title: Collision of two balls

text: Two balls, with masses $m_1$ and $m_2$ are falling from the heigh $h$. All the collisions are perfectly elastic. Do not consider the size of the balls. $m_1 < m_2$ and lighter ball is on the top. ![enter image description here](http://i.stack.imgur.com/DZxbV.png) What heights $h_1$ and $h_2$ will the balls jump to? * * * And so I have many equation like: $V(-m_1+m_2)=m_1v_1+m_2v_2$, $(m_1+m_2)gh=m_1gh_1+m_2gh_2$ and $\frac12 V^2(m_1+m_2)= \frac12 m_1 v_1^2 + \frac12 m_2 v_2^2$ (Where $V$ is velocity of both balls right before collision. I consider lighter ball is falling down the same velocity as heavier ball is flying up, so I put opposite sign in the equation of momentum. $v_1$ is a velocity of lighter ball after collision and $v_2$ velocity of heavier ball). But I have no idea how to get $h_1$ and $h_2$ independent of each other. Please don't describe me "you have to do this and this" and "think about what happen when...". I need to see the progress. I need to see these equations and what's happening.


title: why can't you use your cellphone on an airplane?

text: why airplanes are banned with use of cellphones? What were the impacts while we answer an call or make an call ,what was the **physical** reason behind the ban of using cellphone inside airplanes ? I expect answer related to physics here


title: velocity in inertial and nontial frames

text: ![enter image description here](http://i.stack.imgur.com/Pik0X.jpg)![enter image description here](http://i.stack.imgur.com/LL2TD.jpg) I got confused about the difference between the last term of both pictures. In the first one, we have w x r, but in the second we have w x r underlined. Does anyone have a better explanation? They should be the same.


title: Is there an explicit angular momentum in Maxwell equations?

text: Electromagnetism implies special relativity and then the universal constant "c". And if we set c=1, the coupling constant has units of angular momentum (so in relativistic quantum mechanics we divide by $\hbar$ and we get the adimensional coupling $\alpha$). Question, loose, is: In which explicit ways does this angular momentum appear in the classical formalism? Has it some obvious, useful meaning? * * * Edit: some clarifications from the comments. More explicitly, I have in mind the following. In classical no relativistic mechanics a circular orbit under, ahem, a central force equilibrates $F = K / r^2$ equal to $F= m v^2 / r$, and then we have $K = m r v^2 = L v.$ Thus when introducing relativity we can expect that the angular momentum for a particle orbiting in a central force will have a limit, the minimum possible value being $L_{min} = K/c$. Note that this limit does not imply a minimum radius, we also have classically $K = L^2/ mr $, but m can be argued to be the relativistic mass, so when L goes towards its minimum, m increases and the radius of the orbit goes to zero. More edit: Given that it seems that my derivation of the Sommerfeld bound $L_{min} = K/c$ risks to be wrong, I feel I should point out that failure, it it is, is completely mine. The original derivation of this relativistic (not quantum!) bound appears in section II.1 of _Zur Quantentheorie der Spektrallinien_ (pages 45-47 here) and also in Kap 5.2 of his book. The usual argument goes about generic ellipses and its stability properties.


title: Relativistic object impacts the earth

text: A familiar trope within science-fiction is that of a large relativistic object hitting a planet such as the earth. This is normally an interstellar spacecraft or a kinetic weapon with a mass in the range 103 \- 106 kg. But what would actually happen? The two scenarios seem to be: (a) the object creates a kind of conical tunnel through the earth with most of the material exiting on the far side; (b) the object dumps all of its kinetic energy within a few tens or hundreds of kilometres of the impact point and we have the equivalent of a conventional asteroid impact. Light transit time through an earth diameter of 12,800 km is just over 40 milliseconds. There’s not much time for lateral effects as the object barrels in. So what would happen if a 1 tonne object hit the earth, square on, at 0.99995c (γ = 100)?


title: Fermi wavelength of graphene

text: Does anybody know the Fermi wavelength of graphene? I searched the Internet for a while without success. I found, by inspection with the Fourier transform of an S.T.M. image $$ 3.84e^{-10} \mathrm{m}. $$ Is this value of the right order of magnitude?


title: Oort cloud blocking light?

text: Is the Oort cloud blocking a substantial amount of light in the visual spectra, making it harder for observers seeing outside the solar system?


