Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Meaning of the subscripts $L,R$ for the two component Weyl spinors $\phi_{L,R}$ For a Dirac spinor $\psi$, its chiral projections are $\psi_{L,R}$ are defined as $$\psi_{R,L}=\frac{1}{2}(1\mp\gamma^5)\psi.\tag{1}$$ Acting with the chirality operator $\gamma^5$, we find $$\gamma^5\psi_L=-\psi_L,~~\gamma^5\psi_R=+\psi_R.... | It's important to distinguish between the Clifford algebra itself versus a matrix representation of the Clifford algebra. The Clifford algebra itself is an abstract associative algebra generated by basis vectors $e^0,e^1,e^2,e^3$ satisfying $e^a e^b+e^be^a=2\eta^{ab}$. The Dirac matrices provide a matrix representation... | {
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"timestamp": "2023-03-29T00:00:00",
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Contravariant metric in Newton-Cartan spacetime I'm interested in the geometrized Newtonian gravitation or Newton-Cartan theory. In every reference that I have found begins saying that a Newton-Cartan spacetime is a manifold $M$ with some structures. Among then, is always pointed a contravariant metric $g^{ab}$ that re... | The signature is (+++) and the metric has rank 3. See e.g.
https://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_R_Andringa.pdf
in which this is motivated by calculating which metrics are kept invariant under the Galilei group.
| {
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"source": "stackexchange",
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Doubt about ray diagrams In a ray diagram, 2 rays are considered enough to locate the image of a point on a given object. But how can we say that the rays other than the one we drew will meet at that same point?
I guess we can justify this by saying that we get only one image of a given object by a single mirror/l... | This concerns what assumptions we are making about our optical system. Consider making a rudimentary lens using a flat slab of glass with a prism glued on the side. A ray going through the center goes straight on through; a ray going through the prism will be bent, so these two rays will meet somewhere, but there is no... | {
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Cylinder vs cylinder of double the radius roll down an incline plane, which one wins? A solid cylinder and another solid cylinder with the same mass but double the radius start at the same height on an incline plane with height h and roll without slipping. Consider the cylinders as disks with moment of inertias I=(1/2)... | The following equation from @R. Romero's analysis is correct:
$$Mgh=\frac{1}{2}Mv^2+\frac{1}{2}I\left(\frac{v^2}{R^2}\right)\tag{1}$$But, the moment of inertia of a cylinder is given by: $$I=M\frac{R^2}{2}\tag{2}$$ So, combining Eqns. 1 and 2 gives:$$Mgh=\frac{1}{2}Mv^2+\frac{1}{4}Mv^2\tag{3}$$Calcelling M from both s... | {
"language": "en",
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Coaxial cable with infinite return conductor If a coaxial cable has a coaxial return conductor with infinite outer radius, will the return conductor experience a voltage build-up due to current flowing through it, or will it stay on ground potential? Here I'm taking infinite to be the zero potential.
I would appreciate... | AC current, due to the skin effect, tends to flow on the inner surface of the outer conductor of a coax cable. The higher the frequency, the thinner the skin depth. For instance, at $1$MHz, most of the current will flow inside a layer of a couple of hundreds of microns.
DC current will spread out much more, but most of... | {
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Is the normal force equal to weight if we take the rotation of Earth into account? In my physics class we were doing problems such that we set $N$ (normal force) $= mg$. I understand that by Newton's Third Law, if I exert a force on the ground, then the ground will exert an equal and opposite force on me. However, the ... | @Aaron has nicely explained using mathematics.
Let me brief it out qualitatively and also give a slightly different way of looking at it.
*
*following the line of thought in @Aaron's answer, the normal force does not equal mass multiplied by g.
*so the difference between gravitational pull and the normal force will... | {
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Lagrangian of EM field: Why the $B$-field term has a minus sign in front of it in the Lagrangian? I know that $L = T - U$ and that, in the non-relativistic case
$$L= \frac{1}2mv^2 - q\phi(r,t) + q\vec{v}\cdot\vec{A}(r,t).\tag{1} $$
My lecturer used the following form of the Lagrangian density to derive Maxwell's equat... | The only reason why Lagrangians are what they are is because they give the correct equations of motion.
In addition you may want to require that certain symmetries are preserved, but there is not much more to that. Actually, one can prove that for many systems there are infinitely many (different) equivalent Lagrangian... | {
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Do centrifugal force and gravity differ in their effects on objects? If the type of object matters, consider the human body. If the situation matters, consider standing on the inside wall of an O'Neill cylinder compared to standing on the surface of Earth.
"Differ in their effects on objects" means: Would the object be... | Yes. An instrument that can sensitively measure the force gradient (for example, the difference between the force at one spot and the force at a nearby spot, say a foot away) could tell the difference. This “tidal” force will be greater for the O’Neill cylinder.
| {
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Why does the warm air rises up? Warm air has more energy than cold air. This means that according to the Einstein equation $E = mc^2$ the warmer air has a greater mass than the cold one. Why is the warm air rising, if it has a greater mass, which means that the attraction of gravity between the Earth and the warm air i... | $E=mc^2$ is only valid for particles that are not moving. The full expression should be
$$E^2=p^2c^2+m^2c^4$$
where $p$ is the momentum of the particle (which is $0$ when at rest, and we recover the famous $E=mc^2$).
The reason warm air rises is to do with the fact that "warm" air has faster moving particles. This me... | {
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Relationship between freefall velocity time dilation and gravitational time dilation in a Schwarzschild metric If you drop an object into a gravitational field, is its final velocity equal to what it would have to be in flat space in order to generate the same time dilation that you get at a given radius for an object ... | The Schwarzschild metric in Schwarzschild coordinates $(t, r, \theta, \phi)$ shows
$ds^2 = -(1 - 2M/r) dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)$
where:
$c = G = 1$ natural units
$M$ black hole mass
$r_s = 2M$ Schwarzschild radius (event horizon)
The gravitational time dilation measured at ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/442032",
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Would a supersonic object without a combustion power source leave behind a contrail? Contrails, as far as I understand them, are caused by either a pressure change that forces the condensation of H2O(g) OR by the release of warm H2O from a combustion engine. Most plane contrails, I would assume, operate largely by this... | The Chelyabinsk meteor left a contrail (https://www.nature.com/news/russian-meteor-largest-in-a-century-1.12438) although it was not reported to carry engines:-)
| {
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If the molecular collisions are elastic will there be any dissipation in a fluid? Viscosity arises due to collisions of the molecules of one layer of a fluid with another in contact. But viscosity is a dissipative element leading to heating and dissipation. Where does it heat come from? Does it come from the molecular ... | Viscosity arises due to collisions of the molecules of one layer of a fluid with another in contact.
Viscosity is due to intermolecular forces that resist relative motion within a fluid. The viscosity of a liquid can be defined as the force of friction between layers of the liquid that move relative to each other at di... | {
"language": "en",
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What is the shape of a gravitational wave form? What is the shape of a gravitational wave as it hits the Earth, particularly the time portion.
Does time start at normal speed, then slow slightly, and then return to normal speed?
