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.

2,872 questions with no upvoted or accepted answers
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48
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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 ...
19
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1answer
413 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 ...
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0answers
224 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 $$ W= \frac12 h_1 X\Phi_1^2 +\frac12 h_2 \Phi_2\Phi_1^2 + fX.$$ ...
14
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1answer
733 views

Deriving the Poisson bracket relation of the Ashtekar variables

I'm trying to figure out how to calculate the orthogonality of Ashtekar variables with respect to the ADM hypersurface metric and conjugate momentum. $$\{{A_a}^i(x), {E^b}_j(y)\} = 8 \pi \beta \delta^...
14
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2answers
1k 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 ...
11
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3answers
8k views

Basic Thermodynamics: Quasistatic Adiabatic Process

I'm going through the exercises in a Thermodynamics book, just to revise and build my intuition. Right now, I'm working on: Show that for a quasistatic adiabatic process in a perfect gas, with ...
11
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2answers
656 views

Force problem related to adhesive and bonding

I have two PCBs (printed circuit board), and they are glued by adhesives, as show in the pictures. And the location of the adhesives are indicated on the picture (please notice that NO adhesive is ...
10
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1answer
753 views

Escape velocity for Schwarzschild metric

I can't fill in the gaps in my solution to this and assistance or a reference would be appreciated. The question begins with the straightforward derivation of the EoM for a massive particle orbiting ...
10
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1answer
1k views

Question about superconductivity

A long cylinder of radius $R$ is made from two different material. Its radius $r<r_0$ $(r_0<R)$ part is a material with superconducting transition temperature $T_1$, and its $r_0<r<R$ ...
9
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278 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) describes the transition amplitude of the matter (for example an atom) to go from the state $M$ to the state $M$ ...
9
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1answer
429 views

How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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3answers
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Drag - Dimensional Analysis / Buckingham $\pi$

I'm working on dimensional analysis and I'm having trouble. Here's a problem from my book I'm working on. I'm supposed to consider a small sphere experiencing acceleration due to gravity $g$. The ...
9
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1answer
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Analytical solution of two-level system driving by a sinusoidal potential beyond rotating wave approximation

A quantum mechanical two-level system driven by a constant sinusoidal external potential is very useful in varies areas of physics. Although the widely used rotating-wave approximation (RWA) is very ...
8
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0answers
634 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 ...
8
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1answer
517 views

Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch ...
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2answers
875 views

Express Laplace transform of voltage across a capacitor in terms of charge

In Charge Tunneling Rates in Ultrasmall Junctions section 2.1, the authors consider the problem of charge relaxation in a simple circuit shown in Figure A. The implicitly use an assumption made about ...
7
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377 views

Kraus operators for two interacting harmonic oscillators: Problem with the calculation (Ex. 8.21 of Nielsen-Chuang)

I'm working with Exercise 8.21 of the Nielsen-Chuang book on quantum information. It illustrates the amplitude-damping quantum channel by the interaction between two harmonic oscillators (the first ...
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0answers
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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 ...
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4answers
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Distinguishable, indistinguishable paramagnetic ideal gas

In the canonical ensemble, the partition function for an ideal gas is given by: $$\frac{Z}{N!}$$ The factor $N!$ accounts for the indistinguishability of the particles of the ideal gas. What ...
6
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118 views

Extra term when calculating variation in Lagrangian density under infinitesimal Lorentz transform

Consider an (active) infinitesimal Lorentz transformation: $$ x^\mu \rightarrow x^\mu + {\omega^\mu}_\nu x^\nu, $$ so that any scalar field is transformed as $$ \phi(x) \rightarrow \phi'(x) = \phi(x) -...
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225 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{...
6
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1answer
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Schrodinger Wave Functional (quantum fields) - Solving Functional Gaussian Integrals

Okay, So i'm doing some research that involves the Schrodinger representation in quantum field theory. The ground state wave functional for the Klein Gordon field is a generalized gaussian in position ...
6
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1answer
424 views

Sum of energy for 2 solids in rotation

I would like to compute the sum of energy of the following case: Two solids are turning (disks). Yellow solid is turning at $w_1$ radians per second around its center of gravity and blue solid is ...
5
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0answers
687 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 ...
5
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1answer
105 views

What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?