title: The velocity of sound in terms of Fermi velocity

text: Assume this question: In a simple model of a monovalent metal consisting of point positive ion cores embedded in a uniform jellium of electrons, a value for the average energy for electron is: $$E = \frac{9}{10} \frac{e^{2}}{r} + \frac{3\hbar^{2}}{10mr^{2}}\left(\frac{9}{4\pi}\right)^{2/3}$$ where r is the radius of a sphere containing one electron. Find the velocity of sound in terms of Fermi velocity. I'm attempting to solve this question, but I only can find the equilibrium vaule of $r$. I cannot understand the relation between the velocity of sound and this question.


title: What is the black stuff blocking our view of the milky way?

text: I know that it is dust blocking the light. But what is this dust made of, gases or more heavy materials. If it is gas why is it not already drawn together in nebulas?


title: Need help visualising transverse waves

text: From studying waves I find that I can visualise longitudinal waves where the wave propagates in the direction of the displacement. However I don't understand what causes the propagation perpendicular to the displacement in transverse waves. How is a perpendicular displacement caused in adjacent particles?


title: Euler's buckling formula applicable for impact calculations?

text: $$F = \frac{\pi^2 EI}{(KL)^2}$$ Is Euler's buckling formula applicable for impact calculations, considering speeds relevant for a car or aircraft crash? If there is a level where the formula becomes inapplicable or inappropriate in impact calculations, what determines this, and what behavior (and hence other formula) will then be relevant?


title: Difference between propeller and fan in pulling force

text: In the context of producing a pulling force perpendicular to the 'spinning plane' of a propeller/fan,   is it correct to say that a **propeller** mainly achieves it's force by being aerofoils producing lift and a **fan** mainly achieves it's force by utilizing Newtons third law?


States of matter in physics are the distinct forms that different phases of matter take on. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. What is physics behind States of matter? **For Example:** I found a Trick to freeze water in about half a second, in this trick, When I open caps of a plastic bottle of cold water, cold Water freezes in a chain process, You can see the freezing process. What is physics of The process of freezing water?


title: What sense can be made of the natural logarithm in physics?

text: What sense can be made of the natural logarithm, When appearing in a physical process? For example, This integral in the thermodynamic $\int_i^f \frac {dV}{V}=Ln\frac {V_f}{V_i}$ when $V$ denotes Volume. in general $Ln\frac {Q_f}{Q_i}$ when the $Q$ denotes Physical quantities. or this one $S=k_BLn\Omega$ Why sometimes natural logarithm can be interpreted as a physical process? What are the odds of that happening?


title: What was Albert Einstein's proof for $E=mc^2$?

text: Most people know the famous equation: $$E=mc^2$$ What were his steps of thinking for this equation that helped us discover so much about our world?


title: $E=mc^2$ explanation needed

text: My question is pretty standard and easy,I wonder all my life how Einstein get energy equation $E=mc^2$. Also I some videos about that equation on YouTube and some of the peoples said there that the equation $E=mc^2$ isn't full. Any help will be appreciated.


title: How does the uncertainty principle make a photon beam spread out?

text: I'm reading about uncertainty principle, and something has been bothering me for quite a while. There is the formula: $$\sigma_x \sigma_p \ge \frac{\hbar}{2}$$ I know what this means: _the more you know about the position, the less you know about the momentum, and vice-versa._ As an implication of this principle, I see this kind of image: ![image](http://upload.wikimedia.org/wikipedia/commons/thumb/6/62/P_vs_Q_2.svg/640px- P_vs_Q_2.svg.png) And thats my point: as we narrow the hole, we know more about the position, so we are uncertain about... about what? The trajectory? But the state is that **we know more about the position, we know less about the momentum**. So that means: _Momentum = Trajectory_ ? I don't think so, once momentum is mass times velocity. The mass must be constant - I assume - so the velocity must be changing, and that will make the bean spread out after passing through the hole? How is that?


title: Why do meteors explode?

text: A report on the Chelyabinsk meteor event earlier this year states > Russian meteor blast injures at least 1,000 people, authorities say My question is   * Why do meteors explode?    * Do all meteors explode?


title: Compact manifold taken as an Einstein Manifold

text: In Kaluza-Klein theories I often see that the compact space is assumed to be an Einstein manifold, that is, its Ricci tensor is proportional to its metric. So, why is this done?


title: Why should one expect closed timelike curves to be impossible in quantum gravity?

