Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Heat Current (Thermal Conduction) Why isn’t heat current written as $\textrm{d}Q/\textrm{d}t$ and why only as $\Delta Q/ \Delta t$?
| There are a number of historical reasons for this, but the reasons the notation has remained is an important mathematical. Writing $\frac{dQ}{dt}$ would imply that there was a state function $Q$, of which that expression was the time derivative. However, such a $Q$ does not exist. Another way to describe this is tha... | {
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Angular momentum commutation relations The operator $L^2$ commutes with each of the operators $L_x$, $L_y$ and $L_z$, yet $L_x$, $L_y$ and $L_z$ do not commute with each other.
From linear algebra, we know that if two hermitian operators commute, they admit complete sets of common/simultaneous eigenfunctions. The way I... | Be very careful about what the theorem is saying! You posted in a comment:
If two Hermitian operators, A and B, commute and if A has no degenerate eigenvalue, then each eigenvector of A is also an eigenvector of B.
This statement is not symmetric in A and B! How does it apply to our situation? $L^2$ and $L_z$ commute... | {
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Can civilizations orbiting black holes keep time with their interstellar travellers? Due to relativistic-velocity interstellar travel time dilation, travellers would return in the far future of any civilization launching such an effort. A chart from wikipedia indicates typical time dilation ratios for various destinati... | If we assume everything takes place in the Schwarzschild metric, then the time dilation factor is obtained from the metric as
$$c^2 d\tau^2 = c^2\left(1 - \frac{r_s}{r}\right)^2dt^2 - \left( 1-\frac{r_s}{r}\right)^{-1} dr^2 -r^2 d\theta^2 - r^2\sin^2\theta d\phi^2\ ,$$
where $d\tau$ is proper time interval and $r_s$ i... | {
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Spectrum of bosonic Hamiltonian in 2nd quantization I have the hamiltonian
$$H=\varepsilon(a_1^\dagger a_1 + a_2^\dagger a_2) + g(a_1^\dagger a_2 + a_2^\dagger a_1)$$
with $\varepsilon>g\ge 0$, $[a_1,a_1^\dagger]=[a_2,a_2^\dagger]=1$ and all other commutators equal to zero.
What is the spectrum of the Hamiltonian?
In t... | It can be diagonalized in the subspace of states $\{\vert n_1n_2\rangle\, ,
n_1+n_2=N\}$. Indeed, in terms of $\hat L_\pm$ and the total number operator $\hat N$, your Hamiltonian is just
\begin{align}
\hat H=\epsilon \hat N + 2g\hat L_x
\end{align}
with $\hat L_x$ connecting states with the same total $N=n_1+n_2$ (as... | {
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Mass Energy Equation Derivation? I am reading the derivation from Einstein's paper DOES THE INERTIA OF A BODY DEPEND
UPON ITS ENERGY-CONTENT?.
In the end of page 2, he derives that
$$K_0 - K_1 = L\{\frac{1}{\sqrt{1 - v^2/c^2}} - 1\}.$$
The next line he immediately has
$$K_0 - K_1 = \frac{1}{2}\frac{L}{c^2}v^2.$$
I gues... | You should use the following Taylor expansion
$$\frac{1}{{\sqrt {1 - {\varepsilon ^2}} }} = 1 + \frac{1}{2}{\varepsilon ^2} + \frac{3}{8}{\varepsilon ^4} + O({\varepsilon ^6}),$$
where $\varepsilon=v/c$. This is a series expansion around $\varepsilon=v/c=0$, meaning that the speed of particle is much smaller than the s... | {
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Is the squared norm of the four-momentum of a system of N-particles equal to the total mass of the N particles? I know that for a particle, $p \cdot p=\eta_{\mu\nu } p^{\mu }p^{\nu }=p_{\nu }p^{\nu }=-{E^{2} \ c^{2}+| \mathbf {p} |^{2}=-m^{2}c^{2}}$ is obviously invariant, where $m$ is the rest/proper mass of the parti... | In special relativity, rest mass of a system is proportional to net rest energy of that system:
$$
m = \frac{E}{c^2}.
$$
If in systems' rest frame system components move faster they have more energy, hence the system has higher rest energy and corresponding rest mass.
One cannot just add rest masses of the components, ... | {
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Why is there a volumetric dimension to pressure-volume work? This concept was a bit hard to grasp, because I'm bad at seeing the real-world implications of multiplication. Division makes sense, but multiplication is harder for me. Pressure-volume work is measured in $1$ Pa $\times$ m$^3$. It says that $1$ Pa $\times$ m... |
Does that mean that no matter what the volume of the thermodynamic
system is, if it exerts $1$ Pa onto its surroundings, its doing $1$
Joule of work?
No. For a closed system, pressure-volume work is the work done on or by the system resulting in contracting or expanding the boundary of the system (decreasing or incre... | {
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Is it possible to stir the tea so that it cools more slowly? I recently learned that stirring the tea with vertical movements speeds up the cooling process. As a result, I had a question, and whether it is possible to stir it so that it cools down, as slowly as possible. Conditions: the spoon is an aluminum tea spoon, ... | Stirring helps heat transfer (i.o.w. cooling) because it decreases the film thickness of the tea boundary layer with the cup and thereby increases the convection heat transfer coefficient $h$.
Stirring also ensures the tea is at homogeneous temperature, which maximises the temperature difference between the tea and the... | {
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Does it make any difference if we switch particles to their antiparticle? The title itself clarifies the crazy question. Does it make any difference if we switch particles to their antiparticle? By difference, I mean, Would everything alright with the laws of physics?
| I cannot comment on your question, so I write this here: Without going into details of parity violation in the weak interaction, that leads to the fact that particles and their respective anti-particles are treated differently in weak interactions, I answered a similiar question here.
Since this question was only about... | {
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Why do we cool water with ice? I've came up with the following question: Why do we cool water with ice rather than with steel given that the last one has a bigger thermal conductivity?
| There are several factors (temperature, specific heat, volumetric heat capacity, thermal conductivity, and latent heat of fusion for ice) to consider. See the data below for steel and ice. Their relative importance depends on exactly what your goal is (lowest final water temperature and/or time to reach that temperatur... | {
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What motivated Einstein to formulate general relativity? I never really fully understood what motivated general relativity or why the Newtonian concept of gravity was considered problematic.
One thing I always hear is that it is because it doesn't address what causes two masses at a distance to be attracted. Maybe qua... |
I never really fully understood what motivated general relativity or why the Newtonian concept of gravity was considered problematic.
