Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Neutrino flavor and mass eigenstates Neutrions are produced and detected as flavor eigenstates $\nu_{\alpha}$ with $\alpha=e, \mu, \tau$. These states have no fixed mass, but are the combinations of three mass eigenstates $\nu_{k}$ with $k=1, 2, 3$, with mass $m_1$, $m_2$ and $m_3$, respectively. My questions are:
a) d... | (a) They start as a flavor eigenstate, which is a super position of mass eigenstates. The mass eigenstates have different time evolution, hence the state is, in general, a mixed state in either basis.
(b) No. As an analogy, consider polarized photons and Faraday rotation--it may start out + polarized, rotate to a mixtu... | {
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"timestamp": "2023-03-29T00:00:00",
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Feynman Diagram with antineutrinos and neutrinos I just recently learned how to draw feynman diagrams by looking at an equation such as one for $\beta-$ Decay:
$$n = p + e^- + \bar{v}_e$$
I was wondering in $\beta-$ why an electron-antineutrino was produced and not just an electron-neutrino? Similarly, why does $\beta+... |
I was wondering in $\beta^{-}$ why an electron-antineutrino was produced and not just an electron-neutrino?
Because lepton number has to be conserved. In the LHS the lepton number is zero so on the RHS the lepton number must also be zero. All leptons have assigned a value of $+1$ and antileptons have $-1$. The conse... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/370216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Wrong sign in Conformal Casimir The quadratic conformal Casimir in $d$-dimensional Euclidean space is given by
\begin{equation}
C = \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - D^2 -\frac{1}{2}\left(P^\mu K_\mu + K^\mu P_\mu \right)
\end{equation}
as given for example in the beginning of lecture 6 here http://pirsa.org/C14038. ... | To do the computation, considering
$$\frac{1}{2}M^{ab}M_{ab}=\frac{1}{2}(M^{\mu\nu}M_{\mu\nu}+M^{\mu0}M_{\mu0}+M^{\mu,-1}M_{\mu,-1}+M^{0\nu}M_{0\nu}+M^{0,-1}M_{0,-1}+M^{-1,\nu}M_{-1,\nu}+M^{-1,0}M_{-1,0})=\frac{1}{2}(L^{\mu\nu}L_{\mu\nu}-\frac{1}{2}(P+K)^2+\frac{1}{2}(P-K)^2-2D^2)= \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/370318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is it feasible/possible to use refraction for x-ray spectrum analysis? In x-ray spectroscopy Bragg reflection off of a crystal is used for spectral analysis. In x-ray diffraction the same principle is used for monochromatizing the x-ray beam from an x-ray tube. For visible light, however, an optical prism is used to de... | There is http://henke.lbl.gov/optical_constants/
But no, prisms are not suitable for x-rays: dispersion is small, there will always be absorption. And there are anomalous effects near absorption edges.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is going on inside this disc I recently saw a video of a copper disc repelling paper pins which I suppose are made up of steel. By watching the video carefully in slow motion, I feel there is something inside this copper disc (As the pins are getting rotated and thrown out from the disc's surface).
My questions ar... | Some comments suggested that the effect is caused by diamagnetism, other comments contained doubts, as diamagnetism is weak. Let me note, first, that copper is indeed diamagnetic (although it is not mentioned in the video that the disk was made of copper, unless I missed something) and that effects of diamagnetism can... | {
"language": "en",
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Does salt affect the boiling time of water? If I have 1 cup of water on the stove and another cup of water with a teaspoon of salt.
would the salt change the boiling time of the water?
| Since you are interested in time two factors have to be considered:
1) The increase in the boiling point temperature for the salt solution. This requires that more energy (aka heat) be transferred to the solution than the pure water to reach boiling. If this were the only consideration then indeed it would take more ti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Force on plate of parallel plate capacitor with dielectric If we have a parallel plate capacitor whose charge is +Q and the polarization charge as Qp as shown in the figure..
then while finding the force acting on the left plate of the capacitor for instance, shouldn't the force due to the polarized charge -Qp and +Qp... | The nett force on the Dielectric due to the involved Electrostatic interactions is zero. There is, however, a nett force on the dielectric on the plates because the forces from both the face of the dielectric are not equal in magnitude and don’t cancel out.
I have assumed the numerator of both F(-q) and F(+Q) to be ... | {
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Rotation in Higher Dimensions In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation.
In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom rotate, and if so does it rotate around a line, a plane, or a subspace of smaller number of dimensi... | *
*One may show that a general rotation $R\in SO(N)$ in $N\geq 2$ spatial dimensions can be composed
$$R ~=~ R_1\circ \ldots\circ R_{k} $$
of at most $k=[\frac{N}{2}] $ pairwise commuting rotations $$R_1,\ldots, R_{k}~\in~ SO(N)$$ that each leaves a co-dimension-2 subspace invariant (although not necessarily the same ... | {
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Increasing distance between Earth and Moon I have a problem where a planet's rate of rotation is decreasing due to tidal effects of the moon. I know that the angular momentum of the system will be conserved. So, in order to conserve that the moon will recede away from the planet.
$$ L = mvr = m\omega r^2.$$
I'm not sur... | Whether the Earth slows down or speeds up isn't something you can just decide from basic logic. It is a mathematical question whose answer depends on the relative phase relationship of the driving force (the moon) and he driven object (the tide). If the frequency of the driving force is less than the natural frequency ... | {
"language": "en",
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Special relativity and spinning tires I apologize for the subsequent headache. There is a person who claims quite adamantly that Einstein is wrong, using the following reasoning:
A clock on a train slows down the faster the train moves. Mechanical clocks are valid clocks too. The wheels of the train are mechanical clo... | One important feature of special relativity is that special relativity does not know bodies of absolute rigidity, the universe is composed of particles which are held together by finite forces.
Example: Einstein's length contraction considers rigid objects for purposes of simplification only, for a better understanding... | {
"language": "en",
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Why does a DC voltmeter show a zero reading when an AC voltage is applied? One of my books says that if an AC voltage is applied across a DC voltmeter, its reading will be zero.
I think that since average value of AC voltage(in a complete cycle) is zero, DC voltmeter measures it as zero.
But I couldn't find a reliable ... | Think of it his way.
Assume you have a really, really fast DC voltmeter (and really fast eyes as well). In that case you will see the voltmeter go up and down, positive and negative, as the input AC voltage fluctuates.
In reality, a DC voltmeter is not that fast, so it will start to move up a very tiny bit, and then i... | {
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Do higher frequency/energy levels in the EM spectrum mean higher temperatures? I am trying to find concrete evidence that for example, light in the optical spectrum would be hotter than infrared light because it has a higher frequency, and that is directly proportional to energy. Is energy directly proportional to temp... | Molecules of matter at given temperature have satisfy below relation,
$\Delta E=NkT\tag1$
where $N$ is number of particles, in this case number of molecules and $T$ is temperature.
