Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
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 $9.8 \ m/s$?
| The other guys here (@Thomas Fritsch and @AWanderingMind) are perfectly right, and just to see that: g is an acceleration, and acceleration is change of velocity with time, or velocity per time. Like velocity itself is distance per time.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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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 not like Newton's cradle. Experiments with electron beams are done in vacuum.
The EM field is responsible for the interaction of the electron with its surroundings (the starting electrode, the slit, the detector, the walls of the chamber, etc). The EM field fills all space! A famous theory is from Richard Feynman... | {
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Does melting a metal affect its electronic band stucture? Given that the band structure of a metal emerges from the periodicity of the crystalline lattice and the corresponding symmetry arguments, what happens to the band structure as the metal is melted into its liquid state? Would some form of a limited band structur... | What really emerges from the periodicity of the crystalline lattice and symmetry arguments is the so-called dispersion of the energy bands, i.e., the introduction of a (vector) parameter for the electronic states, the wavevector ${\bf k}$, labeling the electronic states that are simultaneously eigenstates of the Hamilt... | {
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How can quantum tunneling happen conceptually? I have read in Griffiths' Quantum Mechanics that there is a phenomenon called tunneling, where a particle has some nonzero probability of passing through a potential even if $E < V(x)_{max}$.
What I don't understand about this is how to conceptualize how this can happen. I... | You're treating Quantum objects as balls. This is misleading. When working with Quantum Mechanics, picture the object as the entire wave. So tunneling happens because parts of the wave pass through the potential. If there isn't a lower energy across the wall, the wave won't pass.
I like to imagine it as a water wave an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Does dusk really remain for a shorter period of time at the equator? It is said that the dusk remains for shorter time at equator than the poles. Because, the equator rotates faster than poles. But it is also true that time is the same in every latitude, and if it's true, then the dusk should remain the same at equator... | It is faster because the sun takes a higher trajectory through the sky typically, and crosses the horizon steeper and thus faster.
| {
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"source": "stackexchange",
"question_score": "16",
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Why doesn't the variation of resistivity with temperature go both ways? I've learnt that the variation of resistivity with temperature for a conductor is:
$\rho=\rho_0(1+\alpha (T−T_0))$
Let's consider resistivity at 0℃ and 100℃.
When heating the conductor from 0℃ to 100℃,
$ρ₁₀₀=\rho_0(1+\alpha (100-0))$
α=$\displaysty... | Maybe we can see this as a purely mathematical misunderstanding, and disregard the discussion about whether such a formula is an approximation (there could in principle exist a material for which the linear relationship was exact, at least in some temperature interval).
So more abstractly, the relation:
$$
y = y_0(1 + ... | {
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If we apply a magnetic field to a core saturated by a permanent magnet, what will happen? If we apply a magnetic field to a core saturated by permanent magnet, will the magnetic field of the permanent magnet and electromagnet get combined?
I mean to say superposition will be applied?
| Yes, they will add up. They will get stronger if they are pointing in the same direction, and weaker if the electromagnet's magnetic field is pointing oppositely to the permanent magnet's magnetic field.
But the material inside won't get magnetized further. The magnetic field from an electromagnet and a permanent magne... | {
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Can the operator field Dirac equation be expressed as Heisenberg's equation? The Dirac equation of the operator spinor field is:
$$(i\gamma ^{\mu}\partial _{\mu} -m)\psi =0$$
where $\psi$ is interpreted to be a quantum field.
I'm wondering, can this be derived from the Heisenberg equation?
$$\frac{d\psi}{dt}=\frac{-i}{... | Suppose you want to change $\psi$ by a "tiny" amount $\delta \psi$. This $\delta \psi$ has to fulfill the same anti-commutation relations as the $\psi$, in particular
$$
\{ \delta \psi, \psi \} = 0
$$
You can generate such a $\delta \psi$ by use of the commutator (not the anticommutator): Suppose for example that
$$
H ... | {
"language": "en",
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Worldsheet constraint Bosonic String I am currently studying David Tong's notes on String theory and there’s a step taken in writing out the worldsheet constraint in lightcone coordinates $\sigma^{\pm}$ for the closed string that I’m not sure about. We have the constraint eq 1.38 written out on page 26 as
$$(\partial_{... | It is just a dummy variable change, from $p$ to $n:=m+p$. Since $m$ and $p$ run through all integers, $n$ also runs through all integers. $\newcommand{\ex}[1]{\mathrm{e}^{#1}}$ Stripping off the physics we have
$$ \sum_{m=-\infty}^\infty\sum_{p=-\infty}^\infty a_m\ b_p \ c_{m+p} = \sum_{m=-\infty}^\infty\sum_{n-m=-\in... | {
"language": "en",
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What's the importance of all four fundamental forces being "curvature"? I've heard about how, in a gauge theory, the gauge covariant derivative of the field around a closed curve is generally not zero, and this is how you can quantify force or field strength. And that this is the same basic idea as curvature, with the... | In the 1920s–1940s, people developed a unified classical theory of gravity and electromagnetism using just this sort of approach. It's called Kaluza-Klein theory. Some aspects of it even generalize to classical non-abelian Yang-Mills theories (R. Montgomery: Canonical formulations of a classical particle in a Yang-Mill... | {
"language": "en",
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"source": "stackexchange",
"question_score": "9",
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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... | Entropy could be minimized in equilibrium. Or it could be 70% of the min. It's not. Instead, it's maximized.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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"answer_id": 5
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Transverse component of distorsion tensor in GR On pages 164-165 of Eric Gourgoulhon's lecture notes on Numerical Relativity, the author introduces the decomposition (9.49) for the distorsion tensor related to a foliation $(\Sigma_t)_{t\in \mathbb{R}}$ with induced metric $\gamma_{ij}$. This tensor is defined as
$$ Q_{... | The paper you are referring to proves that any symmetric tensor field can be decomposed into transverse-traceless, longitudinal and trace parts, eq. 2 and 7.
$$ \psi_{ab} = \psi_{ab}^{\rm TT} + \psi_{ab}^{\rm Tr} + \psi_{ab}^{L}$$
In the above,
$$ \psi_{ab}^{\rm Tr} = \frac{1}{3}\Psi g_{ab} = \frac{1}{3}\psi_{cd}g^{cd}... | {
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Product notation for operators If I have a Hamiltonian
$$\mathcal{H} = \prod_j^N Z_j$$
where $j$'s are different sites on a lattice and $Z$'s are Pauli $Z$ operators does that mean that the Hamiltonian can also be written as
$$\mathcal{H} = Z_1 \otimes Z_2 \otimes \cdot \cdot \cdot Z_N$$
and if they are all Pauli opera... | Short answer: yes. Long answer: yes.
| {
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Differential charge existing We define current by $I=\frac{\mathrm{d}q}{\mathrm{d}t}$. Here, $\mathrm{d}q$ is the infinitesimal element of charge. But again,we know that charge is quantised meaning there is a finite value to the smallest amount of charge which is $e$. Since $\mathrm{d}q$ is infinitely small, $\mathrm{d... | You're mixing up two descriptions that are, in practice, separate.
$i=dq/dt$ is usually used in macroscopic physics, when it is understood that you don't study actual individual electrons. In fact, most of the corresponding physics laws predate quantum mechanics, even predate the discovery of the electron.
In other wor... | {
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Can plasmas be black bodies? I have recently heard the claim that sun can not be composed of plasma because plasma can not be a black body.
