Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What happens to the period of a pendulum if a spherical bob were to spin around the axis of the string? Consider a normal pendulum with a spherical bob oscillating back and forth. Would the period of the pendulum be longer, shorter or unchanged if the bob were to spin around the axis of the string that holds it?
| You say the pendulum has a bob "on a string". That's a complicated geometry to analyze, because even the spherical bob that is not spinning will have a complex motion - if you consider the angle of the string to the vertical $\theta_1$ and the angle of the bob to the vertical $\theta_2$, then these two can oscillate ei... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/359279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can a measurement partially "collapse" a wavefunction? Let's say I have a wavefunction $\Psi$ which can be decomposed into a sum of it's energy eigenstates:
$$ \Psi = a|1\rangle + b|3\rangle + c|8\rangle + d|10\rangle$$
Where, of course, $|a|^2 + |b|^2 + |c|^2 + |d|^2 =1 $.
And let's say I have a device which can measu... | A Positive Operator Valued Measurement (POVM) to describe your measurement could be given by elements $$M_{i}=\frac{1}{3+\min\{i,4\}}\sum_{|j-i|\leq 3} |j\rangle\langle j|$$ for $i=1,2,3,...$, which are positive semidefinite and sum up to the identity. Maybe in practice not all outcomes within your $\pm 3$ uncertainty ... | {
"language": "en",
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Do fields describing different particles always commute? Is it true that field operators describing different particles (for example a scalar field operator $\phi (x) $ and a spinor field operator $\psi (x) $) always commute (i.e. $ [\phi (x), \psi (y) ]=0, \forall x,y $) in interacting theory?
Or is it true only at eq... | No.
In full generality, the super-commutator of two fundamental fields is identical to the Dirac bracket of the corresponding classical variables (modulo the standard obstructions). If the system is unconstrained, then the Dirac bracket agrees with the Poisson bracket, which means that two independent fields super-comm... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/359610",
"timestamp": "2023-03-29T00:00:00",
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Question about Lorentz scalar I have a simple question about Lorentz Scalars.
In my course they are introduced like that.
$\phi$ is a scalar of Lorentz if it follows the following property :
A function $\phi$ is a scalar of Lorentz if it follows the following rules :
$\phi(x)=\phi'(x')$ and $\phi'=\phi$
But what would ... | Lorentz scalars are a subset of Lorentz invariant quantities. A Lorentz scalar is a scalar that is invariant under Lorentz transformation. For example, the dot product of a four-vector with itself is a Lorentz scalar. The 4-velocity is defined as:
$$\textbf{U}=\gamma(c,v)$$
So the dot product of 4-velocity with itself ... | {
"language": "en",
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Wick contraction corresponding to a connected diagram in $\phi^4$-theory to second order I am trying to understand the diagrams that comes from a two-point correlation function, $$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle$$, in $\phi^4$-theory. The zeroth order contribution, i.e. $\lambda^0$, is simply $D_F(x-y)... | Although the third $\lambda^2$ diagram is a little bit unusual in that there are three lines going between the two vertices, the rule continues to apply that there is a factor of $D_F$ for each line, meaning that the third $\lambda^2$ term is
$$-P\frac{\lambda^2}{4!4!}\int d^4z d^4w D_F(x-z)D_F(y-w)[D_F(z-w)]^3\ .$$
(I... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are the real life examples of Double Dirac-Delta Potential barrier/well?
My question is do we see in nature any potential which is close to Double Delta Potential barrier/well? If yes then which are those?
Thanks in advance.
| This is often used as a simplified model for a diatomic molecule. Each $\delta$ represents an atom, and the distance between the wells is the nuclear separation.
The solutions can be identified with the so-called bonding and anti-bonding wave functions. The strength parameter for the potential is often set by conside... | {
"language": "en",
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If I say time is the fourth dimension am I wrong? As far as I know the prevailing view is that time is the fourth dimension, but I've read there is also a spatial fourth dimension and even higher spatial dimensions after that so I hesitate to say that time is the fourth dimension. So, if I say time is the fourth dimens... | You can invent any number of mathematical spaces. There's no reason why time can't be the first dimension of the one you invent, nor why it couldn't be the seventh or that you just have no time dimensions, and infinite other dimensions.
All that matters for whether it makes physical sense to do so or not is whether you... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/360621",
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Planetary motion: with a different nature of potential. Condition for circular orbit
Consider a particle moving in the potential $U (r)= -A/r^n$, where $A>0$. What are the values of $n$ which admit stable circular orbits?
I tried to solve by putting $dr/dt=0$ in the total energy equation $E= T + U_\mathrm{eff}$, but ... | I'm not sure where you got this idea:
when $n=1$, where circular orbits are possible, $U_\mathrm{eff}=L^2/2mr^2+ U (r)$ does not have a minimum since it varies with $1/r$.
Here, have a look at that function:
At small $r$, the $+1/r^2$ dominates and the function is positive and monotonously decreasing. At large $r$, ... | {
"language": "en",
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Terminology confusion - "particle" I am confused about the word "particle" being used in academic contexts. Some professors at my university are adamant on the fact that particles do not exist, and only fields, as per QFT. One of them even showed me a citation from one of Julian Schwinger's QM books where he himself st... | This is an ontological question. Different people may express different point of views on the subject. There is some truth to both particles and field being fundamental.
In a sense all our experiments involve particles. We accelerate and collide hadrons, leptons, and are generally interested in the particles that they ... | {
"language": "en",
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Is energy $E$ in Schrödinger equation an observable/ Can $E$ be measured? Take this quantum approach to estimate mean energy of a molecule:
$$\langle\psi|H|\psi\rangle=\overline E$$
Question:
Is $E$ an observable? How we can compare it to an experimental value? i.e how to experimentally measure it and what are the stat... | Absolute energy is observable since it is the source of gravity. In practice this is hard to do but its observable status is indisputable. The Schrödinger energy is not absolute but relative to the rest energy of the particles that make up the system. Relative energy is observable by observing the products of a transit... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/361132",
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Do gravitational sources move along ‘geodesics’? Assume we have a system of say two bodies which are orbiting each other. Now assume that we wish to find an equation of the orbits of the two gravitational sources. Do they follow a ‘geodesical’ path, if we assume that the sources may or may not be singularities, which i... | I think this is largely a matter of terminology.
If we take for example a test particle moving in a Schwarzschild metric then we can calculate the geodesics in the usual way. However an real particle has a non-zero mass (or energy) and therefore it perturbs the metric. So the particle is not moving in a Schwarzschild m... | {
"language": "en",
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How exactly does vapor pressure relate to saturation pressure? I'm confused as to what exactly is vapor pressure and saturation pressure. From what I understand, vapor pressure is just the equilibrium pressure of a vapor above a liquid at some temperature. Is this not also the definition of the saturation pressure (ie.... | You can have vapour when there is no liquid present and that vapour would exert a vapour pressure.
