question stringlengths 44 16.2k | answer sequencelengths 2 2 |
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It is implied, per QM, that the behavior of subatomic particles cannot be precisely predicted. However, these indeterministic effects do have defined probabilities. By the law of large numbers, they can “average” out and result in approximately deterministic laws.
For this reason, I presume, we can predict with pinpoint accuracy whether or not atleast some kinds of events will happen in the macro scale even if we can’t know their minute details on a subatomic level.
The question then is how fine or loose grained of an event is predictable given all knowledge about antecedent conditions. And how antecedent must these conditions be?
Suppose I woke up today at 9 AM and ate toast for breakfast. If I were to know **everything** that could be possibly known about the configuration of the universe right after the Big Bang, is this event predictable? Can one say, given that knowledge, with assuredness whether or not this will happen?
| [
"\nWhen thinking about the entirety of the Universe in terms of QM you will very quickly run into paradoxes. That's why I don't think we are at a point when your question can be meaningfully answered. For instance, the Universe is by definition a closed system (there is nothing else but it). So it must be in a pure... |
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In papers and books, I find that many objects can be local or global, e.g., local interaction, global operator, local field, etc. What do "local" and "global" mean in the context of physics?
Personally, I think "local" means that an object $F$ depends on spacetime coordinates, i.e., $F(x^\mu)$, while "global" means the opposite, i.e., $F$ is the same on every point in the spacetime. Is this correct?
| [
"\nThis answers only address part of the problem, i.e., assuming that local and global stand for properties of function/vectors/tensorial obejects defined on spacetime. If you look at spacetime as a [manifold](https://en.wikipedia.org/wiki/Manifold) $M$, then local means that the property/definition under considera... |
I understand that superconductors have zero resistivity, however, I wonder if there is a relation between resistivity and the work function of a specific material so that a superconductor has zero (or a low value) work function. If so, photons with very long wavelengths can release electrons from the surface of a superconductor. Am I right?
| [
"\nThe work function is a measure of how much energy it takes to completely remove an electron from a material and send it into **free space**, and costs on the order of 1000-5000 meV of energy. The resistance is a measure of how much energy is needed to move electrons infinitesimally ($\\omega\\sim0$) **within the... |
In quantum mechanics, the probability, say, that a radioactive atom will decay is well defined. By the Born Rule, it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. However, the actual decay point and the actual measurement outcome are individually unpredictable even if one knew everything there is to know about the world.
But if, say, these things were unpredictable and happening for no reason, why does the Born’s rule exist in the first place? Why is it the square of the amplitude and not 1/4 or 1/8?
Secondly, why does the way this work stay constant? If for example, a radioactive atom’s decay point is truly happening for no reason, why doesn’t it just suddenly start decaying with different probabilities?
But because this does not happen, does it not still indicate some level of order? If there is order, where does this come from without determinism?
I am aware that “randomness” can create order through the law of large numbers but that originally interpreted randomness as a mere function of ignorance going back to Laplace’s time. This doesn’t fully explain how or why indeterminism can lead to specific probability functions over others and why they generate particular kinds of order over others.
If one cannot in principle predict what will happen, then presumably, what is happening is occurring for no reason. But if what is happening is occurring for no reason, why are there certain probabilistic laws in the first place?
Given the lack of explanation here, how are we sure that the universe is indeterministic? Is it possible that our theorems deciding that no local hidden variables can exist are simply incorrect?
Of course, even if the universe was deterministic, it would beg the question of how those laws came about. But laws always existing seem to be more explainable than laws ultimately appearing without cause while also staying constant.
| [
"\nAlthough the outcomes of the measurement of a quantum phenomenon might be **probabilistic**, that **does not mean that the outcome could be anything**. For a large enough ensemble, one can predict with a high degree of accuracy what the outcome of the experiment will be. E.g., the spin of a fermion in the [Stern... |
[](https://i.stack.imgur.com/JJpO9.jpg)
I just start learning Physics, and so many strange questions come across my mind. Here is one:
The picture shows the ISS orbiting Earth. Suppose now we (or rather say God) put a baseball behind ISS, Will this ball fall directly to Earth while things in ISS are floating?
| [
"\nIt is all about the *sideways speed* of the object.\n\n\n* The ISS and also your baseball is falling towards Earth constantly. If you just let go of the ball up there, then it will fall down and crash on the ground underneath.\n* Push it a bit sideways while letting go and it still crashes but a bit more to the ... |
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I had this great idea to crank out a lot of ice in my freezer by getting a plastic bin where I could place two stack of ice trays side by side (5 trays high on each stack).
I placed only a single stack 5 high in the freezer on the top shelf in a bin and 24 hours (approximately) later I go to move ice from the trays and into a storage bin... To my surprise the top tray (or maybe two) were the only trays that had froze.
My brain started spinning thinking how this could be so. My best guess is that the bin is preventing the cold air from reaching the trays (insulating the trays).
The next evening I did another experiment. I placed a bin with 5 trays in a single stack on the top shelf in the freezer, another bin with 5 trays in a single stack on the bottom shelf in the freezer, and then as a control, I placed a stack of 3 (ice) trays on the middle shelf with no bin holding them.
The next morning (approximately 6-7 hours later) the stack of 3 on the middle shelf was completely frozen. The top tray on each of the stack on the bottom and the top shelf was completely frozen but the other trays below were still in the process of freezing.
It is quite surprising to me that after 24+ hours some of these ice trays would not completely freeze over. I'm really thinking that the cold air is not getting down into the bins. Makes me hesitant to store any food in bins in either the freezer or the fridge.
If tonight all the trays are not completely frozen over then my next experiment is to take one stack of five and place it in a bin on the top shelf, then in front of the bin, place another stack of five on the lid for the bin (had not been using the lids at all to this point)... Maybe the ice trays would do better on the lid where the freezing air can more easily reach the sides of the ice trays. I'll do the same thing on the bottom shelf, and then place another 3 as a control on the middle shelf...
My mind is just blown that water can sit in a freezer on the coldest setting for more than 24 hours and didn't freeze.
Will try to report back with observations.
Yes, my life is really exciting to be so fascinated by the idea that water didn't freeze in my freezer. Also, this is filtered water (Berkey type filter) but all the water in the trays is the same filtered water... and the middle shelf is freezing (not placed in bins).
The only reason why I was using bins is to try to contain any spills and make it easy to move ice trays from the freezer to place ice in bins and refill and place back in freezer.
Really thinking that I'm going to be better off just placing the ice trays on the lids to the bins and maybe not using the bins at all. This would probably allow for the cold air to reach the ice trays better.
| [
"\nIn a refrigerator or freezer without a circulating fan to stir the air around, it isn't possible to get a uniform temperature in it except by waiting a long time. however, there is another effect which may be important here, as follows.\n\n\nIf you cool down a very pure water sample in a very clean container whi... |
It is implied, per QM, that the behavior of subatomic particles cannot be precisely predicted. However, these indeterministic effects do have defined probabilities. By the law of large numbers, they can “average” out and result in approximately deterministic laws.
For this reason, I presume, we can predict with pinpoint accuracy whether or not atleast some kinds of events will happen in the macro scale even if we can’t know their minute details on a subatomic level.
The question then is how fine or loose grained of an event is predictable given all knowledge about antecedent conditions. And how antecedent must these conditions be?
Suppose I woke up today at 9 AM and ate toast for breakfast. If I were to know **everything** that could be possibly known about the configuration of the universe right after the Big Bang, is this event predictable? Can one say, given that knowledge, with assuredness whether or not this will happen?
| [
"\nWhen thinking about the entirety of the Universe in terms of QM you will very quickly run into paradoxes. That's why I don't think we are at a point when your question can be meaningfully answered. For instance, the Universe is by definition a closed system (there is nothing else but it). So it must be in a pure... |
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Closed 1 hour ago.
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In papers and books, I find that many objects can be local or global, e.g., local interaction, global operator, local field, etc. What do "local" and "global" mean in the context of physics?
Personally, I think "local" means that an object $F$ depends on spacetime coordinates, i.e., $F(x^\mu)$, while "global" means the opposite, i.e., $F$ is the same on every point in the spacetime. Is this correct?
| [
"\nThis answers only address part of the problem, i.e., assuming that local and global stand for properties of function/vectors/tensorial obejects defined on spacetime. If you look at spacetime as a [manifold](https://en.wikipedia.org/wiki/Manifold) $M$, then local means that the property/definition under considera... |
I understand that superconductors have zero resistivity, however, I wonder if there is a relation between resistivity and the work function of a specific material so that a superconductor has zero (or a low value) work function. If so, photons with very long wavelengths can release electrons from the surface of a superconductor. Am I right?
| [
"\nThe work function is a measure of how much energy it takes to completely remove an electron from a material and send it into **free space**, and costs on the order of 1000-5000 meV of energy. The resistance is a measure of how much energy is needed to move electrons infinitesimally ($\\omega\\sim0$) **within the... |
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Closed 22 mins ago.
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In quantum mechanics, the probability, say, that a radioactive atom will decay is well defined. By the Born Rule, it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. However, the actual decay point and the actual measurement outcome are individually unpredictable even if one knew everything there is to know about the world.
But if, say, these things were unpredictable and happening for no reason, why does the Born’s rule exist in the first place? Why is it the square of the amplitude and not 1/4 or 1/8?
Secondly, why does the way this work stay constant? If for example, a radioactive atom’s decay point is truly happening for no reason, why doesn’t it just suddenly start decaying with different probabilities?
But because this does not happen, does it not still indicate some level of order? If there is order, where does this come from without determinism?
I am aware that “randomness” can create order through the law of large numbers but that originally interpreted randomness as a mere function of ignorance going back to Laplace’s time. This doesn’t fully explain how or why indeterminism can lead to specific probability functions over others and why they generate particular kinds of order over others.
If one cannot in principle predict what will happen, then presumably, what is happening is occurring for no reason. But if what is happening is occurring for no reason, why are there certain probabilistic laws in the first place?
Given the lack of explanation here, how are we sure that the universe is indeterministic? Is it possible that our theorems deciding that no local hidden variables can exist are simply incorrect?
Of course, even if the universe was deterministic, it would beg the question of how those laws came about. But laws always existing seem to be more explainable than laws ultimately appearing without cause while also staying constant.
| [
"\n\\*\\*\\* Why is it the square of the amplitude and not 1/4 or 1/8?\n\n\nIt's essential that the description of quantum phenomena use probability amplitudes that have the feature that they are complex numbers and can be negative. The amplitudes allow for interference, etc. We take the square modulus for the prob... |
[](https://i.stack.imgur.com/JJpO9.jpg)
I just start learning Physics, and so many strange questions come across my mind. Here is one:
The picture shows the ISS orbiting Earth. Suppose now we (or rather say God) put a baseball behind ISS, Will this ball fall directly to Earth while things in ISS are floating?
| [
"\nIt is all about the *sideways speed* of the object.\n\n\n* The ISS and also your baseball is falling towards Earth constantly. If you just let go of the ball up there, then it will fall down and crash on the ground underneath.\n* Push it a bit sideways while letting go and it still crashes but a bit more to the ... |
**This question already has an answer here**:
[If I jump will I land in the same spot? [duplicate]](/questions/80090/if-i-jump-will-i-land-in-the-same-spot)
(1 answer)
Closed 3 hours ago.
