title: Uncertainty Principle Intuition

text: So, as your usual physics undergrad, I read Griffiths's derivation of the general uncertainty principle. I understood it but there was no physical intuition given behind it in the book. It was basically a bunch of manipulations and inequalities used to get just the right result. Other than "It's because of the statistical interpretation." is there a more intuitive reasoning behind uncertainty. Like commutators, behavior of particles, etc. Thank you for the replies and as always, much obliged.


title: Can a human size object move so fast that it ceases to be observable?

text: In many anime, comics, movies, etc, we see a lot of super human beings moving and fighting at such high speeds that a regular human being cannot see that they are fighting or moving pass by. In particular, in the first battle between Goku and Piccolo, those two are able to fight in a confined environment (when Piccolo first appears in the comics, stage fight, if I remembered correctly), and regular audiences are unable to _see_ them when they battle. Is it physically possible for a "human" (human size object) to move in a _confined_ environment (let's say, $20~\text{m} \times 20~\text{m} \times 10~\text{m}$ ) so fast that a normal person would be unable to _see_ it? If it is possible, how fast must this "person" be? Assume that this "human" cannot move faster than, or even close to the speed of light. _Some clarifications:_ First, please ignore the strength/physicality of the object, and consider it an object that can can move "freely" in this confined space without causing any "side effects" (such as heat, sounds, etc) Second, please take in note that the object is moving in a confined, rather small 3D space as mentioned in the question, and the observer would be always able to see the entire space. And last, a _blurred_ image would be considered as "able to see".


title: Can lightly-ionized atoms be accelerated to relativistic speeds with current technology?

text: I know there're lots of various particle accelerators, which can accelerate particles to TeV's of energy, but it seems they all work on electrons or nuclei or other elementary of tightly-bound particles. Is it possible, using current technology, to accelerate lightly-ionized atoms, like e.g. $\text{Ca}^+$ to speeds of the order of $\frac34c$? If yes, then is it done somewhere? If no, what are limiting factors? Apart from just possibility to accelerate, would such ions be left intact after acceleration? Or would the accelerating field strip off or considerably excite some of the remaining electrons?


title: Finding the metric tensor from the Einstein field equation?

text: I have have set my self a challenge to learn all the maths behind the Einstein field equation (EFE), and from reading it seems that the Metric tensor is the thing we are trying to find (from the 10 equations that forms the EFE). but since nearly all the terms in it are functions of this metric tensor, this seems very hard. What type of maths do we use to do this? (I have copied and pasted the EFE just for reference):![enter image description here](http://i.stack.imgur.com/imBQZ.jpg) **Edit:** Thanks for your comments. Just as I have no one else to ask, is this the equation that we try to solve: ![enter image description here](http://i.stack.imgur.com/Oaijw.jpg) with the $R_{μν}$ and $R$ terms expanded and where $g_{μν}$ and $T_{μν}$ are the μν th components from the related tensors? (a yes or no answer will be fine, thanks).


title: A problem with supersymmetry transformation invariance

text: Consider the following transformation of the integration measure $dX d\psi_1 d\psi_2$: where $\psi_1$ $\psi_2$,$\varepsilon^1$, and $\varepsilon^2$ are Grassman variables. $\delta_\varepsilon X= \varepsilon^1 \psi_1+ \varepsilon^2 \psi_2$ $\delta\psi_1= \varepsilon^2 \partial h$ and $\delta\psi_2= -\varepsilon^1 \partial h$. The problem asked to show that the above integration measure is invariant under this transformation. My approach to the first one was to just use the fact that $\int \psi_1 \dots \psi_n d \psi_1 \dots d\psi_n=1$, and relate it to the bosonic "standard" variables which return $\int dX= \int 1 dX=X$. However, for the other two, I know that since $\int \psi d \psi=1$, it should follow easily. But what does the $h$ denote in the differential? and is this argument headed in the right direction? Many thanks to all of you.


title: What is different between resolvent and green function

text: I bumped into a book, where Resolvent $R^{\pm}(E)$ is defined as $e^{\mp iHt/\hbar}=\pm\frac{i}{2\pi}\int_{-\infty}^{\infty}dER^{\pm}(E)e^{\mp iEt/\hbar}$ and $R^{\pm}(E)=\frac{1}{\pm i\hbar}\int_0^{\infty}dte^{\mp iHt/\hbar}e^{\pm iEt/\hbar}e^{-\eta t/\hbar}$. It is easy to show that $R^{\pm}(E)=\frac{1}{E-H\pm i\eta}$. Here H is the full Hamiltonian. So can anyone tell me the difference between it and Green Function?


title: helmholtz free energy of a polymer

text: You have a polymer chain of $N$ units, which is represented by $N$ independent springs in series. The springs are Hookean, with spring constant $L$, and the end to end vector is $\mathbf r$. So the energy of the spring is $$\frac{L\mathbf r\cdot\mathbf r}{2}.$$ Neglect kinetic energy contributions. There is one state per volume $W$ in $\mathbf r$-space. Calculate the Helmholtz free energy of the polymer (you need to integrate over all possible values of the three-component end to end spring vectors.) I have no idea how to set this up, what three integrals? How do they lead to the free energy?


title: What conductance is measured for the quantum spin Hall state when the Hall conductance vanishes?

text: It's probably just a definition, but what did König et al. actually measure when he confirmed the existence of surface states in CdTe/HgTe/CdTe quantum wells (see http://arxiv.org/abs/0710.0582)? According to most reviews (e.g. http://www.annualreviews.org/doi/pdf/10.1146/annurev- conmatphys-062910-140538): > "However, because the Hall conductance of the QSH state vanishes, it is > clear that the TKNN or Chern number discussed above, which corresponds to > the the value of the Hall conductance in units of e2/h, cannot provide a > useful classification of the QSH state. This issue has been addressed within > both the topological band theory (23) and the topological field theory (23). > Accordingly, the proper topological invariant is valued in the Z2 group > containing only two elements, 0 or 1, with 1 corresponding to the > topologically nontrivial QSH insulator and 0 corresponding to a > topologically trivial insulator with no robust gapless edge states." and > "Because spin-orbit coupling destroys spin conservation, there is no such > thing as a quantized SH conductance in the QSH effect. This is another way > to understand why the correct topological invariant for the QSH effect is Z2 > and not Z. Finally, the BHZ Hamiltonian predicts a single helical edge state > per edge." I don't understand this. So there is no charge conductance, but we measure charge conductance? What is the difference between spin and charge conductance? I thought König did measure a charge conductance which was exactly twice the Hall conductance (e^2/h) (For me that's quantized...). Does this mean there actually are only two states that lead to twice the Hall conductance? Also: Why is there only a single helical edge state per edge? Why must we have at least one and why can't we have, let's say, two states per edge?


title: Does a charged or rotating black hole change the genus of spacetime?

text: For a Reissner–Nordström or Kerr black hole there is an analytic continuation through the event horizon and back out. Assuming this is physically meaningful (various site members hereabouts think not!) does this change the genus of spacetime? In the 2D rubber sheet analogy a loop drawn around the event horizon wouldn't be contractible because it would go through the event horizon and back out and still have the black hole in it's centre. I don't know if the same argument applies to 4D spacetime.


title: Ziolkowsky Rocket Equation

text: I've looked at multiple sources on the internet and just can't wrap my mind around it. If I reference the wikipedia page on it; http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation I am getting stuck on certain places. $V_e=V-v_e $ Why does this occur? What integration limits do we use to derive this and why? $\Delta V\ = v_e \ln \frac {m_0} {m_1} $ (I am only concerned with this equation up to this part) If someone could explain it from the beginning in an easier fashion that would be great but solving these doubts for me would also help.


title: How does the proof of operator commutativity work with non-continuous operators?

text: In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\\{\phi_n\\}$, then they commute is given as follows : > For any $\phi_n$, $$AB\ \phi_n = a_n\ b_n\ \phi_n = BA\ \phi_n,$$ and so $AB > - BA = 0$. > > Then, for any vector $\psi = \sum\limits_{n=1}^\infty c_n\ \phi_n$, > > $$(AB-BA)\ \psi = (AB-BA)\sum\limits_{n=1}^\infty c_n\ \phi_n= > \sum\limits_{n=1}^\infty \ (AB-BA)c_n \phi_n = 0$$ But how is the third step on the to last line valid, where the operator $C = (AB-BA)$ is moved across the summation sign? $C (\lim f_n) = \lim (C f_n)$ is only valid if $C$ is continuous. But here, while $C$ is linear and self- adjoint, it may not be continuous!


title: How does one interpret the 'per unit area' in numbers like tensile strength?

text: How does one interpret the numbers when reading data about tensile strength, yield strength, and the likes? Say for example reinforcing bars. Grade 40 Rebars are rated at 70,000 PSI for its ultimate tensile strength and 40,000 PSI for its yield strength. I know a material's tensile strength is the point in the stress-strain curve at which the material will ultimately fail while the yield strength is the point on the curve at which the material will permanently deform (correct me if I'm wrong). What I don't understand though is where this stress (pressure?) is supposed to be applied on the object for it to break. What does the "per square inch" (or whatever unit area) point to, the cross section? How should I interpret this data if I plan on hanging some load directly on the center of a rebar whose ends are securely mounted by the ends? What if the load is evenly distributed throughout the bar? What if I use a pipe whose center is hollow? I need to understand this in a practical sense and not just numerically since I'm planning on building some equipment where structural integrity is a must. I'm pretty sure interpreting this should be fairly simple, but I can't seem to get a grasp on the 'real' and 'practical' side of this, regardless on how much I use Google and Wiki for this.


