Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Anisotropic vs Isotropic Harmonic Oscillator IS there any quantum mechanical process which can take over an anisotropic commensurate harmonic oscillation to an isotropic one?
Mathematically, this kind of transformation is available.
http://dx.doi.org/10.1063/1.1666379
But, is there any physical situation where this kin... | Ultracold atoms, and in particular Bose-Einstein condensates, are frequently trapped in magnetoptical potentials which have a quadratic form. If you had a two-dimensional system and changed the trap frequencies in the two directions you would be able to control the anisotropy of the potential very precisely (and of cou... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/520524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On the Euclidean action for QCD The Euclidean action for QCD reads, (see e.g., Eq. (45) in "ABC of instantons" by Novikov, Shifman, Vainshtein, and Zakharov)
$$S_E=\int d^4 x\left[\frac{1}{4}G^a_{\mu\nu}G^a_{\mu\nu}+\psi^\dagger(-i\gamma_\mu D_\mu-im)\psi\right].\tag{45}$$
Here $\gamma_\mu$ are gamma matrices in Euclid... | I think you might be missing something due to the notation used in that article. In Lorentzian signature the quantity $\psi^\dagger \gamma_0 \psi$ is a scalar. Going to Euclidean signature with $\gamma_0 = \gamma_4$, we have that $\psi^\dagger \gamma_4 \psi$ is again a scalar. However, note that the paper defines a new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Will state of water change in certain condition? Imagine I have an iron tank with a $20~\mathrm{pm}$ hole on it. Then I completely fill it with water and use a pump to get the water out of that hole. What will come out, water or gas?
| Pressure alone does not break chemical bonds. Expose liquid water suddenly to a vacuum and it will vaporize as a molecule; it will not dissociate into atoms. This is true regardless of the initial pressure of the water.
The hole that you have made is too small for water molecules. It is even too small for a hydrogen at... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/521139",
"timestamp": "2023-03-29T00:00:00",
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Are there any satellite orbit earth perpendicular to the sun and what is the name of this kind of orbit? I think we could put satellite to orbit earth in such a way that it always see the sun. Which is orbiting along the path of earth orbiting the sun, like a wheel perpendicular to the sun
I don't know the specific nam... | This is called a “sun synchronous” orbit.
If the Earth were a perfect sphere, orbits around it would have a fixed orientation in space. As the Earth goes around the Sun, an orbit would have the right orientation only twice a year.
But the Earth isn’t a perfect sphere: it bulges in the middle. The gravity of that bulg... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/521224",
"timestamp": "2023-03-29T00:00:00",
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Why are topological insulators interesting? Why are topological insulators interesting? Meaning, why should an undergraduate or graduate student start working on this? What are the technological applications? I am not sure how to answer these questions and wikipedia does not help since it does not explain why so many p... | This is to complement yu-v answer.
As this is not an "easy" topic (I mean, the effects are not quite seductive at simple sight compared to the sexy name and invariants are always presented in research papers in, to my taste, an obscure manner), a good reference is an appropriate starting point.
One suggestion to start... | {
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A doubt in trigonometric approximation used in the derivation of mirror formula The following text is from Concepts of Physics by Dr. H.C.Verma, from chapter "Geometrical Optics", page 387, topic "Relation between $u$,$v$ and $R$ for Spherical Mirrors":
If the point $A$ is close to $P$, the angles $\alpha$,$\beta$ an... | For the angle $\beta$ the author is apparantly thinking of AP as being the arc length along the circle rather than the length of the straight line joining A to P. Then
$$
{\rm arclength}({\rm AP}) = \beta R
$$ exactly.
| {
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What determines if a photon is displayed as light or not? A photon is the particle of light. But a photon is also a force carrier and plays a role in many other situations. What I don’t understand is what determines if it shows up as light or not? Is it just a high concentration or pattern? Why don’t photons show up in... | Photons are present in all electromagnetic interactions. Sometimes they are virtual (e.g., in the case of electrostatic interactions), and sometimes real (e.g., in the case of electromagnetic radiation). For a photon to be perceived as light, it must a) be real; b) have the right range of wavelength (or frequency, or e... | {
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Why does light spread out? So we know the light that's emitted from a torch (flashlight) must be moving in straight lines, so why does it spread out when moving? Why does it cover larger area?
|
So we know the light that's emitted from a torch (flashlight) must be moving in straight lines, so why does it spread out when moving? Why does it cover larger area?
As the comments says, straight lines are not parallel lines. They can have different directions, and they do:
The flash light, (or any light source from... | {
"language": "en",
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What are $a$ and $a^*$ called in the context of a classical harmonic oscillator? Consider a harmonic oscillator defined by the coupled differential equations
\begin{align}
\begin{split}
\dot{X} &= \omega Y \\
\dot{Y} &= - \omega X \, .
\end{split} \tag{1}
\end{align}
Defining new variables $a = X + i Y$ and $a^* = X - ... | I would call $a$ and $a^*$ the complex amplitude of the oscillator. Or I guess $a$ is the complex amplitude itself and $a^*$ is the complex conjugate of the amplitude but the distinction is unimportant as they carry the same information (just like in the quantum case).
| {
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Internal force disintegrating a solid body? Let $M$ be a block on a frictionless surface. Now let us mentally divide (not physically) the block into 2:1 ration (i.e $1/3$ of the left be called $M_1$ and $2/3$ right be called $M_2$). So $M_1$ applies force $F_1$ on $M_2$ and $M_2$ applies force $F_2$ on $M_1$ and by 3rd... | Actually you haven't truly addressed as to what kind of material we are dealing with over here, so let me dissect it into two types:
*
*Totally rigid
*Jelly like
Totally Rigid Bodies
For objects made up of materials behaving as such as soon as an internal force tries to deform the object an opposite restoring force... | {
"language": "en",
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How can atomic configurations represent excited states of atoms? My lecture notes on condensed matter physics talk about pseudopotentials of atoms where the core electrons are replaced by an effective potential. This is in the context of DFT. In the lecture notes, my lecturer talks about transferability, i.e. the abili... | It is true that in the context of pseudopotential theory (which has a much broader scope than DFT) people use the term "atomic configuration" in a slightly ambiguous way.
Strictly speaking what they are speaking about is the electronic configuration ot the atom. For instance, the ground state of a neutral sodium atom ... | {
"language": "en",
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How do gluons bind the quarks together within the hadrons and mesons? I was trying to know about the strong nuclear force within the nucleus and the books and websites told me that it is the gluons which carry the force, and now I am curious how the gluons carry this force
| As in your profile you say you are a ninth class student in India, it means that your physics background is still at understanding the classical physics level.
The strong nuclear force and the strong force of particle physics belong to the quantum mechanical studies, which I doubt it is taught in your year.
In classic... | {
"language": "en",
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How was the mass of the moon first calculated? How was the mass of the moon first calculated? How do we measure it now?
| Although this might be more suitable for the History of Science and Mathematics page, I thought I would add a few brief details on the physical details behind these measurements.