Or does it start at a normal speed, slow down slightly, then speed up slightly, and then ... | This is written as if the metric for a gravitational wave was something like $ds^2=(1+f(t))dt^2-dx^2-dy^2-dz^2$. It isn't. A metric of that form is just Minkowski space described in unusual coordinates. General relativity doesn't really even offer us any way of describing the notion of whether time slows down or speeds... | {
"language": "en",
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Can a pair of frictional force have positive work? My teacher told me that friction can have positive work which is true. But he told that a pair of friction forces can never have positive work. I am not able to think the reason for this statement.
Any help will be really appreciated.
| Imagine one block $T$ on top of another block $B$ with both blocks moving with a velocity $\vec v$ to the right ie not moving relative to one another.
A force $\vec F$ is being applied to the bottom block $B$ to cause an acceleration of both blocks to the right.
The friction force on the top block due to the bottom b... | {
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What is the meaning of the negative sign in $\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - (c\Delta t)^2$? In the equation of the spacetime interval formula $\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - (c\Delta t)^2$ is there meaning for the minus sign before the $(c\Delta t)^2$ or is it just a pure mathe... | If we measured distance in light-seconds instead of meters,
the constant c would be 1, and the metric distance element would simply become
$Δs^2 = Δx^2 + Δy^2 + Δz^2 - Δt^2$, or
$Δs^2 = Δt^2 - Δx^2 - Δy^2 - Δz^2$
(both forms are equivalent, because multiplying the vector by -1 does not change its squared length)
This m... | {
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Why do we study the Ising model on $\mathbb{Z}^d$ for $d > 3$? I'm a beginner in statistical physics and I'm reading some stuff about the Ising model. So this might be a silly question. My question is: why we study the Ising model for high dimension cases, despite that our physical world has only dimension $2$ or $3$?
| Let us start with a quote from a paper by Michael Fisher and David Gaunt in 1964 (Phys. Rev. 133, A224), at a time when it was still necessary to justify such studies:
To elucidate the general problem of dependence on dimensionality and coordination number, it seemed worthwhile to investigate the Ising model and self-... | {
"language": "en",
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How should I imagine a multi-particle state in a free QFT? It is reasonable to think of single-particle Focks states as of plane waves. Indeed, since $|p\rangle=a^\dagger_p|0\rangle$ and $\langle x|p\rangle\sim \operatorname{e}^{ipx}$, we conclude that the state $|p\rangle$ can be thought of as a plane wave in the posi... |
Is the matrix element ⟨x|p 1 ,p 2 ⟩
⟨x|p1,p2⟩
really equal to the sum of plane waves?
Nope, it's the product of plane waves, not sum.
Of cause, you have to symmetrize or anti-symmetrize the product according the statistics (boson/fermion) of the particle. In the end, it's a symmetrical/anti-symmetrical sum of ... | {
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Minimum separation from the spacetime interval I've been working through invariant spacetime interval questions recently, and I came across a question in my lecture notes where;
$$\Delta s^2=\Delta x^2 -(c\Delta t)^2 > 0 $$
Now it is clear to me that there is no frame where $\Delta x' = 0$ which I have already proven a... | Let me restate the problem the way I understand it:
we have 2 events A and B separated by a space-like interval
$$\Delta s^2=\Delta x^2 -(c\Delta t)^2 > 0 $$
now, different observers will measure these 2 events A and B and come up with different $\Delta x$ and $\Delta t$, but what will be the minimum possible $\Delta x... | {
"language": "en",
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Is there a simple way to calculate Clebsch-Gordan coefficients? I was reading angular momenta coupling when I came across these CG coefficients, there is a table in Griffith's but doesn't help much.
|
Is there a simple way to calculate Clebsch-Gordan coefficients?
No, or at least nothing that any working human would qualify as "simple" and that will work for any general Clebsch-Gordan coefficient.
The closest you can get in the general case is given in this Wikipedia page, which puts them as
\begin{aligned}
\langl... | {
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Why can't we see images reflected on a piece of paper? Why can't you see a reflected image on a piece of paper? Say you put a pen in front of the paper, even when light rays are coming from other sources, hitting the pen, reflecting back, and hitting the paper, there is no reflection.
What's wrong with the following "... | Light falling on a mirror is reflected in such a direction that the angles of incidence and reflection are the same. Light falling on a white sheet of paper is reflected but scattered, going in all directions from the point where it hits the paper.
| {
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Could a microwave oven be tuned to defrost well? Typical microwave ovens do a lousy job of defrosting because liquid water absorbs their radiation far better than ice. So once a spot melts, it will quickly rise to cooking temperature while the rest of the food remains frozen. Would it be possible to build an oven that ... | It would be very difficult to do so. Microwaves heat by adding energy at resonant frequencies of the molecules. Ice and water have very different ranges:
The ease of the movement depends on the viscosity and the mobility of the electron clouds. In water, these rely on the strength and extent of the hydrogen-bonded n... | {
"language": "en",
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When to use sine or cosine when computing simple harmonic motion For simple harmonic motion (SHM), I am aware you can start of using either sine or cosine, but I am a bit confused as to when you would start off with sine rather than cosine. I know that a sine graph starts at $y=0$ and a cosine graph starts at $y=1$. So... | The function $x(t) = A \sin \omega t$ starts from zero with maximum speed, while the function $x(t) = A \cos \omega t$ starts from $x=A$ (the amplitude) with zero speed, and starts to move towards $x=0$. Starting from rest doesn't imply starting at the equilibrium position: if you start from rest at $x=0$, nothing move... | {
"language": "en",
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Does a rock use up energy to maintain its shape? A rock sitting on land, the ocean floor, or floating in space maintains its shape somehow. Gravity isn't keeping it together because it is too small, so I'm assuming it is chemical or nuclear bonds keeping it together as a solid. If not it would simply crumble apart. So,... | No, the exact opposite is true.
The molecules in a rock don't stay together because they're spending energy. They stay together because of attractive chemical bonds. The molecules have lower energy when they're together than when they're not, so you have to spend energy to break the rock apart, not to keep it together.... | {
"language": "en",
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Can light be compressed? What if we take a cylindrical vessel with an inside surface completely reflecting and attach a piston such that it is also reflecting. What will happen to light if we compress it like this?
| Suppose there is an amount of light (electromagnetic radiation) inside the cylinder. Note that electromagnetic radiation is composed of particles called photons, and if we consider that there is a very large number of photons inside the cylinder, we may use statistical mechanics to create a model of a photon gas. Yes, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Dielectrics and capacitors Does the electric field after passing through the dielectric reduce. What I mean is
If there is a dielectric which is in between the plates ( but NOT IN contact with them )
Suppose field begins from the positive side, then passes through the dielectric ( where the field intensity reduces ) a... | A dielectric is a nonconducting, polarisable material. This means that, when an electric field is applied to it, the molecules and atoms which comprise it shift to align with the electric field. This creates another electric field which partially cancels the original one, thus reducing the overall electric field, insid... | {
"language": "en",
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Is time reversal symmetry true on the microscopic level? I often hear that on the microscopic level, time-reversal symmetry is true for all physical processes. However, I can easily come up with counterexamples that seem to disprove this:
*
*Two particles of opposite charge being attracted by each other and accelera... |
Furthermore, even barely observed on a microscopic level, the gravitational force surely defies time-reversal symmetry. A movie of an apple accelerating away from the ground is immediately recognizable as the reversed version of the true process.
