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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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\...
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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 ...
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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) ...
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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 {\...
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$\int_{-\infty}^{\infty} |\psi(x)|^2 ~ dx = 1$ 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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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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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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 ...
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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 ...
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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) ...
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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 ...
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1answer
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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 ...
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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 ...
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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} ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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 \...
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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 ...
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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, ...
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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^{\...
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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 + ...
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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-...
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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 &...
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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 $k$ ...
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1answer
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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 ...
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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}{...
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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 ...
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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\...
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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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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, ...
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1answer
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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 ...
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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 ...
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1answer
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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-...
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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} \...
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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 ...

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