Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Are two states with the same measurement probabilities necessarily equal up to unitary equivalence? Let $\rho$ and $\rho'$ be $n\times n$ density matrices, and suppose that for every observable $A$ and every $\lambda$ in the spectrum of $A$ we have
$$
\text{tr}(\rho P_{\lambda})=\text{tr}(\rho' P_{\lambda}),
$$
where $... | The answers from Jason Funderberker and Valter Moretti are perfect at explaining mathematically how the equivalence of all possible measurements requires the two density matrices to be the same.
However, something I think is important to highlight is that this is not some kind of mathematical accident, it is an essenti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Do protons or neutrons oscillate? (inside nucleus in atom) Do they oscillate relative to each other? What is the frequency? What is the amplitude? I would think they oscillate since electrons move all over the place at high speeds and there is attractive Coulomb force between electrons and protons.
| Given that nucleon is bound to nuclei radius,- this per Heisenberg uncertainty, gives us uncertainty in nucleon speed, which is :
$$ \Delta v = \frac {ℏ}{2~m_n R_0 A^{1/3}} $$
Where $m_n$ is nucleon mass, $R_0 = 1.2×10^{−15}~m$ and $A$ is nucleon amount in specific atom.
If we would calculate speed uncertainty for neu... | {
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Is it possible to derive the Weyl tensor from the Ricci tensor or Ricci scalar? If so, how? I understand that the Ricci tensor is derived from the Riemann tensor, but I know that the Weyl tensor is also derived from the Riemann tensor. So, is it possible to calculate the Weyl tensor from the Ricci tensor or Ricci scala... | No. The Weyl tensor and the Ricci tensor correspond to different "degrees of freedom" of the Riemann tensor. As an example, notice that there are many Ricci-flat solutions to the Einstein equations, such as
*
*Minkowski spacetime
*Schwarzschild spacetime
*gravitational waves in either of the previous spacetimes
an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor? I understand that the Einstein field equation uses the Ricci curvature tensor, Ricci curvature scalar, and stress-energy momentum tensor. But is there a way to form an equation that uses the ... | Well, as you know the Ricci tensor/scalar are built from the Riemann tensor, so it's not entirely clear to me what you're asking. You can of course write
$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} = R^\alpha_{\ \ \mu\alpha\nu} - \frac{1}{2} g^{\beta\gamma}R^\alpha_{\ \ \beta \alpha \gamma} g_{\mu\nu} = \frac{8... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/716623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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When will the velocity of oscillating particle become zero in simple harmonic motion? This question came in the Jagannath University admission exam 13-14
Q) In simple harmonic oscillation, the velocity of an oscillating particle becomes zero-
(a) when acceleration is maximum
(b) when displacement is minimum
(c) when di... | What a nice exam. All answers are correct. Presuming we are talking about signed quantities and not their amplitude, which seems fair from the wording.
Say we start with maximum displacement at phase angle 0. Then at this point we also have minimum acceleration (maximum absolute value, but with a negative sign). And at... | {
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When is it admissible to neglect the pressure term to use the Burgers' equation instead of the Navier-Stokes equation? The Burgers equation can be understood as a simplification of the Navier-Stokes equations when the pressure term is neglected:
$$
\frac{\partial u_i}{\partial t}+\ u_j\frac{\partial u_i}{\partial x_j}=... | Two straight-forward scenarios I can think of off the top of my head are:
*
*The flow is horizontal (probably also narrow?) such that the pressure gradient is sufficiently small as to be zero (i.e., $\nabla p\approx0$ as in Burgers' equation). A constant flow in a pipe would be one example here.
*The pressure gradie... | {
"language": "en",
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How is time measured in particle experiments? I was reading about the half life measurements and was curious to understand the experimental setups that allows so minute measurements to be captured. Specifically looking into half life of Higgs boson. I am looking to understand one example method of measuring Higgs boson... | The Higgs is a challenging example because the tabulated quantity is the decay width $\Gamma$, from which a mean life $t≈\hbar/\Gamma$ is inferred. That is, nobody starts a clock when the Higgs is born and then stops it $10^{-22}$ seconds later. Instead, the short lifetime of the Higgs contributes an intrinsic uncert... | {
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"source": "stackexchange",
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Why do we need Legendre transformation for thermodynamic potentials? I get the idea that thermodynamic potentials are introduced because it is not always easy to describe a system's energy as a function of variables like $S,V,N$ as we normally do with internal energy $U=U(S,V,N)$. For example, temperature is much easie... | If we didn't use Legendre transform, we would waste precious information. Consider the case of a function of one variable for simplicity: $y=g(x)$. Now what you propose to do is to define $t=\frac{dg}{dx}(x)$ and obtain $x=h(t)$. So you would now have a $y$ expressed as a function of $t$, namely $y=g(h(t))$.
But what ... | {
"language": "en",
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How does entropy work with mixing? Imagine a hypothetical box filled with water, with two equal-volume partitioned sections: one at 40 degrees, and the other at 60 degrees. There is thus an energy associated with the difference in temperature that could be harnessed.
In this first box we remove the partition, the water... | When they reach equilibrium both sides will be 50 degrees. We arrive at this simply by using conservation of energy. We can do that because your heat engine spent its work-energy back into the water. If it has spent it outside the system then Mr. Chemomechanics answer would be relevant.
| {
"language": "en",
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Will quantum events ever occur on a macro-scale rather than a vacuum? Michio kaku says there's a chance we'll wake up on Mars tomorrow https://www.theatlantic.com/science/archive/2018/10/beyond-weird-decoherence-quantum-weirdness-schrodingers-cat/573448/
In this post, it is shown that quantum decoherence in the macro w... | Waking up on Mars is impossible for any reasonable interpretation of impossible. However, macro-scale events that are prohibited in classical physics but predicted by QM, in environments far from vacuums, are commonplace. For example, the temperature predicted for the center of the sun is too low for molecular kinetic ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why magnetic field doesn't do any work on moving charge? As we know the Lorentz force $F=qv\times B$ never does work on the particle with charge $q$. But it does work on a dipole.
My question is, doesn't a moving charge behaves like a dipole? I mean an electron moving around nucleus behaves like magnetic dipole then wh... | An (non-zero) electric charge is Lorentz-invariant. So in motion an (non-zero) electric charge cannot become a dipole since a dipole has 2 opposite charges, therefore has total charge zero. If it could become a dipole during motion, it would violate Lorentz-invariance of the charge.
