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Questions tagged [homework-and-exercises]

Applies to questions of primarily educational value - not only questions that arise from actual homework assignments, but any question where it is preferable to guide the asker to the answer rather than giving it away outright. Please READ THE GUIDANCE IN META before asking homework-like questions.

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1k views

How to apply the Faddeev-Popov method to a simple integral

Some time ago I was reviewing my knowledge on QFT and I came across the question of Faddeev-Popov ghosts. At the time I was studying thеse matters, I used the book of Faddeev and Slavnov, but the ...
12
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0answers
320 views

The dual of a surface element in 4-space

In reading the classic text, "The Classical Theory of Fields", Third Edition, by Landau and Lifschitz, I found an "obvious" statement not so obvious to me. It is on p.19, the statement of the ...
11
votes
0answers
192 views

Simple argument for unexpected behavior in SUSY model

Consider a supersymmetric theory with 3 chiral superfields, $X, \Phi_1$ and $\Phi_2,$ with canonical Kahler potential and superpotential $$ W= \frac12 h_1 X\Phi_1^2 +\frac12 h_2 \Phi_2\Phi_1^2 + fX.$$ ...
10
votes
0answers
840 views

Can spheres leaking charge be assumed to be in equilibrium?

I am struggling with the following problem (Irodov 3.3): Two small equally charged spheres, each of mass $m$, are suspended from the same point by silk threads of length $l$. The distance between ...
7
votes
0answers
229 views

Path Integral on Feynman Hibbs: Interaction of EM field and matter, how can we get to equation (9.68) from (9.67)?

On Feynman Hibbs "Quantum Mechanics and Path Integrals", the equation (9.67) describe the transition amplitude of the matter (for example an atom) to go from the state $M$ to the state $M$ when it ...
6
votes
0answers
123 views

Sphere half-submerged on a dielectric

This is a classic problem in electrostatics with a twist, so I thought it could be useful for others here. I did have a problem with the integration at the end, but on general I think the idea can ...
5
votes
0answers
477 views

Two-point function of a free massless scalar field in Euclidean space-time

Let $\phi(x)$ be a free massless scalar field on $d$-dimesnional space-time with Euclidean metric. I am interested in the operator formalism, i.e. $\phi(x)$ is an operator satisfying $\Delta \phi=0$ ...
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0answers
419 views

Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( \...
5
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0answers
2k views

How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
5
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0answers
846 views

Goldstone modes and Heisenberg model

The ideia is to show that, because of Goldstone modes, 2d systems are quite different from 3d ones. So, considering the Heisenberg model, I'll post here what I'm asked to and my current thoughts on ...
4
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0answers
122 views

Understanding conformal mapping in electrostatics

I'm trying to understand the use of conformal mapping to solve problems in electrostatics. To better understand the idea, I'm trying to learn how to solve this example (but you can propose any other ...
4
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0answers
147 views

Free Field Realization of Current Algebras and its Hilbert space

I have some conceptual confusion regarding the interplay between current algebras, their free field representation and the Hilbert space generated from it. Let's sketch a simple example, $\mathfrak{...
4
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0answers
65 views

Equilibrium points of three masses on a rigid spring ring with gravity

I'm trying to find the equilibrium points of a given system using Lagrangian mechanics (the system is still not rotating at the beginning). should I find the diagonal matrix for the characteristic ...
4
votes
0answers
303 views

Goldstein Classical Mechanics exercise 5.17 (3ed) conceptual

I am having difficulty understanding a concept in the "Heavy symmetric top" type of problems. I will include all of my efforts as to hopefully have someone easily point out what it is that I'm missing....
4
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0answers
186 views

Feynman rules for this perturbative expansion in Grassmann variables

I'm given the integral $$ Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2} w_i(x) ...
4
votes
0answers
287 views

How can I see where this formula for a general vertex factor comes from?