From planck's law, energy difference of higher frequency level is given by,
$\Delta E=Nhf\tag2$
where $N$ is number of particles, here it i... | {
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Mercury's precession I read in an article about Mercury's precession that Newton's law of gravitation predicts such precession of planets ;but fails to caluclate the precession of Mercury.But most of popular science books or other articles on the internet suggest that Newton predicts identical ellipses whereas the rea... | Newton's law of gravitation predicts a perfect ellipse for the orbit of a planet orbiting a star, only in the idealized case that there a just those two bodies (the planet and the star). However, in reality there will be many bodies orbiting the star, each with its own gravitational field that slightly perturbs the orb... | {
"language": "en",
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Clarification about what makes a system isolated I am a grade 12 physics student and I just need some clarification about what makes a system isolated. I've read the definitions online, but they still don't make a lot of sense. For example:
When people jump on a rotating merry-go-round, the angular momentum decreases. ... | It all depends on the system under consideration (that's fancy text for the system you're thinking of). In the first part, you could argue that when people jump off, the momentum of the entire Earth (which is what you landed on) is conserved because it's an isolated system. Your physics teacher is assuming that you are... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Physical meaning of gauge choice in electromagnetism In electromagnetism, it is often referred to gauges of the electromagnetic field, such as the radiation or Coulomb gauge. As far as I know, the definition of a gauge helps us to redefine the problem in terms of a vector potential and a scalar potential that, since we... | In classical physics, and also quantum gauge field theory with an abelian gauge group (like QED), the choice of gauge has no physical significance whatsoever. It's basically just like choosing where to place the origin of your coordinate system. In nonabelian gauge quantum field theory the situation is a bit more subtl... | {
"language": "en",
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"source": "stackexchange",
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Quantum State Representation with Commuting Operators Let $[A,B]=0$. Then, we can find a set of eigenvectors $\{|a_n,b_n\rangle\}$ common to both $A$ and $B$. According to this, and my own understanding, it makes sense to write an arbitrary quantum state as
$$\tag{1}|\Psi\rangle=\sum_n \sum_i c_n^i |a_n,b_n,i\rangle,$$... | In the first equation, n is the index of the pairs ${a_n,b_n}$ of simultaneous eigenvalues, and $i\geqslant 1$. In the Cohen's book equation, he has two indexes, n for $A$ eigenvalues and p for $B$ eigenvalues.
If we have a system with four states $|a>$, $|b>$, $|c>$, $|d>$, that have the property that:
$A|a>=|a>$, $A|... | {
"language": "en",
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Will current flow if there's no return path? Here is the problem I was trying to solve:
Find the potential difference between the points A and D
I used Kirchhoff's voltage law for the left loop and right loop and found out the current through the left loop to be $\frac{10}{2+3}$ A (2A) and for the right loop $\fr... | If there were a constant current between B and C, then the left and the right part of the circuit would charge indefinitely with the charges of opposite signs, the potential difference between the left and right parts would increase and eventually the current between B and C will stop.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Trajectory of a rolling ball with uneven weight distribution A perfect ball is rolling on a plane. Without further forces, it would roll in a straight line, and that's it.
What, however, if the ball's weight distribution is uneven? For example, the ball might have a higher-density smaller ball placed within it, but bei... | Well, as the center of mass is not coinciding with the normal force(vertically upwards), there will be a net torque about the point during the motion of the ball. When the heavier ball is towards the front, it will support rolling and increase the rotational speed of the ball. When it will go to back, it will give the ... | {
"language": "en",
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"source": "stackexchange",
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How do electron microscopes not get obstructed by atoms in the air? How do different electron microscopes avoid just scanning the atoms between the probe and the surface of the object that is actually being scanned?
| Electron microscopes, Transmission Electron Microscope (TEM) and Scanning Electron Microscope (SEM), are both operated under high ($10^{-6}-10^{-8}$ Torr) or ultra high vacuum ($<10^{-9}$ Torr) conditions. There are two reasons for this:
*
*to keep the specimen clean
*to avoid scattering of electrons by residual ga... | {
"language": "en",
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"source": "stackexchange",
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Transition amplitude of n-vacuua in QCD The vacuum state of QCD is a superposition of different ground states with non-tirvial topological charge $\propto n\in \mathbb{Z}$. We lable this vacuum configurations with $|{n}\rangle$. The transition amplitude is given by:
$$ \langle n|m\rangle = \int \mathcal{DA}_{n-m}\ \ ... | The path integral starts from a configuration $A^{(m)}$ of a topological charge $m$ and ends in a configuration $A^{(n)}$ of topological charge $n$.
The reason that it depends only on the difference is that both the QCD Lagrangian and the path integral measure are invariant under large gauge transformation, i.e., gauge... | {
"language": "en",
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Frames of reference and inhomogeneity and anisotropy of space I was reading mechanics by Landau and Lifshitz where I encountered this statement, "If we were to choose an arbitrary frame of reference, space would be inhomogeneous and anisotropic." I tried to think of random frames but couldn't come up with something whi... | Consider a noninertial frame (one accelerating in an arbitrary direction for instance). In this frame the expansion of space would appear different in different directions and would change with time.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/374423",
"timestamp": "2023-03-29T00:00:00",
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Boltzmann's equation and Liouville's theorem in curved spacetime I have two related questions:
How Boltzmann equation can be written in a covariant (using differential forms, connections, and etc.) way in a classical (not quantum) but curved system?
How does the Liouville's theorem change in a curved background?
Any an... | Liouville's Theorem is pretty easy because it's the same in curved spacetime as in flat spacetime. MTW gives "Liouville's theorem in curved spacetime" as
The volume $\mathscr{V}$ occupied by a given swarm of $N$ particles is independent of location along the world line of the swarm.
See section 22.6 for the discussi... | {
"language": "en",
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How does an electromagnetic wave move? Somewhere I found the explanation that the EM fields create and destroy each other during the oscillation (I suppose by Faraday's law) and this makes the wave "move".
I can't imagine this because unitary vectors in E,B and k directions are a right hand ordered set of vectors and ... | An electromagnetic wave is a change in the electromagnetic field of an object. The change in its electromagnetic field can only propagate at the speed of light. If the sun disappeared, it would take 499 seconds for the change in the gravitational field to propagate to earth and earth to go off at a trajectory; at the s... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/375052",
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"source": "stackexchange",
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Extreme life - energy source for living tens of kilometers underground? Living cells were found up to at least 12 miles underground (article), and in other extreme places (BBC survey article), for which beside the problem of just surviving in such extreme conditions, a basic physics thermodynamical question is: what en... | Chemical.
As the Wikipedia entry on Lithoautotroph puts it (restricting ourselves to the deep underground forms):
derives energy from reduced compounds of mineral origin
which they do through inorganic oxidation (see, e.g., Lessons from the Genome of a Lithoautotroph: Making Biomass from Almost Nothing) or other reac... | {
"language": "en",
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Why is the normal force $(M+m)g$? I am trying to understand the solution to this problem. The problem asks to find F such that m stays fixed relative to M. In the solution, it is mentioned that the normal force for block M is (M+m)g, I don't understand that. I thought it is supposed to be only Mg.
