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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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 $d=\frac{1}{2}at^2$?
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5
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How to get distance when acceleration is not constant?
I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question.
The equation for distance of an accelerating object with constant ...
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Why we use $L_2$ Space In QM?
I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
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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:
$$\int_{-\infty}^\infty dk_0 \...
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3
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Understanding the differential in integrals
After years, I still find myself having trouble really internalizing the meaning of various differentials in integrals—specifically, when they come about via reasoning regarding physical phenomena. ...
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Basic Grassmann/Berezin Integral Question
Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
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Integrating radial free fall in Newtonian gravity [duplicate]
I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving ...
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What is the meaning of the double complex integral notation used in physics?
In Altland and Simons' condensed matter book, complex Gaussian integrals are introduced. Defining $z = x + i y$ and $\bar{z} = x - i y$, the complex integral over $z$ is
$$\int d(\bar{z}, z) = \int_{-\...
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2
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Gaussian integral with imaginary coefficients and Wick rotation
Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
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When is Lebesgue integration useful over Riemann integration in physics?
Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesgue integration can be more ...
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Why is the functional integral of a functional derivative zero?
I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e.
$$\...
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Wrong calculation of work done on a spring, how is it wrong?
So I would have thought that this would be how you derive the work on a spring: basically the same way you do with gravity and other contexts, use $$W=\vec{F}\cdot \vec{x}.$$ If you displace a spring ...
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4
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Transformation of $d^4x$ under translation disregarded?
Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
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2
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Derivation of $f(R)$ field equations, problem with integration by parts
I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity.
But I do not understand the following step:
$$
\delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\...
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answer
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Newton's original proof of gravitation for non-point-mass objects
Suppose we have two bodies, one very large (Earth), and one very small (a cannon ball). If the cannon ball is some distance away from the Earth, to find out the force produced on the cannot ball, we ...
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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.
2....
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2
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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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S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. (9.2.15)
In Weinberg's book The Quantum Theory of Fields, Volume 1 on p.388 (Chapter 9), the following identity is used (with $f$ being any "reasonable" function):
$$f(+\infty) + f(-\infty) = \lim_{\epsilon \...
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Recovering QM from QFT
Reading through David Tong lecture notes on QFT.
On pages 43-44, he recovers QM from QFT. See below link:
QFT notes by Tong
First the momentum and position operators are defined in terms of "...
user56963
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3
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How is the Saddle point approximation used in physics?
I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative ...
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3
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Why is the $dx$ right next to the integral sign in QFT literature?
I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?
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Principal value of $1/x$ and few questions about complex analysis in Peskin's QFT textbook
When I learn QFT, I am bothered by many problems in complex analysis.
$$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$
I can't understand why $1/x$ can have a principal value because ...
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The invariant measure on an energy surface of a Hamiltonian system
Consider a Hamiltonian system with a time-independent Hamiltonian $H (p, q )$. By the Liouville theorem, the measure $d^np d^nq $ is conserved.
However, one should also notice that the energy is ...
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7
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What does it mean to integrate with respect to mass?
I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution.
I'm not sure ...
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2
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Complex Gaussian integral with different source terms
Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is
$$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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3
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Electric fields in continuous charge distribution
My question may be very basic, but I can't think of a reasonable explanation for this.
Consider a solid charged sphere. Now, we have an electric field inside the solid sphere, but at any particular ...
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Why are the number of magnetic field lines finite in a particular area?
One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the ...
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How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
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Gravitational potential energy of any spherical distribution
The general formula to get the potential energy of any spherical distribution is this :
\begin{equation}\tag{1}
U = - \int_0^R \frac{GM(r)}{r} \, \rho(r) \, 4 \pi r^2 \, dr,
\end{equation}
where $M(r)$...
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Exponential decay of Feynman propagator outside the lightcone
In Chapter three (I.3) of A. Zee's Quantum Field Theory in a Nutshell, the author derives the Feynman propagator for a scalar field:
$$
\begin{aligned} D(x)&=\int \frac{\operatorname{d}^4 \mathbf{...
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4
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A four-dimensional integral in Peskin & Schroeder
The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660:
$$\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2)^2}e^{ik\cdot\epsilon}=\frac{i}{(4\pi)^2}\log\frac{1}{\epsilon^2},\...
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Moment of inertia: why $\mathrm{dI}=r^2\mathrm{dm}$ instead of $\mathrm{dI}=m\mathrm{dr^2}$?
When computing the moment of inertia, I observed that people usually use the following logic:
$$d I=r^2 dm,\\
\therefore I=\int r^2 dm$$
My question here is, why not use $dI=m ~d(r^2)$?