Newton
The first to calculate (or, estimate) the mass of the moon was probably Newton himself. As mentioned by Baso, Newton used measurements... | {
"language": "en",
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What is the cross-section size of a photon?
How "wide" is a photon, if any, of its electromagnetic fields? Is there any physical length measurement of these two orthogonal fields, $E$ and $M$, from the axis of travel? When a photon hits a surface, and is absorbed by an electron orbital, this width comes into play, as ... | For a back of the envelope calculation, it is often useful to associate with a particle it's Compton Wavelength. This is generally the most accurately you can know an object's position due to the uncertainty principle.
Other related length scales are usually more useful, factoring in features of the particular interac... | {
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Is Griffiths simply wrong here? (Electrostatic Boundary Conditions)
In the above illustration, shouldn't $E_{above}$ and $E_{below}$ be in opposite directions? If not, how did Griffiths end up the following equation? From the above directions, shouldn't the flux add up?
| To support that Griffiths is correct, I would start with quoting from Bob Jacobsen's answer:
He’s picked a sign convention where the field upward is positive everywhere. That means that $E_{below}$ is defined such that a positive value means "the E vector points up" and a negative value means "the E vector points down... | {
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What is baking and what are the effects? In their experiment, Davisson and Germer had to bake the nickel mass because it was oxidized.
What is baking and what does it do to the lattice of the metal?
| The primary effect of the baking was basically to boil off the oxygen that had built up at the nickel surface. When the metal is exposed to air, the oxygen reacts with the surface atoms, and this eventually produces a layer of metal oxide that is typically tens or hundreds of atoms thick. However, the oxygen remains ... | {
"language": "en",
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I dont understand the work equation I don't understand how work = force * displacement as if a force of say 1 Newton was to be applied to two objects of different mass until the object reached a displacement of say 1 meter, surely the object of less mass would displace 1 meter in less time (due to faster acceleration) ... | Let’s take a little look at the definition of work.
The work done by a force in moving some system through a certain displacement is defined to be the force times displacelment.
Let’s take gravity as an example.
If you are holding a rock at a certain height and you let it fall, the gravitational force will be acting du... | {
"language": "en",
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How are topological qubits encoded in the Majorana-fermion-based platform for quantum computing? Where is “the two level system” of a topological qubit encoded in the Majorana-fermion-based platform of quantum computing?
If the Hamiltonian in a topological quantum field theory is absent (H=0), then what dictates the ev... | There are many proposals for using Majorana zero modes to encode a qubit; one of the most popular is called the hexon, discussed in this paper: https://arxiv.org/pdf/1610.05289.pdf.
The main idea here is that a one-dimensional topological superconductor hosts Majorana zero modes at its ends. Two Majorana modes combine... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/524126",
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Is the car braking time formula $ T = v / (\mu_s \, g) $ valid only for uniformly accelerated motion? I'm wondering if the car braking time formula is valid only for uniformly accelerated motion.
$$ T = \frac{v} {\mu_s \, g} $$
with $ v $ average speed, $ \mu_s $ static friction coefficient between the wheel and the gr... | While the derivation you've used assumes uniform acceleration, it is also possible to show that the $T$ you have found is a lower bound on the stopping time of the car, even without assuming uniform acceleration. Roughly speaking, even if the acceleration varies with time, its magnitude can be no greater than $\mu_s g... | {
"language": "en",
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Why does an open ended wire cause the load to the oscillator to increase? Using an ammeter between the wall and signal generator, I noticed that when the lead from a single channel is attached to the signal generator, the power into the signal generator increases by about 100 milliwatts.
When the wire leads are removed... | It appears that someone who is familiar with RF answered my question elsewhere. His answer was that the attached BnC cable to the signal generator is not an open circuit as I has assumed but because of the Mhz frequency, it actually appears to the signal generator as a mismatched circuit (impedance mismatch) drawing a ... | {
"language": "en",
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Why does a weather vane arrow point in the direction of the wind? It seems that a weather vane will rotate in order to minimize energy and thus orient itself parallel to the wind.
What I do not understand is why it is implied that the weather vane arrow should point in the direction of the wind.
I do not understand wh... | When the wind blows perfectly parallel to the wind vane's long axis, there is no rotational force on the wind vane. When the wind direction is not parallel to the long axis of the wind vane it will exert a turning force on the vane until the wind vane is parallel to the wind direction.
The force of the wind on any part... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/524846",
"timestamp": "2023-03-29T00:00:00",
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Properly reporting instrument readings As a first approximation, the uncertainty ($\delta X$) associated to a mensurand can be expressed as $\delta X= \Delta X / 2$ with $\Delta x$ being the resolution of the instrument.
There is also a recommendation indicating that the numerical expression of the result and its uncer... | This is because the first part of the value represent the significant digits that are certain. The answer must contain extra significant digits on top of the ones that are certain. Having taken the reading, you are 100% sure that the ones digit is $3$ and not $4$ or $2$. Writing $3.0$ would mean that you are sure of th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is Snell’s Law valid in this case? When light travels in a perpendicular path from one medium to another medium of different optical density, is Snell’s law valid?
$\sin i$ and $\sin r$ are both 0, right?
So it isn’t valid.
Is this correct?
| In this case where the incident angle is $0^\circ$ to the interface and thus from $$ n_{1}\sin{\theta_1}=n_2\sin{\theta_2}$$ we get that $\theta_1 = \theta_2$ I'd simply instead reason and say that Snell's law is not applicable in this case as it'd result in the forbidden $\frac{0}{0}$ formulation. Instead as Snell's l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A formula which relate spring constant to its physical properties like length and area of cross section I know spring is a mechanical analogue of capacitor. And length of a spring is equivalent to plate seperation of capacitor. So what is the mechanical equivalent of area of cross section of capacitor in spring? And is... | A simple example is that the stiffness of an elastic rod is
$$
k = \frac{EA}{L_0},
$$
where $A$ is the cross-sectional area of the rod, $L_0$ is its unstretched length, and $E$ is the Young's modulus of the material of the rod.
Note, however, that this is only for a rod that stretches uniformly along its length; thi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/525274",
"timestamp": "2023-03-29T00:00:00",
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Problem involving gravitational potential energy
Two uniform solid spheres of equal radii $R$, but mass $M$ and $4M$ have a centre separation of $6R$. The two spheres are held fixed. A projectile of mass $m$ is projected from the surface of the sphere $M$ and towards the second sphere along the line joining the centre... | The null point...
... it is unstable i.e. the force on that point is indeed zero but what if we displace the object by $\mathbf{\delta x}$?
What I mean to say is that the system is in unstable equilibrium.
The force on null point $N$ may be zero but the potential energy at the point is maximum suggesting unstable equ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is there any book that treat time-dependent perturbation theory with rigorous mathematics? I am searching for rigorous mathematics books or notes for time dependent perturbation theory.