In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as: $a(t) = a_0$ $v(t) = a_0t + v_0$ $x(t) = \frac{1}{2}...
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331 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) ...
5
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1answer
1k views

Solving the Lindblad quantum master equation in matrix form

I have just started learning density matrix and quantum master equations, and I am given a problem set that asks to find the solution to the Lindblad equation with $H$, $L_+$, $L_-$, $L_z$, and $\rho(...
5
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598 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 ...
5
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1answer
645 views

How to understand Bloch sphere representation?

I'm really new to quantum computation. Now, I'm going through a tutorial article Quantum Computation: a Tutorial (NB: PDF). I was confused by certain points over there. So, on page 5, when the ...
5
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1answer
839 views

Deriving Yang-Mills Equations

On the same spirit of this unanswered question I am proposing this question which I have been trying for some time now. Here I'm working with dimension $n = 4$ (identifying $\mathbb H = \mathbb R^4$) ...
5
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820 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 $\Delta \phi=0$ ...
5
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4answers
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How to prove Bloch function is periodic in reciprocal lattice?

How to prove Bloch function is periodic in reciprocal lattice? I saw in some textbooks this formula: $$ \Psi_{\mathbf{k}} (\mathbf{r}) = \sum_{\mathbf{G}} c_{\mathbf{k}+\mathbf{G}}e^{i(\mathbf{k}+\...
5
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489 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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2answers
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Smallest number of quantum gates to simulate other gates?

What is the smallest number of Fredkin gates needed to simulate a Toffoli gate? What is the smallest number of Toffoli gates needed to simulate a Fredkin gate? Where the Toffoli's gate is the CCNOT ...
5
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0answers
924 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 ...
5
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2answers
322 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). ...
5
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1answer
907 views

Ray tracing in General Relativity

I would like to find out what one would see at the Schwarzschild radius of a massive non-rotating black hole, if the black hole is surrounded by a bright ring. For that, I would place the observer at ...
5
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1answer
2k views

Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)

1. The problem statement, all variables and given/known data Consider a time-dependent harmonic oscillator with Hamiltonian $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$ $$\hat{H}_0=\hbar \omega \left( \...
4
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1answer
99 views

Reaction force exerted by pulley's support

Consider an in-extensible string passing over an ideal pulley connected to a block of mass $m$ as shown: The ends are pulled with a force $F$ Am I correct in saying that the block will experience a ...
4
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0answers
82 views

Time-reversal symmetry for spinful fermions

For a continuum model of non-interacting spinful fermions with many-body Hamiltonian $\hat{\mathcal{H}}=\int_{\mathbb{R_2}}d\vec{k}\begin{pmatrix} \hat{c}^{\dagger}_{\vec{k}\uparrow} & \hat{c}^{\...
4
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0answers
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Verify Furry theorem in scalar QED

I want to verify Furry theorem in scalar QED. Consider a process with N photons. Is it correct to say that at 1 loop the two classes of Feynman diagram that contribute are the following? with $n+2m = ...
4
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0answers
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Planar Wheeled Inverted Pendulum EOM Derivation

I am building a two wheeled robot as a personal project and am having some trouble with the equations of motion. I derived a state space model but when I simulate it in scipy I get some odd behavior ...
4
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0answers
125 views

Solving scalar quantum field in 1+1D Milne space

So our line element is \begin{equation} ds^2=dt^2-a^2t^2dx^2 \end{equation} doing following coordinate transformation \begin{equation} y^0=t\hspace{2pt}\cosh ax, \hspace{2pt}y^1=t\hspace{2pt}\sinh ...
4
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0answers
411 views

Weyl transformation of Ricci tensor

We define the Weyl transform as, $$ \tilde{g}_{\mu\nu}=\Omega^2g_{\mu\nu}, $$ wherein $\Omega^2$ is a scalar function of space-time $x$. The Weyl transformed Christoffel symbol can be obtained by ...
4
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0answers
156 views

Importance of an extra total derivative term in Liouville theory

In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
4
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0answers
273 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
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1answer
447 views

Why does a salt water solution have a higher coefficient of thermal expansion (i.e. expands more) than distilled water?

I have found several different sources of data which show that a solution with a higher concentration of salt has a higher coefficient of volumetric thermal expansion than a lower concentration salt ...
4
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0answers
911 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
314 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 ...
4
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0answers
230 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{...

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