e^{ip_0x'/\hbar} \frac{e^{-x'^2/2\Delta^2}}{(\pi\Delta^2)^{1/4}}.$$ Using the propagator $U(t)$ in the ...
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0 votes
1 answer
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Loop Integrals and Dimensional Regularization

I want to calculate the divergent part of a Feynman diagram using the Feynman parameters: $$\frac{1}{A_1 A_2 \ldots A_n} = \int_0^1 dx_1 ... dx_n \delta (\Sigma x_i -1) \frac{(n-1)!}{[x_1 A_1 + x_2 ...
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1 vote
1 answer
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Momentum integral yielding $\delta$ function

I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will ...
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Doubt regarding the calculation of first Born approximation of Yukawa potential

The expression for the scattering amplitude upto first Born approximation for Yukawa potential is $$f^{1}(\mathbf{k},\mathbf{k'})=\int_0^\infty r^2 dr \int_0^{2\pi} d\phi \int_0^{\pi}\sin{\theta}\frac{...
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Integrating by parts differentiated vector fields in Lagrangian [duplicate]

when surface terms being ignored, from p.556, ch.26 QFT lectures of Sidney Coleman To get this result, do I have to integrate by parts twice? Do I have to switch the derivatives around and how? ...
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1 answer
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How to deduce the energy of a pair of vortices the classical XY model?

Consider a pair of oppositely charged vortices with unit strength, we estimate the energy of a pair of vortices as: $$ E_{\text {pair }}-E_{0} \cong \frac{J}{2} \int d^{2} r(\nabla \theta)^{2}=\frac{J}...
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2 votes
0 answers
121 views

How to solve this integral?

I am following some notes from a physics class on the Ising model. At some point we get to this integral \begin{equation} \frac{1}{2} \int_{\Omega_B} \frac{\text{d}^Dk}{(2\pi)^D} \ln \left[ \tau + \...
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0 votes
0 answers
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What is the internal potential for a spherical mass for a $r^\alpha$ potential?

Newton's shell theorem makes it easy to find the internal gravitational potential in a spherical body for a standard gravitational $\Phi(r)\propto 1/r$ potential. But the shell property does not apply ...
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1 answer
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Question regarding vector calculus [closed]

Question regarding vector calculus: Mathematically, what properties does this vector field have, in order for its line integral to be equal to the area enclosed by that curve?
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1 answer
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How do you integrate by parts when you have a triple integral?

I'm studying how particles of equal mass behave in a spherical cluster held intact by gravity. I will assume that the mass density $\rho(R)$ of the cluster is a function of the magnitude of the ...
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3 votes
3 answers
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How do you find the final velocity when acceleration is changing between two values over some distance? [duplicate]

How do you calculate a final velocity of an object when given its initial velocity and the object is accelerating between an initial and final acceleration over some given distance?
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Integral divergent for all $d$ in dimensional regularization

Is it possible to have a $d=4-2\epsilon$ dimensional integral which diverges for all $\epsilon$? For example, if you get something like $$\tag{1} \int_0^1 \text{d}t\,t^{-1-\epsilon}(1-t)^{-1+\epsilon},...
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2 votes
0 answers
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Solving a complex Gaussian Integral for Path Integral Formalism [duplicate]

I am trying to solve the following integral, $$\int dz_1d\bar{z}_1\cdots dz_nd\bar{z}_n\:\exp(-\sum_{i,j}\bar{z}_i A_{ij}z_j),$$ where $A_{ij}$ is an $n\times n$ hermitian matrix. I know how to do ...
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Differentials of 4-vectors with broken Lorentz invariance

I am computing a Feynman integral of a fermion bubble in an external field. The field is in the z-direction so Lorentz invariance is broken. I need to break up the $\int d^4 k$ integral into a $\int d^...
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0 votes
1 answer
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Solving the integral for box of radiation

https://www.youtube.com/watch?v=cU7bj4dNMwY&t=3914s Susskind lecture 7 on statistical mechanics show how to calculate energy for box of radiation. After model description and some passages (at 1:...
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