I am an uneducated layman, I've seen a lot of people (laymen) deviate from accepted scientific consensus. I am skeptical and I don't have enough knowledge about physics to argue it... | Plasma in many concrete cases often is not a black body, e.g. plasma in Earth's ionosphere, or in a discharge lamp, or in tokamak. This is because plasma in these cases is very thin (rarified gas), and not a good absorber of radiation, as there is not enough layers to make it absorb close to 100% of incoming radiation ... | {
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Would water flow from the higher container to the lower one? Two identical open-topped containers contain identical amounts of water, and they are connected by a tube at their bases. Container A is higher than Container B. How much water, if any, will flow from Container A to Container B? How does this change if the he... | I like the willingness to experiment!
The result of the experiment is indeed expected. Basically, because there is a connection this is all one body of water. If the surface of a body of water is higher on one side then the water will flow downhill until the surface is level. So here the water will continue flowing unt... | {
"language": "en",
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Why does the graviton polarization satisfy $\epsilon_{ij}(\mathbf{k},\lambda)\epsilon^{ij}(\mathbf{k},\lambda') = 2 \delta_{\lambda\lambda'}$? I am reading the paper ``Graviton Mode Function in Inflationary Cosmology'' by Ng (link here). The graviton $h_{ij}$ is here expanded (in the TT gauge) where
$$
h_{ij}(x) \sim \... | The 2 is conventional, but it makes sense when you think of the simplest form these tensors take when $\hat{k} \propto \hat{z}$:
$$\epsilon_{ij}^+ =\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}
$$
and
$$\epsilon_{ij}^\times =\begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & ... | {
"language": "en",
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Why flapping rudder produce net thrust if one half-stroke produce thrust and second half-stroke drag? In small sailing boat like optimist is well know technique when there is no wind, rudder pupming which push boat forward.You just need push-pull rudder stick left to right with fast movement.
Rudder works complety unde... | Below the horizontal line is my original answer, submitted 5 hours ago, but there is a better explanation that I overlooked.
In a comment to another answer Gordon McDonald points out that since the rudder hinges right at the stern the rear edge of the rudder sweeps out a sector of a circle. That alone will tend to resu... | {
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Problem trying to graphically represent a 2D vector given angle and intensity I'm very new to physics so forgive me if it's a trivial question, but it's something I have trouble figuring out.
I'm trying to solve an exercise, and the exercise says that we're given
$|\overrightarrow{v}| = 2.8 N$, where the vector and the... | *
*The angle of a vector is usually measured from the positive x-axis to the vector, with clockwise angles counting as positive. Thus a vector in the same direction as the positive y-axis has an angle of 90 degrees and a vector in the same direction as the negative y-axis has an angle of 270 degrees (or -90 degrees).
... | {
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Computing the maximum force a rod can bear Suppose I had a rod of diameter $d$ composed of some material with tensile strength $T$. If I then exterted a pulling force $F$ on the ends of the bar, how do I compute the force $F$ for which the rod will break apart? Is there some general equation that I can use to compute... | If you know ultimate tensile strength $T$ of material, then knowing breaking force of rod is trivial,
$$ F_{~br} = T \cdot A $$
,where $A$ is rod cross-section area.
However there is no way to compute ultimate tensile strength of materials, it can only be known from Tensile testing of materials, with some exceptions. F... | {
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Integral of time dependent Hamiltonian Computing the time evolution of a quantum system described by a time-dependent Hamiltonian, $H(t)$, amounts to constructing the time evolution operator
$$U = \mathcal{T} \exp \Biggl( -i \int_{0}^{t} \mathrm{d} \tau \ H(\tau) \Biggr) \ . $$
What if the time-dependence in $H(t)$ can... | You have to be careful about the expression of $U$. The expression you put down mathematically means
\begin{align}
U & = \lim_{n \rightarrow +\infty}{\big(e^{-{i \over \hbar}{t \over n}H(t)}\big)\big(e^{-{i \over \hbar}{t \over n}H(t(1-{1 \over n}))}\big)\big(e^{-{i \over \hbar}{t \over n}H(t(1-{2 \over n}))}\big)\cdot... | {
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Why are bloch factor orthogonal? The Bloch wave can be expressed as:
$$
\psi_{n\mathbf{k}}(\mathbf{r}) = u_{n\mathbf{k}}(\mathbf{r})\,e^{i\mathbf{k}\cdot \mathbf{r}} \tag{A1}
$$
In this problem Bloch wave they say that $u_{n\mathbf{k}}(r)$ is orthogonal. I would like to ask whether $u_{n\mathbf{k}}(r)$ itself can be no... | Bloch's theorem tells us that the energy eigenvectors of a Hamiltonian with a periodic potential can be written
$$\psi_{n\mathbf k}(\mathbf x) = e^{i\mathbf k \cdot \mathbf x} u_{n\mathbf k}(\mathbf x)$$
where $n\in \mathbb Z$, $\mathbf k\in \mathrm{BZ}$ (the first Brillouin zone), and $u_{n\mathbf k}(\mathbf x)$ is pe... | {
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Is there any problem in having a stress-strain constitutive relation that relates time-derivative of stress with strain? We usually use two empirical laws to model viscoelastic behaviour:
*
*Hooke's law of elasticity that relates stress with strain
*Newton's law of viscosity that relates stress with time-derivative ... | In short
Proving such model impossible is probably too ambitious as it does not seem to violate thermodynamic requirements in absolute.
It is nonetheless absent from the literature.
This makes sense as it enables a variety of unusual behaviors.
Examples
For the example below, consider a 1D material that behavior follow... | {
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How can the energy-momentum tensor influence the metric outside an energy-momentum distribution? The elements in the energy-momentum tensor are determined by the mass-energy-impuls distribution as viewed from an inertial frame.
So, if you see a collection of masses with different momenta you can fill in their values in... |
... how that gives you the values of the metric outside of the masses.
Outside the masses the components are all zero, so you would expect a
flat metric, which obviously is false. Is there a kind of analytical
continuation going on?
Metric is spacetime. Its existence presuppose a presence of matter there. A solution ... | {
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Brightness of bulbs in Parallel When adding bulbs in parallel, the brightness is brighter than that of series. But does that mean adding bulbs in parallel will increase the brightness of the other bulbs?
My intuition is as follows: When adding a bulb in parallel the current doubles, but that current splits between the ... | You are correct.
When you put them in parallel, each bulb is seeing the full supply voltage. Hence each bulb will get the same current as it did on its own. So, each bulb shines with the same brightness it would have if there was only one bulb. Of course this assumes the supply is able to provide twice the current.
Whe... | {
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Why does the opposing force differ in when falling on concrete vs on water in spite of Newton's third law? If a person jumps from the first floor of a building and lands on a concrete surface, they will suffer serious injury because of Newton's third law.
If the same person jumps the same distance and lands in swimming... | Lets look at the energy conservation
$$\frac{m}{2}\,v_i^2+m\,g\,x_i=\frac{m}{2}\,v_{f}^2+F_{f}\,x_{f}$$
where f is the final state ans i is the initial state
if both case is the final velocity $~v_f=0~$ but the distance
$~x_{fc} \ll x_{fw} $ this means that the force that injured you $F_{fc} \gg F_{fw}$
where "c" for... | {
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If Aristoteles was right and heavier objects falled faster towards the ground how would be Newton's Laws of Motion described? It seems like it would be like:
a(m)=km
and may be a(m1,m2)=K(m1-m2)
Am I doing any sense?