No. When an apple falls from a tree, it starts motionless at a high po... | {
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What is the function of this complicated tensioning system? I saw this arrangement for tensioning overhead cables from my train window (schematic below). Why not just have one pulley wheel leading directly to the weights? What function do the additional pulleys serve? For that matter, what are the cables for? They're c... | It's to maintain the tension in the overhead powerline. The line acts like violin string, with the collector on the train acting as a bow. If the train is travelling faster than the wave in the power line, then a standing wave may be induced in the power line, causing it to snap. The line will contract and expand with ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Which of the two events will occur first Consider a bottle partially filled with water and it is sealed from everywhere so that no air can enter or exit from bottle.Now make a small hole at the bottom of the bottle and hang it vertically so that hole faces downward direction. As we know bottle is sealed from everywhere... | It depends on the diameter of the hole and the surface tension of the fluid in contact with that hole. For a small hole and/or high surface tension, the liquid sags out the hole and breaks off as a droplet; the remaining liquid recoils back into the hole and folds itself into a bubble which floats upwards. For a large ... | {
"language": "en",
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Movement of fluid in a container filled with that same fluid If a cylinder with the bottom end closed and the top end open was filled with water and then dropped in a pool of water. Would the water inside the cylinder stay in the cylinder?
| If both liquids, the one inside the recipient and the pool are at the same temperature, once the cylinder reaches the bottom, the liquid near the opening will start to diffuse into the rest of the liquid, but the part at the bottom will mostly stay inside.
While dropping, the liquid near the opening will mix into the p... | {
"language": "en",
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Examples of central forces on the path of orbit? In solving a problem from Goldstein (3.13), I solved for multiple properties of a circular orbit with the attractive central force where the path of orbit crosses the point of the force (at origin).
The solutions were simple enough to find, but what's been in the back o... | The orbit here can be taken as the limit as $r_0 \to a$ of the case where the orbit is an eccentric circle with radius $a$ and center a distance $r_0$ from the origin. I solved for the potential and the force law for this general case in this answer. In the limit of $r_0 \to a$ the results simply become
$$
U(r) = -\f... | {
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Momentum Conservation for ARPES I have a question about the principle of momentum conservation the modeling of ARPES:
https://en.wikipedia.org/wiki/Angle-resolved_photoemission_spectroscopy#Theory
We split the initial momentum of the electron ${\displaystyle \hbar k_{i}}$ into the component $ \hbar k_{i\parallel }$ pa... | The kind of classical idea is that the electrons need to gain a certain velocity away from the surface in order to break free of the crystal. This is the 'work function' of the material.
In terms of symmetry, you can think like this: inside the crystal the potential is periodic (i.e. like an array of valleys) and if th... | {
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Why are protons and neutrons the "right" degrees of freedom of nuclei? This question may sound stupid but why do we visualize nuclei as composed of a bunch of neutrons and protons?
Wouldn't the nucleons be too close together to be viewed as different particles? Isn't the whole nucleus just a complicated low energy stat... | This is basically a matter of energy scales. By analogy, you could ask why we don't take into account nuclear structure when we talk about chemistry. The answer is that the eV energy scale of chemistry is mismatched with the MeV energy scale of nuclear structure.
Nuclear matter has two phases. One is the phase we norma... | {
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Santa and conservation of momentum. How is it possible that conservation of momentum doesn't hold here?
Santa wants to deliver his presents but his reindeers are on strike. In order to still be able to get somewhere he decides to sacrifice some of his presents to gain speed. He decides to throw them straight off the b... | The problem is that by throwing only half of the presents, Santa needs to accelerate himself, the sleigh, and the other half of the presents that are still in the sleigh. If he throws them all at once, he only needs to accelerate himself and the sleigh. Santa is losing out on velocity by accelerating his propellant!
| {
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"timestamp": "2023-03-29T00:00:00",
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Is there an intuitive reason the resistance that maximizes power dissipation in this simple circuit has a simple form? Consider the following circuit:
A textbook problem[2] asks to find the resistance $R$ such that the power dissipated in $R$ is maximized (assuming $R_1$ and $R_2$ are fixed).
I found that $R$ should b... |
This is a simple enough result that it seems it might have some
intuitive explanation that doesn't require calculation.
In the circuit given, the equivalent resistance seen by the load $R$ is $R_{eq} = R_1||R_2$. For a Thevenin (Norton) equivalent circuit, this equivalent resistance is in series (parallel) with th... | {
"language": "en",
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Can the second law of thermodynamics be derived from Quantum randomness? The second law of thermodynamics says that the entropy of an isolated system continuously increases. Can we say that this is due to Quantum mechanics, which continuously increases the information contained in the system by producing random numbers... | One of the most catchy Gedankenexperiment of an isolated system with constant entropy is a photon which gets reflected from the surfaces of two perfect mirrors.
At a first glance it looks like the photon gets re-emitted with the same wavelength as it was absorbed. But beside the presumption of such a ideal absorption-e... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/446092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How much energy is wasted by a noisy refrigerator? I recently bought a new refrigerator for my kitchen. The feet are adjustable but I've been lazy.
Whenever the motor runs and the feet aren't all touching the floor there is a loud buzzing noise. As soon as I move the fridge around to level it up the buzzing stops. One ... | One horsepower represents 746 watts. A refrigerator motor develops (typically) 1/4 to 1/3 horsepower of which only a tiny fraction of wattage is dissipated as vibratory noise.
The leakage of heat into the refrigerator through its walls is a far more significant loss mechanism than noise generation.
By the way, the fro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does Ehrenfest's theorem actually mean? I am told that Ehrenfest's theorem, applied to a physical observable $\hat A$, is:
$$\frac{d\langle\hat A\rangle}{dt}= \frac{i}{\bar h}\langle[\hat H,\hat A]\rangle$$
I don't understand how to use this equation or what it means intuitively.
| The equations describes the time evolution of an expectation value of an operator (which is the expectation value of the value measured when doing many measurements on identically prepared systems). The expectation values are taken with respect to the state of the system at time $t$. This answer will show how to use th... | {
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Can we ever "measure" a quantum field at a given point? In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is possible for an electrical field), or the only thing we can obser... | The comment by flippiefanus is curious. First, that contributor criticizes that the answer given is - if I understand correctly - too formalistic, to then go on and refer to the path integral formulation using hypothetical paths which are even more formalistic. Especially if one takes into account that the paths (or he... | {
"language": "en",
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"source": "stackexchange",
"question_score": "8",
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Is spin 1 described by $SO(3)$ or $SU(2)$ What spin is described by which rotation group? I always only find information about spin-1/2
| Quantum spin in nonrelativistic Quantum Mechanics is generally associated either with the projective unitary representations of the rotation group SO(3) or with the vector unitary representations of the special unitary group SU(2). To be more precise, spin comes naturally from the projective unitary representations of ... | {
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Can we weight dark matter? We can observe the effect of light as it gets bended in the presence of dark matter, and I wonder how is it possible to measure their mass given that we can't see them and they don't interact with ordinary matter? I know because the speed of the stars at the edge and the center of our galaxy ... | By definition of dark matter, it is not possible to determine specific masses with astrophysical means, only the gravitational effect they have. The masses may be composed of various things, as this table shows.