text: From the Wikipedia article, it seems that physicists tend to view closed timelike curves as an undesirable attribute of a solution to the Einstein Field Equations. Hawking formulated the Chronology protection conjecture, which I understand essentially to mean that we expect a theory of quantum gravity to rule out closed timelike curves. I am well-aware that the existence of closed timelike curves implies that time travel is technically possible, but this argument for why they should not exist isn't convincing to me. For one, if the minimal length of any closed timelike curve is rather large, time travel would be at least infeasable. Furthermore, this is essentially a philosophical argument, which is based, at least in part, on our desire to retain causality in studying the large scale structure of the universe. So far, the best argument I've heard against CTCs is that the 2nd law of thermodynamics wouldn't seem to have a meaningful interpretation in such a universe, but this isn't totally convincing. A good answer to this question would be some form of mathematical heuristic showing that in certain naive ways of combining quantum mechanics and gravity, CTCs are at least implausible in some way. Essentially, I'm trying to find any kind of an argument in favor of Hawking's conjecture which is not mostly philosophical. I realize that such an argument may not exist (especially since no real theory of quantum gravity exists), so other consequences of the (non)existence of CTCs would be helpful.


title: What kind of invariants are proper time and proper length?

text: Under the Lorentz transformations, quantities are classed as four-vectors, Lorentz scalars etc depending upon how their measurement in one coordinate system transforms as a measurement in another coordinate system. The proper length and proper time measured in one coordinate system will be a calculated, but not measured, invariant for all other coordinate systems. So what kind of invariants are proper time and proper length?


title: R-symmetry commutator

text: I've seen the claim made several placed; Terning's "Modern Supersymmetry" p. 5 on N=1 SUSY algebra states it as well as anyone: > The SUSY algebra is invariant under a multiplication of $Q_\alpha$ by a > phase, so in general there is one linear combination of $U(1)$ charges, > called the $R$-charge, that does not commute with $Q$ and $Q^\dagger$: > >> $[Q_\alpha,R] = Q_\alpha, \;\;\;[Q^\dagger_\dot{\alpha},R]=-Q^\dagger_\dot{\alpha}$ The first statement is straightforward to see. But (1) Why is there is one linear combination of charges that does not commute? (2) How do we arrive at these commutators? (I imagine that the generators can be rescaled to give the coefficient $\pm1$, but I would like a clearer explanation.) I spoke to a peer who said that the commutation relations could be found in a very general, mathematically heavy treatment of the most general possible SUSY algebra. Is there some easier way to understand?


title: "Imaginary" Perfect Time

text: In the definition (in one spatial dimension) of $\Delta \tau$ there is the relation: $(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2$ which is invariant. If $(\Delta x)^2 > (\Delta t)^2$ then there is the characterization "spacelike." In this case $\Delta \tau$ will be an imaginary number. My question is: what is the intuitive physical or geometric meaning of imaginary in this context? (Just a guess; is it related to hyperbolic functions and the Lorentz transformations?) Thanks


title: Quantum eraser double slit experiment

text: In the quantum eraser double slit experiment, does the photon (or wavefunction) pass through one slit or both slits when different polarizers are placed over the slits?


title: Is it possible to focus the sun in such way?

text: ![enter image description here](http://i.stack.imgur.com/XvaWP.gif) Imagine the sliding part of the mirror is controlled by computer and opens on intervals. Is it possible to increase the power of the beam by making it bounce between the mirrors thus going through the lens and then releasing it resulting in beam with more power ?


