instruction stringlengths 31 24.3k |
|---|
Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\cdot) U^\dagger$ by a unitary $U$ operator acting on the mode vector as $U \mathbf a^\dagger U^\dagger = G \mathbf a^\d... |
NASA have a tendency to arrive at targets that are so far away, within seconds of ETA, and statements like " it's like throwing an object from New York and having it hit a particular key on a keyboard in San Francisco" (link) are made to describe how amazing that is.
Now, it's clear that it's literally many many orders... |
The photon spin-1 has two states, $\pm\hbar$, just like the spin qubit ($\pm\frac{\hbar}{2}$).
From a quantum information point of view, they can encode the same amount of data.
However, I am confused about the usual qubit representation:
\begin{equation}
|q\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\gamma}\sin\frac... |
Watching the Starship launch today I noticed it's actually doing some full rotations. Why isn't it controlled? and why does it happen?
I have read that it's the same for military missiles too
|
As the question asks, I am dealing with a problem where I'd like to simplify
$$\sum_{a=1}^3 \sigma^a_{\alpha\beta} S^a_{mn}$$
where $\sigma^a$ are the spin-1/2 Paulis and $S^a$ are some higher-spin representation of $SU(2)$. For the case where $S^a = \sigma^a$, we know
$$\sum_{a=1}^3 \sigma^a_{\alpha\beta} \sigma^a_{\g... |
I'm hoping to find a textbook (or other type of reference) which discusses an introduction to simulating quantum systems with classical computers. To clarify, I'm interested in finding a resource that goes through the more fundamental matrix mechanics/manipulations, and various tips and tricks on a practical level, aim... |
The Proca Lagrangian is
$$\mathcal{L}=-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\mu_0\Lambda^2}A_{\mu}A^{\mu}+A_{\mu}J^{\mu}$$
Where $\Lambda=\frac{\hbar}{m_{\gamma}c^2}$.
The symmetric energy-momentum tensor that can be drived from Proca Lagrangian is: $$\theta^{\mu\nu}=\theta^{\mu\nu}_0+\frac{1}{\mu_0\Lambda^2}... |
Suppose a bob or a ball is tied to a spring which in turn is pivoted to a certain point on a table as shown, if the ball is given velocity $v_o$ perpendicular to spring it moves in circular motion and also the spring elongates(even if the spring is in natural length initially), what force causes it to move out? I beli... |
I'm grappling with understanding Gauss' Law as applied to charged sheets and oppositely charged plates.
From what I've gathered, when using a Gaussian pillbox encompassing both sides of an infinite sheet, we can derive the electric field to be $2EA=\frac{\sigma A}{\epsilon_0}$, which simplifies to $E=\frac{\sigma}{2\e... |
I am interested in solving the dual (adjoint) Lindblad master equation for a time-dependent operator $O(t)$ as follows
\begin{equation}
\dot{O}(t) = i[H, O(t)]+\sum_{\alpha\in I} L_\alpha ^\dagger O(t) L_\alpha -\frac{1}{2}\left\{L^\dagger_\alpha L_\alpha, O(t)\right\}
\end{equation}
where the Hamiltonian $H$ is quadra... |
Consider an Hydrogenic Atom (no relativistic corrections and no reduced-mass effects) in a Quantized Electromagnetic Pulse given by the wave-packet:
$$
\underline{\hat{A}}(\underline{\hat{r}}, t) = \int_{[w_0 - \Delta w/2, w_0 + \Delta w/2]} \mathop{dw} \frac{\hbar}{2 \epsilon_0 V w} \underline{\epsilon} e^{i \delta_w}... |
We can think of Furry's theorem as the consequence of $CP$ invariance of $QED$. For parity, the vector bilinear changes sign, hence, under charge conjugation, it should. The vacuum is $C$-invariant, hence odd number of vector bilinears vanish. This is true for the quark-quark-gluon interaction as well. Hence, for trian... |
I am trying to create a vortex in a closed tube, and I am wondering if there are any resources with research on what shape would be most effective for creating this vortex. Literally any shape, paddle, whisk fan, just which would create the strongest/fastest vortex and equal angular velocities. If the viscosity matters... |
Context
I am trying to understand irradiance calculations for interior lighting in Blender software and got stuck with inputting light strength in lumens, because the lights in the software take watts per square meter per steradian. We are talking only about visible light. Virtual lights in the software have strength t... |
This is a relatively simple question that I just want confirmation on. In literature, I have seen 2 ways of writing the Heisenberg XXZ Chain:
1.) $H = -J \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+1}^y + \Delta S_n^zS_{n+1}^z\right)$
2.) $H = \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+1}^y + \Delta S_n^zS_{n+1}^z... |
On top of my washing machine is an aluminium tray, which holds various objects (used in the making of hot drinks). When the machine is on the final spin phase of its programme, it vibrates strongly. Then, all the objects on the tray steadily rotate in the same direction.
