Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Does work done on a spring = elastic potential energy? So the UK exam board specifications (AQA GCSE) clearly state
"...the work done on the spring and the elastic potential energy stored are equal"
Here's my problem,
So work done = Force x displacement
Force = Spring constant x extension
Elastic potential energy = 0.5... | The work is $Fx$ if the force is constant during the displacement from 0 to $x$.
In the case of an elastic spring, the force is a function of the displacement: $F = kx$
So the work for a small displacement when the spring is streched in a given $x$ position: $\Delta W = kx\Delta x$
Integrating from 0 to x: $W = \frac{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/579993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can Schrödinger's cat be filmed? Before opening the box, the observer does not know if the cat is alive or dead, however, a camera placed internally "knows" all the time, which is really happening.
Does this camera cancel the result of the experiment, even if it is not consulted?
EDITING:
I made this edition just to cl... | Yes, it is possible to put a movie camera in the box with the cat. But when you open the box and look at the movie, you will see the cat die or not die, but not both. And that will agree with what you see when you then look at the cat: it will be either dead or alive. The movie film will be entangled with the cat. B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Null geodetic on different metric relation between affine parameter If i have a null geodetic on a $g_{\mu \nu}$ metric, it's also null in a generic $\hat{g}=\exp{2\omega(x)}g_{\mu \nu}$ for any $\exp{2\omega(x)}\geq 0$?
What's the relation between affine parameter?
| Consider a null geodesic on $g$ with affine parameter $\tau$. We have the following equations
$$
\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\rho} \frac{d x^\nu}{d\tau} \frac{d x^\rho}{ d \tau} = 0 , \qquad g_{\nu\rho} \frac{d x^\nu}{d\tau} \frac{d x^\rho}{ d \tau} = 0 .
$$
For the metric ${\tilde g} = e^{2\omega} g$, w... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/580200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there such a thing as a white laser pointer? I can't seem to find a white laser pointer for sale anywhere. Is it just me or is it somehow not possible to make it? I see only color laser pointers like red, green, blue, etc.
| It's not possible to make.
white is a mixture of all the visible spectrum whereas a laser can only be of a single wavelength (ie a single color in the rainbow)
See Wikipedia white light article for more details on white light
| {
"language": "en",
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Are humans special in that they collapse wave functions? First of all, I don't really believe that humans are special. So I know the answer must be that they are not.
But the way quantum mechanics is described is that all particles exist as clouds when humans are not looking but exist as particles when humans look at t... | What is really meant by an observer in quantum mechanics is not an actual conscious human, but a macroscopic object, contact with which causes collapse of the wave function. That is, the collapse is supposed to occur, even if there are no human observing it - in fact, this happens most of the time, since even in a lab,... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/580444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Inverse-square laws and point particles It's my understanding that many inverse-square laws can be explained as a central point emitting "interaction rays" in all directions equally. And that when another object with some area is "impacted" by those rays, it will then feel an effect proportional to the amount of rays t... | The language of "interaction rays" is wrong and conceptually misleading, but there is just enough validity in the question to be worth an answer, at least in so far as quantum electrodynamics is concerned (gravity is understood as a geometric effect, and the idea has no application there). The electromagnetic force is ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/580711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does the view of an observer near an event horizon change as a function of the observer's position, velocity, and acceleration? I see that this topic has been well covered from the perspective of a) a stationary observer dangling on the end of a rope and and b) an observer free-falling from infinity. What I'm stil... | What you see is what lies on your past light cone. The past light cone only depends on spacetime position, not velocity or higher derivatives.
What you see also depends on your velocity in the sense that it is "distorted" in different ways by Doppler shift and aberration. But given a (infinite-precision, omnidirectiona... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/580848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Imaginary time & predictions Is the imaginary time just a different convention to express the time evolution to make the calculations easier? Hawking also said that
"It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able... | The main things that imaginary times gives us are "instantons". These are solutions of the classical equations that exist only in imaginary time. They describe tunneling processes that are classically forbidden, but quantum mechanically allowed.
For simple systems we can derive these tunneling processes by other means ... | {
"language": "en",
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"source": "stackexchange",
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What’s the motivation for using Ricci scalar in action term for spacetime? Why do we use Ricci scalar in the action equation for the spacetime? Why don’t we use other functions? Is it just intuition? What forces us to use that?
| $R$ is the lowest-order (and thus simplest) scalar that can be formed from the Riemann curvature tensor. That plus a constant term (the cosmological constant) seem sufficient to explain what we observe. It is possible that the Lagrangian density is more complicated, but physicists choose the simplest explanation that w... | {
"language": "en",
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"source": "stackexchange",
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Is equation of continuity valid, while dealing with vertical pipes? Let try an experiment.
If water enter through an end $A$ with some velocity say $v_1$,and leaving end $B$ with speed $v_2$ in a UNIFORM cylindrical tube $AB$ (which is completely filled with water).
If we consider 3 cases.
*
*tube is horizontal
*tub... | If the tube is vertical, the fluid can't be filling the tube, and the area of the stream must be changing, thus allowing a velocity change to be consistent with mass conservation. In the case where A is at the bottom, if the fluid fills the tube, the pressure can't be atmospheric at the bottom. Therefore, you can't ass... | {
"language": "en",
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"source": "stackexchange",
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Lagrange equation for the pendulum This is Wilberforce Pendulum that has spring & weight:
https://faraday.physics.utoronto.ca/IYearLab/WilberforceRefBerg2of8.pdf
Wilberforce pendulum is a system of a spring hanging vertically, and a weight
with moment of inertia is hanging.
The system keeps transferring vibration betwe... | The Lagrangian in question contains the potential energy,
$$U(\theta, z) = \frac12 k~ z^2 + \frac12 \delta~ \theta^2 + \frac12 \epsilon~z~\theta.$$
If we simultaneously add offsets $z = \bar z + \alpha, \theta = \bar \theta + \beta$ and absorb any constant terms into the arbitrary choice of zero for the energy, this be... | {
"language": "en",
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"source": "stackexchange",
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Anomalous global symmetry in non-gauge theories I’m a bit confused on the effects of anomalous global symmetries. So take for instance the following theory
$$\mathscr{L}=\partial_\mu\phi\partial^\mu\phi^*+i\bar{\psi}\gamma_\mu\partial^\mu\psi-y \phi\bar{\psi}\psi+\text{h.c}-V(\phi)$$
with $V(\phi)=m^2|\phi|^2+\lambda |... | There is no anomaly problem with this system --- except that as written it does not have a continuous $U_A(1)$ symmetry. You need to include a term $i\bar\psi \gamma^5 \psi$ term in addition to the $\bar\psi\psi$ term. With that included it is a simple model that can be be used for illustrating chiral symmetry breakin... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/581687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why can we not take gravity to act at the center of mass in this case? I found this problem when I was trying to prove that the gravitational attraction on an object is the gravitational attraction on the center of mass of that object (I had doubts on it). And in doing so I ran into this problem which shows that the ef... | I didn't look over your work (check-my-work questions are off topic), but the conceptual error is still obvious. The gravitational field isn't uniform, so the force acts on the center of gravity, not the center of mass. For uniform fields these are the same thing, but in general they are not.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/581904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's exactly the new definition of kilogram, second and meter? Could one explain this?
