Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
970
questions
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2answers
24 views
Gaussian integral with respect to Grassmann variables
Let $A$ be an antisymmetric matrix of even dimension $n$ and $\theta$ be a column vector consisting of $n$ Grassmann variables $\theta_i$. Then the solution of the integral
$$\int d\theta_1\dots d\...
1
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0answers
32 views
Electric field created by sphere (without Gauss law)
I'm trying to get the electric field from a sphere (charge Q and radius a) in a point on z axis (z>R) beginin from:
$$\int\frac{\rho(r')}{|r-r'|^2}\frac{r-r'}{|r-r'|}d^3r' $$
So i used spherical ...
-1
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0answers
39 views
Integration with respect to Haar measure and reduced density matrix
Consider a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$, with $|A|,|B|>>1$ and not necceserly $|A|=|B|$.
Following Jerusalem Lectures on Black Holes and Quantum Information (eq. 5.8) ...
0
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0answers
28 views
Moment of inertia of cylinder drilled with conical hole [closed]
I want to find the moment of inertia of the following solid: Cylinder drilled with a conical hole.
It is given that density is constant $\rho_0$. So, what I did:
$${I_z} = {\rho_0}\int\limits_U {\...
2
votes
1answer
37 views
$\int_{-\infty}^{\infty} |\psi(x)|^2 ~ dx = 1$ when $\psi(x) = C\exp\left(\frac{x^2}{2a^2} + \frac{ix^3}{3a^3}\right)$
The information given is:
Consider a state $|\psi\rangle $ describing a quantum particle on a line, whose position representation $\langle x|\psi\rangle = \psi(x)$ is given by:
\begin{gather*}
\...
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0answers
56 views
Evaluation of Gaussian integral $\int^\infty_{-\infty}dx\;\exp(A(x-B)^2)$ with $A$, $B$ complex [migrated]
Does
$$I = \int^\infty_{-\infty}dx \;\exp(re^{i\theta}(x-B)^2), \quad B \in \mathbb{C}$$
have a known standard result? I am hoping to use the result for exercises in the path integral formulation of ...
0
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1answer
41 views
Is calculus necessary for finding the difference in entropy?
My book uses calculus to find the difference in entropy of an object with mass m and specific heat capacity s between two states with temperatures $T_1$ & $T_2$.
$dS=\frac{Q}{T}$
$\implies \Delta ...
0
votes
1answer
34 views
How this equation is attained of the sum of the vertical forces of charges?
The conductive sphere which has the radius $a$.
Nextly the sphere has been given the charges of $Q$
And nextly we assume that the sphere was cut at the middle(shown as the red line).
We want to ...
0
votes
2answers
42 views
Expression for the causal retarded potential for $t<0$ must give $0$ but my calculation produces a nonzero result. What's the mistake?
This question was previously asked here in the Mathematics StackExchange but using a slightly different notation. But I did not find the answer I was looking for or rather got two very different ...
0
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3answers
47 views
Integrating drag force
I need help with an integration problem. We have $F_d = \frac{1}{2} \rho c_d Av^2$ and $$W = \int F_d\cdot dx = \frac{1}{2}\rho c_dA\int v^2\cdot dx = \frac{1}{2}\rho c_dA\int(\frac{dx}{dt})^2\cdot dx....
0
votes
2answers
96 views
Direction of $ds$ in $E\cdot ds$
I have started a course on electromagnetism (EM) online, but I have some confusions in the math related to the work or potential energy.
Say we have a force field, and two positions A and B, and a ...
0
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1answer
33 views
Why have we assume that the there should be no acceleration in deriving the Electrostatic potential energy
In the NCERT textbook grade 12, it is mentioned before deriving the work done in moving a charge $q$ from point $R$ to $P$ in presence of an electric field at Origin, $O$ by charge $Q$
Two remarks ...
2
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0answers
32 views
Can current density $J$ for a thin wire by written in terms of the current $I$ and Dirac delta function?
If there is a thin wire with current $I$ flowing through it, could I write the current density at all points in space of a horizontal 2D slice of the wire as $I \cdot \delta^2(\vec r)$ ?
I'm a bit ...
0
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2answers
37 views
Why is $d^3\vec{v} = 4\pi v^2dv$ in integration if the speed is isotropic?
in my galaxies textbook it says that $d^3\vec{v} = 4\pi v^2dv$ for integration if the speed is isotropic, where does this come from?
example: In my statistical mechanics book it was said that the ...
0
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1answer
32 views
Integral involving unit vectors in spherical coordinates
I'm puzzled about the value of the following integral, that I've tried to evaluate after turning the unit vectors in spherical coordinates into cartesian coordinates: $I=\int\frac{\hat{r}}{(r-r')^2}d\...
2
votes
2answers
89 views
How to integrate a tensor in curved spacetime?
I've read "We can only define the integral of a scalar function. The integral of a vector or tensor field is meaningless in curved spacetime" on many books and lectures on General ...
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0answers
34 views
Showing an identity involving Grassmann variables
Given a collection of independent Grassmann variables $\{\theta_i,\bar{\theta}_i, \eta_i, \bar{\eta}_i \}$, where $i \in \{ 1, ..., n\}$, and an invertible $n \times n$ matrix $B_{ij}$. We want to ...