The objects also glide across the surface of the... |
I'm trying to understand the eigenmodes of the following operator:
$$(\Delta_{(1)}^{L L}-\frac{2}{3} R)V_\nu \equiv -\nabla^\mu \nabla_\mu V_\nu+R_{\nu \mu} V^\mu -\frac{2}{3} RV_\nu $$
Where $R_{\mu\nu}$ is the Ricci curvature tensor and working in 2+1 dimensional gravity with a Euclidean signature. Using $R_{\mu \nu... |
I'm studying about short circuit phenomenal. From the information I have searched, given a small battery (AAA), if we connect the 2 poles of the battery directly with a wire, without load (i.e. the light bulb), the circuit is short.
If there is a load, the circuit isn't short.
Could you please explain me why if there i... |
I am struggling to get the same result as this paper (eq. 3.10) for my ghost field when gauge-fixing diffeomorphisms in linearized gravity. I would appreciate it if someone could point me in the direction of some resources that could guide me through this method. I have currently tried it following the outlines from Pe... |
So my doubt involves the massive multiplet of $\mathcal{N}=2$. I am not being able to deduce what particles does the states represents. For example,
The $\mathcal{N}=2$ massive short hypermultiplet $s=0$ is given by
$$(2 \times (-1/2), 4 \times 0, 2 \times (1/2))$$
Consisting of a $1$ Dirac fermion, and two massive com... |
I know the Schrödinger equation is bascially the "quantized" Hamiltonian formalism from classical mechanics, and the Dirac equation is the special-relativistic version. But these equations do not describe deterministic "eye-visisble hard steel-ball-like" objects anymore, like the classical hamiltonian, but just probabi... |
The strength of a magnetic field is proportional to current and decreases with distance based on the permeability of the medium it travels through. For small electromagnets, it seems it's obvious you want to minimize resistance to maximize current. But as an electromagnet gets larger, outer layers have a larger distanc... |
For single-mode Gaussian channels $\mathcal C$, it holds that $\mathcal C = \mathcal A_G\circ L_\eta$, where $A_G$ is an amplifier channel described by the action $$\chi_{\rho'}(\xi)=\chi_\rho(\sqrt G\ \xi)e^{-(G-1)\frac{|\xi|^2}{2}}$$ on the characteristic function $\chi_{\rho}(\xi)=\text{Tr}\rho D_\xi$, where $D_\xi$... |
If I have a pair of eyeglasses that are perfectly parallel to my face, does curving the frame such that it wraps around my face better increase or decrease the h-prism?
|
My understanding is that what we call gravity is masses moving along paths in curved space -- it is not that gravity causes masses to move but they are already moving.
Why are they already moving? And is the speed of this motion the same for every mass, but only greater curvature seems to make them move faster near mor... |
Assume that an inertial observer $S'$ is moving with a velocity $\mathbf u$ relative to another inertial observer $S$ along the $x-$axis.
Now, imagine that the two observers observe a force being applied on an object, also along the $x-$axis.
