Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
0
votes
1answer
31 views
Is the Jacobian different for different ${\cal L}^p$ norms?
(I posted this to the math stackexchange, but I've yet to receive an answer so I figured I should post here too, as this forum seems faster to respond and is full of knowledgable people.)
Because the ...
0
votes
0answers
39 views
What kind of integration is used in different areas of physics? [on hold]
I'm aware that there are multiple types of integration in mathematics, e.g., there's the Riemann integral, the Riemann-Stieljes integral and the Lebesgue integral, but I'm not privy to their ...
-4
votes
1answer
38 views
Electric field at center of a uniformly charged hemispherical shell [closed]
My attempt :
I divided the shell into charged rings by whom I know the electric field at the center by the following relation :
$\hskip2in$
Integrating the electric fields by all the rings I get ...
1
vote
2answers
26 views
Anomalous value of electric field due to a uniformly charged disc at a point on its central axis
Upon trying to find the electric field using integration (as done below) we get the following result :
According to me there are two problems with this result :
This function is not of odd parity.
...
0
votes
0answers
22 views
The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]
I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
-1
votes
1answer
23 views
Kinematics Problem requiring Calculus [closed]
Let the Instantaneous Velocity of a rocket just after launching be given by
v={ 3t for 0<= t <2
2t+ 3t^2 for 2<= t<=3
t^3 for t>3
Find the ...
2
votes
1answer
57 views
How is the complex integration done for the Wigner function in coherent state representation?
$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
0
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0answers
22 views
Manipulation of momentum factors in Feynman integrals
My current understanding:
Consider a massless Feynman graph with propagators of the form $1/p^2$ and some momentum factors in the numerator. Now consider a vertex, which for simplicity of notation is ...
4
votes
1answer
84 views
Complex Gaussian integral with different source terms
Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is
$$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
4
votes
2answers
168 views
Explicit computation of singular part of two-loop sunrise diagram
For $\phi^4$, there is two-loop self-energy contribution from sunrise (sunset) diagram.
The integration is
$$
I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[...
0
votes
2answers
72 views
How is the wave function Lebesgue integrable?
Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
2
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0answers
46 views
Polylogarithmic integrals
NB: I was sent here from Math.SE, stating that polylog integrals are more common in physics and someone here might have an answer. I have asked in Phys.SE chat whether it was okay to post here but no ...
0
votes
0answers
27 views
What is the simplest way of getting the solid angle $\Omega_d$ in a space of $d$ dimensions? [migrated]
It is known that the solid angle in a flat space of $d$ dimensions ($d = 2 n$ or $d = 2 n + 1$) is given by these formulae:
\begin{align}\tag{1}
\Omega_{2 n} &= \frac{1}{(n - 1)!} \, 2 \pi^n, \...
0
votes
1answer
50 views
How to know what the area under curve represents?
Is there a way to find out what the area under the curve represents? For eg. If i gave you a graph of $v$ with respect to $t$ would you be able to tell me what the area under the curve represents ...
0
votes
2answers
26 views
Trying to get moment of inertia of a disc using moment of inertia of a rod
I know how moment of inertia of a disc is calculated using the usual way, but just for fun, I tried this way which is rather giving incorrect answer. I don't know what's the flaw in this and thus ...
2
votes
0answers
75 views
An involved Feynman integral
Working with QCD, I have found the following integral from Feynman diagrams to solve
$$
I(p)=\int\frac{d^4p_1}{(2\pi)^4}\int\frac{d^4p_2}{(2\pi)^4}\frac{1}{p_2^2-m_0^2} \left(\frac{p\cdot p_1-p_1\cdot ...
1
vote
2answers
82 views
How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant?
When writing integrals that look like
$$
\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0)
$$
it is often said ...
2
votes
1answer
62 views
Solving the rocket differential equation
I'm trying to derive the rocket equation.
I'm pretty sure that the differential equation for the rocket equation is
$$v(t)\delta t =\frac{m(t)\delta t }{m(t)} V_e$$
where
$v(t)\delta t$ is the ...