The main problem, and certainly the issue which most motivated Einstein, is that Newtonian gravity is not consistent with relativity. Relativity places an upper limit on all causal inf... | {
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Why is Sellmeier's equation an even function of $\lambda$? According to Sellmeier's formula, the dispersion formula of a transparent material can be written as
$$n^2(\lambda) = 1 +\sum_i\frac{B_i \lambda^2}{\lambda^2 - C_i},$$ where $B_i$ and $C_i$ are empirically-determined material constants. When written this way t... | What would a negative value of the wavelength mean? Well, frequently we define
$$\lambda=\frac{2\pi}{\left|\vec{k}\right|},$$
in terms of the wave vector $\vec{k}$, which would make $\lambda$ nonnegative by defintion. However, if we restrict attention to waves propagating along a single axis, say $\vec{k}=k\hat{z}$, ... | {
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Radiation damage in a crystal Take a single crystal as close to the ideal as possible, say, Si, GaAs, or SiGe, etc. Immerse the crystal in some fixed rate ionizing radiation and measure the radiation damage it has caused to the crystal per unit time. How does the resulting radiation damage depend on the temperature of ... | The colder the crystal gets, the more difficult become diffusion processes in the crystal. Since radiation damage to crystalline materials often involves atoms getting knocked out of their positions in the lattice, and since healing that damage requires diffusion to be enabled, exposing the crystal to radiation at cryo... | {
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What is the proof of $C_{V} = \frac{fR}{2}$? I came across this formula in thermodynamics. Please give me a rigorous proof to this formula. My teacher did not even give any proof neither do any of my books. The formula is :
$C_{V}=\frac{fR}{2}$ where $C_{V}$ is the molar heat capacity at constant volume, $f$ is the tot... | Not really sure if this amounts to a "proof", but, from the equipartition theorem, we know that each degree of freedom has an average KE = $KT/2$ associated with it.
So if you have a system of $N$ particles, you have an average total KE of $fNKT/2=fN(R/N_{a})T/2 =fnRT/2$. (Where $n=N/N_{a}$ is the number of moles, and ... | {
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Do observers near a black hole see far-away onlookers speeding up? I recently watched the movie Interstellar, and I have a question about time dilation, as it's a huge part of the movie. I know that to an onlooker, someone going near a black would appear to slow, due to time dilation, but what would the person nearing ... | Yes, the person near the black hole will see everything else speeding up. We can imagine two twins A and B. One of them, B, goes to the black hole, spends there a long-long time, and then comes back to A. The time loss (or gain) effects from flying back and forth are finite, but the time loss from being near the black ... | {
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Can a body float in the middle of a fluid? Let's say we have a cubic body of side $a$ and made of a material with density $\rho$ and we measure its immersed height in a fluid of density $\rho_f$ by the variable $y$. Then, its potential energy (and considering a gain of potential due to buoyancy) can be written as:
$V =... | Fundamentally, I think it's because you are treating your equation as broader than it really is. It's pretty obvious that it does not apply properly if the RHS is such that V<0 which is the case where the object is floating in the air above the liquid. It very specifically only applies to scenarios where the object act... | {
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Why are constant volume and constant pressure heat capacities basically the same for solids? Are degrees of freedom involved? I knowv that $C_V=\frac{\frac{f}{2} Nk_B}{m}$ and $C_P=\frac{(\frac{f}{2} +1)Nk_B}{m}$. Since for solids their values are very close to each other, I would assume $\frac{f}{2} +1$ is very close ... | To complement the qualitative answers, here's the math. We know that $$C_P-C_V=\frac{VT\alpha^2}{\beta_T},$$ for all materials, where $C$ is the heat capacity (at constant pressure $P$ or volume $V$), $T$ is the temperature, $\alpha$ is the thermal expansion coefficient, and $\beta_T$ is the isothermal compressibility.... | {
"language": "en",
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In special relativity, how do we know that distance doesn't change in the direction perpendicular to velocity? In the theory of special relativity it is said that the distance in the direction of the speed changes by a factor of
$$\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$
How do we know that the distance perpendi... | The answer is straightforward. The formula shows that there is no dilation when your speed is zero, which is what you would expect, otherwise length dilation would be observable in connection with stationary objects. The component of your speed perpendicular to your direction of motion is always zero, hence dilation mu... | {
"language": "en",
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How to find the falling time of an object when acceleration is not a constant? Let's say we are throwing an object from the surface of the earth, this object reaches 70,000km with initial velocity of $10713 \mathrm{m}/\mathrm{s}$ until it reaches the peak high , the g value at 70,000km is $0.068 \mathrm{m}/\mathrm{s}^2... | Even when acceleration is not constant one can apply conservation of energy.
Let the gravitation potential energy of the earth-object system at the highest point be $-C_1$.
By conversation of energy
$$\frac{-GMm}{x} + \frac{mv^2}{2} = -C_1\tag{1}$$
(where $G$ is universal gravitational constant; $M$ is mass of earth; $... | {
"language": "en",
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Problem calculating the ground state radius of Toponium using strong force potential for short ranges If it were true that the top quark were stable we could, in theory, have a hydrogen-like bound state consisting of toponium $T \bar{T}$. Looking at the strong force, for small ranges it takes the form of a coulomb pote... | If you suspect a unit conversion error, the problem-solving strategy is to check the units in every step. This is particularly important if you are working in “natural units” with $\hbar = c = 1$. There are people who insist that natural units are great because you can ignore all of the $\hbar$ and $c$ and stick them ... | {
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Types of light-matter interaction I'm going to study the photoionization, that is the emission of an atom's electron because of it's interaction with EM radiation.
To contextualize it, I would like to know what other effects can the light-matter interaction lead.
| without being exhaustive, here are a few examples:
*
*Reflexion on a metallic surface: interaction between the free electrons of the metallic body and the incident light wave. Basis of the part of optics dealing with mirrors.
*Bragg diffraction: interaction of a light wave with the electrons of atoms regularly space... | {
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Problem with derivation of formula for change in entropy for latent heat We know, $dS=\frac{Q}{T}$[when $Q\to 0$]
$\implies \Delta S=\int\frac{Q}{T}$
$\implies \Delta S=\int \frac{ms}{T}$[m= mass, s= latent heat]
$\implies \Delta S=\frac{ms}{T}\int$
This is where I get stuck, this is clearly due to my lack of knowledge... | You can take out dQ out of the integral when is constant with respect to T.
This happens for a T range near a central vale (say 99-101 C for latent heat of water condensation at around 1 atm).
In case you want a general formula, you could compute dQ as function of the heat capacities
For this, subtract the heat capacit... | {
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Must observables be Hermitian only because we want real eigenvalues, or is more to that? Because (after long university absence) I recently came across field operators again in my QFT lectures (which are not necessarily Hermitian):
What problem is there with observables represented by non-Hermitian operators (by observ... | At the simplest level it is because Hermitian, or more precisely self adjoint, operators have a complete set of eigenstates. The existence of a complete set is essential for the propability interpretation of QM. That the eigenvalues are real is of less importance.
Consider, for exmple, the operators $X,Y$ of the $x$... | {
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Why do colliding magnets stick intead of colliding elasically? In my experience, when two macroscopic magnets collide they stick together. That is, they don't collide elastically. When two magnet-shaped masses collide they will emerge from the collision with (approximately) the same energy. Is there an internal mechani... | One of the major differences between colliding steel balls and colliding steel magnets is the energy that goes into deforming the items as they collide.
