Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why is the density matrix of a system has this block form? In Ficek's paper (http://zon8.physd.amu.edu.pl/~tanas/spis_pub/pdf/04-joptb-S90.pdf), the density matrix of a two two-level atom system has a
block form like this. Why does it make sense to assume this ?
| The basis from the paper is
\begin{align}
\vert 1 \rangle = \vert g_1 \rangle \otimes \vert g_2 \rangle, \\
\vert 2 \rangle = \vert e_1 \rangle \otimes \vert e_2 \rangle, \\
\vert 3 \rangle = \vert g_1 \rangle \otimes \vert e_2 \rangle, \\
\vert 4 \rangle = \vert e_1 \rangle \otimes \vert g_2 \rangle.
\end{align}
Th... | {
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What is positive rotation direction of a pulley in the Atwood machine? In the Atwood machine the mass $m_1$ hangs on the left and $m_2$ hangs on the right, with $m_2 > m_1$. When released from rest the system accelerates clockwise which we define to be the positive direction.
The pulley has non-negligible mass and also... |
The tension on m2 (T2) is in the negative direction
Tension is a scalar. Assuming a "light" string, the tension is uniform and becomes a force (of equal magnitude) at both ends.
The force from the segment attached to m2 on the pulley is clockwise. The force from the segment attached to m1 on the pulley is countercl... | {
"language": "en",
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Why can’t quantum randomness be understood as epistemic? I often hear people say that quantum randomness is “true randomness”, but I don’t really understand it. Please bear with my question.
Before the development of quantum physics, randomness is understood as being “epistemic”. That is, things appear random because w... |
My understanding is that quantum physics can also be described using standard measure-theoretic probability theory, or, in other words, an theory with merely “epistemic” randomness.
Take the simple probability of a dice to come up with one of the six numbers. In principle if the distribution from 1 to 6 is not flat,... | {
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Does the normal reaction on pull up bar change during the pull ups? Intuitively, I know the answer but I can't think of the right math.
I found this question but none of the answers were satisfying enough for me. Human body is not a rigid body so do how do we even apply $\Sigma F=ma_{net}$ over it?
|
Human body is not a rigid body so do how do we even apply $\Sigma
> F=ma_{net}$ over it?
It doesn't matter that the human body is not a rigid body. You only you need to apply Newton's second law to the acceleration of the center of mass (COM) of the human body. To simplify the problem, consider only vertical accelera... | {
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Mental model of general relativity I am trying to visualize the curvature of space-time.
In almost all of the Yt videos on the topic, it's shown as depression in the space-time fabric. But what does the dimension into which space-time curves represent? If some mass creates more curvature in the fabric, what does it sho... | Unfortunately that "model" is complete and utter rubbish. You cannot learn anything from it, so please do not try!
Nobody who understands General Relativity uses it. It does not represent any equations or allow any calculations.
It needs to go away, now. Unfortunately (again) the internet will preserve it forever . ... | {
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How to express $|m\rangle\langle n|$ in terms of ladder operators? Let us consider the Hamiltonian of a single harmonic oscillator, which is expressed in terms of creation/annihilation operators as
$H=\hbar \omega (a^{\dagger}a+1/2)$. The eigenstates of this Hamiltonian are the number states ($n\geq 0$)$$a^{\dagger}a|n... | Combining the existing answers, since $|0\rangle\langle 0|=\prod_{k\ge1}(1-k^{-1}a^{\dagger}a)$ and $|n\rangle =\frac{1}{\sqrt{n!}}a^{\dagger n}|0\rangle$,$$\sum_{mn}c_{mn}|m\rangle\langle n|=\sum_{mn}\frac{c_{mn}}{\sqrt{m!n!}}a^{\dagger m}\prod_k(1-k^{-1}a^{\dagger}a)a^n.$$
| {
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Energy of a system executing forced oscillations In L&L's textbook of Mechanics (Vol. 1 of the Course in Theoretical Physics) $\S 22$ Forced oscillations, one finds the following statement:
\begin{equation}
\xi = \dot{x} + i \omega x, \tag{22.9}
\end{equation}
\begin{equation}
\xi = e^{i \omega t} \int\limits_{0}^{t} ... | *
*The Lagrangian term $xF(t)$ is an interaction term between the system and the environment. The energy of the oscillator system itself is given by eq. (22.11).
2+3. L&L assume that the oscillator is at rest at $t=-\infty$.
| {
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Finding the Hermitian generator of a representation of a Symplectic transformation Consider a set of $n$ position operators and $n$ momentum operator such that
$$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$
Lets now perform a linear symplectic transformation
$$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$
$$p'_{i} =C_{ij}q_{j}+... | *
*Classically, the symplectic group $Sp(2n, \mathbb{R})$ of dimension $n(2n+1)$ is the group of all linear time-independent canonical transformations (CTs)
$$z^{\prime I}~=~\sum_{J=1}^{2n}M^{I}{}_Jz^J.\tag{1}$$
The corresponding symplectic Lie algebra $sp(2n,\mathbb{R})$ is the set of all linear time-independent infi... | {
"language": "en",
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Find coefficient for pure and mixed states Consider a generic $2\times 2$ Hermitian matrix written as $$\rho =\alpha\sigma_0+\beta\hat{\vec n}\cdot\vec\sigma\quad ,$$ where $\hat{\vec n}$ is a unit vector and the coefficients are real numbers.
My question is this; how do I figure out what the coefficients have to be fo... | That seems to be right to me, but don't forget that $\beta = -\frac{1}{2}$ because you had absolute value in your equation and $\beta$ can be any real number. Also, another way to think about pure states is that they are points on the edge of the unit ball; that is to say, if when the density matrix is written in the f... | {
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Are water molecules at the surface closer or farther apart than the molecules inside? My lecturer says that since the energy of the molecules on the surface is higher (less negative), then at equilibrium there will be less molecules on the surface, hence the molecules on the surface are farther apart.
On the other hand... | The water molecules on the surface have a vast empty space to move in on one side. This causes them to move further apart than inside the bulk molecules. This larger separation causes a force between surface molecules that makes the surface act like an elastic container. Hence the droplets on a window. Introducing poll... | {
"language": "en",
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Possible to stir a drink while maintaining a convex shape? The Wikipedia page for the Brouwer fixed point theorem has a cute example:
"no matter how much you stir a cocktail in a glass ... when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took a... | I would suggest putting a lid on the cocktail that leaves not free surface.^^
If you have a free surface, you must make sure that it does not move in vertical direction at any point (that's what I want the lid for) because if it did, there would be a minimum of its height profile, where indeed the convexity of the flui... | {
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Why do we consider special relativity for electrons consider their drift velocity is really small? I know that when looking at the electron and a current carrying wire in different inertial frames there will be different kind of forces acting between these two objects. And I know that it is due to length contraction wh... | (a) If you ignore special relativity altogether, there is no length contraction and so no electrostatic force in the electron frame. Therefore the special relativistic effect, even if it is small, is larger than the (zero) non-relatvisitic effect. This is a general phenomenon: small effects can still be important if th... | {
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$(α|0⟩ + β|1⟩)|0⟩$ in matrix/vector form I am currently working through superdense coding with bell states and have a question regarding this value:
$$(α|0⟩ \ + \ β|1⟩) \ |0⟩$$
I understand that $α|0⟩ \ + \ β|1⟩$ can be represented in vector format as $(α \ \ β)$ (vertically)
However, I'm not sure how to apply the o... | This operation is what's called the tensor product. More formally it is denoted by
$$|\psi\rangle\otimes|\phi\rangle$$
but often the $\otimes$ is left out. When calculated numerically it is called the Kronecker product which can be calculated as follows for a 2 state system
$$\pmatrix{a\\b}\otimes\pmatrix{c\\d}=\pmatri... | {
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How does a gravity defying stand comes in equilibrium? The gravity defying wine bottle stand are now becoming very common these days.