Technically a kilogram (kg) is now defined:
[…] by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second a... | The SI system is now defined entirely by physical constants. There are no more “prototype” artifacts. How it works is thus:
https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9-concise-EN.pdf
The SI is the system of units in which:
• the unperturbed ground state hyperfine transition frequency of the caesi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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How does property of material effect it's interaction in Faraday's Law? We all know that faraday law mathematically states:
$$ V = - \frac{\partial \phi_B }{\partial t}$$
But I think this law is a bit weird because it says that the voltage developed across any conductor due to changing flux is same.. which is a bit cou... | The Faraday law is no more than a different formulation of the Maxwell-Faraday equation: $$\oint_{\partial\Sigma}\mathbf{E}\cdot d\mathbf{l} = -\int_\Sigma\frac{\partial\mathbf{B}}{\partial t}\cdot d\mathbf{A}$$
where $\Sigma$ is any closed surface, $d\mathbf{A}$ a vector element of this surface; $\partial\Sigma$ the c... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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How does the mere movement of gluons keep quarks together? You know the strong force (the one that keeps quarks together). Well it works by exchanging gluons right?
So how does that force keep the quarks together? I mean you can imagine that process as three people passing balls between them right? Well as far as I kno... | You have stumbled into one of the most interesting questions of QED and QCD, that is, how can we model the attractive and repulsive forces by the exchange of the massless mediators (photon and gluon respectively)? The answer is mathematically very complicated and when we look for an explanation in our everyday classica... | {
"language": "en",
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Is string theory the boundary theory of M-Theory? Looking at various AdS/CFT correspondences, we find that some (n-1) dimensional field theories on the boundary of $AdS_n$ with $N=\frac{8}{n-3}$ supersymmetries are equivalent to M-Theory in $AdS_n \times S_{11-n}$.
(e.g. for $n=7$ we get 6D $N=(2,0)$ superconformal CFT... | Heres a simple way to see that the answer is no. The theory on the boundary has to be superconformal, but the classification of superconformal symmetry says that SCFTs terminate in dimension 6. For M-theory there can only be 3d and 6d SCFTs duals because these theories are the conformal fixed points of the worldvolume ... | {
"language": "en",
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"source": "stackexchange",
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What is the analogue for symplectic structure in case of spin variables? According to some (e.g. Haroche and Raimond in Exploring the quantum: atoms, cavities and photons), the quantum world consists (mainly) of spins and harmonic oscillators.
For harmonic oscillators (i.e. bosons), it is well known that they can be ap... | The phase space for spin is the two-sphere $S^2$ with the symplectic form being the area 2-form
$$
\omega= J \sin\theta d\theta\wedge d\phi.
$$
Here $\theta$ and $\phi$ are the polar angles. Then, with
$$
S_x= J \sin\theta \cos\phi,\\
S_y= J \sin\theta \sin\phi,\\
S_z= J \cos\theta,
$$
we have $\{S_x,S_y\}= S_z$ etc.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "6",
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Find the potential difference given the function of electric field I have the following problem:
Suppose that, as a function of x, an electric field has an x component
$E_x=6x^2y$
where the electric field is measured in volts per meter and the distances are measured in meters. Find the potential difference between the ... | If you are moving along the x-axis then $y=0$ and so $E_{\rm x}=0$.
This means that the potential along the x-axis is a constant and so the potential difference between any two points on the x-axis is zero.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What if Jupiter or the Sun was made of rock, like Earth and Mars, rather than gas? Jupiter is a "gas giant". If it was (significantly) bigger the pressure from gravitation would ignite a fusion process and it would become a star, which is basically what happened to the sun.
However, what if a body the size of Jupiter o... | I hope it is ok to link to other stackexchange communities, as there is an excellent answer to be found in the worldbuilding community: Is there a theoretical maximum size for rocky planets. The consensus seems to be that the maximum size for Earth-like planets is at around twice the radius of Earth.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When excess electrons are in an insulator they don't move. Why is this? Let's say you have an insulator that is electrically neutral(has no net charge). Let's say you are able to add additional electrons into the same insulator resulting in the insulator having a net negative charge. These electrons that were added to ... | Consider the following diagram of the quantum mechanical densities of electron states in different types of material:
The Fermi energy $E_F$ is the energy level new electrons are added at. Loosely said, the states below this energy are filled, the states above are empty, with a smooth transition around $E_F$. The cond... | {
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Question from Relative Error Q -A satellite orbits Earth at a height of 560 km. In many computations, the Earth-Sun distance of 1.5 X 10⁸ km is used to approximate the distance of satellite from the Sun. What is the maximum relative error of this approximation?
Please explain the main concept in detail.
| If the distance between the earth and the sun is given as $1.5 \times 10^8$ km then you can assume this figure has a precision of $\pm 0.05 \times 10^8$ km, which is $\pm 5$ million km. If the satellite height is stated to be $560$ km then you can assume this figure has a precision of $\pm 5$ km. I'll let you take it f... | {
"language": "en",
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Does light have a mass? Could you please argue with work and gravity? Or however you like. I just don't get it. Thanks.
A physics professor talked about if light had a mass, it should do work ($W=N\cdot m = \frac{kg\cdot m}{s^2}\cdot m$) to get out of the gravitational field. But because we somehow can see it doesn't, ... | Mass is defined in relativity by $m^2=E^2-p^2$, where $E$ is the mass-energy and $c=1$. A ray of light or an EM plane wave has zero mass. However, mass is not additive, and a collection of light rays, or a more complicated wave pattern can have nonzero mass.
It's not really correct to argue about whether light has mass... | {
"language": "en",
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Sand leaking out of bag and skater "A person skating on a frictionless icy surface is holding a sandbag. The sandbag has a small hole at the bottom, from which the sand starts to leak. As the sand leaks from the sandbag, the speed of the skater..."
The answer was that the speed of the skater stays the same. I thought t... | A given force will propel a lighter body to a greater velocity.
Once moving, if we could just vanish part of the body's mass, it might speed up. But we can't. The sand, although jettisoned, still exists and takes its kinetic energy along with it.
So no speed up.
This ignores air resistance of course. If the leak ch... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/583709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is there a special connection between entropy and heat? As I understand it:
$dS = \frac{1}{T}dU + \frac{p}{T}dV$
(for a thermodynamic system where $dN=0$) and since for an ideal gas $pV=Nk_BT$ and $U=C_VT$ we can say
$dS = \frac{C_V}{U}dU + \frac{Nk_B}{V}dV$
so ultimately entropy changes are caused by changes in pr... | Let the work done on the system be $\delta W$ while its internal energy change be $dU$, assume that the system may also exchange energy with a reservoir that is at temperature $T_r$. Then for an arbitrary process the entropy change $dS$ of the system satisfies $dS \ge \frac{dU-\delta W}{T_r}$. The equality sign holds f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating with creation and annihilation operators I'm currently taking a quantum mechanics lecture and am having trouble with the mathematical formalism.