EDIT
The best known experiment of a ... | {
"language": "en",
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Moment of Inertia Tensor and Center of Mass Can the Moment of Inertia Tensor be diagonal in a reference frame where the Center of Mass of the system is not on some of the axes?
| Yes, we can have a system whose CM is not on a coordinate axis which also has a diagonal inertia tensor. As an example, consider a system consisting of four point masses $m$ at the points $(1,1, 1)$, $(1,1, -1)$, $(-1,1, 1)$, and $(1,-1,1)$. Then the CM of the system lies at
$$
(x_{CM}, y_{CM}, z_{CM}) = \left( \frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Orthocomplement space of $a^\dagger a$? In this note$^1$ page 4 the author states that suppose $a_i$ acts on the whole vector space $V$ where $a_i$ is the annihilation operator of fermions. And since $[a_i^\dagger a_i,a^\dagger_j a_j]=0,\forall i,j$, and $a_i^\dagger a_i$ are hermitian operators, so we can construct a ... | I think you are assuming that $\mathrm{dim}(V)=2^n$, which is not necessarily the case. As a somewhat trivial example, imagine you have an $8$-dimensional space $V$ spanned by orthogonal vectors $|\alpha_1,\alpha_2,\alpha_3\rangle$ with $\alpha_i \in \{0,1\}$, and consider the fermionic raising/lowering operators $a_1... | {
"language": "en",
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If water is nearly as incompressible as ground, why don't divers get injured when they plunge into it? I have read that water (or any other liquid) cannot be compressed like gases and it is nearly as elastic as solid. So why isn’t the impact of diving into water equivalent to that of diving on hard concrete?
| In simple terms, water (or any fluid) will move out of the way; concrete won’t (unless it is hit very hard). The important properties are viscosity and elasticity rather than compressibility.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/718786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can we still see the CMB? May seem stupid but i cant wrap my head around it. if a star explodes we eventually see it when the light gets here. but once its got here we see the event and the star is now gone, we cant see it anymore because its not there anymore. so how can we still see the CMB if all the light has r... | The CMB happened everywhere in the universe. As time goes on we see it from further and further away.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/719177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mass flux in a multicomponent mixture with a concentration-dependent density This question concerns how to properly calculate the change in mass in a multicomponent mixture when the mass density is concentration-dependent.
For a 1D rod of length $L$ that has a mass density $\rho$ and contains $N$ species with mass conc... | First, recognize that because the densities are always uniform, the diffusive mass fluxes are always zero. Therefore, the continuity equation for species $j$ in this special case is:
$$
\frac{\partial \rho_j(t)}{\partial t} + \frac{\partial}{\partial x} \left(v(x,t) \rho_j(t) \right) =0
$$
Using this equation and foll... | {
"language": "en",
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Relation between Rayleigh scattering intensity and Poynting vector of an oscillating dipole In Rayleigh scattering by a molecule the intensity of the scattered light is:
$$I \propto I_0({1+\cos^2(\theta))}$$
while the time-averaged Poynting vector of an oscillating dipole is:
$$\langle S \rangle \propto \sin^2(\theta)$... | The formula you give for Rayleigh scattering is for unpolarized light, consisting of equal amounts of two perpendicular polarisation states. These two states excite two perpendicular oscillations in the dipole moments of the scattering objects.
If you Rayleigh scatter linearly polarised light then you will get the $\si... | {
"language": "en",
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On average, how often does any given hydrogen nucleus run into another hydrogen nucleus in the Sun? I think there is a misprint in an article. I will include the link if you don't mind and cut paste the sentence that I think is a misprint. Here is the link
https://www.abc.net.au/science/articles/2012/04/17/3478276.htm#... | This statement is indeed misleading.
For a given hydrogen atom, the frequency of collisions in the center of the Sun is enormously high, some $10^{17} s^{-1}$ or thereabout. So it will "run into and collide with other protons" all the time.
In contrast, the Sun's livetime is of order $10^{10}$ years, so it will burn th... | {
"language": "en",
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How can Entropy be maximal when it is undefined everywhere else? This question is about classical thermodynamics.
I learned that when an isolated system is not in equilibrium, its thermodynamic variables such as Entropy are undefined.
I also learned that when an isolated system is in equilibrium, its Entropy is maximiz... | Your question is an interesting one. One way of addressing this is to subscribe to the idea that, even in non-equilibrium situations, the state variables such as U, H, S, etc. can be defined and calculated locally (per unit mass or unit volume) based on the local conditions of temperature, pressure, species concentrat... | {
"language": "en",
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Normal ordered exponential of one-body operators Let $\{a_i\}_{i=1}^N$ be a set of annihilation operators (they are either all bosons, or all fermions) satisfying the canonical commutation or anti-commutation relation. In the book Quantum Theory of Finite Systems by Blaizot and Ripka, Problem 1.6 claims that (summation... | Hints: First try to show it for a single bosonic oscillator (for fermions this was done by the OP already). To this end, define the following functions:
\begin{align}
f(M)&:=\exp{a^\dagger a M} \tag{1} \\
L(M)&:=N[\exp{a^\dagger a (e^{M}-1)}] \quad \tag{2}.
\end{align}
Then show $f(0)=L(0)=\mathbb I$ and that $f$ and ... | {
"language": "en",
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How conservation of energy looks like in the moving frame? It is obvious that a car accelerates by converting its chemical energy in the fuel to produce kinetic energy to accelerate itself.
If the energy is lost into friction and heat, the car will slow down.
But from the point of view of the car, it always appears to ... | From the car's point of view, there is a fictitious force, which does work. Therefore, the car sees extra potential energy. You can think of the fictitious force as weak constant gravity. Similar to how you can see gravitational potential energy, in the car's perspective, there is the fictitious force's potential energ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/720630",
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"source": "stackexchange",
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Plane wave in a finite conductor I am questioning the validity of the
standard formula for EM waves in conductors
The standard treatment of EM waves in conductors are:
$$\nabla \cdot \vec{E} = 0$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla × \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$
$$\nabla × \vec{B} = \mu_0 [\sigma... | The case of a finite conducting medium is no different from the case of a finite dielectric medium, and the approach is the same in both cases: we have to solve the Maxwell equations in the two regions and then connect them correctly at the bounding surface using the boundary conditions derived from the Maxwell equatio... | {
"language": "en",
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Why is climate change triggering faster rotation? On July the 29th 2022, the Earth finished its rotation about 1.5 milliseconds earlier than the entire 24 hours. Scientists link this to climate change, saying that a possible reason could be due to the melting of polar glaciers.
I do not know for sure what dictates this... | Another possible contributing factor is the recent drought in many mid-latitude areas of the world, which would reduce the amount of water in dams, lakes, and aquifers. If these are, on average, farther from Earth's rotation axis than the ocean average, that would speed up Earth's rotation.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/721809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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Why do raindrops look like sticks? I always thought that raindrops look like this emoji .
But today, I shot it in slow-mo (see on YouTube), and they look more like sticks.
Was it some light effect of my camera, or do they really look like that? And if it's the real deal, why do they look like that?
Edit: I found anoth... | Photographic sensor, camera sensor, film... all integrate the light received during the exposure time. The brightness and color of each pixel depends on the sum of all the light received during the exposure. So if an object moves during the exposure, it will look blurred, like this car. The background is stationary, so... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/721935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Is spin of elementary particles same as the rotation of a planet? By the word spin of elementary particle, one would imagine the particle to be rotating around its own axis, just like a planet rotates, but is it actually true? While spinning does an elementary particle actually rotate around its own axis just like a pl... | No, for several reasons. For instance, it is possible to stop the rotation around its axis or, more precisely, to decrease or stop the corresponding intrinsic angular momentum of a macroscopic object by exerting a torsion. This is not possible for an elementary particle: the value of the spin is constant.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/722115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the meaning of a thermal equilibrium between matter and radiation? I understand that the thermal equilibrium between two bodies means that the two bodies attain the Same temperature. Therefore,there is no flow of a thermal energy between them. However, I don't know how i should understand the meaning of it when... | There are many questions here. I will answer the one in the title.
