do highly active users on SE write long-form notes or anything, and if so, where would you host them? personal website seems like the obvious place but something like centralised similar to arxiv or SE would make things easier to find. the ability to upload pdf files to a SE profile would be kinda cool...
@qwerty You seem to have some kind of idea that "highly active SE users" form a homogeneous group with broadly similar habits, but they're not. They might be currently professional academics, ex-academics or amateurs with a lot of time and so their habits will vary a lot. Obviously someone who is a professional academic physicist regularly writes "long-form notes" - either lecture notes for their teaching or research articles for publication -
but this has nothing to do with their SE activity, and I don't understand why someone who is not a professional academic might be caused by their SE activity to write such notes.
@ACuriousMind apologies if I caused offence. I did not mean to imply all active users have similar habits, but I did assume there would be an underlying broadly-defined goal of disseminating and discussing physics on some level. My point was that the Q and A format is only one short-form way of doing so.
I am aware that there are professionals, ex-professionals and amateurs. I tend to write things down when reading, digesting or thinking about them and I assumed that other people did the same.
@qwerty Well, for me personally, everything I know is already written down somewhere - that's how I learned it, after all. So I personally don't produce any texts except the answers I write. Over the years I have sometimes had the idea to write a textbook-like document in my personal style, but this would take a lot of time and effort I probably will never get around to.
It takes sooooo much effort. And now that I'm starting work on mine, it just keeps ballooning; to understand the unorthodox way im doing quantum, I would have to write the classical mechanics text first...
For elastic collisions of n particles, we know that momentum in the three orthogonal directions are independently conserved:$$ \frac{d}{dt}\sum\limits_i^n m_iv_{ij} =0,\quad j=1,2,3$$
From this, it follows there's also a corresponding scalar quantity conserved:$$\frac{d}{dt}\sum\limits_i^n m_i(v...
@ACuriousMind interesting! i never feel I have understood something until I have done it in my own words/ own way. tbh, I dont know I'm going to make it in physics professionally (on the postdoc application cycle without serious prospects imo). actually i felt users (like yourself) on this site and wiki who I feel do a lot of good for physics/ people in general in that way, made consider me that SE would be a serious alternative way to contribute if I leave academia.
@naturallyInconsistent it really does! I have run into similar problems...
@Bml you should not state it as "most voted". You should just refer to the name, or just use the link that you get from sharing that particular answer, so that then we will all be on the same page
And you should formulate a question at least to a point whereby someone may guess what it is you are confused about.
@qwerty It is! physics.SE is definitely also a good way to "stay in touch" - knowledge you don't use tends to fade away, and the continuous engagement with physics here keeps it fresh
The factor of 1/2 is required from Galilean invariance, so that the energy mixes up with the momentum without a factor. This was understood before relativity, but it is largely conventional before relativity, since you could make the energy mix up with the momentum using some coefficient. Once yo...
@naturallyInconsistent OK. The answerer says: "In Newtonian mechanics, the energy and momentum transform together after a Galilean boost. If you have a closed system with momenta $p_i$ that add up to zero (center of mass frame), the change in the kinetic energy after a boost, which shifts $p_i\rightarrow p_i - m_i v$ is $\sum_i {p_i^2\over 2m_i} \rightarrow \sum_i m_i~(v_i - v)^2 = \sum_i {p_i^2\over 2m_i} - \sum_i p_i \cdot v + \sum_i m_i {v^2\over 2}$.
@Bml He's starting from that to show it results in no coefficient for the $(\sum p)\cdot v$ term. He could as well have started with an arbitrary prefactor $k$ with $\sum_i k \frac{p_i^2}{m_i}$ and found $2k (\sum p)\cdot v$ for that term, then concluded that $k = 1/2$ for the term to have no additional prefactor. Like many of Ron's answers, I don't find this argument convincing at all on closer inspection because there's no explanation here why we'd want that term to have no prefactor.
In the end I find this entire question somewhat pointless: Obviously the 1/2 is only a convention, since any multiple of energy is still a conserved quantity, and the useful thing about energy is that it is a conserved quantity, not its exact numerical value.
I prefer Qmechanic's answer, which is short and to the point: If you choose prefactors for all the other quantities as usual, then this 1/2 is what drops out of the work energy theorem as the natural conserved energy-like quantity to consider.
