Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to make the Moon spiral into Earth? I recently watched a video of what would happen if the Moon spiraled into Earth. But the video is pretty sketchy on the physics of just what would have to happen for that to occur. At first I thought I understood (just slow the Moon down enough), but my rudimentary orbital mechan... | Another way: maintain a large electric charge on the moon so that it radiates away low-frequency radio waves and looses energy that way.
This method is impractical; but so are all methods. This method is even more impractical than most, but I add it for the interest of the physics. The idea is that any accelerating cha... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/694535",
"timestamp": "2023-03-29T00:00:00",
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If force depends only on mass and acceleration, how come faster objects deal more damage? As we know from Newton's law, we have that $\mathbf{F} = m\cdot\mathbf{a}$. This means that as long as the mass stays constant, force depends solely on acceleration. But how does this agree with what we can observe in our day-to-d... | the equation for your example (Newton second law) is:
$$m\,\ddot h=m\,g$$
thus the acceleration $\ddot h~$ is the same but
with the solution
$$h=\frac{g\,t^2}{2}\quad \Rightarrow\\
v=g\,t$$
eliminate the time t you obtain that
$$v(h)=\sqrt{2\,g\,h}$$
thus the impact $~m\,v~$ is depending on the height $~h~$ from wher... | {
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Time constant versus half-life — when to use which? In some systems we use half-life (like in radioactivity) which gives us time until a quantity changes by 50% — while in other instances (like in RC circuits) we use time constants. In both cases the rate of change of a variable over time is proportional to the instan... | It is just a matter of taste
whether you prefer to write an exponential decay
with the time constant $\tau$ and powers of $e$
$$N(t)=N_0\ e^{-t/\tau} \tag{1}$$
or with the half-life $t_{1/2}$ and powers of $2$
$$N(t)=N_0\ 2^{-t/t_{1/2}}. \tag{2}$$
Both ways are equivalent
and you can switch between them by using
$$t_{1... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/694850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove time dilatation from the Lorentz transform? How to prove time dilatation from the Lorentz transform formula:
$$ t' = \gamma\left(t-\frac{Ux}{c^2}\right) $$ (U: the velocity of the referential R' relative to R)
So far I've found this formula :
$$ \Delta t' = \gamma\left(\Delta t-\frac{U\Delta x}{c^2}\right)... | The starting point is to be clear on what time dilat(at)ion means. Here it is...
The time interval between two events as found in an inertial frame of reference where the events occur in different places is greater than the time interval ($T$, say) in the inertial frame where they occur in the same place.
We assume tha... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/695031",
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What is the acceleration of a ramp on a table when a body slides on it? I found an Olympiad problem:
Find the acceleration of a ramp on a table when a body slides on it. Assume there is no friction between the body and the ramp, and between the ramp and the table.
I found the final solution to this problem but I do n... | TL;DR The free-body diagram should not include the resultant forces, at least not in the way to make you think those are the forces acting on the body in question. The following vectors should be removed from diagrams in your question: (i) $m \vec{a}$ and $m \vec{a}_1$ for the body, and (ii) $M \vec{a}$ for the ramp.
A... | {
"language": "en",
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Subscript $Q$ in $U(1)_Q$ In most quantum field theory books, you read something like
QED is a QFT from Abelian $U(1)_Q$ that describes the electromagnetic interaction ...
But what does the subscript $Q$ stands for in the $U(1)_Q$ group?
| It specifies that one is referring specifically to the symmetry associated with electromagnetism ($Q$ stands for electric charge, but some texts also write $U(1)_{\text{EM}}$, for example). This is meant to distinguish from other possible $U(1)$ symmetries present in the theory or in related theories that correspond to... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/695345",
"timestamp": "2023-03-29T00:00:00",
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Why should a clock be "accurate"? Having read that atomic clocks are more accurate than mechanical clocks as they lose a second only in millions of years, I wonder why it is necessary for a reference clock to worry about this, if the definition of the second itself is a function of the number of ticks the clock makes.
... |
why it is necessary for a reference clock to worry about this, if the definition of the second itself is a function of the number of ticks the clock makes.
The concern is that somebody else (say a scientist in France or China or Botswana) needs to be able to build a clock that measures seconds at the same rate mine d... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/695789",
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Does AdS/CFT help solve the singularity of a black hole? How does thinking of a black hole as encoding its information in its surface help solve what happens inside it, more specifically geodesic incompleteness. Doesn't it tell us that if we can see how light (or matter) behaves around the horizon we can predict what i... | It does not solve what happens inside. The information on the horizon, by the entanglement of virtual particles with the particles that passed, keeps track of the state of the particles inside. The momenta of the infalling particles are entangled with these virtual ones on the horizon. So in a sense the inside physics ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/695830",
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Schwarschild radius and paramaterizing path Consider the metric $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.$$
Suppose a particle very large starts at the initial radius $R$ and then radially infalls in a Schwarzchild manifold
My text then states... | Start with your equation, which reduces to:
$$-1 = -\left(1 - \frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^{2} + \frac{1}{1-\frac{2M}{r}}\left(\frac{dr}{d\tau}\right)^{2}$$
Now, you can leverage the fact that $\partial_t$ is a Killing vector to show that $E = \left(1-\frac{2M}{r}\right)\frac{d t}{d \tau}$ is a cons... | {
"language": "en",
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Lorentz Four-force in General relativity? In special relativity, it's normal to define Lorentz four-force density as $$f_\mu = F_{\mu\nu}\nabla _\lambda F^{\nu \lambda},$$
having Maxwell EM tensor $F_{\mu\nu}$. Can we do it in General relativity? Does "force" even have a meaning?
| It should be valid applying the equivalence principle: , --> ;
Meaning every regular derivative becomes a covariant derivative.. so that ∆ should be uses with the correspondinh Christoffel symbols.
| {
"language": "en",
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Is there any exact law in physics? Even for maybe the most fascinating computation in physics, about anomalous magnetic moment of the electron, theoretical physics gives:
$$a = 0.001 159 652 164 ± 0.000 000 000 108$$
which is not "exactly" true:
$$a_{exp} = 0.001 159 652 188 4 ± 0.000 000 000 004 3$$
so even laws of qu... | Any theory or law in physics is only as good as its experimental validation. Every experimental measurement has a non-zero experimental error. Therefore no theory or law in physics can be validated exactly.
The speed of light in a vacuum may vary in its $20$th digit. The law of conservation of energy may only be correc... | {
"language": "en",
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Are one-dimensional tensors of arbitrary rank just scalars? Consider a tensor of arbitrary rank (2 for this case) $A_{ij}$, and dimension one. Granted there are two indices to specify a component, but since each index can only take one value, there is only one component in this entire tensor: $A_{11}$. So, are all one ... | Tensors of rank $k$ over a vector space $V$ form the space $\otimes^k V$. When the vector space is 1d, then we may as well take it to be the ground field $\mathbb{K}$.
But $\otimes^k \mathbb{K} \simeq \mathbb{K}$.