We imagine a system consisting of a chunk of matter surrounded with a bath of photons. The atoms forming the skin of the chunk of matter are being bombarded with photons; with each collision, energy is being shared back and forth between the vibrational... | {
"language": "en",
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Why don't our eyes get overexposed like a camera with too slow a shutter speed? Perhaps I'm just not understanding the exact mechanism of how light interacts with our eyes and is interpreted as an image, but for cameras light goes through the lens and hits the image sensor/film and the longer the shutter is open, the m... | The sensitive cells in our eyes respond to flux (photons/second) while photographic sensors respond to fluence (photons). It's easy enough to make a sensor that responds to flux: measure the current generated by a photovoltaic. But if you block the current, so that charge builds up in the sensor, you can then measure f... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/722510",
"timestamp": "2023-03-29T00:00:00",
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The double slit experiment from the path integral approach I am reading the book Topics in Advanced Quantum Mechanics by Holstein. In Chapter 3, section 3 he discusses the Aharonov-Bohm effect, but before doing so he discusses the ordinary double slit experiment. This book is based on the path integral approach and wha... | *
*It seems Holstein is conflating their set-ups/sources. Before eq. (3.3) Holstein talks about a single particle time-dependent picture using classical action $$S~=~\frac{m}{\color{red}{2}}\frac{d^2}{t}~=~\frac{pd}{\color{red}{2}},\qquad p~=~\hbar k;$$ and afterwards a time-independent plane wave picture, where only ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/722648",
"timestamp": "2023-03-29T00:00:00",
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Time Dilation Explanation This might be a dumb question but I'm still going to ask it.
So basically I've been trying to learn Special Relativity through this lecture by Brian Greene, and the way he justifies time dilation is through a Light Clock. I think I understand this, but I just don't know how I could picture thi... | I suggest you look into Lorentz transformations in detail. Let's assume the reference frame of a still observer with the 2 axes: the x-axis and time axis t. Let's say there's another observer within that frame moving with a constant velocity v with respect to the first observer (who's still). The two transformed coordi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/722828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Compactification in String Theory and Compactification in Topology are they the same thing? In topology, there is a concept of compactification which is defined as follows.
A space $Z$ is a compactification of $X$ if $Z$ is compact Hausdorff and there exists an
embedding $j:X \rightarrow Z $ such that $\overline{j(X)} ... | I don't know much about compactification in string theory, but the idea is based on the compactification in (4+1) D Kaluza-Klein theory, proposed in the years after the theory of general relativity was published to unite gravity and electromagnetism. (The weak and strong interaction were not discovered yet.)
The idea p... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Besides traveling at the speed of light, how can we be sure that it is possible to have energy and momentum without mass? How can we be sure that it is possible to have energy and momentum without mass? If something were to continually lose energy, would it not also lose a corresponding amount of mass? I understand tha... | Photons are not " predicted" to have no mass, they have no rest mass, a photon with not light speed does not exist. You can measure its momentum $p=\frac{hf}{c}$ and thereby its mass . A particle with rest mass≠0 can never reach the speed c, approaching c its mass increases. If a particle losses energy it also looses m... | {
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Is the equal sign in "$0 ^\circ \mathrm{C} = 273.15\, \mathrm{K}$" fair? I'm in doubt whether the equal sign in an expression like "$0 ^\circ \mathrm{C} = 273.15 \,\mathrm{K}$" is fair because, normally, if $A=B$, then, say, $2A=2B$, which is hardly applicable to "$0 ^\circ \mathrm{C} = 273.15 \,\mathrm{K}$".
So is it ... | This is comparing apples & oranges in general.
Example 0 : Water & Tumbler
A tumbler having 50% water is half-full & half-empty :
0.5 full = 0.5 empty.
Multiply each side by 2 to get :
1.0 full = 1.0 empty !
What is wrong here is that we equated the water Part with the empty Part. Naturally, we will get contradictory a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In a ion source, how do we make the ions get out of the cavity? Ion sources are devices that allow creating ion beams (e.g. argon ions) and to project them outside the device, for example to be further processed by a particle accelerator, or to irradiate materials or biological tissues etc.
The ions are usually created... | Since you want to use the ions in an accelerator, the outside is vacuum or almost vacuum , so the small pressure and the electrostatic force throws them out.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do you account for the weight of the lever? I have a 6 foot lever where the fulcrum is 2 feet from the input force. In an ideal situation, this gives a mechanical (dis)advantage of 0.5. However, let's just assume there is no weighted object being lifted, but the lever itself weighs 50lbs. How do you calculate how m... | Assuming you're considering a static, condition, you can solve the equilibrium of the moments (sum of external moments equals zero) around the fulcrum without considering inertia.
Assuming:
*
*the lever is orthogonal w.r.t. the gravity and the force applied
*mass distribution is uniform along the lever
you can:
*
... | {
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How does pilot wave theory explain "identical particle" interference? Pilot wave theory says that there exist waves in 3D space which carry particles. This explains, say, the double slit experiment.
But this does not explain the behavior of identical particles. According to standard QM, a system of two identical partic... | To quote Bohm's 1952 paper:
In the two-body problem, the system is described therefore by a six-dimensional Schroedinger wave and by a six-dimensional trajectory, specifying the actual location of each of the two particles. The velocity of this trajectory has components $\nabla_1 S/m$ and $\nabla_2 S/m$, respectively,... | {
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Why do you use $n_r = n -\ell -1$ as quantum number instead of $n$ for hydrogen atom? I got two different quantum numbers for the same problem: Hydrogen atom without any interaction.
Then, my energy is $$ E_n = -\frac{R_y}{n^2} $$ with the quantum number $n = 1, 2, 3, ....$
In another version it is $$ E_{n_r} = -\frac{... | Because an atom is three dimensional and the nucleus-electron interation is spherically symmetric - thus the resulting potential is spherically symmetric. The wavefunction can have angular components in addition to radial components in that case.
All this means is that in addition to $n$, you also get $l$ as a quantum ... | {
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Current loop and the associated magnetic field under mirror reflection Let a circular current loop in the $xy$ plane carries a current in the counterclockwise sense. Therefore, it produces a magnetic field in the $+z$ direction (think of it as an arrow parallel to the $+z$-axis). Now consider reflecting this by placing... | It is your case II.
You stumbled upon the fact that
the magnetic field $\vec{B}$ is not a vector, but a pseudovector.
That means it flips direction when going to the mirrored situation.
(image from Wikipedia - Pseudovector)
So you can't take the intuitive picture of magnetic field lines
(like being arrows made of meta... | {
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What is the matrix representation of a function of an quantum operator? If we know the matrix representation of a quantum operator, say $J$, will the matrix representation of any function of the operator i.e$f(J)$, same as acting the function on the matrix of the operator?
| Formally/heuristically, for sufficiently nice operator
$$\begin{align}\hat{A}~=~&\sum_{i,j\in I}|i\rangle A^i{}_j \langle j|\cr
~=~&\begin{bmatrix}|1\rangle & |2\rangle & \cdots \end{bmatrix} \stackrel{=}{A}\begin{bmatrix}\langle 1| \cr \langle 2| \cr \vdots \end{bmatrix}:~~{\cal H}\to{\cal H},\end{align}\tag{1}$$
func... | {
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Do Neutrons Have a Charge Radius? The radius of a proton is described as a "charge radius", about 0.84 fm. The neutron is about the same size, 0.8 fm, but has no measureable charge. Is this a contradiction? Are the two radii measured in the same way?
| The neutron is a quantum mechanical entity, and as well as the proton it is a complex bound state of valence quarks and a sea of virtual quark antiquark and gluons, similar to the proton picture.
