For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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Showing $\frac{d^3\underline{k}}{2\omega_{k}(2\pi)^3}$ is Lorentz invariant [duplicate]

Show $\frac{d^3\underline{k}}{2\omega_{k}(2\pi)^3}$ is Lorentz invariant. Hint: try to evaluate $\int dk_0\delta(k_0^2 - M^2)\theta(k_0)$ where $M^2 = \underline{k} + m^2$ My attempt is as ...
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48 views

Asymptotic behaviour of the propagator for a scalar field

When discussing causality in Chapter 2 of Peskin & Schroeder a couple of equations giving the asymptotic behaviour of the propagator for a scalar field appear: $$ \text{If} \,\, x^0-y^0=t, \, \, \...
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105 views

Electric field due to charged disc, on the plane of the disc [closed]

A standard problem in finding the field is that of a uniformly charged disc, on its axis, but for this problem I'm supposed to find the potential and the field on the edge of the disc, i.e. in the ...
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177 views

Recovering QM from QFT

Reading through David Tong lecture notes on QFT. On pages 43-44, he recovers QM from QFT. See below link: QFT notes by Tong First the momentum and position operators are defined in terms of "...
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4answers
88 views

Proving that the Center of Mass of a solid sphere is at the origin [closed]

For my own knowledge and to understand why. I am trying to convince myself that the center of mass for a rigid solid sphere is at the origin (0,0,0). I begin with the basic definition of CM $$\...
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48 views

Integration to find general solution of free particle [closed]

I was attempting problems in Griffiths Intro to QM when I came across the following: A free particle has the initial wavefunction: $$\Psi(x, 0) = Ae^{-ax^2} \, .$$ Find $\Psi(x, t)$. I ...
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74 views

How does Galitskii's integral converge?

In V M Galitskii's 1958 paper "$\textit{Energy spectrum of a non-ideal Fermi Gas}$," he builds the following integral as part of a longer expression for the real part of the self-energy (eqn 26'). It ...
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90 views

Do wave functions really belong to $L^2$ space, or do we need to restrict our physical Hilbert space even further?

I am beginning to study quantum mechanics and I got stuck right at the beginning. I am trying to prove that the time derivative of the expected value of momentum of a particle is the (negative) ...
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13 views

Calculate distance with variable acceleration [duplicate]

I know the time and I know the force that acts horizontally on a particle. The problem is that the force is F=k(D-V), where k and D are constants. So at the beginning where the particle has zero ...
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75 views

Cancelling the partial of a coordinate, $\partial q$, with the element of a coordinate, $dq$ in Physics [closed]

I've seen in many books, things like this ( I will be simple ): $$\int \frac{\partial f}{\partial q} dq=\int df$$ where $f$ is a function of $q$ and other coordinates. I just axiomatically assumed ...
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27 views

How to intergate the cross section over the surface of a detector?

My beam moves along the $X$ axis. I know the cross section $\frac{d \sigma}{d \Omega}$. My rectangular detector is perpendicular to the $XY$ plane and its surface is perpendicular to the line ...
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61 views

Fraunhofer diffraction at circular aperture - integration of bessel function

I'm trying to understand the calculation here: http://en.wikipedia.org/wiki/Fraunhofer_diffraction_(mathematics)#Circular_aperture for the solution by integration, but I plain and simple fail to see ...
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57 views

How do you interpret definite-integrating of both sides of an equation?

Consider a particle is moving at an acceleration of $a=f(s)$ [$s$ stands for particle's location) if we have the initial velocity of a particle then what is the final velocity? So... we know that $...
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46 views

Avoiding a singularity in the simulation of a spherica pendulum

I didn't know whether to put this here or in StackOverflow - so I open to answers just telling me to go there! I am looking to simulate the motion of a spherical pendulum. The Lagrangian is $$ \...
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30 views

Calculate the parabollic trajectory

In order to calculate the parabollic trajectory (or ballistic) I need intrinsic equation. Now I need to integrate this equation but I don't know how. I would appreciate so much if you help me!
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36 views

Why does the angular average of the divergence result in a factor 1/3?

In going from eq.(6.13) to (6.14) in http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf it is assumed that $$\displaystyle\int\dfrac{d\Omega_{\vec{p}}}{4\pi}\,(\hat{p}\cdot\hat{p})\...
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211 views

Can you take the integral of $ d^2x\over dt^2$? [closed]

I am messing around with physics problems, and as silly as this maybe how do you take the integral of $$\int_0^\infty xd^2x$$ For example taking Newton's Second law $F=ma$ $$ F=m{d^2x\over dt^2} $$...
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344 views

Gauss's law… if the integral defining $\boldsymbol{E}$ diverges?