A tensor is an element of the left side, so equivalently it is also an element of the right side and so i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/696747",
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Are black holes spinning balls of quark-gluon plasma? I had this idea a few days ago that the Higgs event might have been a naked singularity, i.e. the colliding protons (very briefly) fall into a state of infinite density and release two gamma-ray photons as decay products. One thing led to another, and I was led to e... | The answer is probably "not for long". When a star collapses the components get squeezed together and the temperature increases, plausibly turning into a quark-gluon plasma... but this does not stop the collapse. Very quickly (from the perspective of an observer falling with the matter) it reaches the singularity and s... | {
"language": "en",
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Why can’t quantum randomness be understood as epistemic? I often hear people say that quantum randomness is “true randomness”, but I don’t really understand it. Please bear with my question.
Before the development of quantum physics, randomness is understood as being “epistemic”. That is, things appear random because w... | There is, in general, no joint probability distribution for the outcomes of quantum measurements, which means that QM, at least as it is usually formulated, is incompatible with Kolmogorov's framework.
| {
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"source": "stackexchange",
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Why is a relativistic calculation needed unless $pc$ is much smaller than the rest energy of a particle? After introducing the de Broglie wavelength equation, my textbook gives a rather simple example where it asks to find the kinetic energy of a proton whose de Broglie wavelength is 1 fm. In the solution to this probl... | In relativity, the energy $E$ of a particle is related to its mass $m$ and momentum $p$ by
\begin{equation}
E = \sqrt{m^2c^4 + p^2 c^2 }
\end{equation}
Now let's think about the non-relativistic limit, $c\rightarrow \infty$. To do this, we will expand the square root
\begin{equation}
E = m c^2 \sqrt{1 + \frac{p^2}{m^2 ... | {
"language": "en",
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Why does an inkjet printer use CMYK ink instead of RGB ink? Currently, I am studying the inkjet printer in detail. I had come across the ink of printers ink. Why do they not use RGB ink? Why do they use CMYK? I have read somewhere if we mix red and green ink together it will become dark, but not yellow because of absor... | Short answer: Because sabtractive and additive color mixing are not the same.
Longer answer:
With additive color mixing (most screens work like that) you get white when you mix all the colors. This means, the more colors you add, the brighter the result.
With subtractive color mixing (like printers do) you start with a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/698045",
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Energy conservation and time uniformity Reading Peter Atkin's book Four laws that drive the universe, I cannot see the link between time uniformity and conservation of energy the way it is tackled in this excerpt:
" Noether’s theorem, proposed by the German mathematician Emmy Noether (1882–1935), which states that to ... |
Energy is conserved because time is uniform: time flows steadily
What I can understand from this is that some periodic movements keep the same ratio of their periods. And exactly for that reason they were chosen historically to measure what we call time.
The ratios between days (earth rotations), weeks (lunar phases)... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/698300",
"timestamp": "2023-03-29T00:00:00",
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How can a quasi-static process be reversible? As I understand it, a reversible process is required to be quasi-static because each infinitesimal step in a quasi-static process generates only infinitesimal amounts of entropy at a time which can be reversed with only an infinitesimal amount of work. But my question is: e... | No real process is reversible, for precisely the reason you mention: a gradient (e.g., in temperature, pressure, or chemical potential) is required to drive a process, but energy moving down that gradient produces entropy. By skilled engineering (to reduce friction, for instance) and by slow operation, we can reduce en... | {
"language": "en",
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Why doesn't temperature decrease with an increase of volume in the syringe? I made an experiment using a closed syringe with the volume marks. At the beginning, the piston is at $5\rm\, ml$, then I move it to $20\rm\, ml$.
Since the change is approximately adiabatic, we can use the adiabatic expressions, one of them be... | In calculating your expectation, you are neglecting the heat capacity of your experiment. Once you take that into account, you will find that your gas will get cold, but there isn't enough energy to cool the syringe by any significant amount; Also, you are neglecting time, which has the effect that the syringe and gas ... | {
"language": "en",
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Does the CPT theorem imply $CP=T$? Does the CPT theorem imply $CP=T$?
That is, does it imply that the action of Charge Conjugation and Parity inversion on some representation of the Lorentz group, is the same as doing a time reversal?
Specifically, given explicit expressions for $P$ and $C$ (in terms of matrices and co... | The assertion of the CPT theorem is that, under natural hypotheses, the Hamiltonian $H$ operator of a theory is invariant under the simultaneous action of the symmetries (in Wigner's sense i.e. unitary/antiunitary operators) C, P, and T.
$$CPT H (CPT)^{-1} = H\:.\tag{1}$$
This action can also be implemented by a direc... | {
"language": "en",
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Necessary and sufficient conditions for operator on $\mathbb C^2$ to be a density matrix Consider a one-qubit system with Hilbert space $\mathscr H\simeq \mathbb C^2$.
Define the hermitian operator
$$\rho := \alpha\, \sigma_0 + \sum\limits_{i=1}^3 \beta_i\, \sigma_i \quad , \tag{1}$$
where $\alpha,\beta_i \in \mathbb ... | Let us first derive necessary conditions on the coefficients, so assume $\rho$ is a density matrix. From $\mathrm{Tr} \rho =1$ it trivially follows that $\displaystyle \alpha=\frac{1}{2}$.
To proceed, let $\lambda$ and $1-\lambda$ denote the eigenvalues of $\rho$. As shown e.g. here, we find $$\det \sum\limits_{i=1}^3... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/699132",
"timestamp": "2023-03-29T00:00:00",
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Can we write the effective field theory for the toric code model? If not, then why not? If yes, then what is the effective field theory?
| No.
Effective field theory only describes system near critical point. Toric code model is far from critical point. Thus "No".
Toric code model realizes a $Z_2$-topological order. When a state with a $Z_2$-topological order is close to a continuous phase transition, then the state will have an effective field theory des... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/699283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Work of a spaceship in circular motion Say a spaceship is traveling though space in a uniform circular motion.
It's not orbiting any planet, it just flies in circles in an empty space.
The only force working on the spaceship would be the centripetal force caused by the ship's engine.
Thus, the work would be $0$, as the... | You are right in saying that the centripetal force doesn't do any work, in fact the kinetic energy of the system doesn't increase as the absolute value of the velocity $|\vec{v}|$ stays constant.
I guess what you find counter intuitive is that the spaceship has to burn some fuel to keep rotating, so where does this ene... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/699425",
"timestamp": "2023-03-29T00:00:00",
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Can I see separation of variables as a tensor product? Can I see separation of variables as a tensor product?
For example, in a radial potential, the separation of variables brings to the solution $R(r)\Theta(\theta) \Phi (\phi)$. This sounds like an element of the space spanned by $|r\rangle\otimes|\theta\rangle\otime... | Yes, that is exactly what separation of variables is in terms of the Hilbert space - generally, we have that $L^2(X\times Y) = L^2(X)\otimes L^2(Y)$, i.e. the space of square-integrable functions on a Cartesian product is the tensor product of the square-integrable functions on the factors of the Cartesian product. In ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/699573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you estimate the core temperature of a star? Given a star's mass, radius and average composition (e.g. 90% H, 10% He), is there a formula to estimate the core temperature of that star?