For example the machos will not have distinctive masses , they would be planet like,
A massive astrophysica... | {
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Physical interpretation of FRW normal coordinates The Friedmann-Robertson-Walker metric (I consider for notational simplicity the flat space case): $$\text d s^2 = \text d t^2 - a(t)^2\text d \boldsymbol{x}^2$$
can be brought to normal (Minkowski) form at the origin by a quadratic change of coordinates (see e.g. Eq. (1... | I calculate the line element:
Case I:
$\left[ \begin {array}{c} t\\ x\end {array} \right]\mapsto \left[ \begin {array}{c} T-1/2\,H_{{0}}{X}^{2}\\ X-
H_{{0}}XT\end {array} \right]
$
Line element with $V=H_0\,X$
$ds^2\mapsto \left( 1-{V}^{2}a \left( t \right) \right) {{\it dT}}^{2}+ \left( -2
\,V+2\,Va \left( t \right... | {
"language": "en",
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Mass versus rotation Curves Is there an equation that describes the relationship between the mass of the Galaxy and rotation curve?
I found V versus R graphs and equations that describe their relationship (kind of). But I wonder how mass would affect the rotation curves. For instance, if the Milky Way had more mass wh... | import math
from scipy import special
import matplotlib.pyplot as plt
#Constants
G = 4.302*(10**(-3)) # in Pc MS-1 (km/s)
R_halo = 30000 #in pc
M_disk = 10**10 # in solar mass
M_halo = 3*10**11 # in solar mass
R_disk = 3000 # in pc
Radius = []
Velocity = []
V_H = []
V_D = []
for R in range(1,30000,100):
y = R/(2*... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/446920",
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Does a particle with infinite energy escape an infinite well? Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling.
What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well? Does it escape? ... | The maths breaks. As said in the comments, it's an "irresistable force meets immoveable object" paradox, and the mathematics concedes and concurs with its paradoxical nature by breaking.
As mentioned in the other comment, the energy wave functions (without normalization) are
$$\psi_n(x) = \sin\left(\frac{\pi n}{L} x\ri... | {
"language": "en",
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Stack/Chimney Effect: a physical explanation on how height of chimney affects $\Delta P$ A fellow engineer told me that there are greenhouses which exploit the stack effect, in order to cover some or all of their electrical energy needs. This is achieved by installing small electrical generators with fans mounted on th... | The pressure difference here depends on the difference of weights of two identical in size imaginary air columns that go from the ground to the top of the troposphere:
one of them includes the ambient air near the chimney,
another includes the air inside chimney and everything above it.
The taller the chimney - the lig... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/447463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the "lowest energy"? In many textbooks I come across the term lowest energy. For example in atomic structures, electrons are placed in orbitals in order for the atom to have the lowest energy. But what is this energy? Potential- or kinetic energy or the sum of the two?
| In QM the Schrödinger Equation gives you the solutions for the wavefunction of a particle with a given potential. Because the energy is quantized you usually find several possible values for the energy that are given by an integer number $ n $ called the Principal Quantum Number. The lowest value of $ E_n $, normally w... | {
"language": "en",
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Can a battleship float in a tiny amount of water? Given a battleship, suppose we construct a tub with exactly the same shape as the hull of the battleship, but 3 cm larger. We fill the tub with just enough water to equal the volume of space between the hull and the tub. Now, we very carefully lower the battleship into ... | The USS Missouri $5.8 \times 10^7\,\rm kg, \, 270\,\rm m$ long with a fully laden draft of $11.5\,\rm m$ has an underwater surface area in excess of $270\times 11.5\times 2 \approx 6200\,\rm m^2$ and needs to "displace" $5.8 \times 10^7\,\rm kg$ of salt water (density $\approx 1020 \,\rm kg \, m^{-3}$) to float.
Assum... | {
"language": "en",
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"source": "stackexchange",
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Neumark's theorem - equivalence of POVM and projective measurements Let's say we have a system coupled with an ancilla $\vert \psi_{SA}\rangle = \vert\psi_S\rangle\otimes\vert\phi_A\rangle$. The unitary evolution of this state is given by $U_{SA}$. If we perform a projective measurement on the ancilla, the $i^{th}$ out... | Without loss of generality, I will assume $|\phi_A\rangle=|0\rangle$ -- otherwise, just rotate your ancilla space into that basis first as part of $U$. Similarly, I will assume $|m_i\rangle=|i\rangle$.
The condition $\sum M_i^\dagger M_i=1\!\!1$ says that the matrix
$$
V=\begin{pmatrix}M_1\\M_2\\\vdots\\M_k\end{pmatrix... | {
"language": "en",
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How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant? When writing integrals that look like
$$
\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0)
$$
it is often said that th... | Use the relation
$$
\int \frac{d^3{\bf k}}{(2\pi)^3}\frac{1}{2\sqrt{{\bf k}^2+m^2}} = \int \frac{d^4k}{(2\pi)^4}\frac{1}{k^2+m^2}
$$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For dimensional regularization, why the arbitrary mass scale $\mu$ has the meaning of UV cutoff? For a sharp cut-off regularization, we introduce the UV cutoff $\Lambda$. When we need to do momentum integral, we integrate the momentum ball with radius $\Lambda$. This $\Lambda$ has the explicit physical meaning of UV cu... | Dude, you are confused! In the dimensional renormalization scheme (Feynman used to call the shell game of renormalization dippy Hocus-Pocus), it's the $\epsilon = 4-d$ which plays the role of UV cutoff.
The renormalization scale $\mu$ is the energy scale ($p^2 \sim \mu^2$) you anchor your renormalized parameters, such... | {
"language": "en",
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Why can't a particle penetrate an infinite potential barrier? I am studying basic quantum theory. My question is:
Why can't a particle penetrate an infinite potential barrier?
The reasoning that I have applied is that particles under consideration have finite energy. So, to cross an infinite potential barrier the parti... | Imagine a finite potential well of the form
$$
V(x) = \begin{cases} 0 & |x| < L/2 \\ V_0 & {\rm otherwise}\end{cases}
$$
You can solve Schrodinger's equation in the usual way, by splitting the domain in three parts, the resulting wave function will look something like this
$$
\psi(x) = \begin{cases} \psi_1(x) & x < L/2... | {
"language": "en",
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How can two electrons repel if it's impossible for free electrons to absorb or emit energy? There is no acceptable/viable mechanism for a free electron to absorb or emit energy, without violating energy or momentum conservation. So its wavefunction cannot collapse into becoming a particle, right? How do 2 free electron... | Let me start with a simple counter-question. How a free electron in a laser cooling process loses kinetic energy? The photon, hitting the compliant electron, gets absorbed and after is re-emitted with a higher frequency (with a higher energy content).