Can we conclude that $\mathbf F',$ the force $S'$ observes is identical to $\... |
To quote Wikapedia: https://en.wikipedia.org/wiki/Quantum_dot
Nanoscale semiconductor materials tightly confine either electrons or electron holes. The confinement is similar to a three-dimensional particle in a box model. The quantum dot absorption and emission features correspond to transitions between discrete quan... |
When you have length contraction in special relativity
$$L' = L/\gamma$$ the interpretation is that $L'$ is the length of an object with rest-length $L$ moving with respect to an observer at rest. Now, similarly in a tachyonic antitelephone we have
$$\Delta t' = \gamma (1- a v)\Delta t$$
where $a$ is the speed of the ... |
I'm interested in understanding the distinctions between non-Newtonian fluids and regular Newtonian fluids regarding their optical properties, including refractive index, nonlinear optical behavior (such as the presence of $χ₃$), birefringence, and more. While results can vary case by case, are there any general rules,... |
I am a highschool student. I recently learned about conductors with cavity in school. How does a point charge outside the cavity, kept at a certain distance from the conductor, affect the potential of any point inside the cavity. I understand that the potential at any point inside the cavity. But how can an external po... |
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done on cartesian coordinate is generalized for all other coordinates. eg: $g_{\alpha\beta;\mu}=0$. My question is why should... |
I've found written in an undergrad textbook on relativity (Barone's, Relativity; italian book, don't actually know if it's been translated in any other language) that proper orthocronous Lorentz transformation (for simplicity sake we'll say SO(1,3) transformation) form a group which is (i think the proper way of saying... |
How to simulate the 2-body decay of some particle with mass $m$ moving with the given 4-momentum $p^{\mu}$ into two particles 1,2 with masses $m_{1}, m_{2}$ such that, say, the particle 1 will have the polar angle from $\theta_{1}$ to $\theta_{2}$ in the lab frame, and also find a probability that the particle 1 will f... |
It is well established that in gauge theory, the Wilson loops of the theory determine the gauge potential up a gauge transformation. That is, two gauge potentials $A_\mu$ and $B_\mu $ produce the same Wilson loops $W[\gamma]$ for all simple closed curves $\gamma$ if and only if there is a gauge transformation relating ... |
I want to know how to model partially coherent/incoherent light for imaging applications.
Usually, you find mathematical treatments of imaging just for the extreme cases.
For the coherent light, there is a mixing term when two light intensities $I_1$ and $I_2$ meet (interference):
$$I_{tot} = I_1 + I_2 + \sqrt{I_1I_2} ... |
As I understand it indium antimonide (InSb) is a non-reciprocal optical material. Having non-diagonal (non-reciprocal) components in its dielectric susceptibility tensor, in the presence of a magnetic field.
Is there some kind of microscopic time-reversal symmetry breaking mechanism, so I was wondering if it had some m... |
I learnt that geodesics parallel transport their velocity vectors. Does that mean a geodesic cannot intersect itself orthogonally?
|
My questions concerns calculations about a Peltier Thermoelectric Module for the cooling of water between an inlet and outlet. The Peltier water cooling module I am referring to is the following: Peltier Water Cooling Module, and looks like this (the small fan is used to cool the heat sink):
Let's say I have informati... |
Consider the two-way tachyonic antitelephone where the speed at which message is transmitted is $a$. A person $A$ sends a message to $B$ which is moving away with a speed of $v$ with respect to $A$. And when $B$ receives a message $B$ transmits a message back to $A$. Since, both of them have the same device, in each of... |
Let's consider a rubber ball like the ones shown in the picture:
When dropped onto the floor, the ball will rebound, but it won't regain its initial height.
The restitution coefficient is defined as the ratio of the speed immediately after the collision to the speed immediately before the collision:
$$R = \frac{v_{\rm... |
From the definition of Work Done, we know it is $\vec F . \vec s$ , where $\vec s$ is displacement of point of application of force.
Everything was alright until I read these texts from Kleppner and Kolenkow:
highlighted texts show clearly, they were considering the displacement of centre of mass. Also, below the high... |
Say I get careless and drop a black hole. It falls through the Earth, passes through the core, and comes out on the other side.