The answer to this question may differ from what we directly observe because the Earth is spinning.
| [
"\nSee the link for the answer.\n[enter link description here](https://zhuanlan.zhihu.com/p/652945463)\n\n\n",
"-1"
] |
I don't have any physics background aside from intro physics so apologies if my question sounds very shallow.
If lens power is based on focal length, but our eyes adjust its focal length based on the distance of the object we are observing, how do we know that our lens prescription is correct for all distances? Most eye tests I have done involve focusing on objects at a far distance. If our lens prescription is based on focal length when we focus on objects at a far distance, is it incorrect to be using this prescription when doing near distance work?
| [
"\nAn \"ideal\" eye can adjust its optical system from the *near point*, $25\\,\\rm cm$ (least distance of distinct vision) from the eye, to the *far point* at \"infinity\" (largest distance of distinct vision).\n\n\nWhen things go wrong that range might change and the optical system has to be adjusted by using ext... |
Consider the following system:
[](https://i.stack.imgur.com/2aksP.jpg)
I am thoroughly confused about certain aspects of the situation described in this diagram in which a block is placed on a wedge inclined at an angle θ. (Assume ***no friction everywhere***)
Let us consider a few different cases:
Firstly, when the wedge is accelerating toward the left, if I were to observe the system from the ground(assumed to be an inertial reference frame), what will I see? Will I see the block stay put on the wedge and accelerating along with it toward the left or will I see it move down the inclined plane, which itself is moving leftwards?
Secondly, in some problems, they have *mentioned* that the block is accelerating "down the inclined plane with acceleration $a$ **w.r.t** the wedge". In such problems, I pick the wedge as my reference frame, introduce a pseudo force and deal with the situation. However, if I were to observe the block from the ground, what would its motion look like to me?
Thirdly, when drawing the free body diagram of a block that is given to be "moving down an inclined plane", in which direction should I assume its acceleration? Directly downward or along the plane?
Fourthly, if given that the block doesn't "slip over the wedge", what is the condition to be used?
As you may see from all this, I am spectacularly confused about all this changing reference frames and accelerations. If anyone could please sum it up concisely, it would be so so helpful for me. I hope that I have conveyed my doubts clearly. If more clarity is required, please let me know and I will edit my question accordingly. MUCH thanks in advance :) Regards.
| [
"\nRather than answer your individual questions I will give you an overview and then discuss some of the points that you have raised. \n\nThere are many ways of tackling such problems but drawing a few FBDs together with some coordinate axes is always a good to start.\n\n\n[![enter image description here](https://... |
Let's consider a satellite which is moving in an elliptical orbit. The velocity of the satellite is $v\_1$ at the perigee, and the distance of the perigee from the centre of the Earth is $r\_1$.
Is it possible to calculate the change in orbital velocity of the satellite required to change its orbit to a circular orbit with radius equal to the apogee distance?
Here's my approach:
1. First, I tried using law of conservation of angular momentum.
$$v\_1 r\_1 = v\_2 r\_2$$
So I could get a relation between the apogee distance and the velocity at apogee.
But I have no clue how to proceed further. Is it even possible?
| [
"\nYour approach is on the right track, and you could use the law of conservation of angular momentum, but you also need to consider the conservation of mechanical energy to *fully* solve this problem.\n\n\nLet's denote the following variables:\n\n\n* $v\\_1$ = Velocity of the satellite at perigee.\n* $r\\_1$ = Dis... |
In many formulations of Bohmian mechanics, researchers seem to claim that 1) measurements of observables such as spin are just measurements of the position of a *pointer variable*, such as the Stern-Gerlach experiment apparatus and 2) measurements of position reveal the position of particles. This effectively removes the 'measurement problem', where superpositions of classical pointers can be explained.
However, this explanation seems to be insufficient. There exists the (philosophical) question of why the position is such an important variable - in classical mechanics, the position *and* the momentum are considered the state of a system. Even worse, position-measuring apparatus, such as the screen in the double slit experiment, obviously perturb physical quantities such as momentum. How is it that we can say measurement reveals the exact position, but perturbs other variables? Isn't it more natural to assume that position measurements are also performed by entangling the system with a classical pointer object (for instance, the screen and the particles become entangled in a double-slit experiment)? For example, weak measurements are performed similarly, by entangling the system with a Gaussian pointer distribution with a high spread. Therefore, position measurement also perturbs particles and therefore cannot reveal the exact position values. A paper by N. Gisin supports this claim, that position measurements in Bohmian mechanics do not reveal exact positions: <https://doi.org/10.3390/e20020105>.
| [
"\nIn Bohmian mechanics, the entire trajectory of all particles through space as a function of time is well-defined. So they have both a position and a momentum at all times.\n\n\nYes, you are correct that measurements never reveal the exact position or momentum. So what?\n\n\n",
"0"
] |
Context
-------
In the derivation of the Boltzmann factor and the canonical partition function based essentially on Lagrange multipliers presented [here](https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)), the equalities,
\begin{align\*}
p\_j &= \frac{1}{Z} e^{\frac{\lambda\_2 E\_j}{k\_B}} \\
Z &= \sum\_j e^{\frac{\lambda\_2 E\_j}{k\_B}}
\end{align\*}
are established, where $E\_j$ refers to the energy of a microstate, $k\_B$ is the Boltzmann constant, and $\lambda\_2$ is the remaining undetermined multiplier. Then, using the Gibbs entropy $S = - k\_B \sum\_j p\_j \ln(p\_j)$ along with some algebra,
\begin{gather\*}
S = -\lambda\_2 U + k\_B \ln(Z)
\end{gather\*}
Finally, the article suggests the derivative,
\begin{gather\*}
\frac{\partial S}{\partial U} = - \lambda\_2
\end{gather\*}
without further work. Note that the article also improperly uses a total derivative instead of a partial here as the whole point is to make the identification $\frac{\partial S}{\partial U} = - \lambda\_2 = \frac{1}{T}$.
My Question
-----------
My question comes from the details of working out the partial derivative. I obtained,
\begin{align\*}
\frac{\partial S}{\partial U} &= - \lambda\_2 - U \frac{\partial \lambda\_2}{\partial U} + \frac{k\_B}{Z} \frac{\partial Z}{\partial U} \\
&= - \lambda\_2 - U \frac{\partial \lambda\_2}{\partial U} + \frac{1}{Z} \sum\_j e^{\frac{\lambda\_2 E\_j}{k\_B}} \left[ E\_j \frac{\partial \lambda\_2}{\partial U} + \lambda\_2 \frac{\partial E\_j}{\partial U} \right] \\
&= - \lambda\_2 + \frac{1}{Z} \sum\_j e^{\frac{\lambda\_2 E\_j}{k\_B}} \lambda\_2 \frac{\partial E\_j}{\partial U}
\end{align\*}
where we have used the statistical definition of $U$ and the equalities above to cancel one of the derivative terms involving $\frac{\partial \lambda\_2}{\partial U}$. It is at this point that I have a question. Clearly, for this to work, we must have $ \frac{\partial E\_j}{\partial U} = 0$. But I cannot seem to really justify this step to myself. Is it simply that the energies of the particular microstates do not explicitly depend on the average energy? How would one justify this?
| [
"\nImo, your doubt is related to the assumption that the partial derivative of the energy of a microstate, $(E\\_j$), with respect to the total energy, $(U$), is zero, i.e., $(\\frac{\\partial E\\_j}{\\partial U} = 0$).\n\n\nIn the context of statistical mechanics, the assumption $(\\frac{\\partial E\\_j}{\\partial... |
I recently encountered this question:
>
> *How long would an someone need to spend on the ISS so that
> their biological clock would be one day younger than their twin who stayed on Earth? The ISS is orbiting the Earth at an altitude $h$.*
>
>
> *Hint: Consider the twins born on a hypothetical stationary Alien ship and sent immediately away: one twin to Earth and the other to the ISS. Then, from the perspective of the Aliens, the twins are both in motion, but at different speeds. Consider all the referentials as inertial (which they are not in reality).*
>
>
>
I have been thinking about it, and I am pretty sure it is too vague to solve. You can't figure out the ISS and Earth frames' relative velocity unless you assume that it is just the orbital velocity. However, the hint says that relative to a stationary alien frame the earth and ISS are both moving. I thought that maybe you need to take into account the Earth's rotation or something, but then you would need to make an assumption about whether they are moving in the same direction or not.
I also thought that maybe it wants us to find when the ISS is one day ahead as measured in the alien frame, but the alien frame is only mentioned in the hint.
I think that maybe when my teacher was writing this she thought that when the ISS is one day younger as measured in the alien frame is the same as when it will be one day younger as measured by the Earth frame. This is wrong though, right?
**Main Question: Is this problem too vague to solve, or is there something I am missing?**
| [
"\n\n> \n> I thought that maybe you need to take into account the Earth's rotation or something, but then you would need to make an assumption about whether they are moving in the same direction or not.\n> \n> \n> \n\n\nSince the hint says both twins are moving, you indeed need to take into account the Earth's rota... |
Applying the Ampere-Maxwell law for a point charge moving:
$$\nabla × \vec{B} = \mu\_{0} \vec{J} + \mu\_{0}\epsilon\_{0} \frac{\partial \vec{E}}{\partial t}$$
Since differentiation at a point requires continuity, and at r=0 jefimenkos equation show that the electric field is not continuous nor defined, how can we even define dE/dt? How does a system of equations lead you to an answer, and then when checking their validity, leads you to be unable to substitute them in a consistent manner?
How can we define the curl of B when the right hand side is undefined? What am I missing?
| [
"\nTerms in this equation need not have finite value at position of the point charged particle, and some of them certainly don't ($\\vec{J}$).\n\n\nThe equation term $\\vec{J}$ becomes singular (of infinite magnitude) at that position. The equation still makes sense at all other points, which is often enough to mak... |
[](https://i.stack.imgur.com/h8luK.png)
The small ball attached by a thin string is in uniform circular motion as shown in the picture (vertical plane). There are two forces acting on the ball, Gravitational force $F\_g$ and Tension force from the string $F\_T$, and the direction of these two forces are as shown. I've learned that the direction of acceleration is pointed to the center of the circle, so does the net force. But The net force F$\_{net}$ I got is shown in the picture, What did I miss?
| [
"\nFor uniform circular motion, acceleration and force are directed toward the center.\n\n\nHere you have another component. The ball will slow as it rises on the circular trajectory. On the other side, it will speed up.\n\n\nAs the speed varies, the component of force and acceleration toward the center will vary. ... |
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in single slit diffraction i understand that by dividing slit into 2 or 4.... equal parts and thereafter equating the path difference to wavelength/2 gives us minima but find it difficult to believe that when the slit is divided to 2 parts then the minima we get is the first minima and when into 4 then is the 2nd minima and so...