The largest Mercury mirror telescope is the Large Zenith Telescope in Vancouver, Canada. When spinning the Mercury is spread out in a layer that is about 2 milimeter thick at every point on the dish.         I want to examine the following simple case: a Mercury mirror located on the equator, spinning counterclockwise. To form a perfect paraboloid surface the Mercury must experience the same gravitational acceleration at every point on the dish. Then the incline of the dish gives rise to just the right centripetal force towards the center of the dish. Given the equatorial location the relevant rotation-of-Earth-effect is the Eötvös effect. Let's distinguish between a 'southern half' and a 'northern half' of the Mercury mirror. The Mercury in the southern half experiences too little gravitational acceleration; it will tend to slide outward, up the incline. Conversely, the Mercury in the northern half experiences too much gravitational acceleration, and it will tend to slide down the incline. To mitigate the distorting effect you could try a tilt of the mirror's spin axis. However, you can counteract the rotation-of-Earth-effect only partially. The magnitude of the effect is proportional to the velocity relative to the Earth, hence the magnitude of the effect is different at different distances to the center of the mirror. A level of tilt that is just right for the outer rim of the Mercury mirror will be too large for the rest. The best you can do is a compromise that minimizes the effect averaged over the entire dish. It seems to me that the above qualitative reasoning is unassailable. While distortion due to the rotation-of-Earth-effect can be counteracted, it cannot be eliminated.         However, Paul Hickson, director of the Large Zenith Telescope, has come to an altogether different conclusion. In a 2001 article titled Eliminating the Coriolis Effect in Liquid Mirrors he states that the rotation-of-Earth-effect can be eliminated entirely. I have difficulty following the steps of Hickson's derivation, so much so that I cannot pinpoint the error that I feel must be there. There is a another article by Hickson on the same subject, published in 2006. Hydrodynamics of rotating liquid mirrors (Unlike the 2001 article not a free download; I don't have that article). In the abstract it is stated that the shape of the mirror is highly sensitive to the tilt of the axis. The question is the title of this post:   Mercury mirror telescopes: is it possible to eliminate the rotation of Earth effect?


title: Slowing down light in an opaque crystal for a whole minute

text: I just read about a team of physicists at the University of Darmstadt, Germany, that managed to completely slow down a beam of light that traveled through an opaque crystal (article here). How is it possible for a beam of light come to a complete stop? In the article they mentioned that they fired a laser at the crystal causing the atoms to go into a quantum superposition. How does this affect the stopping of the light? Also if the uncertainty principle applies to photons (which I do not know if it does), how does this not violate the uncertainty principle if the photons aren't moving?


title: What is the difference between baryonic and gravitational mass?

text: I was reading a webpage on neutron stars, and it mentioned that a neutron star's gravitational mass is about 20% lower than its baryonic mass due to gravitational redshift. I understand the basics of what the terms mean, but I do not see why gravitational redshifting would cause the gravitational mass to be reduced.


title: image method to calculate the surface charge density on a conducting plate

text: There is a charge q at a perpendicular distance z = d from an infinite conducting plate z=0. We use the image method and place -q on the other side of the plate and calculate the field. This field points outwards from the charge q and inwards towards the -q and the field at z = 0 on the x-y conducting plane is $E_{x,y,z=0}=0$ . By this, I mean to say, that if I draw a Gaussian pillbox near the plate, the field points downwards for both the upper and lower sides of the pillbox and thus the closed integral of $E.dA$ goes to zero ? Is that not true ? What I saw that textbooks use the formula $E = \sigma/\epsilon$ and calculate the surface charge density ? How does that work out ? Is it also possible to say intuitively that the net charge induced on the conductor is -q ?


title: Where do mass polarization terms come from in many-body Hamiltonian? Why are they sometimes omitted?

text: This question is about the Hamiltonian for more than one particle (non- relativistic). Griffiths (Introduction to Quantum Mechanics, 2e) seems to imply that it is $\displaystyle H=-\frac{\hbar^2}{2}\left(\sum_{n=1}^N\frac{1}{m_n}\nabla_{\mathbf{r}_n}^2\right)+V(\mathbf{r}_1,\dots,\mathbf{r}_N,t)$, but wikipedia isn't so clear: https://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29#Many_particles. Initially the article quotes that formula, but it quickly gets confusing: > However, complications can arise in the many-body problem. Since the > potential energy depends on the spatial arrangement of the particles, the > kinetic energy will also depend on the spatial configuration to conserve > energy. The motion due to any one particle will vary due to the motion of > all the other particles in the system. For this reason cross terms for > kinetic energy may appear in the Hamiltonian; a mix of the gradients for two > particles:   >  $-\frac{\hbar^2}{2M}\nabla_i\cdot\nabla_j$   >  where _M_ denotes the mass of the collection of particles resulting in this > extra kinetic energy. Terms of this form are known as _mass polarization > terms_ , and appear in the Hamiltonian of many electron atoms (see below). Unfortunately, the author wrote nothing below which might explain where the mass polarization terms come from.   Could I get some mathematical background, maybe a derivation, of why these terms are present in the Hamiltonian, and why they are sometimes omitted/forgotten?


title: Will this achieve linear movement of a closed container?