1
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3answers
89 views
Meaning of the fermion path integral?
I'm trying to understand fermion fields with the Feynman integral. Is there an explicit matrix representation of the Grassmann numbers used in the field integral? Is there a Grassmann-valued measure ...
1
vote
2answers
62 views
Conversion of 1D charge density to 2D charge density via integration
I'm self-studying EM (using the third edition of Griffiths) and have a quick question. Problem 2.41 states:
Find the electric field at a height $z$ above the center of a square sheet (side a) ...
3
votes
1answer
51 views
Problem with the sign of work
I have been stuck in a conceptual problem about the sign of the work. For example, suppose that we have a mass $m$ on a spring such that the equilibrium position is at $x=0$, and we stretch the spring ...
0
votes
1answer
31 views
Distance as a function of time in the case of an increasing Hubble value
I am trying to deduce the traveled distance of, e.g. a galaxy, as a function over time $D(t)$ in scenarios in which the Hubble value stays fixed or increases proportionally with time.
For the fixed ...
1
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2answers
55 views
Troublesome integrals in Hamiltonian matrix elements for a system of two interacting electrons
I have a quantum mechanical system of two interacting electrons in one spatial dimension. The Hamiltonian of the system is of the form $H = h + \frac{1}{|x_1 - x_2|}$, where $h$ is a one-electron part ...
4
votes
1answer
101 views
Dimensional regularization: order of integration
This is a two-loop calculation in dim reg where I seem to be getting different results by integrating it in different orders. I am expanding it about $D=1$. What rule am I breaking?
$$\int \frac{d^{D} ...
2
votes
1answer
41 views
Moment of Inertia of an arbitrary shape represented by a finite number of spheres
Quick Background
I'm writing software for my research (primarily focused around orbit determination and gravity field estimation around irregular bodies). For generating the truth reference models, I ...
0
votes
1answer
62 views
Why does Gauss' Law derive a different electric field from the integration method? [closed]
I was working on a very simple E&M problem and took two different approaches to it but got different answers - I'd be grateful if someone could spot the error in my logic!
A non-conducting solid ...
1
vote
1answer
39 views
Intensity of light transmitted by a polarizer when the incident light is unpolarized
my question is similar to this older one, but I have not enough privileges to answer it or comment on it.
I do realize that the question has received detailed answers by Selene Routley, and I sort of ...
0
votes
1answer
38 views
Derivation of Archimedes principle for a sphere
I am trying to derivate the Archimedes principle for a sphere with direct hydrostatic pressure calculation. I started with the asumptions that: $F_b=F_2-F_1$ and $F_2=2F_1$, where $F_b$ is a buoyancy ...
0
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0answers
16 views
Doubt regarding solving an integration for radial flow of matter around a star in Newtonian gravity
The spherically symmetric flow of matter around a star in Newtonian gravity is governed by the equation
$$v\frac{dv}{dr}+\frac{1}{P+\rho}\frac{dp}{dr}+\frac{1}{r^2}=0$$
The equation of state is chosen ...
0
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0answers
23 views
Are gaussian integrals also valid for complex constants? [duplicate]
In the WP article about propagators, there is an integral solved as:
$$K(x,x';t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk\,e^{ik(x-x')} e^{-\frac{i\hbar k^2 t}{2m}}=\left(\frac{m}{2\pi i\hbar t}\right)...
0
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0answers
35 views
Convergence of off-shell loop integral
Given the following theory
$$\mathcal{L} = \frac{1}{2}\dot\varphi^2-\frac{1}{2M^2}(\Delta\varphi)^2-\frac{m^2}{2}\varphi^2-\frac{g}{4!}\varphi^4$$
it's easy to show that the free propagator ($g=0$) in ...
2
votes
1answer
78 views
Transition rate derivation in non-relativistic quantum scattering
I am reading Principles of Quantum Mechanics by Shankar, here's a derivation I am puzzled.
To evaluate probability of particle entering detector in some solid angle, using $S$-matrix and Fermi's ...
6
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0answers
349 views
Upper bounds on phase space momenta
Suppose I wish to calculate the phase space volume for the process $\overline{X}X \to A_1 A_2 A_3 A_4 A_5$ in the CM frame of $\overline{X}, X$ so that $\sqrt{s} = 2m_X$. The volume is given by
$$
V \...
3
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0answers
68 views
Dark matter thermally averaged cross section
I'm trying to rederive the results of this classic paper:
https://doi.org/10.1016/0550-3213(91)90438-4
to find the thermally averaged cross-section. I am struggling with a change of variables and ...
0
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0answers
15 views
Proving that the potential is continuous across a dielectric interface
Panofsky and Phillips' Classical Electricity and Magnetism. Chapter 2, section 2.
It is proven for an interface between two dielectrics 1 and 2, with $\mathbf n$ being the normal vector from 1 to 2, ...