There is no acceptable/viable mechanism for a free electron to abso... | {
"language": "en",
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"source": "stackexchange",
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Violating Newtons First Law! Suppose you are inside a very large empty box in deep space , floating ( i.e not touching the box from anywhere initially).The box is at complete rest.
Now you push the box forward from inside.
Now you would go backwards but the box will move forward to conserve momentum.
However since you... | You are right, the force is internal to the system and not just the box. Overall the combined center of mass will remain fixed as you and the box exchange momentum.
There is no paradox here because you are either considering the entire system of box + human with no external forces, or you are considering the box by it... | {
"language": "en",
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Is it the current which create magnetic field, or vice versa, or both? Talking about stationary magnetic field, it is said that if a conductor rotates inside the field, a current is induced. Also, I read that current (moving charges) generate magnetic field, too. How are these connected, and what's the best approach to... |
Is it the current which create magnetic field, or vice versa, or both?
There are involved always three components.
*
*The most known case and the easiest imaginable is the Lorentz force in its primordial meaning $ \vec F = q \vec v \times \vec B $. A charge, moving nonparallel to an external magnetic field undergo... | {
"language": "en",
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What was wrong with the old definition of temperature scale in kelvin? Wikipedia's article on the recent change to the definition of the SI base units states, as the reason for changing the definition of the kelvin:
A report published in 2007 by the Consultative Committee for Thermometry (CCT) to the CIPM noted that t... | I think that the problem is that the temperatures that you quote are a long way away from the triple point of water and no one thermometer can accurately span to your quoted temperatures from the triple point of water.
A definition of the kelvin that fixes the value of Boltzmann’s constant makes it possible to desig... | {
"language": "en",
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From where electrons flow to make a bulb light? Suppose we have the "basic" stuff like a battery 2 piece of wire and a bulb. Battery has a potential difference. But from where electrons flow to make the bulb light? from wire or from battery or from both? also if electrons flow from battery and they go through the wire ... | Every piece of the circuit has the molecular structure in which the electrons can either be bound to its atoms nuclei or they have enough energy to detach from their atom and roam in the field of metallic bond. Conducting metals have low energy threshold that electrons need in order to detach. A force caused by a poten... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the second law of thermodynamics a "no-go" theorem? As defined here, there are several no-go theorems in theoretical physics. These theorems are statements of impossibility.
The second law of thermodynamics may be stated in several ways, some of which describe the impossibility of certain situations.
The question i... | Consider for concreteness the Kelvin-Planck statement that 'you cannot extract net average work in a closed cycle from a single heat-bath'. This certainly has a flavour of a no-go statement. To call it a theorem we normally demand that it is derived (non-trivially) from some other definitions/axioms. One can indeed der... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/450313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Inverse of a matrix in a Path Integral Good morning! I can't make sense of an inverse of a matrix appearing in a calculation for a Wiener Path Integral. In discretized form:
$$\int \prod_{i=1}^N \frac{dx_i}{\sqrt{\pi \epsilon}} e^{-\frac{1}{\epsilon} \sum_{i=1}^N \left( x_i-x_{i-1} \right)^2-\sum_{i=1}^N p_i x^2_i} \de... | It's a tridagonal matrix, so the inverse is found as the Green function of the associated three-term recurrence relation. There is no closed-form solution for general $p_i$, but a detailed description of the related math is in the excercises starting on page 86 of my lecture notes at https://courses.physics.illinois.e... | {
"language": "en",
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Where does the momentum go when atom absorb a photon? Imagine an electron around an atom absorbs a photon and becomes excited, it has now jumped to a higher orbital. At this point in time, where does the momentum of the photon goes?
| Emilio's answer (How does one account for the momentum of an absorbed photon?) is great but quite technical. The simplified version is some of the momentum is accounted for by the different linear momentum associated with different orbitals. From the Bohr model we have $$p_n=\dfrac{\hbar}{a_0 n}$$ where $n$ is the prin... | {
"language": "en",
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Why do macro black holes take ridiculous amounts of time to evaporate—unlike micro black holes which dissolve in even less than a second? Why do macro black holes take ridiculous amounts of time to evaporate, considering that micro black holes dissolve in even less than a second?
Does this mass-based behavior imply tha... | One way of looking at Hawking radiation is to think of the usual representation of virtual pairs as oysters, upon which the BH can feed (losing weight in the process, of course). But it is the tidal effect, the gravitational gradient (GG), which allows the BH to separate the virtual pair, eat one and spit the other out... | {
"language": "en",
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Does the half-life of an element mean it will never decay completely? Example:
Half life of Polonium-194 is 0.7 seconds. If we supposedly take 50g of Polonium, there will surely be a time when no more of this Polonium will be left because if we consider the decay discretely, in the form of individual atoms, won't there... | There will certainly come a time at which we can say "it is more likely than not that not even one atom of the original Polonium sample is left". So, yes, the sample can decay completely.
The fact is, the earth is running out of natural radioactive elements. Most of what is left are Uranium, Thorium and Potassium bec... | {
"language": "en",
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Is there a way to separate 2D from 3D? When we see object around us in space, we can always interpret those in 2D, by considering them to pass through a plane, its only when we interact with those objects do we realise that it is 3D, is there any significant way of knowing this difference, using mathematics?
| Using mathematics can mean different things. I will start explaining with a trivial example.
*
*I give you a two dimensional image(or even series of them ) and ask you to reconstruct a 3-D object from it. In this case, there is no inherent information in any two dimensional image that tells you what the object could... | {
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Fermionic ghost path integral results in $\delta$ function? This is related to a statement in pg 20 of hep-th/9408074 formula (2.39).
Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to my guess is ghost field as it is non-dynamical and assume $\cd... | Eq. (2.39) is a $t$-dimensional Grassmann-odd delta function$^1$
$$\prod_{x=1}^t \delta(\frac{i}{\lambda^{\prime}} g_{ij}U_x{}^i\psi^j)~=~\prod_{x=1}^t \int \! d\eta^x~\exp\left\{ \frac{i}{\lambda^{\prime}}\eta^x g_{ij}U_x{}^i\psi^j\right\}~=~\prod_{x=1}^t \frac{\pm i}{\lambda^{\prime}} g_{ij}U_x{}^i\psi^j .\tag{2.39... | {
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The reasoning behind doing series expansions and approximating functions in physics It is usual in physics, that when we have a variable that is very small or very large we do a power series expansion of the function of that variable, and eliminate the high order terms, but my question is, why do we usually make the ex... | In general, one uses whatever works to learn something about the system.
Get an exact exact solution if you can. But too often that is not possiblr.
So, use any technique you like to learn something about the behaviour.
Turns out that the perturbative expansion can often be used, is usually meaningful around a stable... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/451588",
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"source": "stackexchange",
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$Q$-factor for damped oscillator (not driven)? How would this be defined?
