# 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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### How should I get the expectation value of $r^2$ in the hydrogen atom? [closed]

I'm having trouble finding this expectation value: $$\langle r^2\rangle=\frac{C_{n,l}}{\alpha^5}\int_0^{\infty}e^{-x}x^{2l+4}[L^{2l+1}_{n-l-1}(x)]^2dx$$ Where $x=\alpha r$ and $\alpha=\frac{2}{na_o}$...
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### Moments of physical qualities other than density

If I have some genuine physical object $\Omega$ (that I can describe mathematically) and want to find its center of mass, I can compute its moments to get each coordinate of that center; for example, ...
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### Spherical harmonics integral

I've been struggling with this integral $$\int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi$$ I've tried to use the definition of the ...
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### Integrate continuity equation in QM

From Shankar's QM book pg. 166: The continuity equation for probability density in QM is $$\frac{\partial P(\vec{r},t)}{\partial t}=-\nabla \cdot \vec{j}(\vec{r},t),$$ where $P=\psi^*\psi$ is the ...
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### Momentum integral yielding $\delta$ function

I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will ...
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### How to solve this integral?

I am following some notes from a physics class on the Ising model. At some point we get to this integral \begin{equation} \frac{1}{2} \int_{\Omega_B} \frac{\text{d}^Dk}{(2\pi)^D} \ln \left[ \tau + \...
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### What is the internal potential for a spherical mass for a $r^\alpha$ potential?

Newton's shell theorem makes it easy to find the internal gravitational potential in a spherical body for a standard gravitational $\Phi(r)\propto 1/r$ potential. But the shell property does not apply ...
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### Question regarding vector calculus [closed]

Question regarding vector calculus: Mathematically, what properties does this vector field have, in order for its line integral to be equal to the area enclosed by that curve?
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### How do you integrate by parts when you have a triple integral?

I'm studying how particles of equal mass behave in a spherical cluster held intact by gravity. I will assume that the mass density $\rho(R)$ of the cluster is a function of the magnitude of the ...
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### How do you find the final velocity when acceleration is changing between two values over some distance? [duplicate]

How do you calculate a final velocity of an object when given its initial velocity and the object is accelerating between an initial and final acceleration over some given distance?
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Is it possible to have a $d=4-2\epsilon$ dimensional integral which diverges for all $\epsilon$? For example, if you get something like $$\tag{1} \int_0^1 \text{d}t\,t^{-1-\epsilon}(1-t)^{-1+\epsilon},... 2 votes 0 answers 34 views ### Solving a complex Gaussian Integral for Path Integral Formalism [duplicate] I am trying to solve the following integral,$$\int dz_1d\bar{z}_1\cdots dz_nd\bar{z}_n\:\exp(-\sum_{i,j}\bar{z}_i A_{ij}z_j), where $A_{ij}$ is an $n\times n$ hermitian matrix. I know how to do ...
I am computing a Feynman integral of a fermion bubble in an external field. The field is in the z-direction so Lorentz invariance is broken. I need to break up the $\int d^4 k$ integral into a \$\int d^...