The solution states ... | The normal force is the force the table (or surface) must exert on the block $M$ in order to keep it stationary in the vertical $y$ direction. This means that the normal force must be equal and opposite to the net downward force that is being applied on M. The net downward force being applied on $M$ in this question is... | {
"language": "en",
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How do you solve the Schrödinger equation with a position space delta function potential in momentum space? I am solving the Schrodinger equation in position space with an attractive delta function potential energy,
$$
-\frac{h^2}{2m} \frac{d^2}{dx^2} \psi(x)-\lambda \delta(x) \psi(x)=E \psi(x),
$$
for a bound state. I... | Start from the position space SE with potential $-\lambda \delta(x)$. Define the Fourier transform as:
$$\tilde \psi(k)=\frac{1}{\sqrt{2\pi\hbar}} \int dx \,\,e^{-ikx/\hbar}\psi(x)$$
The Fourier transform of the product is given by:
$$\mathcal{F}[\delta(x)\psi(x)](k)=\frac{1}{\sqrt{2\pi\hbar}} \int dx \,\,e^{-ikx/\hbar... | {
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Nuclear Fusion: Why is spherical magnetic confinement not used instead of tokamaks in nuclear fusion? In nuclear fusion, the goal is to create and sustain (usually with magnetic fields) a high-temperature and high-pressure environment enough to output more energy than put in.
Tokamaks (donut shape) have been the topolo... | Background: confining a plasma requires controlling all of an enormous spectrum of possible instabilities. The tokamak does a good job of stirring the flows so that no individual instability can grow so much that the plasma rushes out and contacts the wall (and thus is quenched). Regarding the particular question of a ... | {
"language": "en",
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Why do fundamental particles have a specific size? If Quantum Field Theory is accurate, all particles are actually just excitations of the field in which the particle interacts.
Therefore, wouldn't it be possible to have particles of any conceivable size, provided the energy, couldn't you have a photon the size of a bu... | Does a single photon have a size?
It depends on what you mean by size. When you look at a basketball and you think of its size, you are looking at the entire space that the entity exists in. But you could cut up pieces of that basketball, you would see that each piece of the basketball takes up space on its own. A phot... | {
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Quantum states in position and momentum phase space While studying introduction to statistical mechanics ,I came across a new idea phase space where we use both position and momentum coordinates to denote a system .In my book the author calculates the number of quantum states within the energy range between E to E+dE.B... | The number of states with an energy between $E$ and $E+dE$ is: $$dN_E=D(E)dE$$ where $D(E)$ is the density of energy states. The number of available states in a system is one, and this number is the same no matter what you consider, whether it's energy or momentum. And it's from here that you impose that the number of ... | {
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In semiconductors when electrons jump from VB to CB, do they leave behind their parent atom's nuclei? In semiconductors/conductors when electrons jump from VB to CB, do they leave behind their parent atom's nuclei?
If yes, when this happens in Si (electron jump from VB to CB) why don't they ionize Si to Si +?
If no, w... | When an electron with its negative elementary charge moves from the VB to the CB it leaves behind a positive elementary charge, a hole. The positive charge of the hole is, of course, related to the positive Si+ ions in the crystal lattice. But the hole is not localized at a specific Si atom. Its wave function is spread... | {
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Inverse Square relationship using paint problem confusion I want to ask a question about the inverse square relationship using an aerosol paint spray mentioned in my book.
I am reading the book Advanced Physics by Steve Adams, and it mentions this in the book.
Imagine you are holding an aerosol paint spray at $50$cm ... | It is square inverse intensity even if you can adjust the spread of spray like most real life paint sprayers to spray wider or narrower, even if the nuzzle has been damaged and sprays a wiggly circle like an amoeba with one or two spots even being sprayed outside of the main circle or holes left inside.
Let's assume th... | {
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Why does gravity need a graviton? Einstein theorized that gravity is a phenomena manifested by the curvature of spacetime, in effect it IS the curvature of spacetime. If this is so, why do we need a graviton to convey the force of gravity? If I have mis-understood Einstein then I would appreciate a little help in gras... | the (mostly) non-mathematical answer is that any time we have a field in physics, there will be defined for that field an associated quantum, which can be considered an excitation of that field. when that field is responsible for transmitting forces between objects, that process can be modeled as the exchange of those ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/377326",
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Solving the Lie algebra of generators: path from algebra to matrix representation Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom representation of groups is more important... | You got outstanding answers, but they are uncharacteristically abstract for physicists; they assume your students are comfortable with basic Lie algebra theory, as taught to mathematicians, by example, in their first week of such courses. Unfortunately, physicists, who are especially used to being taught by example eve... | {
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Why does Griffiths's book say that there can be no surface current since this would require an infinite electric field for an incident wave? In sec. 9.4.2 Griffiths shows the well known boundary conditions for E and B fields, one of them is this:
$$\frac{1}{\mu_{1}}\textbf{B}_{1}^{\parallel}-\frac{1}{\mu_{2}}\textbf{B}... | While Griffiths typically writes surface currents exclusively as $\mathbf{K}_f$, one can also write them as volume currents as $\mathbf{J}_f = \delta(s) \mathbf{K}_f$, where $s = 0$ corresponds to the surface being considered. For an ohmic material, this means an electric field $\mathbf{E} = \frac{1}{\sigma} \delta(s) ... | {
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Unitary Transformation of Eigenstates Suppose I have two operators, $A$ and $B$, with eigenstates $A \lvert a \rangle = a \lvert a \rangle$ and $B \lvert b \rangle = b \lvert b \rangle$, where $a$ and $b$ are all unique. Furthermore, suppose that $A$ and $B$ are related by a unitary transformation $$A = U B U^{-1}.$$ T... | I’m not sure what you mean by “$a$ and $b$ are unique” but clearly if $A=UBU^\dagger$ and $U$ is unitary, $A$ and $B$ have the same eigenvalues but it doesn’t mean $U$ doesn’t do anything.
For instance, the Pauli matrices $\sigma_{x,y,z}$ all have the same eigenvalues, are related by a unitary transformation $U$, but... | {
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Experiment on friction coefficient Here you can see the results of the experiment about a friction coefficient:
The mean of the friction coefficient becomes 0.262 but when I do a linear regression in the form of y=mx the slope is 0.31. Shouldn't it be the same? I used $F_N$ as x values and $F_D$ (friction force) as y ... | You have a $F_D$ measurement issue because from the numbers it appears the linear regression gives you a negative y-offset. This means you are missing some force that is not measured.
Go with the linear regression slope, removing this error as any constant value in the measurements is taken out. This would be the 0.38... | {
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Validity of Boltzmanns Equation and $H$-function theorem? A while ago I came across a resource (which I have forgotten) on the validity of Boltzmann's equation. It talked about the fact that the Boltzmann's equation is valid at the extrema of the $H$-function. In the discussion there was a graph that looked similar to ... | Very late answer, but it may be Kerson Huang "Statistical Mechanics" fig. 4.7 pag. 89 (in second edition).