Quarks and antiquarks are charged and so a scattering of electrons on neutrons should show up a radius due to the charged ... | {
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Uncertainty formula for division $$(A \pm a) \times (B \pm b)=AB\left(1 \pm \frac aA \pm \frac bB\right)$$
Here is the formula im struggling with, I'm just trying to apply this formula for division (i.e for speed, $\frac dt$, instead of momentum $mv$). How would I apply this formula for when $B = \frac 1B$ ? I have dis... | Let $f(X,Y)$ be a function of two random variables $X$ and $Y$. Let $X_m$ be the mean of $X$ and $S_X$ the standard deviation of $X$. Let $Y_m$ be the mean of $Y$ and $S_Y$ the standard deviation of $Y$. The mean of $f(X, Y)$ is $ f_m = f(X_m, Y_m)$. Assuming $X$ and $Y$ are independent, retaining the lower order ter... | {
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Effect on vertical launch of ball from a rotating body If a ball is launched vertically from the surface of the earth at the equator, assuming a vacuum, will the ball land in exactly the same place on the surface of the earth? Or does the fact that the earth is rotating change the place on the surface where the ball w... | Everything on earth is already accelerated to the rotation of the earth, so the ball will land where it was thrown.
Like how if you throw a ball up in a car, it does not move relative to the car as it had been accelerated to the car's velocity
| {
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Can magnetic Susceptibility be complex? Can the magnetic susceptibility of a material be complex? Then what will that complex term represents? does it represent any kind of energy loss in the material?
| The AC Magnetic Susceptibility, has a real and an imaginary part.
The imaginary component, χ" , indicates dissipative processes in the sample.
In conductive samples, the dissipation is due to eddy currents.
Relaxation and irreversibility in spin-glasses give rise to a nonzero χ" . In ferromagnets, a nonzero imaginary ... | {
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Doubling the length of a solenoid doubles its inductance. Two identical solenoids in series have up to four times the inductance due to M. Why? The inductance of a long solenoid with $N_o$ turns and a length $l_o$ is
$$L_o=\pi r^2 \mu_0\frac{N_o^2}{l_o}$$
If I now make a new solenoid, $L_{new}$, with double the length ... | The formula you're using is an approximation that assumes $l>>r$. But in that approximation, there is negligible mutual inductance, since the flux spreads out at the end on a scale of $\approx r$.
| {
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Synthetic aperture for visible light camera It seems to me a camera array for visible light could be constructed to synthesize a large effective aperture to achieve high angular resolution just as a synthetic aperture radar does. This could be used for example on a satellite with several small cameras mounted on extend... | The major difference is that the antennas in RF domain capture amplitude and phase of the incident EM wave but the photodetectors in optical domain are only square law detectors, i.e. they detect power and not phase of the EM wave.
However, the information on EM wave phase is crucial for synthetic aperture methods.
Wha... | {
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How much energy one would get by tethering an imaginary rope to a galaxy and let it unwind until reaching the Hubble horizon? Based on the tethered galaxies gedankenexperiment 1, let's imagine we attach an imaginary cosmologically-long string to a galaxy that is at a sufficiently long distance to be drawn away by the H... | Let's assume that galaxy is in distance L from us. The average velocity of this Galaxy can be explained by the initial universe expanding and is equal:
$$v\sim HL\Rightarrow W_{kin}= M_{galaxy}v^2/2\approx M_{galaxy}H^2L^2/2$$
Considering the probable infinite accelerating of expanding (we cannot be sure in it), we can... | {
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Inonu-Wigner contraction in Weinberg Volume I In volume I of Weinberg quantum theory of fields, on page 61, Weinberg derived the commutation relations of the generators $H,P_i, J_i,K_i$ of the Poincare algebra, then he tried to take the nonrelativistic limits by doing the Inonu-Wigner contraction on page 62. He claims ... | This is necessary since $$[\underbrace{K_i}_{{\cal O}(v^{-1})},\underbrace{P_j}_{{\cal O}(mv)}]=i\underbrace{M}_{{\cal O}(m)}\delta_{ij}.$$
| {
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Why is the monopole problem a problem? According to this Wikipedia article on cosmic inflation:
The magnetic monopole problem, sometimes called the exotic-relics problem, says that if the early universe were very hot, a large number of very heavy, stable magnetic monopoles would have been produced.
Inflation solves t... |
But we have no evidence that such exotic particles even exist!
Exactly. We don't see them, despite the fact that they are predicted by a lot of beyond-the-Standard-Model theories. This means there is a very good chance they might exist, in which case it would be a problem. It is not a critical problem like horizon pr... | {
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How do Parabolic Flights exactly work? I understand how one can feel weightless in an elevator - in the person's reference frame, they are not being pushed up by the floor.
However, I don't understand how this can occur in parabolic flight. A textbook I found said that "passengers will be objects in free fall and will ... | Imagine you throw a ball upward into the sky. The instant it leaves your hand, it is coasting ballistically along a parabolic path through the air. An ant clinging to the ball will not experience its own weight as the ball follows that path, as the ball and the ant are both in free-fall the whole time.
The pilot in the... | {
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I'm having trouble understanding exactly what $δ$ represents in thermodynamics I know that $δ$ sometimes represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not change the free energy since it is at a minim... |
I know that $\delta$ represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not change the free energy since it is at a minimum that is δA=0" but in terms of this what exactly does it mean.
Here, the symbol ... | {
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$Z_n$ gauge theory from $U(1)$ In Appendix A of the paper, "Generalized Global Symmetries" by Gaiotto et al., they have considered an action, which for the purpose of the question, can be taken to be
$$S=\frac{n}{2\pi}\int B dA$$
where $B$ is a periodic scalar and $A$ is a $d-1$ form. Then they went on to construct th... | You need to think about what "$B$ is a periodic scalar" means: When $B$ is $2\pi$-periodic, then since $V$ acts on $B$ as $B\mapsto B + \frac{2\pi}{n}$, you necessarily have $V^n = 1$ since $B+2\pi$ is identified with $B$ so $V^n$ is a "do nothing" operator, i.e. the identity.
Similarly, when Gaiotto et. al. say that t... | {
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Current through a capacitor in AC Circuits I'm a little confused on the equation for the instantaneous current through a capacitor in AC circuits.
My textbook has it as:
$$i_C = \omega CV \ cos(\omega t + \pi/2) = -\omega CV \ sin(\omega t)$$
where $\omega$ = angular frequency, $C$ is the capacitance, and $V$ is the vo... | The pi/2 term comes about when converting angular measure from degree units to radian units, where 2(pi) radians equals 360 degrees.
BTW here is a simple way to visualize current flow through a capacitor:
Imagine two flat metal plates facing each other with nothing but air between them. Each plate is connected to a wir... | {
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Why does particle leave circular motion after string slacks? If a particle is attached to a string and made to move in a vertical circle with initial velocity of $\sqrt{4gl}$ $m/s$ where l is the length of string, at some angle (approx $131°$ with the initial position), the string slacks and the particle leaves the cir... |
Why does this phenomenon occur even though the component of weight can provide centripetal acceleration?