I have to calculate the following:
$$\langle n+2|b^\dagger b^\dagger |n\rangle$$
and
$$\langle n+1| \left(b^\dagger bb^\dagger + \frac {\hat{1}}{2}\right) |n \rang... | Just continually act the operator on a bra or ket. The first one goes as follows:
$$\begin{align}
\langle n+2 |b^\dagger b^\dagger |n\rangle &= \sqrt{n+1}\,\langle n+2|b^\dagger|n+1\rangle \\
&=\sqrt{(n+1)(n+2)}\, \langle n+2|n+2\rangle \\
&= \sqrt{(n+1)(n+2)}
\end{align}$$
because, in this case, $\delta_{mn}=\langle m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof that the electric field in a dielectric decreases by a factor We have a linear homogeneous dielectric material half filling a parallel plate capacitor. It's said that the field inside is reduced by a factor ,so we have
$\mathbf
{E}=\frac{1}{\epsilon_{r}}
\mathbf{E}_{\mathrm{vac}}$.
What's its proof?
( If a... | Gauss Law suggests that $ \nabla . D = \rho $. When there is no charge in a defined boundary. Ingoing and outgoing $D$ are equal. In your case, there are no charges outside of electrodes, thus, there will be no change of $D$ along the paths between the electrodes. Since $D = \epsilon_r . \epsilon_0 . E$ . As a result... | {
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Is there any physical interpretation of the constant which is seen in the constraint curve of an adiabatic process? What is the $C$ in $ PV^{\gamma} = C$? I always saw it as a result out of the mathematical calculations that we do but I recently saw this video which made me think that the constant may have more meaning... | Start your process with the gas having volume and pressure $V_0, p_0$. If the process is adiabatic and reversible then $pV^\gamma = p_0V_0^\gamma$, in other words $C=p_0V_0^\gamma$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If objects in motion experience time differently, how does my body stay synced when I move my legs or arms? If I move my swing my arm really fast, the matter in my arm should experience time slower than the matter in my body.
So how does my body still sync with each other?
And a more general question that derives from ... | Your arm is traveling at a tiny percentage of the speed of light. The speed of light 300 million meters per second, so even if your arm were moving at a ridiculous speed of 300 m/s, it would still be only one millionth of the speed of light. So even if there were a linear effect, it would be tiny. But the effect isn't ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/584228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 9
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The motion of an object moving up and down through an infinite plane Consider an infinitely large plane which exerts gravity onto a point mass with mass $m$ and located at height $h_0$. The point mass moves perpendicularly through the plane without undergoing any friction.
It can be easily derived that the force exerte... | You are obtaining non-harmonic oscillations in potential of type $V(x) = \alpha |x|$. This is correct, and the problem makes perfect sense. I do not want to go through the details of your calculations though - I think it is against the rules of this site.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Antiproton synthesis For a pion minus hitting a stationary proton, what are the other particles if an antiproton is to be created among them? A positive pion is possible but the total rest mass energy of the final state is comparable to the initial particles.
| Begin with the incomplete reaction
$$p + \pi^- \to \bar{p}\ +\ ?$$
Then write the hadrons as their quark-composites.
$$uud + \bar{u}d \to \bar{u}\bar{u}\bar{d}\ +\ ?$$
By counting the quarks and anti-quarks you find,
3 $u$ and 3 $d$ quarks need to be added on the right side.
There are many ways to do this.
Just play ar... | {
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What happens to $A$'s magnitude change in uniform circular motion?
Looking at this picture from Kleppner's mechanics text, Δa is added to $A$ and it is suggested that:
"This causes a change of direction but leaves the magnitude practically unaltered if $ΔA$ is small."
What is practically happening in this sense? Whe... | Uniform circular motion dilemma
I think this link answers my question and provides some good examples of effects of force as well.
| {
"language": "en",
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What does the orbit of Pluto look like from the perspective of Neptune? Pluto is locked in 3:2 resonance with Neptune, so that means that it should be possible to generate a simple looping animation that shows what movements Pluto would make given 2 orbits of Neptune.
I can find some animations of Pluto's orbit, but no... | I do not know if this is what you are looking for, but I thought it might be helpful.
https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/Animation_of_Pluto_orbit.gif/440px-Animation_of_Pluto_orbit.gif
| {
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What does it mean for particles to "be" the irreducible unitary representations of the Poincare group? I am studying QFT. My question is as the title says. I have read Weinberg and Schwartz about this topic and I am still confused. I do understand the meanings of the words "Poincaré group", "representation", "unitary",... | I apologize for not reading all of the answers. In "classical" QFT (e.g., QED and the current standard model that extends QED to electroweak theory -- except that real neutrinos observed by experiment flavor mix and thus effectively are not massless -- as well as QCD), the Poincaire group describes the requirement of ... | {
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Transition between 2 energy levels - wave function picture Suppose we have a system that has discrete energy levels (e.g. hydrogen atom, potential well) and the stationary solutions for the wave function are $\psi_n$. I would assume that there should be a way in which one can model the transition $2\rightarrow 1$ by us... | On place to look for information is by reading about Rabi oscillations, which arise when a two-level system is driven by a near-resonant electromagnetic field. For some situations one could find solutions in terms of wave functions that are exact or nearly exact.
More generally, the way to approach this problem is by c... | {
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Magnetization for Paramagnetic Materials If we consider the magnetization for paramagnetic materials, then we obtain $$M = -n\frac{\partial F}{\partial B} \propto B_J(x),$$ where $$x \equiv g\left( JLS\right) \cdot J \cdot \frac{\mu_B\cdot B}{k_B T}$$ is an auxiliary variable and $B_J( \ . )$ the Brillouin function. No... | Well, I can't see why it bothers you. Mathematically we have all these combinations, with different degeneracies, of S (1/2, 1,...)
Physically, integer S can be achieved either by decimating pairs of spin-1/2 particles (where a m=0 sector is accessible) or by considering a bosonic paramagnet, obtained by analyzing the ... | {
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Direct and in-direct measurments I have seen and heard this many times that a certain (physical) measurement is "direct" or "indirect". What is the characteristic or definition that sharply separates these two notions?
| There is no sharp distinction. The usage of these terms simply is one of context to indicate that one measurement requires less calculation to infer the desired quantity from the experimental data when compared to the other method.
The reason for that is a deeply philosophical one: Every scientific measurement is alway... | {
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Current induced in a superconducting coil Does the magnitude of current induced in a shorted superconducting coil depend on the external flux's rate of change $\frac{d\Phi_{EXT}}{dt}$ ?