Some of the Q-factor definitions I have encountered include:
$$Q=2\pi\frac{\text{Energy stored}}{\text{Mean power per cycle}}\\Q=2\pi\frac{\text{Energy stored}}{\text{Energy lost per period of
oscillation}}\\Q=2\pi\frac{1}{\text{Fractional powe... | A practical way to measure the Q factor for a non-driven oscillator is to measure the logarithmic decrement of the amplitude as the response decays after an impulse, and use that to find the damping ratio and hence Q.
Note that the value of Q is only a constant for linear systems. For a nonlinear oscillator, in general... | {
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"url": "https://physics.stackexchange.com/questions/451690",
"timestamp": "2023-03-29T00:00:00",
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Hamiltonian description of a system I know that phase space is the Hamiltonian description of a system, where we deal with position and momentum in equal footing. My question is in this phase space are those position and momentum are basis for that system?
As far as I know they are independent, in Hamiltonian dynamics,... | I'm afraid you're making a soup of concepts which have nothing to do
with one another. My suspicion derives e.g. from your use of words
like "orthogonal" and "basis", which I would see better in context
about QM.
As to drawing "position and momentum line perpendicularly" you're
attributing weight to an innocent practic... | {
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Different predictions from differential vs integral form of the Maxwell–Faraday equation? Assume a toroidal solenoid with a variable magnetic field inside (and zero outside) and a circular wire around one of the sides.
Because there is no magnetic field outside the solenoid, we have
$$\nabla \times E = - \frac{\partial... | Your conclusion that the electric field is conservative is wrong; from Stokes' theorem,
$$\oint_{\partial \Sigma} \mathbf{E}\cdot d\mathbf{l} = \iint_\Sigma \nabla \times \mathbf{E}\cdot d\mathbf{S},~~~~~~$$
and the curl of $\mathbf{E}$ is not zero everywhere on $\Sigma$.
| {
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How can we determine if poles are North or South when viewing magnetic field lines formed by iron filings? Iron filings can be used to visualize the magnetic field lines of a magnet.
For example, from Wikipedia:
$\hspace{175px}$,The magnetic field of a bar magnet revealed by iron filings on paper. A sheet of paper is ... | We can't. A magnet's North and South poles don't have any particular intrinsic property by themselves, only in how they interact with another North or South pole. If two pole attract, they are opposite, but we can't say which one is which. If two poles repel, they are either both North or both South, but again, we can'... | {
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Influence of Lorentz force on Eddy Current A magnetic field exerts a force on a moving charge called Magnetic Lorentz force. How does this force work in case of eddy currents? The following is an extract from Wikipedia:
"Another way to understand the current is to see that the free charge carriers (electrons) in the m... | One idea may be that the magnetic term $\overrightarrow{v}\wedge \overrightarrow{B}$ is not the only one to put the charges in motion. In an electrical circuit, the electric field associated with the surface charges accumulating at the edge of the circuit must be taken into account. The generalized Ohm's law is written... | {
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Where does it getting wrong , when using $v^2 - u^2 = 2as $ down the incline, for different object having different moment of inertia? Well, Consider a situation there is a sphere and a ring, of same mass $M$ and radius $R$. They both starts rolling down the inclined plane. We know moments of them as well, $$I_\text{sp... |
I get this solution:
The equations of motion are:
$I\,\ddot{\vartheta}=F_c\,R\qquad (1)$
$M\,\ddot{s}=-F_c\,R +M\,g\,\sin(\alpha)\qquad (2)$
and rolling without slipping
$\ddot{s}=\ddot{\vartheta}\,R\qquad (3)$
with $F_c$ is constraint force.
We have 3 equations for 3 unknowns $\ddot{s}\,,\ddot{\vartheta}$ and the ... | {
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Hamiltonian formulation of general relativity Why is it not possible to find a Hamiltonian formulation of general relativity as easily as in classical mechanics? There was a remark to this in my lecture but no real explanation as to why this is.
What stops us from creating a Hamiltonian formulation of GR?
| The basic Hamiltonian formulation of GR is the Arnowitt-Deser-Misner (ADM) formalism from 1959. The Legendre transformation of the Einstein-Hilbert Lagrangian density is singular, which leads to constraints.
References:
*
*ADM, arXiv:gr-qc/0405109.
| {
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Is there anything special about ebonite and fur? I'm from Czech Republic, born 1980. From elementary school, we all remember this mantra:
When ebonite rod is rubbed with fox fur, electrostatic charge is created.
Electrostatic charge is created by rubbing ebonite rod with fox fur.
Rubbing ebonite fur with fox fur ... | The electrons in fur are much less tightly bound than electrons in ebonite (very strong relative bond, ebonite is at the bottom of the negative Triboelectric series, see [1]) and hence ebonite gets a strong relative negative charge [1].
"A material towards the bottom of the Triboelectric series table, when touched to a... | {
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Why do mirrors not follow brewster's angle? Normally a material will have an angle where the reflected light is completely polarized. Now say we have a mirror (implemented by a conductive silver coating) that reflects most of it's incident light. https://physics.stackexchange.com/a/10925 says that this imperfect mirror... | Brewster's angle relates the index of refraction to a polarization phenomenon
in reflection from a dielectric (insulator) material. Most mirrors are
silvered (have a metal coating), and the equivalent dialectric constant
for a metal is ... infinity. That predicts a Brewster's angle
of 90 degrees, which is geometric... | {
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How is quantum mechanics consistent with statistical mechanics? Let's say we have an harmonic oscillator (at Temperature $T$) in a superposition of state 1 and 2:
$$\Psi = \frac{\phi_1+\phi_2}{\sqrt{2}}$$
where each $\phi_i$ has energy $E_i \, .$
The probability of finding each the $i$ state would be 50% in this case. ... | You're comparing the wrong things - you must take the appropriate limits when using statistical physics (or statistical mechanics as you called it) to recover quantum and classical results. Also, recall that when we use statistical physics, we must consider the system in an ensemble formalism, i.e. the canonical ensemb... | {
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Is the Higgs boson an elementary particle? If so, why does it decay? The Higgs boson is an excitation of the Higgs field and is very massive and short lived. It also interacts with the Higgs field and thus is able to experience mass.
Why does it decay if it is supposed to be an elementary particle according to the sta... | Most fundamental particles in the standard model decay: muons, tau leptons, the heavy quarks, W and Z bosons. There’s nothing problematic about that, nor about Higgs decays.
Your question may come from a misconception about particle decay: that it’s somehow the particle ‘coming apart’ into preexisting constituents. It... | {
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Why can blue LEDs be used for generating white light, but red LEDs cannot LEDs consist of pn-junctions, so why can blue LEDs be used for generating white light, but red LEDs cannot
| "White" light consists of a mixture of at least three colors that should be blue(ish), green (ish) and red(dish).
The most common way to get white from basically monochromatic LED light is to use fluorescent material.