Why is the decay of a neutral rho meson into two neutral pions forbidden? (Other modes of decay are possible though.) Is it something with conservation of isospin symmetry or something else? Please explain in a bit more detail.


title: Could compression of electromagnetic waves cause red shift

text: One argument against "tired" light as a possible cause of red shift is that supernovae judged to be closer to us, based on luminescence, appear to be moving away from us faster than distant supernovae judging by red shift. According to the inverse square law, the density of emitted electromagnetic radiation is greater nearer the source. Is it possible that electromagnetic waves close to the source may self compress laterally, thus reducing amplitude and extending their wavelength, then, when further from the source resume their respective amplitudes in proportion to their spectral energies? If so, would not an observer of a relatively close supernova, being positioned in more dense electromagnetic radiation, see more "compression" red shift than for a more distant supernova, thus creating an illusion of faster movement away from us by the former?


title: Why doesn't Length of cycloid depend on speed of rolling?

text: We know that a point on the rim of a rolling body traces out a cycloid during one turn. Why is the length of the cycloid a constant = 8r. Shouldn't length of cycloid depend on speed of rolling body. For example, a faster body should trace out a cycloid which has a longer length than a slower body. What is the flaw in my reasoning?


title: Is it possible for a material to have a polarization field greater than the applied field?

text: In the case of a dielectric (LIH at least, since that is all I've studied), the polarization field is always less than the applied field. In the case of a conductor, the polarization field is equal to the applied field, which is why there is no electric field inside conductors. Is there any sort of material which has a polarization field greater than the field applied to it? My initial thought is no, since you would be creating a more powerful electric field than you started out with, and that would be a way of creating energy, right?


title: How is the operation of a Goldleaf Electroscope explained in terms of virtual particles?

text: If an electroscope is charged negatively the electrons on the leaves will repell each other and stand apart. It is clear than there is a steady force between the leaves that counters gravity. How is this force explained in terms of string theory or quantum field theory and invoking the concept of virtual particle exhange between electrons lodged on the leaves?


title: p-adic quantum mechanic

text: i got a degree on physics so my question is ?as a physicist could i learn P-adic analysis or p-adci quantum mechanics ??   * is there any good book on the subject ? as an introductory level   * How the p-adic schoridignuer equation is defined   * waht are the p-adci eigenfunctions and eigenvalues ??   * how can i define a p-adic derivative ¿are there tables for p-adic derivatives?   * how i must define a p-adic integral ??? are there tables for it   * what is the p-adic Melli transform and why is equal to the euler factor $ (1-p^{-s})^{-1}$   * how the Schrodinguer equation is SOLVED for the p-adic field? -what does it mean 'Nonarchimedean field'   * how is the quantization carried out in the p-adic field   * what are the adeles ? have some meaning in physics ??   * how is the p-adic classical mechanics made ?? .. Newtonian Lagrangian and Hamiltonian mechanics thanks in advance


title: Entropy of radiation emitted into space

text: In several papers I see something equivalent to the following expression for the entropy of radiation given by an astronomical object such as the Sun (assuming the object can be approximated as a black body): $$ s = \frac{4}{3}\frac{u}{T}, $$ where $u$ is the total flux of radiation energy through a spherical shell surrounding the object, $s$ is the entropy flux through the same imaginary surface, and $T$ is the black-body temperature of the object (and hence also of the radiation). It is also generally stated that no entropy is produced during the process of emitting radiation. In a much smaller number of papers I see a formula that corresponds to just $s=u/T$, which is what I would expect. So the short version of my question is, which of these is correct for an astronomical body? But please read the rest, so you can understand why I'm so confused about it. I understand the derivation of the 4/3 formula by considering a photon gas enclosed in a piston (see below), but in the context of a continuously emitting body like the Sun it doesn't seem to make sense. The issue is that if, over a given time period, the body loses an amount of heat $Q$ then this must balance the increase in energy of the radiation field $U$, i.e. $U=Q$. The body loses entropy at a rate $Q/T$, and if radiation is really a reversible process then this should equal the gain in entropy of the radiation, which should therefore be $U/T$. But according to the above formula it's actually $4U/3T$, meaning that the total entropy increases by $Q/3T$. The 4/3 formula above was derived by Planck (in his book "the Theory of Heat Radiation", of which I've read the relevant chapter), who considered a photon gas in a sealed cylinder of finite volume. At one end of the cylinder is a black body and at the other is a piston. The radiation comes into equilibrium with the black body and exerts a pressure on the piston. If one reversibly (i.e. slowly) allows the piston to move then this causes some heat to be lost from the black body. In this case it turns out that $U=3Q/4$, with the discrepancy being due to the fact that the radiation field loses energy when it does work on the piston. The entropy balance then requires that the entropy of the photon gas increases by $4U/3T$, as above. The point is, I can't see how the radiation being emitted by the Sun can be seen as doing work on anything. At first I thought it might be doing work on the outgoing radiation field. So let's draw an imaginary shell around the Sun again, but this time let the shell be expanding at the speed of light. Perhaps the radiation inside the shell is doing work on the radiation outside it, and that's what's "pushing" it away from the Sun? But it seems to me that for anything inside the shell to have an effect on anything outside it, some kind of influence would have to travel faster than light, so I don't think that can be right. In any case it's well known that for a normal gas (made of matter), expanding against a piston is quite different from just expanding into a vacuum. In the former case the temperature and internal energy decrease, because the molecules lose energy in pushing the piston, whereas in the latter case they both remain constant. I haven't found any source that addresses the question of why this would be different for a photon gas. So it seems like the emission of radiation from a body like the Sun into space is quite different from the emission of radiation into a sealed piston, and I'm puzzled as to how the same formula can apply. Below I've listed some possible resolutions to this which seem plausible to me. I haven't been able to find any sources that directly address this issue, or state any of these positions.   1. The emission of radiation into space is an irreversible process after all. $U=Q$, and the total entropy increases by $Q/3T$. (But then, what happens when radiation is absorbed by a body at a similar temperature? Surely the entropy doesn't decrease.)   2. There is some weird sense in which the outgoing radiation can be thought of as doing work on something, so $U\ne Q$. (If so, why does nobody ever explain this subtle and important point?)   3. $Q=U$, but the entropy of radiation emitted into space is actually different from the entropy of a photon gas in a sealed cylinder, such that its entropy is given by $U/T$, not $4U/3T$. (This one actually seems the most reasonable to me. The radiation in the closed cylinder has rays travelling in all directions, whereas the radiation coming off the Sun only has rays travelling in directions away from its surface, so it seems reasonable that they would have different entropies. This would mean that the 4/3 formula doesn't apply to the radiation emitted by astronomical bodies after all - but if that's the case then it's an extremely widespread mistake.) Any insights into which, if any, of these is correct would be greatly appreciated - as would any references to sources that directly address the relationship between Planck's photon-gas piston and the emission of radiation into empty space.


title: Do neutrinos 'become' other types of particles between oscillations?

text: In light of the recent kurfuffle of FTL neutrinos I wonder, do neutrinos 'become' other particles between their oscillations between the different flavors of neutrino? I'm slightly familiar with Feynman diagrams and wonder if there's some state between, say, electron and muon neutrino that may account for a neutrino's apparent FTL properties over a short distance, but would be averaged away over longer distances. In short is the transition from electron -> muon -> tau neutrino more like electron neutrino -> (tachyon) -> muon neutrino -> (slower particle) -> tau neutrino?


title: How can I determine density of a gas only given temperature?

text: I have a homework problem where exhaust is traveling through an exhaust system (assumed to be air for simplicity) from and engine and then released into the atmosphere. The exhaust is at a temperature of 250F. I need to find the pressure at the beginning of the exhaust system. I've been able to set this up and created an equation to solve for the pressure. I assumed that the pressure of the atmosphere where it it exits is 0 atm. But the problem is that the pressure depends on the density of the exhaust (air). I was only given temperature of the exhaust in the problem and I need a second property to determine density. Is there another way I can determine density? I am also given the volumetric flow rate and the area of the piping system


title: What is the answer to Feynman's Disc Paradox?

text: _[This question is Certified Higgs Free!]_ Richard Feynman in _Lectures on Physics_ Vol. II Sec. 17-4, "A paradox," describes a problem in electromagnetic induction that did not originate with him, but which has nonetheless become known as "Feynman's Disc Paradox." It works like this: A disc (Feynman's spelling) that is free to rotate around its axis has set of bead-like static charges near its perimeter. The disc in it also has a strong magnetic field whose north-south axis is parallel to the rotation axis of the disc. The disc with its embedded static charges and magnetic field is initially at rest. However, the magnetic field is was generated by a small superconducting current. The disc is permitted to warm up until the magnetic field collapses. The paradox is this: Conservation of angular momentum says that after the field collapses, the disc must of course remain motionless. However, you could also argue that since the collapsing magnetic field will create a strong circular electric field that is tangential to the perimeter of the disc, the static charges will be pushed by that field and the disc will necessarily begin to rotate. Needless to say, you can't have it both ways! Feynman, bless his heart, seemed to have an extraordinarily optimistic view of the ability of others to decipher some of his more cryptic physics puzzles. As a result, I was one of many people who years ago discovered to my chagrin that _he never bothered to answer his own question,_ at least not in any source I've ever seen. In the decades since then, that lack of resolution has produced a surprisingly large number of published attempts to solve the Feynman Disc Paradox. Many of these are summarized in a paper that was written and updated a decade ago by John Belcher (MIT) and Kirk T. McDonald (Princeton) (Warning: I can see the paper, but it may have access restrictions for others.) My problem is this: I more-or-less accidentally came up with what seems to be a pretty good resolution of the paradox, and it ain't the one described in any of the papers I've seen on it. But I can't easily back off, because the solution is a bit too straightforward once you look at it the right way. I think! I also think that Feynman's solution was very likely to have been relatively _simple_ , and not some kind of tremendously detailed exercise in relativistic corrections. He was after all trying to teach freshmen, and he honestly seemed to think they would all figure it out with a bit of thought! So, help me out here folks: Does anyone know for sure what _Feynman's_ solution to this little puppy was? Along those lines, is Laurie M Brown from Northwestern by any chance linked into this group? I can't imagine anyone who knows more about Feynman's published works! I will of course explain why I think there's a simple solution, but only after seeing if there's something simple (but apparently hard to find) already out there. **Addendum: The Answer!** I am always delighted when a question can be answered so specifically and exactly! @JohnMcVirgo uncovered the answer, right there in Volume II of the Feynman Lectures... only 10, count 'em 10, chapters later, in the very last paragraph of FLoP II 27, in Section 27-6 ("Field Momentum"), p 27-11: > Do you remember the paradox we described in Section 17-4 about a solenoid > and some charges mounted on a disc? It seemed when the current turned off, > the whole disk should start to turn. The puzzle was: Where did the angular > momentum come from? The answer is that if you have a magnetic field and some > charges, there will be some angular momentum in the field. It must have been > put there when the field was built up. When the field is turned off, the > angular momentum is given back. So the disc in the paradox _would_ start > rotating. This mystic circulating flow of energy, which at first seemed so > ridiculous, is absolutely necessary. There is really a momentum flow. It is > needed to maintain the conservation of angular momentum in the whole world. Feynman hints at the above answer in earlier chapters, but never comes right out with a direct reference back to his original question. John McVirgo, again, thanks. I'll review FLoP II 27 in detail before deciding whether to post that "other viewpoint" I mentioned. If Feynman already covers it, I'll add another addendum on why I think it's important. If the viewpoint is not clear, I'll need to do some simple graphics to explain how it may add some clarity to how the angular momentum conservation part works. **Addendum 2012-07-08: Not The Answer!** In the comments, @JohnMcVirgo has very graciously noted that I read more into his answer than he had intended, and for that reason he did not feel he should receive the answer mark. By finding that bit of text at the very end of the chapter John mentioned, I may in fact have answered my own question, at least in the literal sense of "what did Feynman say about it?" But John also points out his own surprise on how Feynman answered, which is different from points made by both him and @RonMaimon. So _for now_ I'm leaving this question open. I will assign an answer eventually, but only after I've read up on FLoP II 27 to the point where I feel I know it inside out. **Addendum 2012-07-08: New Answer!** Well that was a short several weeks! @RonMaimon's additions to his initial answer, combined with his latest comment clarifying the difference between field momentum and "mechanical" momentum, demonstrate a deep understanding of the issues. Since @JohnMcVirgo already suggested the updated Ron Maimon text as the answer, I agree and have so designated it. I am still deeply grateful to John for pointing me to FLoP II 27, since without that clue I never would have found Feynman's answer in his own words. I will at some point bring up my "other view" of Poynting problems as a new question. I now have two of those pending, since I am also still planning an updated Dual Cloud Chamber problem at some point.