I have been told (here) that, under particular conditions, the electric field produced by a charge present in space $D$, defined by $$\boldsymbol{E}(\boldsymbol{x})=k\int_D\frac{\rho(\boldsymbol{y})}{\...
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43 views

Why is the angular average of the direction of the momentum squared a third instead of one?

In the article http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf (in going from eq. (6.13) to eq. (6.14)) it is stated that $$\langle\hat{p}\cdot\hat{p}\rangle_{\hat{p}}=\dfrac{1}{3}...
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134 views

Mathematical confusion in quantum mechanics

During a class about Ehrenfest theorem, my teacher use an equation to proceed its derivation (to prove $\frac{d<r>}{dt}=\frac{<p>}{m}$ ) and that is: $$\int{x\psi\nabla^2\psi^*}d\tau=\int{\...
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51 views

Calculating force repulsive between a line segment and a point [closed]

As shown in the diagram below, let there be a line segment $L$ and a point $p$ such that they both repel each other electrically. Let $\vec{v} = \vec{p} - \vec{q}$ be the vector from $p$ to the point ...
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166 views

Book on gamma functions with applications in physics

I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions. Does anybody know ...
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100 views

Angular frequency integral to wavelength integral

I have this integral $$ \bar{\omega}^2 = \int\limits_{-\infty}^{\infty}\omega^2|\tilde{F}(j\omega)|^2\frac{d\omega}{2\pi} $$ and I want to convert it to wavelength domain $\lambda $. I know that the ...
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48 views

An impossible relationship between variables

This book chapter defines $CV$ as follows: $$ CV^2 = 2\pi\nu^2\int^\frac{V_{th}-V_{ss}}{\sigma_V}_\frac{V_{r}-V_{ss}}{\sigma_V}dxe^{x^2}\int^x_{-\inf}dye^{y^2}(1 + erf(y)) $$ However, Figure 15.2 in ...
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42 views

Solving for the firing rate of a model neuron

I'm trying to decipher Figure 15.2A on page 442 of this book chapter. It plots the firing rate $\nu$ of a model neuron against its mean current $\mu_c$. The equation used to calculate the firing ...
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79 views

How to approach proofs in Electricity and Magnitism that involve integrals?

I have read through both Franklin and Jackson's Electromagnetism books and I am able to understand the different proofs involving integrals but when I try to re-derive them on my own later I am always ...
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112 views

Using the dipole moment to calculate the electric potential

I have the following defintion of dipole moment given: $$ \vec{p} := \int\vec{r}'\rho(\vec{r}')dV $$ Where $\rho$ is a charge density function and $\vec{r}'$ traces out the body of charge. I am ...
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73 views

Why don't both equivalent forms of this delta function give the correct answer?

I am a bit confused on a basic problem involving a Dirac delta function being integrated over in a multiple integral. The original problem is to find the probability distribution in position-momentum ...
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Formula relating sum of values of a function to its integral

I came across the above formula in some quantum mechanics lecture notes explaining the Casimir effect. Anyone seen it before if so could you please tell me its 'name'. B refers to the Bernoulli ...
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118 views

Where is error in this method of finding volume of sphere using integration? [closed]

Where is error in this method of finding volume of sphere using integration?
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325 views

Why is the $dx$ right next to the integral sign in QFT literature?

I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?
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68 views

Integral over a surface in kinetic theory [closed]

I am working with some kinetic theory. I have the distribution function $\Psi (\vec{r},\vec{p},t)$, Where $\vec{r}$ is the radius vector, $\vec{p}$ is the unit vector of orientation, and $t$ is the ...
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114 views

Convert an equation for acceleration into one for velocity

A sledge is pushed in a straight line. Assume surface is smooth. When the sledge is x distance away from the start the magnitude of its acceleration is given by $0.08e^{-4x}$ and is going in the same ...
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How to arrive on the diffraction pattern for the double slit experiment using path integrals for the Gaussian slit case?