If however you have liquid and vapour present in dynamic equilibrium with one another then the pressure exerted by the vapour is the saturated vapour pressure.
So start off with a container with only vapour in it.
The... | {
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Electric field charged disc and L'Hôpital's rule I have been looking at the electric field of a charged disk and have a question about the use of l'Hopital's rule for the limiting case of electric field at points along the axis $z\gg$ disc radius $R$.
$$E = \frac {q}{2\pi\epsilon R^2} \left(1 - \frac {z}{\sqrt{z^2+R^... | Update as the result of a comment from @garyp this time using L'Hôpital's rule.
$$E = \frac{q}{2 \pi \epsilon}\frac{(z^2+R^2)^{\frac 12}-z}{R^2 (z^2+R^2)^{\frac 12}}$$
Now differentiate twice with respect to $R$ the numerator and the denominator individually to get something like
$$ \frac {(z^2+R^2)^{-\frac 12} + R(...... | {
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Distribution of loss in a transmission line to minimize power dissipation This post will ask how to distribute loss in a transmission line so that the line has a known total loss, while dissipating the least amount of power.
We'll refer to "gain" of a transmission line, but we're thinking of the case where the line is ... | The solution is to take $\ln g(x)$ to be equal to a delta function concentrated at the point where $A(x)^2$ is minimal.
| {
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Use of negative frequency for the sake of simplifying mathematics? How can we use the idea of negative frequency for the sake of simplifying mathematics if negative frequency does not exist (to my knowledge) in nature ? For example, when plotting the spectra of a Fourier series.
| The only time I’ve seen negative frequency used is in Quantum field theory where it is proportional to the energy of a particle (i.e. $E=\hbar\omega$) In QFT, Feynman interpreted the negative frequency/energy results of the Klein-Gordon equation (used to find the field for a relativistic particle) as corresponding to a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/362719",
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Physical explanation of Joule heating The heat $Q$ generated in a wire, for a current $I$ flowing through a wire of a given resistance $R$, for a time $t$ is given by $Q=\mathscr{k}I^2Rt$ where $\mathscr{k}$ is the proportionality constant. For a given wire the resistance R is fixed. Is it possible to explain physicall... | Joule heating occurs when the electrons carrying current in a wire lose energy to the metal atoms of the lattice. If there is a current $I$ being driven by a potential difference $V$, that power is $P=VI$.
For a linear resistor, by Ohms law, $I=V/R$ captures how much of the electrical energy is "lost" to heating, thr... | {
"language": "en",
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In physics sometimes we find energy that is negative. What does the negative sign indicate? Sometimes we see energy that is negative, for example, the energy of an electron in orbit. We know energy is something that can do something. In this view does negative energy mean something opposite someway?
| Negative sign simply means that the system is releasing energy. For instance, take the case of gravitational force. The potential energy function of gravity is,
$$U_g(r)=-G\frac{m_1m_2}{r}; U_g(\infty)=0$$
Gravitational potential energy is described as the work done in bringing a body of mass $m_2$ towards $m_1$ from $... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/363216",
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Are there elementary forces acting in directions than 0 or 90 relative to their fields? Some forces act in the same direction as their field orientation, like a gravitation. Other forces, for instance the force acting on a charged particle in a magnetic field, are perpendicular to that their field orientation.
Are ther... | The question is not well adapted to modern treatment of general relativity and quantum theory. In these theories, the notion of "force" is not very useful. Also, it supposes that the fields are characterized by a vector. This is not always the case. For example, general relativity uses a tensor field to describe space ... | {
"language": "en",
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How does one calculate frictional force for an object sliding down a wall? Let's say that there is a book sliding down a vertical wall such that the only fundamental force acting on it is gravity. I want to say that there is a frictional force slowing the book down; however, I can't find a normal force acting on it, so... | If there is no normal force to the surface there is no friction. The book will just fall by gravity.
In real life if starting pressed on the wall and released it may hit a small extrusion from the wall and get a rotation by scattering off it.
| {
"language": "en",
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What determines the energy of a photon emitted by an electron when it changes it's energy level? A nuetral helium atom in is in the excited electronic state, with one electron in the (2p) level and the other in the (3d) level. The atom is initally in a region with zero applied magnetic field.
(a) Can the electron in th... | The n is called principle quantum number because it gives the basic energy level. l and m
give corrections on this basic energy level, and this allows for the possibility of many more transitions, as long as quantum numbers are conserved.
If you look at the hydrogen energy levels at extremely high resolution, you do... | {
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With a very intense light on a black object, will it reflect? I was wondering about the nature of object's colour. I know that an object get its colour from the absorption of visible electromagnetic radiation, reflecting all the other wavelength. But if we take the case of a black object that absorbs every visible ligh... | As I understand your question, you are asking about an ideal black body. Keep in mind that such a thing does not exist in nature. But as long as we know we are talking about an ideal, first, the body will never reflect any light. What will happen is that the intense light falling on it will drive up the temperature ... | {
"language": "en",
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How torque about every point on Axis is same? I read somewhere that torque about every point on an axis is same. But I am really confused about how this can be. Please help me and give an satisfactory answer
| I believe you are referring to the following: if the nett resultant of a system forces on a body is zero, then we can say that the moment of that system is independent of the point about which the moment is calculated. In symbols, suppose we have a system of forces $\vec{F}_i$ acting at positions $\vec{r}_i$, relative ... | {
"language": "en",
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Co and contravariant: tensors or components? I am learning Special Relativity and have a question: given a four vector $\vec{x}$ whose contravariant components are $x^\mu$, do the covariant components $x_\mu = g_{\mu\nu}x^\nu$ make reference to a different physical/geometrical object other than $\vec{x}$?
I mean, for ... | Since we are dealing with a finite dimensional metric space, it's fine to just think of a field $\mathbf v$ and two bases $\mathbf e_\mu$ and $\mathbf e^\nu$ such that $\mathbf e_\mu \cdot \mathbf e^\nu = \delta_\mu^\nu$. Then $v^\mu$ and $v_\nu$ are just the coefficients for $\mathbf v$ in the two bases: $\mathbf v = ... | {
"language": "en",
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"source": "stackexchange",
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What is the form of the $n$-th order term of the perturbation series of an eigenvalue? Suppose I have a matrix given by a sum $A=D+\epsilon B$, where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as power series in $\epsilon$. The leading order is just the eigenvalues of $D$, the first corr... | The answer can be found in Kato's book Perturbation theory for linear operators. I will use Kato's notation. In fact I will answer a more general question where you have an operator which depends analytically on a parameter $x$ (your $\epsilon$). Let such operator be $T(x)$ and let
$$
T(x) = \sum_{n=0}^{\infty} x ^n ... | {
"language": "en",
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What if the Sun became a black hole? Upon talking to someone about the concepts of black holes, a question arised that I did not know the answer to.
If the Sun became a black hole, but the mass remained the same as it is now, the Earth would orbit in the same manner that it currently does because of the reason that the... | Assuming nothing else changed about the body then the change will have no effect on the curvature of spacetime outside the radius that the sun had had previously.