When steel balls collide, the colliding surfaces become compressed, which results in forces tending to push the balls back apart.
When magnets collide, it is not merel... | {
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Why force between charges increases when it moves, instead of decreasing? Imagine two positive charges in a space ship moving with a velocity,v with respect to an observer on earth.
according to the person in the spaceship,the electrostatic force between the charges is $F'=(\frac{1}{4π\epsilon_0})\times \frac{q_1q_2}{r... |
But if you are the observer on earth,then the equation becomes
$F=\gamma \times F' $
The above is not correct. The actual formula in this example would be
$F= F' / \gamma$
In the frame of the person in the spaceship the force will indeed be as you have written
$F' =(\frac{1}{4π\epsilon_0})\times \frac{q_1q_2}{r^2}$
E... | {
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Non-equilibrium phase transition I have come across the term Non-equilibrium phase transition. And unfortunately I can't find any examples of such a phenomenon.
What examples of nonequilibrium phase transitions are known? Are there some analogies with "Ising-like" models, which clarify the main peculiarity of such tran... | Good morning:
I came across the same question a while ago. Let me share some insights with you.
When referring to equilibrium based phase change, that would be having the species freely partitioning towards one single equilibrium stage. There, Gibbs Free Energy of the System is minimum (for constant P and T).
However, ... | {
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Why is the power of a car limited? As an introduction this is what's written in my textbook:
$Power=\frac{work\ done\ by\ driving\ force}{time}$
$=\frac{Driving\ force×distance}{time}$
(as distance over time = speed)
$= Driving\ force×speed$
Note: If the power is constant and the body is accelerating then the speed is... | 1.Consider for a moment a car applying a force by the virtue of its internal mechanism, let just about an equal amount of force be applied on it by the force of an external factor say 'wind'. Now the car just about moves with some small speed and as it keeps covering the distance, the work done by the car keeps increas... | {
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If I pull a metal bar for long enough with a constant small force, will it eventually break? Let's say I have a strong metal bar. I pull it apart with a very small constant force -- obviously it doesn't break. However, this would disturb the internal configuration. If I let go, then eventually the internal configuratio... | If the metal and external forces attain an equilibrium state (the forces pulling on it are equal in strength as but oppositely directed to the internal forces generated by the pulling, or vice-versa) then there is no reason to think that it will stretch further. Maybe in time, the metal bar will get tired (metal fatigu... | {
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Is friction always opposite to velocity? Let's say an object is sliding on a slope and is the object has a velocity of $(0,0,5)$. The friction would be acting in the opposite direction of motion, being $(0,0,-1)$.
However, gravity is also affecting the object on the slope. The gravity is exerting a force of $(-1,-1,0)$... | Kinetic friction always acts opposite to relative motion, while static friction acts opposite to the tendency of motion. Kinetic friction is a dissipative force that turns kinetic energy into non-useful waste heat. If kinetic friction could act in the same direction as relative motion, that would imply that friction co... | {
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How to know which closed loop is valid to take when calculating current using Kirchhoff's 2nd law?
Using Kirchhoff's 2nd law, taking closed loop $\underline{A}BD\underline{A}$,
$$2i_1-3i_2+i_3=0 .....(i)$$
Taking closed loop $\underline{D}BC\underline{D}$,
$$-6i_1+10i_2+17i_3=0 .....(ii)$$
Taking closed loop $\... | It is not solvable because you allow the current to enter the points $A$ and $C$, but don't have any equation connecting the current $I_{A/C}$ and voltage $U_{AC}$. In other words, you haven't chosen a loop that comes through $E$ (the battery).
In Kirchhoff equations, it's important to choose independent loops. But in ... | {
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Coulomb Repulsion and Spin
Because of the coulomb interaction the energy of electrons of the same
spin is lower - the average potential energy is less positive for
parallel spin than for antiparallel spin.
This is quoted from Kittel, Ch 11, under Hund Rules.
I can't understand how Coulomb repulsion is related to spin... | Two electrons with the same spin projections have a symmetric spin-part of their wavefunction and therefore an antisymmetric spatial wavefunction - to be totally antisymmetric. An antisymmetric spatial wavefunction goes to zero in between the electrons, and so same-spin pairs are farther apart on average. Being far... | {
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Infinity in two-point correlator of $\phi^4$ theory I’m reading Peskin & Schroeder as a first intro in QFT. The first half of chapter 4 is spent on calculating the 2-point correlation function in $\phi^4$ theory:
$$\langle \Omega | \phi(x) \phi(y) | \Omega \rangle$$
So far, P&S have shown that to first order the only c... | Yes it is. It is the one-loop correction to the two-point function and needs to be regularised. This is explained in detail in P&S.
| {
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Does universal speed limit of information contradict the ability of a particle to pick a trajectory using Principle of Least Action? I'm doing some self reading on Lagrangian Mechanics and Special Relavivity. The following are two statements that seem to be taken as absolute fundamentals and yet I'm unable to reconcile... | The two statements
*
*Principle of Least action states that the particle's trajectory under the influence of a potential is determined by minimising the action that is $\delta S =0$.
*The universal speed limit of information propagation is $c$.
are not in contradiction with each other. If you take the relativistic... | {
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Where is the potential energy? I read this question What is potential energy truly? and I find the answers not really satisfying.
When I move an object upward in a gravitational field, I have to work against that gravity. There is not really any energy mysteriously stored inside the object. Conversely, if the object is... |
Where is the potential energy ?
Potential energy does not have a location. It is an attribute of the system - specifically, it is a function of the configuration of the system - but it does not have a location any more than the total mass of the system or the distance between the two objects has a location.
| {
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Coefficients in Feynman Diagrams Consider the 4-point correlation function, $G(x_1,x_2,x_3,x_4)$, in $\phi^4$ theory. Let us consider the term which is $\propto \lambda^2$, represented by the following Feynman's diagram:
According to symmetry rules, I should get a $1/2$ symmetry factor, due to the middle lines. Hence,... | The factor is $\frac{1}{2!} \times \frac{1}{4!} \times \frac{1}{4!} \times 8 \times 3 \times 4 \times 3 \times 2 = \frac{1}{2}$.
The $\frac{1}{2!}$ comes from expanding $e^x$ to second order. The two factors of $\frac{1}{4!}$ come from the interaction term (which is $\frac{\lambda}{4!}$). The remaining factors come fro... | {
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What is a non-fundamental force? We all know of the four fundamental forces, gravity, electromagnetism, strong, and weak. However, is there such a thing as a non-fundamental force, and if so, what is the definition of such a thing? Can anyone give examples of non-fundamental forces?
| For this you should first know the fundamental forces (gravitation, weak, strong and electromagnetism). All other forces are manifestations of these four fundamental forces. Tension, friction, and air resistance are some examples of non-fundamental forces.
All the fundamental forces appear to be conserative but most o... | {
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What is the formula to determine the change in pressure when there is a change in flow? [Updated to help clarify my question]
I have a current water flow of 9 GPM (gallons per minute) at 50 PSI through a 1/2 inch diameter pipe pouring out at the end.