For introductory quantum mechanics there is the excellent book spectral theory and quantum mechanics by Valter Moretti and for time-... | During my bachelor, I found quite useful the book Quantum Mechanics Vol.2 by Cohen-Tannoudji, which treates the basics of time dependent perturbation theory really well in my opinion, and with some examples. However, there are other books more focused on a time dependent analysis of Quantum Mechanics like Introduction ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it possible to land on sun? The Sun is, of course, quite hot. A space probe built by normal materials probably would melt or burn even it’s still quite far from the Sun.
However, from a theory point of view, is it possible to build a space probe that could land on the Sun?
| If, by "land on the sun", you mean "land of the surface of the sun", then you might consider the following from https://solarsystem.nasa.gov/solar-system/sun/in-depth/:
Surface
The surface of the Sun, the photosphere, is a 300-mile-thick
(500-kilometer-thick) region, from which most of the Sun's radiation
escapes ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/525818",
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Why does $\phi=\phi^*$ imposed on complex scalar field Lagrangian miss out $1/2$ factors? If we require the reality condition $\phi=\phi^*$ on the Lagrangian for a complex scalar field is $$\mathcal{L}=(\partial^\mu\phi^*)(\partial_\mu\phi)-m^2(\phi^*\phi),$$ two degrees of freedom $\phi$ and $\phi^*$ is reduced to one... | The standard convention is to divide each term in the Lagrangian with its symmetry factor. Therefore the kinetic term for a real (complex) scalar field is
with (without) a symmetry factor $\frac{1}{2}$, respectively. A complex scalar field $\phi= \frac{\phi^1+i\phi^2}{\sqrt{2}}$ can be viewed as 2 real scalar fields.... | {
"language": "en",
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60 kg on earth is 60 kg on the moon I'm writing a trivia quiz and intend this question, which dates from a high school physics test I took in 1972.
An astronaut tips the scale at 60 kg while on earth, what will she be if she steps on the scale on the moon? Answer 60 kg.
Kg measures mass, which is constant.
The questio... | Unfortunately, scales do not measure mass. They measure the force applied by the mass, under the influence of the local gravity. The fact that they are labelled in "mass" is irrelevant. All that happens is that the readout is "adjusted", so that 588 N is shown as 60 kg.
Hence, on the moon your scales will show about 10... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/526210",
"timestamp": "2023-03-29T00:00:00",
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Derivation of the Euler-Lagrange equation for fields In the derivation of the Euler-Lagrange equation in Peskin and Schroeder, p.15, we have:
$$\delta S = \int d⁴x \left[ \frac{\partial L}{\partial \phi}\delta \phi -\partial_\mu \left(\frac{\partial L}{\partial (\partial_\mu) \phi}\right)\delta \phi + \partial_\mu \le... | The last term is a total divergence. The second one is not.
| {
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"timestamp": "2023-03-29T00:00:00",
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From photon flux to number of photons inside a cavity Consider a light beam with photon flux $\Phi$ entering a cavity. The latter consists of two mirrors, from which photons can enter/exit at rates $\kappa_1$, $\kappa_2$, respectively. How do I obtain the number of photons inside the cavity in the stationary state? Wha... | In the interaction picture, the equations of motion for the cavity (assumed single mode by the OP) read:
$$ \dot{a}(t) = -(\kappa_1 + \kappa_2)a(t) + \eta(t). $$
The constants are defined as in the question and $\eta$ is the drive strength, which is related to $\Phi$ (if I understand ''photon flux'' correctly it is $\e... | {
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"url": "https://physics.stackexchange.com/questions/526451",
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Mode expansion of string On Polchinski, the mode expiation of an open string is given as
$$X^i(\tau,\sigma)=x^i+\frac{p^i}{p^+}+i \sqrt{2\alpha'}\sum_{n\neq 0} \frac{1}{n}\alpha_n^ie^{-\frac{\pi i n c \tau}{\ell}}\cos\frac{\pi n \sigma}{\ell}\tag{1.3.22}$$
Giving the B.C. $\partial_a X^i=0$ at $\sigma = 0, \ell$.
Howev... | 1) $\sigma^\pm = \tau \pm \sigma$ so you get something of the form
$$
\sum_n e^{-in\tau}\Big(e^{-in\sigma} + e^{in \sigma} \Big)
$$
which gives you the cosine.
2) The $\alpha'$ (and $l$) is just a convenient normalisation.
| {
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Is the equation for dispersive power applicable for all dispersive elements? The following text is from Concepts of Physics by Dr. H.C.Verma, from the chapter "Dispersion and Spectra", page 434, topic "Dispersive Power":
The dispersive power of a material is defined as the ratio of angular dispersion to the average de... | Dispersive power is used to study dispersion of visible light in optical instruments. The lack of sources indicate that this is a quantity that isn’t used nowadays.
That being said, the above derived definition is a good one because it purely depends on the material and not its geometry. And you’d want to find the max... | {
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"url": "https://physics.stackexchange.com/questions/526796",
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Photon vs electromagnetic waves Suppose an electron makes a single transition from higher to lower energy level releasing energy. Would that energy be released in exactly one photon equal to $h\nu$? Also, is saying "one photon is released" equivalent to saying "one electromagnetic wave of $\nu$ frequency is released"... | We know that the ocean is made of water molecules. We also see that ocean has waves. Now does that mean that each water molecule is a water wave?
Physically realisable EM waves are analogous to the water waves where many photons collectively behave in a manner where EM waves emerge. But what is astonishingly peculiar ... | {
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What is the reason to believe that the laws of physics are same in all frames of reference? The first postulate of Special Relativity is that the laws of physics must be the same in all frames of reference i.e. invariant of coordinate transformations. I know this might be moot to ask but after reading a critique's pape... | If special relativity were not true, the everyday world we experience would be very different, in very obvious ways. I will provide just one simple example:
Special relativity is embedded in Maxwell's equations and tells us (among other things) that if we insert a magnet into a coil of wire, so as to induce a current t... | {
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Gauge ghosts & unphysical states in gauge theory I have a general question about a statement from Wikipedia about ghost states as occuring in gauge theory:
"In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an unphysical state in a gauge theory."
I learned gauge theory up to now wi... | A third view on this. Ghosts are fields that are introduced when you remove the gauge redundancy in the path integral.
In a path integral of a gauge theory you integrate over equivalent fields due to the gauge redundancy. You can fix the gauge and isolate the infinite volume of the gauge group but you are left with a ... | {
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Entropy change in the free expansion of a gas Consider the adiabatic free expansion of a gas since there is no external Pressure hence Work done on the system is 0 and since the walls are insulated (hence adiabatic) the heat absorbed is 0. However since this is a irreversible process then entropy change > 0 hence dQ > ... | The reason is explained by others. Mathematically, for free expansion:
$ΔS_{system}=nR\ln\left(\frac{V_{2}}{V_{1}}\right)\ ;\ ΔS_{surroundings}=0$
Hence overall entropy of universe increases.
| {
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Conservation of Energy stored in electric field? Let's say due to some particle process, an electron is created at time $t >0$. And from this moment on, the electric field will start to propagate to infinity at the speed of $c$. But we know that the energy stored in the electric field is proportional to the volume inte... | Lets say ... an electron is created - there's your problem; you just violated charge conservation.