Inside the sun’s radius, however, is a different story. As it currently stands, the curvature of spacetime increases as one approaches the sun’s radius and t... | {
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Why is the Lorentz transformation's time transformation not just the time dilation? In Taylor's Classical Mechanics text, he derives the Lorentz transformation from length contraction which, in turn, uses time dilation. But doesn't the use of length contraction necessitate that the time transformation is just the time ... | If the time coordinate transformation was given simply by
$$
t' = \gamma t,
$$
all events simultaneous in the $x,t$-frame would be simultaneous also in the $x',t'$-frame. That would be in contradiction to relativity of simultaneity, which is an important part of special relativity.
For example, imagine a spherical expa... | {
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Motional EMF and the flux rule contradiction I have a metallic rod which is being rotated in a constant magnetic field. The EMF is produced in it as per motional EMF and can explained using the Lorentz force. But how can we explain the production of EMF in it using Faraday's flux rule. In this case the rod is in consta... |
... how can we explain the production of EMF in it using Faraday's flux rule? In this case the rod is in constant magnetic field and even though rod is rotating, the flux is not changing.
Faraday’s law is derived from the observation of changing electric or magnetic fields like in transformers.
The induced electrom... | {
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Kepler's genius, How? I have a very simple question. How Kepler knew that orbits are elliptical, say I was living in his time. How would Kepler explain that the orbits are elliptical (since none of his 3 laws explain why orbits are elliptical; I assume he must have had other reasons to believe why orbits are elliptical... | An ellipse was the only thing which fitted the data (without adding the circles within circles special fixes needed for Ptolemy's epicycles)
I suppose his (Kepler's) genius was in trying different mathematical shapes to fit the data rather than arguing from Divine Insight or Ancient Greek authority that orbits must be ... | {
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How to find the direction of velocity of a reference frame where two events are simultaneous in case of a space-like interval Suppose in a inertial reference frame $S$, an event $A$ occurs at $(ct_A, x_A, y_A, z_A)$ and event $B$ occurs at $(ct_B, x_B, y_B, z_B)$.
Now the invariant interval of these two events is,
$$I... | Here is an image which should correspond to the graphical solution of dmckee:
You have to proceed as follows:
*
*Put event A into the coordinate origin (by coordinate displacement).
*Then connect both events by a line x' which is the space axis of the researched new reference frame
*Then draw accordingly the tim... | {
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How does a Higgs triplet transform under $SU(2)_L \times U(1)_Y$ when written as a $2\times 2$ matrix? I recently learned that a Higgs triplet can be written as a $2 \times 2$ matrix:
\begin{equation}
\Delta=\begin{pmatrix} \frac{\Delta^{+}}{\sqrt{2}} & \Delta^{++} \\
\Delta^0 & - \frac{\Delta^{+}}{\sqrt{2}}\end{pmatri... | Indeed, a triplet $\vec{\phi}$ transforming under the triplet representation matrices T can be doted to a Pauli vector to yield a formal adjoint, so a traceless 2×2 matrix $$\Phi=\tfrac{1}{2}\vec{\phi}\cdot \vec{\tau}~,$$ transforming as the doublet representation, conjugately on both sides. This is dubbed adjoint acti... | {
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Books on non-perturbative phenomena in quantum field theory I am looking for any good places (preferably textbooks) to study about introductory non-perturbative phenomena in Quantum field theory.
Any suggestion will be appreciated.
| The OP did not explain what "nonperturbative" means, which can vary. So I will go with beyond perturbation theory, i.e., not just talking of correlations of a QFT as formal power series in $\hbar$ or the renormalized coupling constant. In that case, the literature on constructive quantum field theory deserves to be men... | {
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Does twisting a wire heat it? I was playing with a key chain loop in a (very boring) chemistry class and then I straightened the loop into a wire keeping two end of the loop (now wire) curved so as to easily twist it. It was more or less a S shaped structure of metal with a longer straight part in the middle of S.
On ... | By twisting and untwisting the wire, you did work (in the physics-specific sense of the word) on the wire. Effectively, by exerting a force on the wire, you transfer energy into it. That energy has to go somewhere, and in this case it ended up as heat.
Some of your body heat was probably transferred into the wire in ... | {
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Linear approximation of time dilation. In what point is it? I have watched a video about linear approximation and there was an example, exactly here: https://www.youtube.com/watch?v=BSAA0akmPEU&feature=youtu.be&list=PL590CCC2BC5AF3BC1&t=32m50s about linear approximation of time dilation. I have started to count and thi... | You effectively killed $f(t,v)$ when you set
$$\frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \approx 1$$
The result can be obtained much simpler by using known approximation rules
$$\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}} \approx\dfrac{1}{1-\dfrac{v^2}{2c^2}}\approx 1+\dfrac{v^2}{2c^2}$$
| {
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What is the Bandgap energy of Rubidium? Could anyone please tell me the bandgap energy for alkali metals like rubidium and cesium?
| The optical spectra of the alkali metals show a kind of gap: interband transitions have an onset. In first approximation this is explained by the empty-lattice band structure where the periodicity makes possible vertical transitions from the conduction band from states with a wave vector smaller than the Fermi vector.
... | {
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What is the equation of relative motion for two objects moving in straight lines? If two objects, A and B, are moving in the same direction along straight lines in a plane, they might be diverging, converging or moving in parallel.
If we wish to describe B's motion with respect to A, what is the equation of motion?
For... | There are two objects that are moving in a straight line. The parametric equation for the first may be:
$ \vec x = \vec a + t \vec b $
The equation for the second may be:
$ \vec y = \vec p + t \vec q $
The relative position from $ \vec y $ watched from the point $ \vec x $ is:
$$ \bbox[5px,border:2px solid red]
{ \vec ... | {
"language": "en",
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Why is flux at an interface purely diffusion? In many textbooks, the flux at the point of interface of two phases/regions is given through Fick's first law, with purely diffusive flux, even when there can be bulk convection in both phases/regions.
Essentially,
$$
N_{A,y}\vert_{\xi=0}=N_{A,y}\vert_{y=\delta}=-D_{AB}\l... | Interface is by definition a separating surface across which there cannot be bulk flow. So if follow an interface (Lagrangian approach, and not Eulerian) the only way mass can be transferred across the interface is by diffusion.
| {
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Heat to work or thermal energy to work? A system consists of different forms of energy like thermal energy, mechanical energy, chemical energy, nuclear energy etc. If these energies are to be transferred to another system (call it system 2), it can either be done as heat or work (or mass but here I take system approach... | The 2nd law implies that heat can't be completely converted to work using a cyclic process. Obviously, heat can be converted to work if the process does not have to be cyclic. An example is isothermal reversible expansion of an ideal gas.
| {
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Fock space with mixed anti-commutation/commutation relations? Let's say we have two modes, with the following labeling of occupation number states:
$ \lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix} $
An example of (what I assume to be) fermionic creation operators for the two modes is
$\hat... | The operators $b_i$ defined by the OP correspond to the algebra of hardcore bosons, that is, bosons that cannot be put at the same place.