Assume that initially the flux through the coil is zero and the initial current circulating in that coil is zero, too, but at some lat... | Since the total flux through the superconducting loop remains constant at $0$ (the reason for this is that any change in flux requires a nonzero emf around the loop, which requires in infinite current, so magnetic flux through the loop cannot change), the flux from the self-inductance $L$ of the loop must be equal and ... | {
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Negative critical exponent $\alpha$ for Superfluid helium at lambda point Due to the positive critical exponent of the transition in liquid helium I would expect there to be no peak at the transition $t=0$. Since the $t$ dependent part of the specific heat should go to 0 as $C \approx A \, t^{0.0127} + B$. What is goi... | The coefficient $A$ satisfies $A<0$. Also note that the background contribution $B$ is different for $T<T_c$ and $T>T_c$.
| {
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Does the time-independent Schrodinger equation in 1D have an exact and general solution? The (time-independent) Schrödinger equation is for sure the most important equation in quantum mechanics:
$$-\frac{\hbar^2}{2m}\nabla^{2}\psi(\vec{r}\,)+V(\vec{r}\,)\psi(\vec{r}\,)=E\,\psi(\vec{r}\,).$$
Let’s consider the one-dimen... | There is no general solution. But this is not only due to different possible potential functions, but also due to the boundary conditions. No differential equation can be solved without boundary conditions, and those can vary according to the problem.
Moreover the case distinction $E<0$ and $E>0$ leads to very differen... | {
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The derivative of the unit velocity vector
The set up:
An intertial frame Y-X used to describes trajectory of an insect on some rigid body using some relative vectors. Symbols: $ \vec{r_a}$ is is the vector connecting the origin to some point on the rigid body, $ \vec{r_b} $ is the vector connecting origin to the ins... | Since $\vec v_{rel}$ is a scalar multiple of $\vec r$ we have $\hat v_{rel} = \hat r$, so
$\displaystyle \frac {d \hat v_{rel}}{dt} = \frac {d \hat r}{dt} = \vec \omega \times \hat r
\\ \displaystyle \Rightarrow |\vec v_{rel}| \frac {d \hat v_{rel}}{dt} = |\vec v_{rel}| ( \vec \omega \times \hat r ) = \vec \omega \time... | {
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Why is the potential not infinite? One way to calculate potential (using infinity as our reference point) is to sum all the contributions of charges that are around.
Let's say I want to calculate the potential at some point on charged surface.
At that point, there is some charge (can be infinitesimal) and that charge s... | Point charges are just an abstraction, a limit model, like surface charge distribution in a infinitely thin layer.
You can think at a point charge as the limit for $r \rightarrow 0$ of a continuous distribution in a volume. As you look inside the small charges, the charge enclosed in the volume decreases and goes to ze... | {
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Is the Von Neumann entropy of two initially non-interacting systems always increasing? Let $A$ and $B$ be two systems that does not interact initially ($t=0$), i.e., the density matrix of the initial total system is given by $\rho(0) = \rho_A (0) \otimes \rho_B (0)$. Suppose that interaction between the two systems is ... | This is not true. The interaction $H$ (unless it has irrational eigenvalues) will have a recurrence time $T$ at which $e^{-iHT/\hbar}=\mathrm{Id}$.
At that time, the entropy will be equal to the one at $t=0$. Thus, it cannot increase all the time - unless it stays constant (which it will generally not do).
As an exam... | {
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How would quarks behave in the event of 'The Big Rip'? I have always heard that you can't get a quark by itself because "the energy required to split them apart is enough to create another." But, in the case of The Big Rip, the idea is that phantom dark energy would tear molecules and atoms apart because it's a much st... | In that scenario, the universe wold probably devolve into a quark-gluon plasma.
I was just reading a paper by Caldwell, Kamionkowski, and Weinberg which explores this very question. The paper is on arxiv here.
Part of the abstract reads
"Here, we explore the consequences that follow if the
dark energy is phantom energy... | {
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Discharging of a capacitor Consider a charged capacitor with its positive plate holding charge Q. Now I join the capacitor to an circuit with resistance R . So the capacitor starts to discharge. Small charge $q$ flows out of positive plate in a small time $dt$ . My textbook says that the instantaneous current that flow... | Firstly, you show some confusion between $Q$ an $q$. They are effectively the same thing, the charge stored by the capacitor. Since it is changing, it is customary to use lower case.
$dq/dt$ is the rate of change of $q$. As the capacitor discharges, $q$ falls; thus its rate of change $dq/dt$ is negative.
However the cu... | {
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How is it that for a closed subshell configuration $L=S=0$? For a closed subshell configuration of a many-electron atom, $M_L=\sum_i m_{\ell_i}=0$ and $M_S=\sum_i m_{s_i}=0$. But I do not understand why does it necessarily mean $L=S=0$. The values $M_L=M_S=0$ are compatible with nonzero values of L and S. Then how does... | As you say, for a closed subshell $M_L=M_S=0$
And this is true whatever direction you happen to have chosen for the $z$ axis.
If a vector has a $z$ component for any $z$ direction this can only be because it has length zero, as opposed to happening to be in a particular orientation where it lies entirely in the $xy$ pl... | {
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Understanding Born's rule for non-hermitian Hamiltonians Say I have a non-Hermitian hamiltonian, such as one might have in an incomplete description of a system where the states are allowed to decay. Then probabilities are not conserved since magnitudes will decrease with time. In particular, say I have an eigenfunctio... | Suppose you have $\langle \psi(t) | \psi(0) \rangle = A e^{-k t}$, for some normalization constant $A$.
Then the probability for the system to be in state $|\psi \rangle$ at time $t$ is
\begin{equation}
P_\psi(t) = |\langle \psi(t) | \psi(0) \rangle |^2 = |A|^2 e^{-2kt}.
\end{equation}
I feel this answer is a little gl... | {
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Contradiction in phase of reflected longitudinal waves While studying the fundamentals of sound waves in organ pipe, I noted that the fact about phase of reflected waves is contradicting while referring multiple sources
This book of mine describes the reflection from a rigid surface/closed end to be in phase
Whereas t... | Its a bit weird to compare the phase of a wave and its reflection, since their directions are different. Both the sources you have put up are saying the same thing: Compressions are reflected as compressions, and rarefactions are reflected as rarefactions.
Now because their directions are different, the phase differenc... | {
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What is the physical meaning of the complex field amplitude? I see that the harmonic field is sometimes written in exponential form. But sometimes the complex amplitude of this form is just a constant and in others (like when talking about modes) is dependent on the position. Why sometimes is dependent and what does it... | *
*As explained in detail in What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$?, when complex field amplitudes like $E(\vec r,t) = h(\vec r)e^{-i\omega t}$ are presented, there is a broad convention that the physical field is obtained as its real part, $E_... | {
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Where is the energy involved in osmosis? Osmosis creates pressure on the side of the membrane with higher concentration. But where does the energy for this come from?
| Consider a U shaped tube with a membrane in the middle, permeable to water but not to salt. It is partially filled with water. The height of the water surface on both sides of the tube will be the same, as they both feel the same atmospheric pressure.