Can someone explain in detail the physics involved here?
Here is a video that gives details about its making.
|
The gravity defying wine bottle stand are now becoming very common these days.
The bottles do not defy gravity. The normal force exerted on the support (contact area between the wood and the table) equals weight of the two bottles plus weight of the wood. The bottles would tip over if there is some torque about one o... | {
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How do you estimate the core temperature of a star? Given a star's mass, radius and average composition (e.g. 90% H, 10% He), is there a formula to estimate the core temperature of that star?
I only found one for a lower bound but that wasn't very accurate.
| There are approximate formulas for this, but to do this right you really need an equation of state for matter in the stellar core. Writing that down is really, really hard, which is why predicting the temperature and density profiles for stellar cores is really, really hard.
| {
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Intuition for the negative sign of gravitational potential energy The gravitational potential energy is introduced to us as $U = mgy$. We usually set $U = 0$ on ground level and hence, for an object at height $y$, we have a potential energy equal to $U = mgy$.
I have adopted this convention when getting the potential f... | This is because, when near earth, at the surface, when we define U=0, as we move some height(h) up, potential energy increases by amount $mgh$.
Similarly, when we go down, potential increases by amount $-mgh$.
So when dealing with Gravitational force, which is attractive in nature.
We define U=0 at infinity (You can de... | {
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How can satellites change direction without any medium in space? How can satellites change direction without any medium in space? How do spaceships move in space if there is no medium?
How does Newton's third law of motion work in space?
| You do not need a medium to push against. Several principles are used to change direction of satellites.
*
*Newton's third law is used by thrusters and cold gas engines. Every action (particles leave thruster of the satellite) has a reaction. E.g. The satellite moves the opposite direction. $\vec p_r=-\vec p_a = m_a ... | {
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Difference between the wave forms in the water and in the Young double slit experiment We can observe when we cause a slight disturbance at two points on the water surface which is intially totally undisturbed , it will form water waves which would look like as shown in below image:
we can observe that there are const... | The screen is used simply because it makes the interference effect easier to see and to record and analyse. As is the case with the picture of the interference of water waves which you included in your question, the light interferes with itself everywhere. The use of a screen allows you to see the interference along a ... | {
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Why do we use different differential notation for heat and work? Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we are told that the '$df$' notation is only for functions... | I believe the good way to present thermodynamics is through the formalism of differential geometry.
When the thermodynamic process is reversible it can be described as a curve on a manifold of equilibrium states (because each intermediate step is equilibrated). Then $\delta W = -p dV$ and $\delta Q = T dS$ are differen... | {
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Principle of Equivalence for non-uniformly accelerated system Consider a frame being accelerated according to the law of gravity, $a=-K/r^2$ where $K$ is some constant and $r$ is measured from some point. Since here the acceleration is exactly the form of gravity, can the equivalence principle be extended to include su... | The equivalence principle holds locally, that is, over domains for which changes in position do not significantly change the reading of an accelerometer. While one can come up with certain arrangements of matter such that certain arrangements of acceleration are indeed indistinguishable, one should be careful with them... | {
"language": "en",
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How to find a bend curve in 2D? I am looking for a solution on how to find the (approxmiate) shape when bending a rigid-flex circuit board. Please see the abstract sketch below. I have two solid objects ($A$, $B$) which are connected by a thin and flexible but non-stretchable strip of material.
Given that I know
(i) th... | The curvature is proportional to the local bending moment.
According to the picture, it seems that the strip is firmly held by the parts A and B, in a way that it can not slide. In that case, the situation is pure bending and the curve is an arc of circle. See http://emweb.unl.edu/NEGAHBAN/Em325/11-Bending/Bending.htm
... | {
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Non-Comulative nature of Mass-Radius curves of Neutron Stars For finding the mass-radius curve of neutron stars, we can solve TOV Equations which are constraint equations got by solving Einsteins equations. The boundary conditions are $m(r=0)=0$ and $\rho(r=0)=\rho_c$. Then we put a physical condition that for $r\right... | Your non-cumulative curves relate the total mass of the star to its surface radius. Different points represent different stars.
Your cumulative curves relate enclosed mass of a star versus radius of the shell enclosing the mass. Different points represent different radii within a given star.
| {
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Do the components of a force written for a purpose actually exist? On an inclined plane if you put a box, the force of gravity $mg$ is written as sum of two forces $mg\sin\theta$ and $mg\cos\theta$ where $\theta$ is the angle the incline is making with earths surface. Do these forces $mg\sinθ$ and $mg\cosθ$ actually wo... |
Do these forces $mg\sinθ$ and $mg\cosθ$ actually work on the object?
Only the component of the force of gravity, $mg\sin\theta$, does work on the object. That work is then
$$W=Fd=mgd\sin\theta$$
Where $d$ is the length of the incline plane traveled by $m$. Since $\sin\theta=h/d$ where $h$ is the height of the plane,
... | {
"language": "en",
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Does the Schrödinger equation apply to spinors? I was reading about Larmor precession of the electron in a magnetic field in Griffiths QM when I came across the equation
$$
i\hbar \frac{\partial \mathbf \chi}{\partial t} = \mathbf H \mathbf \chi,
$$
where $\mathbf\chi(t)$ is a 2D vector that represents only the spin st... | The correct wave equation for an electron is the famous Dirac equation (in fact it is correct for all spin $\frac{1}{2}$ particles).
The non-relativistic limit of the Dirac equation is called Pauli equation. The derivation can be found here. In this equation $|\psi \rangle$ is a two-component spinor wavefunction.
Pauli... | {
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Hall sensor for electric(!) field? Is it (in principle) possible to measure the strength of an electic(!) field with a hall sensor?
I think so, for the following reasons:
*
*The hall sensor is a conductor. If we place an conductor in an electic field, charges will rearrange so that there will be no electric field in ... | It seems you do not actually consider a Hallsensor wich needs current flowing trough, but just a conducting sheet put in an electric field measuring the difference between two sides. But first the sheet will disturb the field, and second how will you measure it without a closed circuit of some kind?
| {
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Why is the force on a rigid pendulum directed radially? Consider a typical presentation of a simple pendulum: a point mass $m$ attached to a rigid rod of length $L$, which is free to rotate around a pivot. Newton's equations are
$$\begin{gather}
mg \cos\theta - F = -m L \dot{\theta}^2 \\
-mg \sin\theta = m L \ddot{\the... |
What is the difference between this case and the pendulum? What condition allows us to assume that the force from the rod is radial for the pendulum?
The condition that explains both results is that the rod is assumed to be massless. Since the rod is assumed to be massless then both the net force and the net torque a... | {
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Mathematical Definition of Point Source Wikipedia describes a mathematical definition of a point source as "a singularity from which flux or flow is emanating".