Hardcore bosons correspond to the limit of infinite interaction ($U\to\infty$) of the Bose-Hubbard model
$$
H=-t\sum_{\langle i,j\rangle}b^\dagger_i b_j-\mu\sum_i n_i+\frac U2 \sum_i... | {
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How to represent a axisymmetric, stationary metric in a coordinate independent way? A classic example of a stationary, axisymmetric metric in GR is the Kerr metric. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ it is obvious that the metric is independent of $t,\phi$ and so is stationary and axisymmetric.
Now, oft... | A spacetime is said to be stationary if it has an (asimptotically) timelike Killing vector.
Similarly, if one has a Killing vector which has closed spacelike trajectories, then we get an ignorable coordinate, which corresponds to the axisymmetry.
In the example of the Kerr metric, the timelike Killing vector is $\frac{... | {
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Why melting temperature of polymers depends on the prior crystallization temperature? This graph should show that the melting temperature of a polymer rises when the crystallization temperature of a polymer was higher (the melting temperature of a 100% crystalline polymer should be equal to crystallization temperature)... | I think you have basically answered your own question. I will expand on it here:
The degree of crystallinity of a polymer will differ, depending on a number of factors. Polymers of higher crystallinity tend to have longer folding lengths.
In general, the longer the length of the individual 'crystals', the higher the me... | {
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Lorentz transform of force
If a particle of mass $m$ and velocity $v$ is moving due to a constant electric force what would the force be in the the frame where the particles velocity is 0?
To try and solve this I used the four force and did a Lorentz transform of the four momentum. However I got different answers in ... | You cannot just transform $d \bf p \rm/dt=q(\bf E + v \wedge B \rm )$, as it is not a tensorial equation. The tensorial form of this equation is
$$\frac {d p^\mu }{d\tau } = -\frac q mp^\lambda F_\lambda^{\; \mu} $$
The tensorial nature of this equation guarantees it is valid in any coordinate system. Turning back now... | {
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Why these patterns form in captured image while zooming?
This is a gif format video that shows zooming of an image of computer LCD screen which i captured using my mobile phone. You can see that some fringes are forming and disppearing and hence some patterns are forming while i zoon in or out. How will you explain th... | what you are seeing is a form of something called a moire' pattern which is created because as Bill Oertell points out, the spacing between adjacent pixels in the screen does not match that of the camera photographing the screen. You can learn more about moire' patterns on wikipedia.
| {
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induced charges inside a capacitor with two dielectrics consider that we have two dielectrics inside a capacitor as shown in the picture, let0s consider also that Q is the charge of the capacitor and d the distance between the two plates , the first dielectric occupy a surface of S/3 with a dielectric constant of er1 ... | You can consider the two dielectrics as two capacitors connected in parallel. Since the two are connected across the same potential difference and the distance between both the capacitors(upper and lower one) are same, E= V/d , the electric field in both the dielectrics is the same.
As the dielectric is neutral, the ch... | {
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Energy gap of $In_{0.53}Ga_{0.24}Al_{0.23}As_{0.77}$ Does anybody know how can I calculate the energy gap of $In_{0.53}Ga_{0.24}Al_{0.23}As_{0.77}$ ?
| The first place to look would be Akio Sasaki et al., "Energy Band Structure and Lattice Constant Chart of III-V Mixed Semiconductors, and AlGaSb/AlGaAsSb Semiconductor Lasers on GaSb Substrates", Japanese Journal of Applied Physics 19(9) 1695-1702 (1980). They provide the equations and parameters to calculate quaternar... | {
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What's the purpose of shorting the base and collector of a transistor in current mirrors? I often see this diagram of a current mirror (as shown below).
As far as I know, the purpose of a current mirror is the ensure that the collector current for both transistors are equal.
This can simply be achieved by making sure t... |
This can simply be achieved by making sure that their base-emitter
voltage is the same.
Not quite. The collector current can be written as
$$I_C = \beta_0\left(1 + \frac{V_{CB}}{V_A}\right)I_B$$
and so depends on the base current and the collector-base voltage (Early effect). Connecting the collector and base toge... | {
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How does friction increase energy of a system? I had this doubt while thinking through a question about centre of mass. Consider a system, consisting of a man standing on one end of a plank which rests on a frictionless surface. Now the man starts running towards the other end of the plank(friction is present between t... | Friction isn't the force doing work. Human bodies are much better than inanimate blocks, it's the man using his internal energy from his muscles to kick the plank behind, the friction will stick his leg to the plank while he kicks and prevent it from slipping behind while he is lifting the other leg, ideally work done ... | {
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Rotation in Higher Dimensions In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation.
In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom rotate, and if so does it rotate around a line, a plane, or a subspace of smaller number of dimensi... | In 2d, a rotation matrix has the form
$$
r(\theta)=\left(\begin{array}{cc}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta\end{array}\right):=
\left(\begin{array}{cc}
c(\theta)&-s(\theta)\\
s(\theta)&c(\theta)\end{array}\right)
$$
and rotates vector in a plane.
In 3d a rotation matrix can be written as a product
$$
r_{... | {
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Water molecules on the Sun? How does that work? I've been reading about water on the Sun. The water they talk about is supposedly in a gaseous state because the Sun is so hot. But I'm wondering how even that could exist. Wouldn't the extreme temperature of the Sun ($> 5000^\circ{\rm C}$ on the surface) split it into hy... | It's true that the temperature on the Sun will split water molecules very rapidly. However, the research you referred to discovered trace amounts of water on sunspots, not an atmosphere of water vapor (or worse: oceans)
The water molecules form then are broken down by the temperature very quickly. It is not that no wa... | {
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Why can the Klein-Gordon field be Fourier expanded in terms of ladder operators? Using the plane wave ansatz
$$\phi(x) = e^{ik_\mu x^\mu}$$
the solution to the Klein-Gordon equation $(\Box + m^2) \phi(x) =0$ can be written as a sum of solutions, since the equation is linear and the superposition principle holds, as
$$\... | Let us start with the ansatz (I'll assume mostly plus metric signature)
\begin{equation}
\hat\phi(x) = \int \frac{\mathrm{d}^3 \mathbf{k}}{(2\pi)^{3/2}}\left(\hat A_\mathbf{k} e^{i k\cdot x} + \hat B_\mathbf{k}e^{-ik\cdot x}\right)
\end{equation}
where $\hat A_\mathbf{k}$ and $\hat B_\mathbf{k}$ are some arbitrary op... | {
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What is so natural and inevitable about the expansion of the universe? The opening paragraph of the section the Introduction to Standard Big-Bang Model in the PDG review claims that
The observed expansion of the Universe is a natural (almost inevitable) result of any homogeneous and isotropic cosmological model based ... | A static universe is not viable in GR. It requires a precarious fine tune of the total mass density and the cosmological constant. Furthermore, this balance is unstable. Any deviation from it will either cause the universe to start an accelerated expansion, or start contracting towards a big crunch. This is why Einstei... | {
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Operation on Complex conjugate Why do we sandwich operators in quantum mechanics in such a way that the operator acts on the wavefunction and not on its complex conjugate?
| Take the kinetic energy operator and ground state wave function for a particle in a box with width of $L$ as an example (in 1D)
$$\hat{T}=-\frac{\hbar}{2m}\frac{d}{dx^2}$$
and
$$\psi=\sqrt{\frac{2}{L}}\sin\frac{\pi}{L}x$$
Obviously, $\hat{T}\psi$ means something and $\psi^{*}\hat{T}$ means something else.