Further on simply extending the approach of YDSE in this case like for first minima the path difference is wavelength/2 and for second the path difference is wavelength\*3/2 and so....
(taking initial phases to be same) the results do not match with the textbook results
| [
"\nIf you think of the minima, you always need two elementary wave starting in the slit and in a direction to have another wave with way difference $\\lambda\\over2$ this is the case if you think of the two rays at the edges of the slit having a difference of $\\lambda$sin so the angle to the slit is $sin(\\alpha)=... |
I used to read the term "pure energy" in the context of matter-antimatter annihilation. Is the "pure energy" spoken of photons? Is it some form of heat? Some kind of particles with mass?
Basically, what does "pure energy" in the context of matter-antimatter annihilation refer to?
| [
"\nIf I ruled the world, I would ban the phrase \"pure energy\" in contexts like this. There's no such thing as pure energy!\n\n\nWhen particles and antiparticles annihilate, the resulting energy can take many different forms -- one of the basic principles of quantum physics is that any process that's not forbidden... |
When a fluid-filled container containing an ideal liquid is rotated about its central axis with angular velocity $\omega$, the surface of the liquid forms a paraboloid shape. Now when we derive the equation of the parabola formed by taking a cross section through the midpoint of the paraboloid, using the standard technique used for deriving this formula:
$$y=\frac{\omega^2·x^2}{2g},$$ assuming the vertex of the parabola to be the origin, we assume a centrifugal force
$$F\_\text{c}= \mathrm{d}m · (\omega)^2 · x$$ to be acting on the liquid particles situated along the parabola.
My teachers said we assume a centrifugal force because we are viewing from the frame of the container.
But, then why cannot the same formula be derived from the ground frame, by taking a centripetal force instead of a centrifugal force?
| [
"\nYes, of cause you can do it from the ground frame. You need a centripetal force to keep the fluid particles on a circle. So on the particle you have the weight $mg$ and the force $m·\\omega^2·r$ perpendicular to the slope; together they give the centripetal force.\n\n\n",
"1"
] |
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The Wikipedia article on propagators mentions
that the harmonic oscillator propagator can be derived
from the free particle
propagator using van Kortryk's identity.
This is followed by the identity itself which
appears formidable.
It is not clear to me how to even begin with this.
How is van Kortryk's identity used?
<https://en.wikipedia.org/wiki/Propagator>
| [
"\nLet $H$ be the harmonic oscillator Hamiltonian.\n$$\nH=\\frac{\\hat p^2}{2m}+\\frac{m\\omega^2\\hat x^2}{2}\n$$\n\n\nLet $K(b,a)$ be the propagator for the harmonic oscillator.\n\\begin{equation\\*}\nK(b,a)=\\langle x\\_b|\\exp\\left(-\\frac{iHt}{\\hbar}\\right)\n|x\\_a\\rangle\n\\tag{1}\n\\end{equation\\*}\n\n\... |
[Homework-like questions](https://physics.meta.stackexchange.com/q/714) and [check-my-work questions](https://physics.meta.stackexchange.com/q/6093) are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. Note that answers to homework questions with complete solutions may be deleted!
---
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Suppose we perform a Carnot Cycle between 2 temperature baths $T\_1$ and $T\_2$ with $T\_2>T\_1$ and both having finite heat capacity $C$. What is the final equilibrium temperature?
I have a vague attempt and want to know if its correct:
$$dt\_1=dq\_1/C$$ and $$dt\_2=-dq\_2/C$$ where $dq\_1$,$dq\_2>0$
$$(dt\_2/t\_2)+(dt\_1/t\_1)=0$$
therefore on integrating initial and final temperatures we get $$T\_f^2=T\_1T\_2$$.
So final temp=$$\sqrt{T\_1T\_2}$$
| [
"\nConsider using MathJax when typing formulas. Yes your reasoning is correct. Another way to get it directly is to notice that the entropy of the system is conserved by reversibility of the Carnot cycle. The total entropy is:\n$$\nS = C\\_1\\ln T\\_1+C\\_2\\ln T\\_2\n$$\nwith $C\\_1,C\\_2$ the respective heat capa... |
I've kept a ball between one fixed and another mobile surface of same height (say two books one which functions as a fixed support while other is moved on about by me). While I pull one of the books, the ball starts coming down which implies that $mg$ overpowers normal by the surfaces.
Well then at some instant, I stop pulling the book, then the ball also comes to rest. Why is it so? Why can't the normal always be enough to keep it rest with respect to the y axis?
| [
"\nI don't think that the normal due to the contact surface prevents the ball from falling, but it is the one which provides necessary friction to balance out the weight of the given ball.\n\n\nWhen you remove the contact surface(book in this case), then the necessary friction required to prevent the ball from fall... |
The no-hair theorem states that we can't detect scalar fields outside a black hole, meaning that the solution for the KG equation is trivial, but in fact, we can solve it (for instance for a Schwarzchild BH) and the solution is not trivial, what am I missing?
| [
"\n\n> \n> … what am I missing?\n> \n> \n> \n\n\nYou are missing the meaning of (various) no-hair theorems. Those theorems are talking about *equilibrium* configurations corresponding to end points of gravitational collapse and thus about *static* or *stationary* solutions to Einstein equations with various kinds o... |
If I understand correctly, the rate of radioactive decay depends on the amount of the radioactive element. How then can it be constant, if it depends on concentration?
| [
"\nThe rate of radioactive decay depends on the statistical probability that any random *single* nucleus will decay in a unit of time. That probability does not depend on the number of atoms present in a sample, which is a piece of information that the nucleus of an atom does not have any sort of access to in the f... |
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---
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Closed 22 hours ago.
[Improve this question](/posts/780270/edit)
I use this USB Internet stick:
[](https://i.stack.imgur.com/ftOpi.jpg)
(Huawei E3372h-153)
To connect to the Internet, I simply insert a SIM card into this stick and then insert the stick into a USB port of my laptop.
Based on my experience of participating in Zoom video conferences with this Internet stick, it looks like the quality of the signal depends on the orientation of the laptop (which determines the orientation of the stick because the latter is inserted into a USB port of the laptop), even when the physical location of the laptop remains the same (for example, on the same table in the same apartment).
So, how should I align my Internet stick in order to have the best signal, from the physics standpoint? In the direction of the nearest cell tower? Or perpendicular to that direction? Or maybe at some angle?
I understand that from the practical standpoint, the optimal direction might also depend on surrounding objects such as neighboring buildings and hills as well as the walls of my apartment and even objects inside it. So, I plan to experiment with different orientations and run connection quality tests (ping, download speed, and upload speed), although drawing conclusions won't be straightforward as the signal strength and the connection quality fluctuate with time. The signal strength will go from five bars to two bars and then back to five bars despite the laptop being still, and sometimes the connection gets slow even when I see five bars.
But I am curious about theory. So, assuming there are no obstacles between the cell tower and the laptop with the stick, what should be the best orientation of the stick?
| [
"\nIt's impossible to say. The unknown geometry of the antenna element inside the stick matters, and the computer and attached wiring are also part of the antenna. The environment is full of reflectors and diffractors, so it's not a clean \"wave in empty vacuum\". situation.\n\n\nThe resulting link signal/noise wil... |
Is there a proof from the first principle that for the Lagrangian $L$,
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?
On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the answer, and if possible, the primary source (who published the answer first in the literature).
---
Notes added on Sept 22:
- Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of.
- We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider.
- But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion.
- Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.
| [
"\nWe assume that OP by the term *first principle* in this context means [Newton's laws](http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) rather than the [principle](http://en.wikipedia.org/wiki/Principle_of_least_action) of [stationary action](http://www.scholarpedia.org/article/Principle_of_least_action)$^... |
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---
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Closed yesterday.
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[](https://i.stack.imgur.com/tR7IW.png)
In the solution to this question, they have first taken the component of downwards-acting gravity(g) along the incline which is the sin component then again they have taken the component of the gsin𝜃 along the x-axis and y-axis.
Now my question is how are we supposed to take the component of a component and why not just g in the downwards direction used for calculation of centre of mass?
| [
"\n\n> \n> How are we supposed to take the component of a component ?\n> \n> \n> \n\n\nThere is no reason why you cannot do this - it is just a mathematical procedure. In this case it makes sense because the component of the block's weight perpendicular to the slope is balanced by the normal force from the slope, s... |
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---
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I know that for a linear vertical spring, the governing equation of motion written in the presence of gravity is the same as the one written in the absence of gravity. We can either undergo a transformation or cancel the terms while balancing the laws using the free-body diagram.
What about nonlinear springs?
Please let me know the possible explanations.
| [
"\nIn the case of a vertical nonlinear spring, the effect of gravity can indeed influence the behavior of the spring and the motion of an object attached to it. Unlike a linear spring, where the relationship between force and displacement is linear, nonlinear springs have force-displacement relationships that are n... |
I found this proof of Kepler's first law of planetary motion online [here](https://radio.astro.gla.ac.uk/a1dynamics/ellproof.pdf). Running through the text I found nothing wrong.
However, a warning message at the top the the page saying that "this proof is not examinable!". So what is missing in that proof?
| [
"\nThe proof you've provided is indeed a valid derivation of Kepler's first law of planetary motion, which states that planets move in elliptical orbits with the Sun at one of the foci. The proof correctly derives the equation of an ellipse in polar coordinates and identifies the key parameters, such as the semi-la... |
**This question already has answers here**:
[What happens to the energy when waves perfectly cancel each other?](/questions/23930/what-happens-to-the-energy-when-waves-perfectly-cancel-each-other)
(14 answers)
Closed yesterday.
I have been wondering if the energy changes when two waves interfer constructivly and destructively and if different types waves are able to interfer.
| [
"\nGiven their fundamental differences(mechanical waves need a medium to interact in, while their electromagnetic counterparts do not), it is not common for mechanical and electromagnetic waves to directly interfere with each other in the traditional sense of wave interference(ie constructive and destructive interf... |
Recently I have been reading Berkeley Physics Course - Mechanics. In Chapter 12 (page 350 - 352), author gave a very good example that original definition of momentum $P = mv$ is not working by showing the y-axis' total momentum is not conserved before and after collision. Because:
$$-Mv'\_y(1) + Mv'\_y(2) \neq Mv\_y'(1) - Mv\_y'(2)$$
where
$$v\_y'(1) = {v\_y\over 1+{V^2\over c^2}}\left(1 - {V^2 \over c^2}\right)^{1\over2}$$
$$v\_y'(2) = {v\_y\over \left(1-{V^2\over c^2}\right)^{1\over2}}$$
[](https://i.stack.imgur.com/OWVQA.png)
I continued to read the book and learnt that the corrected definition should be $P = \gamma mv$.
