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4
votes
4answers
261 views

Evaluation of expectation values

I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the ...
0
votes
1answer
66 views

How to integrate two-electron interaction in He? (variational method)

A probe wavefunction in the variational method is $$\psi(r_1, r_2) =\frac{\alpha^6}{\pi^2}e^{-\alpha(r_1+r_2)}$$. In $\left<\psi \right|H\left|\psi\right>$ with $$H = \frac{p_1^2+p_2^2}{2m} - ...
3
votes
1answer
77 views

What does $\int_C V \,d\mathbf{l}$ mean?

What does $$\int_C V \,d\mathbf{l}$$ mean? I initially thought it was simply a line integral around $C$, that is, if $\mathbf{r}: [0,1] \longrightarrow \mathbb{R}^3$ is a paremetrization of $C$, then ...
4
votes
0answers
95 views

Lebesgue integration [closed]

I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about ...
0
votes
2answers
62 views

Non-uniform circular motion, computing the angle

There is an object moving in circle, with this law: $$ \alpha = -k^2 \theta $$ With $ \alpha = \frac{ d \omega }{dt} $, and $ \omega= \frac{d \theta}{dt} $, $\theta$ = angle, $k$ positive ...
0
votes
1answer
45 views

Movement with non-constant acceleration [duplicate]

Suppose we have a material point. If it is moving from position $X_0$ with initial velocity $V_0$ and constant acceleration $A$, then from elementary physics course I remember that its movement is ...
1
vote
2answers
218 views

Position dependent speed, how to compute position

I can't solve a problem: $ A= 0.5 (ms)^{-1}$, $ x_0 = 0.5 m $, $v(t)= A \cdot x^2 $, I have to compute the position at $t=3$ ($x_0$ is the initial position). So my guess is that I should be able ...
5
votes
4answers
558 views

How are electric flux calculations not double integrals?

A disk of radius 0.10 m is oriented with its normal unit vector $\hat{n}$ at 30$^{\circ}$ to a uniform electric field $\vec{E}$ of magnitude 2000 N/C. What is the electric flux through the disk? ...
1
vote
1answer
74 views

Moment of inertia of a sphere

I'm looking at sample calculations of moment of inertia of a sphere here. In the first example (disc method), it has the integral as $dI = \frac{1}{2}r^2 \,dm$, while in the second example (shell ...
0
votes
0answers
31 views

Self-inductance of a toroidal inductor

I am trying to determine the self-inductance of a toroidal coil of mean radius $R$ with $N$ loops of radius $a$ with a current $I$ flowing within them. I have calculated the magnetic field by noting ...
5
votes
3answers
201 views

Integral form of Gauss's law for magnetism from Stokes' theorem?

How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?
1
vote
1answer
41 views

Integrating for velocity [duplicate]

Trying to determine velocity of a falling body with respect to traveled distance and initial speed. I've been provided with the following equation for acceleration as a function of distance and the ...
0
votes
1answer
80 views

Line Integral Parameterization

In math, I was taught to parameterize a scalar/vector line integral. In physics, I remember doing problems where I didn't parameterize the problem and it still came out correct. So, by ...
3
votes
1answer
153 views

Free particle propagator - Evaluating Integral [closed]

In path integral formalism, when evaluating the free particle propagator, we obtain the functional integral of the form, $$ K_0 = \lim_{n\rightarrow\infty} \bigg( \frac{m}{2\pi ...
3
votes
1answer
263 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
1
vote
1answer
37 views

When is the speed specified for an object experiencing an exponential force?

So this is the question given in my text book: A particle of mass m is at rest at the origin at time $t = 0$. It is subjected to a force $F (t) = F_0e^{–bt}$ in the $x$ direction. Its speed ...
1
vote
3answers
194 views

Integration constants in Maxwell's equations (ambiguousness?)