1
vote
1answer
83 views
Gaussian integrals with gamma matrices in their exponents
I should evaluate Gaussian integrals in the 1+1 Minkowski space, which read
$$
I_{1}= \int d^{2}k \, {\rm Tr}\big[ \gamma^{5} \gamma^{\eta} \gamma^{\kappa} e^{\alpha k^{\mu}k_{\mu} + \beta \gamma^{\...
2
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0answers
64 views
Loop integrals in particle physics: Odd number of momenta and tensor decomposition
Given the following two-loop tensor integral,
$$
\int \frac{\text{d}^D k_1}{(2\pi)^D} \int \frac{\text{d}^D k_2}{(2\pi)^D} \frac{k_1^\mu k_1^\nu k_2^\rho}{(k_1^2-a)^\alpha (k_2^2-b)^\beta ((k_1 + ...
1
vote
1answer
113 views
$φ^3$ theory: three-point, one-loop diagram ($D=4$)
I am attempting to calculate the one-loop diagram for the $φ^3$ three-point function, which is equal to (dropping a factor of $-i$): $$g^3 \int_E \frac{1}{(2 \pi )^4} \frac{1}{(k+q)^2+m^2} \frac{1}{(k-...
0
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0answers
35 views
How to know if a measure is Lorentz invariant? [duplicate]
In a Pr. Badis Ydri's book, 'A Modern Course on QFT', the following integral is said to imply that $d^3p/2E_p$ is indeed a Lorentz invariant. How so ? Does it have to do anything with the notion of &...
0
votes
2answers
88 views
Deriving the time needed until Thermal Equilibrium
Two materials of mass $m_1$ and $m_1$, specific heat capacity $C_1$ and $C_2$ and temperature $T_1 > T_2$ are separated by a conductive solid with thickness $L$, area $A$ and heat conductivity $k$ ...
0
votes
1answer
40 views
Change in temperature of overflowing container
Mixing of identical fluids at different temperatures is simple, as per here:
https://physics.stackexchange.com/a/24433/290018
We have a slightly different situation in that the container is ...
1
vote
2answers
93 views
Why does $k_\mu k_\nu \to k^2 \eta_{\mu\nu}/d$ work in QFT calculations?
When doing calculations of Feynman diagrams in QFT I've seen a trick used that goes something like this
$$\int \frac{d^dk}{(2\pi)^d} \frac{k_\mu k_\nu}{f(k^2)}\quad\longrightarrow\quad\int \frac{d^dk}{...
2
votes
1answer
106 views
Transition amplitude integral and causality
I was trying to prove that Quantum Mechanics violates causality.
To do that, I started by computing the transition amplitude between the fixed position $x_0$ and an arbitrary position $x$, during a ...
1
vote
1answer
89 views
Operator integral [closed]
Consider the following integral:
\begin{equation}
L =2 \int_{0}^{\infty} d t \exp\left\{-\hat{\rho}_{\lambda} t\right\} \partial_{\lambda} \hat{\rho}_{\lambda} \exp\left\{-\hat{\rho}_{\lambda} t\right\...
1
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0answers
36 views
Infinite dimensional generalization of the fundamental theorem of calculus
Can we generalize the fundamental theorem of calculus
$$\frac{d}{dx} \int_0^x f(t) dt = f(x)$$
to path integrals
$$\frac{d}{dx} \int_{q(0)=0}^{q(1)=x} \mathcal F[q] \, D q = \, ?$$
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0answers
34 views
Understanding Variational Integrators for simulating Lagrangian mechanics
so I'm mostly a self-taught physicist, hence my general knowledge/understanding is a bit lacking.
I'm trying to understand a bit better, on the intuitive level especially as well on the algorithmic, ...
0
votes
1answer
21 views
Why do I get different results in the draining tank problem?
This is a sketch of the situationWe have to do a small mathematical paper for our school in which we wanted to describe the water that flows out of the cylinder with a differential equation. We also ...
1
vote
2answers
61 views
Finding the centre of mass in polar coords with double integrals
The centre of mass of a body can be found using the general formula:
$$
\bar{\boldsymbol{r}} = \frac{1}{M} \int \boldsymbol{r} \ \mathrm{d}M
$$
(RHB, p. 195)*. When I try to use this method with polar ...
1
vote
1answer
24 views
Euler-Maclaurin-Formula and Finite Size Scaling
I am reading the book "Quantum Inverse Scattering Methode". In this book and in many other papers one looks at finite-size scaling. In this methode one uses often the Euler-MacLaurin-...
1
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0answers
53 views
Integration by parts of covariant derivatives in QED
I am reading Sidney Coleman's QFT ch. 27 (in particular Eq. (27.73)) where he said that we can use integration by parts to write the term in the action
\begin{equation}
\int d^4x (\mathcal{D}^{\mu} \...
0
votes
0answers
31 views
Aharanov-Bohm Effect Gradient of Line Integral
In Griffiths' Quantum Mechanics 2nd edition section 10.2.3 the phase
$$g(\mathbf{r}) = \frac{q}{\hbar}\int_{O}^{\mathbf{r}}\mathbf{A}(\mathbf{r}')\cdot d\mathbf{r}'$$
is defined. It is noted that this ...
