darklight03/StackMIAsub huggingface dataset:
I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have two multiparticle states - a multiparticle state being, in my mind, a complex probability amplitude for each possible configuration of particles, as a function of time - then, is a normalized linear combination of these two states still a valid multiparticle state? Keep in mind they are both functions of time that obey the equations of QFT, so the linear combination is also a function of time, and I'm asking if it still obeys the equations of QFT. I'm trying to think about it in terms of Feynman diagrams. In particular, I'm pretty sure you can linearly combine two multiparticle states at one time with no problem - you just get a superposition rather than a pure state. And since the amplitude for a final configuration is essentially the sum of all propagators from the initial state, well, this sounds linear enough to me. I think you would just sum up the independent contributions from all the pure states of which the initial state was composed. What else could it be? What throws me for a loop is that I've seen several posts here and elsewhere talking about the inherent non-linearity of QFT. But I think they might be talking about linearity with respect to combination of single particle states. I'm not worried about that, however, since I can see that multiple identical particles really form a single mathematical entity (a product rather than a sum, e.g. the Slater determinant for fermions), so linearity wouldn't have much meaning in this context anyway. Still, the whole thing appears rather murky so I'd really like to clear up this point. To put it another way, I know that there is interaction within the evolution of a pure multiparticle state, and this leads to entanglement, which, mathematically, is just the inability of the final state to be factored into a single tensor product of one-particle states. But, is it fair to say there is no interaction between the pure multiparticle components of a mixed multiparticle state? (at least, as long as we don't classify summation of complex amplitudes as "interaction")
Context: I ask this as a school teacher reaching past the boundaries of my expertise. A colleague was talking about the standard model with an advanced student, explaining how particles interact by exchanging gauge bosons, asking the student to imagine gauge bosons as little spheres. Of course they got to the issue that this mental model doesn't add up, since in classical mechanics such a process always leads to repelling forces, while the forces in electron-positron-scattering are attractive. He asked me on my opinion on how to rectify his explanation in such a way that a student could still create a mental image of the process. This is what I came up with, and while I'm sure that it can't do the actual physics full justice, I would like you to point out precisely in what ways this is inaccurate, so that I can improve it in such a way that an advanced student can still have a mental image of the process. The mental model: In classical physics, waves typically obey the superposition principle, that is, two waves don't interact, but can pass through each other without influencing each other. This is true for electromagnetic waves, but also all other waves the students might have encountered in class. It is not always true for water waves, however. For instance, two tall waves crashing into each other won't pass through each other unchanged since they lose energy due to turbulences. The correct description of their interaction requires an additional term besides just the sum of the two wave functions. We can imagine something similar to be true for wave functions in QFT: As a particle like an electron is described as a wave function, the interaction of, say, two electrons is similar to that of two water waves in the sense that they do actually interact, yielding more complicated wave effects. A main difference is that unlike classical fields, which don't interact with other fields, the electron field interacts with the electromagnetic field as well, and the nonlinear effects come from one electron wave packet interacting with the em. field, which then interacts with the other electron wave packet. And since photons are electromagnetic wave packets, we can think of the em. part of the interaction as being comprised of photons. So what do you think? How (in)accurate is this mental model of interactions mediated by gauge fields.
I am currently thinking through evaporation over lakes, specifically the Laurentian Great Lakes (a complex subject, I know). Particularly, I am trying to wrap my head around why evaporation peaks in the fall and winter. Based on what I have read, this fact is due to the vapor pressure gradient that exists between relatively warm water and dry air (dry in an absolute sense because the air is cold) and the high winds which continually replace that dry air over the water. I have also read that this evaporation from the Lakes has a cooling effect on the Lakes themselves, causing a temperature decrease in the Lakes. I have seen it insinuated that over-lake evaporation is synonymous with a latent heat-flux. Based on what I have read, latent heat is an exchange of energy to a substance without a change of temperature of the substance. For water transitioning from a liquid to a gas, the required amount of energy to cause this phase change is called the enthalpy of vaporization or the latent heat of vaporization. When the air is colder than the water, where is the energy coming from to supply the latent heat of vaporization that causes evaporation (assuming its a cloudy day)? If the vapor pressure gradient is the main driver of the rate of the evaporation, how can the evaporation still be said to be a latent heat flux, i.e., how is the transfer of particles based on a pressure differential called a transfer of heat, albeit a latent transfer of heat? Does a latent transfer of heat mean that the water particles which evaporate do not change temperature, though the liquid water which they leave behind decreases in temperature? Sorry, that's more like three questions! Hopefully they are not stupid ones :) it seems this is a complex topic, and my initial intuition that evaporation is always higher when the air is warmer than the water (thus transferring heat energy to the water, increasing the energy of the molecules, leading to increased evaporation rate since the water molecules are now higher energy) is simplistic and even wrong, as it depends on so many more factors than that. Our world is not a simple one, physically at least!