I understand that if I reduce the flow at the end (exit) of the pipe,... | I misunderstood the original question, here is a corrected version of my answer.
First of all I will assume that the source pressure in your plumbing system is 50 PSI. This means at the head end of the pipe the pressure is 50 PSI and at the outlet end of the pipe, the pressure is 0 (i.e., atmospheric).
This means that ... | {
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Why is it that in general relativity an object in spacetime takes the path of longest possible proper time? Why is it that in general relativity an object in spacetime takes the path of longest possible proper time?
Please answer in simple words...
|
Why is it that in general relativity an object in spacetime takes the path of longest possible proper time?
What you stated is not quite accurate. An object that experiences "weightlessness" takes the path of longest possible proper time. If there were no gravity, one could say that an object which is in "inertial mo... | {
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Why $\frac{1}{E-H_0+i\eta}|n\rangle \stackrel{?}{=} \frac{1}{E-E_n+i\eta}|n\rangle$? A Hamiltonian $H_0$ is diagonalized in $\{|n\rangle\}$ i.e. $H_0|n\rangle=E_n|n\rangle$. Why can we write
$$\frac{1}{E-H_0+i\eta}|n\rangle \stackrel{?}{=} \frac{1}{E-E_n+i\eta}|n\rangle$$
$H_0$ is in denominator, how come we can apply ... | The denominator is nowhere zero due to the imaginary $i\eta$ term, so the Taylor series in $H_ß$ and $E_n$ converge everywhere. So you can express the terms as power series where it is clear that each summand of the form $c_k H_0^k|n\rangle$ is equal to the corresponding summand $c_k E_n^k|n\rangle$.
| {
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Is there an alternative to radio waves that can go through metallic objects? Radar can pass through materials such as paper, wood, glass, brick, and concrete, but it reflects off of metal. Is there an alternative to radar that can pass through metal substances? If not, is it likely that we will ever find such a wave?
|
Radar can pass through materials such as paper, wood, glass, brick, and concrete, but it reflects off of metal.
One wants radar to reflect off metal. That way, metalic objects can be picked up by radar. If the microwaves in radar passed through an airplane, or a boat, or a missile, without being reflected, then they ... | {
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Physical interpretation of relativistic synchrotron radiation When an electron approaches the speed of light, the emission pattern is sharply collimated forward, in the direction of motion. I can see how this is mathematically true from taking the relativistic limit of the angular distribution of the radiated energy (a... | This collimation of light in the forwards direction is in fact also what you would expect if non-relativistically.
From the electron's perspective, light is radiated uniformly in all directions. However, the electron is itself moving with respect to an observer in the laboratory.
If we didn't know anything about specia... | {
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Why is there no kinetic term in the Hamiltonian of the Ising model? I am used to the Hamiltonian formalism in the context of (quantum) field theory, where as far as I can remember it always has the form of a kinetic term + a potential term. For me the absence of kinetic terms means a theory without dynamics. In Wikiped... | Just a complement to the other answers, as they do not seem to touch on this issue.
People have of course been studying the dynamics of Ising models for a long time, but they usually (always?) use stochastic dynamics rather than a Hamiltonian evolution. That is, they consider a Markov chain on the set of configurations... | {
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Absolute Directions and Coordinate Systems Do truly absolute directions exist? If not, what directions exist independent of coordinate systems? I feel that, until I understand the answers to these questions, I will not fully understand how vectors exist independent of coordinate systems.
| One example is the direction between two stars in a constellation. The relative position between then changes so little in a timelife that can be taken as fixed.
But for that ones like ursa major in the northern hemisphere or crux in the southern one, all the group rotates along one night. A direction between 2 stars p... | {
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Proton to neutron mass ratio while free versus when bound inside the nucleus? Is the proton to neutron mass ratio different while they are free versus when they are bound inside a nucleus?
|
However, it is hard to say anything about the masses of individual protons/neutrons in the nucleus, because you can only determine the total mass of the nucleus, but not the mass of a single proton/neutron inside the nucleus.
The above quote states the truth, there is no way to measure individual nucleons in the nucl... | {
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How is momentum conserved in this situation? Let’s say we have two inactive electromagnets spaced one light-minute apart, with north poles pointing toward each other. One of the electromagnets is turned on for ten seconds, and then turned off again (or destroyed). Once the magnetic field from the first electromagnet ha... | The turned off field will be a very long wavelength electromagnetic field, with its momentum, which will hit the magnet and transfer its momentum to the solid state lattice of the magnet. The second magnet will send into space, the field meeting nothing on its way .
Edit after comment:
The basic frame of all theories... | {
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Is a small fragment of a white dwarf stable? Assuming it is somehow possible to get a small piece of a white dwarf (maybe a dice) and this piece escapes into free space. Would that piece of white dwarf matter keep its density/state, or would it expand, explode, or transform into another state without the huge gravitati... | The answer is similar to What would happen to a teaspoon of neutron star material if released on Earth? but the energetics are not so extreme. The material consists of fully ionised carbon and oxygen nuclei plus a gas of highly degenerate electrons.
At typical densities for white dwarfs of $10^{9} - 10^{11}$ kg/m$^3$, ... | {
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Expressing vectors as a unit vector I just started vectors and was solving some problems. One of them says
A car is moving at the speed $10\,\mathrm{m/s}$. Write the velocity vector of the car in unit vector notation if the car is moving towards North East.
The problem seems fairly easy.
I solved it like this:
$$ \h... | We can easily check which one is correct by calculating the norm explicitly. Because $\hat{i}$ and $\hat{j}$ are orthonormal, the norm of a general vector
$$ \vec{V} = A\hat{i} + B\hat{j}$$
is
$$ |\vec{V}| = \sqrt{A^2 + B^2} $$
which is really just the Pythagorean theorem.
Doing this for the book answer we get
$$ |\hat... | {
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Why do we factor out a momentum-conserving delta function in a 1-loop 1PI diagram in $\phi^4$ theory?
I do not understand the meaning of factoring out $(2\pi)^4 \delta ^4(p_1 - p_2)$, surely the delta function will have argument zero ($p_1 = p_2$) and should consume one integration and $(2\pi)^4$ should also cancel th... | It just depends on what you want to compute. In the text they are focused on the loop, which is the part that contains divergences, the external legs have been amputated if you want. All they mean by the sentence you highlight is: "we are not computing the exact amplitude for the diagram, but just extracting the diverg... | {
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Non-constant wavelength of particles Is it possible for a particle's wavelength to change with respect to time? I'm thinking of a massless particle like a photon, but as it evolves through time its wavelength changes. I'm aware photons wavelengths do change from the expansion of spacetime, but I'm thinking on a smaller... | The wavelength of a photon does not mean a single photon is spread in space. At our level of experiments and theories photons and the other elementary particles are point particles at (x,y,z) at time t, described by four vectors, whose "length" gives the invariant mass of the particle.