title: Does relativity apply to something experiencing Centripetal forces?

text: If I tie a rope to a clock and spin it over my head, would it lose synchronization with a clock sitting at my feet?


title: Does field energy create rest mass in QFT theories?

text: The central importance of Higgs boson would be that the Higgs mechanism gives rest mass to fundamental particles. It seems like a very natural argument that fundamental particles need to be given rest mass by a field interaction because as something fundamental (or a perturbation in a field as its often described), it can't have any properties beyond what a QFT can explain through interactions with a field. What I don't understand is the role of matter-energy in this story. If the interaction of a particle with a field gave that particle rest mass, it seems logical that the field in the vicinity of the particle is in an excited state, which, integrated over the area of influence, is exactly equal to the rest mass of the particle. Why wouldn't this happen with interactions with other fields? Why does the (baseline) interaction of fundamental particles with the Higgs field create mass while interactions with the electric field, for instance, won't? In fact, it seems like we have lots of fields that could account for particle mass. Are those all just zero net-energy fields? What makes the Higgs capable of this while the others are not? On this point, if the electric field is mediated by the photon force carrier, then how can the electric field from a charged particle have anything less than infinite energy? I believe it has zero energy, but I lack the tools to argue this. Are the photons mediating an electrostatic interaction countable? Or do they exist in a less real sense? The magnetic field, on the other hand, obviously has energy as evidenced by the existence of inductors. The gravitational field is the strangest of all since gravitational flux is proportional to mass itself. Rest mass is proportional to interaction with the Higgs field, and mass interacts proportionally with the gravitational field, but yet the Higgs field is not the gravitational field. Can't mass come from any of the fields? Surely, if negative charge orbits closely to a positive charge, the system's mass is less due to a phenomenon associated with the electromagnetic field. Is this all the correct perspective, or close?


title: Jarlskog Invariant and its mathematical origin

text: CP violation is present in the weak interactions if   1. There are no degeneracies in the up-quark/down-quark matrices   2. The Jarlskog invariant $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$ is nonvanishing Furthermore, all CP violating effects are proportional to $J$. ~~I am getting stuck on showing how _all_ CP violating effects are proportional to $J$. Also, is the Jarlskog invariant a well-known mathematical property of a unitary matrix? What is it quantifying? I'd like to know this to the extent I can generalize this to larger CKM matrices.~~ **Edit:** I did a bad job writing my question. I rewrite it here: Question:   1. How do I constructively derive $J=Im(V_{us} V_{cb} V_{ub}^* V_{cs}^*)$, and how do I generalize this to arbitrary $n\times n$ unitary matrices ?   2. The Jarlskog invariant is invariant under a change of basis. What is the elegant way of showing this?


title: Virasoro Operators commutation relations

text: For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum state i.e. $\boxed{\left<0,0|\left[L_n,L_{-n}\right]|0,0\right>=\frac{D}{2}\sum_{m=1}^{n-1}m(n-m)\left<0,0|0,0\right>=\frac{D}{12}n(n^2-1)\left<0,0|0,0\right>}$. The second expression I can't seem to show explicitly although I know we can can write $\left<0,0|\left[L_n,L_{-n}\right]|0,0\right>=\left<0,0|L_n,L_{-n}|0,0\right>=\frac{1}{4}\sum_{m=1}^{n-1}\sum_{p=1}^{n-1}\left<0,0|\alpha_m\alpha_{n-m}\alpha_{n-p}^{\dagger}\alpha_p^{\dagger}|0,0\right>$ and then it's surely something to do with how these $\alpha$'s act on the vacuum state. But why the $m(n-m)$ and where's the $\frac{D}{2}$ come from? Finally, the last term in the boxed calculation is obviously found from the commutation relation of $\left[L_n,L_{m}\right]$, but therefore $\left<0,0|L_0|0,0\right>=0$. Why is this?


title: How long was a day at the creation of earth?

text: Since the earth is slowing its rotation, and as far as I know, each day is 1 second longer every about 1.5 years, how long was an earth day near the formation of earth (4.5 billion years ago)? I wouldn't assume to just do 4.5b/1.5 and subtract, because you would think the rate of change is changing itself, as seen here from wikimedia. It is a graphical representation of data from INTERNATIONAL EARTH ROTATION AND REFERENCE SYSTEMS SERVICE. They decide when its time for a leap second (the last one being on Jun 30, 2012) The data can be found here.


title: What instrument will measure voids in a brick wall non-destructively?

text: Let's take what's ostensibly a solid brick wall. It looks solid from the front and the back. It's typically at least two widths of brick thick, plus some. What techniques could we use to non-destructively establish the location and dimensions of any voids (larger than, say, $10 cm^3$) within the brick wall? So, the subject of scanning is:   * brick & mortar   * 9 - 50cm thick   * with a surface area of several square metres Options could include:   * instrumented hammer & sound measurements   * x-rays   * ultrasound   * ground-penetrating radar   * other The objective is to find these things out about the voids larger than $10 cm^3:   * dimensions   * locations Indications as to the resolution (accuracy / precision / confidence intervals) of the measurements, appropriate frequencies, and any constraints re penetrating depth would be very welcome


title: Gravitational waves as dark energy?

text: Is the energy carried by gravitational radiation a viable candidate for $\Lambda$ / dark energy?


title: Seeking a specific quantum spin system of interacting spin 1/2 particles

text: Is there a system of interacting quantum spin 1/2 particles (of any topology) whose the states where all spins are up or down are eigenstates of its Hamiltonian and yet does not conserve the total spin polarization (in the z-direction) ?


title: Hollow Wheels Down a Hill

text: Lets say I have a hoop of mass $M$ and radius $R$. It is rolling down a hill without slipping, so I don't need to worry about friction doing work on it. Lets say the angular speed is $\omega = \frac{v_{CM}}{R}$. It's center of mass of the hoop and the tangential speed is $v$. I want to know how to go about solving for the wheels final speed at ground level with an initial speed $v_i$ and initial heigh $h$. I try to go an choose a system: the hoop and earth. Then I have $$\Delta E_{sys} = W + Q + T_{light}$$ With my defined system, there is no external work, (also we assume no heat), and it doesn't given off light. so $\Delta E_{sys} = 0$, and $\Delta E_{sys} = \Delta K_{transitional} + \Delta K_{rotational} + \Delta U + \Delta E_{internal}$ Crossing out change in internal leaves us with $$\Delta K_{transitional} + \Delta K_{rotational} + \Delta U = 0$$ $$ K_{transitional_f} - K_{transitional_i} + K_{rotational_f} - K_{rotational_i} + U_f - U_i = 0$$ I am having trouble trying to figure out whether we have $.5mv^2$ for transitional kinetic energy and $.5m(\omega R)^2$ for rotational, but is not $\omega R = v_{CM}$ so basically rotational and transitional energy have the same magnitude for final and initial?


title: What is a dynamical variable

text: I see the phrase 'dynamical variable' in several of my texts and they don't really define it. A google search doesn't yield much either. I have gathered that a dynamical variable can be represented in terms of position and momentum, and examples are velocity and kinetic energy, but what is the definition?


title: Signal-to-noise ratio of the difference between two signals

text: Something tells me this must be a fairly simple question, but I have somehow been unable to find an answer to it. In short: I need to calculate the _difference_ between two signals, A and B, each one of them with its own signal-to-noise ratio. **What is the SNR of the resulting value**? * * * As for the context: we are doing differential photometry, generating the light curve of some celestial bodies by comparing their instrumental magnitudes (a logarithmic measure of its brightness) to those of others. These magnitudes are derived from the flux of each star, which is also used in order to estimate the signal-to-noise ratio, given by the following formula [ _Astronomical Photometry, A. Henden, 1982_ ]: $$ SNR = \frac{(star\;counts - sky\;counts)}{\sqrt{star\;counts}} $$ Note that, in case the background were neligible, the SNR could be obtained as $star\;counts / {\sqrt{star\;counts}}$, as for photon arrivals the statistical noise fluctuation is represented by the Poisson distribution, according to Henden. Also, _counts_ are a measurement of the flux, as we are using a CCD camera in order to conduct photometry. * * * ## **SNR to error (in magnitudes)** A S/N of 100 means that the noise causes the counts to fluctuate about the mean by an amount equal to one hundredth of the mean value. To compute this error in magnitudes, we compare the mean number of counts, $c$, to the maximum or minimum values induced by noise, that is $$\Delta m = -2.5 \log{( \frac{c\pm\frac{c}{100}}{c})}$$ $$\Delta m = -2.5 \log{( 1\pm\frac{1}{100}})$$ $$= \pm 0.01 ~magnitude$$ In other words, a S/N of 100 implies an observational error of 0.01 magnitude. [From Henden's _Astronomical Photometry_ , pages 77-78]


title: At the molecular level, how is the pressure at the bottom of a lake higher than at the top?

text: Surely the temperature of the molecules is the same throughout the water. Using $p = \rho g h$ seems to assume a constant density as well. But then how is it that the force per unit area on an object placed at the bottom of the lake will be higher than that on an object near the surface? My first thought is that the incompressible assumption is the approximation at fault, but it doesn't seem that a minute increase in the density of molecules at the bottom of the lake could account for _many times_ more bombardments per square centimeter. What am I missing?


title: Given two events such that either one of them is 'on the light cone' of the other, do they constitute a 'null interval'?