The usual definition in Physics describes it just as a source whose dimensions are negligible in comparison to another variable you're relating it to, which le... | "Point" sources are typically represented by $\delta$-functions.
For example, an electron with charge $e$ located at $\vec{r}_e$ must have charge density
$$\rho(\vec{r}) = \left\{{0,\, \vec{r}\not=\vec{r}_e\atop\infty,\, \vec{r}=\vec{r}_e}\right.,$$
and we therefore write that
$$\rho(\vec{r})=e\,\delta^{3}(\vec{r}-\vec... | {
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Is there a quantum of entropy - and how large is it? Google Scholar lists quite a number of papers when searching for "Quantum of entropy" or "entropy quantization". Most are old.
Some of these papers mention k as quantum of entropy, others k/2, still other log 2 k (see H.S. Leff, https://link.springer.com/article/10.1... | In physics, we normally define entropy as $S = k \ln \Omega$, where $k$ is Boltzmann's constant and $\Omega$ is the number of microstates consistent with the observations you've made of the system. For the rest of this answer, I'll work in units with $k=1$ (you can multiply all the expressions below by $k$ if you want)... | {
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Would our orbit really remain the same if the sun were a black hole of equal mass? There seems to be an idea floating around that the sun could be replaced by anything of equal mass with no consequence to our orbit.
It seems to me that if the mass of the sun were confined to a single point that the local geometry of sp... | Yes, the number of gravitational degrees of freedom remains the same if the Sun were magically turned into a black hole. Now apply that to disk-like interaction, and you have focused, super strong gravitation. Count.
| {
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Is angular momentum just a convenience? I'm wondering whether angular momentum is just a convenience that I could hypothetically solve any mechanics problems without ever using the concept of angular momentum.
I came up with this question when I saw a problem in my physics textbook today. In the problem, a puck with kn... |
I'm wondering whether angular momentum is just a convenience that I could hypothetically solve any mechanics problems without ever using the concept of angular momentum.
If your criterion for something being a convenience is that you could solve problems without it then everything in physics is just a convenience.
Th... | {
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"source": "stackexchange",
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Barometric equation with different species The barometric equation gives the pressure dependence of a perfect gas in a gravitational potential. In particular
$$
P(h) = P_0e^{-\frac{mgh}{k_BT}}
$$
where $m$ is the mass of molecules and $T$ the temperature of the gas.
What would happen if we had an atmosphere composed of... | If we suppose ideal gases, remember that the pressure of an ideal gas depends on the number of molecules, however, it has an influence on the distribution of the gas with a gravitational potential. However, you can substitute in your equation the molar mass with the average molar mass. This molar mass $M$ can be found ... | {
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How to calculate velocity vector from scalar angular velocity and position vector in 2D? I would like to know, if I have an angular velocity as scalar, how can I calculate the velocity vector.
I know that the product of angular velocity and the length of the distance gives the speed, but I would like to know vector of ... | I assume you want to find the linear velocity if you know the angular speed. You need to realize that all the points on a rotating rigid body have the same angular speed and angular velocity but different linear speeds and velocities. A point on a rigid body that describes a circle of twice the radius as another point... | {
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How can light produce electric and magnetic field when there are no accelerating charged particles? If we see light as a wave, especially in vaccum, there is nothing there, no particles, yet light has an electric and magnetic field. How can this be possible?
| As pointed out in a comment by @PhysicsDave, there is the supposition of the existence of a field acting as a mediator of electric interaction and magnetic interaction.
The fields extends over spatial distance. That is how the interaction is envisioned. The interaction being mediated does not require particles in the i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Has it been experimentally proven that energy causes gravity? I know that under general relativity energy and mass are equivalent under $E=mc^2$. But has it been experimentally proven that energy alone causes gravity, for example, does a nuclear reaction generate gravity independent of the mass of the reactor alone? I... | The parametrized post-Newtonian (PPN) formalism is a generalized way of exploring gravity theories, including general relativity. In the older "beta-delta" parametrization, three of the parameters ($\beta_1$, $\beta_2$, and $\beta_3$) describe how much gravity is produced by kinetic energy, gravitational energy, and i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/703853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does amplitude really go to infinity in resonance? I was recapping the forced oscillations, and something troubled me. The equation concerning forced oscillation is:
$$
x=\frac{F_0}{m(\omega_0^2-\omega^2)}\cos(\omega t)
$$
I don't understand why this equation predicts that the amplitude will approach infinity as $\omeg... | transfer this differential equation
$${\frac {d^{2}}{d{t}^{2}}}x \left( t \right) +{\omega_{{0}}}^{2}x
\left( t \right) ={\frac {F\cos \left( \omega\,t \right) }{m}}$$
to Laplace domain (with the initial conditions $~x(0)=0~,\dot{x}(0)=0~)$ you obtain
$$x(s)={\frac {Fs}{m \left( {s}^{2}+{\omega}^{2} \right) \left( ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/704112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 9,
"answer_id": 4
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Maxwell's eq-meaning of del's cross and dot product? In maxwell's eq there is del whose cross and dot products exist.
So what is del in cross vs dot product.
What's the difference when it's just a partial differential operator.
| They aren't real cross/dot products, it is a notational trick.
$\nabla × $ is called the curl.
$\nabla \cdot$ is called the divergence.
$\nabla \cdot \vec{F} = \frac{\partial F_{x}}{\partial x} \hat i + \frac{\partial F_{y}}{\partial y} \hat j + \frac{\partial F_{z}}{\partial z} \hat k$
$\nabla × \vec{F} = (\frac{\part... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/704273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lagrangian first integral I want to extremize $$\int dt \frac{\sqrt{\dot x ^2 + \dot y ^2}}{y}.$$
I have thought that, since the Lagrangian $L(y, \dot y, \dot x)$ is $t$ dependent only implicitly, that i could use the fact that $$L(z,z') \implies L - z' \partial L / \partial z' = c.$$
So $$L - y' \partial L / \partial... | with
$$L=\frac{\sqrt{\dot x^2+\dot y^2}}{y}$$
and because L is not a function of x you obtain that
$$\frac{\partial L}{\partial \dot x}=\frac{\dot x}{\sqrt{\dot x^2+\dot y^2}\,y}=\text{constant}$$
from here
$$\frac{\dot x}{\sqrt{\dot x^2+\dot y^2}\,y}\mapsto
\frac{1}{\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,y(x)}=\text{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/704460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is torque defined as $\vec{r} \times F$? Here I cannot convince myself myself that it is units because the torque is defined to be in units of Newton meter is a reiteration of the law stated above. Why was it not $r^2 \times F$ or $r^3 \times F$ or $r^2 \times F^2$ etc. The argument "in our experience how much some... | Torque is change of angular momentum: $$ \vec{\tau} = \frac{d\vec{L}}{dt}$$
Angular momentum is defined as $$ \vec{L} = \vec{r} \times\vec{p} $$
Using the chain rule:
$$ \vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d}{dt}(\vec{r} \times\vec{p})$$
$$\frac{d}{dt}(\vec{r} \times\vec{p}) = \frac{d \vec{r}}{dt}\times \vec{p}+ \... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/705214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Space-time continuum expansion I still don't understand how the expansion of the universe works.
