Skip to main content

Questions tagged [integration]

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

Filter by
Sorted by
Tagged with
55 votes
3 answers
40k views

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 ...
BeauGeste's user avatar
  • 1,621
37 votes
3 answers
202k 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?
user avatar
32 votes
3 answers
6k 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 ...
WedgeAntilles's user avatar
32 votes
4 answers
7k 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 \...
user2582713's user avatar
31 votes
5 answers
5k views

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 ...
Tim Crosby's user avatar
  • 1,323
30 votes
3 answers
5k views

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 ...
Larry Harson's user avatar
  • 5,318
28 votes
3 answers
1k views

Heisenberg's uncertainty principle for mean deviation?

The Heisenberg uncertainty principle states that $$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$ However, this is only for the standard deviation. What is the inequality if the mean deviation, defined as ...
Zach466920's user avatar
  • 1,117
28 votes
2 answers
13k views

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 ...
Adomas Baliuka's user avatar
25 votes
4 answers
7k views

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. $$\...
user22208's user avatar
  • 453
23 votes
4 answers
2k views

Non-unique zero function in the function space (Hilbert space)

I have just started studying about quantum mechanics, and I was studying the definition of the inner product for functions; I am also quite new to linear algebra. While studying I think I encountered ...
Soroush khoubyarian's user avatar
22 votes
1 answer
2k views

Derivation of KLT relations

The KLT relations (Kawai, Lewellen, Tye) relate a closed string amplitude to a product of open ones. While I get the physics behind this I don't really understand the derivation in the original paper (...
A friendly helper's user avatar
20 votes
5 answers
131k views

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 ...
ben's user avatar
  • 1,517
19 votes
4 answers
6k views

Is dimensional analysis valid for integrals

Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no ...
rathankar's user avatar
  • 307
19 votes
3 answers
2k 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 $d=\frac{1}{2}at^2$?
Mark's user avatar
  • 331
17 votes
2 answers
926 views

Cutoff-Scheme Renormalization and Order of Integration in QFT

The following is the result of Fubini's Theorem, describing when you can replace a double integral with an iterated integral safely: For a set $X \times Y \subset \mathbb{R}^2$, if $\iint |f(x,y)| d(...
QuantumEyedea's user avatar
16 votes
3 answers
4k views

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_{-\...
knzhou's user avatar
  • 103k
15 votes
5 answers
2k views

Why doesn't this way of calculating the moment of inertia of a sphere work?

Instead of the usual approach of integrating a bunch of discs, I do it differently. I integrate by subdividing the sphere into a bunch of concentric spheres containing mass on their surface. The ...
user2714980's user avatar
15 votes
3 answers
3k views

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?
Craig Feinstein's user avatar
15 votes
3 answers
3k views

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 ...
TMS's user avatar
  • 2,071
15 votes
1 answer
1k views

What does it mean for a field to be defined by a measure?

In Quantum Physics by Glimm and Jaffe they mention on p. 90 that The Euclidean fields are defined by a probability measure $d\mu(\phi) = d\mu$ on the space of real distributions. Here $d\mu$ plays ...
CBBAM's user avatar
  • 3,340
15 votes
3 answers
4k 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. 2....
346699's user avatar
  • 5,971
15 votes
4 answers
3k views

Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
DFJ's user avatar
  • 339
15 votes
1 answer
1k views

Why isn't the Gear predictor-corrector algorithm for integration of the equations of motion symplectic?

Okumura et al., J. Chem. Phys. 2007 states that the Gear predictor-corrector integration scheme, used in particular in some molecular dynamics packages for the dynamics of rigid bodies using ...
F'x's user avatar
  • 1,799
14 votes
2 answers
2k views

Unfamiliar Notation in Sakurai

In chapter 5 section 9 of Sakurai, 2nd edition, he uses some notation that I am unfamiliar with. This may be suited for Math.se but I figured it could be peculiar physicist notation. Anyways it is ...
ClassicStyle's user avatar
14 votes
4 answers
21k views

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 ...
nagendra's user avatar
  • 325
14 votes
1 answer
2k views

Divergent integrals in QFT

I am starting to learn about QFT and something that I noticed is that integrals who would diverge otherwise are assigned a value if we do it by contour integration using the residues theorem and the ...
Slayer147's user avatar
  • 1,045
14 votes
1 answer
300 views

Monte Carlo integration over space of quantum states

I am currently facing the problem of calculating integrals that take the general form $\int_{R} P(\sigma)d\sigma$ where $P(\sigma)$ is a probability density over the space of mixed quantum states, $...
Juan Miguel Arrazola's user avatar
13 votes
1 answer
8k views

Propagator of a scalar in position space

In his lecture on Supersymmetry and Grand Unification, Leonard Susskind "derives" the propagator for a scalar field from dimensional analysis. He says for a particle going from $x$ to $y$ (where x and ...
jdm's user avatar
  • 4,207
12 votes
4 answers
3k views

Integrating acceleration - wrong choice of bounds in textbooks?