Please look a... | {
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Why is this system isolated?
(Image updated for clearance)
Hello,
Assuming that there is no friction anywhere and both box 1 and ramp 2 start at rest, I was wondering why this is an isolated system in terms of momentum calculation. My professor approached calculating the velocities of the box and the ramp assuming tha... | I am sorry but I have had to write this as an answer as it is too long as a comment.
There is no external horizontal force acting on the system comprising the two blocks so for the system of two blocks the total horizontal momentum must stay constant and it is zero if the blocks start from rest.
There is no horizontal ... | {
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Physical meaning of gauge choice in electromagnetism In electromagnetism, it is often referred to gauges of the electromagnetic field, such as the radiation or Coulomb gauge. As far as I know, the definition of a gauge helps us to redefine the problem in terms of a vector potential and a scalar potential that, since we... | Physical observables in a gauge theory$^1$ are independent of gauge-fixing choices$^1$. Conversely, gauge-fixing choices are unphysical.
--
$^1$ Here we have applied a narrow definition of a gauge theory where gauge symmetry represents a redundant description of a physical system, cf. e.g. this Phys.SE question. In oth... | {
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Quantum State Representation with Commuting Operators Let $[A,B]=0$. Then, we can find a set of eigenvectors $\{|a_n,b_n\rangle\}$ common to both $A$ and $B$. According to this, and my own understanding, it makes sense to write an arbitrary quantum state as
$$\tag{1}|\Psi\rangle=\sum_n \sum_i c_n^i |a_n,b_n,i\rangle,$$... | If $A$ and $B$ are commuting self adjoint operators (more precisely operators with pure point spectrum whose spectral measures commute), then the Hilbert space is decomposed into a direct orthogonal sum of common eigenspaces, where $A$ and $B$ are trivially represented as multiplicative operators: $aI$ and $bI$.
The cr... | {
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For all practical purposes can light be bent (without the help of gravity) or just reflected? For example, if a single beam of light was directed directly at the tangent of a semi circular mirror, would it be considered bending or redirecting many times to form a near circular pattern? When I say bend I mean in a curve... | The phenomenon you're describing, of light being bent, is observed when light passes through a medium with progressively increasing, or decreasing refractive indexes.
You actually observe it when you see a mirage. When it's very hot, the temperature being progressively higher as you approach the ground, the refractive ... | {
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What if cosmological constant was zero? Physicists always ask why the cosmological constant is not exactly zero!
I would ask here, what if cosmological constant was zero? The universe wouldn't expand and matter would exert gravitational force and shrink the universe into a big crunch!
So, why physicists want the consta... | The universe can expand just fine without a cosmological constant. In fact, it was this fact that made Einstein originally add it to the equations when he was making his first cosmological model: he did not know space-time is expanding, so he used a constant as an allowed but ugly fudge-factor to make it static in his ... | {
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Reason for 6π factor in Stokes' law According to Stoke's law, the retarding force acting on a body falling in a viscous medium is given by $$F=kηrv$$ where $k=6π$.
As far as I know, the $6π$ factor is determined experimentally. In that case, how is writing exactly $6π$ correct since we obviously cannot experimentally ... | It is not determined experimentally, it is an analytical result. It is verified experimentally.
As @Mick described it is possible to derive the velocity and pressure field of a flow around a sphere in the Stokes flow limit for small Reynolds numbers from the Navier-Stokes equations if the flow is further assumed to be ... | {
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Why is the helium atom wavefunction a product of the two 1s wavefunctions? From S.M. Blinder's Introduction to QM book, p. 116:
In seeking an approximation to the ground state, we might first work out the solution in the absence of the $1/r_{12}$ term. In the Schrodinger equation thus simplified, we can separate the v... | Because if
$$\require{cancel}\hat{H}=\hat{H}_{1}+\hat{H}_{2}=\left[-\frac{\hbar^{2}}{2m}\nabla_{1}^{2}-\frac{e^{2}}{4\pi\varepsilon_{0}}\frac{1}{r_{1}}\right]+\left[-\frac{\hbar^{2}}{2m}\nabla_{2}^{2}-\frac{e^{2}}{4\pi\varepsilon_{0}}\frac{1}{r_{2}}\right]+\cancel{\color{red}{\frac{e^{2}}{4\pi\varepsilon_{0}}\frac{1}{r... | {
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Matter effects in neutrino oscillation The neutrino oscillation probability in matter is given as:
where
Now what I do not understand is that "As the energy increases, the probability
of oscillation within the sun through the matter effect increases, so the survival probability decreases". I have read this (page 2... | I think the crucial point here is that in the sun you have two variables, the neutrino energy $E$ and the electron number density $N_e$. These two enter the matter potential. In order to run into the MSW resonance (and therefore effectively oscillate into muon neutrinos) these two variables have to combine in a way tha... | {
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Wave function in tensor product of Hilbert spaces If I had the wave function
$$\Psi\equiv\psi(r,\theta,\phi)\otimes\chi \in \mathscr{L}^2(\mathbb{R}^3)\otimes\mathbb{C}^{2S+1},$$
where $S$ is the spin of the state, is it correct to normalize the spin part of $\Psi$, namely $\chi$, regarding the spatial parameters $(r,\... | The norm squared of your wavefunction is
$$\left<\Psi\big|\Psi\right>=\left<\psi\big|\psi\right>\left<\chi\big|\chi\right>$$
and this should be $1$. In particular, you can normalize both $\psi$ and $\chi$ separately and your $\Psi$ will be also normalized.
| {
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Strange interference pattern of light on top of tower, pattern was seen on air. What was it? I was just looking out of window at night when I saw a tower with a light on top. It had a red light.
When I looked at it through my curtains with net on, I saw an interference fringe, one is the main light itself and band of ... | Regarding the "no screen" part: Your retina can be a pretty good "screen" in this case.
| {
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Will movement of charged gaseous particles constitute electrical current? As electrons flow, they constitute current.
So if we manage to supply an extra electron to each of the gaseous molecules present in an enclosure and make the gaseous molecules travel in that enclosure, will that constitute current?