Now we add salt to the left side of the tube, which fully dissolves ... | {
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Operator norm and Action We define the norm of the operator as $\left\lVert A \right\rVert = \sup \frac{\left\lVert A\psi \right\rVert}{ \left\lVert A \right\rVert} = \sup \left\lVert A\psi \right\rVert$ for $A ∈ L(H)$. It is said that $||A||$ measures the magnitude of the action of $A$.
What is meant by the action of... | The sup in the equation is a supremum over all states out of the Hilbert space $\mathcal{H}$. In other words, you pick the state $|\psi\rangle$ out of $\mathcal{H}$ for which the number $\frac{\mid\mid A|\psi\rangle\mid\mid}{\mid\mid |\psi\rangle\mid\mid}$ becomes the largest. It can be shown that the operator norm of ... | {
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Why is the path of light not visible in vacuum? When light is directed towards our eyes we detect the source and locate it. In Huygens' principle it is said that every point on wavefront acts as a source of secondary waves which again travel in all possible directions.
If light is directed not towards our eyes, like pa... | The following may be useful to consider.
It is true that Huygen's principle states that every point on a wavefront acts as a source of secondary waves which travel in all possible directions. However, those individual waves are not seen as separate, but only as a superposition.
In other words, say you are looking at a... | {
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Confusion on wave packet and creation operator in Mark Srednicki's book In Mark Srednicki's QFT book, section $5$, he mentions following things:
$a^{\dagger}({\bf k})$ creates a particle with momentum $k$ and is given by
\begin{equation}
a^{\dagger}(k)=-i\int d^3x [e^{ikx}\partial_{0}\phi(x)-\phi(x)\partial_0(e^{ikx})]... | $a^{\dagger}_k$ creates a particle with a definite momentum $k$; definite in the sense of Dirac delta function.
While $a^{\dagger}_1$ creates a state in which the momentum is not definite but almost smeared over $3\sigma$ range about $k_1$. Wave packet is proper term to refer such kind of state since this is how we cre... | {
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If gravitational field is not real, then am I gaining energy? I don’t know much about General Theory Of Relativity but I have heard that it does not consider gravitational fields like Newtonian Mechanics.
If an object were to be free falling, then according to general relativity that object would be in an inertial fram... | In the frame of reference of a freely falling body, you are gaining speed or energy(assuming you are standing on the ground), because the normal force is accelerating you and doing work on you(in that frame there will be no gravitational force). If that normal force was not there then you to would be in free fall.
Now ... | {
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How does the whole universe agree on the laws of physics? How is it possible that the every particle in the universe agrees on the laws of physics?
What enforces those laws? Might the laws change slightly across the universe in the same way the cosmic microwave background radiation (CMBR) does?
| You seem to attribute sentience and intention to lifeless particles - which they do not possess (or need not posses to describe the universe in a meaningful way, c.f. Occam's razor).
A "law" in physics is kind of the opposite of a "law" in the legal sense. There isn't any "physics police" enforcing obedience of the Uni... | {
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Can a black hole be characterized by an observer who is inside the black hole? According to the no-hair theorem a black hole can be completely characterized by three parameters: mass, electric charge, and angular momentum. Can any of these parameters be determined by an observer who is inside the event horizon of the b... |
Can any of these parameters be determined by an observer who is inside the event horizon of the black hole?
Yes. As long as they can make non-local observations (i.e. they are not confined to a small windowless spaceship) then an observer inside the event horizon can in principle gather just as much information about... | {
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Why is the electrostatic force felt in straight lines? When two positive charges are kept close, they get repelled in the direction of a line joining both the charges. Why is it so?
Also, why is the repulsion in a straight path?
In both the cases, the potential energy of the charge which gets repelled decreases. What... | Following R.W. Bird answer:
Consider an isolated system of two particles:
Since system is isolated, angular momentum (and linear) is conserved.
I.e. $\vec\tau_{net}=\Sigma\space \vec r\times\vec F=0$
But clearly from figure $\vec\tau_{net}=\vec r\space\times\space\vec F_y\ne 0$
Thus as R.W. Bird noted, the system viol... | {
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Dirac delta equalities in physics Earlier I asked this question on the Math Exchange but I'm looking for a physics point of view. How do you interpret an equation like $$x^n \delta(x) = 0, \qquad n\in \mathbb{N},$$ around $x=0$? Why does it suffice to show the integral of this expression is zero around the singularity ... | There may be many different definitions of the Dirac delta, but they all share the common feature that they only make sense inside an integral, and that
$$\int_{-\infty}^\infty f(x)\delta(x-a)dx = f(a)\qquad (*)$$
My preferred perspective is that the collection of symbols on the left-hand side of $(*)$ is defined to me... | {
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Why does the $\mathbf{B}$-field of a cylindrical magnet have no $\phi$-component I have a cylindrical permanent magnet with uniform magnetization $\mathbf{M}=\mathbf{a_z}M$, length $L$ and Diameter $D$.
I'm wondering why the $\mathbf{B}$-field created by this uniform magnetization has no $\phi$-component, that is, the... | A possible way to see it is the following. A standard result from electromagnetic theory is that a magnetization is equivalent to a current distribution, generating the same field, given by a volume current density
$$\boldsymbol{J} = \boldsymbol{\nabla}\times \boldsymbol{M}$$
and a surface current density
$$\boldsymbol... | {
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Normal ordering of hamiltonian
I came across this in the lecture notes of quantum field theory by David Tong. Inside time ordering interactions aren’t taken to be normal ordered. Interaction hamiltonian should be normal ordered otherwise it is not well defined (due to ordering ambiguity and related singularities). Mos... |
Most standard QFT textbooks don’t address this issue. Am i missing something here or normal ordering was assumed?
What? No. What could you possibly mean by "most standard QFT textbooks..."?
Check out the section titled "Wick's Theorem" in Peskin and Schroeder's textbook titled: "An Introduction to Quantum Field Theor... | {
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Galileos statement about object thrown in horizontal direction arriving on the floor at the same time as if free fall Is this statement correct? I intuitively believe it is wrong. Simplest example would be shooting a bullet.
I just read this from my little brothers textbook. Is the textbook garbage or did i miss out on... | The statement is correct as long as the ground is horizontal and aerodynamic forces such as drag or lift can be ignored (so it doesn't apply when throwing a frisbee, for example). The effect of gravity on a bullet's path is known as bullet drop.
| {
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The depicted shape of subatomic particles I physics books and such, I understand why they show atoms as spheres because they have the electron cloud. But why are protons, neutrons, and electrons spheres? I guess people say that because of calculations in nuclear physics, they turn out to be a sphere, but why naturally.... | By machine-gunning protons with electron bullets, you can look at the angles at which the electrons bounce off them and determine something called a structure function for the proton. If there are no preferred directions for the electrons to bounce off in, then you are justified in modelling a proton as an object with ... | {
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Does sugar have a quasi-liquid film on its surface at room temperature? Background
According to an article from Physics Today, ice is slippery because there is a “liquid or liquid-like layer” on its surface.