text: Can I create a device that moves in one direction with a design like Figure 1? I suspect that the forces cancel each other by the recirculation. It is a simple exercise of action and reaction, but I can not figure out the math to explain. Maybe my problem is of type XY problem. My intention is to create a device that produces movement from a **closed container** , something like a black box. (which would be added energy) Figure 1. ![enter image description here](http://i.stack.imgur.com/dt1NI.png)


title: Why does the colour of a thing change when under huge magnification?

text: For instance, this image: ![enter image description here](http://i.stack.imgur.com/6uN1dm.jpg) shows human eyelashes close up. The lashes look green, in fact the whole surface area has a strange tint of green Why is this?


title: De Broglie wavelength of slow moving macroscopic objects

text: I've seen numerous examples where the De Broglie wavelengths of macroscopic objects such as bullets and baseballs have been calculated. However, in each case, the objects are moving fast and the corresponding momentums are large, resulting in tiny De Broglie wavelengths. What happens if you slow the bullet down to almost a stop. It's momentum is now small, so why doesn't it have a measurable wavelength?


title: Atoms and their bonds in an explosion

text: I was watching a video and when the car did blow up I asked to me... what happens with the atoms and their bonds when an object blows up of this way? what is the behavior of the atoms and their bonds when an object is on fire? what is the atomic structure of the fire? thanks in advance :)


title: Asymmetry in magnetic field direction of an electric wire

text: The simplest magnetic field is that of an infinitely long wire with uniform current. It does enjoy radial symmetry about the wire and has the variation as 1/r. To find the direction of the resulting magnetic field you use the right hand grip rule (for conventional current). This rule repeats the experimental fact. But it is a asymmetry. We get a bit less asymmetry calculating the magnetic field for a "anti"wire with positrons. Now we have to use the left hand grip rule. Where this asymmetry comes from?


title: Proof that total derivative is the only function that can be added to Lagrangian without changing the eom

text: So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any. So, if I have a Lagrangian and ADD an arbitrary function of q', q and t in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative? **EDIT** Ok, I feel really dumb now. I guess the most voted answer of the question I posted was kinda wrong. So, I changed my question a little bit: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? This was used in the most voted answer of this other question: Deriving the Lagrangian for a free particle , equation 7. Also, why people only talk about things that change the lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau's mechanics book only consider this kind of change in the lagrangian?


title: Spherical coordinate versor problem

text: I have to calculate   $$ i_\rho \times i_\phi $$ it should be $$ i_\theta $$ but in my notes I have $$ - i _\theta $$ Which one is correct? How can I do this kind of operations without mistakes?


title: Any ket as a sum of eigenkets of a given observable

text: Most quantum mechanics texts include a phrase such as 'any ket can be written as a sum of eigenkets of a given observable'. I have problems with the generalities of _any_ ket. Does this literally mean _any_ ket, or does it mean any ket of the same observable? It seems odd that any ket, say an eigenket of position, should be expressible as a sum of eigenkets of a different observable? Edit: Unless of course a general state is dependent on more than one summation of eigenkets of more than one observable, but the measurement of one of those observables does not depend on the other summations.


title: What are the implications of running a motor well above its rated frequency?

text: Plenty of material on the web will tell you what to expect when you run a 50/60Hz 3-phase induction motor at 60-80Hz, but what if you would like to run it at 140Hz? or 250Hz? Or at any frequency? Currently we have a 50Hz motor which drives a conveyor via a chain arrangement that gears up by a factor of 2.8 (not by design; more by putting together what was readily available). Since our conveyor represents (as we understand it) a constant torque load, would we not be better off eliminating the chain arrangement with its friction losses, and just directly driving the motor at 140Hz? Would the loss of torque at higher frequency be cancelled out by the elimination of the gearing? Is there anything (dangerously) wrong with trying this?


title: What's the difference between running up a hill and running up an inclined treadmill?

text: Clearly there will be differences like air resistance; I'm not interested in that. It seems like you're working against gravity when you're actually running in a way that you're not if you're on a treadmill, but on the other hand it seems like one should be able to take a piece of the treadmill's belt as an inertial reference point. What's going on here?


title: Is the braket notation of the Dirac delta function symmetric?