0
votes
1answer
33 views
Follow up on understanding path integral measures
A while ago I asked the following question:
Understanding Measure in Path integrals
and got to the conclusion that path integral measures are infinite products of $d\phi(x_i)$ for some scalar field $\...
3
votes
2answers
156 views
Problem with loop Integral (HQET)
I have come across the Integral:
$$ \int_0^{\infty}dx [x^2-ixa+c]^{n-\frac{d}{2}}e^{-bx},$$
where $n = 1,2 ; a,b,c,d \in \mathbb{R}; b,d > 0$.
This integral should contain some divergences for $d ...
1
vote
0answers
31 views
What is the magic behind Sector Decomposition?
I have a question regarding Sector Decomposition, which is briefly introduced in this paper arXiv: 0803.4177.
So far I played around with a toy example and applied the Sector Decomposition method to ...
0
votes
1answer
48 views
Problem with converting Integral to Gamma functions (from HQET heavy quark self-energy diagram)
In the calculation of HQET radiative correction, I came across the Equation:
$$\int_0^{\infty}d\lambda ~ \lambda^{-\epsilon}(\lambda+\omega)^{-\epsilon} = \frac{1}{2\sqrt{\pi}}\Gamma(\epsilon-\frac{1}{...
1
vote
1answer
59 views
How to find gravity field of a solid square body?
I was programming gravity simulation and stumbled upon a problem, that Newton's formula for point masses is not enough for me, I need gravity field formula of a solid square body (2D).
To simplify ...
0
votes
2answers
24 views
Moment of Inertia equation for small volume
Below is the equation of the moment of inertia for small volume elements, $\Delta m$
$$I = \lim_{\Delta m_i \to 0} \sum_{i} r^2_i \Delta m_i = \int r^2 dm$$
Can someone please explain it to me on ...
0
votes
1answer
19 views
Scalar field and 2 types of line integrals
Consider the line integral,
$\int _ c$f(x,y)$\vec dr$ , where $f(x,y)$ is a scalar field, and it is evaluvated on a curve $c $. After integration we get a vector let it be $\vec I$ .
$\int _ c$f(x,...
0
votes
1answer
28 views
Meaning of flux 2-integral
Can someone please explain the meaning of flux 2-integral in this sentence:
Mass is evaluated as a flux 2-integral at the asymptotic infinity.
For asymptotic infinity, I believe it is as ...
-1
votes
1answer
64 views
Calculating power with force as a function of time [closed]
While doing an AP question in Physics C today I answered the question differently from the professor but I'm not sure what part of my reasoning is incorrect. The question gives the position of a block ...
4
votes
1answer
40 views
What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?
In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as:
$a(t) = a_0$
$v(t) = a_0t + v_0$
$x(t) = \frac{1}{2}...
1
vote
0answers
91 views
Deriving moment of inertia of a solid sphere [closed]
I have been trying to calculate it on my own, but the answer I get is different to the one I can find everywhere else, so I have to be wrong.
My attempt was a very straightforward one. I used ...
-2
votes
1answer
47 views
Integral over an area of spacetime [closed]
Is it possible to evaluate this integral in spacetime?
$$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$
If it is ...
-2
votes
1answer
55 views
Integration Using Spherical Coordinates [closed]
So I had to find the moment of inertia of a hollow sphere of mass $M$, radius $R$, and negligible thickness.
$dI=R^2 \cdot dm$
where $dm = \dfrac{M}{4\pi R^2}\cdot R^2\sin(\theta)\cdot d\theta\cdot ...
3
votes
1answer
124 views
The acceleration of the particles by finite difference [closed]
I would like to approximate the acceleration of a molecular dynamics system.
I'm following an online tutorial to solve a set of equations for molecular dynamics. I can use $F=ma$ to calculate the ...
-2
votes
1answer
52 views
Given the parameters of the electrostatics problem, is this integral possible to evaluate analytically? [closed]
A cone with apex at the origin has a height $h$ and a top radius $h$, a uniform charge density with no charge on the top face.
I need to find the potential $V$ at a position $z$ on the cone's axis ...