I only found one for a lower bound but that wasn't very accurate.
| Yes, the formula you quoted in comments is an application of the viral theorem, that says for a star in equilibrium that twice its internal kinetic energy plus it's potential energy is zero. This can be written
$$\Omega = - 3 \int P\ dV = -3\int P\ \frac{dm}{\rho}$$
where $\Omega$ is gravitational potential, $P$ is pre... | {
"language": "en",
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Partial trace of local operators applied to maximally entangled states I was looking at a problem where two invertible local operators were applied to a maximally entangled state, and didn't quite understand how some of it works out. We have local operators $A \otimes \mathbb{1}$ and $\mathbb{1} \otimes B$ with $Tr(A^{... | This is not true. A simple counterexample is $A\propto I$ and $B=\lvert0\rangle\langle0\rvert$.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/700310",
"timestamp": "2023-03-29T00:00:00",
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What is the meaning of an object having an uncertainty of velocity of 2 $\rm m/s$? In several questions we are given the uncertainty in velocity of an object and are asked to calculate the uncertainty in position of an object?
Well my doubt is that,as when we say that the uncertainty in position (of an let's say electr... | As Marko points out, this question is unclear, but I see you have the uncertainty principle as a tag.
If you mean that the electron has a velocity with uncertainty $\Delta v=2\ ms^{-1}$ and you want to find its corresponding position uncertainty, then you need to solve the following relation $$\Delta x\Delta p\ge \frac... | {
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Force between two protons Yesterday my teacher was teaching about the production of photons, he told that photons are produced when the electron move from a higher energy level to a lower energy level then suddenly a idea struck in my mind that if electrons are responsible for photons and photons are responsible for el... | Photons are the quantum elementary particles of the electromagnetic force . In the table there are elementary particles with charge other than electrons so photons can be produced at the basic level by other charged particles too.
One way they are produced is the way you have been taught at present, by changes in the ... | {
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Difference between the wave forms in the water and in the Young double slit experiment We can observe when we cause a slight disturbance at two points on the water surface which is intially totally undisturbed , it will form water waves which would look like as shown in below image:
we can observe that there are const... | You need the screen because you do not want to stare in the sun with your eyes even if you were doing it through a peep-hole. Imagine adjusting the size of the diaphragm (of the slits) to see if there is already enough light has already fallen on your retina.
| {
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Why do we use different differential notation for heat and work? Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we are told that the '$df$' notation is only for functions... | Notation
Sometimes heat and work are marked by special signs to underscore that they are not real differentials, such as differentials with a stroke, as shown here or something like
$$\text dU = \delta Q + \delta W.$$
However, there is no single established notation here, and most of the time one simply does not bother... | {
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What does it mean mathematically the position of the center of mass relative to other particles is the same? My understanding is that the center of mass of a system of particles is that an imaginary point that travels in space and has position vector $$\vec r = \frac{\sum_{i=1}^{n}m_i\vec r_i}{\sum_{i=1}^nm_i}.$$
My te... | look at this figure
The vector u is the position between the center of mass and the particle position.
you can describe this vector in a body fixed coordinate system that located at the center of mass ( blue one). Now if you use other intertial system $I_2$ the center of mass vector and the particle position changes, ... | {
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When solving problems on linear momentum, when can external forces be neglected? I was recently solving a problem in which one end of a massless string (in vertical orientation) was tied to a block of mass $2m$ and the other end to a ring of mass $m$, which was free to move along a horizontal rod. The block is then giv... | Remember that momentum and force are both vectors, and when we write Newton's second law to relate force to the rate of change of momentum this is a vector equation:
$$ (F_x, F_y, F_z) = \left( \frac{dp_x}{dt}, \frac{dp_y}{dt}, \frac{dp_z}{dt}\right) $$
which is a set of three equations:
$$\begin{align}
F_x &= \frac{dp... | {
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Different formal definitions of Lorentz Transformations The formal definition for Lorentz Transformation is a matrix $\Lambda$ such that $$\Lambda^\mu_{\ \ \alpha}\Lambda^\nu_{\ \ \beta}\eta_{\mu\nu}=\eta_{\alpha\beta.}$$
In some books I have found a definition that use the transposition: $$(\Lambda^T)\eta\Lambda=\eta.... | This has to simply to do with matrix multiplication. If you have a matrix $A$ that multiplies a vector $x$, this can be written as
$$ A_{ij}x_j = A x$$
where summation over double indices is assumed. Of course you can flip the expression around, as in
$$ A_{ij}x_j = x_jA_{ij}$$
Similarly, a vector can multiply a matrix... | {
"language": "en",
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Do the components of a force written for a purpose actually exist? On an inclined plane if you put a box, the force of gravity $mg$ is written as sum of two forces $mg\sin\theta$ and $mg\cos\theta$ where $\theta$ is the angle the incline is making with earths surface. Do these forces $mg\sinθ$ and $mg\cosθ$ actually wo... | If you put two force meters on the block, one in direction of the incline, one orthogonal to it the two meters show the two forces you calculated and stay so if you take away the inclined plain, So the two forces really exist not just in theorie .
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can they estimate exoplanet radial velocities using Doppler considering spectrograph resolving power? I read that spectrograph resolving powers, the ratio of wavelength uncertainty to wavelength are like 1000 or 10000. Plugging this into the non relativistic Doppler formula gives a velocity uncertainty like 30000 m... | Resolution tells you how well the spectrometer can separate lines with wavelengths close together, but not how precisely it can measure the wavelength of a single line. Measurement precision can be much better than the resolution. Then, techniques like template correlation can effectively average measurements of many l... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/701868",
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How to derive the Unruh effect (or the thermofield double state) from the path integral? I have been reading about the path integral approach to deriving the thermofield double state for the Minkowski vacuum in terms of the Rindler states:
\begin{equation}
\left|0_{M}(t=0)\right\rangle=\sum_{n} \frac{e^{-\frac{\beta}{2... | It seems I was confused by notation here. After all in the first expression the $| \phi_L \rangle $ can only be evolved by $H^R$ and projected onto $| \phi_R \rangle$ if it is in the right-Rindler wedge states. So $| \phi_L \rangle $ must live in the same space as $| \phi_R \rangle $ which can be achieved by applying $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Hall sensor for electric(!) field? Is it (in principle) possible to measure the strength of an electic(!) field with a hall sensor?
I think so, for the following reasons:
*
*The hall sensor is a conductor. If we place an conductor in an electic field, charges will rearrange so that there will be no electric field in ... | The only suggestion that I have seen for measuring an electric field (like the one at the surface of the earth) is to look at the AC output from a small, rapidly rotating, dipole antenna. An un-powered DC circuit goes quickly to zero current, field, and voltage difference.
| {
"language": "en",
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Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model? We know that the Standard Model is a theory about almost everything (except gravity). So it should be the basis of fluid mechanics, which is a macroscopic theory from experiences. So is it possible that we can derive equation... | From your comment :
So is it possible to prove the consistence of fluid mechanics with the Standard Model?
The standard model is consistent with special relativity and quantum theory. We know those explain everything our normal fluid equations deal with because it just atoms, ions and electrons, so it's a very safe ... | {
"language": "en",
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Example for a physical distribution without a well-defined standard deviation Is there a physical example of a distribution that has a diverging standard deviation (like the Cauchy distribution) and is there an intuitive reason for the standard deviation diverging?