Btw I'm no negationist, nor I'm trying to create a negationist movement here, I just wonder how physics... | You can define it in multiple ways. Let's say near the earth, the weight is not
$$-mg$$
(pointing downwards)
rather:
$$-m^2g$$
Then from Newton's second law:
$$ma=-m^2g \implies a=-mg$$
So objects with greater mass accelerates more.
But you can also define the gravitational force as:
$$-me^{m/k}g$$
for a constant k, an... | {
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Why is it easier to raise AC current to high voltage than DC? In my country (and maybe all around the world I don't know) once electricity has been generated, it is then raised to 200k Volts for transportation.
I know this is to reduce the loss. Given $P=U.I$ and $P=I^2.R$, raising U will lower I and so limit the loss ... | Changing the Voltage of AC can be done with a simple iron core transformer. That's a simple device without moving parts that only consists of a magnetic core, copper wire and some isolation (optionally a cooling fluid). Almost nothing that can break. Good transformers can have amazing efficiency of way more than 95%.
T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dispersion equation with variable wavenumber The wave equation
$$u_{tt}=c^2 u_{xx}$$ is known to have a simple wave solution $u(x,t)=Ae^{i(kx-\omega t)}$ where the dispersion equation is simply $c=\omega/k$. Yet, let the wavenumber be a function in $x$, then the independent variable $x$ will appear in the dispersion so... | Since you have only one wave $k=\frac{2\pi}{\lambda}=cst$, so $k_{x}=0$ , your formula is simpler : $$c^{2}=\frac{-\omega^{2}}{(ik)^{2}}$$
i.e.$$k=\frac{\omega}{c}$$
Note: the last relation gives:
$$(2+x^{2})k_{x}i=k+xk_{x}-\frac{\omega^{2}}{c^{2}}$$
$k(x)\in \mathbb{R}$, for the equation to be homogeneous,
in the lef... | {
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What is the fractional frequency stability of a thermal damped harmonic oscillator? Suppose I have a lightly driven (classical) damped harmonic oscillator at temperature $T$. Suppose $\omega$ and $Q$ are specified as well as the mean energy $\bar{E}$ in the oscillator due to the driving/dissipation equilibrium. What wi... | The by-now ancient work/publication but also time-tested by billions of well-designed oscillators and probably will answer your question is Leeson: A Simple Model of Feedback Oscillator Noise Spectrum, Proc. IEEE, 1966 pp329-330.
| {
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Friction coeffecient between two stacked blocks moving at constant velocity So I came across a problem, it says that there are two masses, $m_1$ and $m_2$, stacked on top of each other, and they are moving at a constant speed. There is also friction between the two blocks, with coefficient $\mu$. It gives us the values... | As you see in the Free Body Diagram, the equilibrium equations are:
$$
F_y=N-m_2g=0 \to N=m_2g \\
F_x=f_r=0
$$
As you see, if your assumption is the block is moving at a constant speed, the friction force $f_r$ over him must be zero.
If $m_1$ start moving from $v=0$ with a positive acceleration, the friction force $f... | {
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What happens to resistance of tap water as voltage is increased? In recent days I have done a few experiments measuring the current of water as it goes up from 9 volts up to 36 volts, and following Ohms law to convert it to resistance. And I discovered a very interesting trend. In between 9 and 18 volts, there is a mas... | You are electrolytically decomposing your test electrodes. They must be made of platinum to prevent this effect.
| {
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Why does a rocket engine that produces a constant thrust over a set period of time have less energy if it has more mass? (Zero-$g$) A rocket engine with the thrust of 1N working for 10 seconds will add more kinetic energy to the rocket if it is attached to a 10kg rocket and less if it is attached to a 20kg rocket. The ... | Because you are ignoring the exhaust. Also, I think you are only looking at this in the frame where the rocket starts at rest. Let's look at that frame first.
When we have a force between two masses (like a bullet and a rifle) then:
*
*The change in momentum between both masses are equal
*The change in KE is great... | {
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What reason/evidence do we have to think that the Planck length is the smallest length possible? From what I've gathered, Planck length is the smallest measurable length, though we do not know whether it is the smallest length physically possible. The Planck temperature is called the theoretically highest temperature, ... |
Or are there only amateurs who think so?
Bingo :) The Planck units don’t give sharp limits on the existence of quantities like length, time, etc (or rather, there’s no reason to think that they do). Instead, they provide a scale at which a more complete theory of fundamental physics (which incorporates quantum gravit... | {
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How is Newton per meter Cubed related to Newton per meter squared (=Pascal)? Is there a way to relate $\frac{N}{m^3}$ to $\frac{N}{m^2}$?
| As already pointed out, this is the unit of pressure gradient. But it could also be a “weight density”.
From the standpoint of a physicist, it’s conceptually cleaner to express the weight per unit volume of a substance as a mass density (which for incompressible substances is invariant, and thus a more fundamental char... | {
"language": "en",
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Limit definition of scalar curvature for flat vs curved space in 2D, 3D and so on in Zee In Zee's book, Einstein Gravity in a Nutshell, p. 6 + p. 77, he says that
\begin{equation}
R = \text{lim}_{\text{radius} \rightarrow 0} \frac{6}{(\text{radius})^2} \left(1 - \frac{\text{circumference}}{2\pi \text{ radius}} \rig... | On p. 6 + p. 77 Ref. 1 is apparently talking about the Gaussian curvature in $d=2$, which is half the scalar curvature in $d=2$.
Later on p. 345 + p. 350 Ref. 1 is talking about the scalar curvature $S=g_{ij}R^{ij}$, so let's do the same here.
The Wikipedia page lists that in $d$ dimensions, the scalar curvature is
$$
... | {
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Friedmann equation I've seen in literature
$$\dot{H} + H^2=\ldots$$
Source: https://en.wikipedia.org/wiki/Friedmann_equations
Defining the LHS. Since
$$H = \frac{\dot{a}}{a}$$
And that
$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}(\rho + 3P)$$
Then replacing gives
$$H^2 = \frac{8\pi G}{3}(\rho + 3P)$$
So my que... | Note that $H=\frac{\dot{a}}{a}\implies\dot{H}+H^2=\frac{\ddot{a}}{a}$. You're asking about the special case $k=0,\,\Lambda=0$, but seem confused about what results we obtain. In this case, the Friedmann equations are$$H^2=\frac{\dot{a}^2}{a^2}=\frac{8\pi G\rho}{3},\,\dot{H}+H^2=\frac{\ddot{a}}{a}=-\frac{4\pi G(\rho+3p/... | {
"language": "en",
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Scaling of thermal resistance to air vs oil For electronic components, thermal resistance from component to ambient(air) is often given, and used with a resistive model of 'thermal resistors' to make temperature rise calculations simple. These values are usually given in some context such as to component to air, to hea... | No, one cannot generally estimate the thermal resistance of one fluid by scaling the value for another fluid by the ratio of their thermal conductivities.
The reason is that heat transfer in fluids is often dominated by convection, and convection is mediated by many more parameters than just the thermal conductivity.
F... | {
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Deriving the Vlasov equation in {$\vec r, v_{||}, \mu, \varphi$} coordinates I'm reading some lecture notes on drift kinetics and I'm having trouble with one derivation. The general idea is changing phase space coordinates from {$\vec r, \vec v$} to {$\vec r, v_{||} \text{ (parallel velocity)}, \mu \text{ (magnetic mom... | Equation (2.9) is a mathematical identity -- it is always true because (in the collisionless limit) $f$ satisfies
$$\frac{df}{dt}=0$$
Fundamental to the calculation in Felix's notes is that
$$\dot{\varphi} = \Omega_s + \text{small corrections}$$
It is not possible to make this derivation comprehensive with any simple t... | {
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Deriving $\langle H\rangle$ from average momentum and position for a LHO Assume that we know the values of $\langle x\rangle$ and $\langle p\rangle$ for a LHO, that is in a random superposition of zeroth and first state. Derive $\langle H\rangle$.