In the case of photons, the in... | {
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Is being "stationary" also an example of a "periodic motion"? I read an idea in mathematics about periodic functions that a constant function is also a periodic function with an undefinable period.
So , suppose a body is at rest (for an observer). This means that it's position doesn't change with time for that observer... | As a specific example of periodic motion, an object oscillating under harmonic motion will satisfy the differential equation (assuming a one-dimensional problem)
$$\frac{d^2x}{dt^2}+Ax=0.$$
A solution of this could be in the form
$$x=a\sin(\omega t),$$
where $A=a\omega^2$. We could define a stationary object to be in t... | {
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Which paper introduced the concept of the "Bloch sphere"? Everyone loves Bloch sphere, but which paper of Bloch was it introduced? The Wikipedia article on Bloch sphere (here), as of 17/May/2021 links to this paper of Bloch “Nuclear Induction” but it doesn’t seem to discuss the Bloch sphere!
I would appreciate if you c... | The wikipedia page cites Bloch's paper "Nuclear induction" in 1946 where he wrote down Bloch's equation for nuclear magnetization. It seems that Bloch didn't explicitly say anything about "his sphere", or anything about geometric representation of quantum states of spin-1/2. He went straight to the equation of motion f... | {
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What happens to entropy when half the particles are removed? Curiosity question. What happens to entropy in the following situation?
A gas fills an entire container and is in equilibrium. Suddenly all particles are removed from half the container. As such, there are now 1/2 the original number of particles, but all ... | Let the container be insulated and rigid, and let the initial entropy of the 2N particles be 2$S_0$. If a rigid partition is first inserted to separate the two sets of particles without letting the remaining N particles expand to the full container, the final entropy of the container contents will be $S_0$ (since entr... | {
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How to identify if a photon comes from the sun? Is there any way to know whether a group of particles is generated from the sun rather from an artificial source?
| There is no physical difference between the photons emitted by the sun and those emitted by an artificial source, so there is no way you can look at a photon and know for sure what generated it. However, sunlight has specific characteristics which means that you can usually distinguish it from e.g. electric lights.
For... | {
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Why is $kx−ωt$ a constant for a travelling wave? In my physics textbook, there is a statement like this:
The motion of a fixed phase point on a progressive wave is given by
$kx−ωt$= a constant.
What does this mean? Why is it a constant?
Does fixed phase point mean that it is a particle of the medium always at the same ... | Suppose the equation of the wave is $y=A \sin (kx - \omega t)$.
If $kx - \omega t = \pi/2$ then $y=A$ and there is a wave peak of amplitude $A$.
How might you measure the speed of a water wave?
What you might do is follow a particular peak and measure how long it took $t$ to travel a given distance $d$.
You might then ... | {
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What is the force pair for the normal force? Clarification on Newton's 3rd Law In the process of trying to wrap my head around Newton's 3rd law I've come across 2 definitive statements.
*
*Forces must occur in pairs
*Forces must act on different bodies
This is confusing to me when applied to the classic box on a fl... | The action-reaction normal force pairs are: the force the table exerts on the box, and the force the box exerts on the table.
The force of gravity ${\bf \vec F}_g$ actually comes from the earth. The earth exerts a force of gravity on the box (and table), and the box (and table) exert an equal and opposite force on the ... | {
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How to measure the speed of an electric current? We all know that the definition of a current is the amount of charge flowing per second, that is often expressed by the equation $i=dq/dt$. But is it possible to measure the speed of an electric current in m/s? And also how can we measure such speed?
| If you mean by speed of electrons then,
drift velocity is $$v=\frac{I}{nqA}$$
where $n$ denotes charge density of a particular substance (it's $8.5\times 10^{28} \text{m}^{-3}$ for copper as an example),
$q$ as you may already know denotes charge of electrons ($1.602\times 10^{-19} \text{C}$),
$A$ denotes the cross sec... | {
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How do you understand stall in terms of Newtonian mechanics (i.e. without Bernoulli's Principle)? Okay, so. I understand how a wing generates life using Newtownian mechanics, to wit: the air molecules crash into wing, which is at an angle to the air molecules. As a result, the air molecules are deflected downward and... | The relevant forces are the weight of the aircraft and the vertical components of the forces on the wings. The sum of the latter forces must equalize the weight to keep the plane in the same altitude.
Following a naive idea, we could model the process as elastic collisions of molecules. They hit the wings horizontally ... | {
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Why the mass of initial particle has to be greater than the sum of masses of final particles? Suppose we have a decay of a rest particle $A$ into other particles $a_1,...,a_n$
\begin{equation}
A \rightarrow a_1+a_2+\cdots+a_n
\end{equation}
It is always stated that in order to this particle decay to be possible, the ma... | Conservation of mass/energy tells us that
$\displaystyle M_Ac^2 + \text { kinetic energy of original particle}= \sum_{k=1}^nM_{a_k}c^2 + \text{ kinetic energy of decay products}$
If the original particle is at rest then its k.e. is zero. Assuming the decay products are not all at rest (if they were, how could we detect... | {
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Could the Michelson-Morley experiment have detected gravitational waves? If everything went perfectly and they had no outside noise while conducting the experiment, could they have detected gravitational waves? What would it have looked like to them?
| In practice if the conditions had been perfect - no, absolutely not. LIGO has the precision of being able to measure a distance change of +/- width of a hair on a scale of a few lightyears. Even if a gravitational wave is larger by a few orders of magnitude the Michelson-Morley experiment was nowhere in the same univer... | {
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Sabine Hossenfelder's argument to rule FTL paradoxes out I have recently seen Sabine Hossenfelder's video on faster-than-light travel, in which she argues no paradoxes would arise if we had something like a FTL warp device, because the trajectories leading to paradoxes are not permitted in the first place. Since I didn... | There is an assumption that the possibility of time travel would allow for paradoxes because a return to the past would allow changes to be made to the conditions that then existed- the famous example being that one might kill one's grandfather.
The scope for such paradoxes disappears if one adopts the presentist view ... | {
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Rainbows and Clouds What is the intensity of sunlight required to form a rainbow. I read in the love of physics by Walter Lewin than
"“For you to see a rainbow, three conditions need to be met. First, the Sun needs to be behind you. Second, there must be raindrops in the sky in front of you—this could be miles or just... | One can always use a garden hose to make a small, but very own rainbow.
This way, it is possible to try and check all of these conditions.
*
*The Sun must be behind you. In theory, higher-order rainbows (created by multiple reflections inside the droplets) can exist in the Sun direction, but they are at least as fain... | {
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Work done to bring a mass from an infinite distance away In the notes I received from high school teacher, it says:
How much work is done by the gravitational field of the planet in bringing a satellite from an infinite distance to a position $r$ away?
In general,
$W=\Delta U=\int_{\infty}^{r} F\,dr= \int_{\infty}^{r}... | Basically, this problem boils down to perspective.
Your high school teacher told you, how much work 'YOU ( The satellite )' did, in bringing the satellite (itself). The negative sign implies that work was done on it instead of it doing any work.