Is there a physical context where I should expect my s... | The standard deviation is one example of a metric to characterize the width probability distributions. There is no particular physical meaning to the standard deviation diverging, in general; you just need to use a different measure of the width of the distribution. It is just a mathematical tool, that is useful in som... | {
"language": "en",
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Why can't I use vector addition in this way here?
I know that this exact question has been asked here a number of times, but none of the answers sit right with me. The question says that the ends of the strings are pulled with a velocity of "u" units. We are to find the velocity of the block in terms of u. The answer ... | The model you are assuming here is not correct for the following reason: Imagine you have a horse carriage that is pulled by one horse with a speed of $v$. Then (if the horse is strong enough) the carriage will travel at a speed of $v$. Now, if you put 10 horses in front of the carriage, the carriage will not travel at... | {
"language": "en",
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Is angular momentum just a convenience? I'm wondering whether angular momentum is just a convenience that I could hypothetically solve any mechanics problems without ever using the concept of angular momentum.
I came up with this question when I saw a problem in my physics textbook today. In the problem, a puck with kn... | Angular momentum is a fundamentally conserved quantity. One does not need macroscopic, bound objects to see this.
Consider two identical particles moving in opposite directions. In the center-of-momentum frame, their trajectories will be parallel. If they are not on a direct collision course, then their trajectories wi... | {
"language": "en",
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Which assumption of the no-hair theorem does the Einstein-Maxwell-Dilaton-Axion (EMDA) black hole violate? I know that generally a black hole should have no hair. For example, as proved by Bekenstein in this paper, we can not couple as massive scalar/vector etc. to a black hole which is stationary.
However, I am aware ... | Specifically the cited Bekenstein's 1972 no scalar hair theorem is evaded because it is for free scalar fields, while in EMDA theory dilaton and axion fields are coupled to Maxwell field and to each other. Moreover, while the EMDA black hole has parameters called dilaton and axion charges, they do not represent a true ... | {
"language": "en",
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How does air cool as it nears the poles? I understand air is heated by the equator causing it to rise towards the poles. But why does air cool and sink after nearing the poles. Shouldn't the air still possess heat, after being heated by the equator?
Could adiabatic expansion cause the cooling of this air as it journeys... | Everywhere on Earth, at all times, the air is cooling by radiating to space. That lost heat is replenished by solar energy. There's more solar energy per square meter at the equator than the poles. "Hot air rises", so, very crudely speaking, the air rises at the equator, with the matching downward flow happening at the... | {
"language": "en",
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How can light produce electric and magnetic field when there are no accelerating charged particles? If we see light as a wave, especially in vaccum, there is nothing there, no particles, yet light has an electric and magnetic field. How can this be possible?
| Something, such as accelerating charge particles, caused the light to propagate. The particles don't have to accompany the light. Consider an analogy - something creates a sound wave that you hear some distance away. You could ask how could sound produce compression and rarefaction of air when there is nothing pushing ... | {
"language": "en",
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Does amplitude really go to infinity in resonance? I was recapping the forced oscillations, and something troubled me. The equation concerning forced oscillation is:
$$
x=\frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t)
$$
I don't understand why this equation predicts that the amplitude will approach infinity as $\omeg... | Your solution
$$x(t)=\frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t) \tag{1}$$
was derived from the differential equation
$$m(\ddot{x}+\omega_0^2x)=F_0\cos(\omega t) \tag{2}$$
So the forced oscillation (1) is indeed a mathematically correct solution of (2).
But for the resonance case ($\omega=\omega_0$) the solution (... | {
"language": "en",
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"source": "stackexchange",
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Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$? I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that way. Could someone provide an expl... | It's the chain rule for partial derivatives under the change of variables
$$
x= x(x',t')\\
t= t'
$$ You need to be careful to specify what is being fixed in each derivative though, so it should be
$$
\left(\frac{\partial}{\partial t}\right)_x = \left(\frac {\partial}{ \partial t'}\right)_{x'}\left(\frac{\partial t'}{... | {
"language": "en",
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Lagrangian first integral I want to extremize $$\int dt \frac{\sqrt{\dot x ^2 + \dot y ^2}}{y}.$$
I have thought that, since the Lagrangian $L(y, \dot y, \dot x)$ is $t$ dependent only implicitly, that i could use the fact that $$L(z,z') \implies L - z' \partial L / \partial z' = c.$$
So $$L - y' \partial L / \partial... | $\newcommand{\bl}[1]{\boldsymbol{#1}}
\newcommand{\e}{\bl=}
\newcommand{\p}{\bl+}
\newcommand{\m}{\bl-}
\newcommand{\mb}[1]{\mathbf {#1}}
\newcommand{\mc}[1]{\mathcal {#1}}
\newcommand{\mr}[1]{\mathrm {#1}}
\newcommand{\gr}{\bl>}
\newcommand{\les}{\bl<}
\newcommand{\greq}{\bl\ge}
\newcommand{\leseq}{\bl\le}
\newcomman... | {
"language": "en",
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Quantum and classical physics are reversible, yet quantum gates have to be reversible, whereas classical gates need not. Why? I've read in many books and articles that because Schrödinger's equation is reversible, quantum gates have to be reversible. OK. But, classical physics is reversible, yet classical gates in clas... | Note that classical computing can also be made reversible. Take for example AND gate. As pointed in one of the answers, if the result is 0 you are unable to decide which of the inputs is 0. However, if you copy input to output, i.e. the gate will have three outputs - copy of two inputs and actual AND output, then the ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "22",
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What does it mean for two variables to be canonically conjugate? The word "canonical" has been used in many of my classes (canonical
ensemble, canonical transformations, canonical conjugate variables) and I am not really sure what it means physically.
More specifically, in the context of the Hamiltonian formulation of ... | In Hamiltonian dynamics, a change of variables $(q,p) \longrightarrow (Q,P)$ that leaves the form of the equations of motion unchanged (the symplectic structure is conserved) is called a canonical transform.
$$\begin{cases} \frac{dq}{dt}= \frac{\partial H(p,q)}{\partial p} \\ \frac{dp}{dt}=- \frac{\partial H(p,q)}{\pa... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/704922",
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"source": "stackexchange",
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Is Polarization Inconsistent with Classical Electricity? I had read that the Bohr-Van Leeuwen theorem shows paramagnetism to be impossible with only classical magnetism. It has been explained to me that magnetism is ultimately a quantum effect. Is there an analogous result with electricity?
The motivation for this ques... | The Bohr-van Leeuwen theorem states that a system of charged particles that obeys Boltzmann's probability distribution won't get magnetized by external magnetic field, diamagnetically or paramagnetically.