Idealised as point masses, the black hole would reach an equal height on the other side of the Earth, and if I phone up a friend on the other side perhaps they could catch it. But if the mass ... |
From what I am able to find, the phase velosity of the wave exceeds c, but the group velosity remains less than c. However, why does the wave form wavepackets after entering a medium with refractive index less than unity?
|
I have the Lagrangian
$$ \mathcal{L} = \frac{1}{2}D_\mu \Phi^\dagger D^\mu \Phi - \frac{m^2}{2} \Phi^\dagger \Phi - \frac{\lambda}{4}(\Phi^\dagger \Phi)^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$
where $\Phi = (\Phi_1, ..., \Phi_N)\in \mathbb{C}^N$ is a vector of $N$ complex scalar fields and
$$ D_\mu \Phi = (\partial_\mu ... |
I've always had a hard time wrapping my head around the 2 below statements being true for vaporizing a liquid into a gas:
When a liquid reaches its boiling point the temperature stops rising (and any further energy added goes into breaking bonds and changing phase)
A specific amount of heat (latent heat) is required f... |
I am working with a lagrangian on a homework problem. I expect it to have some gauge invariance. I can show that the Lagranian is invariant under those (gauge) tansformations but I have to use equations of motions to prove so. In typical textbook examples, I have seen gauge invariance established without using equation... |
Imagine two drilled magnets threaded onto a frictionless upright shaft, set N-N. One fixed down, the other floating above.
Now, if I move an iron gate in and out of the gap (the shaft has a gap), the upper magnet will move down and up.
How would you express the force exerted on the iron gate by the magnets, on the in a... |
Given the geodesic equations for a photon in a Schwarzchild or Kerr metric (provided by a near BH for example), the radial equation has usually two possible signs:
\begin{equation}
\dfrac{dr}{d\tau}= \pm \sqrt{R(r;b)}
\end{equation}
where $\tau$ is an affine parameter and $R(r)$ is also called radial potential and dep... |
Above is Tong's notes which shows how the Klein-Gordon equation is derived from Dirac equation. But I don't get why:
$\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \frac{1}{2} \{\gamma^{\mu},\gamma^{\nu}\}\partial_{\mu}\partial_{\nu}$
Because
$\frac{1}{2} \{\gamma^{\mu},\gamma^{\nu}\}\partial_{\mu}\partial_{\... |
Consider a Lagrangian $L(\phi)$ for a field $\phi$ (assume it is a free real scalar for simplicity). Then the time ordered propagator can be expressed as a path integral
$$
\langle\Omega|T\{ \phi(x) \phi(x') \}|\Omega\rangle = \int D\phi\ \phi(x) \phi(x') e^{i \int L} .
$$
This is a standard result from Zee's textbook ... |
I know the two-body problem has a stable solution and the three-body problem does not.
In the case that there are two comparable large bodies (twin planets) in a stable mutual orbit, what happens to a small, light particle's trajectory in that system?
If there were a number of tiny moons around either of these planets,... |
E. R. Tuttle in 1967 proposed a suitable method to determine the spectral terms for equivalent electrons in an atom in LS coupling condition. You may download the paper here, or
E. R. Tuttle; Terms Obtained from Configurations of Equivalent Electrons. Am. J. Phys. 1 January 1967; 35 (1): 26–29.
I am having the difficul... |
I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure:
I wonder what symmetry or current (in case there is a continuum infinity of non-diffeomorphic differential structures just like that of $\mathbb{R}^4$) can be... |
I suspect it is an easy problem for those who know quantum mechanics well. But web search and reading several links it found could not help. As of now I got only that at orthogonal direction to one that has been measured spin is superposition of "up"/"down" with equal values. But what is the formula for arbitrary direc... |
I am trying to compute the Fourier transform of the 2D $t$-$V$ model for the case $t=0$.
\begin{equation}
\hat H = -t \displaystyle \sum_{\langle i,j\rangle} ( \hat c_i^{\dagger} \hat c_j + \hat c_j^{\dagger} \hat c_i) + V \sum_{\langle i, j \rangle} \hat n_i \hat n_j.