Here comes my question. I now tried to use the $P = \gamma mv$ to go back to the case mentioned above in the hope that it could yield a conservation in y-axis.
$$ {-M\over \left(1-{v\_y'(1)^2 + v\_x'(1)^2 \over c^2}\right)^{1\over2}}v\_y'(1) + {M\over \left(1-{v\_y'(2)^2 + v\_x'(2)^2 \over c^2}\right)^{1\over2}}v\_y'(2) = {M\over \left(1-{v\_y'(1)^2 + v\_x'(1)^2 \over c^2}\right)^{1\over2}}v\_y'(1) - {M\over \left(1-{v\_y'(2)^2 + v\_x'(2)^2 \over c^2}\right)^{1\over2}}v\_y'(2)$$
But very unfortunately they don't equal, since
$${M\over \left(1-{v\_y'(1)^2 + v\_x'(1)^2 \over c^2}\right)^{1\over2}}v\_y'(1) \neq {M\over \left(1-{v\_y'(2)^2 + v\_x^{'}(2)^2 \over c^2}\right)^{1\over2}}v\_y'(2)$$
The left side contains $v\_x'(1)$ which is something I am not able to eliminate, while the right side $v\_x'(2) = 0$.
So I'm wondering what's wrong with my deduction. Is my understanding about the y-axis momentum wrong or purely calculation mistake?
| [
"\nIn the original analysis, when you're using the non-relativistic momentum definition $P = mv$, you indeed encounter a problem with momentum conservation in the y-axis, which indicates that the classical momentum definition is not adequate in special relativity. This leads to the introduction of the corrected def... |
Last winter I started toying with the galaxy gravitational rotation curve graphs. I started modifying the exponent of $r$ that in effect change the $1/r^2$ law and therefore correct the mismatch, just to see where it will lead me.
[](https://i.stack.imgur.com/CX9uA.png)
Using Newtons law of gravity and centrifugal force equation I was able to find the constant GM (representing the interior mass for that distance) for a certain radius and then solve the reverse task of what the approximate power of r should be in the inverse gravitational law for a certain distance to align the gravitational equation with observation.
There are several limitations while trying to fit different functions that adhere to physical phenomena. The power must equal 2 for distance 0 to achieve spherical symmetry, and furthermore should be limited by value 1 since this corresponds to a planar symmetry.
After trying exponent functions (that violate the first principle) and log functions (indeterminate at 0) I came up with the form $r^{2-C \sqrt{r}}$. This violates the second principle but since the effect should be local and limited to the galaxy, it works as the first approximation.
Fitting the function I get the graph
[](https://i.stack.imgur.com/lrfx3.png)
Repeating this procedure for a multitude of galaxies I got a larger database, with which to ascertain the constant in the equation. Thinking that the more important and readily available data for galaxies is mass and radius I redefined the constant to be $C\_1 r^\alpha m^\beta = C$. This pushed the constant definition just further but hopefully gives an accurate enough model of calculations.
Using galaxies NGC3521, NGC 5055 and NGC 4736, NGC4826 that have similar radius to define the power of the mass dependence (came out as 0.027 and 0.022, averaged 0.025) and doing the same with NGC 7793, NGC 2976 and NGC 7331, NGC6946 to get two estimates for radius (-0.5778, -0.40694 averaged to -0.5). The weak relation of mass seems insignificant but with galaxies of equal radii, mass has always an increasing effect on the constant, weak as it is.
After knowing the constants alpha and beta we can use a statistical average to get the approximate constant C\_1 for our datagroup giving us the empirical relation
[](https://i.stack.imgur.com/oo7yZ.png)
[](https://i.stack.imgur.com/NTeVp.png)
Where r is the distance from the galactic center in meters, $r\_g$ is the radius of the entire galaxy in lightyears and $m\_g$ is the mass of the galaxy in solar masses.
Using this function to backwards plot the exponent equation we get a collection of all other graphs confirming the general fit of the model (atleast with the calculated constant from data, not empirical model).
[](https://i.stack.imgur.com/E17uy.png)
[](https://i.stack.imgur.com/P4jz7.png)
The above relation should be a somewhat good empirical equation on its own. Probably the constant (if it even has a functional relationship to reality) is still varied by other possible parameters.
Is this something that is known or have I just been dealing with numerology?
While I am studying physics, I am in a completely different area with no aquaintances in the relevant field. Perhaps it will just have been an interesting 5 minute read.
| [
"\nBy eye this curve doesn't seems like a good one. Try to study the residues of the $\\chi^2$.\n\n\nIn principle for a good fit they should be gaussian, but since this is an approximation you can partially ignore that.\n\n\nNonetheless they should at least be symmetric about zero and, at least by eye, that's not y... |
Let's consider a glass slab ABCD.
Now a light ray is incident on the near end of the air-glass interface AB and is emerging from the glass-air interface BD. In such a manner:
[](https://i.stack.imgur.com/G5N3D.png)
Is there something wrong how I have made my figure. Can light emerge in such a manner from a glass slab.
This is my first question and English is not my first language I hope you understood my doubt. I feel like the figure I have drawn does not look like the conventional ray diagram of light entering a glass slab.
Also I am not an expert I am just a student who wants to know the answer to my question.
| [
"\nThere is a potential problem.\n\n\nThe critical angle $c = \\sin^{-1}\\left ( \\frac 1 n\\right)$, where $n$ is the refractive index of the glass.\n\n\nAt the first interface $r\\le c$ and at the second interface $i\\_2 \\le c$ for there to be refraction\n\n\nHowever, $i\\_2 + r= 90^\\circ$, and so only if $n\\l... |
Let's consider a glass slab ABCD.
Now a light ray is incident on the near end of the air-glass interface AB and is emerging from the glass-air interface BD. In such a manner:
[](https://i.stack.imgur.com/G5N3D.png)
Is there something wrong how I have made my figure. Can light emerge in such a manner from a glass slab.
This is my first question and English is not my first language I hope you understood my doubt. I feel like the figure I have drawn does not look like the conventional ray diagram of light entering a glass slab.
Also I am not an expert I am just a student who wants to know the answer to my question.
| [
"\nThere is a potential problem.\n\n\nThe critical angle $c = \\sin^{-1}\\left ( \\frac 1 n\\right)$, where $n$ is the refractive index of the glass.\n\n\nAt the first interface $r\\le c$ and at the second interface $i\\_2 \\le c$ for there to be refraction\n\n\nHowever, $i\\_2 + r= 90^\\circ$, and so only if $n\\l... |
Boltzmann's entropy formula:
$S=k\_{\mathrm {B} }\ln \Omega$
where $\Omega$ is the number of real microstates corresponding to the gas's macrostate.
Let's assume that we are talking about an ideal gas in a fixed close isolated space, with $n+1$ atoms.
It seem that the number of possible microstates of the following two situations are the same. Is their entropy the same ($x>y>0$)?
* 1. $n$ atoms have energy $x$, and one atom has energy $y$,
* 2. $n$ atoms have energy $x/2$, and one atom has energy $y + nx/2$
There is a bijection between the microstates of 1 and 2. So the number of possible microstates for these two are the same.
However intuitively, I would expect that 2 should be more likely as I would expect that each time there is an interaction between two atoms, some energy from the higher energy atom would be transferred to the lower energy atom.
Where I am getting things wrong?
And generally, how do we compute the number of possible microstates of a system as in the two cases above? (I wonder how experimental physicists use the formula in practice.)
| [
"\nYou're comparing the two microstates incorrectly.\n\n\nScenario Uno\n\n\n* There are $n$ atoms with energy $x$ each.\n* There is one atom with energy $y$.\n\n\nScenario Dos\n\n\n* There are $n$ atoms with energy $x/2$ each.\n* There is one atom with energy $y + nx/2$.\n\n\nNow, let's analyze these scenarios in t... |
In Polchinski's exposition of the RNS formalism for the superstring (String Theory: Volume II, chapter 10), in page 8, he mentions the worldsheet fermion number operator, which he calls $F$. He then goes on to define
\begin{equation}
(-1)^F = e^{i \pi F}.\tag{10.2.19}
\end{equation}
To my best knowledge, this object is meant to commute with the worldsheet bosonic operators $X^\mu(z)$ and anticommute with the fermionic ones, $\psi^\mu(z)$ (this section of the book is focused only on the open superstring, so the antiholomorphic fermion is abscent). Since the fermions can be expanded as
\begin{equation}
\psi^\mu(z) = \sum\_{r \in \mathbb{Z}+ \nu} \frac{\psi^\mu\_r}{z^{r+ 1/2}},\tag{10.2.7}
\end{equation}
where $\nu =0$ in the R sector and $\nu=1/2$ in the NS sector, $e^{i \pi F}$ can only anticommute with $\psi^\mu(z)$ if it anticommutes with each one of the modes. This requirement can be written in the form
\begin{equation}
e^{i \pi F} \psi^\mu\_r = - \psi^\mu\_r e^{i \pi F} = \psi^\mu\_r (-1) e^{i \pi F} = \psi^\mu\_r e^{i \pi(F+1)},
\end{equation}
which, upon expanding both sides, means that $$F \psi^\mu\_r = \psi^\mu\_r(F-1).$$ Here comes the problem: if he were to define the worldsheet fermion operator as
\begin{equation}
F^\prime = \sum\_{r \in \mathbb{Z}+ \nu} \psi^\mu\_r \psi\_{-r \mu},
\end{equation}
is is easy to see that indeed $$F^\prime \psi^\mu\_r = \psi^\mu\_r(F^\prime-1).$$ However, he defines it as
\begin{equation}
F = \sum\_{a=0}^4 i^{\delta\_{a,0}} \Sigma^{2a,2a+1},\tag{10.2.21+22}
\end{equation}
where
\begin{equation}
\Sigma^{\mu \rho} = - \frac{i}{2} \sum\_{r \in \mathbb{Z}+ \nu} \left[ \psi^\mu\_r , \psi^\rho\_{-r} \right]\tag{10.2.20}
\end{equation}
are the spacetime Lorentz generators for the spinors. What is the purpose of this definition? This thing doesn´t even satisfy $[\psi^\mu\_r,F]=\psi^\mu\_r$, as is shown in equation (10.2.23), so I don't understand how this could be the fermion number operator.
| [
"\nRecall that\n$$\\{ \\psi\\_r^\\mu , \\psi\\_s^\\nu \\} = \\eta^{\\mu\\nu} \\delta\\_{r,-s}. \\tag{1}$$\nLet's start by trying your definition,\n\\begin{equation}\n\\begin{split}\n[ F' , \\psi\\_r^\\mu ] &= \\sum\\_s [ \\psi\\_s^\\nu \\psi\\_{-s,\\nu} , \\psi\\_r^\\mu ] \\\\\n&= \\sum\\_s \\psi\\_s^\\nu \\{ \\psi... |
Let the differential cross section of a scattering experiment given by $\frac{\text{d}\sigma\_{c}}{\text{d}\Omega\_{c}}(\vartheta\_{c})$, where $\vartheta\_{c}$ describes the scattering angle in the center of mass frame. The relation between $\vartheta\_{c}$ and the scattering angle in the laboratory frame is given by
$$
\tan\vartheta\_{L} = \frac{\sin\vartheta\_{c}}{\frac{m\_{1}}{m\_{2}}+\cos\vartheta\_{c}} \qquad (1).