Is this kind o... | Any spatial movement of charge constitutes an electrical current. Thus also the movement of gas ions, whether positively or negatively charged, produces an electrical current. Currents produced by ionized gas molecules are very common. For example in gas discharge tubes used for illumination.
| {
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Is a canonical transformation equivalent to a transformation that preserves volume and orientation? We have seen the reverse statement: Lioville's Theorem states that canonical transformations preserve volume (and orientation as well). Is the reverse true?
If I demand a map from the phase space to the phase space to pr... | *
*Counterexample: The transformation
$$Q^1~=~2q^1 ,\qquad P_1~=~p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~p_2 $$
preserves phase space volume & orientation, but is not a symplectomorphism.$^1$
*For 2D phase space, the canonical phase space volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n}$$
is the symplectic ... | {
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Extreme life - energy source for living tens of kilometers underground? Living cells were found up to at least 12 miles underground (article), and in other extreme places (BBC survey article), for which beside the problem of just surviving in such extreme conditions, a basic physics thermodynamical question is: what en... | One article to which your article points to is this one they state:
How would these microbes have survived? Counterintuitively, the exceedingly high pressure in a miles-deep habitat — in the neighborhood of 5,000 times the pressure exerted by the atmosphere at sea level — could have helped. High pressures actually can... | {
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What are the Feynman diagrams for neutrino oscillations? Which Feynman diagrams are at the basis of neutrino oscillations? I find no clear explanation via Google.
| There are none, and the question isn't really even sensible.
Neutrino oscillation is not mediated by force carrying particles any more than any other change of quantum basis is.
This is similar to asking "What are the Feynman diagrams for the uncertainty principle?". In both cases we're talking about what happens when ... | {
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"url": "https://physics.stackexchange.com/questions/375905",
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"source": "stackexchange",
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Venus, Earth and Mars Magnetic fields Why does Earth have a magnetic field, while it appears that Venus and Mars have none or very little?
| The most accepted theory for the existence of a magnetic field in our planet is that the field is generated by electric currents due to the motion of convection currents of molten iron in the Earth's outer core driven by heat escaping from the core, a natural process called a geodynamo.
As you can see here https://nssd... | {
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Do bubble universes in eternal inflation have edges? I've been having a hard time understanding bubble universes in eternal inflation. So they are just finite regions of space where inflation has stopped and Hubble expansion has taken over? I just can't understand how a finite universe can work with out current underst... | The bubbles do have edges, because bubble collisions have been sought after (thus far unsuccessfully, I believe) as evidence of field-based inflation, and the collision of objects without any sort of edge would be indeterminate. The terminology varies with the model: Guth, who's usually credited with originating infla... | {
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What are the initial conditions associated with solving the geodesic equation in General Relativity? Can we say that initial conditions for solving the geodesic equation in general relativity be intial velocity of a particle?
| The most practical way to solve the geodesic equations is using a dynamical systems approach, where one re-writes the 2nd-order ODEs as a coupled system of first-order ODEs. Let the geodesic equation be given as:
$\ddot{x}^a + \Gamma^{a}_{bc} \dot{x}^{b} \dot{x}^{c} = 0$.
Then, this can be written as a system of first... | {
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What is the relation of the mass of vector field in bulk and the scaling dimension of current operator in CFT? In AdS/CFT correspondence, we know that,
$$m^2=\Delta(\Delta-d)$$
where m is the mass of a scalar field and $\Delta$ is the scaling dimension of the dual operator in CFT. What about the relation of the mass o... | A vector field $A_{\mu}$ with spin-1 in the bulk has a dual spin-1 operator on the field theory, let's call it $J_{\mu}$.
If the vector field is massless, then $\Delta_{J}= d-1$ and $J_{\mu}$ is a conserved current.
In any other case, the current is not conserved.
The relation you wanted
$$m^2 = R^2(\Delta − 1)(\Del... | {
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What is the actual reason for the effects of fictitious forces? Suppose a person is standing in a bus moving with constant velocity. Assume that static friction between his feet and bus surface is very low (you could assume he is on roller skates) and we are observing this situation from the frame of reference of the b... | The simple answer is the the person on the roller skates is not accelerating and no force is acting on them.
Suppose I am standing on the pavement outside the bus when the bus starts to move. Clearly I am not accelerating because I am just standing there. If I watch the roller skater through the bus windows then when t... | {
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What is a destructive measurement? What are destructive measurements or incomplete measurements, and what is the (conceptual) difference between them and a usual measurement?
I read somewhere that destructive measurements consume their qubit.
reference: measurement calculus
p.6 we simplified equations 2 and 3 to equ... | *
*Destructive measurements are processes that completely destroy the system they are measuring, and they are primarily used when detecting light. As an example, to detect the polarization of a photon you can pass it through a polarizing beam splitter and put detectors on either output port: you get complete informati... | {
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Why can large objects at greater distance be treated as a point particle? Why can large objects at greater distance be treated as a point particle?
"The bodies of our solar system are so far apart compared with their diameters that they can be treated as particles to an excellent approximation."
was the statement wr... | You can expand the gravitational field of an arbitrarily shaped body systematically in something called the multipole expansion (this is also done in electromagnetism). The lowest order term will be the slowest to decrease with distance and that lowest order term is just that of a point particle.
For instance say you h... | {
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Where does the fine structure constant come from? I have this question: Where does the fine structure constant come from? Is it derived? Is it assumed? I will be most thankful if you will also include other detailed info that you think is also good to know, or just suggest a reading on it.
| The fine structure constant is one of the fundamental constants in nature, just like the speed of light or Planck's constant. It is there, and that's all we know for sure. We don't really have a compelling theory on its origin, nor a mechanism that explains its value.
In short, the fine structure constant is not a deri... | {
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Why does Griffiths's book say that there can be no surface current since this would require an infinite electric field for an incident wave? In sec. 9.4.2 Griffiths shows the well known boundary conditions for E and B fields, one of them is this:
$$\frac{1}{\mu_{1}}\textbf{B}_{1}^{\parallel}-\frac{1}{\mu_{2}}\textbf{B}... | I think this is so because for finite conductivity and for ohmic conductors, J=$\sigma$E would require that the current density be parallel electric field. Since for conductors, electric field is perpendicular to the surface, so J (current) would also be normal to surface. But the boundary condition n $\times$ H = K re... | {
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Small inter nuclear separation limit for Diatomic molecule Let’s take the a simple $H_2^+$ molecule, where there is only electron which is $r_a$ away from the first proton and $r_b$ away from the other one.
Let’s call the separation between the two protons $R$.
As $R\rightarrow \infty$, the electron will stick to one o... | Think physically about the shape of the odd state. It has a nodal plane at the midpoint of the line connecting the two nuclei; this is the only node to the wave function. As the nuclei become coincident, that nodal plane ends up passing through the combined nucleus; since this is the only node, the resulting wave fun... | {
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Can humans hear transverse sound waves? According to Wikipedia and validated by a clever thought experiment here, sound waves can be transverse as well as longitudinal, if they're propagating through a solid. Consider my mind blown and my curiosity piqued. However, is this a phenomenon we can hear? And is there any rea... | sound transmission through a solid can occur by either compressive waves or shear (transverse) waves because a solid is capable of sustaining shear stresses. sound transmission through air is exclusively by compression waves because air cannot sustain shear stresses.