If the universe is made up of an infinite number of points that make up space-time, then how can space expand or stretch. Common sense dictates that in order for space to be stretched, the length between points must increas... | An infinite number of points is not a definite number. Don't think of it like a normal number.
No matter how small a distance you make between two any points there is always an infinite number of points between them. There's no cutoff for that. So when you imagine two points spreading apart there are an infinite num... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the angular momentum of a particle rotating around an axis in 3D?
What would be the angular momentum of the particle at position $r_i$ in the diagram above?
The vector from the axis of rotation is $R_i$ and the tangential velocity is $v_i$ so the magnitude of angular momentum should be $|R_i|m_i|v_i|$ which g... |
they all take each particle's angular momentum to be ri×(mivi) instead of Ri×(mivi). Which is correct and why?
I can understand your confusion since both terms do not seem to be the same. I think the main point can be extracted, for example, from the second ressource you cite
The total angular momentum of the body (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What frequency of cord shaking maintains the same vertical motion for a point on the cord after increasing the wave speed on the cord? I'm studying for my upcoming AP Physics 1 exam but can't figure out this problem
A student shakes a horizontally-stretched cord, creating waves. The graph above shows the vertical pos... | You already have a nice, brief and concise answer but I'd like to add more to it.
From the "fundamental" equation for waves
$$c = \lambda f \tag{1}\label{1}$$
where $c$ is the speed of wave propagation, $\lambda$ the wavelength and $f$ the temporal frequency, we can see that if you change the speed of propagation then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Clarification on Searle's method to determine thermal conductivity
Here In Searle's method to determine the thermal conductivity of a good conductor, the heat transfer rate of the conductor along a displacement(thickness) d is given by:
Q1/t=KD(T1-T2)/d.
=>Q1=KD(T1-T2)t/d
And the heat transferred to the cold water is... | Consider a proof by contradiction: $Q_1\neq Q_2$, such that $Q_1=Q_2+Q_3$, where $Q_3$ must be stored on the right side of the rod. (We can show this with an energy balance of the right side of the rod.) We consider the sensible heating of that region: $Q_3=mc\frac{dT}{dt}$, where $m$ is the mass, $c$ is the specific h... | {
"language": "en",
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In general relativity, assuming a spherical uniform mass distribution, what is the total energy value of the gravitational field inside the sphere? In Newtonian mechanics, assuming a spherical uniform mass distribution, the total gravitational potential energy (gravitational self-energy) inside the sphere is
$$U_{gs}=-... | In general relativity, there is no gauge invariant local definition of the energy of a gravitational field. In simpler terms, there is not a well-defined way to define the total energy contained in a finite region of space. So, while you can take a non-relativistic limit and recover Newtonian gravity, it is not possibl... | {
"language": "en",
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Most likely velocity vector of ideal gas?
$$\left( \text{probability of a molecule having velocity }\vec{v}\right) \propto e^{-mv^{2}/2kT}$$
So the most likely velocity vector for a molecule in an ideal
gas is zero. Given what we know about Boltzmann factors, this result should hardly be surprising.
Schroeder's Therm... | The distribution is
$$
w(v_x,v_y, v_z)=Ae^{-\frac{m(v_x^2+v_y^2+v_z^2)}{2k_BT}},
$$
where A is the normalization constant that can be easily found by integration (see Gaussian integral):
$$
\int dv_xdv_ydv_z w(v_x,v_y, v_z) = 1.
$$
The average velocity along any direction is zero, e.g.,
$$
\langle v_x\rangle = \int dv_... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/707071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Expressing Maxwell's equations in tensor form using Electromagnetic field strength tensor I have yet another derivation question from Carroll's General Relativity textbook. Given the electromagnetic field strength tensor is of the form: $$ F_{\mu\upsilon} =
\left(
\begin{matrix}
0 & -E_1 & -E_2 & -E_3\\
E_1 & 0 & B_3 ... | Since $F_{ij} = \epsilon_{ijk}B^k$ one has $\epsilon^{lij} F_{ij} = \epsilon^{lij}\epsilon_{ijk}B^k = 2\delta_{lk} B^k$, hence $B^l = \frac{1}{2} \epsilon^{lij} F_{ij}$.
The third Maxwell's equation in OP's question can be expressed with the field strength tensor $F_{\mu\nu}$ according to
\begin{align}
0 &= \epsilon^{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/707341",
"timestamp": "2023-03-29T00:00:00",
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Is it possible to visualise red shift? If a picture of a star or galaxy hurtling away from Earth is taken, does it appear red despite it being a different colour? Would a blue coloured star moving away from us appear red to us or vice versa? If so how do scientists understand if say, the red colour of a star is due to ... | To detect the redshift of distant objects, we can use the fact that (to the best of our knowledge) the laws of physics are the same everywhere. This means the spectral lines of elements (both absorption and emission) will be the same at the location of the star as they are on earth.
We can measure absorption and emissi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Conservation of energy and work done by a torque Suppose you let a solid roll down an incline without slipping, from height $h$. My textbook gives the following conservation of energy relation
$$mgh = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2.$$
Why do we not have to include the work done by the static friction (nonc... | You're correct it does zero work because before the motion and after the motion the incline is in the same location. There can be no energy extracted from it. The only energy available is the gravitational position energy from the height of the ball.
The rotational energy added to the solid comes at the expense of th... | {
"language": "en",
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What happens when the universe runs out of fuel? After some X billion years, one would think the stars in the entire universe will run out of hydrogen. What would happen next? Is there any way to get hydrogen out of heavy metals (extreme fission)? Just curious.
| Well, fate of the universe has a lot of possible scenarios, from which periodical expansion/contraction is very unlikely, because universe expands in an accelerated fashion, and it doesn't looks that it will change, unless dark energy will run-out too, which is highly unlikely due to quantum mechanical laws. So most pr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Partition Function Question I've been looking at calculating the Internal energy of a non-isothermic van de Waals gas, and in doing so have been researching the free energy and hence the partition function necessary to calculate that.
The problem is that the partition function cannot be computed in any formulae that I ... | In my experience, any useful thermodynamic quantities involve taking the natural log of $Z$ rather than the absolute value of $Z$ itself. For example the Helmholtz free energy is
$$F=-kT\ln Z$$
and internal energy is
$$U=kT^2\frac{\partial\ln Z}{\partial T}$$
It's fine that $Z$ is usually incommputably large. It is a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/708452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complex Coupling Strength in Light-Matter Interaction Hamiltonian The quantised electric field operator is given by :
$$ \hat{\mathbf{E}}(\mathbf{r},t) = i\sum_{\xi}E_{\xi}\left(\mathbf{u}_{\xi}(\mathbf{r})a_{\xi}-\mathbf{u}^*(\mathbf{r})a_{\xi}^{†}\right) $$
where the complex pre-factor is sometimes absorbed into the ... | You can redefine the ladder operators to absorb the phase. The mapping
$$a \rightarrow e^{i\phi} a$$
is canonical, since it preserves the commutation relations.