The relativity argument is just a red herring: We might just as well multiply the relativistic energy expression conventionally by 2 to get no 1/2 at first order, this is not a different explanation than "it's a convention" at all.
@ACuriousMind I guess he says we'd want that term to have no prefactor for reasons of simplicity, but in the end it is a matter of convention. As he says:
@Bml The part of the answer that is tolerable, is the link to SR. Either you work with $$\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}=m_0c^2+\frac12m_0v^2+\cdots$$ and convert that to momentum form as per usual, or you work with $$\sqrt{(m_0c^2)^2+(pc)^2}=m_0c^2+\frac{p^2}{2m_0}+\cdots$$ to obtain the $\frac{p^2}{2m_0}$ term.
"It is simplest if you take the coefficient $1/2$, ultimately because this is the natural nonrelativistic limit of relativity [but although it is the natural limit, this does not exclude it being a convention]. You would have a factor of $2$ in the $p$ mixing term if you didn't take the coefficient $1/2$. It is actually a testiment to the insight of the 19th century physicists that they adopted the most natural convention before relativity was discovered.
@Bml For centuries, people worked with $2E$ everywhere. It is cumbersome and irritating to have to remember where the 2 goes for all the integrals that would have to deal with the difference, but otherwise, the exact same physics happened. So, yes, it is purely a convention.
@naturallyInconsistent In what sense "it is cumbersome and irritating to have to remember where the 2 goes for all the integrals that would have to deal with the difference"?
@Bml Where you had some $\text{PE}=\int\vec F\cdot\vec{\mathrm dr}$, you will then have to do $2\text{PE}=2\int\vec F\cdot\vec{\mathrm dr}$. It is cumbersome, but people will make do
@ACuriousMind Actually, no, I disagree with this. The rest mass v.s. energy conversion rate would be altered; the factor of half will still appear to be the natural comparison between the rest mass energy contribution v.s. the non-relativistic kinetic energy contribution, and so I think Ron is correct.
@naturallyInconsistent Sure, but the explicit 1/2 would vanish, to be counterbalanced by $E=2mc^2$, no? What you're saying is just a demonstration that "there has to be a 2 somewhere", but it is still convention whether you put it on a bunch of other stuff as a 2 or on kinetic energy as a 1/2. I would even agree the 1/2 is the better convention, but I don't think relativity completely removes this being a convention.
@Bml Because in the old times before classical physics was this neat assembly of well-trodden paths it is today, people simply noticed that a quantity proportional to $mv^2$ was useful because it was conserved. You don't need to put a 1/2 there until you have developed the rest of classical mechanics so far that you get stuff like the work-energy theorem that puts the 1/2 there as a consequence of all your other conventions.
for a time, people weren't even really sure what "kinetic energy" really was supposed to mean - you can read up on the Vis viva debate where (bizarrely, from a modern viewpoint) people debated whether $mv^2$ or $mv$ was "the" conserved quantity to consider.
you can also see the interesting historical remark there that the 1/2 was "standardized" around 1820, pretty long after both energy and momentum had generally been established as useful
@ACuriousMind No, I'm saying that it would have been $2m_0c^2+\frac12(2m_0)v^2+\cdots$ and thus the factor of half will still be natural. Note that once we have SR, the natural thing to do is to set $c=1$, and then the $2m_0$ will really suggest that either we redefine energy units, or mass units, to get rid of this factor of 2. Because one object should have one object's worth of rest mass energy that is in agreement with each other.
oh, you perhaps mean the typo; yes, there should be a 1/2 there
The kinetic energy after the boost is $\sum_i \frac{m_i}{2}(v_i - v)^2$. This is then equal to the final expression for the energy in the answer, there's just a 1/2 in that intermediate step missing
which is pretty confusing since that's the entire point of the answer, but I guess 32 upvoters didn't care :P
@Bml That $1/2$ is there so that we get $F = ma$, not $F = 2ma$, the relativity argument is not correct, one fixes the relativity coefficient so that it reproduces the usual non-relativistic case, somewhere it amounts to a convention on the way that $m$ appears, and $F = ma$ is the standard convention
If you send $m \to 2m$ then it goes away from the KE but arises in other places like $p = mv \to 2mv$ and $F = ma \to 2 ma$ and in the relativity formula etc
@ACuriousMind OK, suppose we don't know that factor $1/2$. The kinetic energy in the frame of reference of the centre of mass is $E_k = k \sum_i m_i (v_i - v)^2$. This is equal to: $E_k = k \sum_i m_i v_i^2 - 2k v \sum_i m_i v_i + k v^2 \sum_i m_i = k \sum_i \frac{p_i^2}{m_i} - 2k (\sum_i p_i) v + k M_{total} v^2$.