There are 3 mechanisms that can cause this layer to exist, each playing a role that depends on the temperature, ... | If so, I don't think it's a significant effect. Otherwise, sugar would not flow as freely as it does. However, sugar is somewhat hygroscopic, and sugar that has absorbed water vapour from the atmosphere can stick together in lumps. In warm humid climates, that happens fairly quickly to sugar that's kept in an unsealed... | {
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Would an astronaut in this spacecraft feel weightless?
A spacecraft is placed in orbit around Saturn so that it is
Saturn-stationary (the Saturn equivalent of geostationary – the
spacecraft is always over the same point on Saturn’s surface on the
equator).
Information the question provided:
mass of saturn = $5.68\tim... | Feeling weightless is like saying that you feel not a normal force on yourself preventing free-falling.
I think that this is the starting point for formulating Einstein Weak Equivalence Principle.
| {
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Can we prove the Earth rotation with a disk mounted in its center on a frictionless axis? Can somebody prove the rotation of Earth if it places an uniform disk with a hole in its the center on an axis and orient it paralelly to Sun ecliptical disk?Just assure himself that the disk on the axis is in equilibrium and the ... | If the disk is spinning then it will act as a gyroscope and will try to keep its axis oriented in a constant direction in space. This is called a Foucault gyroscope and it can be used to demonstrate the rotation of the earth, although great care must be taken to minimise friction and allow the axis of the gyroscope to ... | {
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How to show connection between measured data (sound in $\rm dB$) and the size of speakers? How can we relate the measured data of speakers with its SIZE (speaker)? I have the data on white noise and pink noise (FFT and 1/3 octave), I'm a bit confused about how to show the relationship (connection) between them.
Here I ... | In general you cannot say anything because there is no direct relashionship between the size of the speaker and the output level. Think for example of a speaker box containing three speakers, the bass, the middle range and high frequency one. If you apply a high frequency signal (in the audio range) the big bass spea... | {
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Why is the $y$-component of acceleration twice the $x$-component?
We were given this question. the answer said that when $m$ released, the $y$ component of acceleration of $m$ should be $2$ times the $x$ component of acceleration of $m$. I can get that the $x$ components of acceleration of $M$ and $m$ are equal but I ... | The pulleys make it trickey. You can see how this works by looking at the lengths of the individual rope pieces. When the cart moves 1 meter to the right the bottom part gets 1 meter shorter. The top part also gets 1 meter shorter. Since the rope can't change length the left part (that is attached to $m$) gets longer b... | {
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Importance of complex functions in quantum mechanics In quantum mechanics, we work with the space $\mathcal{H} = L^2(\mathbb{R})$ of functions with complex value square integrable. Thus Hermitian operators will play a central role since they have a real mean, to which we can give a physical interpretation:
$$\overline{... | Since the solutions to the time-dependent SE are of the form $\psi(x)e^{-iEt/\hbar}$ it is difficult to avoid complex wave functions.
Moreover a coherent state
$$
\vert \alpha\rangle=\sum_{n=0}^\infty e^{-\vert\alpha\vert^2/2}\frac{\alpha^n}{\sqrt{n!}}\vert n\rangle
$$
for instance is a complex combination of eigenfunc... | {
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Might the Kaluza-Klein scalar provide a solution to the dark puzzles? Kaluza-Klein theories of a five-dimensional spacetime yield not only the equations of general relativity and electromagnetism, but also a scalar field. This scalar field, sometimes quantised as the radion or dilaton, is thought not to exist.
Given to... | Just wanted to add to the other answer, that the current outlook for a Kaluza-Klein-type theory being the answer for dark matter, and dark energy also, isn't compelling. There are of course people who've considered such models because they're phenomenologically interesting, but they're either under stringent constraint... | {
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Why is this paper stack not flat? So we have lots of paper for photocopy. They come in a bundle. What I have noticed is that There are sine curves or maybe cosine when I saw them from the side. That's Look like:
Is there any explanation for these curves?
| Let's assume that they were originally perfectly stacked. If you apply some nonuniform external force to the whole stack, some sheets might slightly slide around (which is true in this case, as visible in the photo; note sides of the stack). This can be viewed as a deformation of the whole stack. Due to being stacked, ... | {
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Any good references on the analytic structure of scattering amplitudes? In papers they often say things about the analytic structure of S matrices - things like resonances are poles on the unphysical sheet, particle channels cause a square root branch cut etc.
I've seen this demonstrated in a couple of simple cases but... | In general analyticity is consequence of causality and unitarity.
A classic text is Eden, Landshoff and Polkinghorne The analytic S-Matrix. It focusses on $S$-matrix elements, but discusses the analytic properties of individual Feynman diagrams.
| {
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Does a rest charge on the surface of the earth emit radiation Suppose we have an observer $O$ in free fall and electric charge $e$ on the surface of the earth. As $O$ is in inertial frame, Maxwell's equations are valid for $O$. However, according to general relativity the electric charge $e$ is accelerated. Does the o... | https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field#:~:text=The%20paradox%20of%20a%20charge,the%20context%20of%20general%20relativity.&text=Maxwell's%20equations%20say%20that%20an,stationary%20particles%20in%20gravitational%20fields.
This will answer fully, but quick answer,... | {
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What does it mean to treat space and time on equal footing? I often read from textbooks that in relativity, space and time are treated on an equal
footing. What do authors mean when they say this?
Are there any examples that show space and time are treated on an equal footing? Conversely, what examples show that space ... | In Special Relativity, there is the invariant interval defined as $$\Delta s^2=c^2\Delta t^2-\Delta x^2$$ (for relative motion in the x-direction only). Here $\Delta t$ and $\Delta x$ are the difference in t and x for two events in some frame of reference. It has the same value in any other inertial reference frame usi... | {
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A problem containing constrained motion of strings and a block Here is the question :-
Two methods came to my mind while trying to solve it, which are:
*
*I assumed the velocity of M as v(upwards). Then, as the strings are inextensible the cosine component of v should be equal to u, if that doesn't happen then the s... | Generally, when it comes to constraint relation of inextensible strings the components of velocity ,acceleration etc,of a particle attached to it are taken along the direction of the string as its length cannot change.It is wrong to take the component of the veleocity of the string in the direction of motion of the ob... | {
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Center Point In Circular Motion Suppose a point initially located at (x,y) moves to (x',y') in a circular motion with angular velocity $w$. Then, the center of this circular motion (x*,y*) can be found by the following:
where $\theta = w \Delta t$.
I really do not understand how this relation holds. Do you have any id... | This isn't a complete answer, but it may help show the formula is reasonable.
First there is a line segment connecting $(x,y)$ and $(x^{'},y^{'})$. The center of the circle is somewhere on the line that bisects that segment and is perpendicular to it.
The center of the segment is $$(x_0,y_0) = \left(\frac{x+x^{'}}{2},... | {
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How exactly do quantum numbers increase in relation to energy levels in more than one dimension? When you have increasing energy levels in 2 or 3 dimensions how does the values of quantum numbers $n$ increase for each dimension?