So there is no mistake in your logic, but the dynamics only depend trivially on the phase. Note, however, that your Hamiltonian may need to be adapted in certa... | {
"language": "en",
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Is it possible the Black Holes to be pure deformations in the fabric of spacetime and not an effect of super-dense matter? Is there any theory in the literature that supports this hypothesis that BHs in their center do not have a super-dense matter singularity but are pure deformations in the fabric of spacetime itself... |
Is there any theory in the literature that supports this hypothesis that BHs in their center do not have a super dense matter singularity but are pure deformations in the fabric of spacetime itself or vacuum space
It's the General Theory of Relativity. The Schwarzschild solution for non-rotation, non-charged black h... | {
"language": "en",
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Non-parallel light diffraction Does light (and in general any kind of wave) diffract only when the wave fronts are parallel? Like if you did the double slit experiment when the waves were coming from a point source close to the diffraction grating, would the light still diffract?
| Yes it does. A common version on the double slit experiment is, in fact, the one you describe. You can, for instance, install a narrow hole with the primary light source behind it. This hole will act as the source for the experiment, and you can put it as close as you like from the double slit, so the wavefront will be... | {
"language": "en",
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Introduction to magnetohydrodynamics does anybody have any reference books for introduction to magnetohydrodynamics? I want to dive into this topic and I don´t know about any good reference.
| If you want a very quick introduction to Magnetohydrodynamics (MHD) I would recommend Reitz, Milford and Christie's 'Foundations of Electromagnetic Theory'.
If you are willing to take a leisurely path that will build your foundations you can choose Arnab Rai Choudhuri's 'Physics of Fluids and Plasmas'. The author does ... | {
"language": "en",
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Angular Momentum and Coefficient of Restitution If there was a situation where you had two rods pinned in the center and the left rod having an initial angular velocity $\omega_1$, and the right rod was at rest. I am wondering what the final angular velocities would be if there was a coefficient of restitution during t... | The problem is that angular momentum is not conserved in this scenario. In order to define angular momentum in a system, there needs to be an origin. If we put the origin at the center of the first rod, $M_1$, then it is clear that any force that the pin at $M_1$ exerts does not result in any torque in the system. Howe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does the standing wave equation proof require $\ell=Nλ$? Consider two identical sources $S_1$ and $S_2$ of waves, separated by a distance $\ell$ (as shown in the figure).
The sources produce waves in opposite directions(and towards each other). Now, suppose we wish to derive the equation for the standing wave produced... | I think that you're missing boundary conditions. Anytime you want to solve partial derivative equations, you have to provide boundary conditions, or the solution is undefined.
To get a standing wave, the most commun situation is to use "strict boundary conditions", where the wave's amplitude has to be zero on both extr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does gravitation really exist at the particle level? As I understand, we usually talk about gravity at a macro scale, with "objects" and their "centre(s) of mass". However, since gravity is a property of mass generally (at least under the classical interpretation), it should therefore apply to individual mass-carrying ... | Here is an easy way to grasp how difficult it would be to make the direct measurement you propose.
Take two protons and place them one centimeter apart. They will exert a certain tiny amount of gravitational attraction, which we measure by some magic means, and a certain amount of electrostatic repulsion, which we will... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/709780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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What is the instant velocity? The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
| Average speed is defined as passed-distance-over-passed-time:
$$v_\text{average}=\frac{\Delta s}{\Delta t}.$$
In other words, choose a point on your path. Then choose one more point. Plug in the difference in distance and time between them.
If your speed varies in-between those two chosen points, then the average speed... | {
"language": "en",
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Can't understand a statement about motion From the book where I am studying motion, It says
Motion is a combined property of the object under study and the observer. There is no meaning of rest or motion without the viewer.
I know that, for an object, it can be said that 'it is moving' in one frame of reference, and ... | I think the second sentence is nonsense if taken literally. The temperature and pressure of the air in my room is the result of the kinetic energy- ie the motion- of the air molecules. You do not need to posit the existence of an observer to understand that.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does a sliding object stop moving by the effect of kinetic friction, if kinetic friction is constant and Fk can't be greater than Fapp on its own The kinetic friction is constant. So if say, I apply a force of 20N on an object and the kinetic friction is 13N, then how does the object stop moving after some seconds.... | There are very tiny bumps and ridges along the surfaces of contact.
What we call "Force of friction" is a sort of overall estimate on a large-scale of how all these bumps colliding with other surface bumps are causing deceleration.
The coefficient of friction of surfaces has been proven experimentally.
But what's going... | {
"language": "en",
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Absence of velocity in energy conservation
Here we see a rod at rest hinged about a point. We want to know the angular speed of the rod when it becomes vertical as shown in the figure.
The solution which is given in the books goes like this.
Taking the black line as reference level,they say that initially,the rod had ... | When you're using the moment of inertia, you have to specify a reference point or axis. Then $\frac{1}{2}I\omega^2$ is the kinetic energy of the system in a rotation around that point.
So you have two equivalent ways to procede:
*
*Use $I_G$, the moment of inertia with the center of mass as a reference point. Since t... | {
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What is the plane of a concave or convex mirror? What is the plane of a concave or convex mirror? Is it along its principal axis or its area? I am convinced that the plane should be along the area of mirror but i am not sure about it. I tried to find it in books but didn't got any answer. Some books have questions like... | There is no such particular definition for plane of mirror, in fact this term is very ambiguous since spherical mirrors aren't planar. "Plane of mirror" - these words are just used to express questions (like plane of paper or plane of the plane mirror). Physics doesn't defines that this particular plane is called the "... | {
"language": "en",
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Showing $[\hat{A},\hat{B}] = i\mathbb{1} \Leftrightarrow [\hat{A},\hat{B}^{\textstyle n}] = i\,n\,B^{\textstyle n-1}$ Actually it is self redundant to show having in mind $[\hat{A},\hat{B}^{\textstyle n}] = \,n\,B^{\textstyle n-1}\,[\hat{A},\hat{B}]$ but I suppose it is not supposed to solve it this way. Instead I gues... | Note that $-\hat{B}^n \hat{A} = -\hat{B}^{n-1}\hat{B}\hat{A} = \hat{B}^{n-1}(i\mathbb{1}-\hat{A}\hat{B})$ and then you can continue with
$-\hat{B}^{n-1}\hat{A}\hat{B}=-\hat{B}^{n-2}\hat{B}\hat{A}\hat{B}=\hat{B}^{n-2}(i\mathbb{1}-\hat{A}\hat{B})\hat{B}$.
Can you see the pattern?
| {
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Eyes shut, can a passenger tell if they’re facing the front or rear of the train? Suppose you’re a passenger sitting in one of the carriages of a train which is travelling at a high, fairly steady speed. Your eyes are shut and you have no recollection of getting on the train or the direction of the train’s acceleration... | The simple answer is no, you would not be able to know. In fact you would not even be able to know that you are traveling at a constant speed. To you, the train could be completely still (assuming that the environment is perfect. Otherwise you could tell by for example sound). Even when the train starts to accelerate o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The effective action in the linear sigma model I am reading the section 11.4 of Peskin and Schroeder's book (page 373), and there is a step I could not follow.
To calculate the effective action of linear sigma model, the determinant of $\frac{\delta^{2}\mathcal{L}}{\delta\phi^{i}\delta\phi^{j}}$, and in peskin's book, ... | Simply read the diagonal N×N matrix $$ -\partial^2\delta^{ij}+\mu^2\delta^{ij} - \lambda[ v^2 \delta^{ij} + 2\phi_{\mathrm{cl}}^i\phi_{\mathrm{cl}}^j],
\tag{11.67}$$
where, to allay your confusion, I call $\phi_{cl}=v$, so $(\phi_{\mathrm{cl}}^k)^2=v^2$.