You can't create a single charge out of nothing.
You might create a pair of charges (e.g., by decay of a neutral boson), one positive and one negative, but then you'd have two opposing fields whose energy content would ex... | {
"language": "en",
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Does friction do work or dissipate heat? I know there are a bunch of similar questions but I read through them all and they don't answer my question.
Let's say I give a box on a floor an initial "kick" of force such that it has kinetic energy $KE$. Due to friction between the box and the floor, the box will slide to a ... | 1) Work is done by the friction forces until the box stops.
2) Box kinetic energy is transformed to increased temperature (internal energy) of the sliding surfaces.
3) The cooling to the neighbourhood is an irreversible process, increasing entropy.
| {
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Why COM and COG are different things? In the formula for Newton's law of gravitation i.e.
$$F = G(m_1)(m_2)/r^2 $$
here $r$ is the distance between COM of the two objects , so we consider that the object is a single point located at it's COM and at this point gravitational force of attraction acts . Then why is somet... | The COM for a rigid body is a fixed position that does not change. The center of gravity can depend on orientation, and can be in a different place on the rigid body, when in a non uniform gravitational field. In such a situation, the COG will always be closer to the main attractive body than the COM. For example, a sa... | {
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Is there some other measurement that describe the ability to absorb some spicified range of frequencies of sound? There are a variety of materials could be used to absorb sound in order to soundproof.
Absorption coefficient is used to describe the absorb sound ability of the material.
Is the following guess true:
some ... | The absorption coefficient is usually specified by the manufacturer over some frequency range, in dB of loss. High frequencies are commonly absorbed by light, fluffy materials with a lot of air in them. Low frequencies are commonly absorbed by heavy materials with little or no elasticity.
| {
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Unique characterization of Ideal gas In Thermodynamic state - Wikipedia, it defines a thermodynamic state as:
A thermodynamic state of a system is its condition at a specific time, that is fully identified by values of a suitable set of parameters known as state variables.
In the part explaining state functions, it s... | Your doubt is well founded. A three-dimensional space of thermodynamic states is justified only in the case of specific classes of systems. Generalizing a result valid for the ideal gas to every thermodynamic system is an unduly step. It is possible that somewhere in the text there was an explicit indication that all t... | {
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Constraining the 2-point correlation function Consider the two-point function
$$
\langle\mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\rangle=f(x_1,x_2)
$$
If the operators are in a CFT, we can constrain this function using the symmetries of the theory. Using translational symmetries and the symmetries of the Lorentz group we ha... | You did all the work but just missed the last step:
As you wrote, from the definition of $k_\mu$ you have
$$
(k_{1\mu}+k_{2\mu})|x_1-x_2|=-(x_{1\mu}+x_{2\mu})|x_1-x_2|,
$$
and so
$$
(k_{1\mu}+k_{2\mu})\frac{C_{12}}{|x_1-x_2|^{\Delta_1 + \Delta_2}}
= (\Delta_1 + \Delta_2) (x_{1\mu}+x_{2\mu})
\frac{C_{12}}{|x_1-x_2|^{\De... | {
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Diffusion equations and classical mechanics In Wikipedia it is stated that the diffusion equation can be derived from the continuity equation. It is not clear to me how the classical mechanics affect the diffusion equation. For example, if the total force on an object would be equal to the second time derivative of mom... | The diffusion equation can be derived from the continuity equation, the equation that reflects the conservation of mass in continuum mechanics. Changes in the momentum would only have an impact on Newton's second law, which in continuum mechanics is given by the momentum equation, and not on the continuity equation. As... | {
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How to calculate the resistance impulse of a wheel on an axle when it receives torque during acceleration? I am programming a vehicle system for a company, but I cannot find the right formulas to be able to have access to the resistance pulse of a wheel attached to an axle during acceleration / deceleration. All I know... | To answer this problem, first do a free body diagram that includes all forces and motions acting.
Now assume that the axle bears some fraction of the total mass of the vehicle, $m_{\rm axle}$.
Also assume no slipping, which means $v - \Omega R =0$ or it terms of accelerations
$$ \dot{\Omega} = \frac{\dot{v}}{R} $$
whe... | {
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Normalization of the action in Special Relativity The action for a massive point particle in Special Relativity is given as
$$A =-mc^2\int d\tau,$$
Where $\tau$ represents the proper time, and $m$ represents the (rest) mass. From what I could understand, the Action must not change with respect to the reference frame, a... | Since action has the units of angular momentum, the proportionality constant needs the units of energy by dimensional analysis. It must also be Lorentz-invariant, so is $mc^2$ times some real number. This number's modulus can be varied without changing the resulting equations of motion, but @Cryo's answer shows a modul... | {
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Can a Black Hole become a normal mass again? It is well known how Black hole forms. But once it is formed are there any circumstances which can lead a black hole to becoming a normal mass again?
| Apart from the possibility of BH relics as outlined above, if it were possible to remove or ‘destroy’ a BH event horizon, then that might qualify. Of course, to avoid the cosmic censorship hypothesis (no naked singularities please) you need:
*
*a BH that doesn’t have a singularity, i.e. something like a Bardeen bla... | {
"language": "en",
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Are reflectance $R$ and transmittance $T$ independent of the direction in which the light travels? If we have two different optical mediums $1$ and $2$, and we know the reflectance and transmittance when the light gets to the interface from the first one to the second, $R_{1 \rightarrow 2}$ and $T_{1 \rightarrow 2}$, w... | This question can be answered by looking at the Fresnel equations for reflectance and transmission, which are shown below in the case of normal incidence and ignoring polarization.
$$r = \frac{n_1-n_2}{n_1+n_2} \\ t = \frac{2n_1}{n_1+n_2}$$
and the total reflectance and transmission are $R=\vert r \vert^2$ and $T=\frac... | {
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If magnetic field lines don't exist, what are these iron filings doing around a magnet? Obviously the iron filings can be seen aligning themselves along the virtual magnetic field lines produced by the permanent magnet, the virtual magnetic field line is made of electromagnetic field due to the alignment of electrons i... | What is happening when iron filings form a "field line" pattern, is actually an energy minimisation process, somewhat akin to the reason that solar systems form out of rotating clouds of dust. Each individual iron particle becomes magnetised by the applied field, much more so than the surrounding medium. There are then... | {
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Boyle's Law and hot air balloon A bit dumb question because it is really difficult to imagine it. :-
Pressure is force per area. Talking about gases, the pressure is said to be the force molecules exert on walls of let's say a balloon.
