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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2
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4answers
27 views

Is work equal to area between graph and $x$-axis in a graph of force vs displacement?

Is this true even when displacement is not in direction of force? $$W = \int (F\cos\theta)\text dx$$ and area is $\int F \text dx$. In my book that area is given as one of the definitions ...
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0answers
24 views

Reciprocity Between $X$ and $K$ operators

Ref.: Shankar 2nd edition page 69. Regarding the reciprocity Between the $X$ and $K$ operators, goes on to compute the matrix elements of $X$ in the $K$ basis : $$\langle k|X|k'\rangle=\frac{1}{2\pi}\...
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2answers
93 views

Integral as summation in quantum mechanics

I have just started QM and one thing that keeps bugging me is that whenever we have a continuous summation we take it as an integral (like in the formula below)... So why can we do this why summation ...
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0answers
23 views

Why are most derivations in physics doable via substitution in the integrals? [on hold]

I have studied almost all mechanics in my curriculum (uptill rotational dynamics) also I studied about waves, electrostatics, magnetics and thermodynamics. I am yet to see and integral which comes up ...
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1answer
20 views

How to recover the potential field from Green's function and Poisson's equation for a point charge

I first find the Green's function for the following PDE in $n=3$ dimensions, where $k:=|k|^2$. $$\nabla^2G(x,x')=\delta^3(x-x')$$ Upon Fourier transforming both sides, and inverting, I find that $$G(x,...
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0answers
28 views

Constant Acceleration and Displacement

How can I conduct an experiment to show that the area under a velocity-time graph equals the displacement when the velocity is changing at a constant rate? I've tried to measure free falling objects, ...
0
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1answer
27 views

Finding suitable element to perform integration upon [closed]

Is there any precise (proper) method or technique to specify the element on which integration will be performed. Is it the same for all properties like moment of inertia, gravitational potential, ...
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0answers
49 views

Absolute Value of a Complex Integral [migrated]

My question may seem naive, but I couldn't find its answer in books, websites, etc. Assume that I want to calculate numerically the absolute value of the following integral $$I = \int_0^T\exp(if(t)),...
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0answers
27 views

Total amount of diffusely reflected light off of a sphere?

I have a numerical simulation that uses ray tracing to calculate the total amount of light picked up by a sensor, after diffusely reflecting off of an object. To validate this simulation, I'd like to ...
2
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1answer
187 views

Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

Consider a momentum integral of the form $$I=\int d^3k\int dk^0\frac{1}{k^4}$$ where $k^2=(k^0)^2-(\vec{k})^2$ and the integral over $k^0$ runs from $-\infty$ to $+\infty$. This integral is common in ...
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17 views

Units after integrating a ratio metric: Point Source Transfer Mobility

I want to calculate the so called Line Source Transfer Mobility (LSTM) from 10 equispaced Point Source Transfer Mobilities (PSTM). The PSTM measures the response to an excitation, and it is defined ...
2
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1answer
54 views

Fractional Fourier Transform and Fresnel Propagation

I am currently trying to wrap my head around Fresnel propagation, and I understand it is mathematically linked to the Fractional Fourier Transform, but I'm having a hard time with the units and the ...
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1answer
55 views

Direction of integration and boundary limits in electromagnetism?

I have encountered several problems regarding the choice of direction of integration and the boundary limits, this semester in electromagnetism. Is there some rule, so I don't do it wrong again. In ...
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0answers
57 views

How can we evaluate the following integral using the tricks of delta functions? [migrated]

I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here. I am starting with ...
3
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2answers
75 views

Contour for integration in 1D scattering problem

A plane wave scattered by a 1D potential can be described by, $$\psi(x) = \begin{cases} e^{ikx} + R e^{-ikx}, & x<0\\ T e^{ikx}, & x>0 \end{cases}$$ where $R$ is the reflection ...
0
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1answer
53 views

Delta function and potential step [closed]

I have a potential consisting of an attractive delta funtion well located at the origin and a superimposed with a potential step at the origin, just like: With $$V(x)=-\lambda \delta (x) +V_0 ...
0
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3answers
70 views

Change of variable in 4-dimensional integral

If I have a measure $d^4 x$ and I want to perform a conformal transformation $x^\mu \rightarrow \frac{x^\mu}{x^2}$, how do I get that the transformed measure is $\frac{d^4 x}{x^8}$? I started by ...
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2answers
28 views

Intuition of Distance covered when accelerating [duplicate]

When you're moving at $5$ m/s for $1$ second, you have traveled $5$ m. When you're moving at $5$ m/s (initial velocity) and you accelerate $2$ m/s for $1$ second, you have traveled $5$ m + $1$ m (...
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0answers
44 views

Computation of Wigner Functions

The Wigner function can be computed as the Fourier transform of the Weyl-ordered characteristic function: $$ W(\alpha) = \frac{1}{\pi^2} \int e^{\lambda^* \alpha - \lambda \alpha^*} C_W(\lambda) d^2\...
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0answers
49 views

How to evaluate the period of a particle in a system with potential energy $U=-U_0/\cosh^2(\alpha x)$?

I am working through the textbook "Mechanics", from the series "Course of Theoretical Physics " by Landau and Lifshitz. In Chapter 3, where the authors talk about integrating the equation of motion $E=...
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0answers
57 views

Can we do one-loop integrals in the unitary gauge?