I wish to take the path integral route to derive the diffraction pattern for the double slit experiment using the Gaussian slits as the nature of the slits. The kernel looks like: \begin{equation} K((...
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68 views

MEMS Accelometer and Gyroscope data fusion for counting cycles

I will simplify my scenario as follows: a 6-DOF MEMS gyroscope and accelerometer unit is placed on the edge of a rotating wheel and the goal is to count the number of cycles using any of the units of ...
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137 views

Problem with the Cooley-Numerov Method for Solving the Radial Nuclear Schodinger Equation in the Born-Oppenheimer Approximation

I have been trying to implement a solver for the radial nuclear Schodinger equation in the Born-Oppenheimer approximation using a similar method to R. J. Le Roy's LEVEL program[1]. I have as input a ...
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45 views

Calculating initial velocity to achieve orbital motion using verlet integration

I'm attempting to build a simple computer programme which models bodies orbiting other bodies. I've implemented a Verlet integrator (https://en.wikipedia.org/wiki/Verlet_integration) and I can ...
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175 views

Composing integrals in physics?

OK... so this problem isn't really specific... it's more of a conceptual puzzle. I've recently started using integrals while solving problems in physics (specifically Newtonian Mechanics and other ...
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47 views

Elementary question about distributive property of variation operator on an exterior product

I am trying to work out the equations of motion of a 11-dimensional supergravity action $$S = \frac{1}{2\kappa^2}\left(\gamma\int d^{11}x\sqrt{|g|}\mathcal{R} - \frac{\alpha}{2}\int G \wedge \star G -...
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Already integrated formula of magnetic field for a finite line?

I was looking for a formula to simulate a magnetic field due to a straight finite line. The closest to what I wanted to find was in these lecture notes (formula 9.11.2 page 9-50), \begin{align} \...
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65 views

Understanding speed distributions: Average speed, RMS speed from a graph?

So taking a look at this graph.... ![http://i.imgur.com/XUwpnSK.png][1] I want to express $A$ in terms of $N$ (total # of particles) and $V_0$. So I've found the piecewise function for $f(v)$ and ...
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143 views

Can you express the Feynman propagator as a limit?

At first I thought that the Feynman propagator was the limit of: $$ G(x) = \frac{1}{x^2 + i \varepsilon} $$ But if you apply the wave equation to this you get: $$ \Box G(x) = \frac{\varepsilon}{(x^...
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Electic potential due to finite rectangular plate

I am trying to find the potential at any point (x,y,z) due to a rectangular plate with a constant surface charge density. Let's assume the plate is centered on the X-Y plane and extends from -n to n ...
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59 views

Importance sampling for Coulomb potential

The integral I have to solve is: $$I=\int\int d \mathbf{r}d \mathbf{r}' \frac{\Phi(\mathbf{r})\Phi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}$$ It is a six-dimensional integral which I am going to ...
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305 views

Integration by parts to derive $d\langle x \rangle / dt$

I am reading "Introduction to Quantum Mechanics" by David Griffiths and I am having trouble understanding part of a derivation of $\frac{d\langle x\rangle }{dt}$ in section 1.5 - Momentum - of the ...
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41 views

Off-axial Field of Finite Solenoid

Regarding the computation of the off-axial field of a finite solenoid: The Radial and Z components of the off-axial magnetic field of a solenoid are given as: \begin{align} B_r &= \frac{\mu ni}{\...
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66 views

Property of the wave functions of a free particle

How can I show that the following holds? $$\langle nlm\mid \partial_z^2\mid nlm\rangle=-\int_0^{4\pi}d\Omega\int_0^{\infty}drr^2\left|\partial_z\psi_{nlm}\right|^2$$ The wave functions of a free ...
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131 views

Why doesn't $\vec{E} =\frac{1}{4\pi\epsilon_0} \int\frac{\rho \hat{r}\;dxdydz}{r^2}$ blow up at $r=0$, when $\rho$ is finite?

Electric field at $(x,y,z)$ produced by a continuous distribution of charges is given by:$$\mathbf{E}(x,y,z) =\dfrac{1}{4\pi\epsilon_0} \int\dfrac{\rho(x',y',z') \mathbf{\hat{r}} \;\mathrm{d}x'\mathrm{...
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An integral involving the Bose-Einstein distribution

I'm trying to reproduce the following calculation from the book by Fetter and Walecka (eq. 55.37 and following ones), which represents the temperature dependance of the non-condensate part of a weakly-...
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51 views

Change of variables for integral operator

One can write the operator $L=(\sqrt{1-i\partial_x^2}-1)$, as an integral, that is $$(\sqrt{1-i\partial_x^2}-1)B(x,t)=\frac{i}{4\pi^2} \int_{-\infty}^{\infty}(\omega(k_o+\kappa)-\omega(k_o))e^{i \...