This is what a spectrum of a blue LED looks like:
This gets converted in the LED to something like t... | {
"language": "en",
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Dirac matrices in 1+1 dimensions Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
| The converse problem, constructing the 1+3 dim $\gamma^\mu$s (4 × 4 matrices) out of the 1+1 dim ones (2 × 2 matrices) is solved systematically here in WP .
The stated one is straightforward, since the standard Dirac rep 1+3 ones amount to just
$$\gamma^0=\sigma^3\otimes I, \qquad \gamma^1=i\sigma^2\otimes \sigma^1,... | {
"language": "en",
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Is there a connection between the energy distribution and time dilation? Can anyone please help me understand what is descibed bellow?
Scenario 1.
We have a pair of atomic clocks. Let's call them clock A and clock B. We switch both of them on at the same time. Clock A will stay on Earth and clock B will go with the ast... |
If astronauts with clok B in scenario 1&2 always spend 22 years, use same energy and reach same maximum speed...But trave distance will be different. Will it have impact on time dilation?
Yes. The proper time experienced by the clock is given by $\int ds$, where $ds^2=dt^2-dx^2-dy^2-dz^2$ (in units with $c=1$). The p... | {
"language": "en",
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Vrms for monotomic, diatomic, and polyatomic molecules in my notes from class I have that
Vrms= sqrt(3kT/m) for a pt molecule
Vrms= sqrt(5kT/m) for a diatomic molecule
and that
Vrms= sqrt(6kT/m) for a triatomic-> higher order molecule
but RMS Speed of Gas Molecule for Polyatomic Molecules
says that Vrms=sqrt(3kT/m) al... | $\frac 12 mv^2_{\rm rms}$ is the mean translational kinetic energy of an atom/molecule and it is the the mean translational kinetic energy which is proportional to the temperature.
Vibrations and rotations may contribute to the total kinetic energy of a molecule but it is the translational kinetic energy which is lin... | {
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Is it possible to have a planet entirely made out of liquid water? Earth is mostly covered in oceans, but they only go a few kilometres deep. It's obviously not possible to have a planet the size of the earth to be made entirely out of water, because of the kind of pressures reached in the interior.
a. But say that we ... | Suppose we assemble a large mass of water in space, then it will form a sphere held together by its gravitational field. The question is then how large this sphere can become before the pressure at the centre causes the water to solidify to ice.
The calculation of the pressure at the centre is straightforward in princi... | {
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What if we set Hamilton-Jacobi equation as an axiom? We usually postulate the principle of least action. Then we can get Lagrangian mechanics. After that we can get Hamiltonian mechanics either from postulate or from the equivalent Lagrangian mechanics. Finally we get the Hamilton-Jacobi equation (HJE).
But what if we... | I) The Hamilton-Jacobi equation (HJE) itself sure ain't enough as an axiom without some kind of context, setting, definitions and identifications of various variables.
II) Let us assume:
*
*that Hamilton's principal function $S(q,P,t)$ depends on the old positions $q^i$ and the new momenta $P_j$ and time $t$,
*the... | {
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Conservation of momentum if kinetic energy is converted to mass There is a moving object. Through an unspecified (science fiction) mechanism its kinetic energy is converted to mass and the object comes to rest.
The mechanism is fictional but in good scifi it is good to adhere to the laws of nature. Does the convers... | Momentum was non-zero before the conversion and zero after the conversion and so conservation of momentum was violated. Therefore, your described conversion is not consistent with the known laws of physics which require conservation of momentum to always hold.
| {
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Dimensional analysis - application to logarithms I read some nice threads about this topic:
physics StackExchange
maths StackExchange
stats StackExchange
However, it still puzzles me that logarithm of some physical quantity has no units.
Example, let's assume we have a collection of values of the distance between two c... | The reason a logarithmic function, or an exponential function can't have dimensions is easy to see if you consider what the expression for a logarithm is in terms of a power series.
$$
\begin{align}
\ln x &= (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} + \cdots\\
&= \sum\limits_{n=1}^\infty \left((-1)^{n-1}\fr... | {
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Possible paradox relating to acceleration, velocity, and work I recently encountered a question that made me think of this "paradox":
Imagine the following situation:
There are two forces: Force 1 and Force 2. They accelerate a mass $m$ from the same initial velocity to the same final velocity in the same amount of ti... | Another way to look at this. You are correct in saying that the work done by each process is equal, since the two processes produce the same change in final kinetic energy. You are also correct in the think that force required to move the particle over the curved path is larger than the force required to move the pa... | {
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Two black holes colliding (classical mechanics) I had this question come up in my exam where two identical black holes are in orbit around each other. There is a loss of energy via gravitational waves : $$\frac{d E}{d t} = kr^4\omega^6$$ where $k$ is a constant, $r$ is the separation between the black holes and $\omega... |
Is it possible that the dependence is $1/r^5$ after all?
Yes. For confirmation, see equation (2.38) in [1], equation (3) in [2], and the combination of equations (41)-(42) in [3]. All agree with your result $dE/dt\propto 1/r^5$.
However, note that the same result can also be written $dE/dt\propto v^6/r^2$ where $v$ i... | {
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Confusion with Virtual Displacement I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs instantaneously in the configuration space, but this doesn't make any mathematical sense to me. I... | *
*Let there be given a manifold $3N$-dimensional position manifold $M$ with coordinates $({\bf r}_1, \ldots, {\bf r}_N)$. Let the time axis $\mathbb{R}$ have coordinate $t$.
*Let there be given $m\leq 3N$ holonomic constraint functions
$$f^a: M\times \mathbb{R} ~\to~\mathbb{R}, \qquad a~\in~\{1,\ldots, m\}. $$
The c... | {
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What is the point of generalizing a more specific result to an order of magnitude? In my textbook, an example wants me to find an estimate of the number of cells in a human brain.
It gives the volume of the brain as $8 \times 10^{-3}\ \rm m^3$ which it then estimates further as $1 \times 10^{-2}\ \rm m^3$. It follow... | An immediate advantage is that the calculation can be done in your head. Quickly, what's
$$\frac{(8\times 10^{13})(3\times 10^{-3})(2\times 10^{5})}{(2\times 10^{7})(5\times 10^{8})}+3\times 10^{-2}?$$
How about
$$\frac{(10^{14})(10^{-3})(10^{5})}{(10^{7})(10^{9})}+ 10^{-2}?$$
In the second example, I can add the expon... | {
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What is the meaning of effective density in porous media? Is the density of air inside the pore space not same as density of free air? I am trying to understand the physical meaning of using effective density in porous media. Is it a fictitious value? Can't I use the density of solid and fluid as it is while modeling p... | Density in a porous medium is not the same as the density of a pure substance.