It occurs precisely because the weight can provide centripetal acceleration. More to the point, the weight provides too much centripetal acceleration.
The force from the string is naturally regulated by the fact t... | {
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Angle in a spacetime diagram
FIGURE 4.13 A Lorentz boost as a change of coordinates on a spacetime diagram. The figure shows the grid of $\left(c t^{\prime}, x^{\prime}\right)$ coordinates defined by $(4.18)$ plotted on a $(c t, x)$ spacetime diagram. The $\left(c t^{\prime}, x^{\prime}\right)$ coordinates are not or... | As @benrg says, the "Minkowskian-angle" (called "rapidity") uses the
hyperbolic functions, and not the circular functions.
With velocity $(v/c)=\tanh\theta$, we have
time-dilation factor $\gamma=\frac{1}{\sqrt{1-(v/c)^2}}=\cosh\theta$ and
Doppler factor $k=\sqrt{\frac{1+(v/c)}{1-(v/c)}}=\exp\theta$.
Geometrically, an a... | {
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Understanding Maxwell relations by considering the differential changes of the enthalpy In my book it talks about deriving the Maxwell relations and how they can be obtained by considering the differential changes of the enthalpy. By doing this from $$H=U+PV$$ we get $$dH=TdS+VdP$$ and from equation $$U=-PV+H$$ we get ... | The starting point is the fundamental differential $$dU = T dS - P dV\tag{1}$$ This is a fundamental result in thermodynamics and is obtained by combining the first and second law on a closed system undergong reversible process.
For enthalpy start with $H=U+PV$ and take its differential:
$$dH = dU + P dV + V dP$$ Apply... | {
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Does the quantum mechanical wave function work as a charge distribution? The quantum mechanical wave function is traditionally interpreted as a probability distribution of the particles position. It is possible to interact with an electron in a conductor, without collapsing the wave function, which can be distributed o... | The charge distribution is related to the probability of the electron being in a given location. That is, the amplitude-squared of the wave fucnction, $\bar{\psi} \psi $.
Under some circumstances this can work as a charge distribution. For example, in an atom the charge distribution is given that way, including the nu... | {
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Can a Matrix Have EigenBras? I only need a yes or no, but I cannot find anything online. I know a matrix $A$ can have eigenkets found by using.
$$A\psi=\lambda\psi.$$
However, I was wondering if my matrix $A$ was Hermitian, could I just apply the Hermitian conjugate to the equation and find eigenbras for the system too... | Yes. In fact: $$A |\psi\rangle = \lambda |\psi\rangle \Rightarrow \langle\psi|A^\dagger = \langle\psi|\lambda^*,$$
So, eigenbras and eigenkets are intimately related, which is why we normally just talk about eigenvectors.
| {
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When building an optical system, how can I increase the size of the useful image? I am trying to put together a system of lenses that has a probe with a 2 mm diameter. The system has a probe with an objective and a relay and a set of lenses with a larger diameter that function as a microscope, magnifying the image gene... | I asked in the Zemax forum and they suggested that the problem was the magnification. And that was it! Magnification was the cause of the problem. I thought by matching the pupils and ensuring an appropriate size for the exit pupil I would have a good image size but that was not the case. To find the best image size I ... | {
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Why do we observe particles, not quantum fields? My understanding is that, in the context of quantum field theory, particles arise as a computational tool. We perform an expansion in the path integral in some parameter. The terms in these expansions correspond to Feynman diagrams which can be interpreted as interaction... | This is more of a side comment, but one way of thinking about quantum field theory is to regard it as a computational tool to compute particle interactions a la Weinberg : see https://arxiv.org/abs/hep-th/9702027 for this viewpoint on QFT. In a nutshell, we construct quantum fields to describe particle interactions obe... | {
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Do theoretical particle physicists really believe the particles they work on do not exist? Inspired by this article in The Guardian.
In private, many physicists admit they do not believe the particles they are paid to search for exist – they do it because their colleagues are doing it
This makes no sense to me - ther... |
Is it really the case that many theoretical particle physicists don't believe the particles they work on exist?
I hope not.
To be an adequate researcher, in physics particularly, you have to focus on what you are researching, the way an artist focuses on his/her work.
I think this is a biased article by a person who ... | {
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Rod in zero gravity pushed Imagine a rod - an ideal line segment of some length $l$ - is in zero gravity and initially no force is acting on it. Then something hits it and it gets some impulse $p$ in some location along the rod at some possible angle (excluding unrealistic cases such as a perpendicular impulse that is ... | If you know the point around which the system would rotate, you can apply the Conservation of angular momentum.Now, there is only one point about which a rigid system can freely rotate, that is the Center of mass.The center of mass rotation about any other point would accelerate and give impulses both in translational ... | {
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What does it physically mean for the inverse of the metric tensor of inertial frames to be the metric tensor itself? The metric tensor of inertial frames in S.R is given by $$g_{\alpha \beta}=diag(1,-1,-1,-1)$$
It's inverse $$(g_{\alpha \beta})^{-1}=g_{\alpha \beta}$$
I was wondering what this means geometrically. I kn... | OP's title condition is not a covariant/geometric/physical condition: it depends on choice of coordinates. E.g. if it holds for coordinates $x^{\mu}$, it will not hold for coordinates $x^{\prime \mu}=2x^{\mu}$.
See also Is it foolish to distinguish between covariant and contravariant vectors? and links therein.