\end{equation}
My attempt:
To compute the Fourier... |
How do mirrors work? How do the atoms in a mirror reflect photons. Does it absorb and reflect? As I understand it, after an atom absorbs a photon it is bound to release that same photon of the same energy. Is this correct?
|
Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\cdot) U^\dagger$ by a unitary $U$ operator acting on the mode vector as $U \mathbf a^\dagger U^\dagger = G \mathbf a^\d... |
I am trying to prove that $$⟨nlm|\frac{1}{^2}|nlm⟩=\frac{1}{^3*^2*(l+\frac{1}{2})}$$
(where $$ is the Bohr radius) for the $|⟩$ state of hydrogen. I know how to do this using Hellmann–Feynman theorem , and I am trying to use the conventional way of evaluating $\int_{0}^{\infty} \Psi_{nlm}^*\frac{1}{^2}\Psi_{nlm} \,dr $... |
I'm reading Schutz's Introduction to GR and came across an exercise problem. The problem is the following:
Show that the vectors $\{\vec e_{\bar \alpha }\}$ obtained from
$$\vec e_{\bar \beta } = \Lambda^{\alpha}_{\bar \beta}(-v) \, \vec e_{\alpha}$$
satisfy $\vec e_{\bar \beta} \cdot \vec e_\bar \alpha = \eta_{\bar \... |
I have been trying to find a mathematical proof (or even from a reliable source) which verifies that/proves that:
A small change in temperature leads to a small change in entropy.
However, I was unable to find one with a relation between change in entropy and change in temperature. Can someone please help me?
Edit: thi... |
Does Rydberg constant vary with mass? I researched a bit and some sources say it doesn't depend on mass of the element. How is this possible? Doesn't the name itself say its a constant?
|
First, is entanglement of three particles in $W$-like state deliberately possible (and not by chance)? Second, is the following statement correct?
In the doubly entangled $W$ state, represented as
$$ |\psi\rangle = \frac{1}{\sqrt{2}}(|100\rangle - |010\rangle), $$
when you measure the spin of one particle (let's say th... |
I understand the argument presented in the answer to this question on how successive redshifts combine, such that $$1+z = (1+z_1)(1+z_2) = 1 + z_1 + z_2 + z_1z_2 \\ \therefore z = z_1 + z_2 + z_1z_2.$$However, this seems to contradict the intuition that (non-relativistically) the velocities $v = zc$ giving rise to the ... |
What I'm trying to derive
The Cartesian position of the lunar ascending node relative to the true equator and equinox of date reference frame. My issue is I'm getting a bit tripped up with reference frames.
My attempt
Consider the moon at a time $t$. Let $r$,$v$ be the position and velocity of the moon at $t$, measured... |
Let the Moon have angular velocity $\omega$ around the Earth. The Earth itself revolves with velocity $V$ around the Sun. The radius vectors are $r_i$ from Sun to a point on the Moon, $r_i'$ from Earth to the point and $R$ from Sun to Earth. $r_i=R+r_i'$
The angular momentum of the Moon with respect to the Sun is there... |
I've been presented the Clausius-Mossotti equation in my Electromagnetic Optics class, but I was not given any derivation of the expression:
$$\vec{P}=N\alpha \left(\vec{E}+\frac{\vec{P}}{3\varepsilon_0}\right)$$
where:
$N$: number of molecules per unit volume in the dielectric
$\alpha$: molecular polarizability
I've... |
In my Electromagnetic Optics class, we tried to reconcile the microscopic (optical) properties of matter with its macroscopic counterparts, and one of the most challenging properties is the (complex) refractive index, $n_c=n+i\kappa$, where $n$ is the refractive index and $\kappa$ the material's absorbance.
After all t... |
When bending sheet metal in a press brake, it stretches slightly. For example, if I had a piece of steel 200 mm long by 1 mm thick and bent it in half, the two legs will each be longer than 100 mm so it’s obvious that the material has deformed and stretched a tiny bit. But what about its cross sectional area? Does that... |
I'm studying Kardar's "Statistical mechanics of particles" book and tackled a problem. After solving it, I checked Kardar's solution and found that he has different approach. I'm interested in the community's opinion on the correctness of my solution compared to Kardar's approach.