For example if you have ground state $E_1$ then you have for $(n_x,n_y,n_z)$ is $(1,1,1)$ i... |
So does this mean for energy $E_2$ you have $(2,2,2)$?
This is wrong! The 3-Dimensional box (and 2-Dimensional) have degeneracy is energy state that there 2 or more level for which the energy is same. Like for $E_2$, We have $(2,1,1),(1,2,1),(1,1,2)$.
Note that for 3-D box, The energy given by
$$E_{n_1,n_2,n_3}=(n^2_... | {
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How to solve this problem without using energy considerations? This is a problem from my introductory physics textbook:
A cylinder is released from rest from the top of an incline of inclination $\theta$ and length $l$. If the cylinder rolls without slipping, what will be its speed when it reaches the bottom.
This is... | You use force and torque relationships to do the evaluation. You can find this approach discussed in the Halliday and Resnick Physics textbooks.
It is important to recognize that for rolling without slipping $v = r \omega$ and that allows for the relatively simple evaluation using force and torque provided by Ali; thi... | {
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Time dependent Schrödinger equation with time independent potential and separation of variables suppose we have a potential that's independent of time $V(x,t) = V(x)$
so in Schrödinger equation we get:
$$i\hbar \frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V(x)\Psi... | Yes, see for example this small set of notes that outlines how the Schrodinger equation comes apart into two separate equations: $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) \tag{1},$$ and $$i\hbar\frac{d\phi(t)}{dt}=E\phi(t) \tag{2}.$$
| {
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Problem with direction convention with angular velocity and angular displacement Example 16.17, on page 380 of the dynamics part of the 14th edition of R.C. Hibbeler's Engineering Mechanics: Statics & Dynamics, states:
The crankshaft $AB$ turns with a clockwise angular acceleration of $20\;\text{rad/s}^2$. Determine th... | After some closer inspection, I finally noticed the mistake! There is an inconsistency with direction. Assume counter clockwise is positive. When your $\theta$ INCREASES in magnitude, then your rod ${AB}$ is moving COUNTER CLOCKWISE, so $\omega$ is POSITIVE.
Now if your rod is to move clockwise, then $\omega$ must be N... | {
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Does spin really have no classical analogue? It is often stated that the property of spin is purely quantum mechanical and that there is no classical analog. To my mind, I would assume that this means that the classical $\hbar\rightarrow 0$ limit vanishes for any spin-observable.
However, I have been learning about spi... | An essential difference is that there is no representation of spin in ordinary $3D$ space$^\dagger$. Unlike the spherical harmonics $r^\ell Y_{\ell m}(\theta,\varphi)$ which can be expressed in terms of spherical (and eventually Cartesian) coordinates, such a representation in terms of "physical" coordinates is not po... | {
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Cylinder’s kinetic energy and the arc-length A uniform cylinder of mass m and radius r rolls without slipping on a parabolic surface which $y=kx^2$
then how to write cylinder’s kinetic energy by arc-length $l$. since it's not a circular surface with constant $R$， I don't know how to work out kinetic energy in this cas... | To get you started.
Start with $dl^2=dx^2+dy^2$, take the square root and factor out $dx$. You can calculate
$\frac{dy}{dx}$.
Integrate to get $l$ in terms of $x$ and use the $y(x)$ equation to get $l$ in terms of $y$. Use the potential and kinetic energy equations from there.
| {
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Can coldness be converted to heat energy? We know that the heat can be converted into heat energy with the help of thermoelectric generators, but why can't we generate energy from coldness?
Like the temperature of the universe in 1 K, can this be used in the near future to be used as an energy resource for probes or sa... |
Edit- here is the link to article that made me think about this.
somewhere in middle it is written that scientists can harness the cold
energy using some active input method.
The following statement from the article is poorly worded:
“Essentially, a sky-facing surface passes its heat to the atmosphere as thermal radi... | {
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Error propagation for cube and then square root I have a variable $z$ and I know its error value $\Delta z$.
So $z = 4.480$ and $\Delta z = 0.168$. I need to find $y + \Delta y$ such that
$$y + \Delta y = (z+\Delta z)^{3/2}$$
So in this case, what is $y$ and $\Delta y$?
I am finding that
$$y = z^{3/2} \tag 1$$ and $$\... | I do not understand why you want to evaluate $y + \Delta y$ as $(z + \Delta z)^{3/2}$? You can evaluate the uncertainty in y(z) where y(z) is a function of the random variable z; for your case $y = z^{3/2}$. If this is the case, the following applies.
I assume 4.480 is the mean for z and $\Delta z$ 0f 0.168 is the st... | {
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Is Light our limit? Suppose something existed faster than light will we be able to perceive it?
And even if we encounter it wouldn't seem to travel with speed of light?
| There exist hypothetical particles named Tachyons that would travel faster than light. They are merely hypothetical since Einstein's relativity does not forbid their existence but there is no experimental support for their existence at all.
If such particles existed they would to us also appear to travel faster than li... | {
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Advanced/Retarded Eddington-Finkelstein coordinates & Black/White Holes The Schwarzschild spacetime is described by $$ds^2=-(1-\frac{r^*}{r})c^2dt^2+(1-\frac{r^*}{r})^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2,$$ where $r^*$ is the Schwarzschild radius.
The advanced Eddington-Finkelstein (EF) coordinate system is a... | The different coordinate systems cover different parts of the full solution, which are relevant to different sorts of objects.
The full Schwarzschild vacuum solution has four 4D regions, and 3D boundaries (event horizons) between them. They're conventionally labeled with roman numerals:
Region II is the black hole int... | {
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Bragg-Williams theory of phase transition
Hi I am currently studying Bragg wiliam theory and I dont understand how they derive the equations f(T,m). Actually i dont understand how they got S/N given by equation 8. Can someone explain?
| Well, as usual the entropy is the logarithm of the number of microstates. If the total number of spins is $N$ and there are $N_{\rm up}$ "$+$ spins", then this corresponds to a total number of possible configurations given by $\frac{N!}{N_{\rm up}!(N-N_{\rm up})!} = {}^NC_{N_{\rm up}}$ (the number of ways of choosing w... | {
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Time dependence of operators In Griffiths's Introduction to Quantum Mechanics, while studying the time evolution of the expectation value of position, the author wrote:
$$\langle x\rangle=\int_{-\infty}^{+\infty}x|\Psi(x,t)|^2\,dx.$$
So
$$\frac{d\langle x\rangle}{dt}=\int x\frac{\partial}{\partial t}|\Psi(x,t)|^2\,dx.$... | The are two formulations of quantum mechanics :
*
*Schrödinger representation. The time evolution is encoded in the state vector, wavefunction - $\Psi(x,t)$, and the observables(operators) are constant in time
*Heisenberg representation. Now the operators evolve in time, and the state vectors are time-independent, k... | {
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Maxwell equations: Lagrangian or Eulerian description I am just wondering the 4 Maxwell equations (i.e Fadaray Law, Maxwell-Ampere) are Lagrangian or Eulerian description? Does it really matter?
| The distinction between the two descriptions you consider assumes that there is a field of velocities describing the motion of the particles of the continuous body.