I ...
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2
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Gaussian integral of a function with non-zero mean (generalizing Wick theorem)
From the wikipedia article, for a Gaussian integral of an analytic function we have that
This is equivalent to the Wick theorem when $f(x)$ is a polynomial.
Now I'm trying to obtain a similar formula ...
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1
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Switching from sum to integral
I'm specifically asking about an equation in An Introduction to Quantum Field Theory, by Peskin and Schroeder. Example from page 374:
$$\mathrm{Tr} \log (\partial^2+m^2) = \sum_k \log(-k^2+m^2)$$
...
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1
answer
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What it means to integrate over $n$ variables out of $N$, where $N>n$?
I was reading Theory of Simple Liquids, when I came across BBGKY hierarchy. In deriving the expression for the hierarchy, they integrate an integration of N variables over N-n variables to make the ...
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0
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Feynman rules for this perturbative expansion in Grassmann variables
I'm given the integral $$ Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2} w_i(x) ...
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Generalising a Dirac Delta function formula in General Relativity
I'm currently stuck on a problem where I have to integrate on a particular set defined through a dirac delta function. If I understood correctly it all boils down to using the curved analogous of
$$
...
user78618
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1
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Fourier transform of the free propagator squared - $\int d^{4}p\ \frac{e^{-i p\cdot x}}{p^{2}+m^{2}-i\epsilon}$
The point of the question is to ask what is the function given by the following integral:
$$
H(x,y) \ \equiv \ \int \frac{d^{4}p}{(2\pi)^{4}} \frac{e^{-i p \cdot (x-y)}}{(p^{2}+m^{2}-i\epsilon)^{2}}
$$...
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3
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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 ...
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Trajectory of a photon around a Schwarzschild black hole?
Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see this for illustration). Its impact parameter is $b$ and its distance ...
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1
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Physical intuition on the integral contained in D'Alembert's Formula for the wave equation
If $\phi(t,x)$ is a solution to the one dimensional wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then D'Alembert's Formula gives
$$\phi(t,x)= \frac 12[ \phi(...
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2
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Integration by parts to derive $d\langle x \rangle / dt$
I am reading "Introduction to Quantum Mechanics" by David Griffiths and I am having trouble understanding part of a derivation of $\frac{d\langle x\rangle }{dt}$ in section 1.5 - Momentum - of the ...
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1
answer
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Gaussian integrals in Feynman and Hibbs
I was going through the calculation of the free-particle kernel in Feynman and Hibbs (pp 43). The book describes
$$
\left(\frac{m}{2\pi i\hbar\epsilon}\right)\int_{-\infty}^{\infty}\exp\left(\frac{im}{...
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1
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Why does the integral symbol disappear when applying a functional derivative?
it is known that variation is defined by following:
but could anyone tell me why the integral symbol disappears after following functional derivative?
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1
answer
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Torque on wire summarized with magnetic moment
The magnetic moment of a current-carrying wire loop $L$ is
$$
\boldsymbol\mu = \frac I2\oint_L\mathbf{r} \times \mathrm{d}\mathbf{r}
$$
so the torque it experiences under a uniform magnetic field $\...
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2
votes
1
answer
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Dirac notation, integral and change of basis
Suppose, I have some operator $\hat{A}$, such that in the $x$-basis, it is written as $f(x).$ I'm trying to calculate the expectation value of this operator in integral form. That is given by the ...
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1
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The computation of the propagator in two dimensions
I did the computation of the propagator in two dimensions at (19.26) in Peskin & Shroeder as follows.
First I performed a Wick rotation.
\begin{alignat}{2}
\int\frac{d^2 k}{(2\pi)^2}e^{-ik\cdot (...
- 515
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vote
3
answers
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How to calculate the moment of inertia of a solid cube? [closed]
How do I calculate the moment of inertia of a uniform solid cube about an axis passing through its center of mass?
I also wanted to know if the moment of inertia ...
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vote
4
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Deriving equations of motion using integration
Please refer to my school textbook pg48 (of the book, and not the pdf counter) here: http://ncertbooks.prashanthellina.com/class_11.Physics.PhysicsPartI/ch-3.pdf
My doubt is in this context: (right ...
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1
answer
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Supersymmetric Localization (Mirror Symmetry)
I'm reading Chapter 9 of Mirror Symmetry book. As you can see in eq. (9.30) his model for SUSY is
$$\begin{align}
\delta_\epsilon X &=\epsilon^1\psi_1 + \epsilon^2\psi_2\\
\delta\psi_1 &= \...
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