The problem asks to estimate length sc... |
Since $[L^2, L_z]=0$, we can say that they share a common eigenbasis, call it $f$, and $L^2f=\lambda f$, $L_z f=\mu f$
The ladder operators for the $z$ component of angular momentum are
$$L_\pm=L_x\pm iL_y$$
Applying the ladder operator to eigenfunction, $f$, we get
$$L_z(L_\pm f)=(\mu\pm\hbar)(L_\pm f) \tag1$$
At this... |
From what I understand about the delayed choice quantum eraser is that (per the diagram below), an interference pattern will be seen on D0 if a photon hits the last detectors D1 and D2 where they are scattered by a beam splitter. This observation occurs even if the photon hits the last detectors (D1 and D2) after the o... |
On page 3 of this paper (https://hal.science/hal-00627906v3/document), the authors say that in the $c\to\infty$ limit only the global generators will survive when computing the conformal block. In particular, they say that because the norm is
$$||L_{-n}|\Delta_s \rangle||^2=2n\Delta_s+\frac{c}{12}n(n-1)(n+1),$$
which ... |
I’m a third year physics student who was relaxing on vacation during spring break, and now am having a crisis because I realized I somehow never came across this problem before.
A vector in polar coordinates can be written as:
$$r \cos\theta \hat{r} + r \sin\theta \hat{\theta}.$$
Position vectors can be simplified to j... |
I am trying to understand some of the dynamics in Stommel's two-box model of ocean circulation in a way that I can explain to someone without too intense of a calculus or physics background. I want to work toward helping students create a mathematical model, but I don't want to get bogged down too much in the details o... |
I'm interested in the general formulas that give the exact uncertainties $\Delta r$ and $\Delta p_r$ (the radial momentum) for all stationary states $|n,l, m \rangle$ (or $\psi_{nlm}(r, \theta, \varphi)$), as functions of the quantum numbers $n = 1, 2, 3, \ldots, \infty$ and $l = 0, 1, 2, \ldots, n - 1$. The radial mo... |
A particle of mass $m$ is moving with initial velocity v. A constant force F acts on the particle in a direction which always points to a fixed point P. If initially the direction of v was perpendicular to F and the displacement between P and the initial position of the particle was d how can I derive the function for ... |
My solution was that I would need to sum the $x$ and $y$ components for the net force. However, the solution manual say that the net force should be calculated by using the Pythogorean theorem. I believe am missing something as I don't really get why we need to do that, so please help me with this. I am attaching scre... |
My question is: are these three statements correct?
(1) The net work on an object that rolls without slipping can be exactly divided into a "work on the center of mass" and a "work causing rotation about the center of mass": $W_\text{net} = W_\text{com} + W_\text{rot}$.
(2) Although it is true that the work done by sta... |
Why is Koopman's theorem a reliable approximation for ionization energy but unreliable for electron affinity? I understand that it has to do with cancellation of errors by orbital relaxation and electron correlation, but why do they cancel more in one case than the other?
|
I want to know the difference between spherical and Boyer-Lindquist coordinates. Don't they both use $r, \theta, \phi$ parameters? I've searched books and sources on the internet and there's none that explain what Boyer-Lindquist coordinate is.
|
I understand that the change in weight would be tiny even to a physicist and nothing for any practical purpose. I am also not talking about smoke and water vapor. I am only referring to energy given off as heat and light leaving the closed system.
As a follow up, would the energy have weight equal to the difference if ... |
I've been researching the concept of Transverse Electric and Magnetic modes, and I've found in this Wikipedia article that TE, for example, is defined when there is just $H$ in the direction of propagation, and no $E$ at all. Nonetheless, my professor keeps referring to TE and TM modes in plane harmonic waves, and it's... |
I think my confusion stems from this: if a book is resting on a table I understand that the force of gravity acts on the book and as it is in equilibrium, the table exerts a force equal in magnitude on the book so that there is no net force. Is it correct to say there is an electrostatic force that the book applies on ... |
As far as I understand, if a contravariant vector transforms in the form:
$$\vec{x}'=A\vec{x}.$$ (Where $A$ is the transformation matrix)
Then the covariant vectors shall transform as
$$\tilde{w}'=(A^{-1})^T\tilde{w}.$$
Now, If I write $A_{uv}={A^u}_v$; where 1st indice denotes row and 2nd one denotes column; we can wr... |
Do all measuring tools work by interacting with the physical property of interest and then changing spatially? Can we only quantify physical properties this way?