text: One basic part of determining a  metric (or applicable generalization) of a given set $\cal S$ of events (up to an arbitrary non-zero constant) is to determine to which pairs among those events, $\mathscr P, \mathscr Q \in \cal S$, to assign the (suitably generalized) distance value $s[ \mathscr P, \mathscr Q ] = s[ \mathscr Q, \mathscr P ] = 0$. In general relativity, is this assignment for two such events $\mathscr P$ and $\mathscr Q$ made if and only if event $\mathscr Q$ is 'on the light cone of' event $\mathscr P$, and vice versa event $\mathscr P$ is 'on the light cone of' event $\mathscr Q$;   i.e. in the (physics related) terminology of H. Minkowski, " _Raum und Zeit_ ", (1909):   if and only if " _one event had sent light towards the other, or had received light from the other_ " ?


title: Is it possible to produce images of pair production in home-made cloud chamber?

text: There are some nice pictures on the web showing the counter-spiralling paths of an electron positron pair produced in a bubble chamber with a uniform magnetic field, for example:- ![e+ e- pair production](http://www.bigganblog.com/wp- content/uploads/2012/03/I15-70-positron.jpg) Would it be practical to produce a cloud chamber and Helmholz Coils capable of imaging such events in an ordinary domestic garage? What sort of source would I need? How often might such events be detected?


title: Why is UV index so high when storms are announced?

text: Everytime I check the weather news and storms are announced, the UV index is very often (if not always) "very high" or "extreme", why is that?


title: Reluctance of torus shaped iron core with embedded wire loop

text: **Imagine a circular wire loop** (r = 50mm), the wire has an assumed diameter of zero, **which is embedded in a torus shaped iron core with a circular cross-section** of R = 10mm. A current in that loop would cause a circular magnetic field around the wire. **Is there any possibility to calculate the reluctance of that core?** I'm looking for a solution for weeks now, without any success. A solution for harmonic currents is desired, but I would even be happy for a DC solution. ![enter image description here](http://i.stack.imgur.com/y65Cw.png) * * * # CONTEXT **to explain what this is all about.** My real geometry looks as following: ![enter image description here](http://i.stack.imgur.com/gVvQ5.png) A torodial coil surrounded by a core with a cross section of a rounded rectangle. So I'm interested in the reluctance of the greyish part (and the other corners). If you put all corners together you'd get the mentioned torus. The green lines are the magnetic flux, the rectangle in the middle the torodial coil. For high frequencies and/or high conductive and/or high permeable materials the influence of the corners is negligible, for my case unfortunately not. I'd guess there is no analytic solution, but any idea which could get me close to it, would help. Thank you! * * * * * * # Attempted solution # Preface If one wants to calculate the permeance $P$ of a rectangular bar: ![enter image description here](http://i.stack.imgur.com/YWrk8.png) it is an easy task: $$P = \frac{\mu a b}{L} ~~~~ \rightarrow ~~~~ P\propto ab ~~~~and~~~~ P\propto\frac{1}{L}$$ where $\mu$ is the material constant. (Permeability) But my geometry is a **torus with just a quarter of its circular cross section** and the field $V$ **passes through it parallel to the circumference** of the (full) cross section: ![enter image description here](http://i.stack.imgur.com/Uq1n1.png) How can I calculate the permeance of this geometry, when there are the same proportional relations as above? * * * # Attempted solution I divide my geometry in $N$ hollow toruses with constant wall thickness $\Delta R$ and medium length element $\Delta L$, so the field passes an area of $\Delta A$: ![enter image description here](http://i.stack.imgur.com/nXbbl.png) A little piece of the radius $R$ is $\Delta R = \frac{R}{N}$. Now one can calculate: $$\Delta P_{n} = \frac{\mu \Delta A_n}{\Delta L_n} $$ with $$ \Delta A_n = \pi \bigg( (r+(n+1) \Delta R)^2-(r+n \Delta R)^2\bigg) $$ (Consider the full torus circumference, not just a quarter as displayed) and $$ \Delta L_n = \frac{\pi}{2} (2n+1) \frac{\Delta R}{2} $$ (but quarter cross section!) follows: $$P = \sum^{N-1}_{n=0} \Delta P_{n} = \mu\sum^{N-1}_{n=0} \frac{\pi(2r\Delta R+(2n+1)(\Delta R)^2)}{\frac{\pi}{2}(2n+1)(\frac{\Delta R}{2})}~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$= 4\mu\sum^{N-1}_{n=0} \frac{2r\Delta R+(2n+1)(\Delta R)^2}{(2n+1)(\Delta R)} $$ $$= 4\mu\sum^{N-1}_{n=0} \Bigg( \frac{2r}{(2n+1)} + \Delta R \Bigg)~~~~~~~~~~ $$ $$= 4\mu \Bigg( R + 2r \sum^{N-1}_{n=0} \frac{1}{(2n+1)} \Bigg)~~~~~~~~~~ $$ **And this series does not converge for $N\rightarrow\infty$.** Which is physically seen not possible, so there must be a problem with the math. Do you see what I'm missing?


title: On numerically solving the Schrödinger equation

text: I just read a paper 'A pocket calculator determination of energy eigenvalues' by J Killingbeck (1979). Link: http://iopscience.iop.org/0305-4470/10/6/001 I have some questions about section 2.   1. Why $R<0$ implies the $E$ estimate was too high and $R>1$ implies too low? And is it possible that $0<R<1$ remains?   2. Why "if the starting point is chosen at $r=h(l+1)$, then $R(hl)$ can be assigned any finite value without disturbing the calculation at larger $r$ values" ?   3. How to guess a trial energy $E$ (especially the ground state energy)? (Actually I'm interested in a later paper by Killingbeck (1982) 'Finite- difference methods for eigenvalues'(Link: http://iopscience.iop.org/0022-3700/15/6/009), in which the author applies the method in 'A pocket calculator determination of energy eigenvalues' to some potentials with highly singularity at origin) * * * _mod note: copying the text of whoplisp's post here since it isn't actually an answer to the question_ They convert the Schroedinger equation into a finite difference equation (instead of a variational approach): $-\frac{1}{2}\Delta\psi +V\psi=E\psi$ $V$ is a central potential Ansatz with solid harmonic $\xi_l$ of degree $l$ (e.g. $x$ or $xy$) $\psi=\xi_l\phi(r)$ leads to $2r(V-E)\phi=rD^2\phi+(2l+2)D\phi$ Notation: $D=\frac{d}{dr}$ finite difference approximation $h^2D^2\phi \rightarrow \phi(r+h)+\phi(r-h)-2\phi(r)$ $2hD\phi \rightarrow \phi(r+h)-\phi(r-h)$ crucial step: **introduce ratio variable** $R(r)$: $\phi(r+h)=R(r)\phi(r)$ convert Schroedinger equation in recursive equation to calculate $R(r)$: $2r[1+h^2(V-E)]=R(r)[r+h(l+1)]+[r-h(l+1)](R(r-h))^{-1}$ > the corresponding 1D Schroedinger equation is: > > $2[1+h^2(V-E)]=R(x)+(R(x-h))^{-1}$ if $r=h(l+1)$ is chosen then $R(hl)$ can be assigned any finite value without disturbing the calculation at large $r$ values to find the ground states with angular momentum $l$ start with an estimate for $E$, some non-zero $R(hl)$ and some small $h$. $R(r)$ as $r$ increases will either become negative or pass through unity from below. first thing happens when guessed $E$ was too high second thing indicates $E$ was too low (because $\psi$ increases with $r$) you just try various values and 'sandwhich' the solution example for $V=\lambda r-1/r$: $\lambda=1$, $h=.1$, state=2p: 1.9759


title: The relativity of simultaneity

text: In his popular book on relativity, in chapter IX, "The relativity of simultaneity", Einstein describes an experiment in which a flash happens simultaneously on A and B, as defined by the fact that an observer at the middle point M can see the light coming from A and B at the same moment:                             --train-->     ---embankment---A-------M-------B----      Then, he proceeds to say that an observer on the train "is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A". But I find the opposite result, from the train point of view:   * the embankment moves to the left (in the diagram) at a speed of v.   * so, after dt, A moved by v*dt to the left, the light coming from A moved by c*dt to the right, and the distance between A and the light increased by (c+v)*dt.   * similarly, the distance between B and the light coming from B increased by (c-v)*dt.   * so, the light coming from A is going to reach the middle point M before the light coming from B, so the observer is going to see the light from A first (if he happens to be at M at the correct moment), which is the opposite of what is said in the book. Another way to state about the same thing is: the light from both points go towards M, but M is moving to the left, so the light from A is going to reach M before the light from B does. Where is the error in my reasoning?


title: Intuitive explanation of Poiseuille's law -- why $r^4$?

text: According to Poiseuille's law, the effective resistance of a tube is inversely proportional to the fourth power of its radius (as given by the following equation). $$R = \frac{ 8 \eta \Delta x}{\pi r^4} $$ I can go through the derivation of the law but is there an intuitive explanation for why it is the fourth power of radius, or a thought experiment that might make it more obvious?


title: Why can we treat quantum scattering problems as time-independent?

text: From what I remember in my undergraduate quantum mechanics class, we treated scattering of non-relativistic particles from a static potential like this:   1. Solve the time-independent Schrodinger equation to find the energy eigenstates. There will be a continuous spectrum of energy eigenvalues.   2. In the region to the left of the potential, identify a piece of the wavefunction that looks like $Ae^{i(kx - \omega t)}$ as the incoming wave.   3. Ensure that to the right of the potential, there is not piece of the wavefunction that looks like $Be^{-i(kx + \omega t)}$, because we only want to have a wave coming in from the left.    4. Identify a piece of the wavefunction to the left of the potential that looks like $R e^{-i(kx + \omega t)}$ as a reflected wave.   5. Identify a piece of the wavefunction to the right of the potential that looks like $T e^{i(kx - \omega t)}$ as a transmitted wave.   6. Show that $|R|^2 + |T|^2 = |A|^2$. Interpret $\frac{|R|^2}{|A|^2}$ as the probability for reflection and $\frac{|T|^2}{|A|^2}$ as the probability for transmission. This entire process doesn't seem to have anything to do with a real scattering event - where a real particle is scattered by a scattering potential - we do all our analysis on a stationary waves. Why should such a naive procedure produce reasonable results for something like Rutherford's foil experiment, in which alpha particles are in motion as they collide with nuclei, and in which the wavefunction of the alpha particle is typically localized in a (moving) volume much smaller than the scattering region?