The dots are to evidence the local maxima in time, related to states of molecular chaos of the gas. The theory states that a system with a distribution function $f$ that satisfies Boltzmann's transport equation te... | {
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Interval Preserving in Minkowski Space The squared line element in any spacetime is given as $$ds^{2}=g_{ab}dx^{a}dx^{b}.$$ The use of tensors helps us to infer that the line element in some other frame would be $$ds'^{2}=g'_{ab}dx'^{a}dx'^{b}$$ where simply $dx'^{a}=\frac{\partial x'^{a}}{\partial x^{b}} dx'^{b}$.
My... | The reason you've been unable to find a derivation of the Lorentz transformation (relating, say, Bob's frame to Alice's) from the usual two postulates of special relativity is that the Lorentz transformation does not follow from those postulates. You're going to need some additional assumption.
You can, for example, ... | {
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$i\varepsilon$ in momentum space propagator; is it actually needed? In (say) phi-4 theory the momentum space propagator is given by:
$$\frac{i}{p^2-m^2+i\varepsilon}$$
where I am using the signature $(+---)$. Now momentum space we can do momentum space integrals using the Schwinger Paramterization etc in which we do no... | Yes, it is necessary. See, the real space propagator isn't actually analytic, and the poles in the momentum space propagator will make of your integrals diverge unless you move them off of the integration axis (equivalently, deform the contour around the poles) and take a limit that moves them back onto the integration... | {
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Does radiation cause a change in temperature? If yes, then is there a limit to the temperature decrease? If no, then can the body which radiates heat attain an absolute zero temperature?
| Yes, radiation can cause a change in temperature. It's a form of heat transfer, after all. Radiative heat transfer can cause an object to warm up or cool off.
The Earth's temperature hasn't changed all that much over the last several million years. (Global warming and ice ages represent smallish temperature changes.) T... | {
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Would "gravity" and the "law of gravity" have a meaning in a universe without matter? I was discussing the fact that if there was no matter in the universe, just vacuum and radiation, can we say that anything called gravity wouldn't exist?
In that universe, the Friedman equations would still be useful, but is it relat... | The Friedman Equations would still exist. In other words, $G_{\mu\nu}=8\pi G T_{\mu\nu}$ would still apply, but the assumption of vacuum means that $T_{\mu\nu}$ would be 0 (we can immediately see that the gravitational constant G falls out due to multiplication by zero). If I claim gravity is the curvature of spacetime... | {
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Physical significance of the zeroth component of 4-velocity and 4-force Is there any physical significance of the zeroth component of the four velocity vector and four force vector? I understand that the space part of u$^\mu$ is related to ordinary velocity and space part of F$^\mu$ is the usual force. But are there an... | The zeroth component of the 4-velocity $u^a=(\gamma ,\gamma \vec v/c)c$ is essentially the time-dilation factor $\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$ (multiplied by $c$ for dimensional purposes). Using the rapidity $\theta$ (the Minkowski angle between two timelike vectors), that zeroth component is essentially $\cosh\th... | {
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Density Functional Theory for Quantum Field Theory vs fixed-particle-number Quantum Mechanics Introductions to Density Functional Theory (DFT) usually discuss the Hohenberg–Kohn theorems which prove that there exist universal functionals of density that can be used to determine ground state properties of a system. This... | I suspect this isn't quite what you're looking for, but it's too long to share in a comment:
Relativistic corrections can be added with augmentation methods. Usually though, relativistic corrections arise in the core of (typically heavy) atoms and may therefore be incorporated with small adjustments to the pseudopotent... | {
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How to measure a static electric field? I looked up google but didn't find any design for measuring electric field that doesn't vary with time.
My own idea is to use two parallel plates (like a capacitor but without the dielectric). In an electric field E a potential difference V = Ed (d is separation between the plat... | An old method to measure an electric field does, ideed, use two thin metal plates of area $A$ held on insulating handles. These metal plates are put in contact and the combined plates are inserted into the electric field so that the surfaces are normal to the field lines. Then a (positive and negative charge) of $Q=A\... | {
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Intrinsic Concurrent Pitch/Yaw/Roll Rotation Between Two Rotation Matrices Given an object with a rotation matrix, how do you calculate the pitch, yaw, and roll velocities that needs to be applied over time for the object to reach a goal rotation matrix given that:
*
*The x-Axis is left, the y-Axis is up, and the z-... | You can find the rotation axis $\vec{z}$ and angle $\theta$ between the two orientations and use this information to apply $\vec{\omega} = \frac{\theta}{\Delta t} \vec{z}$.
Given two 3×3 rotation matrices $\mathrm{R}_1$ and $\mathrm{R}_2$ the relative rotation matrix is
$$ \mathrm{R} = \mathrm{R}_1^\top \mathrm{R}_2 $$... | {
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What does Mobius group/transformations have to do with special relativity? The group of Mobius transformations, denoted by ${\rm Mob}(2,\mathbb{C})$, is isomorphic to ${\rm SL}(2,\mathbb{C}))/\mathbb{Z}_2$ which in turn is isomorphic to the Lorentz group ${\rm SO}^+(3,1)$.
This connection, to me, seems very intriguing... | You'll find an intuitive way into your question via the Wiki article on Möbius.
"In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativist... | {
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Is this "Permanent magnet gun" real or fake? There are several videos on youtube describing linear accelerator built solely from permanent magnets, put like this:
I find it hard to believe that this can work because if it did, it would be exploitable to gain energy. Where would that energy come from then?
I think it's... | In the video I see that the author needs quite some force to load the gun. So here he puts the projectile in a high potential state, i.e. he provides the energy during the loading process. When released this energy accelerates the projectile. Hence, no violation of energy conservation.
Edit
Here a small representation ... | {
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Strouhal number motivation I am looking for a nice way to motivate the Strouhal number definition. Let me illustrate what I mean on the Reynolds number. (As ususal, $\mathbf{u}$, $p$, $\rho$, $\nu$ denote the flow velocity, pressure, density and kinematic viscosity respectively.)
Sure, there are multiple good ways to s... | In your definition of the dimensionless time you have assumed that the characteristic scale for time is $\frac{L^2}{\nu}$. If you instead assume that the characteristic scale is the inverse of the vortex-shedding frequency $f^{-1}$ and redo the analysis you will retrieve the Strouhal number. You will need to rescale th... | {
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What does covariance/non-covariance mean in QFT? I'm studying QFT using the book of Mandl and Shaw. In the first chapter they start by quantising the electromagnetic field, but in a "non-covariant" way. What do they mean by that?
They have a chapter about the covariant theory of photons (chapter 5). They say using the... | Covariance means Lorentz invariance in explicit form. For example, you may work with a specific coordinate system, and derive expressions in terms of these coordinates, but Lorentz invariance will no longer be obvious.