I have been reading Srednicki from the beginning and doing all the exercises, and I hit a big roadblock at Q10.4, as I can't seem to figure out what Srednicki is doing in his solution. Luckily, I ...
4
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0answers
214 views

1D Quantum scattering from $V(x) = e^x$

Define $V(x) = e^{x}$, $x \in \mathbb{R}$ and consider the Hamiltonian $H = - \frac{d^2}{dx^2} + V(x)$. The eigenvalue problem is $$ -\psi''(x) + e^{x} \psi(x) = E \psi(x)\,, \quad x \in \mathbb{R}\,. ...
4
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0answers
92 views

Constructing the Kruskal diagram for a 2-dim metric of the following form

We are given $ds^2 =- \frac{du\,dv}{M - uv},$ where $$v=t+x\, \qquad u= t - x\qquad -\infty < t, x < \infty$$ and $M$ is a positive constant. The Riemann curvature tensor is proportional to $\...
4
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0answers
188 views

Generating function of point transformation

I am asked to show that the generating function corresponding to a point transformation in Lagrangian mechanics can be taken as null. The point transformation consists of $$ Q_i=Q_i(q,t), $$ and ...
4
votes
0answers
132 views

Where is wrong in the derivation about adiabatic theorem

There is a homework in Quantum Mechanics which is about adiabatic theorem. Let us argue where is wrong in following derivation. $$i \partial_t |\psi(t)\rangle = H(t) |\psi (t)\rangle \tag{1}$$ if $\...
4
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0answers
945 views

The most probable value for x for an electron in the ground state in H atom

To find the most probable radius of an electron in the ground state of a H atom you simply take the derivative of the probability distribution to find the value of r such that p(r)=max; this occurs ...
4
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0answers
55 views

LFRW Universe - equivalence principle and Hubble flow

After a suitable change of coordinates, the metric for the (flat) FLRW universe becomes $$ ds^2 = (1 + 2 \Phi(\vec{x},t))dt^2 - (1-2\Psi(\vec{x},t))d\vec{x}^2, $$ where $$ \Phi(\vec{x},t) = -\frac{1}...
4
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0answers
157 views

How does one extract classical results from QED Lagrangian?

A well known result from classical electrodynamics for the total energy density of a charged spherical shell is, $$ u = \frac{3e^2}{32 \pi^2 \varepsilon_0 r^4}, $$ How would you extract that result ...
4
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0answers
632 views

Landau quantization: degeneracy of first level

In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way: Note: The electron spin is not considered. Number of states of a free ...
4
votes
0answers
614 views

Scalar Curvature of a Conformally Flat Metric

Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
4
votes
0answers
120 views

Vacuum expectation value for 2 point fermionic field

I was trying to compute $\langle 0|T(\psi_\alpha(x)\bar\psi_\beta(y))|0 \rangle$, i.e., the 2-point function for the Dirac field. While, I could easily compute, $\langle 0|\psi_\alpha(x)\bar\psi_\beta(...
4
votes
0answers
343 views

Material preventing nucleation, why is it not used for soda container?

I would like to know the answer to the question "why do materials preventing heterogeneous nucleation of $CO_2$ aren't used for soda bottles and glasses?". Two possible answers so far that I thought ...
4
votes
0answers
165 views

Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
4
votes
0answers
461 views

How to get the inverse of the propagator?

For a free EM Lagrangian, the propagator is as below in momentum space: $$ S[A]=\int d^4kA_{\mu}(k)\underbrace{[-k^2g^{\mu\nu}+k^{\mu}k^{\nu}]}_{M}A_{\nu}(k). $$ It is easy to calculate the $\det(M)$ ...
4
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0answers
185 views

Angle sum of triangle in Schwarzschild solution

Curvature of space is often intuitively explained as angles of a triangle not adding up to 180 degrees. My questions concerns that. Suppose you have a perfectly spherical star of uniform density - so ...
4
votes
0answers
104 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
4
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0answers
182 views

Axion Model Field Theory Problem

This is a homework problem for a field theory class dealing with an axion model. Originally, we are given that $$S[a]=\int_Md^4x \frac{1}{2}(\partial_{\mu}a(x))^2$$ has a continuous global ...
4
votes
0answers
203 views

S-Matrix Generating Functional (Problem 4.1 in Weinberg)

I'm currently working through Weinberg's QFT book, but I'm somewhat stuck at problem 4.1, which states: Define generating functionals for the S-matrix and its connected part: \begin{equation} F[\...
4
votes
0answers
332 views

Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ d}...
4
votes
0answers
362 views