$$
Now the differential cross section in the laboratory reference frame can be calculated via
$$
\frac{\text{d}\sigma\_{L}}{\text{d}\Omega\_{L}}(\vartheta\_{L}) = \frac{\text{d}\sigma\_{c}}{\text{d}\Omega\_{c}}(\vartheta\_{c}(\vartheta\_{L}))\cdot \frac{\sin(\vartheta\_{c}(\vartheta\_{L}))}{\sin(\vartheta\_{L})}\cdot \frac{\text{d}\vartheta\_{c}}{\text{d}\vartheta\_{L}}(\vartheta\_{L}) \qquad (2)
$$
For Rutherford scattering, the differential cross section in the center of mass frame is given by
$$
\frac{\text{d}\sigma\_{c}}{\text{d}\Omega\_{c}}(\vartheta\_{c}) = \left( \frac{1}{4\pi \epsilon\_{0}} \frac{Z\_{1}Z\_{2}e^{2}}{4E\_{c}} \right)^{2} \frac{1}{\sin^4\left( \frac{\vartheta\_{c}}{2} \right)} \qquad (3)
$$
In the laboratory frame it is apparently given by
$$
\frac{\text{d}\sigma\_{L}}{\text{d}\Omega\_{L}}(\vartheta\_{L}) = \left( \frac{1}{4\pi \epsilon\_{0}} \frac{Z\_{1}Z\_{2}e^{2}}{4E\_{L}} \right)^{2} \frac{4}{\sin^4\vartheta\_{L}} \frac{\left( \cos\vartheta\_{L}+\sqrt{1-\left( \frac{m\_{1}}{m\_{2}}\sin\vartheta\_{L} \right)^{2}} \right)^{2}}{\sqrt{1-\left( \frac{m\_{1}}{m\_{2}}\sin\vartheta\_{L} \right)^{2}}} \qquad (4)
$$
I tried for hours to derive formula (4) using (1),(2), and (3). I didnt get anywhere. Could you help me? Thanks in advance
| [
"\nAfter an Odyssey up the reference tree i found the answer in\n\n\n\n> \n> W. K. Chu, J. W. Mayer, and M. -A. Nicolet, Backscattering Spectrometry (Academic Press, New York, 1978)\n> \n> \n> \n\n\nSo you don't have to search it here it is:\n\n\nStarting with the formula:\n$$\n\\frac{\\text{d}\\sigma}{\\text{d}\\O... |
From the following [image](https://www.eigenplus.com/wp-content/uploads/2022/12/isotropic_material-2048x1444.jpg), why do we still call it isotropic? if the density at A and B differ, I don't think it's enough to call it isotropic. In my opinion, material is only isotropic if when we choose some point, around it in all directions, properties are the same throughout the end of the material from that point, but image taken from [here](https://www.eigenplus.com/homogenous-and-isotropic-material/), still says material is isotropic.
| [
"\nFor a system to be isotropic w.r.t. a point of reference it has to appear the same in all directions. Say if I'm located at a point $p$ and from there I find the system isotropic then this just means that the system has the same property at a fixed radius $r$. In particular, it does not need to have the same pro... |
I don't understand why the nuclear decay formula is derived with logarithms.
if $t=T\_{1/2}$ then
$N=N\_0\cdot\frac{1}{2}$
if $t=2T\_{1/2}$ then
$N=N\_0\cdot\frac{1}{2^2}$
if $t=3T\_{1/2}$ then
$N=N\_0\cdot\frac{1}{2^3}$
and so on... so the formula can be expressed in this form
$N=N\_0\cdot \frac{1}{2^{\frac{t}{T\_{1/2}}}}$
$N=N\_0\cdot 2^{-\frac{t}{T\_{1/2}}}$
So why is it derived further with natural logarithms? why doesn't the derivation just stop there?
the next derivation is
$\ln \left(\frac{N}{N\_0}\right)=\ln \left(2^{-\frac{t}{T\_{1/2}}}\right)$
and finally
$N=N\_0e^{-\frac{\ln \left(2\right)}{T\_{1/2}}t}$
| [
"\nThe decay graph can be described in a number of ways each with a parameter which is a characteristic of the decay.\n\n\nYour description the characteristic parameter is the *half-life*, $\\tau$ and so $N(t) = N\\_0\\,2^{-t/\\tau}$.\n\n\nAnother often used parameter is the *decay constant*, $\\lambda$ which is th... |
[Homework-like questions](https://physics.meta.stackexchange.com/q/714) and [check-my-work questions](https://physics.meta.stackexchange.com/q/6093) are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. Note that answers to homework questions with complete solutions may be deleted!
---
Please help the asker edit the question so that it asks about the underlying physics concepts instead of specific computations. If there are any complete answers, please flag them for moderator attention.
Closed yesterday.
[Improve this question](/posts/780104/edit)
I have to find the magnetic field generated by an infinite wire carrying current by using the curl formula:
$$\nabla \times B=\mu\_0J$$
Let's say the wire is along the z axis, we can use cilindrical coordinates and rewrite the equation; according to symmetries:
$$\frac{1}{r}\frac{d}{dr}\left( rB\_\theta(r) \right)=\mu\_0J(r)$$
Now, the idea my professor wants us to follow in this case is solving the differential equation by separating variables.
However, what he does in his notes is considering the equation:
$$\frac{1}{r}\frac{d}{dr}\left( rB\_\theta(r) \right)=\mu\_0J$$
when $r\le R$, and
$$\frac{1}{r}\frac{d}{dr}\left( rB\_\theta(r) \right)=0$$
when $r\ge R$.
I know how to solve the exercise once this distinction is made, but I don't understand why it is done like this: why is $J=0$ outside the wire? I don't understand which $J$ I have to consider in this formula.
| [
"\nThe reason your professor considers 𝐽 = 0 outside the wire (𝑟 ≥ 𝑅) is related to the assumption that the current is confined to the wire itself. In the context of solving for the magnetic field generated by an infinite wire carrying current, there's typically an implicit assumption that the current is flowing... |
Do [Boltzmann brain](https://en.wikipedia.org/wiki/Boltzmann_brain) thought experiments suggest literally anything can form randomly?
What are the limitations to what random fluctuations can form? Literally any physical, material object?
Lastly, I am curious as to how this compares to an object with a high degree of complexity always existing. For example, is a bike more likely to always exist for no reason than it being created by random fluctuations? Or is it less likely?
| [
"\nA very handy way of looking at the statistical mechanism of a system is by looking at its so-called [phase space](https://en.wikipedia.org/wiki/Phase_space) (read as state space).\n\n\nConsider a [microstate](https://en.wikipedia.org/wiki/Microstate_(statistical_mechanics)#:%7E:text=Treatments%20on%20statistical... |
This is my first post and also the first time I've ever tried understanding a physics concept by approaching it with my own math logic so tips on trying to do this more effectively are also welcome
I’m trying to understand the link between a force experienced by a wire in a homogenous magnetic field and the magnetic field created by the current in the wire.
I noticed that the force vector is always directly towards the spot on the circle where the homogenous magnetic field and the magnetic field created by the current "cancel each other out" and I’m trying to figure out how they relate to each other.
I added together the vectors of the homogenous magnetic field and the wire’s magnetic field (the wire’s magnetic field vector is the tangent of the circular magnetic field around the wire) and drew the resulting vector lines when the homogenous magnetic field and the wire’s magnetic fields are equal. Just to visualize the interactions between the magnetic fields somehow.
I noticed that they only cancel each other out in 1 specific spot but I don’t quite understand the link between the force, the direction of the force vector and the magnetic fields "cancelling each other out" (or the vectors just pointing in opposite directions). you can strengthen the homogenous magnetic field which results in a stronger current and therefore force or you can increase the current which results in a stronger magnetic field of the wire and thereby also force.
So how is the energy converted from magnetic fields "cancelling each other out" in a specific point to a force and why is the force vector always pointing to this point?
[](https://i.stack.imgur.com/9DWjh.jpg)
| [
"\nThe combination of magnetic fields is sometimes called the *catapult field* as illustrated below.\n\n\n[](https://i.stack.imgur.com/Qquir.jpg)\n\n\nThe name catapult field comes from the idea that magnetic field lines are always in a state of te... |
I am trying to understand the implementation of POVMs on a Hilbert space by using unitary operations and projective measurements in a larger Hilbert space. In A. Peres' *Quantum Theory: Concepts and Methods*, Section 9.6, he claims that a POVM on a Hilbert space $H$ of dimension $n$, consisting of rank-1 elements
$$ {\{ A\_{\mu} = |u\_{\mu}\rangle \langle u\_{\mu}| \} }\_{\mu=1}^N $$
(with $|u\_{\mu} \rangle$ subnormalised) can be implemented on a Hilbert space $K$ of dimension $N$.
With $ |u\_{\mu}\rangle = \sum\_{i=1}^n u\_{\mu i} |i\rangle $, he claims that the following orthonormal basis for $K$ can be constructed:
$$ |w\_{\mu}\rangle = \sum\_{i=1}^n u\_{\mu i} |i\rangle + \sum\_{j=n+1}^N c\_{\mu j} |j\rangle, $$
by considering that the vectors
$$ {\left\{ |u'\_{\lambda}\rangle = \sum\_{i=1}^N u\_{i\lambda} |i\rangle \right\}}\_{\lambda=1}^n $$
are orthonormal due to the constraint
$$ \sum\_{\mu=1}^N A\_{\mu} = I \Rightarrow \sum\_{\mu=1}^N u\_{\mu i} u\_{\mu j}^\* = \delta\_{ij} $$
and noting that completing this into a basis and setting the vectors as columns of a unitary $U$ would yield $\{ w\_{\mu} \}$ as the rows of $U$.
He then argues that, because the projection of $|w\_{\mu}\rangle$ onto $H$ as a subspace of $K$ would yield $|u\_{\mu}\rangle$, a projective measurement in this basis implements the POVM.
However, it is not clear how this can actually be implemented. Peres states that, if the physical system's state is actually $K$, and that it just happens that the system of interest is confined to a subspace $H$, then this implementation is possible. He then moves on to describe a more general construction involving a separate ancilla system. So what I want to ask is, **what are the general conditions under which this construction can implement a POVM**? And, as a follow-up, given a POVM for which the construction works, **is this the smallest possible Hilbert space that can implement the POVM**?
| [
"\n### Naimark dilation for arbitrary POVMs\n\n\nAny POVM $\\{\\mu\\_b\\}\\_{b=1}^{m}$ can be implemented as a computational basis projective measurement in an enlarged space via the Naimark's construction. Restricting our attention to finite-dimensional spaces, let's assume without loss of generality that the rele... |
I recently came across a paper that argues that we can substitute a qubit for arbitrarily large number of bits in a physical system (<https://arxiv.org/abs/quant-ph/0110166>).
The notion of bit represent classical information. This then implies qubit can hold a huge amount of information. On the other hand, we have Helovo bound that we can access (or extract) the information from qubit up to two bits only. That is, Helovo bound is about the amount of information that we can extract from a superposition representing states of qubit, it is not about the information that qubit is carrying with it.