title: Length contraction and simultaneous length measurements

text: I am just working through an argument from Halliday Resnick to derive the Lorentz contraction (see quote below). Some paragraphs before this, the authors note that: > If the rod is moving, however, you must note the positions of the end points > _simultaneously_ (in your reference frame) or your measurement cannot be > called a length. A paragraph later they invoke the following argument: > Length contraction is a direct consequence of time dilation. Consider once > more our two observers. This time, both Sally, seated on a train moving > through a station, and Sam, again on the station platform, want to measure > the length of the platform. Sam, using a tape measure, finds the length to > be $L_0$, a proper length because the platform is at rest with respect to > him. Sam also notes that Sally, on the train, moves through this length in a > time $\Delta t = L_0/v$ where $v$ is the speed of the train; that is, $$ L_0 > = v \Delta t \quad \text{(Sam)} $$ This time interval $\Delta t$ is not a > proper time interval because the two events that define it (Sally passes the > back of the platform and Sally passes the front of the platform) occur at > two different places, and therefore Sam must use two synchronized clocks to > measure the time interval $\Delta t$. > > For Sally, however, the platform is moving past her. She finds that the two > events measured by Sam occur _at the same place_ in her reference frame. She > can time them with a single stationary clock, and so the interval $\Delta > t_0$ that she measures is a proper time interval. To her, the length $L$ of > the platform is given by $$ L = v \Delta t_0 \quad \text{(Sally)}. $$ Then they conclude by dividing the two equations above: > $$ \frac{L}{L_0} = \frac{v\Delta t_0}{v \Delta t} = \frac{1}{\gamma}$$ or $$ > L = \frac{L_0}{\gamma} $$ > > which is the length contraction equation. However I don't see in what sense the length was measured simultanous in the derivation above, how is the detailed connection between the statement that the length measurement has to be simulanous and the quoted derivation?


title: how is feshbach resonance potential term physically produced?

text: In Feshbach resonance model, a 2*2 potential term with space dependent diagonal and non-diagonal terms is written $\left(\begin{array}{cc} V_{11}(\mathbf{r}) & V_{12}(\mathbf{r})\\\ V_{21}(\mathbf{r}) & V_{22}(\mathbf{r}) \end{array}\right)$ How are these terms physically produced ? An example of Feshbach resonance that can be taken is of two atoms far apart from each other in a singlet or triplet state when close together, are coupled via a hyperfine interaction of one's electron's moment with the nuclear moment of another. What I think is that these terms are a combination of the 2*2 Leinard-Jones potential and the magnetic energy (due to non-zero magnetic moment) term combined but would like to get a detailed picture of generating such a 2*2 potential interaction.


title: Antimatter Propulsion System

text: Is there a possibility ( in theory ) to build a antimatter propulsion system, if so how can we control the collision of matter-anitmatter, will humans be able to control this force just like electricity ?


title: Does time stand still at a phase transition?

text: For second order phase transition thermodynamic properties can be described in very general terms by their critical exponents. So at every transition the correlation length $\xi$ should diverge as $$\xi \propto |T-Tc|^{-\nu},$$ where the exponent depends on the specifics of the system. Is the same true for the correlation time $\tau$ or only in special cases? Additionally if the correlation time diverges how can there still be fluctuations, isn't everything infinitely strongly coupled together as $\tau \rightarrow \infty$?


title: Can Werner states have bound entanglement?

text: Let us consider the maximally entangled state \begin{equation} |\psi\rangle=\frac{1}{\sqrt{n}}(|0,0\rangle+\cdots+|n-1,n-1\rangle) \end{equation} and construct the pseudo-pure state \begin{equation} \rho_\lambda=(1-\lambda)|\psi\rangle\langle\psi|+\lambda\frac{I_{n^2}}{n^2}, \end{equation} where $I_{n^2}$ is the identity matrix and $0\leq\lambda\leq1$. I was told that for any dimension $n$ and for any $\lambda$, $\rho_\lambda$ is either separable or entangled which can be determined by partial transpose. It can be rephrased as > There does not exists any dimension $n$ and any $\lambda$ such that the > corresponding $\rho_\lambda$ is a bound entangled state. I could not find out the proof and could not make one by myself. Can someone give me a proof or at least refer a paper containing the proof. Advanced thanks for any suggestion. ADDITION: Also the same question can be asked for the case, when the coefficients of $|j,j\rangle$ are non-uniformly distributed nonzero complex numbers (such that the sum is $1$).


title: Fundamental Representation of $SU(3)$ is a complex representation

text: Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:   $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now if $-(T^a_R)^* = T^a_R $, the representation $R$ is real. Again if we can find a unitary matrix, $V(\neq I)$ such that $ \qquad \qquad \qquad -(T^a_R)^*=V^{-1} T^a_R V \quad \forall a $ then the representation $R$ is pseudoreal. If a representation is neither real nor pseudoreal, the representation $R$ is complex. **Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus- minus pairs.** Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by $T^a =\lambda^a/2; \quad a=1,...8$,   where $\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1/\sqrt{12}, 1/\sqrt{12}, -1/\sqrt{3} )$. My doubt is: **According to the claim, is the fundamental representation of $SU(3)$ a complex representation?**


title: Are there old aged particles?

text: To measure the lifetime of a specific particle one needs to look at very many such particles in order to calculate the average. It cannot matter when the experimentalist actually starts his stopwatch to measure the time it takes for the particles to decay. If he measures them now or in 5 minutes makes no difference, since he still needs to take an average. If he measures later there will be particles out of the picture already (those who have decayed in the last 5 min), which won't contribute and the ones his measuring now behave (statistically) the very same, of course. I have just read the following in _Introduction to Elementary Particles_ by Griffiths: > Now, elementary particles have no memories, so the probability of a given > muon decaying in the next microsecond is independent of how long ago that > muon was created. (It's quite different in biological systems: an 80-year- > old man is much more likely to die in the next year than is a 20-year-old, > and his body shows the signs of eight decades of wear and tear, But all > muons are identical, regardless of when they were produced; from an > actuarial point of view they’re all on an equal footing.) But this is not really the view I had. I was imagining, that a particle that has existed for a while is analogous to the 80 year old man, since it will probably die (decay) soon. It just doesn't matter because we are looking at a myriad of particles, so statistically there will be about as many old ones as babies. On the other hand it is true that I cannot see if a specific particle has already lived long or not; they are all indistinguishable. Still I am imagining particles as if they had an _inner age_ , but one just can't tell by their looks. So is the view presented in Griffiths truer than mine or are maybe both valid? How can one argue why my view is wrong?


title: physical intuition behind quasi-bound state formation in feshbach resonance

text: In Feshbach resonance, by scattering theory formalism it is found that the resonance in cross-section happens when bound state energy of the closed channel is near to the scattering state energy of the open channel. This is explained for the case of 2 atoms interacting via a Lienard Jones potential (for far away distances) and insertion of an extra term in the potential due to hyperfine interaction at close distances. Explanation for resonance is given as follows :- the scattering state continuum energy of the 2 atoms on addition of the hyperfine interaction at small separation distances goes into the excitations of the internal states leading to a lowering of the kinetic energy of the 2 atoms and formation of a quasi-bound state. Can this point be explained in more detail ? how are the excitations in the internal states coming from the hyperfine interaction term and leading to a quasi-bound state (bound state with a finite lifetime) is not clear to me.


title: 17 Joules of Energy From a Mouse Trap

text: Do you think it would be possible to get 17 joules out of a standard size mouse trap. By my math, it is a torsion coefficient of 3.45 or so out of the spring.


title: Finding distance when the force is a function of time

text: I'm having trouble with this homework question > A mysterious rocket-propelled object of mass 49.0 kg is initially at rest in > the middle of the horizontal, frictionless surface of an ice-covered lake. > Then a force directed east and with magnitude $F(t) = (16.3\text{ N/s})t$ is > applied. > > How far does the object travel in the first 5.50s after the force is > applied? For some reason I'm not getting the correct answer. I think maybe I'm not understanding how to use the magnitude of the force they are giving me. I know how to use a constant force, but is this different because the force is a function of time? I tried starting it like this: $$\begin{align}F &= ma\\\ 16.3(t) &= (49)(a) \\\ 16.3(5.5) &= (49)(a) \\\ 89.65 &= (49)(a) \\\ a &= 1.82959\ \mathrm{m/s^2}\end{align}$$ So now we know that: $$\begin{align}t &= 5.5\text{ s} \\\ a &= 1.82959 m/s^2 \\\ V_o &= 0\end{align}$$ So I plug it into my equation: $$\begin{align}\Delta X &= V_o t + 1/2 a t^2\\\ \Delta X &= (0)(5.5) + (1/2)(1.82959)(5.5)^2\\\ \Delta X &= 27.7\text{ m}\end{align}$$ But that's not the right answer.


title: "Redshifting" of forces in stationary space - times

text: Here's the problem statement: Let $(M,g_{ab})$ be a stationary spacetime with timelike killing field $\xi ^{a}$. Let $V^{2} = -\xi _{a}\xi ^{a}$ ($V$ is called the redshift factor). (a) Show that the acceleration $a^{b} = u^{a}\triangledown _{a}u^{b}$ of a stationary observer is given by $a^{b} = \triangledown^{b}\ln V$. (b) Suppose in addition that $(M,g_{ab})$ is asymptotically flat. Then, the "energy as measured at infinity" of a particle of mass $m$ and 4 - velocity $u^{a}$ is $E = - m\xi _{a}u^{a}$. Suppose a particle of mass $m$ is held stationary by a (massless) string, with the other end of the string being held by a stationary observer at infinity. Let $F$ denote the magnitude of the local force exerted by the string on on the particle. According to part (a), we have $F = mV^{-1}(\triangledown ^{a}V\triangledown _{a}V)^{1/2}$. Use **conservation of energy** arguments to show that the magnitude of the force exerted on the **other end** of the string by the observer at infinity, is $F_{\infty } = VF$. My attempt: (a) A stationary observer's 4 - velocity must be proportional to the time - like killing vector and it must be normalized to -1 so we find that for a stationary observer, $$u^{a} = \frac{\xi ^{a}}{(-\xi ^{c}\xi _{c})^{1/2}}$$. Now we compute, $$\triangledown _{b}u^{a} = \frac{\triangledown _{b}\xi ^{a}}{(-\xi ^{c}\xi _{c})^{1/2}} + \frac{\xi ^{a}\xi ^{c}\triangledown _{b}\xi _{c}}{(-\xi ^{c}\xi _{c})^{3/2}}$$ so $$u^{b}\triangledown _{b}u^{a} = \frac{\xi^{b}\triangledown _{b}\xi ^{a}}{(-\xi ^{c}\xi _{c})} + \frac{\xi ^{a}\xi^{b}\xi ^{c}\triangledown _{b}\xi _{c}}{(-\xi ^{c}\xi _{c})^{5/2}} = -\frac{\xi^{b}\triangledown ^{a}\xi _{b}}{(-\xi ^{c}\xi _{c})}$$ where the second term vanishes because it is a contraction of a symmetric tensor with an anti - symmetric one and I have swapped the indices in the first expression using killing's equation. Therefore, $$a^{a} = u^{b}\triangledown _{b}u^{a} = \frac{1}{2}\frac{\triangledown ^{a}(-\xi^{b}\xi _{b})}{(-\xi ^{c}\xi _{c})} = \frac{1}{2}\triangledown ^{a}\ln V^{2} = \triangledown ^{a}\ln V$$ as desired. (b) This is where I'm totally stuck. As far as conservation of energy goes, we know that $E$, as defined above, is a conserved quantity **along** the worldline of the stationary particle; physically $E$ is the energy needed to bring in the particle from infinity to its orbit. Here we have a stationary observer at infinity holding this particle stationary by a long thread. There is a tension force at the end the observer holds and at the end the particle hangs by. Let's say the observer exerts the force $F_{\infty }$ at event $P_{1}$ and the particle feels the local force $F$ at event $P_{2}$. As far as I can tell, all we know is that in between these events, $E$ is constant. But how do I relate $E$ to $F_{\infty}$ and how do I do this using the conservation of energy explained above? I should note that I tried something on a whim and looked at $\triangledown _{b}E$. We know that for the stationary particle hanging from the string, to which this total energy is attributed, $$E = -m\xi _{a}u^{a} = -\frac{m\xi_{a}\xi^{a}}{(-\xi _{c}\xi^{c})^{1/2}} = m(-\xi^{c}\xi_{c})^{1/2}$$ so if we compute the derivative we get $\triangledown _{b}E = mV\triangledown _{b}\ln V$ thus $(\triangledown ^{b}E\triangledown _{b}E)^{1/2} = mV(\triangledown ^{b}\ln V\triangledown _{b}\ln V)^{1/2} = VF$ but I have NO idea how this quantity is related to $F_{\infty }$, if at all. If it is related somehow I have no idea how one uses conservation of energy to arrive at the relation. Thanks in advance!