Usually in examples of Boyle's law, our teachers mention hot air balloon, that the s... | There are three forces at play here caused by the internal pressure, external pressure and the elastic balloon. The internal pressure balances out the other two to maintain equilibrium. For ease of calculations, I am assuming that no air is able to diffuse through the rubber and escape. Thereby the internal pressure st... | {
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Why isn’t the center of the Earth cold? If the pressure of the Earth is keeping the inner core solid, keeping it rigid to take up the least space, and temperature is dependent on how much the atoms are moving, why isn’t the inner core cold? If the pressure is so high that it’s forcing the inner core to be solid then th... | Your argument would require that all solids must be cold, because all solids have constituent atoms that are constrained to remain in their solid lattice positions. But clearly not all solids are cold, so there is something wrong with your argument. That thing is that atoms or molecules constrained within a solid struc... | {
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Are there any black hole neutron star binary? Has a black hole-neutron star binary aver been observed?
I mean observed in any way: gravitationally, through eclipse, or any other means.
EDIT Thanks to the comment to this question, we know that some system are known from GW observation.
Are these observations the only ... | A quote from the website of the BlackHoleCam European Research Council Project (as of 17/2/2020):
Although pulsar-BH systems can provide unique benchmarks of theories of gravity, they are expected to be very rare and to date not a single pulsar-BH system has yet been found.
In principle, a pulsar-BH system could be u... | {
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Electron absorption by materials Is it possible to define whether or not a material has absorbed electrons after it has collided with an accelerated electron beam? If so is it possible to calculate where they gathered after the collision?
|
Is it possible to define whether or not a material has absorbed electrons after it has collided with an accelerated electron beam?
It will depend on a large number of conditions: energy of beam, the type of material, in general the boundary conditions of the experiment.
If so is it possible to calculate where they g... | {
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If sound passes through material, vibration is produced. So are electromagnetic waves produced too? Sound means vibration of molecules and vibration produces electromagnetic waves. So, this means that sound produces electromagnetic waves directly.
Is this possible?
| I am deeply surprised by the neglect of the energy balance. In the previous answers. All sound is dispersed within a body. The energy difference between the incoming sound and the outgoing sound turns into heat.
The increase in temperature of the body is accompanied by an increase of electromagnetic radiation. So your ... | {
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Is tension a conservative force? Are forces such as tension (from an in extensible string), normal reaction, and applied force from us, non conservative forces? If so why?
I have read in few books that these forces are labeled as nonconservative, but most of the time they are internal and do zero work and thus mechanic... | First, you need to know how a conservative force is defined. A force is conservative if the work done by it is path independent.
Now coming to your doubt, according to the work-energy theorem,
$$
\text{work (done by all the forces} = \Delta\left(\text{kinetic energy}\right).
$$
If you know, potential energy is defined ... | {
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Dynamicity inside a stationary water drop I was doing some experiments with water drops on lampblack when I saw this.
You can see the full video here.
Inside a water drop which is perfectly still from the outside, you can see some moving things, which I suppose are some lampblack flakes from the surface. I am not both... | Based on the similar movement of inhomogeneities on soap bubbles, I'd say the flakes' movement is caused by air currents around the drop and perhaps thermal convection.
Source: https://youtu.be/LM3p3X92mWI
| {
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Why does the same proportion of a radioactive substance decay per time period? (half life) Just wondering, if decay is random, why does the activity half every half life, as in, why does it have to reduce by the same proportion in the same time period?
| A couple of answers above hit it well. Here is a slightly different perspective.
From an visual standpoint, consider a pointillist painting. If you look at any single dot up close, the painting makes no sense. Stand back, and order falls into place.
The term “random” does not mean without order. It means that nothing w... | {
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How long will last the CBMR? I am confused about the duration of the cosmic background microwave radiation.Will it eventually be replaced by same radiation from discrete directions?
| The current model of the expansion of the universe is dominated by $\Omega_{\Lambda}$. In the distance future
1-$\Omega_{\Lambda}$ << 1.
At this era the expansion of the universe will become exponential, and at some point the temperature of the CMB will become less than the temperature of Hawking radiation of black hol... | {
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How do I decompose my overall rotation curve for M31 into a rotation curve for the disk and halo? How do I decompose my grand rotation curve into rotation curves for the disk and halo, like the image below?
The picture is from this link pages.pomona.edu/~tmoore/RotCurve
The data points are for NCG 2403, my rotation... | You cannot without additional information - at a minimum, how much lumnosity comes from with radius $r$ in the galaxy studied.
The rotation curve arises from the sum of all gravitating matter that is present in the system.
In principle you can estimate the contribution of the disk to the rotation curve by measuring ho... | {
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Help using the definition of Hermitian operator in $\int\psi^*(\hat F-\left)^2\psi dr$ In my lecture the professor said that the mean value of a physical quantity- since it must be real- must satisfy the following condition:
$$\begin{align}
\left<F\right>=\left<F\right>^*
\end{align}
$$
$$\begin{align}
\int\psi^*(\hat ... | It is usually assumed that $\psi$ can be expressed as a linear sum $\sum_i a_i\psi_i$ of eigen function $\psi_i$ with eigen values $\lambda_i$. So $F \psi = \sum_i a_i\lambda_i\psi_i$. This converts the problem into complex number algebra.
In effect, this is a rephrasing of - turn the operator into a matrix - but I hop... | {
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Why can terms of a spacetime metric only have 2 differential factors? In my general relativity textbook, the following equation is given:
$$ds^2 = g_{\alpha \beta}(x)dx^{\alpha}dx^{\beta}$$
Which describes the line element $ds^2$ for a metric. $g_{\alpha \beta}$ is a matrix that is the metric itself. This equation impl... | The expression for the line element is a generalization of the Pythagorean Theorem of 2D Euclidean space,
$$c^2=a^2+b^2,$$
to 4D curved spacetime, on an infinitesimal scale. So the squares shouldn’t be surprising.
In 3D Euclidean space, you have probably seen that infinitesimal arc length along a curve is just
$$ds^2=d... | {
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Due to incompressibility of water same volume of water should flow through pipe of different cross section . is that really true This seems true when I slightly compress the opening of pipe the speed of water flowing through pipe increases and same amount of water flows out but as I further decrease the opening of pipe... | Yes, due to incompressibility, the same amount of water flows into and out of any given section of a tube - that is correct. But due to viscosity and a no-slip condition on the tube wall, the pressure of the water drops downstream (if the tube width and flowrate is constant).