As is well-known, classical conserved systems have conserved quantities by virtue of continuous symmetries, which can be derived from Lagrangian mechanics. For example, two masses on a spring can swap momentum between the two, but translational invariance ensures that the total momentum is conserved. But what if we introduce damping, specifically viscous damping, into the equations? If we consider a fluid medium, our two-mass system surely loses momentum to the fluid. But suppose that the spring itself is viscoelastic, so that it generates equal and opposite forces on the two masses proportional to their relative velocity. Then, the system is translationally invariant in some sense and conserves momentum, but is not Lagrangian (at least not in a simple way). Is there a theory of how to calculate conserved quantities based on symmetries in damped systems, analogous to how it is done in undamped systems? I can do this for specific cases, but I'm not sure how to do it in general. A few attempts: Constructing a complicated Lagrangian consistent with damping Declaring that damping is a purely phenomenological effect coming from many microscopic degrees of freedom behaving conservatively, and so concluding that the system should have the same conserved quantities as if there were no damping (problem here is that some of the conserved quantities, like energy, must leak into the unobservable microscopic degrees of freedom). Considering the corresponding conservative system and showing that it has conserved quantities. Then arguing that this is true for each frequency even if the "spring constants" are complex and frequency-dependent, and so concluding that damping doesn't really change anything. Again, though, this seems to prove too much, since energy shouldn't really be conserved. By way of motivation, I'm working with a system that has an unusually large and complicated number of symmetries, and I'm trying to determine how worried I should be about damping killing the conservation in a real experimental system.
Imagine we have a hollow metallic toroid, with copper wire winded around it, which carries electric current. That implies we have magnetic field inside the hollow toroid. The toroid has vacuum inside. We have a set up of a high voltage supply and an electron gun that takes the free electrons from the metallic toroid and shoots them inside it. The velocity of electrons shot inside the toroid is low enough so the magnetic field will bend its trajectory in a complete circular path within the boundaries of the toroid. Now the electrons are flying in circular trajectory inside the toroid. But this situation can't be hold forever, as the electrons are centripetally accelerated the irradiate photons and thus loose kinetic energy. As they loose kinetic energy and velocity, at some point they will stop orbiting and will stay still. But unless they still exactly in the geometrical center of the section of the toroid (which is highly unlikely), they will be attracted to the boundary of the toroid due to Coulomb forces. And as they move towards the boundary, they will regain velocity and start orbiting again. Resuming, the electrons will loose energy due to their accelerated motion, then regain energy, and then loose it again. Apparently this cycle will repeat endlessly, meanwhile they will radiate photons as they loose kinetic energy. If my analysis is correct, how the energy conservation principle will be applied here? Radiating photons endlessly means giving endless energy. The second part of my question is as follows. Suppose the electrons doesn't radiate photons, due to some arbitrarily stated postulate (like the Bohr's explanation on why electrons doesn't fall on the nucleus of an atom). Apparently there is no obstacle to hold an infinite amount of electrons inside the toroid. The only limiting factor will be the amount of the voltage applied to fire new electrons inside the torus, as previously fired electrons will create repelling Coulomb force for the new incoming electrons. But there will not be such thing as "dielectric rupture" as in the case of an ordinary capacitor, so hypothetically an infinite amount potential difference can be set between the hollow toroid and its inside. Is this assumption correct?
I'm having trouble finding a good phrasing to describe a component of a system that is too important -- in a sense that it distracts a person from all the other components, that should have been, or were formerly, important aspects of the experience. The case where I'm trying use it is something like this: We tried adding a new super-queen to our variant chess game that could move anywhere -- but we found that it was ______, and that none of the other pieces were important to the gameplay as a result. Another example, possibly from academia: We recommend that teachers not offer students extra-credit work during a semester. We have found this to be _____, in that the option routinely distracts students from the main body of work and preparation for exams. The key thing I'm trying to highlight is that participants in the system become hyper-focused in their attention on this one component, to the detriment of the other parts. (E.g., while the chess example indicates an overpowered piece, that's not what I want to emphasize here; it could also be the result of a super-weakness that becomes the only thing you'd want to attack, say.) I could also imagine this being used in a work of art: Say, an element of a painting that is too bright or oddly placed. Or a breakout character in a TV show who takes over what was meant to be an ensemble story (i.e., they "may... overtake the other characters in popularity, including the protagonist"). Is there a good word or phrase to indicate this state of being too important, and attracting too much attention?