Where to find cross section data for e- + p -> p + e-? PDG's cross section data listing does not include it.


title: How does enthalpy help us to find the state of equilibrium?

text: For an adiabatic system with constant external (generalized) forces, in a state out of equilibrium we have: $$\delta H\leq 0 $$ If I understand correctly, derivation of this inequality does not rely on the second law of thermodynamics, which states how systems evolve toward equilibrium. Here is the derivation: Having constant external forces $\mathbf{J}$, the work input to the system is $\delta W \leq \mathbf{J}.\delta \mathbf{x}$ , where $\mathbf{J}$ and $\mathbf{x}$ are generalized forces and displacements respectively. Equality is achieved for a quasi-static change with $\mathbf{J_{internal}}=\mathbf{J}$, but generally there is some loss of the external work to dissipation. Now the first law gives $\delta E\leq \mathbf{J}.\delta \mathbf{x}$, or $\delta H\leq 0$, where $H=E-\mathbf{J}.\mathbf{x}$ . **My question is** what is the underlying physical principle behind this inequality, and how can we deduce the direction of evolution toward equilibrium without drawing on the second law?


title: Is the "Doctor Who" spacetime affected by Hawking's chronology protection mechanism?

text: Recently, there has been a paper1 (and an accompanying layman-ized white paper2) on "Traversable Achronal Retrograde Domains In Spacetime", TARDIS for short. It proposes a spacetime geometry that contains closed timelike curves. Now, Hawking once proposed3 a mechanism that apparently causes all closed timelike curves to more or less destroy themselves. Basically, quantum fluctuations cycle through the curve and build upon themselves (in a sense, they overlay with their "past selves"), leading to a divergent expectation value for the energy-momentum tensor. The media (which has dubbed it the "Doctor Who spacetime") seems to have caught on to this paper as the next time machine. Usually, the term "closed timelike curve" is associated with time machines because of the causality violations a CTC can cause. Is this really possible? Or does Hawking's mechanism protect this system from a causality violation, destroying the CTCs in it? 1\. arXiv:1310.7985 [gr-qc]; "Traversable Achronal Retrograde Domains In Spacetime", Benjamin K. Tippett, David Tsang 2\. arXiv:1310.7983 [physics.pop-ph] 3\. Hawking, S. W. (1992). Chronology protection conjecture. _Physical Review D, 46(2)_ , 603.


title: Usage example of stabilizer codes QEC

text: This question directly follows the previous one about $X$ stabilizers and phase-flip errors: Practical example of stabilizer codes Let's now consider a second part of the quantum circuit that is able to detect and correct bit-flip errors: ![enter image description here](http://i.stack.imgur.com/A38HS.png) Just a recall of the Steane code and its stabilizers used in this example: $$ \left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle) $$ $$ \left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle) $$ $$ K^1 = IIIXXXX $$ $$ K^2 = XIXIXIX $$ $$ K^3 = IXXIIXX $$ $$ K^4 = IIIZZZZ $$ $$ K^5 = ZIZIZIZ $$ $$ K^6 = IZZIIZZ $$ The state of both parts, individually considered, is: $$\left|\psi\right\rangle_F={1\over 2}(\left|\psi\right\rangle_I+U\left|\psi\right\rangle_I)\left|0\right\rangle + {1\over 2}(\left|\psi\right\rangle_I-U\left|\psi\right\rangle_I)\left|1\right\rangle\;\;\;\;(1)$$ where $U \in \left\lbrace K^1,K^2,K^3\right\rbrace$ for the first part and $U \in \left\lbrace K^4,K^5,K^6\right\rbrace$ for the second part. Determined how does the first part of the circuit work, the second part uses $K^4, K^5, K^6$ generators to check whether the input has a bit-flip error or not. $K^4, K^5, K^6$ act checking the parity of the input; if the parity checking operation gives back an odd result, then one qubit flipped. Through syndrome measurement we are able to detect where the error occurred and correct it by triggering the right $X_i$ gate. Is this how detection and correction work for this part of the circuit? Further trouble: considering the syndrome measurement about $(1)$, how does it come? Since, in this case, ancilla qubits are used as control qubit for $K^i$ generators and not vice versa, then is it thanks to the superposition nature of the system that, if we measure $\left|1\right\rangle$ from the syndrome, we project the input to the state $\left|\psi\right\rangle_F = \left|\psi\right\rangle_I - U\;\left|\psi\right\rangle_I$, so the error occurrence? Thank you


title: How can enzyme/substrate reactions that adhere (largely) to quantum theory also require 'Newtonian' consideration of gravity?

text: I'd just like to ask for a little clarification here due to confusion from interdisciplinary studies. I'm currently reading the 1976 paper related to the recent 2013 Nobel Prize for Chemistry, by 2 of the laureates Warshel and Levitt — _Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and Steric Stabilisation of the Carbonium Ion in the Reaction of Lysosyme_ — and have recently been reading into quantum chemistry, as well as studying biochemistry and have had further lectures in Physics to boot. I'm wondering if the points made here are comparable to points I've heard in mechanics lectures, about a cannonball not being modelled as a quantum object, since every subatomic particle would be too computationally intensive, so a coarse-grain model is used (i.e. Newton's "classical" laws). I wouldn't be so quick to draw parallels between mechanics lectures and enzyme biochemistry but it seems to be in the language they are using here (see highlights below). From what I understand (in mechanics) only 3 of the 4 'fundamental forces' (strong, weak, and EM) only strictly apply at the quantum/microscopic scale, and scientists are as it stands struggling to work gravity into the setup. Is this worth considering in this paper - if an enzyme is considered all of a sudden due to its size as a quantum object does this entail ignoring gravity QM now in favour of 'Newtonian', inertial frame-of-reference type mechanics, or considering it in some other 'Newtonian' manner? The introductory section is as follows (full paper linked above): > The conformation of the enzyme-substrate complex can be adequately studied > using empirical energy functions based on the “classical” contributions of > bond stretching, bond angle bending, bond twisting, non-bonded interactions, > etc. The mechanism and energetics of the enzymic reaction can only be > studied using **a quantum mechanical approach**. Previous quantum mechanical > calculations on enzymic reactions have been limited in several respects. In > the first place, they deal with an over-simplified model system consisting > of only small fractions of the atoms involved in the real enzyme-substrate > interaction. Second, all available quantum mechanical methods treat the > reaction as if in a vacuum and are, therefore, unable to include the > dielectric, which has a very important effect on the energy and charge > distribution of the system. This limitation does not depend on the actual > quantum mechanical scheme, being equally valid for the simplest Hiickel > treatment and most extensive _ab initio_ calculation. > > Because the **treatment of the whole enzyme-substrate complex quantum > mechanically is computationally impossible** , it is necessary to simplify > the problem and use a **hybrid classical/quantum mechanical approach**. The > method described here considers the whole enzyme-substrate complex: the > energy and charge distribution of those atoms that are directly involved in > the reaction are evaluated quantum mechanically, while the potential energy > surfac e of the rest of the system, including the urrounding solvent, is > evaluated classically. An important feature of the method is the treatment > of the “dielectric” effect due to both the protein atoms and the surrounding > water molecules. Our dielectric model is based on a direct calculation of > the electrostatic field due to the dipoles induced by polarizing the protein > atoms, and the dipoles induced by orienting the surrounding water molecules. > With the inclusion of the microscopic dielectric effect we feel that our > model for enzymic reactions includes all the important factors that may > contribute to the potential surface of the enzymic reaction. Because we use > analytical expressions to evaluate the energy and first derivative with > respect to the atomic Cartesian co-ordinates, it is possible to allow the > system to relax using a convergent energy minimization method. Here we use > the method to study the mechanism of the lysozyme reaction, with particular > emphasis on factors contributing to the stabilization of the carbonium ion > intermediate. We first outline our theoretical scheme, paying attention to > the **partition of the potential surface into classical and quantum > mechanical parts and to details of the microscopic dielectric model**. Then > we use the method to investigate factors that contribute to the > stabilization of the carbonium ion intermediate and deal with the relative > importance of each factor. The effect of steric strain is considered for > both the ground and the carbonium states and found to be of minor > importance. **Electrostatic stabilization** is treated at some length with > special emphasis on the balance between stabilization by induced dipoles > (polarizability) and stabilization by charge- charge interactions. The > results demonstrate the importance of electrostatic stabilization for this > reaction.


title: How Come string theory is based on the fact that a "string", a theoretical dimension, exists?

text: I See no explanations beyond concise explanations that a "string" is nothing more than a hypothetical, one-dimensional subatomic particle. If so, why imagine that they are "strings" instead of just "subatomic particles", or the such? I don't understand the idealization behind string theory at all ... adding the fact that strings are coexistent with theoretical physics, that says a lot right there. String theory just seems, to me at least, like a delving dream in to a complex system to model something already existing and know, just in a different way.