If you have a constant pressure potential ... | {
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Why do springs have a linear relationship? Why does:
F = k*(change in position)
Why can't the relationship be quadratic or higher ordered?
| It's an approximation motivated by calculus. Since by definition force is $0$ in the equilibrium position we label as $x=0$, the lowest-order approximation of $F$ as a Taylor series of $x$ is of the form $F=-kx$, unless $k=0$. (The sign indicates a force opposing the perturbation from a stable equilibrium with $k>0$.) ... | {
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Is there an approximation for the Lorentz factor for very large velocities? I am aware of the approximation generally used for low speeds to calculate the Lorentz factor, that being,
$$\gamma \approx 1 + \frac{1}{2} \left(\frac{v}{c} \right)^2$$
But I need the exact opposite thing -- is there any suitable approximatio... | For ultra-relativistic particles, $c-v$ basically stops being an experimentally accessible observable. Unless you are extremely careful about timing, you assume that the beam is traveling at $c$ and measure the Lorenz factor by comparing the kinetic energy per particle to the particle mass, $\gamma = E/mc^2$. (If you c... | {
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How is a 25-year-old can of soda now empty without having been opened or poked? I just discovered in my parents' basement a Sprite can from 1995* and also a Coca-Cola can probably from the same year.
Both cans are unopened and have no visible damage or holes. The Coca-Cola can feels "normal", but the Sprite can is emp... | As others have said, more than likely it was never full in the first place. My grandfather worked for a can manufacturing company and had a large collection of empty sealed beer cans of all different brands. Your can could have been a reject or more likely just a sample.
| {
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"source": "stackexchange",
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Splitting a single particle wave function The wikipedia article on the double slit experiment contains the following animation:
https://upload.wikimedia.org/wikipedia/commons/transcoded/a/a0/Double_slit_experiment.webm/Double_slit_experiment.webm.180p.vp9.webm
Here we can see that part of the wavefunction is reflected ... |
Does this only happen when there are multiple particles?
No. Wave function reflection can happen with single-particle wave functions.
what happens when a wave function of a single particle splits in two parts, with each part propagating in a different direction? Is this even possible?
Yes, this is possible. It jus... | {
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How to find the magnetic field of a current using the differential form of Maxwell's equations? To find the magnetic field produced by a long straight wire, one would ise either Biot-Savart law or Ampere's Law in integral form. How do you find this simple result starting from $\nabla \cdot \vec{B} = 0$ and $\nabla \tim... | Edited in the light of first comment below answer
Your problem arises because Maxwell's equations are local equations, applying at a point. Thus the four equations in your question, judging by the last one, apply to a point at which $\mathbf J$ is non-zero (that is a point in your wire). But you wish to find $\mathbf B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why doesn't Rayleigh scattering happen at low altitude in air? An answer I found online was: At high altitude where molecules are far apart, scattered photons can travel without interfering with each other, thus they fill the sky with blue light. But at low altitude, molecules are so close to each other that scattered ... | The assumption is wrong. Rayleigh-scattering does happen independent of altitude. That it happens at sea level, too, becomes clear when you look at the other bank of a large river mouth or a nearby coast over a bay: they loose much contrast and everything is blue-ish; the same effect happens in the mountains looking fr... | {
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"source": "stackexchange",
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Clarification of the concept "less resistance means less heating" in a wire So my textbook says that the reason cables that are suppose to carry high currents, are thicker that those that are meant to carry lesser current, is that "less resistance (of the wire) means less heating..."? Is this even true?
Isn't CURRENT ... |
Isn't CURRENT the reason wires heat up? If we decrease resistance, more current flows, and that should produce more heating!
The key thing you're missing here is that current is (almost) constant and is (almost) not dictated by the wire. The "ideal" wire has no resistance and the current through it is solely set by ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Post-measurement $\psi$ in quantum mechanics I have a question regarding the wave function after a measurement. Everything I found online says that this is the following formula:
$\psi = \frac{M_m\psi}{\sqrt{P(m)}}$
Where $P(m)$ is the probability of observing m, the $\psi$ on the left is the wavefunction AFTER the me... | It is instructive if you work with some simple setups. Consider the example of a spin-half system where you’re measuring the spin along the z-axis. We know that the possible outcomes are either up or down. So in general your Say your pre-measurement state is a normalised linear combination:
$$|\psi\rangle=\alpha|u\rang... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\mathcal{N} \ge 2$ Supersymmetry massive supermultiplets In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined:
$$
\{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24}
$$
$$
\{Q^I_\alpha,Q^J_\beta\}=\epsilon_{\alpha\beta}Z^{IJ}
$$
$$
\{\bar{Q}_{I\d... | The point is to devise an algorithm that constructs supermultiplets given only the commutation relations.
Since you have a nice basis of harmonic oscillator esque creation/annihilation operators, you can define a supermultiplet by postulating a vacuum state $|s\rangle$ annihilated by $a,b$, so that every state in the s... | {
"language": "en",
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Gravity, matter vs antimatter I have a simple question regarding matter-antimatter gravity interaction.
Consider the following though experiment:
If we imagine a mass $m$ and an antimass $m^-$, revolving around a large mass $M$
the potential energy of mass $m$ should be:
$$ U_1=-\frac{GmM}{R} $$
and the potential energ... | In addition to John's answer:
There is a subtlety in antimatter. In the standard model it is axiomatic that matter and antimatter have the same sign mass. But as long as gravity is not quantized in a theory of all four forces, it is possible that antiparticles,even having a positive mass, instead of being attracted gr... | {
"language": "en",
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"source": "stackexchange",
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Velocity after applying a force in the vacuum I’m sorry for so simple question, but I just need to be sure.
I understand, that the changing of the speed occurs only when the force is applied, I understand that if one punch a ball in the free space it will infinitely move with a constant velocity
Some point-like body wi... | In order to determine the velocity as a result of the application of the force, you need to know the duration (time) of the application of the force on $m$, or the displacement $x$ of the mass $m$ during the application of the force, and the force as a function of time if not constant.
Your value of $v_2$ is based on a... | {
"language": "en",
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The Enigma of Universal Gravitation Forces This is taken from a book called "Physical Paradoxes and Sophisms" by V. N. Lange.
1.22. The Enigma of Universal Gravitation Forces
The law of gravitation can be written $F=\gamma\frac{m_1m_2}{R^2}$.
By analyzing this relationship we can easily arrive at some interesting ... | Answers which merely refer to the distance between the centre of mass of one thing and the centre of mass of another are missing the point being made by Lange.
When we sit on a chair, the distance between part of your body and part of the chair does tend to zero, so what happens to the force? If our atoms had point-li... | {
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Geodesics through the wormhole To define a traversable wormhole, there should be some conditions on the metric components, such as:
I) No event horizon,
II) Minimum value for the shape function (considering a spherical symmetric solution, we should have $b(r_t)=r_t$ ),
III) The flare--out condition ($b'(r_t)<1)$ ans... | There's a variety of metrics used for the Morris-Thorne metric. The two most common are the Schwarzschild coordinates and the proper radial coordinateS.