You can certainly hear both sorts of waves by pressi... | {
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Experiment on friction coefficient Here you can see the results of the experiment about a friction coefficient:
The mean of the friction coefficient becomes 0.262 but when I do a linear regression in the form of y=mx the slope is 0.31. Shouldn't it be the same? I used $F_N$ as x values and $F_D$ (friction force) as y ... | Expounding on sammy's comment, when you divide $F_D$ by $F_N$, since $F_D$ only is measured to one significant figure you can only report $\mu_S$ to one significant figure.
Value of $\mu_S$ from calculating the mean of all experiments: 0.262 = 0.3
Value of $\mu_S$ from linear regression: 0.311 = 0.3
So for your case ... | {
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Which vacuum is the Universe really in? There ate two types of vacuum of the Standard model-the vacuum of the Higgs potential and that of the vacuum of the Yang-Mills fields labelled by the Chern-Simons number. See the figure 5 here.
The Lagrangian of the Standard electroweak theory contains both the gauge fields and H... | A "pure" Higgs theory (i.e. containing only the Higgs field) has a vacuum labeled by the VEV of the Higgs field, a pure YM theory has a vacuum labeled by the $\theta$-angle, and the combined theory, i.e. a YM theory with a Higgs field as we find it in the standard model, has a vacuum labeled by both the Higgs VEV and t... | {
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Divergence of Electric Field Due to a Point Charge I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law.
$$ \nabla \cdot \mathbf E = \frac {\rho}{\epsilon_0}.$$
That's fine and all, but I run into what I believe to be a co... | What you want to compute is essentially $$\vec\nabla \,\frac{\vec x}{\left|\vec x\right|^3}$$
at the origin. Of course, that doesn't exist as a function since the field is singular. On the other hand, you have already shown that it vanishes everywhere else.
So you need to interpret the expression in a weak sense, i.e.... | {
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Does radiation cause a change in temperature? If yes, then is there a limit to the temperature decrease? If no, then can the body which radiates heat attain an absolute zero temperature?
| Radiation obviously causes a change in temperature. Sit in front of a fire for a while.
The upper limit of the decrease in temperature is the temperature of the cold source which is acting as the counterpart.
A body which only radiates heat could only attain absolute zero if heat were being pumped out of it somehow and... | {
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Weird sound coming from thermos flask I had a thermos flask that I had never used since I had bought it. Today, I decided to take it out and use it for storing hot water, so that I don't have to heat water every time I feel thirsty.
An interesting incident occurred. I heated water to near about boiling temperature, and... | What is probably taking place is this: when something hot is inside the thermos, the air trapped between the outside of the thermos liner and the inside of the plastic sleeve surrounding it begins expanding due to (slow) heat loss from the thermos liner. where the liner meets the sleeve around the upper end of the ass... | {
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How to measure a static electric field? I looked up google but didn't find any design for measuring electric field that doesn't vary with time.
My own idea is to use two parallel plates (like a capacitor but without the dielectric). In an electric field E a potential difference V = Ed (d is separation between the plat... | You can't measure the voltage between two plates in a static electric field because the field will also exist within the wires of you meter. You need to rotate the plane of the plates and use slip rings to feed the voltage to an AC meter. As I recall, there is often a vertical electric field near the surface of the e... | {
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Can we consider non-inertial frames in Lagrangian dynamics formulated through d'Alembert's principle? When we derive Euler-Lagrange equations from an action principle, there is no explicit mention of a reference frame, so I assumed that the formulation is correct even in non-inertial frames (is this true?).
But I have ... | Yes, you can safely use non-inertial reference frames provided the reactive forces due to contraints are ideal and provided you include all inertial forces in the set of non-reactive forces.
| {
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Can we prepare a superposition of two many body state efficiently using quantum circuit? Let's say we have two quantum many body states $|\psi_1\rangle$ and $|\psi_2\rangle$(or equivalently, two quantum circuit $U_1$ and $U_2$ ), also an ancilla qubit $\alpha|0\rangle+\beta|1\rangle$. The goal is to prepare a state $\a... | Yes.
Given a circuit for a unitary $U$ on $\mathbb C^d$, we can always build a circuit for a controlled-$U$ gate acting on $\mathbb C^2\otimes C^d$, this is, a gate acting as
\begin{align}
|0\rangle|x\rangle&\mapsto |0\rangle |x\rangle\\
|1\rangle|x\rangle&\mapsto |1\rangle (U| x\rangle)
\end{align}
(i.e., $U$ acts on... | {
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Finding time period of SHM from equation of displacement Say for example I've got the equation of a SHM as: $$x = A \cos (\omega t + \phi)$$ where $A$ is the amplitude.
How do I find the time period of this motion?
I tried by finding the second order differential of the given equation.
$a = \dfrac {d^2 x}{d t^2} = - A ... | There is a very simple mistake in your math. Notice $A$
is part of $x$, it is factored so you'll get to $\omega=\omega$ again. If you want to find a meaning to $\omega T = 2\pi$, consider the fact that $\cos$ (or $\sin$) are periodic functions with period $2\pi$. Hence, every time you have a time difference such that $... | {
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How do anomalies work in the causal formulation of QFT? In the Epstein-Glaser formulation of a QFT, the would-be divergences are taken care of by meticulously splitting the distributions that appear in the construction of the $S$-matrix (or correlation functions). As a result, there are no divergences anywhere and the ... | Anomalies may (or may not) appear as obstructions in the proof of the Ward-Takahashi identities, which provide gauge invariance. See
D.R. Grigore, The structure of the anomalies of gauge theories in the causal approach, J. Physics A: Math. Gen. 35 (2002), 1665.
See also Chapter 15 (Interacting quantum fields) from the... | {
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Why do Newton's laws have to be used only when working with a particle? I have a small understanding of physics but I am not studying the subject.
Whilst trying to model a plane landing in Differential equations (an A-level maths module), we were told that you have to assume that the plane is a particle to be able to a... | Because Physics, though a precise discipline, often works with approximations.
In the question you're concerned with the plane can be modelled accurately by thinking of it as a particle. Nothing essential to the question is lost by this approximation.
If we, on the other hand, were concerned about the aerodynamics of... | {
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Commutator identities and Fourier transform Is it possible to derive one side of the arrow below from the other by using only the Fourier transform and its reciprocal?
$$[\hat{p},f(\hat{x})]=-i\hbar f'(\hat{x}) \leftrightarrow [\hat{x},f(\hat{p})]=i\hbar f'(\hat{p})$$
| Under some hypotheses on $f$ the answer is positive. I consider the simplest case below.