If the work done by a body is negative, work is being done 'on' the body, ... | {
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Capacitor reactance with frequency variation Consider that I have a capacitor and resistance connected in series, and I have a voltage source $V_0 \cos(\omega t)$ or some alternating source of voltage connected.
I do understand that the reactance varies with frequency in fact it decreases as frequency increases.
But wh... | The maximum charge on the capacitor is determined by the amplitude of the applied voltage, but at a higher frequency it comes and goes more rapidly. More rapid charge movement means more current and less reactance.
| {
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Relation between critical temperature and density of states The BCS theory predicts that the critical temperature of the superconducting transition is given by
$$
T_c \approx \theta \exp \left (- \frac{1}{U D(\epsilon_F)} \right )
$$
where $\theta$ is the Debye temperature, $U$ is the coupling constant of the electron-... | I think it is misleading to try to separate the DOS and the interaction strength. Rather one should think of $UD(\epsilon_F)$ as a dimensionless coupling strength. This view is further reinforced, if one looks at the problem from the point of view of the renormalization group, where reducing the energy cutoff (i.e., ch... | {
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Electromagnetism and CPT simmetry I'm having some trouble understanding the CPT theorem and in particular the time reversal part.
Let's consider for example an electron and a positron on the x axis, with the electron on the left and the positron on the right, attracting each other.
If we apply charge conjugation we swa... | When you apply time-reversal you should take your initial conditions as your final conditions, and vice-versa, and reverse the velocities.
So let's first start with electron and positron far apart at some distance $d(0) = d_0$ and at rest, $\vec{v}_e(0) = \vec{v}_p(0) = 0$. You roll the movie, and they will attract eac... | {
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Relation between central and mean density in a White Dwarf (polytropic model) I'm currently studying the radius of a white dwarf and, in deriving some useful equations, I've seen my professor use the following relation: $\bar\rho=\frac{\rho_c}{6}$ when using a polytropic model $P(r)=\rho(r)^\gamma$ with $\gamma=\frac{5... | This is a standard result obtained by numerically solving the Lane-Emden equation for a polytrope with $n=3/2$ (equivalent to an adiabatic index of $\gamma = 5/3$).
It can be found in any textbook or set of lecture notes that treat polytopes as a means of solving the stellar structure equations (e.g. these lecture note... | {
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$∆H$ when non-expansion work is done I have learnt that at constant pressure, enthalpy change $∆H=q$ provided no non-expansion work is done. Why is this the case? What will be relationship of $∆H$ with $q$ if non-expansion work is done? Again, Gibbs free energy measures the maximum non expansion work that can be done b... |
I have learnt that at constant pressure, enthalpy change $∆H=q$
provided no non-expansion work is done.
What you refer to as "expansion work" is normally referred to as boundary (expansion or compression) work, or $pdV$ work, which is applicable to closed systems (no mass transfer involved). The other possible type o... | {
"language": "en",
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What is the basic physics of current-electricity? Why current decreases when length of resistor increases and How the speed of electricity is almost $c$ (speed of light)?
| Answer for question 2:
The speed of electricity is in fact the speed of electromagnetic wave propagation or the spread of electromagnetic field inside the conductor. That is not the speed of the electrons, no object having mass can reach the speed of light.
But why the speed of light? It's because the light is an elect... | {
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Why is this result no contradiction to length contraction? Let's consider 2 events in an inertial system $S$:
$$x_1=(0,0,0,0) \quad \quad \quad x_2=( \Delta t, \Delta x,0,0)$$
If we assume the two events occur in the same place we have: $\Delta \bar{t}= \gamma \Delta t$ for every other inertial system, also known as ti... | The biggest problem with understanding special relativity is to understand that events that are simultaneous in one frame are not simultaneous in any other frame. Length in $S$ is measured by taking two measurements at the same time in $S$. The phrase "same time" should immediately alert you that it will not be the sam... | {
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How gauss law is not applicable here? I am a high school student and I am very confused in how to use Gauss law, when we have to calculate electric field due to infinite sheet or wire, and for spherical surfaces, I know how we do that, we select any point nearby the sheet because at nearby points the components of ele... | When we say Gauss's Law is applicable for infinite bodies it implies that Gauss's Law can be applied only to find electric field at a point from where the body appears to us to be infinitely large.This distance is only a few atomic radii for a rod.If you are at a larger distance from the rod you will be able to see the... | {
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Conjugate nuclei I'm trying to find an explanation for the difference in binding energies between conjugate nuclei being governed by the Coulomb interaction. I was looking at the semi-empirical mass formula and trying to draw some conclusion.
We have that:
$$ R = R_0 A^{1/3}$$
and:
$$ Z = A-N$$
so for the Coulomb term ... | In the semi-empirical mass formula
$$E_B(N,Z)=a_VA-a_SA^{2/3}-a_C\frac{Z(Z-1)}{A^{1/3}}-a_A\frac{(N-Z)^2}{A}+\delta(N,Z)$$
the volume and surface terms only depend on $A$, so they give the same value for a nucleus and its mirror nucleus, since the mirror nucleus is obtained by changing $N\leftrightarrow Z$, so $A$ is f... | {
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Are the properties of anti-hydrogen opposite to those of hydrogen? In the series Crisis of Infinite Earths, the whole story is that there is a wave of antimatter rampaging through the multiverse. So I got interested and googled "how to create antimatter", and I found out that when an antiproton and a positron are prese... |
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge)
Bold mine. Note physical charges. There are specific numbers associated with elementary particles that define them, and those are the ones reversed, as in the... | {
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What would be the most UV resistant rubber color? I'm looking for the longest lasting speargun bands. They come in a variety of colors so I'm wondering if a certain color of rubber would help them last longer. They will be exposed to a lot of sunlight and saltwater. I think it's the UV exposure that causes the most dam... | Welcome to Physics.SE.
This is not an easy question because while considering damage to an object, the energy absorbed is just one of many factors (to really answer the question one should know the atomic composition of the material, calculate the normal oscillation modes and their frequency, search for extinction rule... | {
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If a photon strikes a perfectly reflecting mirror, it is essentially at rest at the instant of collision. So why does the photon exist? Photons can only move at the speed of light.how do we define its existence at the instant when it collides with a perfect mirror?
| The photon is never at rest. It always moves with the speed of light. Before the collision, it moves at the speed of light. Then it's absorbed with no intermediate state of lower velocity. After the absorption, the photon just doesn't exist anymore.
So it never is essentially at rest. Not even at the time of absorption... | {
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Why do we experience time flow? I had read that a photon travelling at the speed of light does not experience time flow (time stops for it) also I had read somewhere that we are travelling at the speed of light. Why do we experience flow of time then?
| Your question raises a common and interesting misconception.
Time does not slow down from the perspective of a moving observer. The time dilation effect means that where a moving observer passes between two stationary clocks, the time experienced by the observer is less than the time difference recorded on the clocks. ... | {
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Are charged particles cold? Are charged particles colder than neutral ones?