$\hspace{5cm}$ Imagine we want ot compute one of the diagrams for the self-energy of the quark $u$, with external momentum $p$. Inside the loop, we would have a $W^+$ and a $d$-quark propagator, with ...
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1answer
48 views

Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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0answers
37 views

Field Theory: Converting $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$

For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
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1answer
164 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 ...
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2answers
83 views

Feynman's proof for Newton's shell theorem [closed]

I have two questions concerning this proof: Firstly, what is the difference between the increments ds and dx? Are they not just the same thickness of the strip? Secondly, why can the integral ...
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0answers
14 views

Magnetic field from a current arc and its limiting behaviour

For my sanity and the sake of book keeping, I have been going over the very well known and documented calculation for the magnetic field of a current loop. I have been trying to verify the limit ...
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1answer
44 views

Problem in the continuum limit of a Kronecker delta

I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta. Imagine to integrate a function depending on $t$ and $t'$, both ...
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1answer
62 views

Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
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1answer
44 views

How to calculate the gravitational binding energy of a uniform cube of length $L$ and mass $M$?

The functional form is known already (as attached). But what is the solution for this integral?
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0answers
19 views

Finding Line Integral Of Magnetic Field over Quarter Circle

This is one of the question in my workbook and I tried solving it in above way. However I am not able to get the answer. Can anyone suggest a method to solve this question ? Am I doing a mistake in ...
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0answers
24 views

Literature result for generic one-loop triangle integral

I've never really asked a literature based question here before, but was wondering if anyone knows where I may find a reliable source for symbolic expressions for generic one-loop triangle diagrams ...
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1answer
47 views

Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
0
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1answer
29 views

What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
0
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3answers
64 views

How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
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1answer
47 views

Calculating electric potential — denominator going to zero [closed]

Calculate the potential inside a uniformly charged solid sphere of radius $R$ and total charge $q$. My attempt: There are several ways to solve this problem but I'm curious as to whether this ...
0
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0answers
22 views

Action variable integral

I am solving an action angle variable problem and I'm stuck at the point where I have the following expression for the integral $$ I = \frac{b\sqrt{mE}}{\pi} \int_\theta^{2\pi-\theta}\sqrt{(1-cos \...
0
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4answers
63 views

Deriving the formula of potential difference duo to a solid conducting sphere

The doctor in university was deriving a formula and I can't understand how it works A sphere with charge Q The Sphere's radius is R, and we are trying to derive a formula for potential difference at ...
0
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1answer
41 views

Nonlinear GPE of a solid block seems wrong

I am trying to calculate the gravitational potential of a solid block, and I have a nonlinear answer which strikes me as wrong. A block with horizontal surface area $A [m^{-2}]$ and uniform density $\...
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0answers
41 views

How to prove that the limit of surface integral exists?

My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$: Question Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains ...
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0answers
12 views

Find the time required to travel pulse from bottom to the top of string .(as shown in the figure)

Given that μ is the mass per unit length of string , $M_{b}$ is the mass of block hanged to the bottom of the string and M is the mass of string . I tried like this we know that $V=\sqrt\frac{T}{μ}$...
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0answers
42 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
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2answers
80 views

Can someone provide to me an intuitive explanation of the second integral of position with respect to time?

I am aware of what the first integral of position, absement means (at least to a very superficial level). However, I can find nothing regarding the physical intuitive meaning of absity, the second ...
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0answers
19 views

Amperian Loops vs Gaussian Surfaces

I am wondering what makes an amperian loops different from a gaussian surface? I know they are used the calculate different things, but when I take the integral how are they different? This is in ...
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0answers
20 views

Problem with an exchange integral

I'm reviewing the chapter on molecules from Gasiorowicz's Quantum Physics book (Ch. 20 of 1st edition) and it gets to a part where it solves the $H_2^+$ ion using the variational principle. There are ...
0
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1answer
38 views

Anomalous magnetic moment of the electron - integration problem

In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated: $$ F_{2}(...
3
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2answers
551 views

Integral Notations in Quantum Mechanics [duplicate]

I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the ...
0
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1answer
78 views

How to calculate the derivative of scale factor as a function of conformal time from the solution of Friedmann equation

For the flat geometry of lamda CDM model, the solution for Friedmann equation is $$ a(t) = \left\{ \frac{Ω_{m,0}}{Ω_{Λ,0}} \sinh^2 \left[\frac{3}{2} \sqrt{Ω_{Λ,0}} H_0(t - t_0)\right] \right\}^{1/3},...
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2answers
24 views

element of surface area versus vector element of surface area

In the context of calculating electric flux, is there a difference between element of surface area versus vector element of surface area? Thanks
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3answers
56 views

Deriving an integral in a mechanics problem (massless string holding up a disk)

I am told that a massless string is holding up a disk of mass $M$ and radius $R$. I want to find out the value of the tension $T$ in the string. The textbook does this trivially by stating that $2T=...
0
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1answer
52 views

UV divergence integral

Could anyone please explain how to calculate integral such as $$\frac{\Omega}{2}\int_{-\infty}^{+\infty} \frac{d^3k}{(2\pi)^3}\ln\left[{1+\frac{a^2}{k^2}}\right]=-\frac{\Omega a^3}{12\pi}+I_0~?$$ ...