If $U$ is a unitary operator on the Hilbert space $H$ and $A: D(A) \to H$ is a self-adjoint operator over the same Hilbert space, form spectral calculus it arises that $$Uf(A)U^{-1} = f(UAU^{-1})\tag{1}$$ for every measurable funct... | {
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How does the expanding universe affect quantum fields? The universe is expanding. It then seems logical to say that the QM fields are expanding as the universe expands. My question is how does this happen? When I consider an expanding field it forces me to consider the actual properties of the field. Are the QM fields ... | Photons are quantum particles you get when the electromagnetic field is quantized. So consider the photons which make up the cosmic background radiation. They were generated when the mean temperature was about 3,000 K, but now represent a temperature of about 2.7 K. They didn't "cool off", but instead their wavelength... | {
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Gravity in vector We know that gravity is a force. But what is it's direction? Can it be expressed by vector and how can we do that? This question can also be asked for Coulomb's Law.
| As a first statement, I like to begin by stating that gravity is always towards the mass (e.g. always attractive). In other words, if mass A pulls on mass B, I would state that the direction of the gravitational force is towards mass A. If you set up coordinate system, you may then put this in by hand. (The same line o... | {
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Energy density in string wave The total energy density in a harmonic wave on a stretched string is given by
$$\frac{1}{2}p A^2 \omega^2 sin^2(kx-\omega t).$$
We can see that this energy oscillates between a maximum and a minimum. So the energy is maximum at 0 displacement when the string is stretched and at its maximum... | In a travelling wave the total energy of a piece of string between $x$ and $x+dx$ is not constant. This is because each piece of of string is doing work on it's neighbour to the right at a rate
$$
P= - T\frac{\partial y}{\partial x}\frac{\partial y}{\partial t}.
$$
The local version of the energy conservation law is t... | {
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Energy-momentum tensor from the variation of action of RNS strings In exercise 4.6 p. 121 of Becker, Becker, Schwarz's book 'String theory and M-theory', they state that under using the variation $\delta_+X=a^+\partial_+X$ and $\delta_+\psi_A=a^+\partial_+\psi_A$ where $A=\pm$, we may identify the components of the ene... | I will just write the variation for Bosonic field, similar logic will follow for $\psi$.
Start with $2 nd $ line of your calculation and substitute the value of $\delta_+ X$. Expression will look like this
$$2 a^+ (\partial_-X) \partial_+ \partial_+ X + 2 a^+ (\partial_+X) \partial_- \partial_+ X $$
Do the integratio... | {
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Where is humidity? During hot and humid weather, we sweat incessantly due to high humidity. But when we sit under a fan, we feel cold and comfortable. Why do we feel cold and chilled? Why don’t we feel the humidity?
| The moving air produced by the fan causes forced evaporation of the sweat secreted by the sweat glands. It takes energy (heat) to change sweat (liquid water) into water vapor and that heat energy is taken from the body thus cooling the body down which makes one feel more comfortable. This is one of Nature’s way of r... | {
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Is coherent stimulated emission possible for particles other than photons? By coherent stimulated emission I am referring to any process analogous to stimulated emission in lasers, where one particle interacts with an excited energy state, which leads to a second photon being emitted with the same phase, frequency, and... | There are "atom lasers", coherent states of propagating atoms that can be emitted from Einstein-Bose condensates. One can quibble about whether it is a laser since the 'L' is for light. In any case, it is experimentally demonstrated. While I have not seen any papers on alpha particles, helium atoms have been used.
Cohe... | {
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Why do gravitational mass and inertial mass appear to be indistinguishable? I have learnt that heavier the object is (the more gravitational mass it has), the more resistance to the change of motion it is (the more inertial mass it has).
I can accept this fact but I can't find out the reason behind it. What dynamic, wh... | The answer is that more mass is defined to provide more inertia.
Newton noticed that, for any given object, $F\propto a$, that is to say the force on an object and the acceleration of that object are proportional. Whenever we find a proportionality like this, we assign a multiplier to turn that $\propto$ into an $=$. ... | {
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Gravitational field strength Can I use $g=GM/r^2$ to calculate the gravitational field strength proton or electron or any other particles? If not then why? If yes then what would be that really mean?
| Usually (that I know of) the noticeable differences between General Relativity and Newtonian gravity only become apparent on a macroscopic, $>=$ planetary scale. For example the precession of Mercurcy, light bending, black holes etc...
So for atomic scales, you can use the Newtonian formula, yes.
It tells you the str... | {
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Time Reversal Operator and Rotations In Sakurai's Modern Quantum Mechanics Chapter 4, in the discussion about the Time Reversal Operator, the following formula is presented
$$\Theta \mathbf{J} \Theta^{-1} = -\mathbf{J} $$
This is a requirement necessary to conserve the canonical commutation relations between the genera... | I am not sure to understand the question. Are you asking why exponent do not change as a consequence of $\Theta S_k \Theta^{-1} = -S_k$?
You are forgetting that $\Theta$ is anti linear, so $$\Theta e^{-i a S_k} \Theta^{-1} =e^{\Theta(-i a S_k)\Theta^{-1}} =e^{-(-i) a \Theta S_k\Theta^{-1}}= e^{-i a S_k}\:.$$
| {
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Probability of measuring eigenvalue of non-normalised eigenstate This came up while working on a question about measuring the angular momentum of a particle in a superposition of angular momentum eigenstates:
Given that:
$$\langle\theta,\phi|\psi\rangle \propto \sqrt{2} \cos(\theta) + \sin(\theta)e^{-i\phi} - \sin(\th... | Yes and no. You can just normalize the results with:
$$ \langle \psi | \psi \rangle$$
but you have computed that incorrectly. Remember, the differential solid angle is:
$$ d\Omega = d(\cos{\theta})d\phi, $$
you have used:
$$ d\Omega = d\theta d\phi.$$
I suggest you verify with a table of $l=1$ spherical harmonics.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Recognizing speech at 1bit quantise depth? i found on german wikipedia an audio example of 1 bit depth quantising, where the speech still can be recognized. how is it possible if at 1 bit depth we have just two values: "signal" and "no-signal"?. here is the examle: https://upload.wikimedia.org/wikipedia/commons/4/43/Am... | A 1 bit depth quantised signal still contains more than one bit of information. The signal level varies from moment to moment, and this provides extra information.
In the case of speech we tend to recognise the rhythmicity and structure as speech even if we cannot make out individual words. Some formant sounds may be ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/382997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is so special about the factor $\sqrt{1-{v^2/c^2}}$ in special relativity? I am studying a book about relativistic equations and special relativity, and I keep seeing $\sqrt{1-{v^2/c^2}}$ everywhere. It is not, as with most of the concepts in special relativity, simply a mathematical construct; it is a logical con... | 1) At least at low speeds, you expect $x'=x-vt$, just from elementary considerations. ($vt$ is, after all, the distance traveled in time $t$, so a person traveling at speed $v$ will have his origin displaced by the amount $vt$.
2) If you believe space and time should be treated symmetrically, then you are led to exp... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/383290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
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