To make (conventionally) the energy mix up with the momentum using no coefficient, we have $2k = 1$, so $k = 1/2$.
From that: $E_k = 1/2 \sum_i \frac{p_i^2}{m_i} - (\sum_i p_i) v + 1/2 M_{total} v^2$, and since $(\sum_i p_i) v = 0$ by definition, we have $E_k = 1/2 \sum_i \frac{p_i^2}{m_i} + 1/2 M_{total} v^2$, where 1/2 \sum_i \frac{p_i^2}{m_i} is the kinetic energy with respect to the inertial frame of reference and 1/2 M_{total} v^2 is the kinetic energy of the centre of mass. Correct, or is there some flaw in the reasoning?
> Everett recounted his meeting with Bohr as "that was a hell... doomed from the beginning". Léon Rosenfeld, a close collaborator of Bohr, said "With regard to Everett neither I nor even Niels Bohr could have any patience with him, when he visited us in Copenhagen more than 12 years ago in order to sell the hopelessly wrong ideas he had been encouraged, most unwisely, by Wheeler to develop. He was undescribably[sic] stupid and could not understand the simplest things in quantum mechanics.
@ACuriousMind So why does kinetic energy with respect to a fixed inertial system (not the CoM frame of reference) have the same expression? Shouldn't they differ by a minus sign?
@ACuriousMind Earlier I calculated the kinetic energy of a system of point masses with respect to the CoM frame, and it turns out to be the sum of the kinetic energy of the centre of mass and the kinetic energy with respect to a fixed inertial system. But shouldn't there be a minus sign instead of a plus sign?
If I choose to calculate kinetic energy with respect to a fixed inertial frame of reference (not CoM frame), I get the same result as if I calculate kinetic energy with respect to the CoM frame. This seems like a contradiction to me.
> Gell-Mann and Hartle, along with a score of others, have been working to develop a more palatable interpretation of quantum mechanics that is free of the problems that plague all the interpretations we have considered so far.
Basically once Bohr and Heisenberg could no longer properly defend proper QM, the casuists seen their chance and started bringing in their own desires again
@ACuriousMind Perhaps I have realised the error. The answerer starts from the CoM frame, with respect to which the velocity is $v_i$, then performs a Galilean boost assuming the velocity of the centre of mass is $-v$ (minus sign). Thus, $(v_i - v)$ is the velocity of a material point with respect to the fixed inertial frame of reference, not with respect to the CoM frame.
Here it all fits: I was not calculating kinetic energy in the frame of reference of the centre of mass, but in a generic fixed inertial system.
Also, @ACuriousMind and @naturallyInconsistent, I cannot understand whether your arguments on special relativity are in disagreement or not. Could you clarify?
I don't think we're disagreeing, I'm just willing to entertain much more stupid choices (i.e. $E = 2mc^2$ and refusing to redefine my mass unit) as "conventions" ;)
@ACuriousMind But even when you take that, you can see from my argument that the natural factor of half will still appear, and so I think Ron's SR argument is the only correct way to argue why the factor of half is natural to physics.
@naturallyInconsistent That's what I meant by "there's a 2 somewhere" - yes, the ratio you're talking about is real and "physical". But that does not force us to put the 1/2 into the expression for the kinetic energy necessarily, it's just the most efficient way to do this. You can say "the kinetic energy has a factor of 1/2 compared to the rest mass energy" or you can say " the rest mass energy has a factor of 2 compared to the kinetic energy", it's the same thing
> In 1970, the Student Mobilization Committee published a set of secret minutes it had obtained on a 1967 Jason seminar on problems of counter-insurgency. The regular Jasonite participating was Dr. Murray Gell-Mann and the main thrust was to find ways of getting social scientists usefully involved in solving problems of interest to the military. Selected quotes:
“Gell-Mann: Can we find out what effect increasing police density or ear cutting, or other negatives have on villager attitudes?”