On the other hand, when all your formulae have is dot products ($p \cdot q$), derivatives ($\partial_... | {
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Approximate Killing vector field in general relativity In this paper the authors consider an approximate Killing field $\chi$. It vanishes on a given 2 surface and its first order part is given.They say that if it obeys the Killing equation $\chi_{a;b}+\chi_{b;a} = 0 $ then its second order part vanishes.
Do you under... | Zhen Lin shows here that
if a vector field $\chi$ obeys the Killing' equation $\chi^i_{;j} +\chi^j_{;i} = 0 $ then $\chi^a_{;bc} = R^a_{;bcd} \chi^d $everywhere.
As $\chi$ vanishes at p, we have at this point $\chi^a_{;bc} = 0. $
The second order part of the taylor serie being quadratic in the covariant derivatives, i... | {
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Why do Newton's laws have to be used only when working with a particle? I have a small understanding of physics but I am not studying the subject.
Whilst trying to model a plane landing in Differential equations (an A-level maths module), we were told that you have to assume that the plane is a particle to be able to a... | Because any bigger system is made up of small particles and if want to apply Newton's equations to the whole system then we have to apply them on every individual constituent particle, that would be very tedious and lengthy. That's why we use them only for particles.
| {
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Difference between vorticity and circulation The definition of vorticity is $\boldsymbol{\omega} = \nabla \times \mathbf{v}$, where $\mathbf{v}$ is the velocity vector field.
Now, if I look at a rotating flow in cylindrical coordinates I find that:
$$\nabla \times \mathbf{v} = \frac{1}{r}\frac{\partial (r v_{\theta})}{... | There is a singularity at the origin: a delta function in the vorticity field. The vorticity is zero (irrotational flow) everywhere but at the origin, where it is infinite. The circulation around any path not enclosing the origin is zero. The circulation around any path enclosing the origin is a constant (non-zero).
| {
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Gravity in vector We know that gravity is a force. But what is it's direction? Can it be expressed by vector and how can we do that? This question can also be asked for Coulomb's Law.
| If we use the centre of the earth as origin, we have
$$\mathbf{F}=-\frac{GM_{\oplus}m}{r^3}\mathbf{r} \tag{$r>R_{\oplus}$}$$
where $\mathbf{r}=(x,y,z)$ and $\displaystyle \left| \frac{\mathbf{r}}{r^3} \right|=\frac{1}{r^2}$.
At the surface of the earth,
$$g=\frac{GM_{\oplus}}{R_{\oplus}^2} \approx 9.8 \text{ m s}^{-2}$... | {
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Energy density in string wave The total energy density in a harmonic wave on a stretched string is given by
$$\frac{1}{2}p A^2 \omega^2 sin^2(kx-\omega t).$$
We can see that this energy oscillates between a maximum and a minimum. So the energy is maximum at 0 displacement when the string is stretched and at its maximum... | PE and KE that we are talking here are of a small part of string. PE and KE are maximum when the element passes through its mean position as velocity is maximum and string part is most stretched. At crest (or trough) velocity is zero and string part is not stretched so both PE and KE are zero of that string part.
Now t... | {
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Does the annihilation of antihydrogen with heavier matter resulting in conversion of heavier elements back to hydrogen? If an antihydrogen atom annihilate with a heavier atom of matter, will the remaining nucleus of the heavier atom be disassembled into individual protons and neutrons?
If so, is this considered to be a... | An antihydrogen beam is a very recent achievement in particle physics
The ASACUSA experiment at CERN has succeeded for the first time in producing a beam of antihydrogen atoms. In a paper published today in Nature Communications (link is external), the ASACUSA collaboration reports the unambiguous detection of 80 anti... | {
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Abrikosov's Vortex Lattice (Beta Parameter) In order to find the correct vortex lattice configuration (i.e. ground state) in Ginzburg-Landau theory (or the Abelian Higgs Model), it is standard practice to minimize the beta parameter:
$\beta=\frac{\langle |\phi|^{4}\rangle}{\langle |\phi|^{2}\rangle}$.
What is the diffe... | The answer can be found in Abrikosov's paper: Sov. Phys. JETP 5 1174 (1957). The free energy for the vortex lattice is
$F=H_{0}^{2}-\frac{(H_{c}+H_{0})(H_{c}-H_{0})}{\beta(2\chi^{2}-1)}$, where $H_{0}$ is the external magnetic field, $H_{c}\equiv\chi$ is the critical field above which the vortex lattice is unstable and... | {
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Work and mechanical energy I have come across the following lines in "Introduction to Mechanics" by Kleppner and Kolenkow.
A peculiar property of energy is that the value of mechanical energy $E$ is arbitrary; only changes in $E$ have physical significance. This comes about because the equation $$U_b - U_a = -\int_{a}... | In this case, the "physical significance" is referred to the evolution of the system: it will be defined only by the differences between certain values of $E$, then $E$ is arbitrary since whatever constant you add to $E$ (i.e. you change $E=K+U$ with $E'=K+U+c$) it will not be relevant since the constant will cancel wh... | {
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dependence of fundamental frequency of vibration of a stretched string on the medium in which it is kept suppose a stretched wire's fundamental frequency in air is 280 Hz. What would be it's fundamental frequency in water ?
(all other conditions of the string remain same)
I looked into the laws of vibrations of stretch... | the surrounding medium has a characteristic acoustic impedance which can be calculated. if that characteristic impedance is close to that of the vibrating string, then two things will occur: first, the string will be strongly damped and second, the mass of the surrounding medium will begin to couple to the mass of the ... | {
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Harnessing permanent magnetism? Putting aside any energy generating schemes that would break the laws of thermodynamics, is it possible or is there a motor which generates power using a permanent magnet? So that the energy wouldn’t be coming from nothing but from the atoms in the magnet being misaligned.
| In my opinion the answer is: energy is conserved, if kinetic energy is extacted using a permanent magnet, it will be at the expense of demagnitization of the magnet. Here is a link with some estimates of the amount of energy stored in a permanent magnet.
There is energy stored in a permanent magnet which slowly become... | {
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Exercise on bosonic vacuum
Consider bosonic canonical transformation, generated by operator $S = e^{\lambda (a^{\dagger})^2}$. Show, that
\begin{equation}
b \equiv SaS^{-1} = a - 2\lambda a^{\dagger}.
\end{equation}
Calculate the norm of transformed vacuum $S|0>$ and show that the norm is finite only if $\lambda... | From $(a^\dagger)^n|0\rangle=\sqrt{n!}|n\rangle$ we have $e^{\lambda (a^\dagger)^2}|0\rangle =\sum_{n=0}^\infty\frac{\lambda^n\sqrt{(2n)!}}{n!}|2n\rangle$, which has norm $\sum_{n=0}^\infty\frac{\lambda^{2n}(2n)!}{n!^2}$. The Stirling approximation gives $\frac{(2n)!}{n!^2}\approx\frac{4^n}{\sqrt{n\pi}}$ for large $n$,... | {
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Gravitational field strength Can I use $g=GM/r^2$ to calculate the gravitational field strength proton or electron or any other particles? If not then why? If yes then what would be that really mean?
| You could use Newton's classic equation, but, as illustrated in an answer here, its effects would be almost negligible.