So I tried solving this problem with writing $H=\hbar\omega (a^*a+1/2)$.... | The system is in a state
$$\lvert\Psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle\qquad \alpha,\beta\in\mathbb{C}$$
The complex constants $\alpha$ and $\beta$ are to be determined up to an arbitrary phase factor using the condition on the average values and the normalization condition
$$|\alpha|^2+|\beta|^2=1.$$
We... | {
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Propagators for a generic Lagrangian density Suppose we have a generic Lagrangian density, for example:
$$\mathcal{L} = \alpha A_{\mu\nu}A^{\mu\nu} + \beta B_{\mu}f_\nu(p^2) A^{\mu\nu} + \gamma B_\mu\partial^\mu h$$
where $A_{\mu\nu}$,$B_\mu$ and $h$ are generic fields, $f_\nu(p^2)$ a generic function and $\alpha$,$\be... | *
*Formally speaking, if the Hessian matrix of the quadratic action is non-degenerate, the free propagator is given by the inverse matrix, cf. e.g. my related Phys.SE answer here.
*Concerning OP's last question: If a matrix element is zero, the corresponding inverse matrix element might not necessarily be zero.
| {
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On the proof that $4$-velocity transforms like vector Let $U$ and $U'$ be the $4$-velocities associated to the coordinates $(t,x)$ and $(t',x')$ related through the Poincaré transformation $P:\mathbb R^4\to\mathbb R^4$, i.e. $(t',x')=P(t,x)$.$^1$
Of course the Jacobian $\Lambda\in\mathbb R^{4\times 4}$ of $P$ is a Lore... | I think you are making this unnecessarily difficult. Here is a proof.
$$
U = \lim_{\delta\tau \rightarrow 0} \frac{X(t+\delta\tau) - X(t)}{\delta\tau}
$$
Now use that $\delta\tau$ is invariant and the difference of two 4-vectors evaluated at a given event is itself a 4-vector (which is easy to prove). It follows that $... | {
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Does the earth’s rotational angular velocity change? This is what is written in The Feynman Lectures on Physics, Vol. 1 (ch.5)
We now believe that, for various reasons, some days are longer than others, some days are shorter, and on the average the period of the earth becomes a little longer as the centuries pass.
Wh... | The Earth is not a single rigid body, but consists of at least five separate regions which can move relative to one another. These are the crust (which is the region that we use to measure day length), the mantle, the core, the oceans and the atmosphere. Although the total angular momentum of the Earth may not change, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If reference frames are equally valid, then why do teachers say the geocentric view is wrong? If all reference frames are valid, then why is the geocentric model taught as "wrong" in schools?
I've checked many websites but none of them clear the issue. Wiki says that in relativity, any object could be regarded as the c... | If 'geocentrism' means that you can regard the Earth as stationary and describe the motion of Sun and planets accordingly, then geocentrism isn't wrong.
But if 'geocentrism' means that The Sun and planets have simple (for example circular) orbits about the Earth, then it is wrong. Almost 2000 years Ago, Ptolemy knew th... | {
"language": "en",
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"source": "stackexchange",
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Why Is Capacitance Not Measured in Coulombs? I understand that the simplest equation used to describe capacitance is $C = \frac{Q}{V}$. While I understand this doesn't provide a very intuitive explanation, and a more apt equation would be one that relates charge to area of the plates and distance between them, I'm havi... | The definition of capacitance, $C=Q/V$, suggests that it should be measured in the units of charge per units of potential.
Remark: What is more amusing is that in some system of units (e.g., in cgs) the units of capacitance turn out to be the units of length.
| {
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Why does particle leave circular motion after string slacks? If a particle is attached to a string and made to move in a vertical circle with initial velocity of $\sqrt{4gl}$ $m/s$ where l is the length of string, at some angle (approx $131°$ with the initial position), the string slacks and the particle leaves the cir... | Because tension and weight works in tandem to break the circle this time, and tension is not supported by any of centripetal forces, so after some time circle collapses :
| {
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Pascal's law further simple proof for students of an high school Pascal's law states that a force applied on a surface of a fluid is transmitted within the fluid in all directions of the fluid with the same intensity on equal surfaces. Similarly, it can be stated that pressure exerted at one point of a fluid mass is tr... | The pressure varies linearly with depth $h$, but the other two pressures are constant without any depth variation transmitted everywhere by Pascal's Law.
School students who understand histograms as changes in height of a graph with respect to a horizontal independent varying quantity may also hopefully follow pre... | {
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The size of the universe and the scale factor of $\Lambda$CDM model I wonder is there a relation between the size of the universe and the scale factor calculated by solving Friedmann equations.
I mean if the volume of the universe nowadays is a round $V= 10^{78} m^3$, does this mean the current value of the cosmologica... | The size of the universe and the size of the observable universe are different things. The radius of the observable universe is equal to the conformal time times the scale factor if they're appropriately normalized.
If $k=\pm 1$, the scale factor is the radius of curvature of the spatial slices. Commonly it's called $R... | {
"language": "en",
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How electrolytes conducts electricity? While studying electrochemistry, I came across two key points that I'm unable to understand.
why does DC alone break down the electrolytic liquid
and
b) Why doesn't AC do the same?
| I'm going to assume breakdown means the reassociation of the dissolved analyte in solution. Suppose the dissolved salt is potassium. The redox potential for K is:
K+ + e− ⇌ K(s) at -2.93 V
Any voltage less than -2.93 V will create solid potassium at the electrode and reduce the number of K ions near the surface. This ... | {
"language": "en",
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Work Integral and its derivation The work integral is something I saw long time ago and in completely understood it.
\begin{align}
W_{12} & =\int F(x)dx=m\int^{t_2}_{t_1}adx=m\int\left(\frac{dv}{dt}\right)dx=m\int\left(\frac{dv}{dx}\right)\left(\frac{dx}{dt}\right)dx\\
&=m\int\left(\frac{dx}{dt}\right)dv=\frac12\left(m... | They're using the $2^{nd}$ principle of dynamics $\frac{d \mathbf{Q}}{dt} = \mathbf{F}$ to replace $\mathbf{F}$ with $\frac{d \mathbf{Q}}{dt} = \frac{d}{dt}(m\mathbf{v})$.
With the assumption $\dot m = 0$, you can further manipulate the expression $\mathbf{F} = m \frac{d \mathbf{v} }{dt} $, before writing the work inte... | {
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Why does a piece of thread form a straight line when we pull it? Experience tells that if we pull a piece of thread, it forms a straight line, a geodesic in the Euclidean space. If we perform a similar experiment on the surface of a sphere, we will get an arc of a great circle, which is also a geodesic.
How to show thi... | As @Fardin pointed out, you only get a straight line in the absence of gravity. While gravity is active, the string will form a catenary.
In general, the string will try to follow the shortest distance between the 2 endpoints. If it didn't do that, the tension on the string would try to shorten it. The definition of a ... | {
"language": "en",
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Sakurai on the time-evolution operator I have three questions about Sakurai's discussion of the time-evolution operator.
*
*First question:
In equation 2.12, Sakurai requires the composition property of the time-evolution operator:
$$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$$
Why is this required?
*Second question: In equat... |
Third question: Why is time evolution represented by an operator at all when, as Sakurai points out, time is not an observable like position or momentum? Observables are represented by operators.