In classical system, such as deterministic finite automata (DFA), state actually represent the memory. The number of states of a DFA is the size of the memory. Here we can't do that. The state space of a qubit is represented by complex linear combination of two basis states (with normalization condition) is of infinite size. Considering each state as a bit is perhaps the reason that we may substitute a qubit for arbitrarily large number of bits, and that I believe we can do before the act of measurement. So if a qubit represent the quantum memory how can we compare its size with classical memory?
Here is another paper that claim for unbounded memory advantage provided by qubit. (<https://iopscience.iop.org/article/10.1088/1367-2630/aa82df>).
Moreover, it seems that many physical system have memory of its own, it may be classical or quantum. But we usually don't take into account, at least explicitly. This is the impression I have after reading Galvao and Hardy paper mentioned above. I am not physicist therefore this impression might well be totally misplaced.
Any help clearing up my confusions (and misconceptions) is greatly appreciated.
| [
"\n\n> \n> So if a qubit represent the quantum memory how can we compare its size with classical memory ?\n> \n> \n> \n\n\nYou can think of a quantum computer as being an analogue computer - the state of a qubit is like the angle of a wheel or the level of water in a tube in an analogue computer. In principle you c... |
If we have a sealed piston-cylinder with gas inside and we heat it from the outside, the temperature changes. At the same time, if we look at it as a control mass system,the volume increases as the gas expands. Mass remains same. Hence specific volume changes too. So how can temperature and specific volume always be independent?
| [
"\nTake gas in a box with a fixed size and heat it. The temperature increases, but the specific volume remains fixed. This is also a perfectly reasonable physical situation thatwe could easily achieve. Comparing this situation with the one in your question, we conclude that while I *can* increases the temperature a... |
Take the following standard setup and calculation for a Faraday disk:
A metal disk of radius $a$ rotates with angular velocity $\omega$ about a vertical axis, through a uniform field of magnitude $B$, pointing up along the vertical axis. A circuit is made by connecting one end of a resistor to the axle and the other end to a sliding contact, which touches the outer edge of the disk.
The speed of a point on the disk at a distance $s$ from the axis is $v = \omega s$, so the force per unit charge is $\mathbf{f}\_{mag} = \mathbf{v} \times \mathbf{B} = \omega s B \mathbf{\hat{s}}$.
The emf is therefore
$$ \mathcal{E} = \int\_o^a f\_{mag} ds = \omega B \int\_o^a s ds = \frac{\omega B a^2}{2} $$
However what if the magnetic field didn't reach over the entire disk? What if only one slice of the disk was in a uniform magnetic field. My intuition would say that because the negatively charged particles in the disk are only periodically exposed to the magnetic field and thus only periodically exposed to the force per unit charge, they will move to the outside of the disk more slowly. So the emf should be less. But by how much? And where in the calculation does this come in?
| [
"\nIf only a sector of the disk is in uniform magnetic field, then there can't be equilibrium of charge in the disk, because in the frame of the disk, the charge experiences time-dependent force, due to moving in and out of the magnetic field. The disk will experience electric currents in its frame, and thus dissip... |
I'm aiming to use differential equations to simulate thermal transfer between different materials in a 3D space, however, my physics knowledge is lacking. Is there a formula that can give the rate of thermal transfer between two different materials (e.g copper, aluminium) with a two-dimensional area of contact?
| [
"\nYou could use a formula known as Fourier's Law, which essentially states that\n\n\n$Q = -k \\cdot A \\cdot \\frac{dT}{dx}$\n\n\nWhere:\n\n\n* Q is the rate of heat transfer (in watts, W).\n* k is the thermal conductivity of the material (in watts per meter-kelvin, W/(m·K)).\n* A is the cross-sectional area throu... |
I think I am misunderstanding something elementary. Supposing a scenario where you have a negative charge enclosed in a Gaussian surface, why would the presence of outside charges not affect the flux?
In scenario 1 there are no outside charges, so there will be field lines coming in from infinity and none going out so total flux is negative, which makes sense.
But in scenario 2 where you have multiple positive charges outside the Gaussian, do field lines from these charges not end in the negative charge in the Gaussian? And the more positive charges you have, the more field lines that end in the negative charge?
I don't understand why the flux is the same in both scenarios.
| [
"\n\n> \n> And the more positive charges you have, the more field lines that end in the negative charge?\n> \n> \n> \n\n\nThe number of field lines chosen to describe the amount of charge is purely arbitrary and doesn't follow any physical laws. They are simply used to depict the magnitude of charge on individual c... |
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I know the Fourier transform is $$
F(\omega)=\int\_{-\infty}^{\infty} f(x) e^{-i \omega x} \,d x
$$
$$
f(x)=\frac{1}{2 \pi} \int\_{-\infty}^{\infty} F(\omega) e^{i \omega x} \,d \omega,
$$
but in quantum mechanics the momentum wave function $\phi(p)$ and positional wave function $\psi(x)$ are related by
$$
\psi(x, t)=\frac{1}{\sqrt{2 \pi \hbar}} \int\_{-\infty}^{\infty} \phi(p, t) \mathrm{e}^{+\mathrm{i} p x / \hbar} d p
$$
$$
\phi(p, t)=\frac{1}{\sqrt{2 \pi \hbar}} \int\_{-\infty}^{\infty} \psi(x, t) \mathrm{e}^{-\mathrm{i} p x / \hbar} d x.
$$
My question are:
* Why can $\phi(p)$ be Fourier expanded in $\psi(x)$?
* where did the $\frac{1}{\sqrt{2 \pi \hbar}}$ come from? Why not $\frac{1}{2 \pi}$？
| [
"\nYou are just asking notation. The Fourier transform comes with a choice of where and how to put the $2\\pi$, unless you start with the most mathematically natural version.\n\n\nFirst, you should rewrite your transformation with the usual symbols, $k$ instead of $\\omega:$\n$\n\\begin{align}\n\\tag1f(x)=\\frac1{2... |
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If there are only two particles at some distance away from each other in a space, say one picometer, with one negatively charged and one positively charged, say with a magnitude of 1.6\* 10^-19 coulombs each, and they are given an initial velocity apart of 10% the speed of light, and they both have a mass of $1.67 \times 10 ^{-27}$ kg, will they ever come to the same spot?
I know the function that, when given the particles's distance and difference in charges, gives the force acting to one towards another: $F\_e=\frac{KQq}{d^2}$. The similar equation for gravity ($F\_g=\frac{GMm}{d^2}$) would also need to be applied (though gravity's effects are much weaker), and then the effect of the forces to be able to alter the initial velocity could be determined with $a=\frac{F\_{net}}{m}$, which could then be used to find velocity and position. The forces and their effect should be applied to both particles, which means doubling the acceleration they come together at. (I am not taking into account relativity, though I wonder if that would play a role since fast speeds are considered.)
However, it seems that a very important part of this that I currently do not know is how to incorporate the idea of time. I am currently taking Calc 3, so the math may be over my head; do I just have to wait and learn the math?
I am also interested in finding out if it would be possible to find the most extreme values of difference in charge, initial velocity, initial distance, and masses, to still be able to have the particles meet again.
| [
"\nIt is important to know that for the data you've provided- there would be more than simply electrostatic and gravitational forces at play. Since the velocity of particles is of the order of 10% of the speed of light, *relativistic* effects would also come into play. Even in the simplest solution (where i will as... |
I don't understand how to explain why kinetic energy *increases* using Faraday's Law as a charged particle reaches the end of a magnetic bottle.
I know that along the field line in the bottle, the magnetic moment of the particle is conserved (first adiabatic invariant) which leads to $v\_{\perp}$ icreasing as $v\_{\parallel}$ decreases, and the kinetic energy of the particle is just equal to $K.E.=\frac{1}{2}v\_{\perp}^{2}$. This makes sense, but I have no idea how this relates to Faraday's Law.
I am tempted to invoke the integral form of Faraday's Law:
$\oint\_C {E \cdot d\ell = - \frac{d}{{dt}}} \int\_S {B\_n dA}$
where the closed loop is the path the particle is gyrating around the magnetic field, but I am not sure that is right. Obviously, as the field line density increases towards the end of a magnetic bottle, the flux through a given area increases.
Any help here would be appreciated. I am just having a hard time contextualizing these principles for a magnetic bottle.
| [
"\nI suspect you know that the total energy doesn't change; the kinetic energy due to the velocity perpendicular to the axis of the magnetic bottle, $1/2 m v\\_\\perp^2$ increases at the expense of the remaining kinetic energy term $1/2 m v\\_{||}^2$. As the particle moves into a region of higher magnetic field, th... |
It's well known that the basic formula for the angular magnification of an optical telescope is
$$ |M| = \frac{f\_{ob}}{f\_{oc}} $$
where $f\_{ob}$ is the focal length of the objective, and $f\_{oc}$ is that of the ocular (the *eye-piece*).
So, if $f\_{ob}$ of the objective increases, so will $M$ of the telescope. Now, based on simple lens maker's formula of thin lens approximation,
$$ \frac{1}{f} = (n-1) \left( \frac{1}{R\_1} + \frac{1}{R\_2} \right)$$
a long focal length lens, has a large radius of curvature; i.e., a less curved surface with less bending of incoming light rays.
My question is, assuming every approximation holds, to get larger and larger $M$, we need flatter and flatter objective lenses, so in the limit does that mean that a flat glass, of (practically) infinite radius of curvature, yields
infinite (very large) magnification? And further, since a piece of flat glass can be replaced with empty space (as it won't bend any light rays), could we then, in principle, get an infinite (very large) magnification by just the ocular alone?
What's wrong in this reasoning?
| [
"\nA couple of problems:\n\n\n1. Your formula assumes the telescope is short compared to the distance to the object you're imaging. An infinitely long telescope violates that assumption.\n\n\nBut more seriously:\n\n\n2. You're using a ray optics approximation, ignoring diffraction. With diffraction, what happens as... |
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I recall reading in Feynman and Hibbs about an integration
trick that can be used to solve certain types of integrals
that involve the Coulomb potential.
What was the trick?
| [
"\nThe trick is,\nmultiply the integrand by $\\exp(-\\epsilon r)$, solve the integral,\nthen let $\\epsilon\\rightarrow0$.\n\n\nFor example, consider the following momentum transfer function\nwhere $\\breve p=p\\_a-p\\_b$.\n$$\nv(\\breve p)=\\frac{4\\pi\\hbar}{\\breve p}\n\\int\\_0^\\infty \\sin\\left(\\frac{\\brev... |
**This question already has answers here**:
[Why does a ball rolling without slipping stop due to friction? [duplicate]](/questions/410708/why-does-a-ball-rolling-without-slipping-stop-due-to-friction)
(2 answers)
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I have been reading about rolling motion and I seem to have a confusion regarding the friction acting on such bodies.
First off for a body to start rolling their has to be some friction b/w the surfaces in contact. (right? Otherwise it would simply slip on the surface.)
I also know that friction acts in a direction which opposes relative motion (correct me if this statement is incomplete.)