Aside from that, we don't know if Newtonian gravity even applies to particles on that scale. To answer that question would require a theory of quantum gravity which, to date, has not yet been develope... | {
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Original 1925 paper by Einstein on Bose-Einstein Condensation? Does anyone know if it is possible to retrieve the original 1925 paper by Einstein on Bose-Einstein Condensation? Possibly a translation into english, but german would be fine if no translation is available. I have managed to find a translation of Quantenth... | For those looking for the original german A Scanned version is provided by the university of Münster
| {
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What prevents two objects from falling toward each other faster than the speed of light? I was thinking about what happens when two objects fall toward each other in space. The farther apart they are when they begin falling, the faster they will be traveling when they hit each other due to gravity's acceleration. So, i... | No, this is not the case at all. And, in fact, I can give you a concrete answer for a specific case (and from this you can extrapolate almost everything there is to it):
If you place a space ship infinitely far away from planet Earth (in an otherwise empty universe) and wait infinitely long, then the space ship will cr... | {
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Applying Kinematics to find retarding force in a medium Question
If an object free falls let's say off a cliff that is 3 meters high, clearly it increases velocity and if at the bottom of the three meter there was a bucket full of jelly which created a retarding force in which the object stops 1 meters in and if the ob... | Both methods will work. It's not uncommon that there are more than one different approaches to a problem and you can choose whatever method seems most convenient.
For example consider the initial drop from rest at height $h$ down to the surface of the jelly. The PE change is $-mgh$ and since the total energy is conserv... | {
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What determines the direction of current in a superconductor Type I superconductors have no electric field nor magnetic field inside of them, when they are in the superconducting state. This means no voltage difference across any two points or regions inside of them. Yet they carry a current. This means the Cooper pair... | There is no electric field in a superconductor, but there can be a voltage across it. Recall that:
$$ \vec E = -\nabla\phi-\frac{\partial\vec A}{\partial t} $$
so the voltage need not be zero for the electric field to be zero.
Any superconducting loop has some inductance, so this voltage is required to get a current go... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does doping a Sodium Iodide scintillator with Thallium result in a higher ratio of Compton interactions to total interactions? We are looking at designing a Compton camera for 662KeV photons and have been told that "the fraction of Compton interactions to total interactions in the photopeak is usually higher than f... | The doping of the NaI crystal with thallium improves the scintillation efficiency by improving the light emission due to the improved recombination by light emission of electrons and holes at the dopant site. Thallium in small concentrations in the NaI crystal is a so-called scintillation activator The effect of thalli... | {
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Moment of Inertia of an Equilateral Triangular Plate I was reading about moment of inertia on Wikipedia and thought it was weird that it had common values for shapes like tetrehedron and cuboids but not triangular prisms or triangular plates, so I tried working it out myself. I will post my attempt below, but for some ... | I can confirm your result. I can also suggest you a neater way to derive it inspired by David Morin - Introduction to Classical Mechanics, check it out in a library if you have access.
The main idea is to use the symmetry of the equilateral triangle and split it into 4 smaller equilateral triangles like this
Now analy... | {
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Is there evidence that a = dv/dt and a = F/m are always equivalent? If the rate of change in velocity in a particle (of mass m) caused due to a force F is dv/dt, then
F = m dv/dt
It may be argued that this is how we define force. But my question is:
Can there be any kind of force, which is so strange that no matter ho... | This formular holds only true for time-constant masses. The original formular is $$\vec{F} = \dot{\vec{p}}$$
which, taking the derivative, leads with to
$$\vec{F} = m\dot{v}+\dot{m}v$$
So $$F=m\dot{v}$$ is only true if the second term is zero, so if $\dot{m}$ is zero.
| {
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Conservative force definition Classical Mechanics, by John Taylor defines a conservative force $F$ as a force that satisfies:
*
*$F$ depends only on the particle's position and no other variables.
*Work done by $F$ is the same for all paths taken between two points
I'm wondering if this definition is redundant. D... | The comment of @probably_someone shows clearly the necessity of (1). It eliminates a possible force dependence on time, velocity or on any other parameters.
(2) does not follow from (1): Consider the force on one pole of a long thin bar magnet which is next to a current carrying wire. The work done moving it in a cir... | {
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Why do we use superposition instead of tensor product in interferometer? In the description of a neutron interferometer here, it says:
In an interferometer the incident beam is split into two (or more) separate beams. The beams travel along different paths where they are exposed to different potentials (which results ... | In a nutshell: these are not two different systems, but the probability amplitudes of the two different states of the same system.
I do agree that simple discussions of two interfering waves (electromagnetic or particle waves) are in practice only complicating the matter, as opposed to thinking of a single wave in a m... | {
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Energy in simple harmonic motion ─ where is the kinetic energy stored, and where is the potential energy? When a mass connected to a spring is in simple harmonic motion and somewhere between the mean and extreme positions the mass is cut from spring. Then instantaneously after cutting the mass will only have its kineti... | While the mass is not cut from the spring, there is transfer of energy. This is the transfer of potential energy into kinetic energy of the mass.
When you cut the spring, the block will proceed to move with the kinetic energy it had before. The potential energy in the spring will not disappear or somehow suddenly trans... | {
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Why do charged particles deflect one way but not the other in a magnetic field? I am well aware that a charged particle moving in a magnetic field will experience a force perpendicular to that magnetic field. But why is it that positive and negative particles experience a force in opposite directions?
What exactly dete... | I've been told I can't just link to another site so I will try to paraphrase the article I linked to.
The magnetic force is perpendicular to the velocity of the particle it is acting on. That causes the direction of the particle to change and travel in a circular motion.
https://cnx.org/contents/bZRPyVNP@2/Motion-of-a-... | {
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How to reconcile infinite cross section of resonances with cross section formula from quantum mechanics? If we consider $s$-wave scattering for two scalar fields $\phi$ and $\chi$ with an interaction $\frac{g}{2}\phi^2\chi$, then the Lorentz-invariant scattering amplitude to second order is:
$\mathcal{M}_{fi} = \frac{-... | The issue is that the $\mathcal{M}_{fi}$ in the question is only calculated to second order. When you include loop corrections and "dress" the propagator, it takes on a different structure and the width of the peak becomes finite.
| {
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Thermal energy generated by collision observed from two different frames of reference An isolated system is composed of two bodies $A$ and $B$, with masses $m_A$ and $m_B$, $m_A \ne m_B$, which are in route of collision. The relative velocity between them is $v$. The collision is inelastic and I want to calculate how m... | The problem is that you assume that the reference frames $S_A$ and $S_B$ stay the inertial frames before and after the collision even though their velocities change abruptly at the collison. It would be better to use the center of mass reference frame which does not change with the collision.
| {
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Understanding the introduction of a symbol in Einstein's paper In Einstein's paper "On the Electrodynamics of Moving bodies" (1905); first he introduces two sets of coordinates, for two inertial frame moving with relative velocity of $v$: ($x$, $y$, $z$, $t$) and ($\xi$, $\eta$, $\zeta$, $\tau$).