In classical electrodynamics, if the electric field (or magnetic field, either of the two) is fully known (for simplicity: in a vacuum with $\rho = 0, \vec{j} = 0$), is it possible to unambiguously ...
2
votes
3answers
115 views

Calculating the Potential from the E-Field

I find that often times I'll be tripped up by questioning whether or not I can do something mathematically, and be unable to come up with a satisfying answer. This is, unfortunately, one of those ...
3
votes
0answers
49 views

The question about MTW 4-momentum integral expression and lorentz nature

In section 5.8 of Misner, Thorne, and Wheeler's "Gravitation" there is a proof that 4-momentum determined as $$ \tag 1 p^{\mu} = \int T^{\mu 0}\,\mathrm{d}^{3}\mathbf r , \quad \partial^{\mu}T_{\mu ...
5
votes
2answers
200 views

Evaluate $1$-loop contribution to the $4$-point Green's function

I am trying to evaluate the following integral \begin{equation} I = \int \frac{d^d p_\text{E}}{(2 \pi)^d} \frac{1}{(p_\text{E}^2+m^2)((q_\text{E}-p_\text{E})^2 + m^2)} \tag{1} \end{equation} where ...
1
vote
1answer
300 views

Superficial Degree of Divergence for Feynman Diagrams

The superficial degree of divergence for a diagram is defined as the power of $k$ in the nominator minus the power of $k$ in the denominator. It is written to be equal to $4\times$ ...
6
votes
1answer
220 views

Integral in $n$−dimensional euclidean space

I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help ...
1
vote
2answers
150 views

Is there a physically motivated “trick” to evaluate this convolution?

I'm working on adapting some of the formalism in this paper to a system I'm working with. The part I'm interested in amounts to convolving a density profile $\rho(r)$ with a smoothing kernel ...
1
vote
1answer
92 views

Prefactor for phase space integration

When calculating the canonical partition sum, we had the following: $$ Z_\text C = \sum_{\vec p} \sum_{\vec x} \exp(-\beta H(\vec p, \vec x)) $$ Now, since $\vec p$ and $\vec x$ are pretty much ...
2
votes
3answers
212 views

Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s ...
0
votes
1answer
107 views

Making a cut trough a center of mass, can the masses of the pieces be equal?

Let's say point $P$ is the center of mass of an irregularly shaped object. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same ...
3
votes
1answer
121 views

Correction to Period of a Pendulum

In one derivation of the corrected period of a pendulum, we started off like so: The mass has a height $y$ given by $l(1-\cos \theta )$. $E = K + E \rightarrow \frac{1}{2}ml^2 \dot{\theta}^2 + ...
11
votes
1answer
573 views

Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
4
votes
2answers
488 views

Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. Example 6.1 During a collision with a ...
5
votes
0answers
194 views

The commutator of scalar field [closed]

I have a real scalar field which is given by the propagator as: $$[\phi(x),\phi(y) ] =\int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_0} (\exp(-ip\cdot (x-y)) -\exp (ip\cdot (x-y)))$$ And I am asked to show ...
2
votes
1answer
236 views

Integrals over grassmann numbers

I want to prove an identity from Peskin&Schroeder, namely that $$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m^* \theta_l \exp(\theta_j^* B_{jk} \theta_k)=\det(B) ...
2
votes
1answer
137 views

Integration on a general equation for instantaneous angular acceleration

An equation for instantaneous angular acceleration is given as: $$ \alpha \equiv \lim_{\Delta t\to0}\frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt} $$ The text I am reading says writing this ...
0
votes
1answer
103 views

Integration with Grassmann variables

How to show that $$ \int d\Psi d\bar {\Psi}e^{i \int d^{4}x\bar {\Psi} \hat {A} \Psi} = det (\hat {A})? $$ $\Psi , \bar {\Psi}$ refers to Dirac spinors (the second is $\bar {\Psi} = ...
4
votes
1answer
81 views

Question about exterior derivatives

I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge ...
0
votes
2answers
88 views

Number of 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$ (μν|λσ)=∫dr⃗_1dr⃗_2ϕ_μ(r⃗_1)ϕ_ν(r⃗_1)r^{−1}_{12}ϕ_λ(r⃗_2)ϕ_σ(r⃗_2) $$ I am told that for a basis set of size $K=100$, there are ...
0
votes
1answer
469 views

Derivation of Moment of Inertia and centre of mass?