Now suppose a body (say a sphere) is performing a pure roll, i.e it doesn't slip on the surface.
At the point of contact between the sphere and the surface the rotational and linear velocities are equal and opposite in directions ($v=wR$) so the point of contact is essentially at rest and there is no relative velocity b/w the the point of contact and the surface,and thus NO FRICTION acts on the sphere. (pure roll.)
So my question is how does the sphere slow down eventually? For it to slow down friction has to act opposite to the motion of the sphere but why does this friction act if there is no relative motion between surfaces in contact (i get that the entire sphere has a relative $v$ wrt to the surface but the friction acts at point of contact which is a rest.)
Maybe I am thinking too ideally and in reality other factors are also at play, if so what is the more realistic system?
| [
"\nIf there were no drag or losses, then you're correct. The sphere would move forward and rotate indefinitely.\n\n\nBut assuming the surfaces are not completely smooth and there is air in the room, then there will be drag forces on the sphere. These forces slow down the sphere relative to the surface.\n\n\nOn a fr... |
Let's assume that a cylindrical magnet is clamped by an horizontal arm extending out of the wall. The top of the magnet is its north pole and the smooth face facing down is its south pole. If I bring another cylindrical magnet and position it perfectly in line below the fixed magnet, with its top being its north pole and the bottom being the south pole, would it be possible that the weight of the magnet going down and the magnetic force attempting to pull the magnet upwards balance out so that the magnet can levitate in mid air?
Conceptually, it seems to make sense as in this case, the magnet is not prone to flipping since it poles facing each other are not repelling. However, I'm not sure if this violates Earnshaw's theorem.
It it's not possible, then why isn't it possible? Did I miss something in my assumptions? And if it is possible, Is it also possible for the levitating magnet to rotate about its central vertical axis and still remain suspended in space?
| [
"\nEarnshaw's theorem doesn't say that you can't have equilibrium; it says that you can't have *stable* equilibrium.\n\n\nThere's a point below the fixed magnet where the attraction it exerts on the \"floating\" magnet exactly cancels out the force of the floating magnet's weight. But if the floating magnet is pert... |
[Homework-like questions](https://physics.meta.stackexchange.com/q/714) and [check-my-work questions](https://physics.meta.stackexchange.com/q/6093) are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. Note that answers to homework questions with complete solutions may be deleted!
---
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I am doing research on the behavior of lightning strikes and how they affect objects they hit. I have found this article that gives some information about lightning, like speed, length, temperature, and width of lightning:
<https://www.metoffice.gov.uk/weather/learn-about/weather/types-of-weather/thunder-and-lightning/facts-about-lightning>
Also, I have found that the electrical charge of lightning is between 14 C and 18 C. What I need is to find the pressure that average lightning exerts on the lightning rod. I am interested in ways to measure the pressure and mechanical stress that a lightning strike can cause in a lightning rod and if that can cause the rod to oscillate vertically or horizontally so that pressure sensor tapes can be used to measure the change in rod tangential and radial pressure after a lightning strike.
I need to know what is the average pressure expected from lightning and how to calculate it.
| [
"\nThe pressure from a lightening strike is not due to a lot of electrons hitting the target (some strikes are opposite polarity), but because of the explosive heating of the air in the discharge.\n\n\nOne way of estimating the pressure is the ideal gas law: $PV=nRT$. In lightening $T$ increases from ca 300 K to ab... |
I wanted to explain to a person knowing some very very basics of physics, why the force of attraction on two charges $Q\_1=3nC$ and $Q\_2=-1nC$ are the same. Of course my interlocutor thought that $3nC$ charges should experience larger attraction force.
Well, how can you explain that to somebody that didn't fully conceptualise Newton's Third Law? Forces come in pairs - ok great - if one accepts that, the work would be over.
Or is it a strict logical consequence of Newton's Third Law and there is no other way?
I am really puzzled by this, the obvious things are the most difficult ones :)
How would you explain the situation between the two charges I mentioned? I found that textbooks give the fact about forces being the same as obvious.
| [
"\nSimple, partial answer\n----------------------\n\n\nOne way to explain this is that there are two contributions to the force, that get multiplied together. A way to express this is:\n$$\nF = q E\n$$\nwhere $q$ is the charge that the force acts on, and $E$ is the (magnitude of the) electric field at the location ... |
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Almost any starting book on quantum mechanics started by studying one-dimensional quantum systems. Other than their simplicity, are there other reasons for which they are important to understand more complex configurations? On the other hand, how can we reduce 3-dimensions Schroedinger equation into a radial and angular parts,hence a one dimensional problem.
| [
"\nHere are some typical one dimensional problems you might solve in quantum mechanics, and some ways they serve as toy models for more realistic problems. The one dimensional problems often display some interesting behavior. When more complex problems can be solved analytically, it is often by transforming them in... |
The Carter constant for the [Kerr Newman](https://en.wikipedia.org/wiki/File:Kerr-Newman-Orbit-1.gif#Equations) metric
$$ \rm C = p\_{\theta}^{2} + \cos^{2}\theta \ \Bigg[ a^2 \ (m^2 - E^2) + \left(\frac{L\_z}{\sin\theta} \right)^{2} \Bigg] $$
with (in $[+---]$ signature)
$${{\rm E = -p\_t}=g\_{\rm tt} \ {\rm \dot{t}}+g\_{\rm t \phi} \ {\rm \dot{\phi}} + \rm q \ A\_{t}}$$
$${\rm L\_z = p\_{\phi}}=-g\_{\rm \phi \phi} \ {\rm \dot{\phi}}-g\_{\rm t \phi} \ {\rm \dot{t}} - \rm q \ A\_{\phi}$$
$${\rm p\_{\theta}}=-g\_{\rm \theta\theta} \ {\rm \dot{\theta} \ } { \color{#aaaaaa}{ - \ g\_{\rm t \theta} \ {\rm \dot{t}} - \rm q \ A\_{\theta}}}$$
(with the terms that are 0 grayed out) does not work in the [Kerr Newman (Anti) De Sitter](https://en.wikipedia.org/wiki/Kerr%E2%80%93Newman%E2%80%93de%E2%80%93Sitter_metric#Boyer%E2%80%93Lindquist_coordinates) metric. In some references on the subject I've seen it mentioned though, but I've never seen its explicit form in any of them. Does it even exist in this case?
I suspect if so, there must be some additional terms dependend on $\Lambda$ and $\theta$, since even on an orbit at constant $\rm r$ (but variable $\theta$) the form that works fine with $\Lambda=0$ gives a sine-like function with $\Lambda \neq 0$ (with an orbit like [this](https://commons.wikimedia.org/wiki/File:Orbit_in_the_Kerr_Newman_De_Sitter_Spacetime.png) where the other constants of motion $\rm E$ and $\rm L\_z$ work just fine and stay constant as they should).
Since Carter seems to have come up with his constant with a lucky guess while no one really knows why it even works, I'm not sure how to derive it by first principles, and before I try to land a lucky guess myself, maybe somebody has already figured it out or knows a reference where it is not only mentioned, but also given (for the KNdS/KNAdS metric).
It doesn't really matter since I use the second order equations of motion anyway, and the two constants of motion $\rm E$ and $\rm L\_z$ are sufficient to monitor the numerical stability, but for the sake of completeness I'm still interested if there is a version of the Carter constant that works with $\Lambda$.
| [
"\n[Here](https://arxiv.org/pdf/0707.0409.pdf) is a paper that gives the Conformal Killing-Yano tensor for any member of the Plebanski–Demianski family of solutions. This is the most general family of type D vacuum solutions to the Einstein-Maxwell equation with cosmological constant, which includes the Kerr-Newman... |
so I recently came across the term 'virtual mass' and when I looked up more about it, it just gave me some stuff about fluid mechanics that I dont understand properly. My understanding of virtual mass is that when a body is pushed through a fluid (let's say water) it moves slower than usual and behaves like hoe a big object with more mass would behave on land but then I realized that what I was thinking of is just upthrust/buoyancy so can somebody please explain virtual mass in simple (layman terms) words. (I just cant wrap my head around it)
| [
"\n*Virtual mass*, or *added mass*, is a concept of fluidodynamics related to the enhanced inertia acquired by a body in a fluid. A short description of the concept can be found on [this Wikipedia page](https://en.wikipedia.org/wiki/Added_mass).\n\n\nIt is an effect due to the coupling between a body in a fluid and... |
I hope that my question will be suitable for this forum: I would like to understand the difference between the so called consistent history approach to QM and several other interpretations. In [this](https://physics.stackexchange.com/questions/3862/is-the-consistent-histories-interpretation-of-qm-a-many-worlds-interpretation) discussion it is remarked that similarly to the many worlds interpretation, there is no wave function collapse in the consistent history approach: the collapse is only apparent and is caused by the interactions with the environment, in the so called *decoherence* effect. But there are big differences between CH and MWI: CH is fundamentally probabilistic while MWI is deterministic; there is one world in CH (but there are many *ways of describing it*) while MWI carries ontological baggage: all branches are no less real than our branch. So I would agree that these approches are quite different.
However, if insetad of MWI I switch to relational interpretation (by Carlo Rovelli) or to QBism, these differences disappear: both interpretations (relational and Qbism) are probabilistic and both assume unique world: in relational approach we speak about properties of the system *with respect to another system*, in QBism we speak about *properties with respect to agents*: in CH the role of the system/agent is, as far as I understood, played by a ,,set of questions which we would like to ask'' (the so called framework). I don't see where the difference lies beside slightly different philosophical flavour of way of putting it. So I would like to ask:
>
> In which sense is consistent histories interpretation different from relational interpretation and from QBism?
>
>
>
Also, I believe that QBism became much more popular nowadays and CH interpretation seems a little bit forgotten:
>
> Why was consistent histories interpretation abandoned by a physics community?
>
>
>
| [
"\nFrom a birds-eye view, all three of these are quite similar. (Consistent Histories = CH, Relational Quantum Mechanics = RQM, and QBism= QB.)\n\n\nWhen Rovelli [first published](https://link.springer.com/content/pdf/10.1007/BF02302261.pdf) his take on RQM (1996), he explicitly compared it to CH, and didn't see mu... |
It is given that $\frac{ds}{dt} = \frac{d\theta}{dt}$, i.e. the time derivative of s and $\theta$ are equal to each other.
Does it follow that for small values of $t$, $\Delta s ≈ \Delta \theta$? My thought process for this was that if I multiply $dt$ on both sides, I am left with just the value of $ds=d\theta$, which I can rewrite as $\Delta s ≈ \Delta \theta$. Would I be correct in thinking this?
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"\nThe answer depends on a necessary clarification: does the equation\n\n\n$$\\frac{ds}{dt} = \\frac{d \\theta}{dt}$$\n\n\nhold for **all** $t$, or just for a **particular** value of $t = t\\_{0}$?\n\n\nIf the equations holds for **all** $t$ then the equation can be integrated and we will have\n\n\n$$ s(t) = \\thet... |
I was reading Fermi's 1932 paper "[Quantum Theory of Radiation](http://fafnir.phyast.pitt.edu/py3765/FermiQED.pdf)". I was able to understand the paper until this particular derivation (*which was excluded*).