The Boltzmann distribution of momenta of charged particles does not allow for preferred direction of electric curre... | {
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Why is torque defined as $\vec{r} \times F$? Here I cannot convince myself myself that it is units because the torque is defined to be in units of Newton meter is a reiteration of the law stated above. Why was it not $r^2 \times F$ or $r^3 \times F$ or $r^2 \times F^2$ etc. The argument "in our experience how much some... | I personally prefer a derivation using the principle of virtual work where the formula of torque directly comes out. While angular momentum is a natural property to consider for a spherically symmetrical problem, this alternative approach shows its relevance for statics of rigid bodies even when this symmetry is not pr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Ehrenfest theorem initial conditions The Ehrenfest theorem states that the expectation value of position $x$ obeys the following equation: $$m\frac{\mathrm{d^2}}{\mathrm{d}t^2}\langle x\rangle=\langle F(x)\rangle$$
But there I only need 2 initial conditions and for the wavefunction I need infinitely many initial condit... |
So if I know the initial position and momentum of the expectation value I know how the system behaves. My question is now why I need more initial conditions in quantum mechanics and where they come from.
That's just it, you really don't know how the system behaves! While, for the quantum oscillator, the first moment ... | {
"language": "en",
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Is the speed of signal transport via electricity as fast as light? Let us assume a time synchronization system that comprises a sender and a receiver. The sender generates and sends an encoded signal which presents the current time to the receiver periodically, and the receiver calibrates its clock according to this si... | The propagation rate of a electrical signal along a wire depends on how much capacitance and inductance it exhibits on a per-foot basis. These parameters vary according to the diameter of the wire, its construction (parallel vs. coaxial), the proximity of other wires, and the type and thickness of insulation it is coat... | {
"language": "en",
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Two questions about the Higgs decay mode graph I have two questions reading this graph which shows the Higgs decay mode:
I know the mass of Higgs bosons is measured to be around 125 GeV, which is the solid line on the graph, so I wonder why could the mass on the x-axis become other values other than the measured resul... | Based on the date (2013), I assume this figure was made before the Higgs was discovered or its mass was published. The curves are based on a theoretical calculation of the various branching ratios as a function of the Higgs mass within the Standard Model. If the branching ratios are also measured then this would be a w... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/705674",
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What frequency of cord shaking maintains the same vertical motion for a point on the cord after increasing the wave speed on the cord? I'm studying for my upcoming AP Physics 1 exam but can't figure out this problem
A student shakes a horizontally-stretched cord, creating waves. The graph above shows the vertical pos... | Since this is a $y$ vs. $t$ graph, the frequency of the wave can easily be picked out as the (inverse) time between matching parts on the wave (e.g. peak to peak, $0$ to $0$, trough to trough, etc.). In this case our analysis does not depend on the wave speed, which relates to the rate at which the wave travels through... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A mysterious phase difference Today my teacher was discussing the Poisson spot and gave a simple explanation for why there must be a bright spot on the axis of the disc when illuminated with parallel monochromatic light.
What he said was:
Say we instead have a circular aperture in an infinite plane, we know there must ... | Your teacher was referring to Babinet's principle. It is often a good idea to fix your ideas on actual computation. You have an incident light field $\phi_i$ on the plane. As it crosses the plane, it either gets multiplied by $h_a=1{[r\leq R]}$ in the case of the aperture of radius $R$ (origin at the center of the aper... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Speed of light and the equivalence principle In 1913 Albert Einstein wrote:
"I arrived at the result that the speed of light is not to to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis."
Is the fir... | In 1913 Einstein was still working on general relativity and it was not complete. Furthermore, it would be decades before the community, including Einstein, really began to understand the important concepts of spacetime geometry. This quote is very early and is not really correct by modern understanding, with a century... | {
"language": "en",
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Confusion with the variational operator $\delta$ and finding variations I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The variation of our action $\delta S$ is just $$\delt... | Yes, that happened.
I guess you meant
$$
\delta f = \sum_i \frac{\partial f}{\partial x_i} \delta x_i
$$ on your third equation. Also you've implicity fixed inital $t_0$ and final $t_1$, so that your action integral really is
$$
S = \int_{t_0}^{t_1} dt L
$$
and therefore, since the limits are fixed, variation "commutes... | {
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Am I in a superposition? Someone looks at me. Now, they know my position and my momentum, with some uncertainty.
Therefore, they haven't measured either my position nor my momentum, since neither is known perfectly. They measured some other observable $O$, and found me in some eigenstate of $O$: $|\psi\rangle$.
$|\psi\... | TL;DR Your position is not well-defined... but for more mundane reasons.
Quantum mechanical view
You are not in a pure state - that is you are not an object, that can be described by a wave function, but rather a collection of zillions of particles in a state of thermal equilibrium. That is, you can be described by a d... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/707721",
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Applications of Signal and System theory I recently heard a lecture about Signals and Systems and find the subject extremely exciting. I would like to do more in this direction, so I would be interested to know in which modern research area of physics one needs a lot of Signal and System theory, since this is not clear... | It's very important in instrumentation and data analysis. This paper used a matched filter method that Bill Wheaton and I came up with to dig a spectrum out of some rather crappy data. This caused a colleague to accuse us of witchcraft (ツ), but she was able to confirm and extend our result using better data and more tr... | {
"language": "en",
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What happens when the universe runs out of fuel? After some X billion years, one would think the stars in the entire universe will run out of hydrogen. What would happen next? Is there any way to get hydrogen out of heavy metals (extreme fission)? Just curious.
| Then star formation ceases and the universe goes dark. At this stage of the universe's evolution, there'll still be plenty of hydrogen, they just don't form stars.
In theory you can create hydrogen out of heavy metals, but it's a process that requires energy. If you have the energy banked somewhere (and you'll need a L... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/708256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using Wick's Theorem in an example with the harmonic oscillator I understand Wick's theorem to be,
$$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$
And I'm researching combinatorics and quantum theory in general.
How would one connect Wicks theorem to the quantum HO, an example would be appreciated.
| I'm not sure what you mean by this notation. I think most readers will be more comfortable with
$$T\{\phi(x_1)\phi(x_2)...\phi(x_m)\}=N\{\phi(x_1)\phi(x_2)...\phi(x_m) + \text{all possible contractions}\}.$$
For general operators $\hat{A}, \hat{B}, \hat{C},...,\hat{Z}$, you could write
$$T\{\hat{A}\hat{B}\hat{C}...\ha... | {
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How does the combination of lens create a sharper image? There's a line in a book which states that the combination of lens helps create a sharper image, but I don't understand how. Does more magnification mean sharper image?
| It's hard to answer without knowing the context of the statement. But generally, multiple lenses can reduce aberrations. Real lenses aren't perfect, and images suffer because of that. Rays originating at a single point hit the lens at different places and at different angles and they do not converge on a single poi... | {
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Is it possible the Black Holes to be pure deformations in the fabric of spacetime and not an effect of super-dense matter? Is there any theory in the literature that supports this hypothesis that BHs in their center do not have a super-dense matter singularity but are pure deformations in the fabric of spacetime itself... | Your explanation in your comment:
My definition is the absence of spacetime or vacuum space inside the event horizon
will not work. If this were correct then the whole event horizon would be a single point in spacetime i.e. it would in effect have zero radius. In this case photons (which follow geodesics in spacetime... | {
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AdS-CFT correspondance from 1D to 4D From what I understand the AdS-CFT correspondence states that the bulk dynamics of a $n$-dimensional gravitational theory are encoded in the degrees of freedom of its dual CFT in the $(n-1)$ dimensional boundary.