Consider the general case when several phases are present in a porous medium. By definition, effective density is $$\rho_i = \lim_{\Delta V \to 0 } \frac{m_i}{\Delta V} \tag{1}\label{dens},$$ where $m_i$ - mass of $i$ phase (for example mass... | {
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How does pressure cooker increase cooking speed? My teacher mentioned something about pressure cookers making cooking faster and I was wondering what exactly is the reason behind this?
| If we heat a liquid, the average Kinetic Energy of the liquid increases and at a certain stage, the energy becomes sufficient to break the molecular attraction. The molecules anywhere in the liquid can form vapor bubbles. These bubbles float to the surface of the liquid and finally come out of the liquid. This phenomen... | {
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Derivation of heat capacity at constant pressure and temperature I have a question pertaining to the following derivation:
For the heat capacity at constant volume part, we apparently have:
$$dQ = C_v dT + P dV$$
But I find this confusing, as if we are to assume volume is constant, then $dV =0 $ so I would say that
... | I call this equation ($C_V=(\partial U/\partial T)_V$) the cruelest equation in introductory thermodynamics because of how often it trips people up.
It looks like the misconception here is thinking that the heat capacity is how much the internal energy $U$ increases for a given increase in temperature $T$. This is not... | {
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If I leave a glass of water out, why do only the surface molecules vaporize? If I leave a glass of water out on the counter, some of the water turns into vapor. I've read that this is because the water molecules crash into each other like billiard balls and eventually some of the molecules at the surface acquire enough... | There's a fundamental difference between a liquid changing to a gas at the surface vs. in the bulk: the formation of new surface area, which costs energy.
Net evaporation from the surface is spontaneous whenever the relative humidity is less than 100% because energy fluctuations enable surface molecules to detach into... | {
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Why in the BCS ground state the probability amplitudes are taken real? In some references (see for example Ballentine ch. 18.5) the ground state of the BCS theory is assumed to be
\begin{equation}
|BCS\rangle = \prod_{\bf k} (u_{\bf k}+v_{\bf k}\hat{c}^{\dagger}_{\bf k,\uparrow}\hat{c}^{\dagger}_{-\bf k,\downarrow})|0\... | I have found the answer to my question on page 86 of the following document
https://www.physik.tu-dresden.de/~timm/personal/teaching/thsup_w11/Theory_of_Superconductivity.pdf
Here is clear that we have no reason a priori to assume that the two parameters are real. Therefore we should start with complex amplitudes.
Sea... | {
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How does the CMB constrain the baryon asymmetry? The CMB contains information about baryon acoustic oscillations in which baryons (I assume protons and electrons) and photons form a plasma exhibiting sound waves. How is information about the baryon asymmetry of the universe contain in this?
Why do we not get some combi... | Here is a description:
The acoustic oscillations arise because of a conflict in the photon–baryon plasma in the early universe. The pressure of the photons tends to erase anisotropies, whereas the gravitational attraction of the baryons, moving at speeds much slower than light, makes them tend to collapse to form over... | {
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What's the difference in a $P$-$V$ diagram that is curved versus one that is straight?
So what would the difference be between the graph above versus one that has the same initial and final points but the path is curved. I'm sure it has something to do with temperature, so does it mean temperature is constant? Or is t... | The only difference between a straight PV diagram versus a curved PV diagram is the work done in both cases (provided the final and initial points are same for both diagrams).
As you might know, the work done in such a case is the area under the curve of the PV diagram. So the work done will be more or less depending ... | {
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Can superconductors undergo a BKT transition? In the article by Kosterlitz and Thouless (1973) they write in the abstract: "This type of phase transition (BKT) cannot occur in a superconductor for reasons that are given". Later in the paper they say that their argument for the BKT transition cannot be carried through b... | I found the same statement puzzling given more contemporary discourse. Tony Leggett provides an explanation for this in his lecture notes on 2D materials (https://courses.physics.illinois.edu/phys598PTD/fa2013/L12.pdf), with the argument of enhancement of the London penetration depth in very thin (and dirty) films, and... | {
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Central Forces: Newtonian/Coulomb force vs. Hooke's law We know that a body under the action of a Newtonian/Coulomb potential $1/r$ can describe an elliptic orbit. On the other hand, we also know that a body under the action of two perpendicular Simple Harmonic Motions can also have an elliptic orbit. Hence I was wonde... | Your two examples are both central forces. For gravity the potential is:
$$ U_g = -\frac{k}{r} $$
while for the simple harmonic motion the potential is:
$$ U_s = kr^2 $$
Both of these allow circular orbits,and for a circular orbit you cannot tell which is which. However for an elliptical orbit you can because with grav... | {
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Why is the far point of human eye infinite? In my exams, the presence of this question, which unfortunately I couldn't answer, made me wonder why is the far point of an eye infinite?
First thing that came into my mind was that how come we can see till infinity?
Far point of eye is sometimes described as the farthest po... |
There's obviously a limit to a distance where the eye can see.
But such limits are not a function of the eye itself. For objects on earth the distance we can see depends on atmospheric conditions, curvature of the earth and size of the object.
For stars in the sky it ultimately comes down to brightness. Distant objec... | {
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"answer_id": 3
} |
"Killing leaves" in General Relativity? I now about Killing vector fields in GR but recently stumbled upon the notion of "Killing leaves" and couldn't find any simple explanation of this notion. For example, this paper writes: "integral submanifolds generated by vector fields of a Killing algebra are called Killing lea... | On page 3, it's defined as:
integral submanifolds of the distribution, generated by vector fields of a Killing algebra $\mathcal{G}$, are called Killing leaves,
A good overview of a (tangent) distribution and how it relates to the foliation of a manifold into "leaves" (hypersurfaces) might be found here, or on notes... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/458858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Do black holes have a limit of mass?
*
*If you have a bunch of gas and increase its mass, gravity combines it into a planet.
*If you have a planet and increase its mass, gravity forces the planet to undergo nuclear fusion, turning it into a star.
*If you have a star and increase its mass, the gravitational collapse... | The rules of classical general relativity say that when you add mass to a black hole, you get a larger black hole. If you add angular momentum to a black hole at a greater rate than that at which you add mass, it would theoretically be possible to get a Kerr black hole with $a \gt M$, which would convert the black hol... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/459067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Creation operator acting on a coherent state. Occupation number operator For a coherent state
$$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}(a^{\dagger})^n}{n!}|0\rangle$$
I want to find a simplified expression for $a^{\dagger}|\alpha\rangle.$ I can only get this $$\begin{align}
a^{\dag... | The following expression can sometimes be useful:
$$ a^\dagger |\alpha\rangle = \left( \partial_\alpha + \frac{\alpha^\ast}{2} \right) |\alpha\rangle . $$
To prove this, just calculate
$$ \partial_\alpha |\alpha\rangle = \partial_\alpha \left( \mathrm e^{-|\alpha|^2 / 2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/459146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How do I derive the angular frequency of a simple pendulum through conservation of energy? Is it possible? I'm not exactly sure what I'm doing wrong.
So far I've gotten:
$mgl(1-$cos$\theta) = \frac12\omega^2l^2$
Which then gives $\omega = \sqrt\frac{2g(1-cos\theta)}{l}$ which is incorrect. Where am I going wrong??
| In your equation, $\omega$ stands for angular speed, not angular frequency. The expression you found gives you the angular speed at the bottom of the swing as a function of the maximum angle to vertical $\theta$.
Note: you forgot an $m$ on the right hand side of your original equation, but I figure that was a typo sinc... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/459498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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