In addition to J. Murray's answer, I'd like to add an answer only to this specific question, because I suspect the OP migh... | {
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Confusion about Transforming Christoffel Symbols I'm trying to understand how transforming Christoffel symbols works. Specifically I'm thinking about the transformation between Schwarzschild and Eddington-Finkelstein coordinates,
$$\Gamma^v_{\;vv}=\frac{\partial v}{\partial x^m}\frac{\partial x^n}{\partial v}\frac{\par... | transformation between Schwarzschild and Eddington-Finkelstein coordinates.
the line element of Schwarzschild metric is:
$$ds_S^2=- \left( 1-{\frac {{\it r_s}}{r}} \right) {{\it dt}}^{2}+{{\it dr}}^{2}
\left( 1-{\frac {{\it r_s}}{r}} \right) ^{-1}+{r}^{2}{d\Omega }^{2}
$$
and of Eddington-Finkelstein metric
$$ds_E^2=-... | {
"language": "en",
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I'm having trouble understanding the intuition behind why $a(x) = v\frac{\mathrm{d}v}{\mathrm{d}x}$ I was shown
\begin{align}
a(x) &= \frac{\mathrm{d}v}{\mathrm{d}t}\\
&= \frac{\mathrm{d}v}{\mathrm{d}x}\underbrace{\frac{\mathrm{d}x}{\mathrm{d}t}}_{v}\\
&= v\frac{\mathrm{d}v}{\mathrm{d}x}
\end{align}
However, this feels... | Writing for a general case, $v$ can be an explicit function of both $t$ and $x$ (for 1D motion along $x$).
$\therefore$ \begin{equation}
dv=\frac{\partial v}{\partial x}dx+\frac{\partial v}{\partial t}dt \Rightarrow \frac{dv}{dt}=\frac{\partial v}{\partial x}\frac{dx}{dt}+\frac{\partial v}{\partial t}=a \quad ...(\star... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cross product and spinor correspondence I wonder if there is a correspondence between a cross product of two vectors $\vec{x}, \vec{y} \in \mathbb{R}^3$ and their associated spinors $\lambda^\alpha, \tilde{\lambda}^\dot{\alpha}$ and $\omega^\alpha, \tilde{\omega}^\dot{\alpha}$.
Here is what I mean by that:
Given two ve... | from the Wikipedia
$$\vec x\mapsto X\quad,\vec y\mapsto Y
\quad,\vec z=\vec x\times\vec y\mapsto Z$$
$$\frac 12\left(X\,Y-Y\,X\right)=i\,Z\quad,\rm det(Z)=0$$
with
\begin{align*}
&X=\begin{bmatrix}
\xi_{x1} \\
\xi_{x2}\\
\end{bmatrix}
\begin{bmatrix}
-\xi_{x2} & \xi_{x1} \\
\end{bmatrix}\quad ,\vec x\cdot \vec x... | {
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A pendulum in a superfluid Imagine to submerge a pendulum in a supefluid. Of course we assume an ideal pendulum, whose joint does not freeze or deteriorate due to the extremely low temperature. We also assume the superfluid to be at zero temperature, so we can neglect its normal component.
What happens to its oscillati... | If the pendulum is fully submerged (and the velocity of the sphere is sufficiently small) then the flow around the sphere is Stokes flow, and the drag is proportional to viscosity. As a result there is no drag in a zero temperature superfluid. If the sphere is only partially submerged, then it can excite surface waves ... | {
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"source": "stackexchange",
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How to derive the $vx/c^2$ term from first principles? In Lorentz transforms, the formula for time transformation is
$$t' = \gamma \left( t - \frac{v x}{c^2} \right)$$
I understand that the term $\frac{v x}{c^2}$ represents "time delay" seen by a stationary observer but I don't understand how to derive it from first pr... | I will try with the diagram below, we suppose that the container ijfg is filled with water, the light crosses this container of the face $f$ towards the face $g$ with a speed $v$ and put a time $t$ to cross it, we have $$l=vt$$
if there is no water, the light will travel a distance L =ct (during the same time t),we hav... | {
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Is there a known closed-form expression for the susceptibility of the 2-D Ising model at $B = 0$? The Onsager solution for the 2-D Ising model allows us to find (among other things) complicated expressions for the internal energy of the system (in the thermodynamic limit and in zero magnetic field):
$$
u \equiv \frac{U... | There are no explicit expressions, as far as I know, only expressions in the form of (complicated) infinite series, originating from expressing the magnetic susceptibility as a sum over 2-point correlation functions and using the exact expressions known for the latter. These have been used to analyze the remarkable ana... | {
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Could you feel your weight falling through the a tube drilled through the center of the earth? Suppose you drill a hole through the center of the earth (assume the earth is uniform and no air resistance) and you jump in. Would you be "weightless" throughout the entire fall?
The reason I ask is that if you jump off a cl... | In order to feel your weight something must be present that prevents you from free falling.
In everyday life the ground we are standing on provides the barrier that keeps us from free falling.
You refer to the implication that an object in free fall inside a corridor straight through the center of gravity of a gravita... | {
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Regarding Lenz's Law presented in hyperphycsics The following diagram is presented in hyperphysics as an introduction of Faraday's Law and Lenz's Law.
If the red arrows represent the direction of current, then what do the positive and negative poles across the resistor means? From my understanding, resistors do not pro... | It seems you got confused by the drawing about what is cause
and what is effect.
Of course the resistor does not produce the voltage.
The voltage is produced by the coil.
And this voltage is then consumed by the resistor.
For reducing confusion let us first consider the situation without the resistor.
The changing mag... | {
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Various expressions for total amplitude in Frederic Schuller's German QM lectures In this lecture at around 53:31, the equation for total amplitude in terms of elementary amplitude along a path is written down.
$$\phi_{\text{total} } ( \overline{x_0} , \overline{x_n} , t_N- t_0) = \lim_{N \to \infty} \prod_{i=1}^n \in... | Yes, $$A(\epsilon)~=~\sqrt{\frac{2\pi i\hbar \epsilon}{m}}, \qquad \epsilon~=~\frac{t_N-t_0}{N},$$ in eq. (3) is the famous Feynman fudge factor, which needs to be included in the path integral measure. For details, see e.g. this, this & this Phys.SE posts.
| {
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Associativity of covariant derivatives I'm having trouble proving that covariant differentiation is an associative operation.
Essentially I'll have to show
$$\nabla_\mu( \nabla_\nu \nabla_\sigma) = (\nabla_\mu\nabla_\nu) \nabla_\sigma. $$
But is it enough to show that both LHS and RHS yield the same result when acted... | Frankly it boils down to function/operator composition, which is associative.
Take a general tensor $T^\alpha_\beta$.
$$(\nabla_\nu \nabla_\sigma)T^\alpha_\beta = \nabla_\nu (\nabla_\sigma T^\alpha_\beta )$$
so
$$\nabla_\mu( \nabla_\nu \nabla_\sigma)T^\alpha_\beta = \nabla_\mu(\nabla_\nu (\nabla_\sigma T^\alpha_\bet... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Is there a conflict, or is there not a conflict between the Pusey-Barrett-Rudolph (PBR) theorem and the information theory interpretation? In the wikipedia article, it says that the PBR theorem sort of rules out the psi epistemic interpretations. I want to know, is this the end of the information theory interpretation ... | OKay so PBR is solely a statement about hidden variable theories, i.e. theories which say that the pure state is an incomplete description of physical reality, and that a hidden state provides the complete description.