In the equation above for the MI for a rod, why are we taking the limits from -l/2 to l/2? And why doesn't the integral doesn't include the centre of mass?
0
votes
2answers
116 views

Function with poles/singularities; Polynomial approximant has no poles

I don't know if i should ask this question or if it makes too much sense. My knowledge of this topic is quite incomplete, so please bear through with me. Any insights are appreciated. A function ...
0
votes
2answers
54 views

How to get $t$ from $a(v)$?

I read what if we have acceleration given as a function of velocity we can calculate time as $$t(v) = t_0 + \int_{v_0}^{v} \frac{dv}{a(v)}.$$ Why?
2
votes
1answer
619 views

Momentum variance in momentum space for particle in a box

My assignment asks me to compute the momentum space wavefunction of the nth energy eigenstate of the particle in a one-dimensional infinite square well, then "show that your result is in agreement ...
2
votes
2answers
84 views

Integration of 3-momentum

During a lecture that I missed, I was trapped when the lecturer uses the relation $$dp_x~ dp_y ~dp_z ~=~d^3\mathbf{p} ~=~ 4\pi p^2 dp.$$ Can I know how is this relation derived please?
7
votes
2answers
201 views

A problematic integral in calculating the entanglement entropy in 1+1 D free massive bosonic field theory

I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the entanglement entropy of 1+1D quantum field theory (arXiv). The identity is given ...
2
votes
3answers
380 views

Electrostatic energy integral for point charges

The electric energy stored in a system of two point charges $Q_1$ and $Q_2$ is simply $$W = \frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$$ where $a$ is the distance between them. However, the total ...
1
vote
1answer
824 views

Gravitational force exerted by a rod on a point mass

I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well. A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
0
votes
0answers
206 views

Making a 3D physics engine, realistic? If so, where do I begin my research?

As a programming (technology), physics and math project in school, I'm considering programming my own 3D physics engine as a learning exercise. The physics engine should then be able to be used in a ...
1
vote
0answers
25 views

Hemisphere irradiance

How do I calculate sky irradiance from radiance (L) from a hemisphere above a surface which is tilted relative to the normal (x=0,y=0,z=1). I have L as a function of zenith (0 to 180deg) and azimuth ...
4
votes
1answer
188 views

How do I calculate integral analytically for small $k$?

In a Heisenberg antiferromagnet, the dispersion relation is \begin{equation} \omega_{\mathbf{k}} =JSz\sqrt{ 1-\gamma_{\mathbf{k}}^2} \end{equation} where ...
7
votes
3answers
739 views

Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. ...
3
votes
3answers
315 views

Physical significance of getting an non-integrable function in an equation

I just found out during my Calculus course in High School, that there exist functions which cannot be integrated. Then I thought that I come across a lot of integrals while solving Physics ...
1
vote
2answers
292 views

Calculating the expectation value for kinetic energy $\langle E_k \rangle$ for a known wave function

I have a wavefunction ($a=1nm$): $$\psi=Ax\exp\left[\tfrac{-x^2}{2a}\right]$$ for which I already calculated the normalisation factor (in my other topic): $$A = \sqrt{\frac{2}{a\sqrt{\pi a}}} = ...
0
votes
1answer
132 views

Centrifugal force on tilted object

The centrifugal force acting on a revolving particle with negligible size is $\frac{mv^2}{r}$. What if the size is not negligible? Say we are talking about a large homogeneous circular disc, so its ...