| {
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Dimensionless power spectrum According to some physics notes I'm studying, this is the dimensionless power spectrum:
$$\Delta^2(k)=\dfrac{k^3 P(k)}{2\pi^2}$$
which is defined that way so that, after integrating out the azimuthal angle, we get:
$$\Delta^2(k)\ d\ln k=P(k)\dfrac{d^3k}{(2\pi)^3}$$
Can anyone explain to me ... | I find this to be an extremely confusing (and misleading) way of writing things down, but here's the best I can do to make sense of it. Multiply both sides by $dk/k$ and pull out a $4\pi$ on the right-hand side, yielding
\begin{align}
\frac{dk}{k}\Delta^2(k) &=\frac{dk}{k}\frac{k^3 P(k)}{2\pi^2}
\Longrightarrow
\Delta^... | {
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At which electric field or voltage does field emission occur? Imagine our environment is a vacuum. At which value of electric field or voltage, does field emission occur? I just want to know what is the maximum electric field we are allowed, to place between two electrodes, without field emission phenomenon. So, you ca... | Field emission is a quantum tunneling process, and as such can occur at any voltage, although for low enough fields it will be exponentially suppressed. Intuitively one should expect the emission rate to go as something like:
$$dN/dt \propto \exp(-eEd/W)\equiv \exp(-E/E_c),\, E_c= W/ed$$
where $e$, $E$ and $W$ are the ... | {
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Unitarity of the time-evolution operator In Sakurai equation 2.15, the infinitesimal time-evolution operator takes the form
$$\mathscr{U}\left(t_0+d t, t_0\right)=1-i \Omega d t $$
where $\Omega$ is a Hermitian operator. Below is a subsequent excerpt:
With $(2.15)$ the infinitesimal time-displacement operator satisfie... | This is not anything special about the time evolution operator. It's simply a way of expressing calculus using infinitesimals. A similar example would be that if $y=x^2$, then $dy/dx=[(x+dx)^2-x^2]/dx=2x+dx$, where we throw away the second term at the end. People used to habitually express calculus this way for hundred... | {
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Complex Grassmann Dirac Functional - How do we integrate over it? I'm following the Book of Brian Hatfield (Quantum Field Theory of point particles and Strings), p.192 here: For real Grassmann numbers (and Functionals thereof):
If $\Phi[\psi]$ is a functional, and $\psi(x)$ is a Grassmann-valued function, we demand tha... | Motivated by the one-dimensional complex delta function, whereby
$\delta_\mathbb{C}(z):=\delta(z)\delta(\bar{z}),$
so that
$$\int \mathrm{d}z\wedge\mathrm{d}\bar{z}\ \delta_\mathbb{C}(z-\zeta) f(z) = f(\zeta),$$
you can define
$\delta_\mathbb{C}[\psi] := \delta[\psi]\delta\!\left[\bar{\psi}\right],$
satisfying
$$\int\m... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/731486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding page 141 of Blundell’s Concepts in thermal physics On this page (in the second edition), there is a figure containing two states A and B of a system:
There are two paths between A and B: one is an irreversible change, and the other is a reversible change. However, on the same page it says that entropy st... | They might be referring to an adiabatic change, in which case, for a reversible path, the entropy change is zero. If B is the final state for an adiabatic reversible change, there is no adiabatic irreversible path starting at A for which B can be the final state. However, if the irreversible path is not adiabatic, th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Boltzmann entropy, what is $\Omega$, in $S=k_B \ln( \Omega )$? The Boltzmann's entropy formula $S=k_b \ln \Omega$ is valid both for equilibrium and non-equilibrium systems. On my textbook, it is written that $\Omega$ is the number of accessible microstates of the system. So, i guess that, considering an isolated system... | In the Boltzmann definition $S = k_B\ln \Omega$, $\Omega$ is not necessarily the number of accessible microstates, but it is the number of microstates compatible with the macrostate $\mathbf X$ (e.g. in equilibrium state, the triplet $E,V,N$, or in non-equilibrium state, $V$ and the density fields $e,n$). Only in equil... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complex Cosmological constant Does a complex cosmological constant ($\Lambda = a + ib,\quad b\neq 0$) exist? If it does exist, what does it represent physically?
For example, we interpret $\Lambda > 0$ as dS space and $\Lambda < 0$ as AdS. These have different cosmological consequences. Hence I am wondering about comp... | The cosmological constant enters the Einstein equations in the $\Lambda g_{\mu\nu}$ term. Classically, the Einstein equations result from differential geometry, where all objects of interest are real-valued, since by definition all charts of a differentiable manifold have values in $\mathbb{R}^n$ (corresponding tuples ... | {
"language": "en",
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Do the parts of proton move around and in that sense all protons, unlike electrons, are not the same? Two different protons could be in a different state because of its parts "moving" in some sense?
| Protons can be distinguished from each other in particular interactions using conservation rules for energy momentum and angular momentum, otherwise they are exchangeable.
Protons are complicated bound states of their constituents. As in all quantum mechanical formulations there are excited states as the answer in the ... | {
"language": "en",
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Conservation of mass in Navier-Stokes equation I just started looking into Navier-Stokes Equations and one of the two equation of Navier-Stokes is:
$$\nabla \cdot \vec u = 0.$$
This equation is said to be the conservation of mass. The concept of conservation of mass in fluid dynamics makes sense. My understanding is si... | The conservation of mass in the Navier-Stokes equations is,
$$\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf v\right)=0.\tag{1}$$
Using the material derivative, Equation (1) can be written as,
$$\frac{D\rho}{Dt}+\rho\left(\nabla\cdot\mathbf v\right)=0.$$
Then by using the condition of incompressibility $D... | {
"language": "en",
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Does a random number generator have real entropy? In thermodynamics, entropy is defined for gases. Of course, my laptop is not a gas. However, it contains a random number generator and I have seen the word ‘entropy’ being used in this context. Is this the same entropy? How can this entropy be linked to the definitions ... | Your laptop likely has a real hardware-based random number generator. Common implementations use electrical noise, caused by the fact that your hardware exists at a temperature above 0K.
However, the entropy of that electrical noise is reduced by many, many orders of magnitude. To name just one simple statistical reaso... | {
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Wrong scale factor on wikipedia From wikipedia we have for Scale-Factor $a(t)$:
".. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time "t" to their distance at some reference time t0". The formula for this is:"
$$ d(t) = a(t) * d_0\tag {F1}$$
"..if at pr... | Formula F2 is correct: the universe was smaller in the past, $a(t)$ for $t$ before the current time ($t_0$) is smaller than $a(t_0)$.
The argument that "the greater the speed, the greater the distance and also the greater the redshift" is not without any merit, but we must be careful about the specific distances we are... | {
"language": "en",
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Center of Gravity derivation question I need a sanity check, please.
When determining the center-of-gravity of a lamina described by $f(x)$, we know that by definition,
$\bar{x} M = \sum_{i=1}^{N} m_i \tilde{x_i}$
where $\tilde{x_i}$ is the location of the centroid of strip located at $x_i$
Assuming uniform density and... | Are you trying to re-invent the integral? When you go from discrete to continuous, you make the intervals infinitely small and you have to sum over a infinite number of "slices", broadly speaking retaining only the terms $\mathscr{O}(\Delta x)$.
x-coordinate of $G$.
In the continuous limit, the distance of the center o... | {
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Asymmetric bar errors Generally, when gaussian or normal distribution is assumed, we can calculate $\sigma$, the standard deviation of the population and $s$, the SD of the sample. When we have a non-symmetric probability distribution, how are asymmetric $\sigma_L$ and $\sigma_R$ calculated? In what cases are preferred... | Asymmetric error bars are common in experimental high energy physics, where the Particle Data Group convention is that uncertainties are defined by 68.3% Confidence Intervals (CI), so the error bars represent the 15.9% and 84.1% CI points. As noted in @basics answer, this or similar CI conventions are probably easier... | {
"language": "en",
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Wave eigenfunction and eigenvalue for step potential Given the Schrödinger equation:
$$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = E\psi$$
where:
$$\left\{ \begin{array}{l} V(x) = V_0 \text{ for }x>a \\ V(x) = 0 \text{ for } 0\leq x \leq a \\ V(x) = \infty \text{ for } x<0 \end{array}\right.$$
... |
We are left with 2 Unknowns, while we have 3 conditions left.
We want $\psi$ to be continuous at $a$,
we want $\psi′(x)$ to be continuous at $a$,
and finally we want it to be normalized.
How is this possible?
You are right, for most values of $E$ there is no solution.
However, for some special values of $E$ there is ... | {
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Zero interference between disconnected and connected diagrams in $2\to 4$ scattering in $\phi^6$ theory In Schwartz's book Quantum Field Theory and the Standard Model Exercise 7.2 is to show that in $2\to 4$ scattering there is no interference between 6-point vertex diagrams and the disconnected diagrams with two three... | Bit of a late response but I was wondering about this myself. I found this thread on PF that explains it quite nicely.