The Schwarzschild coordinates are just done using the classic spherical symmetric coordinates,
$$ds^2 = -e^{2\phi_{\pm}(r)}dt^2 + \frac{dr^2}{1 - b_\pm(r) / r} + r^2 ... | {
"language": "en",
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What does this notation for spin mean? $\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$ In my quantum mechanics courses I have come across this notation many times:
$$\mathbf{\frac 1 2}\otimes\mathbf{\frac 1 2}=\mathbf{1}\oplus\mathbf 0$$
but I feel like I've never fully understood what this nota... | This is actually the decomposition for the tensor product of irreducible representation of SU(2). We can set your $1/2$ as $j$, which means the (2j+1) dimimension irreducible representation of SU(2). Generally, Clebsch–Gordan series gives:
$$D^{\left(j_{1}\right)} \otimes D^{\left(j_{2}\right)}=\bigoplus_{J=\left|j_{1}... | {
"language": "en",
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Intuitive explanation why rate of energy transfer depends on difference in energy between two materials? The temperature of an object will decrease faster if the difference in temperature between the object and it's surroundings is greater.
What is the intuitive explanation for this?
| I think it’ll be helpful to think in terms of the kinetics of the constituent particles. When an object is at a higher temperature, the kinetic energy of its constituents is higher. They are more in motion when compared to the ones with a lower temperature.
Now if there’s a high temperature object in contact with a lo... | {
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Quantization of complex scalar field I'm learning Peskin's qft now and I'm a little confused about problem 2.2 .
Suppose I write the field $\phi(x)$ as:
$\phi(x) =\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{p}}} (a_{p}e^{-ipx}+b_{p}e^{ipx})$
I know that $b_p$ should be written as $b_p^\dagger$ because it annihilate a... | I am afraid what is wrong is in your initial writing.
A complex scalar field should be written as
$\phi(x) = \int \frac{d^3p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_p}} (a_p e^{-ipx} + b_p^\dagger e^{ipx})$
And, by complex conjugation,
$\phi^*(x) = \int \frac{d^3p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_p}} (a_p^\dagger e^{i... | {
"language": "en",
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Must a rainbow's observer also be able to observe the sun? I think I may recall sometimes a rainbow going away when a cloud comes between myself and the sun. I know that the appearance of the rainbow is location dependent. But do we need the sun, rainbow, observer triangle for the effect to manifest?
| While observing a rainbow your back is on sunside, but you dont face the sun (into your eyes). Because light undergoes total internal reflection within each raindrop. What you see is the refracted ray undergoing dispersion.
| {
"language": "en",
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Where does energy go in joining capacitors of different capacitance charged by different potential, hypothetically assuming no resistance in circuits? I don't understand why there is any change in initial and final energy since we have already assumed a perfectly conductive circuit. I mean, theoretically at least, ther... | There is a short period of time through which the electrons move from the charged capacitor to the uncharged capacitor. Through this time, assuming the wire has zero resistance, the energy is lost through the magnetic field generated by the accelerating electrons.(We know the electrons accelerate because the potential ... | {
"language": "en",
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Why do positive charges attract negative charges? From school and/or university we know that there is Coulomb's law that allows us to calculate the amount of force between two electrically charged particles.
However, I have never found any information on the origin of this force.
Is there any explanation as to why this... | The electromagnetic interaction is one of the four fundamental forces of the universe. It just so happens that there are two types of charges that interact under this interaction. If you put two particles with the same type of charge together, they repel each other. Two with different types of charge, they attract each... | {
"language": "en",
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Has a 3D chart of nuclides ever been published or proposed $(N, Z, S)$? What information might it show? Phys.org's 'Strange' glimpse into neutron stars and symmetry violation leads to the new Nature Physics Letter Measurement of the mass difference and the binding energy of the hypertriton and antihypertriton and that ... | (This anecdote kind of straddles the line between a comment and an answer.)
I saw the beginning of such a table in a conference presentation a decade ago. The format was the same as the usual (Z,N) chart of nuclides, but the data were measured lifetimes for hypernuclei where one baryon was a $\Lambda$. The heaviest nu... | {
"language": "en",
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Can momentum never be zero in quantum mechanics? I have seen Zetilli's QM book deals with $E>V$ and $E< V$ (tunnelling) in case of the potential wells deliberately avoiding the E=V case, so I thought maybe something is intriguing about this and made this up.
Suppose the total energy of the particle is equal to its pote... | You are not wrong, but it is worth noting that the same thing is true of any momentum eigenstate (or closely related unbound eigenstate of a Hamiltonian with a potential well in it). Explicitly
$$
-i\hbar\frac{\partial}{\partial x} \psi(x) = p\,\psi(x)
$$
then
$$
\psi = A e^{i \frac{p}{\hbar}x}
$$
which is not normali... | {
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How do we know not all photons are absorbed? Only those of specific energies? When a photon hits an electron in an atom, its energy has to be equal to the difference in energy between the current shell and a shell with a higher energy level, otherwise it is not absorbed at all.
How do we know not all photons are absor... | Here is the spectrum of light coming from the sun.
Solar spectrum with Fraunhofer lines as it appears visually.
The spectrum seen has all the wavelengths ( frequency = c/wavelength), and the dark lines are the absorption lines. Absorption and emission lines are one of the reasons quantum mechanics had to be invente... | {
"language": "en",
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BEC in two dimensions with interactions For noninteracting 2D system of bosons, many textbooks have a statement that no BEC exists as the system is capable to accommodate infinite number of bosons when chemical potential $\mu \rightarrow 0 $. But what happens if interaction is taken into account?
| Bose-Einstein condensation per se is a non-interacting effect, solely driven by particle statistics. In $d=2$ dimensions, free space, the energy density is such that particles can still be accommodated in the excited states for $T \neq 0$ thereby not triggering the macroscopic occupation of the ground state.
This does ... | {
"language": "en",
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Outside temp of vessel of is 40 Deg C and Water inside it is 86 Deg C.. Why? I was boiling water in a steel utensil.
Used Fluke's IR Gun (59 Max) to measure the temperature of Water inside AND the surface temp of vessel outside.
The boiling water read 86 Deg C whereas the outside surface temp of vessel read just 40 D... | When the surface is reflecting (like the metal) you are actually measuering then temperature of what can be seen in the reflection, not the reflecting body.
| {
"language": "en",
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Why is the speed of light in vacuum a universal constant? While getting familiar with relativity, the second postulate has me stuck. "The speed of light is constant for all observers". why can't light slow down for an observer travelling the same direction as the light?
| The constancy of speed of light was first predicted by Maxwell. He had discovered for equations, which we now call Maxwell equations.
Maxwell equations
$$\nabla\cdot{E}=\frac{\rho }{\epsilon_{o}}$$
$$\nabla\times{{E}}=-\frac{\partial B}{\partial t}$$
$$\nabla\cdot B=0$$
$$\nabla\times B=\mu_o j+\mu_o\epsilon_o \frac{\... | {
"language": "en",
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Why dont we consider the cohesive forces between the surface and inner molecules while formulating excess pressure inside a droplet? As the surface of a liquid will have imbalanced cohesive forces leading to the phenomenon of surface tension,we just consider the forces among the surface molecules which act tangentially... | The surface tension is due precisely to the cohesive forces that you say are ignored. Surface tension is defined to be the surface free-energy $F=E-TS$ per unit area of the surface, and so takes into account the unsatisfied bonds that are present because the inside and outside environments of a surface molecule are di... | {
"language": "en",
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Two versions of Diffusion coefficient I found two versions of the Diffusion coefficient, first:
$$D=\frac{\pi \lambda }{8}\overline{c}$$
Where $ \overline{c}$ ist the particles mean thermal velocity and $\lambda$ the particles mean free path.