Adopting one or the other description respectively means (a) to use the initial position of the particles to label the integral curves of the field (Lagra... | {
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Are the potentials of the electrolytes equal in Galvanic cell? My book defines Electrode Potential for a Galvanic Cell as follows:
A potential difference that develops between the electrode and the electrolyte is called Electrode Potential
$$E= V_{electrode} - V_{electrolyte}\tag1$$
It further defines cell potential ... | Yes, the potentials of electrolytes are equal. If they are of different type and connected by a salt-bridge then they are at same potential, because we assume that the salt bridge offers no resistance to electrons, it is essentially short-circuiting the two electrolytes. If the electrolytes are the same then no salt br... | {
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Is information conserved in quantum mechanics (after wave function collapse)? I have heard in popular science that there is a law of "conservation of information." Some times this is described as: for any event that happens, there is enough information to reconstruct the original state. So, for example, if you knew the... | Short answer: the collapse of a wavefunction destroys information.
As you correctly said, as long as the quantum state evolves according to the Schrodinger equation, information is conserved.
If we adopt an interpretation of quantum mechanics in which collapse happens upon measurement (the Copenhagen interpretation), ... | {
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Averge of the mean field theory hamiltoninan So I am trying to solve a problem where I have to calculate E and I am given the hamiltonian so I can compute the averge of H which is then E.
H is defined as: $ H = -J*d*m\sum{\sigma_i} + \frac{JdNm^2}{2}$
Here J is a constant d is also a constant, N is number of particles ... | Taking average on both sides of the equation we get:
$<H> = -Jdm<\sum \sigma_i> + \frac{JdNm^2}{2}$
By definition $m = \frac{<\sum \sigma_i>}{N}$. Substituting it in the above equation we get:
$<H> = -\frac{JdNm^2}{2}$
This is the average energy.
| {
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Can two photons excite an electron consecutively? I know that a photons energy is quantized and that it can excite a bounded electron from one energy state to the other whic depends upon the energy the photon carries my question is that can two photons consequtively excite an electron from an initial state (say E1) to ... | To add to the answer by @CrazyGoblin. Every state can be characterized by a lifetime, $\tau_i$, which characterizes the rate of relaxation $\Gamma_{i\rightarrow 0}=1/\tau_i$ to the ground state. Inducing an absorption to a higher energy state requires that the absorption happens faster than the relaxation. In other wor... | {
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The Dirac comb with one $\delta$-function removed Has anyone ever encountered the Dirac comb/Shah function with one removed $\delta$-function,
$$
V(x)=\frac{\hbar^2\kappa}{m}\sum_{n\neq0}\delta(x-an),
$$
in any literature? I want to find the solution of the Schrödinger equation with such potential but currently, I'm ex... | This is an interesting, and likely solvable, generalization of scattering from a delta -potential. I would suggest formulating it differently:
*
*Taking unmodified Dirac comb as the potential for calculating the Basis functions (i.e., the Bloch waves):
$$
V(x) = \lambda\sum_{n=-\infty}^{+\infty}\delta(x-an)
$$
*and ... | {
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What is angular velocity in 3-dimensional space? (Revised) If I'm not mistaken, there are analogies between the translational dynamics of a rigid body and the rotational dynamics of that body. For example, the position of a rigid body is analogous to the orientation of that body. What property of a rigid body, from a r... | If I understood, you are making a kind of analogy table, where position in translational dynamics is equivalent to angular orientation in rotational dynamics.
Following this line, what is the analogy pair of linear velocity?
There is a problem because velocity is conserved for translational movement without applied for... | {
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Why is it that we ignore height difference when applying Bernoulli in an airfoil I learn physics myself and sorry if this is a very simple question
*
*Why is it that we can apply Bernoulli on above and below the plane even if the are not in the same streamline?
*Why do we ignore height difference when doing so?
Any... | Your first question finds an answer in this Wikipedia article :
if the fluid flow is irrotational, the total pressure on every
streamline is the same and Bernoulli's principle can be summarized as
"total pressure is constant everywhere in the fluid flow". It is
reasonable to assume that irrotational flow exists in any... | {
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Justification for "reducing integrals" in the virial expansion for gases I have been following this document (https://sites.chem.utoronto.ca/chemistry/jmschofi/chm427/gases.pdf) regarding the virial expansion of gases and on finding the virial coefficients.
On Page 7, they do the following:
$$Z_3 = \int _V d\mathbf{r}_... | This is not a single variable integral and you should recall what you learn in multiple variable calculus --- the change of variable should follow by Jacobian. Obviously here the Jacobian is 1. Then after change of variable $f_{12}(r)=f_{12}(r_1-r_2)=f_{12}(r_{12})$ would depend on $r_{12}$ only and has nothing to do w... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is are induced electric field's non-conservative while static electric fields are conservative? I have learned that the $E$-field induced by changing magnetic flux, such as in 'motional emf', is non-conservative in nature.
I am also aware that static $E$-fields are conservative in nature.
What is the reason for thi... | The reason static field is conservative is this: static field means, by definition, total Coulomb field of any static configuration of charges. We know Coulomb field is conservative (because of Coulomb's law).
If tommorow there was a non-conservative field discovered that is due to and associated with static charges, t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/597687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Is $(L^2, L_z)$ a complete set of commuting observables? According to the main definition we define a (C.S.C.O.) complete set of commuting observables $(A,B,C, \dots)$ if:
*
*Every commutator between the operators of the list is $0$
*If we fix the eigenvalues of the operators there exists a unique eigenvector with t... | Regarding the Hydrogen atom, being in an eigenspace of both $L^2$ and $L_z$ means knowing the type of orbital the electron is in ($s$, $p$, $d$, etc.) - this gives the $l$ label - and also which specific orbital it is in ($p_x$, $p_y$, $d_{x^2-y^2}$, etc.) - this gives the $m$ label - see here.
However, every shell (la... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/597950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Ball rotating in a circle without rope/string - why doesn't the ball fall after the peak? I am currently learning about the problem of a ball rotating in a vertical circle - where we are interested in determining the minimal height $h$ required in order for a ball sliding from rest down a frictionless track (which cons... | Suppose the car reached the top point with a velocity $v$. The centripetal acceleration must be $g \implies g = \frac{v^2}{r}$
If the track was removed, the problem was to find the trajectory of a particle with an horizontal initial velocity $v$, having a constant acceleration $g$ downwards. We know that the solution i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/598060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Show that the initial speed of gun and the shell are in the inverse ratio of their masses A shell of mass $m$ is fired horizontally by a gun of mass $M$ which is free to recoil
and which is on horizontal ground. Show that the initial speed of gun and the
shell are in the inverse ratio of their masses.
Here is my attem... | Both velocities (shell and gun) have to be with respect to an observer stationary on the ground (an inertial reference frame). So your use of (v - u) for the shell for the conservation of momentum is correct, since this is the velocity of the shell in the inertial frame.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/598156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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