title: The virial theorem and a delta function potential

text: So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = V_0\delta^{(D)}(\textbf{r}-\textbf{r}')$, where $\delta^{(D)}(\textbf{r}-\textbf{r}')$ is the delta-function in D dimensions. I'm not sure why, but I think that I should get that: $\langle \textbf{r}\cdot\nabla V\rangle = \frac{1}{D}\langle V_0\rangle$ since the delta written out as a product of different components is: $\delta^{(D)}(\textbf{r}-\textbf{r}') = \frac{1}{\sqrt{det(G)}}\prod\limits_{i=1}^D\delta(x_i-x_i')$, with $x_i$ the different components of the vector $\textbf{r}$, given in the base with metric G, where $\sqrt{det(G)}$ gives the D-dimensional volume-element in the basis $\textbf{e}_i$. I don't know wether there is a more rigorous reasoning for this? Or wether this is even correct ? **Addendum: a different perspective:** Another way to look at it, is that if I rescale my vector $\mathbb{r}$ bij a factor $\lambda$, I get: $\delta^{(D)}(\lambda\textbf{r}-\lambda\textbf{r}') = \frac{1}{\lambda^D}\delta^{(D)}(\textbf{r}-\textbf{r}')$. This makes me also think that i should get the above relation for the virial theorem. But still I'm not sure of my reasoning ! > **Extra demand on potential (necessary for finite system)** > > Next to my delta-potential, I also have an extra confining potential to keep > the particles together. For simplicity I'll take an harmonic trap $V(r) = > \frac{1}{2}m\omega^2r^2$ which keeps the particles together! So this is the > other term of the potential, but this one I didn't consider in my question > because that one posed no problem to my calculations!


title: Anomalous dimensions in the $O(N)$ model

text: * Is there any statement known about the anomalous dimensions of the $O(N)$ model in various dimensions and/or in the large-N limit?    * If a $\phi^4$ ("double-trace") term is coupled to an $O(N)$ model then is there an argument as to why this quartic term is ignorable?  [..I believe that there are analogous statements known for higher bosonic spin fields too - at least for the second question of mine..] I would be happy to see some pedagogic references which hopefully derive these.


title: Electric potential in a capacitor

text: Im a writing a school report regarding the electric energy and potential of a capacitor. In short: An experiment was carried out where we charged a capacitor, and then decharged it with an electric engine, lifting up a predefined weight. We then measured how high the weight was lifted with different voltages. i then found the effiency of the system by comparing the mechanical energy with the electrical energy. The second half of the report is then to find the electric potential of the capacitor, but I'm not to strong at electrics, and can't find anything about this in my physics book. My question is now; how can i find the electric potential of the capacitor?


title: What is the Principle of Maximum Conformality?

text: I'm trying to understand this article about an advance in the theoretical understanding of QCD which centers on the Principal of Maximum Conformality. What is this Principle? In other words, what is being maximized and what does this tell us about the structure of QCD? Also, is this a new principle or a new application of an old principal? Here's the full paper on the principle's application to top physics.


Does the volume of a thermodynamic system always have to change for it to do work? If yes,could you explain why? And if no, could you provide the example of a system, where it is not neccesary.


title: How to recover units?

text: Theorists frequently use convenient units like $\hbar=1$ or $m=2$ or whatever is useful to simplify the notation in the problem. And after all the calculations are done the units are recovered based on what the unit of the answer needs to be. I can definitely see why those units are convenient, but I really don't feel comfortable with the recovering step. So far I have only seen one example and it is not quite enough. If somebody could provide some examples of how to recover the units that would be great. As for as I understand that should only need to involve the initial assumptions and the final answer. Or if somebody knows a good explanatory text that would also be very much appreciated. Are there any caveats/limitations while using convenient units? EDIT: To further clarify why I am confused about this entire procedure:   * Let's assume that I have a problem involving a trap length $L$ and a wave length $\lambda_0$. For convenience I set $L=1$ and $\lambda_0=1$. My final answer needs to be in dimensions of meters. How do I know whether my final answer is supposed to be multiplied by $L$ or divided by $\lambda_0$?   * Lets say I need my final answer in units of angular momentum times capacitance per volume (just as purely hypothetical example). And I started of by setting constants like $\epsilon_0=1$,$L=1$(some length scale),$p0=1$(some momemtum scale). This problem is easy enough that I can still figure it out. But what if I had to deal with constants like the bohr magneton or the conductance quanta. It could become very hard to figure out how I need to combine certain constants to produce the right units. Especially once the number of constants increases. Is there some kind of procedure one can follow that will always spit out the right combination of constants?


title: Why do quantum physical properties come in pairs?

text: Why do quantum physical properties come in pairs, governed by the uncertainty principle (that is, position and momentum?) Why not in groups of three, four, etc.?


title: Simple Relativity Question

text: I've been taking a course in modern physics and we've spent several weeks on relativity now, but I still am confused how to deal with this problem, which has shown up in several different ways throughout the course so far. Keep in mind that the units are SR, not SI. If an astronomer wants to travel to a planet that is 200 years away, and he wants the journey to take him 10 years, how fast must he travel? I'm mostly confused about how to begin, because I fully understand Lorentz transformations and spacetime/Minkowski diagrams. I just don't know how to figure out how fast he needs to go. If I knew how fast he was going, I could reverse the process of this question, but most special relativity problems begin by giving velocities and other quantities are then solved for.


title: How many atoms in the particle (AuNP)

text: Let's say I have to determine number of atoms of gold in ball-shaped nanoparticle with diameter of 75 nm. I need general method of solving such problems more than specific answer for only this question.


title: Minimal voltage in linear accelerators to achieve relativity?

text: Could anyone tell me what is the minimal voltage between anode and cathode in a linear accelerators to achieve speeds where relativity starts to show? Let me ask in a different way: "What is the minimal voltage scientists work with in linear accelerators?"


title: Is the zero acceleration path also the shortest path between two points?

text: In flat, free, Euclidean space, the shortest path and the zero acceleration path are the same path, which is a straight line. However, in general relativity, is the zero acceleration path also the shortest path between two points? I am assuming that free fall is zero acceleration.


title: James Webb Space Telescope's halo orbit at Lagrange point L2

text: The James Webb Space Telescope (JWST) is expected to be launched in 2018 and operate in the L2 vicinity, about 1.5 million km from Earth. ![L2 orbit](http://i.stack.imgur.com/Df7OU.jpg) It will be placed in a halo orbit around the unstable L2 saddle point with a semi-major axis of about 500,000 km and a period of about half a year. I thought it interesting to determine the hypothetical mass at L2 to produce those orbital factors ignoring all other bodies. Using Kepler's third law, ${p^2} = {{4{\pi ^2}} \over {G({M_1} + {M_2})}}{a^{^3}}$, resulting in a mass of over ${10^{23}}$ kg, similar in mass to our Moon. Of course, JWST's halo orbit is determined by the complex interactions of the Sun's and Earth's gravitational fields and not by a lunar sized body located at L2. What are the semi-major axis size limits that can be achieved in L2 halo orbits, and is halo orbit stability affected by these sizes?


title: Energy-time uncertainty and pair creation

text: Usually, the energy-time analogue of the position-momentum uncertainty relation is quoted as $\Delta E \Delta t \geq \frac{h}{4 \pi}$. This has interpretational issues and such. But, with a suitable definition of $\Delta t$, this can be derived. See The energy-time uncertainty relation for instance. The interpretation for $\Delta t$ is then the amount of time needed for an arbitrary observable to change by one standard deviation. When popular science writers try to heuristically explain Hawking radiation, they mention the same uncertainty relation and say that for time scales $\Delta t \sim \frac{1}{\Delta E}$, particles with energy $\Delta E$ can be created. This happens even in vacuum. Can someone please explain how the second relation follows (heuristically, at least) from the first?


title: Why does a tire need to be filled with air?

text: The **Mini 4WD** 's tires aren't full of air, and it can run. Also, the tank doesn't have tires with air. So, the question is: why do real cars on the road need to be filled with air? What is the idea behind it? ![](http://i.stack.imgur.com/0ciAs.jpg)


title: Why is storage of spent nuclear fuel dangerous?

text: Just what the title states; there's a good deal of noise made about transport, and storage of spent nuclear fuel. Why all the hullabaloo when the fuel is all spent?


title: Bending a flat sheet

text: I am interested in the following problem: starting from a flat sheet, once can bend it into a (sector) of a cylinder isometrically. If further (orthogonal to the first plane) curvature is induced, the sheet will necessarily be stretched. The question is, how much stretch is locally caused by the increase in curvature? I could attempt to answer the point by classical continuum mechanics tecniques, but I am told differential geometry allows an answer to be given immediately. Could somebody please point me towards the answer, or suitable material to study? Many thanks


title: what is the physics just at bubble formation.

text: ie do a couple of molecules with more energy bang out a volume in the liquid where molecular attraction is overcome and the energetic molecules can move at random? where can I find out more about this subject?


title: Heisenberg's uncertainty principle - $ \Delta p $

text: So I was reading this paper, "Limits to Binary Logic Switch Scaling—A Gedanken Model". The following is the paper's abstract: > In this paper we consider device scaling and speed limitations on > irreversible von Neumann computing that are derived from the requirement of > "least energy computation." We consider computational systems whose material > realizations utilize electrons and energy barriers to represent and > manipulate their binary representations of state. So, logically, a bit of physics is used in the paper. What the author does on the second page is rewrite $ x = \dfrac{\hbar}{\Delta p }$ to $ x = \dfrac{\hbar}{\sqrt{2mE_{\text{bit}}}}$. I am aware that $ p = \sqrt{2mE}$, but why can you use $\Delta p = \sqrt{2mE}$? Is this allowed or does the author make a mistake?


title: Hubble Constant and Planck Constant

text: Has there been any effort to understand Hubble's limit as a quantum phenomenon? It seems like they are suggesting the same thing.


title: Rotation of bodies about centre of mass

text: Imagine two masses connected by a massless, inextensible string kept on a smooth, horizontal plane. Now suppose one of the masses is given a velocity perpendicular to the string. So the masses will rotate about their centre of mass, which lies on the string. My question, how can we determine the velocity of the other block during motion of system? Consequently, how to find the tension present in the string during the motion of the bodies?


title: Stable and Unstable Orbital Resonance

text: I was wondering if anyone can show me why some orbital resonances are unstable. For example in the asteroid belt there is a depleted distribution at 3:1 resonance with jupiter. What is the cause of this depletion and is it likely that other planets play an effect in determining the width of the depleted region or is it from some collisional process in the belt ( defined as full width half maximum of the depleted region)? A mathematical argument would be preferred but any ideas on the FWHM of the depletion would be appreciated. ![enter image description here](http://i.stack.imgur.com/WhNi2.png)


title: Are there any "Neutrinian" standard candles?