(Found in W. C. Hinds, Aerosol Technology. Wiley Interscience (1999). S.156)
... | So to start out, you're looking at the RMS and mean of the Maxwell-Boltzmann distribution; you can look them up on Wikipedia, where they match Hinds' expressions.
In the two expressions you have, you see they'd cancel out were you to get rid of the square root. So, I believe $\lambda$ to be defined differently in your... | {
"language": "en",
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Conservation of energy of 2 identical Rolling Disks with and without friction My physics book claims that if two identical disks moving at the same velocity travel up nearly identical hills, with the second hill not having friction, then the disk rolling up the first hill will travel to a greater height. Given that the... | Cars can drive up hills. Their tires' rotation combines with friction to move forward.
When the friction's removed, e.g. due to the road being icy, the tires' rotation doesn't have the same effect.
That said, the textbook's a bit off: friction means that the disc's rotation matters, but this doesn't necessarily mean t... | {
"language": "en",
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Decay of electron? Have we detected any decays of electrons to an electron neutrino and $W$-boson in Fermilab or in CERN? Are neutrinos the only possible stable leptons inside an electroweak field?
| A very interesting question. If we ignore charge conservation law, then electron could decay into neutrino and photon :
$$ e^− \to ν_e + \gamma $$
Current estimates gives that life time of electron $\gt 10^{26} \,\text{years}$. Feynman diagram of such electron decay :
It is calculated that energy needed to break elect... | {
"language": "en",
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Why complex numbers are used in electronics? The impedance of a capacitor or an inductor is imaginary. How do we know these quantities are imaginary?
| A capacitor (with capacitance $C$) is fully described by
the differential equation between current $I(t)$ and voltage $V(t)$:
$$I(t)=C\frac{dV(t)}{dt} \tag{1}$$
Suppose you have an AC voltage with frequency $\omega$
connected to the capacitor. By using the complex calculus this is
$$V(t)=V_0 e^{j\omega t} \tag{2}$$
The... | {
"language": "en",
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Why can vector components not be resolved by Laws of Vector Addition? A vector at any angle can be thought of as resultant of two vector components (namely sin and cos).
But a vector can also be thought of resultant or sum of two vectors following Triangle Law of Addition or Parallelogram Law of Addition, as a vector ... | The vectors can be arbitrary and there are good reasons explained in the other answers. But I would like to approach this form another angle.
Often you as the person doing the problem setup can choose whatever coordinate system you like. Then, unless there is a apparent benefit of not choosing a orthogonal basis, you w... | {
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Why is the electric potential at infinity zero? As per net results, the potential at infinity is considered to be zero. Apart from considering this as a physics law, is there any proper reason why we consider potential at infinity to be zero?
| By definition the potential energy is chosen to be zero at infinity. It can also be defined to be zero at the ground. Generally speaking the work $W$ done for moving a body against a force $\vec F(\vec r)$ from point $A$ to $B$ is given by the difference of potential energies
$$
W = U(\vec r_A) - U(\vec r_B).
$$
That i... | {
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How can you (computationally) calculate the halfchain entanglement entropy of a spinchain? I am simulating a (small) spinchain with exact diagonalization and dynamics. I would like to track the entanglement entropy of half the chain with the other part of the chain.
I have the vectors of my state $|\Psi(t)>$ in the ba... | It all boils down to a two-partite system,
$$
\vert\psi\rangle = \sum c_{ij} \vert i,j\rangle\ ,
$$
where $i$ and $j$ are all indices in the left and right part, respectively. Then, the reduced density matrix is given by
$$
\rho = \sum_{ii'} \left(\sum_j c_{ij}c^*_{i'j}\right)\vert i\rangle\langle i'\vert\ .
$$
That i... | {
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Why can a partial derivative be added to a hamiltonian in canonical transformations? In canonical transformations, how come we allow hamiltonian to change by a partial derivative of time?
$$H'(P, Q, t) = H(p, q, t) + \frac{\partial F}{\partial t}.$$
Here $F$ is the generating function.
I mean geometrically that is not ... | In a canonical transformation, the new hamiltonian could have nothing to do with the initial hamiltonian, it just have to preserve Hamilton's equations. So in the new variables $(Q,P,t)$ you have to have that
$$\dot{Q} = \frac{\partial K}{\partial P} \qquad \dot{P} = -\frac{\partial K}{\partial Q}$$
where $K$ is the n... | {
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Why do antiferromagnets occur at lower temperature than ferromagnets? The minimal model for describing magnets is the Heisenberg Hamiltonian
$$H = -\frac{1}{2}J\sum_{i,j} \mathbf{S}_i \cdot \mathbf{S}_j$$
Where $i,j$ are nearest neighbors and the factor of $1/2$ is for double counting.
If $J$ is positive, spins will w... | It is the value of the exchange parameter that is smaller for antiferromagnets than for ferromagnets.
For example, Iron has $J$ of roughly 0.3 eV and La$_2$CuO$_4$ of 0.13 eV. Iron has a 1000K transition temperature and La$_2$CuO$_4$ of about 325K.
| {
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Chromatic aberration in lenses
We all know that concave lenses act as diverging lenses, but while searching for remedies for chromatic aberration, I observed a concave lens (in the picture see flint glass) acting as a converging lens, could someone please explain why?
| Actually, a lens does not necessarily behave as diverging or converging under all conditions.
In the picture you have posted, if you notice carefully, the incident ray is converging.
Usually, we consider diverging incident light rays as "real" objects (because the rays will emerge from a point in front of the lens) an... | {
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} |
Collision Thought Experiment I am thinking of a system of isolated, moving, and colliding particles in a frictionless box. The collisions were inelastic. My question is:
Supposing that the initial total momentum of the system is not zero. Since the collisions are inelastic, kinetic energy will not be conserved (convert... | In many-particle problems one has to distinguish the constants characterizing the movement of the system as a whole (it energy, the three components of the center-of-mass momentum, and the three components of the angular momentum), and the relative movement of the parts in the system.
So, indeed, the relative movement ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/539987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Interplay of energy between photon, electronic energy levels and kinetic energy I can understand that atoms have quantized energy levels for its electrons, but an atom's translational kinetic energy is continuous. As such, why is the absoprtion spectrum not continuous? That is to say, why can't the excess energy from t... | Because both momentum and energy have to be conserved when the atom absorbs the photon.
Suppose we have an atom with a mass $m$ and the energy difference between the initial and final levels is $E$. The photon energy is $hf$, so conservation of energy gives us:
$$ hf = E + \tfrac12 m v^2 \tag{1} $$
where $v$ is the spe... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/540103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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