The non-gradient part of this diagonal matrix is, for $i,j= 1...,... | {
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Does it take light longer to reach me if I'm moving away? I'm having trouble grasping the intuitions behind the second postulate of special relativity, particularly what it implies.
*
*For example, imagine that a laser pointer is aimed at me at 1 lightsecond away. Then, I move away from it at a speed of 0.5c. Does th... |
For example, imagine that a laser pointer is aimed at me at 1 lightsecond away. Then, I move away from it at a speed of 0.5c. Does the laser still take 1 second to reach me?
It takes 2 seconds according your buddy that you leave behind standing still. Because the buddy sees the distance shrinking at rate 0.5 c.
Accor... | {
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Could a glass cup containing a vacuum rise into the air? https://what-if.xkcd.com/6/ has been mentioned here before, but I'm questioning whether or not the glass cup with the bottom half as a vacuum would rise at all.
To start with, a vacuum exerts no force. Any perceived "sucking" is actually external pressure pushing... | It's not just buoyancy because the slug of liquid can move. Your "force field" at the top of the glass modifies the premise of the XKCD scenario. If you look closely at the illustrations, his slug of liquid moves downward as the glass moves upward. The atmosphere pushes down on the slug of liquid and up on the glass... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is sand in a vacuum a good thermal insulator? My reason for thinking that sand in a vacuum would be a good insulator is that heat cannot be conducted in a vacuum, and the area of contact between adjacent grains of sand is very small, which means heat would transfer between grains relatively slowly. Is this correct, or ... | Powder filled vacuum (I don't know if sand per se is used for this purpose) is in fact used in cryogenic insulation, and can be better than vacuum alone, because vacuum alone is susceptible to radiant heat transfer and also a type of heat conduction that happens because the mean free path of the remaining molecules (in... | {
"language": "en",
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"source": "stackexchange",
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Matrix form of fermionic creation and annihilation operators in two-level system I'm trying to find the matrix form of fermionic creation and annihilation operators in two-level systems from this text. I understand that for one site, the operators take the form:
$$
f_{0}=\left(\begin{array}{ll}
0 & 1 \\
0 & 0
\end{arra... | Briefly: you have to order your sites and add a string $\eta_{\alpha}$ of operators in front of the creation and annihilation operators
$$
\overline{f}_{\alpha}=\eta_{\alpha}f_{\alpha}, \qquad \overline{f}_{\alpha}^{\dagger}=\eta_{\alpha}f_{\alpha}^{\dagger}, \qquad \eta_{\alpha}=\prod_{\beta=1}^{\alpha-1}\begin{pmatr... | {
"language": "en",
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Points where electric field is zero when charges are present at vertices of a regular polygon There is a $n$-sided regular polygon with a charge $q$ at each vertex. I know that there are $n$ points, other the center of the polygon, where the electric field is zero. But why is this so? Is there a general way to prove it... | *
*For what it's worth, 2D Morse theory (with the assumption that all critical points for the electric potential are non-degenerate) yields that
$$c_1-c_0~=~n-1\qquad\text{and}\qquad c_2~=~0,$$
where
$$\begin{align} c_0~:=~& \#{\rm minimum~pts}, \cr
c_1~:=~& \#{\rm saddle~pts}, \cr
c_2~:=~& \#{\rm maximum~pts},\end... | {
"language": "en",
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"source": "stackexchange",
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How are these Covariant Derivative Identities found? In David Tong's Gauge Theory notes on page 137 near eq. (3.30) he makes use of the following expressions for the covariant derivative $D_{\mu}$
$$\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]D_{\mu}D_{\nu}=\frac{1}{4}[\gamma^{\mu},\gamma^{\nu}][D_{\mu},D_{\nu}]\tag{1}$$
and... | By Leibnitz' rule
$$
e^{-ikx} \partial_x \{e^{ikx} f(x)\} = e^{-ikx}\{f(x)(\partial_x e^{ikx})+ e^{ikx}(\partial_x f)\}\\
= e^{-ikx}\{ f(x) (ik e^{ikx})+ e^{ikx}(\partial_x f)\}\\
= ik f(x) + \partial_x f(x)
=(\partial_x +ik)f(x).
$$
As $f(x)$ can be anything, we have
$$
e^{-ikx}\partial_x e^{ikx}= \partial_x+ik.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why do not all the semimetals turn into the excitonic insulators? I want to know the condition for forming the excitonic insulator.
When the binding energy of the exciton, $E_b$, overcomes the band gap, the system becomes the excitonic insulator. If so, all semimetals should meet this condition and be supposed to turn ... |
When the binding energy of the exciton, Eb, overcomes the band gap, the system becomes the excitonic insulator.
What is meant here is the direct gap, i.e., the gap between the conduction and valence band states located at the same point of the Brillouin zone (having the same quasi-momentum). On the other hand, in mos... | {
"language": "en",
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On the equivalence of the Schrödinger and Heisenberg (and all other) pictures The Schrödinger and Heisenberg (and, indeed, infinitely many other pictures) are often referred to as equivalent descriptions of quantum dynamics in a given system. I'm wondering two things in particular:
*
*What exactly do we mean by equiv... | (1) Indeed, you say it correctly: both representations ("pictures") are equivalent because there is a one-to-one correspondence between their predictions. More precisely, there is an isomorphism whereby any solution in the Schrödinger picture admits one solution and only one solution in the Heisenberg picture.
(2) Rega... | {
"language": "en",
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In Hamilton-Jacobi theory, how is the new coordinate $Q$ time-independent when Hamilton's principal function separates? Following the notation in Goldstein, the solution to the Hamilton-Jacobi equation is the generating function $S$ for a canonical transformation from old variables $(q,p)$ to new variables $(Q,P)$ wher... | The culprit of OP's question seems to be that Goldstein considers 2 different generators for a type-2 canonical transformation (CT) in Hamilton-Jacobi theory:
*
*Hamilton's principal function $S(q,P,t)$. Here the Kamiltonian $K\equiv 0$ is zero. Therefore all the new phase space variables $(Q_S^i,P_j)$ are constants... | {
"language": "en",
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Spin Connection Vanishes? I'm trying to reproduce a result for the components of the spin connection in FRW spacetime. The formula for the spin connection $\Gamma_{\mu}$ is
$$\Gamma_{\mu} = \frac{1}{2} \Sigma^{a b} L_{a}^{\nu} L_{b \nu, \; \mu}$$
Where $\Sigma^{ab} = \frac{1}{4}[\gamma^{a},\gamma^{b}]$ are Lorentz grou... | I think this is good exercise for me. Please tell me if I made any mistakes.