The question is the following: Suppose we start with a $1D$ CFT theory... | A $d$-dimensional conformal field theory with the right properties is `holographic', meaning that it's dual to a $(d+1)$-dimensional gravitational theory in AdS. But that $(d+1)$-dimensional theory cannot itself be holographic in the same sense, for a few reasons:
*
*It has gravity! The original $d$-dimensional theor... | {
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Introduction to magnetohydrodynamics does anybody have any reference books for introduction to magnetohydrodynamics? I want to dive into this topic and I don´t know about any good reference.
| Goedbloed and Poedt's Principles of Magnetohydrodynamics (Amazon link) is a good primer on MHD.1 If I recall correctly, the book is primarily aimed at the physics of tokamaks, but many of the principles found therein can be applied to other aspect of MHD (e.g., astrophysics). It has been some time since I last looked a... | {
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Angular Momentum and Coefficient of Restitution If there was a situation where you had two rods pinned in the center and the left rod having an initial angular velocity $\omega_1$, and the right rod was at rest. I am wondering what the final angular velocities would be if there was a coefficient of restitution during t... | No the coefficient of restitution does not apply to with rotational velocity. In fact conservation of momentum will not apply either because the bodies are pinned and can transfer momentum to the earth or receive momentum from the earth as needed.
In addition the contact between the bodies is still done through a force... | {
"language": "en",
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Are curvilinear coordinates inertial? At 1:46:34 of this lecture by Frederic Schuller, Inertial coordinates are defined as ones which satisfy the following equation:
I am confused by the above equation because it would imply any curvilinear coordinate eg: polar is not inertial. I thought 'inertial' meant the frame i... | Reference frames can be inertial or non inertial. Coodinate systems are not reference frames unless the frame is somehow being tied to the coordinate system. Does the book explain how the frame is attached? If not, the book is making a non-standard definition.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does gravitation really exist at the particle level? As I understand, we usually talk about gravity at a macro scale, with "objects" and their "centre(s) of mass". However, since gravity is a property of mass generally (at least under the classical interpretation), it should therefore apply to individual mass-carrying ... | FWIW, small particles react to the big ones:Experiments have been done with neutrons in a gravity field. The phase of their wavefunction was shifted, as was shown by interference. If the neutron didn't have it's own gravity field, would it react? Would an electron without charge accelerate in an electric field? Food fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Confusion in the showing EM wave exist from Maxwell equation When deriving the mathematical description of a field, we set the current density and charge to zero in Maxwell's equations.
However, this condition is not absolutely true anywhere on earth.
Yet, we are able to apply EM waves for problems in communication, me... | First of all, a general solution of a system of inhomogeneous linear equations (such the Maxwell equations with sources) can be always decomposed into a particular solution of the inhomogeneous equations and the general solution of the homogeneous ones (i.e., the Maxwell equations without sources, of which EM waves are... | {
"language": "en",
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What is the instant velocity? The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
| Let's say your position is given by a function $x : I \to \mathbb{R}^3$, where $I \subset \mathbb{R}$ is an interval of time. Then the velocity is defined by
$$x'(t) = \lim_{h \to 0}\frac{x(t+h) - x(t)}{h}.$$
If $I$ is just a single point, this limit does not really make sense, because there is only one $t$ we are allo... | {
"language": "en",
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What does Specific Heat capacity of a material depend upon? I have read in many places that specific heat capacity is an property of the material but I haven't really understood what it depends upon, as in what factors affect that specific heat, I have thought of different things but none of them are consistent, especi... | This is a complicated business which I will simplify a bit.
Heating an object causes its constituent atoms to randomly vibrate more vigorously. In so doing, the atoms are continuously exchanging kinetic energy back and forth between each other and so the bonds that connect those atoms are continually being exercised as... | {
"language": "en",
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Do-It-Yourself physics experiment Is there any simple experiment in physics of the first half of the 20th century I could do at home? I tried to make a cloud chamber, but it didn't work at all... A spectrometer from a CD disk is already being implemented. I also thought about the double-slit experiment, but it was set... | The first cyclotron was a tabletop device.
Then again, a cyclotron requires a source of ions, I don't know whether that is doable as a home project.
Also, I don't know how high of a vacuum is required.
The difficulty, of course, is that the experiments that have gone down in history are the ones that in their time wer... | {
"language": "en",
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The electron's charge/mass ratio In my physics laboratory class, we were discussing the following results after conducting experiments to determine the charge-mass ratio, $e/m$, of the electron. In the experiment, electrons were accelerated and executed circular motion perpendicular to a homogeneous magnetic field whic... | It is most definitely the magnitude of charge that matters, reversing the charge all it does is make a mirror image circle, so since in your experiment, you are measuring inherently sign-less/directionless parameters like radius, time, and magnetic field, speed etc. You would get a positive value for e/m. However, when... | {
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Is there a mixed field-particle representation of QED? In QFT, we can write the amplitude for a field to be in configuration $A_{in}(x)$ at time $0$ and end in configuration $A_{out}(x)$ as $K_t[A_{in},A_{out}]$, for example. Alternatively we could expand this in terms of photons and calculate the amplitudes of photons... | Sure; this amounts to a choice of a basis in the space of states in which your basis states are eigenstates of the electric field operator and have a definite electron number. This can be done for a free theory, or for an interacting theory you can at least do it approximately.
| {
"language": "en",
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Showing $[\hat{A},\hat{B}] = i\mathbb{1} \Leftrightarrow [\hat{A},\hat{B}^{\textstyle n}] = i\,n\,B^{\textstyle n-1}$ Actually it is self redundant to show having in mind $[\hat{A},\hat{B}^{\textstyle n}] = \,n\,B^{\textstyle n-1}\,[\hat{A},\hat{B}]$ but I suppose it is not supposed to solve it this way. Instead I gues... | It's quite straightforwards operator manipulation.
$we \ have: \ [\hat{A}, \hat{B}]= \hat{A}\hat{B}-\hat{B}\hat{A}=i \mathbb{I} \implies \hat{B}\hat{A}=\hat{A}\hat{B}-i\mathbb{I}$
$THUS:$
$$[A,B^n] = AB^n - B^n A = AB^{n}-B^{n-1}(BA) \\
=AB^{n} -B^{n-1} (AB - i1)\\
=AB^{n} -B^{n-2}(BA)B + iB^{n-1} \\
=AB^{n} - B^{n-2... | {
"language": "en",
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Eyes shut, can a passenger tell if they’re facing the front or rear of the train? Suppose you’re a passenger sitting in one of the carriages of a train which is travelling at a high, fairly steady speed. Your eyes are shut and you have no recollection of getting on the train or the direction of the train’s acceleration... | I never seen a train overtaken by another train in a parallel adjunct track lane. Usually there are trains bypassing opposite direction to each other from parallel track lanes.
I think it would be easy to distinguish with confidence if the passing train sound comes initially from your back or from your front.
Additiona... | {
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"source": "stackexchange",
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The effective action in the linear sigma model I am reading the section 11.4 of Peskin and Schroeder's book (page 373), and there is a step I could not follow.