Then he introduces one... | It seems to be a Galilean transformation between two coordinate systems with relative velocity v in x direction.
The Galilean transformations form a part of the symmetry group of Newtonian mechanics.
But is it fine to assume the Galilean transformations in order to derive that Galilean transformations must be replac... | {
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Electrons diffusion in gas with present of electric field For research purposes, I am looking for a way to calculate how far the electrons "spread" perpendicularly to the electric field in a chamber of gas.
For example, if a beam of alpha particles ionize the gas in a chamber and leave a track of electron-postive ion ... | I have found the answer in the W. R. Leo's Techniques for Nuclear and Particles Physics Experiments. The equations that I needed are Eq. 6.18
$D=\frac1 3 v \lambda $ and Eq. 6.46 $\sigma=\sqrt{2Dx/\mu E}$.
| {
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Does a gas of neutrons obey the ideal gas equation much better than hydrogen gas Consider a gas of neutrons. Does it obey the ideal gas equation much better than hydrogen gas at the same temperature and pressure?
| It will not be easy to establish a pure neutron gas in thermal equilibrium to test the ideal gas law.
(1) Free neutrons have a life time of only 14.7 minutes. They decay (predominantly) into a proton, an electron, and an electron antineutrino. So you will always have positive protons and electrons and thus hydrogen aro... | {
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Is there a SI unit for space-time? Space and time are routinely combined into space-time nowadays, which implies that the SI meter and second should be combined into a single SI unit such as [meter-second]. So far, I haven't come across such a SI unit.
|
So far, I haven't come across such a SI unit.
And you won't find one. The problem is that meters and seconds are inconsistent with one another, much as are the customary units of mass and force. With mass in pounds mass, force in pounds force, and acceleration in feet per second squared, one has to use the ungainly $... | {
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Gravitational potential at the center of a uniform sphere Feynman in second Messenger Lecture said that potential at the center of ball with small radius $a$ is equal to average potential on surface of ball minus $G$ times mass inside the ball divided by $2a$. You can see it on the picture. I don't understand that. I h... | Feynman's answer does not refer to a sphere filled with uniform mass density. It is a completely general statement about the potential at the center of a (small enough) sphere given an arbitrary mass distribution. It is a different way of stating Newton's law of gravitation. It can be derived from the differential form... | {
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Trouble in understanding AM modulation Amplitude modulation is in fact, superimposing the low frequency transmission signal into a high frequency carrier signal, right?
So, if the transmission signal can be represented as $c(t)=A_c \sin (\omega_ct )$ and the carrier wave can be represented as $c(t)=A_m \sin (\omega_mt)... | The low frequency transmission signal means slowly changing $A_c(t)$, not fixed $A_c$ with fixed $\omega$ which does not carry any information.
| {
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Are the electromagnetic waves transverse? The em waves are said to be the oscillations o electric and magnetic field perpendicular to each other and to the direction of propagation of wave and hence transverse.
However consider a charged particle oscillating along x axis with no motion along y and z axis.
Let it be at... | You are right in your observation of the electric and magnetic fields at a point P. This is, however, a consideration of the so-called near-field of an oscillating charge. The near field doesn't constitute a freely propagating electromagnetic field. To get the freely propagating (far field) electromagnetic field, you h... | {
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Why do the following Network Transformations give different answers? I did a Star to Delta Network Transformation and a Delta to Star Network Transformation on different parts of the original circuit as shown in the image below.
It gave me two new circuits. On solving those circuits, I get different answers for the equ... | Because you've made a mistake in your application of the formalism.
Both of the transformations you have performed are guaranteed, when done correctly, to produce completely equivalent circuits. The only thing that can break here is the 'done correctly' part ─ so double- and triple-check all your algebra.
Ultimately, i... | {
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Definition of stress-energy tensor
The image from the wiki article on the stress energy tensor gives $T_{00}$ as $1/c^2$ times the energy density. I believe this is incorrect and that the $1/c^2$ factor should be dropped. Am I missing something?
| You are correct and Wikipedia is wrong. Energy density is $Nm/m^3 = N/m^2$ in SI units. Pressure is also $N/m^2$. Dividing $T^{00}$ by $c^2$ makes no sense.
| {
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Are all waves either transverse or longitudinal? So I recently searched up "em wave transverse proof", and I understood it pretty well enough I think.
After that, I just started to wonder if all waves are either transverse/longitudinal. If there are waves that are neither one of them, how do we put that in mathematica... | I was wondering if there is a difference between a transverse and longitudinal wave...
Imagine a rubber rod, it flexible to the sides so you can bend it and oscillate it like a string if you do it fast enough. That would be its transverse wave behavior.
Now the same rubber rod can be compressed or decompressed and that... | {
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Is the drag force on a thrown object higher in hot or cold air? Increased temperature lowers viscosity of gases like air but also decreases density.
So then drag force would be lower if an object is thrown in higher temperatures but what about viscosity? I thought viscosity was drag.
| Depends on how fast the object is traveling through the fluid a.k.a. the value of the Reynold's number.
If the flow around the object is turbulent (high Reynold's number) then density is the key fluid property. If Reynold's number is small then viscosity becomes relevant.
| {
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Von Karman mixing length $l=k \frac{du/dy}{d^2u/dy^2}$
In a fully developed turbulent flow of an in-compressible fluid inside a pipe of radius $R$, velocity at the center is $U_m$. If we define $U^*=\sqrt{\tau_0/\rho}$, where $\tau_0$ is the wall shear stress and $\rho$ is the density, then find the velocity distribut... | try to write $$ y = k(R-r)(1-\frac{R-r}{R})$$ now find when y is maximum then you graph this function :$$u^+ = \frac{1}{\kappa} \ln\, ((R-r)y_m) + C^+$$ you sould get something like this 1) REICHARDT 2) PRESENT 3) UNIVERSAL
| {
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Why is the internal energy of a real gas a function of pressure and temperature only? While studying thermodynamics, I read that the internal energy of an ideal gas is a function of temperature only. On searching the internet, i found an article which stated that the internal energy of a real gas is a function of tempe... | In real gas Pressure and volume change when temperature changes so work and heat exchange with surroundings then internal energy changes during this process
| {
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What is the shortest distance between electron and positron before they are annihilated? I just want to know how does an electron felt the presence of a positron before they are converted into energy? Also how does the electron tell if it is positron or proton if this makes any difference?
| As a greasy handed experimenter I have a simple answer to this. Look through all the high-energy, exclusive data you have available on $e^+ + e^- \to 2\gamma$ (or possibly $e^+ + e^- \to \mu^+ + \mu^-$), pick the event with the largest squared four-momentum transfer $Q^2 = -t$ (or invariant mass $s$), and use that to c... | {
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Mass-Energy equivalence in case of minimal coupling The energy-momentum relation of a free particle is (in SI Units):
$$
m^2c^4 =- c^2 \vec{p}^2 + E^2
$$
Minimal coupling is a way to fix a gauge freedom for the choice of canonical momentum (which I can in special relativity give as $ p_{\mu} = \left( \begin{matrix} \fr... | Yes, indeed it is the expression of the energy of a relativistic massive charged particle in the presence of an electric potential. Consistently, if the expression is expanded in the limit $cp\ll mc^2$, it reduces to the known non-relativistic formula
$$
E = \sqrt{m^2c^4+c^2p^2} + e\Phi \simeq mc^2+ \frac{p^2}{2m}+e\... | {
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Blocking WiFi with Faraday cage As part of a project I'm trying to prevent WiFi transmission of frequency 2.4 GHz from reaching a Raspberry Pi via a Faraday cage.