@Slereah lol... that's very little to go on but yes, something was certainly going on... but maybe Murray was actually being the heroic type and decided not to sell us off immediately to the secret race of aliens that the Pentagon aligned itself with... at least not until the end of the current millenium. He showed the aliens we still have potential by discovering quarks, so they decided to give us another shot for the next 1000 years. At least that's my theory but it's open to critique 🤣
is there a paper describing the basic mechanism of an atomic bomb? I am curious because it seems like quite a distinct example of a "micro- to extremely macroscopic" evolution of a system
@SillyGoose The simplest kind, pure fission bombs, are just bunches of subcritical radioactive material that you then bring together with some mechanism to achieve supercritical mass
i think i am interested in the theory of how a (sustained) nuclear chain reaction is modeled (presumably using the framework of textbook qunatum mechanics). not necessarily how to start one
@SillyGoose What exactly is there to model? When some uranium atom splits, it releases on average X neutrons. Each neutron then has some probability per other uranium atom it encounters to induce fission of that atom. When this leads to each fission producing on average more than 1 neutron that hits another atom and splits it, you get a chain reaction - supercriticality. There isn't really anything particularly quantum about this except that there are probabilites :P
i guess i am conceptually thinking that the time evolution of an atomic bomb (from pre-explosion to post-explosion) could be an interesting place to see a quantum-to-classical transition of dynamics
as I just said, you don't really need to understand much about quantum mechanics to understand an atomic bomb, you just need to know that one particular kind of uranium releases a lot of energy and more neutrons when you fire neutrons at it. Then you bring enough of that together to be dangerous and - boom
certainly there are some more detailed models of what exactly happens during fission, but they are entirely irrelevant for the workings of an atomic bomb
@RyderRude How can he be conscious, being part of the eternal quantum whole, there is only a single unitary evolving consciousness we are taking part in... $|\text{OM}\rangle$....
separately, how does an atomic clock work? to my current understanding, we might have (ideally) a two state system $\lvert 1 \rangle$ and $\lvert 2 \rangle$. then we can pump in energy to the system to induce transitions $\lvert 1 \rangle \mapsto \lvert 2 \rangle$ in a precise way (e.g. after 2 seconds of energy pumping my $\lvert 1 \rangle$ will transition to $\lvert 2 \rangle$). Then, do we let the atom naturally decay $\lvert 2 \rangle \mapsto \lvert 1 \rangle$ at some natural rate
I'm going to dump some notes of interest as I learn more and more about it here.
https://www.youtube.com/watch?v=eOti3kKWX-c Inside the HP 5061A Cesium Clock by CuriousMarc has some good information.
Launched in 1964, the HP 5061A is a very portable atomic clock similar to the ones that were take...
@Amit it says e.g. if u give a powerul ai an image and all the knowledge about how the human brain works, the ai can work out what the image looks like to a human subjective experience
@Slereah so ive seen before conceptually saying that atoms have stable half lives and that this is used. going off of this post, it seems the time electron takes to reverse its spin direction would be a very "set" time, like that process takes that amount of time with certainty. is this so?
only now I realized you're talking about a film. No I don't watch that kind of genre.. I thought you were asking me about my experiences of the spirit world, lol
@Amit if you want to learn actual philosophy of consciousness, most sources like Wikipedia would be better than trusting what Ryder Rude says. Mary's room is not an argument about "illusionism", it's an argument about physicalism. Eliminative materialism and illusionism are not synonyms, and while all eliminative materialists tend to be physicalists, physicalists do not need to be eliminative materialists.
@ACuriousMind Thanks. Honestly I'm just going along for the Ryde 😉 and I won't try to learn anything seriously from a chat conversation with anyone. Nonetheless your care regarding accurate info is duly appreciated and commendable
@RyderRude It's fine as I said, it's not really a subject I'm very much into apart from a casual interest
@RyderRude Then why are you using those discrete categories? This is exactly the same thing as with mathematical technical terms: If you do not want to talk at the level of accuracy these terms imply, don't use them.