To calculate the effective action of linear sigma model, the determinant of $\frac{\delta^{2}\mathcal{L}}{\delta\phi^{i}\delta\phi^{j}}$, and in peskin's book, ... | Hint: Eqs. (11.67) & (11.68) yield
$$ \frac{\delta^2\mathcal{L}}{\delta\phi^i\delta\phi^{j}}=-\partial^2\delta^{ij}+\mu^2\delta^{ij} - \lambda\phi_{\mathrm{cl}}^2 [\delta^{ij} + 2\delta_N^i\delta_N^j].$$
| {
"language": "en",
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"source": "stackexchange",
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Could a glass cup containing a vacuum rise into the air? https://what-if.xkcd.com/6/ has been mentioned here before, but I'm questioning whether or not the glass cup with the bottom half as a vacuum would rise at all.
To start with, a vacuum exerts no force. Any perceived "sucking" is actually external pressure pushing... | Only if the cup is big enough and resistant to breaking, it will fly up. The volume, with zero mass inside, grows faster than the glass volume (assuming constant glass thickness). So the upward buoyancy force will exceed the total mass at some point.
I only now saw the real problem. If you pull a vacuum in a cylinder w... | {
"language": "en",
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What does general relativity's metric tensor have to do with quantum electrodynamics? Sabine Hossenfelder recently posted a YouTube video titled, The Closest we have to a Theory of Everything.
At 9:15, she shows the action $S$ for electrodynamics and, immediately after, the Einstein-Hilbert action for general relativit... | The action for QED in flat spacetime is
$$
S = \int d^4 x \left( - \frac{1}{4e^2} F_{\mu\nu} F^{\mu\nu} + i {\bar \psi} \gamma^\mu D_\mu \psi + m {\bar \psi} \psi\right) , \quad D_\mu = \partial_\mu + i A_\mu , \quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu .
$$
Here $\gamma_\mu$ is a $4\times 4$ Dirac ma... | {
"language": "en",
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Electric potential generated by spherical symmetric charge density I know this question is pretty basic but I found a supposedly wrong formula in my notes and I'm trying to understand where this is coming from. Suppose we have a spherically symmetric charge density $\rho({\boldsymbol{r}})=\rho(r)$, then the formula I w... | The formula
$$\phi(r)= \frac{1}{r}\int_0^r 4\pi\rho(r')r'^2dr' \tag{1}$$
for the potential is indeed wrong,
as you have already proven by checking
$\mathbf{E}(\mathbf{r})=-\nabla\phi$.
It misses the contribution of charges outside of
radius $r$ to the potential $\phi(r)$.
While these outside charges have no effect on t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/712081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is sand in a vacuum a good thermal insulator? My reason for thinking that sand in a vacuum would be a good insulator is that heat cannot be conducted in a vacuum, and the area of contact between adjacent grains of sand is very small, which means heat would transfer between grains relatively slowly. Is this correct, or ... | Comparing sand in air to sand in vacuum:
The sand in vacuum is less conductive than sand in air in all reasonable conditions.
The reason is, the air conducts some heat and vacuum doesn't, air can even facilitate convection, on the other hand, air does not interfere much with radiative heat exchange between the sand gra... | {
"language": "en",
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Table of integrals for dimensional regularization Is there any reference (book or paper) that contains a list of integrals useful for dimensional regularization?
I would need it to approach integrals like these
$$
\int d^dx \frac{x^\mu}{|x|^{2d-4}}, \qquad \int d^dx \frac{(x-y)^\mu}{|x-y|^d |x-z|^{d-2}}.
$$
Any suggest... | *
*Appendices of QFT textbooks usually have tables of DR integrals used through the textbook, for example in Peskin and Schroeder Appendix A.4, and Schwartz Appendix B.
*For general one-loop integrals, there are easy to use computer packages like Package-X on Mathematica, that can deal with massive propagators.
*For... | {
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Is the gluon also a repulsive force?
In the picture of a proton we see 3 quarks, held together by gluons. But the two $u$ quarks repel each other , so the gluons act through the strong force, whereas the $u$ and $d$ quarks attract each other, if all this is correct, then what keeps them a part? can a gluon also act as... | Lets start from experimental facts. The proton exists and is stable. This means that at the quantum level there must be one stable wave function a solution of the complicated potentials of both the electromagnetic and the strong force. The proton is even more complicated than your picture as there are innumerable quark... | {
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Approximation of Spherical Bessel function I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fou... | This is too long for a comment so I wrote this answer.
I looked in the obvious place,
G. N. Watson,
"Treatise on the Theory of Bessel Functions",
(Cambridge University Press,Cambridge,1980), second edition,
in section 8.12 he gives an expansion first derived by
Meissel for large order and $x$ times the order large.
Wat... | {
"language": "en",
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Pressure inside a gravitationally bound sphere of uniform density I've looked in many places to find an equation for the pressure inside a sphere of uniform density, but didn't find any, so I decided to take a stab at it. I first found the equation for gravitational acceleration inside a sphere,
$$g(r)=\frac{GM\left(\f... | The expression given in the question is correct. Maybe for some context: the relevant equation for the pressure balancing gravity inside a static, spherically symmetric body is given by Newton's hydrostatic equilibrium:
$$
\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2}=-g(r)\,\rho(r)\longrightarrow dP = - \rho(h)\,g(h)\, dh.
... | {
"language": "en",
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How does current flow in a purely inductive circuit if the net voltage is zero? Considering the equation,
$$E=−L\frac{di}{dt}$$
The negative sign in the above equation indicates that the induced emf opposes the battery's emf.
If we're talking about a purely inductive circuit, the induced emf is equal and opposite to ap... |
How does current flow in a purely inductive circuit if the net voltage is zero?
The problem in this question is that it is based on a completely wrong assumption. This concept of “net voltage” isn’t really a thing. In fact, by Kirchoff’s voltage law your “net voltage” is guaranteed to be zero. So the net voltage bein... | {
"language": "en",
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How you calculate the age of the observable Universe if the acceleration expansion is not constant? What makes us believe that the Cosmological constant was the same in the past?
And if there is no way to prove this then could the age of our Universe be different from the current calculated value since the Universe cou... | The time evolution of a universe in which the cosmological constant is actually a variable can indeed be modeled on a computer, and the results compared to observational data. If we imagine a finite-element expansion model where the time variable is divided into discrete slices, then for any time slice the CC correspon... | {
"language": "en",
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Why is the net flux the same for both spheres?
There are 2 spheres both of radius "r" and "R" respectively. Using Gauss' law to find the net flux from the surface, we use:
$$ = \frac{Q(charge \ enclosed)}{\epsilon_0} = E × ∮ds × cos(0°)$$
Here, when we use the first formula, we get the same answer for both of the sphe... | The net flux through any closed surface surrounding a point charge $q_{in}$ is equal to $\frac{q_{in}}{\epsilon_0}$, and is independent of the shape of that surface. The following illustration should make things a bit clearer:
Clearly all the field lines passing through $S_1$ also pass through $S_2$ and $S_3$.