Would a 20 micron aluminum foil do the job?
| Yes.
The quantity that measures how far an electromagnetic wave of frequency $f$ can pass through a conductor is called the skin depth:
$$ d = \sqrt{{\rho \over \pi f \mu}} $$
Where $\rho$ is the resistivity and $\mu$ the magnetic permeability of the conductor. This equation is true at frequencies lower than $1/\rho\ep... | {
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Is potential energy a type of energy at all? Is potential energy, whether it be that of a charge in an electric field or a mass in a gravitational field or anything like that, actually an energy that the particle itself contains, like kinetic energy? Or is it just a measure of its ability to do work?
Is it the case th... | Well, as the name suggests, its the potential of the body to store energy inside it or some other way (whatever you like), potential energy is actually potential to do work. But having a constant potential energy means nothing.
as F = - Del (V)
This means , its something , not nothing.
Just think of Gravitational poten... | {
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Why is the singlet state for two spin 1/2 particles anti-symmetric? For two spin 1/2 particles I understand that the triplet states ($S = 1$) are:
$\newcommand\ket[1]{\left|{#1}\right>}
\newcommand\up\uparrow
\newcommand\dn\downarrow
\newcommand\lf\leftarrow
\newcommand\rt\rightarrow
$
\begin{align}
\ket{1,1} &= \ket{\... | Let's temporarily forget that the two $m=0$ states exist, and consider just the two completely aligned triplet states,
$\newcommand\ket[1]{\left|{#1}\right>}
\newcommand\up\uparrow
\newcommand\dn\downarrow
\newcommand\lf\leftarrow
\newcommand\rt\rightarrow
$
$\ket{\up\up}$ and $\ket{\dn\dn}$.
There's not any physical d... | {
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Adding energies vs. adding momenta Suppose we have a (time and space) translationally invariant system with the Fock space for a Hilbert space. Temporal and spatial translation invariance implies that energy and momentum are good quantum numbers. In the free theory, the momenta and energies of one-particle states simpl... | In quantum mechanics, the linear momentum is represented by a one-body operator: the momentum of the $i$-particle only depends on the variables of this particle, $\hat{\bf p}_i = -i\hbar \nabla_i.$ So, the total momentum of the system is simply defined by the sum of the momentum of each particle. Of course, if the part... | {
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What is the definition of the charge conjugation? I seem to have troubles finding definitions of the charge conjugation operator that are independant of the theory considered.
Weinberg defined it as the operator mapping particle types to antiparticles :
$$\operatorname C \Psi^{\pm}_{p_1 \sigma_1 n_1;p_2 \sigma_2 n_2;... | All of your fields naturally lie in some representation of the group of all symmetries (these include gauge symmetries, global gauge transformations and global Lorentz transformations). Charge conjugation is simply passing to the conjugate representation of that group.
E.g. complex scalars are 1d irreps of $U(1)$, and ... | {
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What experimental bounds do we have on big $G$? I know that there has been a large amount of controversy surrounding the exact value of the gravitational constant $G$, but I know that there is not a substantial difference in the measured value. So I was wondering what experimental bounds we have on it so far?
| According to NISTconstants, as of 2017,
$$G = 6.674 08(31) \times 10^{-11} \space {\rm m}^3 {\rm kg}^{-1} {\rm s}^{-2} $$
which means the range is 6.67377 to 6.67439
It's not the easiest measurement to make, large forces from electrical, magnetic, and other sources have to be considered.
| {
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Potentials and position uncertainty In the Schrodinger equation, we have some potential $V(x)$. But generally, there is some uncertainty in the position with solutions to Schrodinger's equation. Classically, we would say that a particle at position $x$ is associated with the potential at that point -- is there a quantu... | The schrodinger equation tells us that a particle interacts with a potential if its wavefunction has a non-zero amplitude in the region enclosed by the potential. That is to say, the wavefunction's time evolution is independent of the potential energy's value in regions where the wavefunction is zero.
However, the wave... | {
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Why change in resistivity is proportional to the original resistivity? When there is a temperature change $\Delta T$, the change of resistivity is
(1) proportional to $\Delta T$
(2) proportional to the original resistivity $\rho_0$
Hence we can define the temperature coefficient of resistivity $\alpha$ so that
$$\Delta... | It's just a matter of definition. If we expand $R(T)$ in a Taylor series about $T_0$, the first two terms are
$$
\rho(T) \approx \rho(T_0) + \rho'(T_0) (T - T_0).
$$
The coefficient $\alpha$ is then defined so that
$$
\alpha = \frac{\rho'(T_0)}{\rho(T_0)},
$$
which yields $\Delta \rho \approx \rho_0 \alpha \Delta T$... | {
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Why does the electric field strength for a dipole go as $1/r^3$? I've been given the following graphic to help wrap my head around this.
If the potential can be shown to represent a $1/r^2$ relation, then I'm more than happy to accept that the electric field is hence a $1/r^3$ relation, but I need to accept the first p... | The important physical interpretation that you need to keep in mind is that the charges are opposite, so the $1/r$ pieces of their potentials cancel out. We'll see this happen explicitly in the math in the correct derivation.
But it also explains why your logic isn't quite enough to understand what's going on: You rea... | {
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What is the 4-spin vector of a photon? The photon, being a vector boson has 2 spin states, $\pm 1$.
In relativity, we can determine the four-spin vector $s^{\mu}$ of a particle (see e.g. Costa et al. 2017).
What would $s^{\mu}$ be for a photon? How does it relate to the 2 spin states?
Thanks
| There is no such thing as a rest frame for a photon, so there is no such thing as a 4-spin in the sense of the Wikipedia article you link.
In fact, there is no such thing as spin for a photon, cf. also this answer of mine for a lengthy elaboration on how the properties of masslessness, being a gauge boson and having no... | {
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"timestamp": "2023-03-29T00:00:00",
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Why do different materials reflect different light? So, as far as I understand, white light contains photons of all energy levels. These hit a material, say iron. The photons that are below the energy level to move electrons just pass through. The others deflect the electrons to another orbit and when the electrons go ... | The photons of visible light do not pass through. The energy that is not reemitted is absorbed as heat. That’s why black surfaces get hotter than white services.
| {
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