4
votes
2answers
233 views
Sum to an integral in deriving equipartition theorem
I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum,
$${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
1
vote
1answer
57 views
Perpendicular weight force on an object that is tipping over [closed]
I'm currently working on a problem I can't seem to find an answer to. I have an object that is hanging over a cliff. This object is exactly 12m in length, and it starts off in equilibrium (6m over the ...
-4
votes
1answer
56 views
No clue about a term [closed]
$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$
I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$
...
2
votes
1answer
116 views
Integration of the splitting function
I have a problem performing the following integration provided in the paper by Catani and Seymour (arXiv: hep-ph/9605323) page 27. Given is the integral
$$
\mathcal{V}=\int_0^1 (z(1-z))^{-\epsilon} \...
1
vote
2answers
100 views
How to determine the units of my integral? [closed]
Given that I have some coefficient(i.e. a number) which is to be determined from a radial integral:
$$b_{n00} = \frac{(2\pi)^{1/4}}{\sigma^{3/2}} \frac{1}{\sqrt{3}} [C(000|000)]^2 \int^{\infty}_{r = ...
1
vote
0answers
40 views
Integration by parts in dimensional regularisation
I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that
$$\int d^Dl_1...d^Dl_L \...
1
vote
0answers
42 views
Explicit integration of the time-dependent Schrodinger equation when eigenvalues are unknown
Let's consider the hydrogen atom Hamiltonian
$$H = - \frac{1}{2}\Delta - \frac{1}{r}$$
The solution for the corresponding time-dependent Schrodinger equation is the following:
$$\psi = \psi (t = 0){...
4
votes
1answer
135 views
Time evolution operator in QM
I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator:
\begin{align*}
i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t)
\end{...
0
votes
1answer
75 views
Integrating rigid body equations for a game engine simulation
I'm a mechanical engineer who's trying to implement a physics engine for a 3D game simulation, so I apologize for being incorrect or simply ignorant of some aspects of computation. I'm implementing ...
0
votes
1answer
31 views
How can I calculate the total rotation a detuned pulse will apply a nuclear spin?
I'm trying to model the effect a radiofrequency pulse will have on a nuclear spin at different detunings. The pulse has a sech lineshape, a pulse area (time integral of the pulse envelope) of $\frac{π}...
0
votes
1answer
39 views
How to combine limits when integrating in the frequency domain
I want to combine the signal of two separate pulses in the frequency domain in order to calculate their overlap (by multiplying the two signals together and integrating). However, one of these signals ...
1
vote
1answer
35 views
Integrating Carnahan-Starling Pressure
Given the Carnahan-Starling equation of state for a solution of hard-spheres,
$$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$
where $\rho = N/V$ is the number density and ...
1
vote
1answer
87 views
Computing integrals for divergent loop amplitudes?
I am trying to compute the cross-section for the diagram below with a divergent triangle loop:
$\qquad\qquad\qquad\qquad\qquad$
where $X^0$ and $X^-$ are some fermions with zero and negative charge ...
0
votes
1answer
33 views
How to find the net electric force exerted on a uniformly charged rod by another, same rod on the x-axis (they don't touch)? [duplicate]
How to find the net electric force exerted on rod 2 by rod 1, both being on the x-axis, both having the same length and constant linear charge density, being some distance apart?
More specifically, ...
1
vote
1answer
48 views
The average velocity of a particle
The Maxwell distribution of velocities is:
$$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$
I want to understand how to obtain the average value of the velocity. The ...
0
votes
1answer
53 views
About the quadratures method
in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75
it says: "Equations 3-18 and 3-20 are the two remaining integrations,
and formally the problem has been reduced to quadratures..."...
1
vote
0answers
21 views
Hylleraas Multi-dimensional integral [closed]
Evaluate the hylleraas integral
$$\int \int \frac{\exp(-ar_{1}-br_{2}-cr_{12})}{r_{1}r_{2}r_{12}}d^{n}r_{1}d^{n}r_{2} $$
with $r_{1}=|\vec{r_{1}}|$, $r_{2}=|\vec{r_{2}}|$, $r_{12}=|\vec{r_{2}}-\...