The ele... | {
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How to find expansion of slightly modified Coulomb's potential? From here I know that,
${{\frac {1}{|\mathbf {r}_1 -\mathbf {r}_2|}}=\sum _{\ell =0}^{\infty }{\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }(-1)^{m}{\frac {r_1^{\ell }}{r_2^{\ell +1}}}Y_{\ell }^{-m}(\theta ,\varphi )Y_{\ell }^{m}(\theta ',\varphi ').}$
... | The usual multipole expansion follows from the Legendre identity
$$
{\displaystyle {\frac {1}{\sqrt {1-2xt+t^{2}}}}=\sum _{n=0}^{\infty }P_{n}(x)t^{n}}
$$
The generalization to arbitrary powers requires the Gegenbauer identity
$$
{\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{{n=0}}^{\infty }C_{n}^{{(\alpha )}}(x)t^{n}
$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Permanent magnets - why spontaneous symmetry breaking? I am not a physicist. I'm curious about the cause of permanent magnetism in ferromagnetic materials. So far, I have formed the impression that a macroscopic net magnetic dipole moment is formed from the collective alignment of electron magnetic dipole moments (in s... | a macroscopic net magnetic dipole moment is formed from the collective alignment of electron magnetic dipole moments (in suitable metals). But why do the electron spins align?
The magnetic dipole is something that the electron has permanently. It is even a constant. If you put the connection with spin in the background... | {
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Why I cannot write the time evolution operator $e^{-i(T+V)t}$ as the product of operators $e^{-iTt}e^{-iVt}$ To calculate the wave equation of a time-independent Hamiltonian we use:
$$
\Psi_{i}(r,t)=e^{-iH^{0}t}\psi_{i}(r,0).
$$
We also know that the time-independent Hamiltonian $H^{0}=T+V$ is given to the sum of kinet... | This is because of the BCH formula
$$\begin{align}e^Z~=~&e^Xe^Y
\cr\Downarrow~&\cr
Z~=~&X+Y+\frac{1}{2}[X,Y]+{\cal O}(X^2Y,XY^2),\end{align}$$
or equivalently, the Zassenhaus formula. But check out the Trotter product formula.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Non-relativistic approximation of the retarded Lienard-Wiechert field I'm working on a revision of the absorber theory of radiation proposed by Wheeler and Feynman on their paper "Interaction with the Absorber as the Mechanism of Radiation". On page 161 they say the retarded field of the source reduces to
\begin{equati... | $-\hat n \times (\hat n \times \vec{\mathfrak{U}})$ is the component of $\vec{\mathfrak{U}}$ perpendicular to $\hat n$ and has a length $\mathfrak{U}\sin(\hat n \cdot \vec{\mathfrak{U}})$, so it appears they are just giving the value of this component.
| {
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I can't seem to figure out a way to compute a gradient without reference coordinates I'm not sure if this question is better asked here or in Mathematics but here it goes:
I'm studying electric dipoles, and this exercise I'm working on asks for the energy between 2 dipoles, given by $$U_{DD}=-\vec{p}_1\cdot\vec{E}_2\,\... | We can use the identity
$$\nabla(A\cdot B) = A \times (\nabla \times B) + B \times (\nabla \times A) + (A\cdot \nabla)B + (B\cdot \nabla) A$$
So,
$$\nabla(-p_1 \cdot E_2) = \nabla[p_1 \cdot (\nabla V)]$$
$$=\nabla\left[p_1 \cdot \nabla \left[K_e \frac{p_2 \cdot R_2}{R_2^2}\right]\right]= K_e \nabla\left[p_1 \cdot \nabl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Will the potential energy is same in both the cases? Suppose there is a charge $Q$. Now bring in another charge $Q'$ from infinity to a position a distance $r$ from charge $Q$. Then the change in potential energy is equal to $kQQ'/r$.
My question is: will the potential energy will be same if the same charge $Q'$ is bro... | It will be the same only if you ignore the electric field of the dQ's that you moved there first, that is, you only consider the electric field of the original charge Q at the origin. Otherwise you would be including in the calculation the self energy of the electric charge Q', which is infinite. the self energy is the... | {
"language": "en",
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How to find the electrostatic potential due to a uniformly charged rod on its axis? I was trying to find the potential at a point (at a distance $r$) on the axis of a rod (length $L$, charge $Q$) and ran into a problem. Let's assume the farther end of the rod(relative to the point) to be $a$ and the other to be $b$. I ... | Let me formulate your problem in my words so that we are sure what we understand each other: we have an infinitesimally thin rod of length $L$ and a total charge $Q$ which is uniformly distributed along the rod. You are interested in the value of the electrostatic potential $V$ in a point on the line on which the rod l... | {
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Issue expanding $\sin \theta$ about $\theta_{eq}$ Quoting a textbook:
$$(m_1 + 2m_2\sin^2\theta)\ddot\theta = m_1\Omega^2\sin\theta\cos\theta - \frac g L (m_1 + m_2)\sin\theta.\tag{10}$$
We can simplify this expression a bit by relating $\frac g L (m_1 + m_2)$ to the equilibrium angle $\theta_{eq}.$
$$(m_1 + 2m_2\bbox... | with
$$f(\theta)=m_1+2\,m_2\sin^2(\theta)$$
hence
$$f(\theta_0+\delta\theta)=m_1+2\,m_2\sin^2(\theta_0+\delta\theta)=
m_1+2\,m_2\,[\sin(\theta_0)\cos(\delta\theta)+\cos(\theta_0)\sin(\delta\theta)]^2
$$
with $~\delta\theta \ll 1\quad \Rightarrow$
$$\cos^2(\delta\theta)=1\quad,\cos(\delta\theta)=1\quad,
\sin^2(\delta\th... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/715750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why are fields described as force divided by mass or charge? I have read that application of force on a body from a distance, like gravitational or electrostatic force is a two-step process, first, the field is created by the body, then, the application of force on the second body by the field. I want to know why the e... | The definition of field, is there to tell us about the effects of the field on an object of unity value. most force fields have the parameter of the object they effect as a multiplier, hence when you set the parameter to unity value, it is the same as dividing the force by that parameter.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/715867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Why, in this solution, acceleration is constant even when it depends on distance between two charges? I used integration of $a=dv/dt$ to solve this Why, in this solution is acceleration constant, even when it depends on the distance between two charges? I used integration of $a=dv/dt$ to solve this.
Question
Two partic... | Actually, acceleration is not constant in this case because in time $dt$ the force would change. So, the acceleration also changes even in time $dt$. I think the solution is wrong but the answer is correct.
If you go for energy conservation, which doesn't depend on acceleration you get the same answer:
$$ K_{1} + U_{1}... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/716100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Yang-Baxter equation for an $S$ matrix depending on total momentum I have a system where the two-particle scattering matrix $S_{12}(p_1,p_2)$ depends on the momentum difference $p_1-p_2$, and also on the total momentum $P=p_1+p_2$ in some non-trivial way. One can use parametrization of energies and momenta $E=m\cosh(\a... | It can certainly not be ruled out, that such scattering matrices exist.
An example:
The YBE (Yang-Baxter Equation) is a matrix equation. So, if you consider a scalar or diagonal scattering matrix (i.e. no exchange of charge/change of particle type), then it is satisfied for any $S(p,q)$ due to commutativity:
$$
S(p_1... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/716268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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