A clock interacts with time and changes spatially.
A mercury thermometer maps the displacement of volume to a unit of temperature.
Or a weight scale on earth... |
I recently came across the concept of the hill sphere radius,where we find the maximum distance from the secondary body where a satellite can stay stationary with respect to the secondary body, without falling into the primary body. I tried solving for the case where the satellite was also revolving around the secondar... |
I am confused about the qualitative behavior of ferromagnetic materials. Referencing the diagram below, I understand that magnetizing a ferromagnet from point A to point C (its saturation magnetization) will result in the shown hysteresis curve, where to demagnetize the material (go from C to E) requires a reversed fie... |
I've been reading Sean Carroll's book on GR and I stumbled upon an exercise on EM using $p$-forms. I think I've solved the problem correctly but I am having problems with my answers. I'll provide the question, my answer and what is troubling me, if anyone could help me it would be much appreciated.
Assuming we are in E... |
I have to find $\phi_0$ from following wave function in the momentum space:
\begin{equation}
\phi(k) = \phi_0 \text{exp}\bigg(-\frac{(k-k_0)^2}{2\kappa^2}\bigg)
\end{equation}
I know that I have to use the normaliazation criteria:
\begin{equation}
\lVert \phi(k) \rVert = 1
\end{equation}
With the $L^2$ Norm:
\begin{equ... |
Do all shockwaves have an expansion fan or expansion wave behind them? Does the air always expand behind a shockwave?
I assume that the strength of the expansion wave depends on the strength of the shockwave, and also the air has to be compressible.
|
Can anyone explain convergence of parallel rays on the focus of a parabolic reflector using Fermat's Principle? using optimization techniques from calculus?
|
Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose surface would be (diffeomorph to) 3D ?
For instance the Robertson-Walker metric for $k=0$ could turn out as a $T^3$ if cer... |
(For the purposes of this question I'm ignoring atmospheric pressure completely)
Consider a hollow cubic box with side length $1\text{m}$ that we fill up with water. Now suppose we make a small opening of $1\text{mm}^2$ and on top of this we attach a square rigid pipe of length $1000\text{m}$ and cross sectional area a... |
Suppose I connect the + (or -) pole of a battery to the earth with a conducting wire, will there be a current flow?
As I understand, the Earth is considered to be "neutral" in the sense that it contains inside it many many positive charged particles and negatives particles which cancel each other.
I understand also tha... |
For example, if there is a tube with a certain diameter and one fan in this tube that generates a flow of air with velocity X, will an additional fan (or several fans) accelerate the air even more? If so, how much?
|
I'm a high school senior planning a mathematical exploration project where I am aiming to answer the question: I have a spherical ball of dough, which is uniformly at temperature 25 degrees Celsius. I put it in the oven at 200 degrees Celsius. Assuming that the thermal conductivity of the dough is consistent all throug... |
Consider a Newtonian fluid. The shear tensor is defined by:
\begin{equation}
\sigma_{ij} = \frac{1}{2}\Big(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\Big) - \frac{1}{3}(\nabla \cdot \mathbf{v})\delta_{ij}.
\end{equation}
Suppose we consider it on a sphere, i.e. $x_1 = r \sin(\theta)\cos(\phi)... |
For the past few days I've been going through Baumann's notes on cosmology, more specifically I'm currently busy with chapter 2 Inflation. I don't get how he comes to the following equation involving a rewriting of the comoving Hubble radius
$$(aH)^{-1} = H_0^{-1}a^{\frac{1}{2}(1+3w)},$$
in which the $w$ comes from the... |
I've recently been reading about what really causes lift on an airfoil and the article linked mentions that even a symmetric airfoil or even a flat plate generates lift as long as the angle of attack is non-zero.
It explains the pressure differential between the top of the airfoil and the bottom of the airfoil being ca... |
I read that it is generally believed that information is preserved in black hole evaporation, and people's views only diverge when it comes to how information is preserved. Is this true?
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.