Scenario Consider an empty universe with just the Earth, Moon and the Sun. The Earth and the Sun will rotate about their center of mass, which is inside the Sun. The Earth and the Moon will orbit their center of mass, which is inside the Earth. The Moon and the Sun will orbit about their center of mass, which is inside the Sun. Assumptions Since the Sun has a center of mass with respect to both the Earth and the Moon, which are moving separately, the Moon pulls on the Sun too, just like the Earth does. This results in the Sun not having a closed loop orbit around their common center of mass. I assume the third body affects the two other bodies in this manner: The center of mass of the Earth-Sun system rotates in the same manner the Earth-Moon system rotates about their center of mass The center of mass of the Moon-Sun system rotates in the same manner the Earth-Moon system rotates about their center of mass These two effects describe the same motion when viewed from different frames (Earth-Sun and Moon-Sun systems) The above scenario is more pronounced if you consider Jupiter instead of the Moon, but I'm looking at the minute details. Question The question is whether all three bodies always revolve around their common center of mass, which also accounts for the motion of all bodies with respect to one another? If I'm wrong, how exactly does motion happen in this three body system? (Explain using simple words) Can this be generalized for an n-body problem? (Although probably not useful for computation but seems good for a mental image) I think this can be consolidated into a two-body problem by considering the Earth-Moon system as a single body at their center of mass, resulting in a single stationary point as the center of mass of the whole system. That is, I'm taking the center of masses of the Earth-Sun system and the Moon-Sun system, and taking their center of mass, and I expect it to be the center of mass of the Earth-Moon-Sun system. Or is it just that bodies only revolve around their common center of mass, and the center of mass of the three-body system has no say in describing the motion, aside from being a static point? References The motion of the Earth and the Moon around the Sun for a month (from a WIRED article) The Solar System's barycenter with respect to the Sun (from the Wikipedia article on Barycenter)
Initially, I was looking for how centripetal force is produced on the surface of the rotating earth for a mass kept at any latitude. I went through the following threads - Which force provides the centripetal acceleration that makes objects on earth's surface rotate about Earth's axis of rotation? Is the normal force equal to weight if we take the rotation of Earth into account? Question about the Normal Force exerted by Planet Earth in relation to centripetal force If the ground's normal force cancels gravity, how does a person keep rotating with the Earth? From there, I understood that the resultant of normal force (N) and gravity (mg) is the required centripetal force. But what is bothering me now is HOW? According to the answers, the normal force is slanted such that it is not exactly opposite to gravity. Thus, they don't cancel out, resulting in a horizontal centripetal force. I'm still confused and have the following questions : Why is normal force slanted in the first place ? (Is it because of the earth's bulge, friction or centrifugal force?) I think there's also a vertically-upwards component of the resultant, why is that? (the resultant of normal force and gravity) This is what I see, what is the reason behind this? Source of the image - https://en.wikipedia.org/wiki/Equatorial_bulge Edit - I've gone through this question Is the normal force equal to weight if we take the rotation of Earth into account? but this doesn't clear my doubt regarding the upward component of the resultant of gravity and the normal force. I posted this question because I wanted some more insight into that poleward force and its relation to the bulge of the earth, which isn't emphasised in that question. kindly reopen my question.
This Q&A did not answer my question. The voltage of a circuit is the difference in each Coulombs potential energy at the negative pole, compared to the positive pole. At the negative pole, there's a whole wire for the electrons to pass through under the influence of the Coulombic forces; a whole wire with atoms between which they can accelerate, before transmitting their kinetic energy into the atoms. The longer the wire, the more durations of acceleration (thus, the more kinetic energy is produced). But, the longer the wire, the more resistance there is, thus, the lower the amperage. A lower amperage means less acceleration between each collision (assuming constant wire diameter). So, as the wire gets longer, there's more durations of acceleration, but the acceleration is lower. The accepted answer to this Q claims that this is the explanation. But I doubt these two factors cancel each other out such as to leave the voltage unchanged with change in wire length. That is, despite this, I still think a longer wire would mean more kinetic energy is produced. More kinetic energy means there must have been more potential energy that was transmitted into it; thus, the voltage must have been higher. But it isn't, so what gives? There's the distance factor; the longer the wire, the further away from the charged poles of the battery the electrons get. However, this would then make the voltage dependent on how close you lay the wire to the poles, which is again contrary to the assertion that the voltage only depends on the battery.