To paraphrase, the idea is that with a total $S$-matrix of
$$S=-i\lambda \ \delta^4(\sum_{\text{total}} p) + -ig\delta^4(\sum_{\text{subset}} p)\delta^4(\sum_{\text{subset}} p),$$ the second term is on... | {
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Experimentally Measuring the Velocity of Water coming out of an Orifice I plan on doing an investigation into Torricelli's Law, where I will be looking at one of the following:
*
*How the cross-sectional area of an orifice affects the velocity of water coming out of it (constant height).
*How the height of an orifi... | With a "high speed" camera, and assuming a near laminar flow, you could cut the flow with something like a fan blade and measure that edge created using a scale behind the water with respect to the camera.
Another is if you used a plunger (imagine a large syringe) then you can measure its position which directly gives ... | {
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How to show Pauli-Villars regularization introduce a momentum cut-off? Pauli-Villars regularization instructs us to do such a replacement:
$$\frac{1}{p^2-m^2+i\epsilon} \rightarrow \frac{1}{p^2-m^2+i\epsilon} - \frac{1}{p^2-\Lambda^2+i\epsilon}$$
And then claim: such a replacement implies a smoothly cut off when $p\geq... | When $p^2\gg \Lambda$, as you can see directly from your "modified" expression, the propagator scales as $\sim 1/p^4$ instead of $\sim 1/p^2$. If you go through arguments about the superficial degree of divergence, you'll find that this extra negative weight to the propagator will remove the divergent behavior you woul... | {
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How does the Pusey-Barret-Rudolph (PBR) theorem not just disprove hidden variables? In Quantum Mechanics, two different wavefunctions can have a non-zero probability of finding a particle at a position $x$.
According to hidden variable theories, if a particle is found at $x$, it was there before the measurement. $x$ wa... | In the post, I assumed that $x$ completely specifies the hidden variable state. But that's not true because the two states can have the same $x$ but differ in $p$. Hence the combined states $(x, p) $ are different and there's no contradiction with PBR.
| {
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Balmer proportionality How did Johannes Balmer arrive at
$$
\lambda \propto \frac{n^2}{n^2-4}, \quad (n=3,4,\dots),
$$
and then how did Rydberg mathematically derive
$$
\frac{1}{\lambda}=R\left(\frac{1}{n^2_1}-\frac{1}{n^2_2}\right)?
$$
I know $n$ stands for the shells but in the textbook, it doesn't define what $n$ is... | I recommend reading Balmer's original paper "Notiz über die Spektrallinien des Wasserstoffs" (1885).
Balmer took the known wavelengths of the visible hydrogen spectrum
($H_\alpha$, $H_\beta$, $H_\gamma$, $H_\delta$) as measured by
Ångström with high precision.
He recognized they are related by certain fractions.
$$\beg... | {
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How did the T-duality change the dimension of the $D$-brane? Quote Clifford Johnson $D$-brane page 125
Observe that, since T-duality interchanges Neumann and Dirichlet boundary conditions, a further T-duality in a direction tangent to a $Dp$-brane reduces it to a $D(p − 1)$-brane, while a T-duality in a direction
orth... | $T$-dualizing a certain direction in the target space changes the Neumann $\leftrightarrow$ Dirichlet boundary conditions of open strings, which is interpreted as the D-brane is filling/tangent $\leftrightarrow$ intersecting that direction, i.e. the D-brane dimension is 1 more $\leftrightarrow$ 1 less, respectively.
Ca... | {
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How can we describe the class of trajectories around a point mass in general relativity? As per the answers to this post, a Newtonian gravitational trajectory of a test particle about an ideal isolated point mass is always a conic section.
An ideal point mass in GR is a black hole, either Schwarzchild or Kerr, right? ... | The timelike geodesics around a Kerr (or Schwarzschild) black hole are not nearly as nice as their Newtonian counterpart, and certainly cannot be described as something as simple as a conic section. Here is an example of a generic bound geodesic in Kerr:
However, the geodesics in Kerr spacetime form an integrable syst... | {
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Expression for energy of a massive radially moving particle in a static spacetime I have become utterly confused about times and velocities measured by different observers. Let us take the specific case of a Schwarzschild spacetime. When we say that there is a particle of mass m moving with 4-velocity $u^a = dx^a/d\tau... | Oops: for the second formula,we have :$$E=-g_{00}mc^{2}u^{0}=-g_{00}mc^{2}\frac{dx^{0}}{ds}=\frac{-g_{00}mc^{2}dx^{0}}{\sqrt{-g_{00}(dx^{0})^{2}+g_{\alpha\beta}dx^{\alpha}dx^{\beta}}}$$
we know that : $dl^{2}=g_{\alpha\beta}dx^{\alpha}dx^{\beta}$ and $\;\;c^{2}d\tau^{2}=-g_{00}(dx^{0})^{2}$
$$=\frac{-g_{00}mc^{2}dx^{0}... | {
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Do solar panels act as an electrical load on the sun? We know that the sun uses nuclear fusion to generate sunlight and heat energy. If we are using solar panels to harvest solar energy, aren't we putting some electrical load(resistance) on the sun? If yes, does it have any effects on it?
Edit: Does the presence of a r... | No, but your intuition that there should be some effect isn't entirely incorrect. The term for the type of phenomenon you're describing is "near field effects," and it comes up in antenna engineering. There are two types of near field effect - radiative and reactive. Radiative effects are caused by EM waves bouncing ba... | {
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How do I derive the Dirac Lagrangian? It's frequently said, that the Lagrangian of a Dirac field is
$$\mathcal{L}=i\bar{\psi}(\gamma^\mu\partial_\mu-m)\psi.$$
Applying the Euler-Lagrange equation we get the Dirac equation. Although, we can get a similar construction of Lagrangian, which leads to the same equation, e.g.... | Both are perfectly valid. I actually prefer the second.
| {
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How to explain that black holes mergers take finite time but black hole forming infinite? Through detection of gravitational waves we observe coalescences of black holes, black holes and neutron stars, or even neutron stars into black hole, although forming of a black hole should take infinitely long time for an outsid... | Technically, the final phase of a compact binary coalescence, the ringdown, takes infinitely long. The ringdown is an exponentially decaying process, and a well known property of a decay exponential is that it never truly vanishes. More practically, the exponential decay is so quick that within a fraction of second it ... | {
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Do all objects at the same temperature glow the same color? Does Kirchhoff's law for heat radiation imply that all objects at the same temperature will glow the same color?
In other words, if a piece of molten iron glows the same color as my body, which radiates the same color as the sun, therefore all three objects -... | Yes, all the bodies glow same color at given temoerature and wavelength at thermal equilibirium according to kirchoff's law. What difference is that intensity of different objects is different, not the shape of spectral curve. So a ceramic glow with same color as iron at same temperature. It is stated as,$$\frac{e_1}{a... | {
"language": "en",
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In firing a single photon at the center divider of a double slit, does it ALWAYS go through the slits? If we think of a single photon approaching the slits as a wave function, and we fire the photons at the midpoint of the two slits, one at a time, then I would think the probability function is highest at this midpoint... | Short answer: When the photon hits the walls or the web of the double slit, obviously an interaction takes place. The photon is absorbed and finally EM radiation is re-emitted.