Since the relational interpretation and the information theoretic interpretation do not assert the ex... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Weight in Interplanetary Space How is weight zero in interplanetary
space? The Moon is orbiting the Earth because of the gravitational pull of earth. Then gravity must exist in interplanetary space too. So any body in space must also have an acceleration due to gravity ($g$) but $g$ must actually be 0 for weight to be ... | Depends how you define weight. Operational weight, (which you measure with weight scales) is zero, of course because body doesn't exert any force on scales/support operating in Earth orbit or space.
However, gravitational weight defined as $$ W = G \frac {Mm}{r^2} $$
is not zero, because body $m$ is attracted gravitati... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Why do we need an earth wire? Apologies if this question has already been asked before.
In this video and other sources, it says that the ground/earth wire is connected to the outside metal casing of an electrical appliance in order to create a low-resistance path back to the live wire in case of a fault. However, if t... | The ground the person is standing on would be the return path. The Earth is an effectively infinite sink for current, at 0 V.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Is the interaction picture in QFT properly used? $\newcommand{\ket}[1]{\left\vert#1\right\rangle}$In Quantum mechanics, we have the interaction picture. When the Hamiltonian is in the form of $H=H_{0}+V$, we can transform the evolution equation into $\frac{d}{dt}\ket{\phi}_{I}=V_{I}\ket{\phi}_{I}$, where $\ket{\phi}_{I... | This confused me as well. Remember that the $\phi$ in $\frac{g}{4}\phi^4$ that we used in the Interaction picture has time dependence. More precisely, the $\phi$ that we use is the free field solution, i.e. it is the solution to $\frac{d \phi}{dt}=-i[\phi, H_{0}]$, which is the same as the value of $e^{iH_0t}\phi e^{-i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Can we just take the underlying set of the spacetime manifold as $\mathbb{R^4}$ for all practical purposes? In mathematical GR and also in some informal GR presentations (eg: MTW), manifolds are always mentioned before talking about GR... but now I am starting to wonder.. if it even actually neccesary?
In this answer, ... | Topological censorship is a theorem from the 1993 paper "Topological censorship" by Friedman, Schleich and Witt. It is a technical statement about certain manifolds (!), and it does not say that "it's apparently impossible to actually check the topology at a global level" as the question claims.
The paper explicitly sa... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/733848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do we prove that the 4-acceleration transforms as a 4-vector in Special Relativity? In order to define the acceleration of a body in its own frame, we need to first prove that the acceleration is a four-vector so that its dot product with itself can then be labeled as acceleration squared in the rest frame. For vel... | Is it not so by definition?
$$
{\bf a}= \frac {d{\bf v}}{d\tau}
$$
where
$$
{\bf v}= \frac{d{\bf x}}{d \tau}
$$
is a 4-vector and $\tau$ is a scalar.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Experimentally Measuring the Velocity of Water coming out of an Orifice I plan on doing an investigation into Torricelli's Law, where I will be looking at one of the following:
*
*How the cross-sectional area of an orifice affects the velocity of water coming out of it (constant height).
*How the height of an orifi... | Place a measurement grid by the stream and start filming. As clear water flows, add food color to the water. Measure the movement of the front of the colored water versus the grid. Preferably, use clear pipes to allow measurement of the water velocity before it leaves the orifice.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Why does a sensitive thermometer absorb little heat? In an experiment to measure the specific heat capacity of water I'm trying to make it as accurate as possible. And somewhere I read that a sensitive thermometer absorbs little heat. By "sensitive" I am referring to the amount of change in thermometric property for a ... | A thermometer that absorbs a lot of heat will change the temperature of what it is measuring and then measure the wrong temperature.
To be sensitive, any sensor must disturb its enivronment in a very predictable way. For a thermometer that is easiest to achieve for one that minimizes heat absorption.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Why does using images that are not really formed work in ray optics? It's all in the title. For instance, if I have two lenses , I have been taught to first find the position of the image formed by the first lens, and then use that image to find the final image formed by the 2nd lens, if the first image is formed beynd... | I addressed this before but will elaborate further. Refer to the diagram here.
Suppose there is an object R on the axis a distance r to the left of a lens with focal length f and r<f. When the rays leave the lens they diverge as if coming from a point P a distance p to the left of the lens. So P is a virtual image. We ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Local $SU(2)$ symmetry breaking and unitary gauge In a $SU(2)$ gauge field theory with scalar field $\phi$ in the fundamental representation of the $SU(2)$ group with lagrangian $$\mathcal{L} = -\frac{1}{2}TrF_{\mu\nu}F^{\mu\nu} + (D_{\mu}\phi)^\dagger(D^{\mu}\phi) + \mu^2\phi^\dagger\phi - \frac{1}{2}\lambda(\phi^\dag... | Your results are right. That is exactly what it should be here. Quoted from Weinberg
$$0=\sum_{nm}\tilde{\phi}_m (t_\alpha)_{mn} v_n\qquad\qquad(21.1.2)$$
Eq. (21.1.2) shows that there are no Goldstone boson fields in unitarity
gauge. Since the theory is gauge-invariant this means that there are no
physical Goldstone ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Definition of momentum We say that momentum is the measure of how a body is moving or the quantity of movement inside a body
But what this definition really mean?
This terms are very vague
$p=mv$,why the movement inside the body depend on it's mass?
| (In classical mechanics), the definition of momentum is $\vec{p}=m\vec{v}$.
The reason this is a good definition is because it is useful. In particular, the momentum of a collection of particles that are not in an external potential is conserved. Conserved quantities make it possible to understand aspects of the behavi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/735142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
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Optical theorem Peskin and Schroeder I'm trying to understand the optical theorem of Peskin and Schroeder
$$\tag{7.50} \text{Im} M(k_1,k_2\rightarrow k_1,k_2)=2E_{cm}p_{cm}\sigma_{tot}(k_1,k_2\rightarrow\text{anything})$$
which Peskin and Schroeder says follows from
$$\tag{7.49} -i[M(a\rightarrow b)-M^\ast(b\rightarrow... | From your comments it looks like you're mainly confused about the $E_1 E_2 | v_1 - v_2 |$ prefactor, so I'll try to be very explicit about that part. Start from Eq. (7.49) and take the initial and final states to be the same (that is, $b = a$). Then we get
$$
2 {\rm \:Im \:} \mathcal{M}(a\rightarrow a) = \sum_f \int d ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/735269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to describe the physics process of scintillation? I want to find some references on describing the physics of scintillation. As we know the lights generated by scintillator through atom activation and de-activation, and each material has a
spectrum and its intensity veries with wave length as shown in the figure be... | To calculate scintillation yield of materials is impractical. It's determined experimentally.
The power of mathematics in physics blinds many to its severe weaknesses. While it may be insightful, even very simple problems are often very difficult to compute. Of course, textbooks avoid these problems, thus perpetuating ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/735790",
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"source": "stackexchange",
"question_score": "2",
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Relation between velocity and mobility of electrons and holes I have been studying band theory and semiconductors in condensed matter physics and I am confused about the relation
between mobility and velocity of electrons and holes in semiconductors.
My standard text book reference, Introduction to Solid State Physics,... | The expression
$$
\mathbf{v}(\mathbf{p}) = \nabla\epsilon(\mathbf{p})
$$
is the velocity of an electron with momentum $\mathbf{p}$. This velocity can be calculated for an electron anywhere in the band.