text: Are there any theoretical models/investigations of object far away that could act as some type of neutrinian standard candles? That is, they spew out neutrinos of with some know characteristics that could be used to obtain information about other observables? See here and here where they discuss how IceCube has detected cosmic neutrinos.


title: Cheap launching objects to orbit

text: Why slingatron hasn't been successful ? Same thing with gigantic nail clipper. Since we dont need great force when arm length increases, why couldn't we build gigantic nail clipper orbital launcher :) ? Provided of course that we had material able to sustain great compression stress. Disclaimer, no im not affiliated in any way with creators, dont know them, not even same country. Just thought that idea was cool.


title: Ba K Lines in a Cs-137 gamma ray spectrum

text: I'm looking at a spectrum for the gamma ray spectrum of a 137-Cs source and I'm not sure where the peak in the lower energies of ~32keV comes from. I understand that it's to do with some sort of Barium x-ray emission but I don't really know how this occurs. If anyone could explain this to me it would be greatly appreciated.


title: Schrödinger's cat and the difficulty of macroscopic superposition state

text: The Schrödinger's cat was regarded as peculiar since we seldom encounter a superposition state in macroscopic scale: $$ | \mathrm{dead \,\,cat} \rangle + | \mathrm{alive \,\, cat}\rangle $$ We more often describe an unknown cat as $$ | \mathrm{dead \,\,cat} \rangle \langle \mathrm{dead \,\,cat} |+ | \mathrm{alive \,\, cat}\rangle \langle \mathrm{alive \,\, cat}| $$ without superposition. I often heard that it is difficult to prepare and maintain a large-scale superposition state. Similar difficulty also occurs in quantum computing. My question is, actually what is the reason for the difficulty to prepare and maintain a large-scale superposition state? If it is decoherence, why decoherence happens? Is that because of entropy?


title: Is it possible to avoid the radiation that caused the American flag turned into white on the Moon?

text: While lunar images have proven that the American flags planted during the Apollo missions are still standing on the moon, lunar scientists have now said that they probably no longer hold the iconic stars and stripes — radiation from the sun most likely bleached out all the colors. The result? The flags are probably completely white by now. (Source) The flags were made by Annin Flagmakers. They are nylon and cost NASA very little. Why so cheap? Didn't NASA know that it would eventually fade away? What other material could NASA have used to avoid this radiation effect?


title: Why is the $SO(4)$ symmetry of the Hydrogen atom called dynamical?

text: Why _dynamical?_ My previous quantum mechanics teacher could not answer it.


title: Mysterious magnetic force!

text: If you have a loop in which current goes clockwise as seen from top, then it forms a north pole if seen from below. Now if a particle with positive charge goes from left to right under it, it experiences and upward force, now if the 1st loop was very large and particle goes from near about the axis of loop, it goes in a circle in anti-clockwise direction as seen from the top. This again generates a north pole if seen from top. Q1. Since we have two north poles facing each other, why don't we get a new repulsive magnetic force? If you closely observe this systen then the negative charges travelling in 1st loop in anti-clockwise direction have synchronized with the positive charge going anti-clockwise below it. Not only this, if you send the particle from right to left, it experiences a downward force and again synchronizes, now again if you try sending negative particles they get synchronized in the same way with the positive particles. Q2. Why is such a synchronizing taking place? I am more interested in the first question for now, as I am trying to work out the second problem myself.


title: Decrease in entropy in a fluid flow

text: Let's imagine a section of a pipe through which a fluid, gas for example, flows. When there is no pressure gradient, there is no flow. However, that does not mean that the molecules are at rest. They are "Maxwell-Boltzmann" distributed but the net macroscopic velocity is zero. When the pressure gradient is turned on, the velocity prefers to be (opposite) to the direction of pressure gradient. Do you agree that this is a more ordered state then a gas at rest and therefore has a lower entropy? Is a lower entropy (one of) the cause of flows getting unstable?


title: LSZ reduction theorem derivation in Weinberg QFT

text: When deriving LSZ reduction theorem Weinberg in his QFT book have assumed n-point generalized Green functions, $$ G(q_{1},...,q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}q_{j}x_{j}} \langle |\hat {T}\left( \hat {O}_{l}(x_{1})\hat {A}_{2}(x_{2})...\hat A_{n}(x_{n})\right) |\rangle , \quad (1) $$ where $\hat {O}_{l}(x)$ transforms under the irreducible representation of the Lorentz group as some free field $\hat {\Psi}_{l}(x)$. By insertion between $\hat {O}_{l}(x_{1})$ and $\hat {A}_{2}(x_{2})$ functional unit $$ \sum_{i, \sigma}\int d^{3}\mathbf p | (\mathbf p , \sigma )_{i}\rangle \langle (\mathbf p , \sigma )_{i}| $$ and by allocation of one-particle states from it he have "reduced" (with some hints) $(1)$ to the form $$ G(q_{1},...,q_{n}) \to f(q)\sum_{\sigma}\langle | \hat {O}_{l}(0)| (\mathbf q_{1}, \sigma )\rangle \times $$ $$ \times \int d^{4}x_{2}...e^{-iq_{2}x_{2}-...}\langle (\mathbf q_{1}, \sigma ) |\hat {T}\left( \hat {A}(x_{2})...\right) | \rangle \delta (q_{1} + ... + q_{n}). \qquad (2) $$ Here $f(q)$ contains the pole of the first order $\frac{1}{q^{2} - m^{2} - i\varepsilon}$ and $q = q_{1} + ... + q_{r}$. After that he says that in $(2)$ there is equality $\hat {O}_{l}(0)| (\mathbf q_{1}, \sigma )\rangle = \frac{1}{\sqrt{(2 \pi )^{3}}}Nu^{\sigma}_{l}(\mathbf q_{1})| \rangle $. So I have the question: why was factor $N$ (in comparison with free field-like expression $\hat {O}_{l}(0)| (\mathbf q_{1}, \sigma )\rangle = \frac{1}{\sqrt{(2 \pi )^{3}}}u^{\sigma}_{l}(\mathbf q_{1})| \rangle $) appeared? What is its physical sense? Is its appearance connected with the fact that $| \rangle$ doesn't refer to the "usual" vacuum? Can you also comment this statement, if you please?


title: What is to be considered a "body" in physics?

text: Well, the question says it all; is there a definition of body in physics? What is to be considered a physical object and what it cannot?


title: How can macroscopic $\bf D$ be related to microscopic $\bf E$ just by a constant $\epsilon$?

text: In electromagnetism, the electric displacement field $\bf D$ and magnetic field strength $\bf H$ are macroscopic functions defined in matter, and $\bf E$ and $\bf B$ are microscopic functions. How can the relation $\bf D=\epsilon \bf E$ hold while on one side is a macroscopic quantity and on the other side is a microscopic one?


title: Why a transistor is activated when I touch its base with my finger?

text: Is my body charged eletrically? If so, how much? Will it work with all types of transistors?


title: 2 dimensional Coulomb's law equation

text: We can notice that in the Coulomb's law equation,   $$\begin{equation}\tag{1}F=\frac{1}{4\pi\epsilon}\cdot\frac{q_1q_2}{r^2}\end{equation} $$ $4\pi r^2$ factor in the denominator expresses directly the surface of a virtual sphere with radius $r$. Actually we can look at this equation as it was for $3$ dimensional objects. If we suppose want to consider for $2$ dimensional objects, can we modify the equation as,   $$\begin{equation}\tag{2}F=\frac{1}{2\pi\epsilon}\cdot\frac{q_1q_2}{r}\end{equation}$$   Here we can think of $2\pi r$ as area of virtual circle. I don't really know whether it works or not. So, can we have equation (2) as the modified equation for electrostatic force between two $2$ dimensional uniformly charged objects?


title: How to find maximum velocity

text: I stack a question about projectile question. The question was > _A projectile is being launched from ground level with no air resistance. > You want to avoid having it enter a temperature inversion layer in the > atmosphere a height $h$ above the ground (a) what us the maximum launch > speed you could give this projectile if you shot it straight up ? Express > your answer in terms of $h$ and $g$.(b)Suppose the launcher available shoots > projectiles at twice the maximum launch speed you found in part (a). At what > maximum angle above the horizontal should you launch the projectile ?_ I could solve the (a) part. How did was following (a) using the following formula to drive $V$ $\delta x$ = $\frac{V^{2}-Vi^{2}}{2g}$$ also we have $Vi = 0$, $\delta x = h$ I got $V = \sqrt{2gh}$ after that I think I need to use some kind of angle relative formula to create $arccosx$ or $arcsinx$ will be equal to some number then find the angle but I still don't have idea which formula do I need to use and find max angle. also Do I need to split $Vx$ and $Vy$ from $V$ ? One more question, I have seen some minimum spe


title: Force and Laws of Motion

text: While catching a fast moving cricket ball, why do fielders gradually pull back their hands backwards with the moving ball? Please advice which among the following is the correct answer. If both are incorrect kindly give the correct answer:      **A** : By pulling the hands backward the fielder increases the time during which the high velocity of the moving ball decreases to 0. Thus, the acceleration of the ball is decreased and therefore the impact of catching the fast moving ball is also reduced.      **B** : By pulling the hand backwards, the fielder gradually and continuously decreases the momentum of the moving ball and thereby reducing the velocity to zero.


title: What name would you give to the method of approximating an arbitrary magnet with many smaller dipoles?

text: Let's say I had an arbitrarily shaped permanent magnet, with total magnetic moment $M_{0}$. Ways to calculate the magnetic field of this magnet include an analytic solution (if one exists), as well as finite element modeling. Now lets say I wish to calculate the magnetic field if this arbitrarily shaped magnet by replacing it with $N$ identical magnetic dipoles, each with magnetic moment $\frac{M_{0}}{N}$. I then calculate the magnetic field at a point by considering the magnetic field from each of these $N$ magnetic dipoles. What would you call such a method? I'm sure someone has described this in the literature, I just can't seem to find any references.


title: Is duplication of nucleus, electron and proton (sub-atomic particles) possible?

text: usually a single cell develops into a huge mass to form a substance or an organism. is this happening because of multiplication of sub-atomic particles? if yes, then is it possible to multiply the cell number, as it is usually happening in our body tissue?


title: Full Width Half Maximum (FWHM) of a transmission spectrum

text: I have a transmission spectrum of a material which has been fit to a Lorentzian. According to Wikipedia here and here, FWHM is the spectral width which is wavelength interval over which the magnitude of all spectral components is equal to or greater than a specified fraction of the magnitude of the component having the maximum value. What I understood from that is the FWHM gives an idea of the frequency components which are greater than or equal to the resonant frequency. I am not sure what 'specified fraction' is. I am also confused of whether the FWHM tells me something something about the material or the source?