Long answer tl;dr: What happens at the boundary regions between the slit walls? The photon passes through unhindered? No, it also has an intera... | {
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When are my fluid approximations wrong? I did some classical approximations of the Navier Stokes equations, fluid is:
*
*non-viscous
*incompressible
*irrotational
When are these approximations wrong? and particularly is there a "general method" to evaluate in a theoretical way "the error" of an approximation?
For ... |
When are these approximations wrong ?
Almost all real fluids will have some viscosity (an exception may be superfluid helium-4) and some degree of compressibility. And there are simple situations where flow is not necessarily irrotational e.g. flow between two concentric cylinders. So one or more of the assumptions o... | {
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How are electrons really moving in an atom? Niels Bohr proposed the solar system model of the atom, which is the most recognizable, and assumed electrons revolve in circular paths called orbits/energy levels. However, we know that the Bohr model is incorrect due to later observations, like the Stark and Zeeman effect.
... | While classical mechanics describes the motion of objects, quantum mechanics describes the object in terms of a probability amplitude density. Thus, quantum mechanics assumes that the object takes all possible paths, which are consistent with the information available at the start of the experiment. Therefore, the mot... | {
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In magnet and conductor problem, what is the source of electric field? I have difficulty understanding forces involved in moving magnet and conductor problem.
When a ring conductor is at rest and an ordinary bar magnet is moving, there's electric force. I have trouble understanding how the electric force arise here. Ba... |
According to this equation $\nabla \times E = -\frac{\partial B}{\partial t}$, it looks like there is electric field flowing clockwise direction like the current. So the clockwise electric field which produces electric force is responsible for creating clockwise current?
Yes.
| {
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Orbital motions Suppose two satellites are moving in circular orbits around the earth and then crash into one another.
My Question is this, how do we know that the bodies move in an elliptical orbit after the crash? Why not just remain in a circular orbit with a smaller radius (r)? Why not instead go into a hyperbolic ... | bevor the collision both masses have the velocity $~\omega\,r~$.
the velocity $~v'~$ after the collision is the center of mass velocity
$$v'=\frac{m_1\,v_1+m_2\,v_2}{m_1+m_2}=\omega\,r\frac{m_1-m_2}{m_1+m_2}$$
with your data $~v'=\frac 35\omega\,r$
using the Vis-Viva equation
ellipse
$$v^2=G\,M\left(\frac 2r-\frac 1a\r... | {
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Does horizontal acceleration affect gravity? If we apply 1G horizontally in some object, will this constant force equal to G affect the time of falling? If the force does not affect gravity, why gravity is prioritized over this force if both are equal?
Edited: For the ones who didn't understand what i mean well, i mean... | In newtonian mechanics atleast we take acceleration to be a vector, and in english we can usually say that gravity acts vertically, so when you have a force that is acting horizontally that is 90 degrees relative to the vertical this leaves no component of the force to decelerate the falling gravitational effect.
I hop... | {
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What is the relation between Newton's third law and law of conservation of energy? Using newton's third law we can find law of conservation of momentum.
*
*$F_{12} = - F_{21}$ $\hspace{2cm}$ Newton's 3rd law
*$F_{12} t = - F_{21} t$ $\hspace{1.8cm}$ Multiplying by "t" o.b.s , it is impulse (this step is important ... | The third law does not imply conservation of energy. Suppose the potential energy of a two particle system is $V(x_1,x_2,t)=\frac{t}{x_1-x_2}$.
Then, the forces on the particles are:
$$F_1(t)=-\frac{\partial V}{\partial x_1}=\frac{t}{(x_1-x_2)^2}$$
$$F_2(t)=-\frac{\partial V}{\partial x_2}=\frac{-t}{(x_1-x_2) ^2}$$
The... | {
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"source": "stackexchange",
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Field renormalization in 4D $\phi^4$ theory from sunset diagram Consider the contribution to the two-point function in (euclidean) 4D $\phi^4$ theory, given by the sunset diagram. It is proportional to:
$$
\int \frac{d^4k}{(2\pi)^4} \frac{d^4q}{(2\pi)^4} \frac{1}{(k^2+m^2)(q^2+m^2)((k+q-p)^2 + m^2)}
$$
where $p$ is an ... | I think the final integral is still divergent. Consider the integral over $q$. It is no less than the integral over $q$ over a ball of radius order 1 centered at $-k$, which in turn is around $k^{-4}$ at large $k$. Therefore the $k$ integral is logarithmically UV divergent, as expected.
| {
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What is the state of an entangled photon after its twin is absorbed? Let's two photons are entangled in polarization after a laser beam passes through a Betha Barium Borate crystal. They take different paths and one of them (1) is absorbed in a black sheet. What is the state of the leftover photon (2)? Is it in superpo... | The state of the remaining photon depends both on the nature of the initial entangled state as well as the nature of the absorption (or detection) of the other photon. To get an entangled photon with a BBO crystal, one needs to use type II phase matching, which would produce a Bell state for a single photon pair, given... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 2
} |
Why there is no $s$-channel for fermion-fermion scattering? I'm learning the Lagrangian for Yukawa theory, where $L_{int} = \phi\bar{\psi}\psi$. For the fermion-fermion scattering, we can draw the Feynman diagrams as
My question is why we can't have $s$-channel here? If it exists, it still seems like we can have a dia... | Because the interaction has to have fermion going in and fermion coming out, or anti-fermion in/anti-fermion out. The arrow shows the direction of fermion propagation. There is no interaction like
so there is no s-channel.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is there a "fundamental problem of thermodynamics"? The "fundamental problem of mechanics" can be boiled down to finding and solving the equation of motion of a system. Similar statements can be said for quantum mechanics for the Schrödinger equation and for electrodynamics and Maxwell's equations, etc. But is there su... | I'm lumping together statistical mechanics with thermodynamics here, since in my mind they are inextricably linked. I would say the fundamental problem of statistical mechanics is to calculate the maximum entropy distribution for a system plus its environment, given a set of constraints and experimental conditions. The... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
} |
Exchange operator with position and momentum I need to prove that for a given exchange operator $\hat{P}_{12}$ such that,
$$\hat{P}_{12}|x_1,x_2\rangle = |x_2,x_1\rangle $$
$\hat{P}_{12}\hat{X}_1\hat{P}_{12}=x_2$ and $\hat{P}_{12}\hat {P}_1\hat{P}_{12}=p_2$ where $\hat{X}_i$ is the position and $\hat{P}_i$ is the mome... | Let us suppose that the vectors $|\alpha_1,\alpha_2\rangle$ form a basis of your Hilbert space and that ${\cal O}_1$ and ${\cal O}_2$ are operators such that $${\cal O}_1|\alpha_1,\alpha_2\rangle=\alpha_1|\alpha_1,\alpha_2\rangle,\quad {\cal O}_2|\alpha_1,\alpha_2\rangle=\alpha_2|\alpha_1,\alpha_2\rangle.$$
You want to... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Is there a relationship between the electrical power supplied to a material and the emitted wavelength? I have a graphite bar that is used in a vacuum furnace that serves as a heating resistor and the heat is provided by radiation, so I would like to know by knowing the electrical input power (240kW), the radiative pow... | The material is not emitting a single wave length, but rather a continuous spectrum, which is described by Planck's law:
The temperature then describes the shape of the curve and the position of the peak. One could the determine the position of the peak wave length in terms of temperature - see also Wien law.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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