Vertex for quartic interaction of complex scalar multiplet

Since I'm new to QFT and I tend to do a lot of errors during calculations, I would like you to tell me if I got the four-point vertex of the quartic interaction with a multiplet of complex scalar ...
4
votes
0answers
2k views

Killing vectors for 2-sphere as generators of $SO(3)$ symmetry

How to get Killing vectors in a form of generators of $SO(3)$ group symmetry? By using Killing equations for metric $ds^{2} = d\theta^{2} + \sin^{2}(\theta^{2}) d\varphi^{2}$ I got $$ \varepsilon_{\...
4
votes
0answers
267 views

About Dirac equation in curved spacetime (spherical)

I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta d\...
4
votes
0answers
144 views

Polchinsky's Evaluation of the One Loop String Path integral

I try to evaluet the matrix M in the Polchinsky's article(Communications in Mathematical Physics,1986, Volume 104, Issue 1, pp 37-47,"Evaluation of the one loop string path integral",Joseph Polchinski)...
4
votes
0answers
332 views

Angular momentum of particle in dipole magnetic field

Basically I'm just trying to find the expression for the angular momentum of a particle of mass $m$ and charge $q$ in a dipole magnetic field. In cylindrical coordinates, $\vec{v}=v_{\rho}\hat{\rho}+...
3
votes
0answers
67 views

A question about writing a wave as the superposition of other waves

In quantum mechanics, if a wave function is the superposition of many wave functions, it can be written as $$\frac{1}{\sqrt{2\pi}}\int g(k)e^{i(kx-\omega t)}dk \, .$$ On page 24 of Quantum Mechanics ...
3
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0answers
26 views

Do the distances from directrix and focus in a parabola shaped trajectory have a physical meaning?

According to Wikipedia, Galileo experimentally discovered that uniformly accelerated motion describes a parabolic trajectory using an inclined plane. Today we are taught at school that a ball thrown ...
3
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0answers
38 views

Star-shaped phase space

I am asked to classify the following phase spaces. The phase spaces 2 and 3 are fairly simple (harmonic oscillator and a elastically reflected particle). However, I fail to classify the phase space ...
3
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0answers
112 views

Examples of central forces on the path of orbit?

In solving a problem from Goldstein (3.13), I solved for multiple properties of a circular orbit with the attractive central force where the path of orbit crosses the point of the force (at origin). ...
3
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0answers
158 views

General Commutator for Spherical Tensors (reformulated)

Edit: I think what I was looking for wasn't very well understood, so I'm reformulating my question to make it clearer. Hope this helps. In the book of "Angular Momentum: An Illustrated Guide to ...
3
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0answers
117 views

Derivation of Equation of Trajectory around a Kerr Black Hole

I was trying to derive equation of motion for test particle around a Kerr black hole. My work is as follows: The Kerr metric is as follows $$ \mathrm ds^2 = -\left(1-\dfrac{2Mr}{\rho^2}\right)\...
3
votes
0answers
100 views

Torricelli's Law and number of holes

Trying to determine if the number of holes at the bottom of a bucket will change time it takes for water to empty the bucket. Looking at the equation, it would seem that as long as the area of the ...
3
votes
0answers
210 views

Gaussian integral in momentum space

My question is related to p. 353 of Altland and Simon (section 6.7) which concerns about the following field integral where $\beta = 1/T$ and $V_n$ is defined in the following way: It seems to be a ...
3
votes
0answers
247 views

Deriving a Massive Propagator from a Massless Propagator

I'm trying compute something I already know the answer to in order to test myself and gain confidence in my QFT computational skills, but I'm not getting the right factors. The text I'm following and ...
3
votes
0answers
115 views

LSZ Reduction Formula and Wavepackets in Peskin

In Section 7.2 of Peskin and Schroeder, the LSZ Reduction formula is discussed. I can follow the discussion leading up to the equation after (7.41) [at the bottom of Pg. 225]: $$\sum_\lambda \int\frac{...
3
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0answers
95 views

Valid Bogoliubov transformation?

Is it possible to diagonalize the Hamiltonian: $H=c_1 c_2^\dagger+c_2 c_1^\dagger +\delta(c_1c_2+c_2^\dagger c_1^\dagger ) $ using the Bogoliubov canonical transformation, where the $c_1,c_2$ are ...