@ACuriousMind i was just about to give the example of Mathematics as something where discrete categories are perfectly well defined and accuracy is good
That's what someone says who does not want to invest the actual effort required to understand the nuances of a topic, but wants to sounds knowledgeable about the topic anyway. The difference in this case matters enormously, since a physicalist may believe that every mental state has a direct physical counterpart: For instance, they may think that there is a specific neural configuration of the brain that corresponds to "imagining the color red".
An eliminative materialist would claim that this mental state is too vague of a concept and does not actually exist; that most of the "mental states" we commonly talk about exist only as turns of phrase and have no direct reflection in any physical substrate.
yes, i was using eliminative materialism in exactly the sense u wrote
Suppose Mary's room were about physicalism, them a physically handicapped Mary would merely get to know the neural configuration counterpart, but not the experience
she would get to know the experience if the experience were the same thing as the neural countepart,
and that is eliminative materialism
@ACuriousMind also, this is not physicalism. Allowing for mental and physical counterparts that are correlated is not enough to make it physicalism. It could be dualism or dual aspect monism
Well, the problem is that you've also incorrectly recounted the standard formulation of Mary's room, she is not physically incapable of perceiving color, the point is that she's trapped in a black-and-white room (hence the name, see Wiki).
A phosphene is the phenomenon of seeing light without light entering the eye. The word phosphene comes from the Greek words phos (light) and phainein (to show). Phosphenes that are induced by movement or sound may be associated with optic neuritis.
Phosphenes can be induced by mechanical, electrical, or magnetic stimulation of the retina or visual cortex, or by random firing of cells in the visual system. Phosphenes have also been reported by meditators (called nimitta), people who endure long periods without visual stimulation (the prisoner's cinema), or those who ingest psychedelic drugs....
@RyderRude Then the intellectually honest way to say this would be e.g. "This isn't exactly the original Mary's room, but I think the following setup captures the same essence with fewer issues:". But you didn't do that, you just told someone else that this was Mary's room. Which is simply factually wrong.
I don't know how you expect anyone to take you seriously when the factual accuracy of your statements seems to fluctuate wildly depending on what you currently find most convenient
@RyderRude I find it very revealing that you consider the literal truth of your statements to be an irrelevant detail, and I'm done with this conversation.
@ACuriousMind I'd vehemently disagree, but it is clear that you aren't paying anywhere near as much attention to that than would be warranted for the amount of disagreement I'm putting in. I am in no way arguing that "KE has the half, compared to rest mass energy" v.s. "rest mass energy has the 2, compared to KE". I am arguing that after SR is known, you can no longer argue that missing this factor of 2 or half is natural any more.
@naturallyInconsistent I understand that you think we disagree, but I don't understand what exactly the disagreement is. Maybe I've expressed myself poorly. I agree that the relativity argument is a good way to argue that calling $\frac{1}{2}mv^2$ instead of $mv^2$ "kinetic energy" is "natural".
I still maintain, as natural as this is, it's still a convention, and an alternative would be to e.g. call $mv^2$ kinetic energy and $2m$ the rest mass energy (and perhaps introduce this factor of 2 in other places, too). This would be consistent, just unnecessarily complicated.
In relativity, you have an action $S = - \alpha \int ds$, you fix $\alpha$ by the non-relativistic limit, which means invoking $T = \frac{1}{2} m v^2$. Alternatively it comes from generalizing $F = ma$ to a relativistic equation, but this also assumes $F = ma$ and $T = \frac{1}{2} mv^2$.
Really as far as thought experiments about qualia go, I prefer the Bridgman version, where he mentions the case of kittens who do not open their eyes for weeks
when we write $\mathbf{L}^2 = \hbar^2\left[N(N+1)+a_k^{\dagger}a_k^{\dagger}a_ja_j \right]$
the number operator defined as $N = a_ja_j^{\dagger}$ is the sum of the three number of operators $N = N_x+N_y+N_z$, which yields $N|n_x,n_y,n_z\rangle = (n_x+n_y+n_z)|n_x,n_y,n_z\rangle = n_{\text{tot}}|n_x,n_y,n_z\rangle$
ok yeah it must be I forgot about repeated indices lol :P