If a charged particle is vibrating due to temperature, it will release some of its energy as electromagnetic waves. So that means it's losing energy, cooling itself off. Is there an error in my logic?
| They are cold and lonely, and other particles of the same charge find them repulsive ;)
More seriously - cold/hot is the concept based on temperature, which is the measure of the average kinetic energy of the particles. As such, the temperature is not directly related to the charge of particles, although in some situat... | {
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If photons have also particle properties why should they not collide with each other? If photons have also particle properties why should they not collide with each other? Collisions between fermions are possible as collisions between fermions and photons(bosons) except collisions between photons (that are described by... | The QED Lagrangian is:
$$ L = \bar\psi[i\gamma^{\mu}(\partial_{\mu}+ieA_{\mu})-m_e]\psi-\frac1 4 F_{\mu\nu}F^{\mu\nu} $$
The $ie(\bar\psi \gamma^{\mu}A_{\mu}\bar\psi)$ terms shows us that photons ($A_{\mu}$) couple to spinor ($\gamma^{\mu}$) electrons ($\psi$) with a strength proportional to charge ($e$).
The portion d... | {
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Gravitational time dilation and power beaming Consider this thought experiment that kept me awake for far too long last night.
Two spacecraft are located near a black hole, and their relative positions are such that, due to the gravitational time dilation caused by the black hole, time goes by 1.1 times as fast for spa... | Consider the laser beam to consist of a stream of photons emitted at a certain rate. The gap between each photon emission has one value according to ship A and another according to ship B because of the relative time dilation.
So yes, ship B receives the photons for a longer period of time, but it receives them at a lo... | {
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When I walk down the stairs where does my potential energy go? When I leave my room I walk down three flights of stairs releasing about 7kJ of potential energy. Where does it go? Is it all getting dispersed into heat and sound? Is that heat being generated at the point of impact between my feet and the ground, or is it... | Make it simple. If a mass of your weight fell down the height of three flights of stairs through the air, when it landed where would the kinetic energy accumulated by falling go?
*
*moving the earth for conservation of momentum
*dissipated in heat on the ground
*deformation of the matter of the weight
*sound... | {
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Angular velocity: why does decreasing the radius affect the tangential velocity? I'm given the following very simple problem:
There is an object of mass $m$ attached to a string (whose mass can be neglected) on a table, whereby the string goes through the hole in the middle of the table and the particle moves on this ... | The point is that the object does clearly not move on a circle if you pull the string and decrease its distance from the center hole. Its path is rather some sort of spiral. Although this will barely be noticeable in a real experiment, the local center of curvature of this spiral will be changing and be different from ... | {
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Thomson scattering from Larmor's formula (Jackson's derivation) The equation numbers below are the same as they appear in Jackson's Classical Electrodynamics.
If a plane monochromatic electromagnetic radiation is incident on a free particle of charge $e$ and mass $m$, the particle will accelerate and emit radiation. Fo... | Imaginary ("complex") representation of oscillating quantity (velocity or acceleration) works straightforwadly only in linear equations. Formulae containing other than constant or linear terms require that the actual physical value of the quantity (acceleration, electric field) is put in.
In your case, the formula cont... | {
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Confusion in derivation of Formula of divergence
This is from the book Mathematical Methods (Arfken),
Can someone explain how did one del(x) and one dx came here,?
| What the LHS of the highlighted equation represents is the change in $\rho v_x \,dy \,dz$ between two $x$ values. Let the rate of change of this wrt $x$ be $$\frac{\partial (\rho v_x \,dy\, dz)}{\partial x}=\frac{\partial (\rho v_x )}{\partial x} dy\,dz$$
Now $\Delta (\rho v_x)=\frac{\partial \rho v_x}{\partial x} \Del... | {
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From the initial condition problem of the Euler-Lagrange equation to the principle of least action I browsed through many similar questions about the Initial Condition Problem (ICP) and Boundary Value Problem (BVP) for Euler-Lagrange equations, here some interesting but (in my opinion) incomplete examples: E.g.1 or E.g... |
Will the first variation of the action functional be zero if we plug the solution that we found from the ICP?
Yes, if we [besides assuming the appropriate initial boundary condition (IBC)] also assume an appropriate final boundary condition (FBC).
Concerning consistent choices of boundary conditions (BCs), see also e... | {
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Position eigenstates in curved space How does one define position eigenstates in curved space (say a manifold $\mathcal{M}$)? Let us say that it is defined as usual $$\hat{x}|x\rangle = x|x\rangle$$ Then how does one define the identity operator in this basis? Is this proposal correct ? $$\int_{\mathcal{M}} d^4x~\sqrt{... | This is more a question of how to normalize the eigenstates. This is just a convention that can choose. Myself, I like to write $\langle x|x'\rangle= \delta_g^n(x-x')$ where the delta function is defined by
$$
\int_M d^nx \sqrt{g} \,\delta^n_g(x-x')=1,
$$
but other choices may be preferable.
What is important that you ... | {
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Does creating creating an unknown quantum state result in a superposition? Suppose a blind man builds a machine that paints three apples with three colors, either red, blue or green. Once the machine has done this, are the three apples in the following superposition:
\begin{align}
&|R\rangle_1\otimes |G\rangle_2\otimes... | Talking of superpositions only makes sense for objects which you can measure in more than one basis - say, the color basis and a basis of "red plus green" vs. "red minus green". This is not possible for apples.
So no, this has nothing to do with superposition. (More generally, incomplete knowledge has nothing to do wit... | {
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How to determine the boson of a given particle decay? How do I know which of the SM forces take part at a given particle decay? For example, look at the following decay:
$$\tau^- + \nu_{\tau} \rightarrow \ell + \bar \nu_{\ell}$$
This decay takes place via a $W^-$ boson. I'm now wondering, if the force carrier must be ... | If you replaced the $W^-$ with a $Z^0$ or $\gamma$, (electrical) charge would not be conserved at each vertex.
Since you have a change in lepton flavour, it has be the weak interaction. If it were just a scattering event, with no charge transfer, then it might be either EM ($\gamma$) or weak ($Z^0$), albeit with differ... | {
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How can melting of ice be a reversible process when it produces an increase in total entropy? The melting of ice is a reversible process according to most sources I have seen. However, the melting of ice increases the total entropy of the universe which shouldn't be possible if the process was reversible.
Can someone h... | Why do you think the entropy increases? The change in entropy of the heat source is $−/_{\rm melt}$
and the change in entropy of the water/ice mix is $+/_{\rm melt}$
with the same $$,
so the sum of the entropy changes is zero.
The water in contact with the ice is always at the melting temperture, so no entropy is gener... | {
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An interesting question about the effects of time dilation Let's say we build a train track around the earth running from the Arctic to the Antarctic and back again. A train is going to run on this track at a speed of almost the speed of light lets say $299,792,457 \;\text{m/s}$ ($1 \;\text{m/s}$ less than the speed of... | What physics says is that what you have proposed is strictly impossible. The energy required to accelerate the train could not be delivered to it. Gravity would not be sufficiently strong to constrain the train to the track. The train would burn-up with friction long before it reached a tiny fraction of light speed. Th... | {
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How to reconcile time dilation in relativity with what you actually see?