I've noticed in my physics textbook (and in a lot of other popular sources), that the process of integrating non-constant acceleration to get to a velocity formula, the integrating bounds imposed on ...
Ius Klesar's user avatar
12 votes
1 answer
16k views

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 ...
346699's user avatar
  • 5,971
12 votes
1 answer
3k views

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.
F R's user avatar
  • 121
12 votes
2 answers
22k views

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 $M(r)$...
Cham's user avatar
  • 7,572
11 votes
4 answers
3k views

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},\...
soliton's user avatar
  • 1,681
11 votes
1 answer
7k views

Integrals over Grassmann numbers

I want to prove an identity from Peskin&Schroeder, namely that $$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m \theta_l^* \exp(\theta_j^* B_{jk} \theta_k)=\det(B) B^{-1}_{ml}...
TheoPhysicae's user avatar
10 votes
3 answers
4k views

Principal value integral

I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has: Meanwhile the principal value integral is defined by: $$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int dx\,...
Virgo's user avatar
  • 2,094
10 votes
4 answers
1k views

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 ...
poisson's user avatar
  • 1,937
10 votes
2 answers
2k views

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{...
alonso s's user avatar
  • 293
9 votes
5 answers
1k views

Is the average force calculated from $F(x)$ the same as that calculated from $F(t)$?

Say a force is doing work on an object in one dimension. I could calculate the average force over the distance with $$\frac{1}{\Delta{x}}\int_{x_1}^{x_2} F(x) \text dx$$ If I also formulated force ...
iRove's user avatar
  • 195
9 votes
7 answers
2k views

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 ...
QED's user avatar
  • 313
9 votes
4 answers
2k views

Are there 'higher order moments' in physics?

This may be a rather noob question but please let me clarify: I'm struggling to understand the use of the word 'moments' w.r.t., probability distributions. It seems after some research and poking ...
PhD's user avatar
  • 193
9 votes
2 answers
1k views

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(\...
kayla's user avatar
  • 141
9 votes
1 answer
2k views

How Do We Define Integration over Bra and Ket Vectors?

I'm having trouble understanding the completeness condition for bra and ket vectors in Hilbert space, especially in the continuous case. The discrete case makes a fair amount of sense; given any ...
FlyingPiper's user avatar
9 votes
1 answer
743 views

What advantages have a symplectic or geometric integrator over a simple one, say, RK4?

I heard that a symplectic integration algorithm has a property related to the phase space of a system, but i don't understand much further than that. I'm interested in applying that method to a non-...
Gabriel Sandoval's user avatar
9 votes
1 answer
1k views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
Jia Yiyang's user avatar
  • 3,995
9 votes
1 answer
783 views

Physical intuition/interpretation of fractional derivatives/integrals?

Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them: Velocity is the derivative of position ...
Ron's user avatar
  • 401
9 votes
1 answer
775 views

Integration of Differential Forms

I want to understand what it actually means to integrate a differential form on a manifold. Being a mathematician, the explanation I always get is that they simply follow the right transformation rule....
ofiz's user avatar
  • 191
8 votes
4 answers
2k views

How are electric flux calculations not double integrals?

A disk of radius 0.10 m is oriented with its normal unit vector $\hat{n}$ at 30$^{\circ}$ to a uniform electric field $\vec{E}$ of magnitude 2000 N/C. What is the electric flux through the disk? I ...
user44557's user avatar
  • 103
8 votes
5 answers
2k views

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 ...
Danny  Han's user avatar
  • 309
8 votes
1 answer
1k views

Does the universe obey the holographic principle due to Stokes' theorem?

Does the universe obey the holographic principle due to Stokes' theorem? \begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation} Can this ...
m0nhawk's user avatar
  • 1,009
8 votes
3 answers
6k views

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 ...
Kam's user avatar
  • 93

1
2 3 4 5
29