On the other hand, the velocity associated with mobility is the drift velocity,
$$v_d = \mu E,$$
which describes the v... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/736250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I catch someone falling from a short height without hurting or bruising them I am the backspot in the stunts for cheer and I keep catching our flyer but I am not absorbing her fall so she has bruises underneath her arms. What is the physics of catching her without hurting her?
| Cheer coaches can give much better practical advice than we can, but from a physics perspective the goal is to have a uniform de-acceleration over a long time and distance instead of an abrupt de-acceleration. Once you receive them on the way down you want to slow them as smoothly as possible over the longest possib... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/736792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Magnetic dipole Hamiltonian from current-current interaction In the Coulomb gauge, we can write the electromagnetic Hamiltonian as
\begin{equation}
\label{eq:em-hamiltonian}\tag{1}
H_\mathrm{EM} = - \int d^3 x \, \mathbf{j}(\mathbf{x}) \cdot \mathbf{A}(\mathbf{x})
+ \frac{1}{2} \int d^3 x \, d^3 x' \, \frac{\rho(\m... | Start with the (particular choice) of $\mathbf{A}(\mathbf{x})$ for a uniform field,
$$
\mathbf{A}(\mathbf{x}) = -\frac{1}{2} \mathbf{x} \times \mathbf{B},
$$
and set $\rho(\mathbf{x}) = 0$. Then $H_\mathrm{EM}$ reduces to
$$
H_\mathrm{EM} = \frac{1}{2} \int d^3 x \, \mathbf{j}(\mathbf{x}) \cdot \big( \mathbf{x} \times ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of free parameters in $SU(5)$ GUT model Lately, I have been studying the potential of scalar fields in this theory. In general, what is the point of this GUT if, there, more free parameters have been added?
The standard Higgs potential in the Standard Model with only 2 free parameters (Higgs mass and self-coupli... | There is no fixed significance to free dimensionless parameters in a theory examined; e.g., m, as used in the SM, is not quite a mass, etc.
The GUT action you wrote is the most general renormalizable SU(5)-invariant potential given the fields involve, a 24 and a 5, and the discrete symmetries excluding the odd terms.
S... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can we say that work done by carnot engine in a cycle equals net heat released into it even when it is operated b/w 2 bodies and not 2 reservoir? When a carnot engine is operated between 2 reservoir then after each cycle it return to its initial state so change in internal energy is zero and so work done by it equa... | I think the question is asking how—if the Carnot engine's original state was at $T_\mathrm{high}$, corresponding to the initial temperature of a high-temperature finite body—can the engine return to this original state after a cycle that removes heat from that body, bringing it to $T_\mathrm{high}-\delta T$. Is this co... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Why does high frequency have high energy? The electromagnetic spectrum's wavelengths all travel at the same speed, $c$. Also, the wavelength $\lambda$ and frequency $\nu$ are related by $c = \lambda \cdot \nu$. Since all moving particles here would have the same speed, why would higher frequencies have more energy?
| Massless and massive particles (like photons and electrons respectively) have different dispersion relations, i.e., the relations between the particle momentum and its energy, $\epsilon(p)$. Thus, for electrons we have
$$
\epsilon(\mathbf{p})=\frac{\mathbf{p}^2}{2m}
$$
whereas for photons
$$
\epsilon(\mathbf{p})=c|\mat... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 8,
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Magnetic field modeling with noises I am trying to make a 3d grid of a magnetic field with some noises (which will be added to the ordinary field) for a computer simulation. I have the formula for the ordinary field, also I am using a fast Fourier transform (FFT) to create Gaussian Random Field for noises.
The problem ... | You can use the fact that the divergence-free constraint, $\nabla\cdot\mathbf{B}=0$, becomes
$$
\mathbf{k}\cdot\tilde{\mathbf{B}} = 0
$$
in Fourier space, where $\tilde{\mathbf{B}}$ denotes the Fourier transform of $\mathbf{B}$ and $\mathbf{k}$ is the wavevector.
To get a vector-valued field, you could first generate a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can both of these equations for pressure be correct? Consider the Gibbs equation:
$$du=Tds-pdv$$
Identifying partial derivatives, one obtains:
$$-p=\left( \frac{\partial u}{\partial v} \right)_T$$
But you can also show that:
$$p=T\left( \frac{\partial s}{\partial v}\right)_T -\left( \frac{\partial u}{\partial v} \r... | Yes, both are true. Let consider this equation first, $$p=T\left( \frac{\partial S}{\partial V}\right)_T -\left( \frac{\partial U}{\partial V} \right)_T $$ For an ideal gas at constant temperature $\frac{\partial U}{\partial V}_T$ is not zero, but this is derived from constant gibbs energy thus it becomes zero.
Now, $\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Peeling theorem for a generic field We know that in asymptotically simple space-times, if the generators of conformal boundary $\mathscr{I}^{\pm}$ satisfies the asymptotic Einstein's condition, then any purely outgoing (incoming) field can be written as polynomial in $1/r$ (Refer to section 9.7 of "Spinors and space-ti... |
Is there a peeling theorem for a generic propagating field <…>?
“Peeling” implies that spacetime is smooth near null infinity (in the sense that it possesses conformal compactification with smooth boundary), but suitably generic spacetimes are not, they develop logarithmic singularities around null infinities. So lit... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/737873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why can the coriolis force potential be written as $ E_\text{cor}=m\dot{\theta}\begin{vmatrix} X & Y \\ \dot{X} & \dot{Y} \end{vmatrix}$? I found the following formula for the Coriolis force written here:
$$ E_\text{cor}=m\dot{\theta}\begin{vmatrix}
X & Y \\
\dot{X} & \dot{Y}
\end{vmatrix}=m\dot{\theta}\ (\dot{Y}X-\do... | the potential energy for Coriolis force is:
$$U=-m\,\vec v\cdot (\vec \Omega\times \vec r)$$
with
$$\vec r=\begin{bmatrix}
{x} \\
{y} \\
{z} \\
\end{bmatrix}\quad,
\vec v=\begin{bmatrix}
\dot{x} \\
\dot{y} \\
\dot{z} \\
\end{bmatrix}\quad,
\vec \Omega=\dot\theta
\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bma... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/738156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it better to average each reading before applying a formula, or apply the formula to each set of readings and then average? If $f$ is some function of independent variables $a,b ...z$ and readings of each of them have some inherent (random and systematic) error, would it better to average the readings of each variab... | Ask yourself: What would I like to know? Usually, we are most interested in the end result $y=f(x)$ and not in the intermediate result $x$. Therefore, we are interested in the distribution/average value/standard deviation etc. of the end result. However, $f(\bar x)$ is not the average value the end result, and $f(\bar... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/738404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does a small puddle of water evaporate faster at the edges than the center? I have read that ceiling tile stains and coffee ring stains are darker on the edges than the center because the puddles evaporate fastest at the point of contact between the surface, air, and water and water that is evaporated leaves behind... | All liquids are not evenly spaced like a rectangular block, but rather like an irregular ellipsoid with a bulge in the center. Its impossible to discern this bulge with the naked eye, however, it is very visible in mercury:
Why this bulge
is created in the first place because of surface tension. The liquid tries to ha... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/738566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Interesting relationship between the 2D Harmonic Oscillator and Pauli Spin matrices I have an isotropic 2D Harmonic Oscillator in cartesian coordinates
\begin{equation}
H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{1}{2} m\omega^2 (x^2 + y^2)
\end{equation}
In terms of the usual creation and annihilation operators fo... | The Hamiltonian in question has cylindrical symmetry, and can be transformed to cylindrical coordinates (i.e. $x,y\longrightarrow r, \phi$). The wave function then decomposes into radial and angular part, and the eigenstates obtained called Fock-Darwin states. Interestingly, the solution is simple also works with magne... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the state of an entangled photon after its twin is absorbed? Let's two photons are entangled in polarization after a laser beam passes through a Betha Barium Borate crystal. They take different paths and one of them (1) is absorbed in a black sheet. What is the state of the leftover photon (2)? Is it in superpo... | What do the experiments say? With a quantum measurement, the measured state depends on what measurement is performed. If you assume that the photons individually have states before measurement, you get Bell's Inequality, and the experiments falsify this. It thus doesn't make sense to ask what the state of the photon is... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conservative Force: Translational Invariance I have a question about the following.