If a traveler is moving at $0.5c$ towards a clock which is located one light year away, his relativistic time dilation is $1.15$. But in the time he takes to arrive at the clock, he must catch up the 1-year lag that he saw when the clock was 1 li... | I suspect that you're trying to calculate this by multiplying time intervals by the Lorentz factor, $\gamma$. And you're ignoring length contraction. To do this properly, you need to use the full Lorentz transformations.
Don't mix up what each observer measures & calculates with what they observe; observations include ... | {
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If $E=mc^2$, then why do different substances have different calorific values? Today during a classroom discussion, I realised that if we consider the equation $E=mc^2$, then we are establishing a relation between energy and mass but we often observe that different substances produce different amount of energy when the... | The mass of your reaction products after burning the two substances will slightly differ, indeed, because a different amount of mass has been converted to energy and passed to the surroundings.
| {
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How to get the term $\hat c^\dagger \hat c +\frac12$ in the derivation of qubit transmons quantization? I'm wondering about qubit transmon quantization derivation, which I found here:
My question is the following:
How can I get this term, while derivation?
I tried to do it by myself, but my result was without this ter... | Since $[\hat{c},\,\hat{c}^\dagger]=1$, the diagonal terms in $(\hat{c}+\hat{c}^\dagger)^2$ are$$\hat{c}\hat{c}^\dagger+\hat{c}^\dagger c=2\hat{c}^\dagger c+1=2\left(\hat{c}^\dagger c+\frac12\right).$$It all stems from that.
| {
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Why is everything not invisible if 99% space is empty? If every object is $99$% empty space, how is reflection possible? Why doesn't light just pass through?
Also light passes as a straight line, doesn't it? The wave nature doesn't say anything about its motion. Also, does light reflect after striking an electron or at... | As joseph h has said, atoms have an electron cloud surrounding them even though most of the space in an atom is empty. This electron cloud interacts, for example, with light and leads to reflection, refraction, dispersion, etc.
| {
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Why doesn't classic mechanics hold the concept of identical particles? I was told that the concept of identical particles is because Hamiltonian is unchanged when permutating two particles, but if you imagine a classic mechanics system of N particles, it is also true that the Hamiltonian is unchanged when permutating a... | Your question 2. raises a profound issue.
The atoms in a crystal can be identified by where they are, so they are usually counted as distinguishable when doing statistical mechanics such as computing the vibrational contribution to the crystal's specific heat.
However, there is nothing in principle to stop them from qu... | {
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Can I apply the Born rule to a Dirac spinor? How does a Dirac spinor such as:
$$
\psi = \pmatrix{a_0+ib_0\\a_1+ib_1\\a_2+ib_2\\a_3+ib_3}
$$
Connect to a probability?
Can one apply the Born rule of this object?
| Sure you can! This is like the whole point of spinors!
For simplicity sake let's think about particles with spin $1/2$ (like electrons): Spinors are vectors in the Hilbert space of spin, you can write them as column vectors
$$\chi = \begin{bmatrix} a\\b\\\end{bmatrix} \ \ \ \ \ \ a,b \in \mathbb{C}$$
or equivalently yo... | {
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Why are $L=S=0$ in full shells? Spectroscopic notation There are some threads on this question but I'm still not understanding this considering the following scenario: If we take for example Helium in its ground state so $1s^2$. Here obviously $L=0$, but coupling the two electron spins $s=\frac{1}{2}$ yields the follow... | Why do full shells have $L=S=0$?
Electrons are fermions, so they obey the Pauli exclusion principle: a shell is not full until electrons are in every possibly allowed state. If they have spin quantum number $s=1/2$, the available states are $m_s = \pm 1/2$, so the shell will be "full" when both these states are occupie... | {
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Do Newton's laws of motion hold true in non-inertial frames of reference? My book derived the formula for the acceleration of a rocket at any instant in the following way:
$v_r=$ velocity of gas released from the nozzle relative to the rocket
$dt=$ infintesimal time interval
$dm=$ mass of the gas released from the nozz... | So, what you need to understand is that the calculations are all being done from any arbitrary inertial frame of reference. We are not calculating $a$ or acceleration of the rocket from the reference frame of the rocket (a non-inertial reference frame) itself. It is just easier to express the velocities of the rocket a... | {
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Derivative of Modified Bessel Function I am working on an exercise in Quantum field theory (problem 2.3 in Peksin and Schroeder's An Introduction to Quantum Field Theory) and am using the solutions: (https://theoreticalmaximum.files.wordpress.com/2017/07/intro-to-qft-solutions2.pdf). On page six of the pdf it seems to ... | It depends on how we define $K_\alpha$. For example, taking$$K_\alpha(\omega):=\int_0^\infty\cosh(\alpha u)\exp(-\omega\cosh u)du$$gives$$\frac{d}{d\omega}K_0(\omega)+K_1(\omega)=\cdots$$
| {
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} |
Magnetic dipole moment of electron I have read an electron has an intrinsic magnetic dipole moment.
Does this mean that because dipole moment can be thought of as a current loop, and a current loop radiates EM Waves due to a changing $dJ/dt$ that electrons must radiate EM waves when they are stationary?
can someone sh... | The electron doesn't radiate because the "current" in the Amperian model of dipoles is constant. Alternatively, the dipole can be represented by two magnetic monopoles separated by an infinitesimal distance (Gilbert dipole), which also confirms the static picture.
In reality, the electron's magnetic moment is a quantum... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/648921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What interval to use when proving orthogonality of wavefunctions? When proving that $\psi_1=\sin(n\pi x/a)$ and $\psi_2=\cos(n\pi x/a)$ are orthogonal to each other in a 1D box, the main problem that I am facing is what to use as the domain of integration. If I take the interval $[0,a]$ as we use in the Schrodinger wav... | There is no rule for determining the interval. It is simply a consequence of the problem in hand. Lets consider a free particle in a 1D box.
Lets consider a particle of mass $m$ moving inside a 1-dimensional potential box, constrained between $x=0$ and $x=a$. To solve this problem we need to solve
$$-\frac{\hbar^2}{2m}... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/649076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Chemical potential from entropy For the internal energy we have the following equation:
dU= TdS -pdV + $\mu$dN.
Now for the derivation of the entropy w.r.t the particle number we have:
$\frac {\partial S}{\partial N}=\frac {\partial S}{\partial E} \frac {\partial E}{\partial N}= \frac {\mu} T $. This would be my result... | Careful practitioners always include the conditions of partial differentiation. If instead of writing
$$\frac{\partial S}{\partial N}=\frac{\partial S}{\partial E}\frac{\partial E}{\partial N}$$
you had written
$$\left(\frac{\partial S}{\partial N}\right)_X=\left(\frac{\partial S}{\partial E}\right)_X\left(\frac{\parti... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/649228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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