The vierbein are
$$
L_0^\mu=\frac{1}{a(t)}\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\quad
L_1^\mu=\frac{1}{a(t)}\begin{pmatrix}0\\1\\0\\0\end{pmatrix},\quad...
$$
The nonzero Chrstoffel symbols are
$$
\Gamma^t_{tt}=3\frac{\dot a}{a},\quad
\Gamma^... | {
"language": "en",
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Relativistic ship and base on Earth talking Let say humans make it possible to travel so close to the speed of light that time dilation for the ship can achieve 100 times the time of a standing still object. So one of that ships starts its journey and after a while something goes wrong and they have to call the base on... | Let's say there are two events:
A. The pilot starts their message
B. The pilot finishes their message
In the rocket frame, which we'll call $S^{\prime}$, the time between those events is $\Delta t_{AB}^{\prime}$ as you've said in the question. To work out the time between the message being first received and ending on ... | {
"language": "en",
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Why I cannot write the time evolution operator $e^{-i(T+V)t}$ as the product of operators $e^{-iTt}e^{-iVt}$ To calculate the wave equation of a time-independent Hamiltonian we use:
$$
\Psi_{i}(r,t)=e^{-iH^{0}t}\psi_{i}(r,0).
$$
We also know that the time-independent Hamiltonian $H^{0}=T+V$ is given to the sum of kinet... | When we put the Hamiltonian $H$ in an exponential like this, we are not assuming the $H$ represents a scalar value. The exponential notation used in this case is just shorthand for the Taylor series equivalent for the exponential function:
$e^{-iHt} \equiv 1 + (-iHt) + \frac{(-iHt)^2}{2!} + \frac{(-iHt)^3}{3!} + \cdot... | {
"language": "en",
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Can we have motion in systems where inertia is neglected? According to Newton's law:
$$ \sum F=ma$$
So, if we have some acceleration, it's because we have a certain amount of motion in our systems.
This makes me confused if inertia was neglected. Are there cases where we can have motion when inertia is neglected? Maybe... | We sometimes assume a body has negligible mass, such as a rope in a pulley system where the pulley and weights on the ends of the rope have much greater mass than the light rope. This simplifies the evaluation of the motion. See a basics physics textbook for examples.
| {
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What is the origin of the inertia? Is there any explanation why it is harder to move an object with more mass than an object with lesser mass? What kind of force is opposing our force? Is it finalized currently and well known what the origin of inertia is, or not yet 100%? I read some explanations that are linked to ge... | Currently, there is no scientific consensus on why inertia exists, or why the geodesic through spacetime is the least path of energy (ultimately leading to: why do we need to expend energy to move an object away).
Although, we know how much force is needed, we don't know why, and so this question may be well suited for... | {
"language": "en",
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Is there a fundamental lower bound to resistor precision? The resistor noise index provides a lower bound to how precise a resistor can be measured, even using ideal instrumentation. Effectively, it even makes no sense to define the resistance any more precise than given by its noise index. Intuitively, this can be und... | Resistor excess noise is poorly understood. There is, as far as I know, no verified model that can yield the answer you seek.
This is a fine example of a physical effect that is observable with simple lab equipment but remains a mystery.
| {
"language": "en",
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Can we say that Black hole is a Black body? I want to know if a black hole is a black body.I got this doubt while studying photoelectric effects.
I am an 11th grader.I would be glad to have your help.
| The Hawking radiation theoretically emitted by a black hole has a black body spectrum. So in a sense we could say that a black hole is a black body, although at a really really low temperature (a tiny fraction of a degree above absolute zero). But there is nothing deep about this - it is really just a coincidence of te... | {
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Why are fields described as force divided by mass or charge? I have read that application of force on a body from a distance, like gravitational or electrostatic force is a two-step process, first, the field is created by the body, then, the application of force on the second body by the field. I want to know why the e... |
Why are fields described as force divided by mass or charge?
Because they follow from the classical universal law of gravitation and Coulomb's law.
The force that each of two masses or charges experience is due to the gravitational or electric field generated by the other mass or charge.
For gravity the magnitude of... | {
"language": "en",
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Are two states with the same measurement probabilities necessarily equal up to unitary equivalence? Let $\rho$ and $\rho'$ be $n\times n$ density matrices, and suppose that for every observable $A$ and every $\lambda$ in the spectrum of $A$ we have
$$
\text{tr}(\rho P_{\lambda})=\text{tr}(\rho' P_{\lambda}),
$$
where $... | Actually $\rho=\rho'$. Indeed, specializing $A= P= |\psi\rangle \langle \psi|$ for $||\psi||=1$, the hypothesis implies $\langle\psi| (\rho-\rho')\psi\rangle =0$. Linearity permits to relax the requirment $||\psi||=1$. By polarization, in turn, it implies $\langle\psi| (\rho-\rho')\phi\rangle =0$ for every pair of vec... | {
"language": "en",
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"source": "stackexchange",
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The value of $g$ in free fall motion on earth When we release a heavy body from a height to earth. We get the value of $g=9.8 \ ms^{-2}$. Now, I'm confused about what it means. For example, does it mean that the body's speed increases to $9.8$ every second? Or, does it mean that the speed of the body is $9.8 \ m/s$?
| It means the speed of the falling body increases with 9.8 m/s each second.
| {
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Solving 3D Kepler Problem substitution goes wrong I'm trying to arrive at the effective potential equation in Kepler Problem using Routh reduction method. We can procede in two ways, either using polar coordinates in the plane where the orbit happens or using spherical coordinates. I'm having trouble with this last one... | Substituting some conserved quantity is not how the Routhian mechanics works. If your Lagrangian is $\mathcal{L}(r,\dot{r},\theta,\dot{\theta},\phi,\dot{\phi})$ (we do not exclude $\phi$ here as generalization for other possible potential), we always have a Routhian $R(r,\dot{r},\theta,\dot{\theta})$ given by
$$R(r,\d... | {
"language": "en",
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Is there a version of the Einstein field equations that uses the Riemann curvature tensor instead of the Ricci curvature tensor? I understand that the Einstein field equation uses the Ricci curvature tensor, Ricci curvature scalar, and stress-energy momentum tensor. But is there a way to form an equation that uses the ... | If you mean an equation such that the stress energy tensor at a point of the spacetime determines the Riemann tensor at that point, then the answer is negative for physical reasons. Such an equation would imply that the spacetime is flat outside the gravitational sources. This makes no sense because the gravitational f... | {
"language": "en",
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Double slit experiment: Are electrons interacting with other electrons to create a wave? Assume a double slit experiment with electrons and no observer (light source). Can the wave-like behavior and resulting interference pattern be explained by the single electron that is being shot, doesn't really travel to the detec... | No, the experimental evidence does not support the idea that the interference patterns are created in the way you suggest. Aside from the fact that experiments have been performed in a vacuum, which rules out the idea that the incoming electrons are interacting with other particles in their path, diffraction experimen... | {
"language": "en",
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Why are positrons traveling backward in time in Feynman diagrams? We draw the positron as if it's traveling backward in time. Why? The momentum of the positron is drawn opposite to the time-direction of the diagram. I don't see this having any effect on the calculation. We still arrive at the result $k=p_1+p_2$, and no... | The arrow direction in a Feynman diagram does not correspond to the time direction. There are conventions in which times flows from left to right, i.e. the time on the left is always earlier than the time on the right, although these are not the most common ones. Regardless of the convention, the arrow direction is jus... | {
"language": "en",
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Breaking down of 2nd law of thermodynamics Do you know a scenario where the second law of thermodynamics breaks down?
| The second law breaks down when some of the assumptions underlying this law (or the thermodynamics itself) are broke.