Why if there are two masses, $m_1$ and $m_2$ respectively, and the only force acting on them is from their mutual interaction which is conservative and central, the following is true?
$U(\vec{r_1},\vec{r_2})=U(\vec{r_1}-\vec{r_2})$
(It ... | Here is a slightly artificial but mathematical proof. I will look at scalars, not vectors, to make the derivation slightly easier. But it should be easily extendable to vectors. Define
\begin{align}
r&=r_2-r_1&\iff &&r_1&=\tfrac 12(R-r)\\
R&=r_2+r_1&&&r_2&=\tfrac 12(R+r)\tag{1}
\end{align}
These quantities have the sam... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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A particle on a ring has quantised energy levels - or does it? The free particle on a ring is a toy example in QM, with the wavefunction satisfying $-\frac {\hbar^2}{2mr^2}\psi_{\phi\phi}=E\psi$ and the cyclic BCs $\psi(\phi+2\pi)=\psi(\phi)$. This problem is easily solved to give $\psi_m=e^{\pm im\phi}$ for $m=0,1,2,\... | $\psi^2$ is most certainly not observable: Observables are the eigenvalues of hermitian operators. For a state $\psi$ to be physical, the expectation values for any observable under this state needs to be physical, ie for instance $\langle\psi|\hat{H}|\psi\rangle<\infty$. The problem is that for the proposed $\psi$ (wh... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/739892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 5
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Paradox regarding Young-Laplace equation Recently I've been working with the Young-Laplace equation and came across the following physical paradox that I couldn't wrap my head around. It goes like this:
Imagine a spherical droplet (filled with water) in air. By the Young-Laplace equation, we know that the pressure in ... | Pressure is uniform all around the droplet. Remember that stress is a vector quantity.
Equilibrium of an elementary part of the interface.
The equilibrium of the elementary surface of the interface reads,
$\mathbf{t_n} dS = \Delta P \ \mathbf{\hat{r}} \ dS + d \boldsymbol{\Gamma}$,
being $\Delta P \ \mathbf{\hat{r}}$ t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is this form of writing the six antisymmetric gamma matrices correct? I encountered the following expression in Ashok Das' QFT Lectures:
$$\sigma_{\mu \nu } =\frac{i}{2}[\gamma ^\mu,\gamma^\nu]=i(\eta ^{\mu \nu}-\gamma^\nu \gamma^\mu)=-i(\eta ^{\mu \nu}-\gamma^\mu \gamma^\nu).$$
I understand why the expression with... | Well, we know that the gamma matrices obey the Clifford algebra $$\{\gamma^\mu,\gamma^\nu\}=\gamma^\mu\gamma^\nu+\gamma^\nu \gamma^\mu=2\eta^{\mu\nu}$$
Because of that we have
$$i(\eta^{\mu\nu}-\gamma^\nu\gamma^\mu)=i\left[\eta^{\mu\nu}-(2\eta^{\mu\nu}-\gamma^\mu\gamma^\nu)\right]=i(-\eta^{\mu\nu}+\gamma^\mu\gamma^\nu)... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is energy "equal" to the curvature of spacetime? When you are solving the Einstein field equations (EFE), you basically have to input a stress–energy tensor and solve for the metric.
$$
R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} = 8 \pi T_{\mu \nu}
$$
For a vacuum solution we have:
$$
T_{\mu \nu} = 0
$$
This yields:
$$
R_... | To provide a more plain answer to your question:
No, energy and the curvature of spacetime are two different things. Energy is a physical quantity that is associated with the motion or arrangement of matter and radiation, while the curvature of spacetime is a property of space and time that is determined by the distrib... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 3
} |
Question about uncertainty principle and attempts at simultaneous measurement of position and momentum Uncertainty principle for position and momentum: $$
\Delta x \Delta p \ge \frac{h}{4\pi}$$
So suppose we have a particle... and we have 2 different measuring devices. The first measuring device measures position. The ... | The question you are asking simply does not make any sense in Quantum Mechanics. Quantum Mechanics says that, upon a position measurement, the particle becomes a position eigenstate. And upon a momentum measurement, the particle becomes a momentum eigenstate.
There is no state that is simultaneously both position and a... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Why do channels arise from "failing to record measurement outcomes"? In Preskill's notes, the need for quantum channels began with the following situation: System A starts out in a pure state and interacts with system B, therefore forming a joint state of system AB. We then imagine measuring system B (the pointer) but ... | Quantum mechanical systems interact with the environment. If they do so, we can think of the environment as effectively measuring the system -- formally, this is completely equivalent. However, the information is "hidden" in the environment, and we don't have access to it (and it is typically very hard to access it, as... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/740892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Given a magnetic field how to find its vector potential? Is there an "inverse" curl operator? For a certain (divergenceless) $\vec{B}$ find $\vec{A} $ such that $\vec{B}= \nabla \times \vec{A} $.
Is there a general procedure to "invert" $\vec{B}= \nabla \times \vec{A} $? An inverse curl?
(I was thinking of taking the ... | No, there is no inverse curl operator. In that way it is just like the ordinary derivative: $f(x)$ cannot uniquely be determined by integrating $f'(x)$. In general the relation $\vec{B} = \nabla \times \vec{A}$ defines a set of differential equations, which don't uniquely determine $\vec{A}$. For example you can add an... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/741557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How to go from Lagrange equations to d'Alembert's principle? All sources I know show how to use d'Alembert's principle and/or Hamilton's principles to derive Lagrange equations. It is also common to use d'Alembert's principle to derive Hamilton's principle (see Lanczos "the variational principles of mechanics", p.112) ... | On one hand Lagrange equations
$$\frac{d}{dt} \frac{\partial T}{\partial \dot{q}^j} -\frac{\partial T}{\partial q^j}~=~Q_j,\qquad j~\in~\{1, \ldots, n\}, \tag{LE}$$
make sense in pretty much any setting, while on the other hand d'Alembert's principle
$$ \sum_{i=1}^N \left(\dot{\bf p}_i-{\bf F}_i\right)\cdot \delta {\bf... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/741701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Impact of distance from galactic centre on the value of energy in the cosmic ray spectrum where knee is observed? This question is based on the recommendation and great explanation by @Kyle_Kanos. Is it known what causes the "knee" in the observed Cosmic Ray spectrum?
Accepting the reason for the occurrence of knee aro... | The position of the knee should shift with your location in the galaxy, yes, but not to first order. Since the galaxy is disk-like, the appropriate Lamor radius is the thickness of the disk, which is very roughly independent of distance from the center. Or at least to a sufficient approximation, given that this anyway ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/741965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
What would a standing wave of light look like? I want to know what a standing wave of light would like and what properties it might have that are interesting.
| This answer is about how a technical diagram of such a wave looks like, since with the naked eye you won't be able to see a standing wave of light for the simple reason that you can only see the wave part travelling in your direction.
If the two wave components of the standing wave are of the same wavelength like the r... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/742157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 3
} |
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