I'm now in the process of writing a report on a lab work i did with an Abbe refractometer. In all sources i found the working principle of this refractometer is described as such: "Light shines into a illuminating prism who's side that contacts the sample is roughed so light scatters uniformly in all direction into the sample. The sample is held between the illuminating prism and the refraction prism (RP). Rays that hit the sample/RP interface at an angle large than the critical angle of the interface suffer total internal reflection (TIR) and thus do not penetrate into the refracting prism. This creates a zone in the refracting prism where there is a shadow. The angle of that shadow depends on the refractive index (RI) of the sample, and it can be measured by adjusting the light/shadow line". Additionally, some sources explicitly say that the RI of both prisms must be larger than the RI of the sample. Critically, i did not find any source that said otherwise. If this is the case, how is it possible for there to be TIR in the sample/RP interface if the RP's RI is larger than the sample's? Furthermore, i used https://phydemo.app/ray-optics/simulator/ to simulate the apparatus. I found that a shadow zone only formed when the condition: "prism's RI > sample's RI" was met. Of course in this situation there was no TIR. In fact, forcing the situation where there is TIR, absolutely no shadow zone was created. Can someone explain to me how this actually works then? Is there something i am getting wrong? I'm almost sure there is because even manufacturer pages say it works with TIR... But still, it is very confusing!
Whenever I Google to try to find an actual formal statement of the first incompleteness theorem (as opposed to all the oversimplified explanations that talk about "true but unprovable theorems" rather than theorems independent of the axioms), the definitions that don't just mention something like a system with "strong enough to do arithmetic on the natural numbers" mention a system with an "a sufficiently expressive procedure" for enumerating theorems, which is reminiscent of terminology used in explanations I've seen of Turing machines, and so I thought perhaps it meant that the incompleteness theorems apply specifically to Turing complete systems. However, when I posted about this on Quora I got responses saying that Turing completeness has nothing to do with it, whereas, on the askcomputerscience subreddit, I got responses saying that yes, the type of systems the incompleteness theorems apply to are mathematically equivalent to Turing complete models of computation. So which is correct? Is a system with a "a sufficiently expressive procedure for enumerating theorems" just a Turing complete complete formal language/model of computation? If not, what exactly does that terminology mean? To be clear, I understand that the first incompleteness theorem states only that a formal language with "a sufficiently expressive procedure for enumerating theorems" must be either inconsistent or incomplete (i.e. must either imply contradictions or contain theorems that are probably true in some models of the language and provably false in such models), so I don't need an explanation of what the theorem says, so much as clarification on what types of formal languages it actually applies to. I did see this answer to a related question, which seems to imply that the types of languages/systems the incompleteness theorem applies to are systems that are "recursively axiomatized" systems, but it's not entirely clear what that means. Based on other simplified explanations I've seen of the the theorem, I guessing it refers to something like "a system sufficiently powerful to talk about itself" which would intuitively seems to match up with the concept of using Godel numbering to construct a statement like, "This theorem is unprovable", but that still seems rather vague and nonrigorous. Surely, there has to be a specific and rigorous definition/description of the sorts of systems the theorem applies to though, otherwise how would it be useful?
I have been using '[...]' to indicate skips in the middle of sentences, '[...].' to indicate that a single sentence has been skipped (or that a middle of sentence skip '[...]' is at the end of a sentence and so is punctuated), and '[....].' to indicate that two or more sentences have been skipped (or to indicate that an end of sentence skip '[...]' bleeds into a '[...].' or '[....].'): [...]. Because of this qualitative simplicity of negation that has returned to the abstract opposition of nothing and ceasing-to-be to being, finitude is the most obstinate of the categories of the understanding; [...] finitude is negation fixed in itself and, as such, stands in stark contrast to its affirmative. [....]. The determination of finite things does not go past their end. [....]. But since it looks clunky, I was wondering if this is correct instead: [...] Because of this qualitative simplicity of negation that has returned to the abstract opposition of nothing and ceasing-to-be to being, finitude is the most obstinate of the categories of the understanding; [...] finitude is negation fixed in itself and, as such, stands in stark contrast to its affirmative. [....] The determination of finite things does not go past their end. [....] I have not been able to find anything on the topic of skipping entire sentences within block quotes that incorporate entire paragraphs. I know it would be preferable to avoid this situation in the first place by paraphrasing or excluding parts (some style guides even say that ellipses should never be used at the start or end of quotes), but I can't do that here for multiple reasons.