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> [](https://i.stack.imgur.com/b8TQ3.png)
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$u\_m,u\_n$ are the eigen-functions of the electron which was initially solved using the unperturbed hamiltonian.
How to derive the ***highlighted-equation***, using the left hand side?
I know that $\hat{p}=-i\hbar \nabla$ in three dimensions. But I never quite understood how to arrive at the energies $E\_m$ and $E\_n$.
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"\nThis is likely using the fact that $[\\hat{H},\\vec{r}] = -i\\frac{\\hbar}{m}\\vec{p}$, which you can verify by direct computation. (Here, I am using $\\vec{r}$ for the position vector, replacing $X$ in the paper.) By replacing the momentum operator with the commutator, you can expand out the commutator and act ... |
**This question already has answers here**:
[Is mechanical energy conserved in all Inertial frames? (Newtonian Mechanics)](/questions/575865/is-mechanical-energy-conserved-in-all-inertial-frames-newtonian-mechanics)
(3 answers)
Closed 2 days ago.
I am aware there are similar questions already asked, however, I find none of the answers satisfactory, they either do not contain any mathematics at all, or mathematics of a level I am not capable of comprehending.
My doubt is that, if the change in kinetic energy is dependent on frame, then how is the law of conservation of energy valid in all frames? One possible explanation I have thought of is that the change of potential energy is also frame dependent, but I cannot think of a suitable example to justify the same.
Please answer only using newtonian mechanics I do not know statistical or lagrangian mechanics.
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"\nIn classical mechanics, Galilean transformations (we'll assume a non-relativistic problem) relate the coordinates of events as measured in two different inertial reference frames that are related by a constant relative velocity $v$ along a straight line. The Galilean transformations for space and time coordinate... |
[Homework-like questions](https://physics.meta.stackexchange.com/q/714) and [check-my-work questions](https://physics.meta.stackexchange.com/q/6093) are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. Note that answers to homework questions with complete solutions may be deleted!
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Closed 2 days ago.
This post was edited and submitted for review yesterday and failed to reopen the post:
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> Original close reason(s) were not resolved
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[Improve this question](/posts/780018/edit)
The free particle propagator is a well known function,
for example, see Wikipedia.
However, I cannot find a source that explains how to derive
the free particle propagator.
Please explain how the free particle propagator is derived.
<https://en.wikipedia.org/wiki/Propagator>
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"\nA propagator is the amplitude for a particle to go from $x\\_a$\nat time $t\\_a$ to $x\\_b$ at time $t\\_b$.\n\n\nLet $K(b,a)$ be the propagator\n$$\nK(b,a)=\\langle x\\_b|\\exp[-iH(t\\_b-t\\_a)/\\hbar]|x\\_a\\rangle\n$$\n\n\nwhere $H$ is the free particle Hamiltonian\n$$\nH=\\frac{\\hat p^2}{2m}\n$$\n\n\nBy the... |
In Griffith's EM, section 7.2.1 includes following pictures to talk about Faraday's law , where by Faraday's law I will be referring to its differential form rather than integral form.
[](https://i.stack.imgur.com/iWRE6.jpg)
In the images (b) and (c) I can understand either the magnet moves or strength of it changes so that
$$\frac{\partial \textbf{B}}{\partial t} \neq \textbf{0} $$
From which a a potential is induced.
But in the picture (a) , the same term is always zero, and i can't say why does electric fields arise.
It is perhaps due to some relativistic effect about current , but I can't do give a technical answer, (if you could pls :) )
But, i can just say that bcs effect is observable in (b), by special relativity same thing must happen. (yet, why i should have started with picture (b) as my reference frame?)
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"\nThe case a) is different: here EMF in the circuit is not due to induced electric field $\\mathbf E\\_i$ (which is negligible here), but due to motion of part of the circuit in external static magnetic field, thus it is better called *motional EMF*. The motional force per unit charge is $\\mathbf v\\times\\mathbf... |
This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to answer the titular question on my own, I was recently led to think that scale transformations and addition by a total time derivative are the *only* transformations of the Lagrangian which give the same Equation of motion according to [this](https://physics.stackexchange.com/questions/131925/) answer by Qmechanic with kind help from respected users in the [hbar](https://chat.stackexchange.com/rooms/71/the-h-bar). I will ignore topological obstructions from now on in this question.
The reasoning was that if we have that
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> Two Lagrangians $L$ and $L'$ which satisfy the same EOM then we have that the EL equations are trivially satisfied for some $L-\lambda L'$. The answer "yes" to (II) in Qmechanic's answer then implies that $L-\lambda L'$ is a total time derivative.
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However I am confused upon reading the Reference 2 i.e. "J.V. Jose and E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998"; (referred to as JS in the following) mentioned in the answer. I have several closely related questions---
JS (section 2.2.2 pg. 67) says
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> For example, let $L\_1$ and $L\_2$ be two Lagrangian functions such that the equations of motion obtained from them are *exactly the same.* Then it can be shown that there exists a function $\Phi$ on $\mathbb{Q}$ such that $L\_1-L\_2 = d\Phi/dt$.
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(i) Why doesn't this assertion not violate Qmechanic's answer to (I) [here](https://chat.stackexchange.com/rooms/71/the-h-bar), namely why should two lagrangians having same EOM necessarily deviate by a total derivative of time, when Qmechanic explicitly mentioned scale transformation to be such a transformation?
Infact on the next page(pg. 68) JS also says
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> It does not mean, however, that Lagrangians that yield the same dynamics must necessarily differ by a total time derivative.
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exactly what Qmechanic asserts with example.
But then JS gives an example
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> $L\_a= (\dot{q\_1}\dot{q\_2} -q\_1q\_2)~~\text{and}~~L\_b = \frac{1}{2}\left[\dot{q\_1})^2 +(\dot{q\_2})^2 - (q\_1)^2- (q\_2)^2\right]$ yield the same dynamics, but they do not differ by a total time derivative.
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Also, this is (atleast manifestly) not equivalent to a scale transformation of the Lagrangian.
So we found a transformation which is not a scale transformation or adding a total derivative but gives the same dynamics contrary to what I was led to think from Qmechanic's answer by the reasoning mentioned in the first paragraph(actually first block-quote) of this question.
(ii) So where was the reasoning in the first para wrong /How is the assertion in the first para consistent with this particular example?
Surprisingly JS pg 68 also says
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> What we have just proven (or rather what is proven in Problem 4) means that if a Lagrangian is
> changed by adding the time derivative of a function, the equations of motion will not be changed.
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(iii) But didn't they just state and prove the converse on pg. 67?
* If "Yes" then
$~~~(a)$ As asked in part (A) of this question how is this consistent with part (I) of the previous answer by Qmechanic.
$~~~(b)$ Did JS make a mistake on pg. 68 then?
* If "No" i.e. what is stated on pg 68 is what is proved on pg 67 then
$~~~(a)$ Did JS make a mistake on pg. 67 then? i.e. How are the statement on pg 67 and the proof consistent with the statement on pg 68?
$~~~(b)$ Why did Qmechanic cite this as a reference to support his answer to (II) which is *almost*(\*) the converse of what is given in JS?
(\*) "*Almost*" because I don't see how the statement on pg. 67 uses the word "trivially" from Qmechanic's statement
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> If EL equations are trivially satisfied for all field configurations, is the Lagrangian density $\mathcal{L}$ necessarily a total divergence?
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Assuming, the answer to the above question is somehow "yes", what I understood from pg. 67 of JS is that they are simply taking two Lagrangians which satisfy the same EOM as the assumptions of the theorem that they move on to prove and what Qmechanic is taking as assumption in his answer to (II) is a single Lagrangian which satisfy the EL equations for any arbitrary choice of $q,\dot{q}$ and $t$ on which the Lagrangian depends. So how are these two assumptions are supposed to be the same? Or more simply put,
(iv) How is Qmechanic's statement in answer to (II) in the previous question equivalent to what is stated in JS?
(v) Finally, is the titular question answerable with or without using Qmechanic's previous answer? If so, How?
If someone is not interested in answering all these questions (labelled $(i)-(v)$ with ($iii$) containing multiple subquestions), an overall explanation/discussion in new words, of the answer by Qmechanic and what's written on JS and their comparison might also help.
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"\nRather than answering the question point-by-point, I am going to cut to the essence.\n\n\nSuppose that we are given $n$ independent variables $x=(x^i)=(x^1,\\dots,x^n)$, $m$ \"dependent variables\" $y=(y^\\sigma)=(y^1,\\dots,y^m)$ and also their \"formal derivatives\" eg. $y^\\sigma\\_i$, $y^\\sigma\\_{i\\_1...i... |
In "conventional" quantum mechanics, in the presence of electromagnetic fields, we use minimal substitution in the Hamiltonian in the simplest case. I.e.
$$H=\frac{\vec{p}^2}{2m}\rightarrow \frac{(\vec{p}+e\vec{A}/c)^2}{2m}$$
What if one wants to consider more complicated gauge fields $$\vec{A}=\vec{A}^\alpha t\_\alpha$$ where $t\_\alpha$ are the generators of the corresponding gauge group? Can we simply minimally substitute that as well?
If so, I guess the gauge transformation for the quantum states changes as well. If so, how?
I've tried searching for this in multiple resources but found nothing. I even tried Quantum Field theory books that treat quantum mechanics as a $(0+1)-$dimensional QFT, but found nothing.
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"\n\n> \n> But I don't know how the wavefunction would transform in QM when we introduce, say, an 𝑆𝑈(𝑁)gauge field. I guess maybe the question should have been \"how do quantum states transform in an 𝑆𝑈(𝑁)\n> gauge theory?\"\n> \n> \n> \n\n\nLook at your hamiltonian in electromagnetism,\n$$H=\\frac{\\vec{p}^2... |
[Homework-like questions](https://physics.meta.stackexchange.com/q/714) and [check-my-work questions](https://physics.meta.stackexchange.com/q/6093) are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. Note that answers to homework questions with complete solutions may be deleted!
---
Please help the asker edit the question so that it asks about the underlying physics concepts instead of specific computations. If there are any complete answers, please flag them for moderator attention.
Closed 3 days ago.
[Improve this question](/posts/779925/edit)
I dont understand the argument on page 38 eq. (C-6) of Cohen's quantum mechanics. Could someone break down for me what is $g(k)$? Is it the initial condition?
[](https://i.stack.imgur.com/AYOu7.png)
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"\nIf you set $t=0$ in C-6, you get\n$$\n\\psi(\\vec{r}) = \\frac{1}{(2\\pi)^{3/2}}\\int d^3 k\\ g(\\vec{k}) e^{i\\vec{k}\\cdot\\vec{r}}\n$$\nThen it is clear that $g(\\vec{k})$ is the Fourier transform of $\\psi(\\vec{r})$, which is the \"initial state\" or if you prefer \"initial condition\" for the wave function... |
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