Among what is typically cited as "violations" of the second law are:
*
*Violation of the laws of thermodynamics in small systems. Thermodynamics and statistical physics are applicable in thermodynamic... | {
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The Absorption of light I just want to know, how can we explain the phenomena of : 'absorption of light' based on the wave aspect of light ( light is an electromagnetic wave). In fact, light has an electromagnetic field, so it has an electric field which interact with atoms's electrons, I think at this point, somethi... | Every light absorption has a preceding light emission. Electrons emit photons after an excitation. And that is where we should start.
These electrons are elementary electric charges and are, last but not least, elementary magnets. To the photons they bequeath both the electric and the magnetic field. And they do so in ... | {
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Will quantum events ever occur on a macro-scale rather than a vacuum? Michio kaku says there's a chance we'll wake up on Mars tomorrow https://www.theatlantic.com/science/archive/2018/10/beyond-weird-decoherence-quantum-weirdness-schrodingers-cat/573448/
In this post, it is shown that quantum decoherence in the macro w... | I will add that there are everyday examples of quantum effects on the macro scale here on earth. My favorite is the laser, where simulated emission is purely quantum but those emission events do not average out to zero; instead they add together and are blindingly obvious on the macro scale.
| {
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Interpretation of Feynman Slash Notation I'm self-learning Relativistic Quantum Mechanics and was playing around with the Dirac equation when I noticed something.
I was trying to interpret the meaning of ${\not} \partial$. So since I can represent a 3-vector by a traceless hermitian matrix: $$\vec{x}=x^i \vec{e}_i \to ... | Yes, of course; people do this routinely. Read your texts on.
The essence of the map is its invertibility through
$$\operatorname{Tr} \left(\gamma^\mu\gamma^\nu\right) = 4\eta^{\mu\nu},$$
so that
$$v_\mu ~~ \mapsto ~~ {\not} v \equiv v_\mu \gamma ^\mu ~~
\mapsto~~ \tfrac{1}{4}\operatorname{Tr} \left( {\not} v\gamma^... | {
"language": "en",
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Why is angular momenum related to the spin? What I know about spin ½ particles is that they are represented by spinors, and thus, you need to apply a 720° rotation in order for the spinor to return to its original value. Spin 1 particles are vectors, and their transformation is trivial.
Also, spin ½ have $\frac{\hbar}{... | A very simplified, hand-wavey explanation is to say that in the same way that linear momentum is related to one over the wavelength of the wavefunction as you move each linear direction, so the angular momentum (spin) is related to one over the 'wavelength' (really we need a new word, like 'waveangle' for this idea) of... | {
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If water is nearly as incompressible as ground, why don't divers get injured when they plunge into it? I have read that water (or any other liquid) cannot be compressed like gases and it is nearly as elastic as solid. So why isn’t the impact of diving into water equivalent to that of diving on hard concrete?
| Well, depends on the way someone dives in and the altitude.
You might have wondered at some point, why does it hurt when you fall into the water with your belly, but when you fall in with your feet almost vertically to the water you feel no pain. On the internet you can definitely find some answer on this question, for... | {
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Finding Locally flat coordinates on a unit sphere I know this is more of a math question, but no one in the Mathematics community was able to give me an answer, and since physicists are familiar with General Relativity, I thought I might get an answer.
Imagine a unit sphere and the metric is:
$$ds^2 = d\theta ^2 + \cos... | starting with components of the unit sphere :
\begin{align*}
&\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}=\left[ \begin {array}{c} \cos \left( \phi \right) \sin \left( \theta
\right) \\ \sin \left( \phi \right) \sin \left(
\theta \right) \\ \cos \left( \theta \right)
\end {array} \right]
\end{align*}
from... | {
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How does a decelerating universe agree with this version of the Friedmann-Robertson-Walker formula? I was watching a video discussing dark matter, and he presented the following simplified version of the Friedmann-Robertson-Walker equation:
$$ \left(\frac{\dot{a}(t)}{a(t)}\right)^2 = \frac{8\pi G}{3} \rho(t) -\frac{k}{... | The video is "Dark Energy" by DrPhysicsA, and the claim in question is at 19:45.
It's a mistake. It would have been correct to say that if $ρ(t)>0$ at all times and $k\le 0$, then the RHS is positive at all times, so there is no time at which $a'(t)=0$, so $a'(t)$ can't switch sign, so if the universe is expanding at a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/719178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why do we give so much importance to energy, i.e., the conserved quantity under time symmetry? In almost all equations—from GR to QFT—energy conservation is a tool for solving those equations, but we know that energy on large scales is not conserved.
Why do we still use this (not) conserved quantity in our fundamental ... | As a complement to Nickolas's great answer, note that throughout physics, conservation laws are so useful that even approximate conservation laws can be of enormous interest. Conservation laws let you deduce aspects of the behavior of a physical system, without needing to actually solve equations. This is tremendously ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/719642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How can Entropy be maximal when it is undefined everywhere else? This question is about classical thermodynamics.
I learned that when an isolated system is not in equilibrium, its thermodynamic variables such as Entropy are undefined.
I also learned that when an isolated system is in equilibrium, its Entropy is maximiz... | As you wrote, entropy is undefined in non-equilibrium conditions. However, one can compare entropy in different equilibrium states of the same system. That is a meaningful comparison, and in particular, one can compare the entropy value in conditions characterized by the same values of the thermodynamic variables but d... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/719989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 6,
"answer_id": 4
} |
Angular momentum of a planet about apogee
Angular momentum of a planet about its apogee is maximum at __________
Now, I do know that
*
*Angular momentum of a planet around the focus of the elliptical orbit (the sun) is conserved due to gravity being a central force. (Side note: this gives Kepler’s second law.)
*Th... | I just found out an explanation: For local extrema of $\vec L$, we must have $\dfrac{d\vec L}{dt}=0$ but we know $\dfrac{d\vec L}{dt}=\vec \tau$ so that $$\tau_{\text{about apogee}}=0$$ which happens at only the apogee and perigee. It is easy to notice that apogee gives the minimum (because $\vec r=\vec 0$) so that $\v... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/720152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How conservation of energy looks like in the moving frame? It is obvious that a car accelerates by converting its chemical energy in the fuel to produce kinetic energy to accelerate itself.
If the energy is lost into friction and heat, the car will slow down.
But from the point of view of the car, it always appears to ... | Let' say there are two earths, on one of those earths a car is accelerating.
Now the car says that the kinetic energy of the earth that the car is not pushing increases by a huge amount, and the kinetic energy of the earth that the the car is pushing increases by the same huge amount plus the energy of the burned fuel.... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/720630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What does an area represent in a spacetime diagram? If I have a spacetime diagram with $t$ on the vertical axis and $x$ on the horizontal axis, does calculating an area in this diagram have any physical significance?
The reason I ask is because I'm trying to figure out if the $u$ in all the $\cosh(u)$ and others hyperb... | The name itself of "hyperbolic functions" is due to the fact that the functions $cosh(t)$ and $sinh(t)$ play, in the parametric representation of the equilateral hyperbola $x^{2}-y^{2}= a^{2}$
the same role as the functions $cos(t)$ and $sin(t)$ for the circle
$x^{2}+y^{2}=a^{2}$
The parametric representation of circl... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/720934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
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