text: I have a book saying, > $\int \delta(x-x')\psi(x)dx = \psi(x')$ where $\psi(x) = \langle > x\lvert\psi\rangle$, so our definition of delta function would be $\langle > x'\lvert x\rangle = \delta(x-x')$. However I could find some documents (example; refer to 3. Position Space) saying, $$\delta(x'-x'') = \langle x'\lvert x''\rangle$$ which corresponds to $\delta(x-x') = \langle x\lvert x'\rangle$. So the result should be $$\delta(x-x') = \langle x\lvert x'\rangle (=) \langle x'\lvert x\rangle \tag{1}$$ I think neither of them is an error, because my book uses the definition many times and I have found many documents explaining as $\delta(x-x') = \langle x\lvert x'\rangle$. Is (1) correct?


title: Relativistic Mass and Dark Matter

text: In a question here Ron Maimon comments that "relativistic mass makes gravity, not rest mass." If so, does that mean that the faster that stars orbit the galaxy the larger the relativistic mass of the galaxy therefore more gravity therefore the stars orbit even faster therefore the gravity gets even more therefore the stars keep on getting faster ... So the speed of stars and mass of galaxy would be correlated with age and history of the galaxy or galaxy clusters and that's why calculations of galaxies' masses don't add up.


title: Light coming from distant galaxies

text: Some of the distant galaxies appear to be receding from us faster than the speed of light due to stretch of the space between us and those galaxies. By an analogy with the ant on a rope paradox, the light emitted from those galaxies can actually reach us. Is it true that, at one point of time, those galaxies suddenly disappear from our view, and then after sufficiently large times (which may be more than the lifetime of the universe), they come into view again? And by the same reasoning, would more previously hidden galaxies will become visible at some time?


title: Quantum Coin Flipping Protocol

text: $\newcommand{\ket}[1]{\left|#1\right>}$ I have the next protocol:   1. $A$ tosses a fair coin $a\in \\{0,1\\}$, if $a=0$, $A$ sends to $B$ $\ket{\psi_0}=\ket0$, if $a=1$ $A$ sends to $B$, $\ket{\psi_1}=\ket{+}$.   2. $B$ now picks randomly $b\in \\{0,1\\}$.   3. $A$ sends to $B$, the value of $a$. After step 1., $B$ measures the state he gets in the basis $\\{\psi_a, \psi^{\perp}_a\\}$, if he doesn't get $\psi_a$ he wins, otherwise the score is $a\oplus b$, i.e if it's $0$ then $B$ wins, if it's $1$ then $A$ wins. The question is to find the best strategy for $B$ to win if he's dishonest and $A$ is honest, and it's probability, and when B is honest and $A$ is dishonest to show that there exist a constant positive prob for which $B$ can win the game. My attempt at solution is as follows: For the first question: a. Well, as far as I can see $B$ can only cheat on choosing between b=0 or 1, but still he's left with 50/50 chance of winning, I don't see better strategy. b. Here when $A$ cheats and $B$ is honest, if $A$ picks $a = 0 \ or \ 1$, and $\psi$ he sends is different than $\phi_a$, then by the criterion of the game $B$ wins. In case $A$ doesn't cheat $B$ has 0.5 chance of winning the game. So the probability should be (prob. A is dishonest) _(prob B to win)+(prob A is honest)_ (prob B to win)= 0.5*1+0.5*0.5=0.75. But then again, I might be wrong here. :-(


title: kitaev-honeycomb : can't get wilson loop squared to yield +1

text: I'm new here, loving this website and I'm having some difficulty with the wilson-loop operator in kitaev's honeycomb model. **problem statement** The Kitaev model (Kitaev, 2006 is the original paper) consists of spins residing at the lattice sites of a honeycomb lattice with separate nn couplings for the three directions that are identified for the bonds. The wilson-loop operator is $w_p=\sigma^x_1 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6$, where the indices $i \in\\{1,...,6\\} $ indicate the $6$ lattice sites involved in the hexagonal loop (see picture). ![Kitaev honeycomb loop](http://i.stack.imgur.com/aiT0X.png) In Jiannis K. Pachos' book (Introduction to topological quantum computation, 2012) the author states that $(w_p)^2=1$, which I'm trying to find myself. Actually this should be not at all hard, but I'm stuck unfortunately. **attempt at solution** I've tried the following $$ (w_p)^2 = \sigma^x_1 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \sigma^x_1 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \\\ = -\sigma^x_1 \sigma^x_1 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \\\ = -\sigma^y_2 \sigma^y_2 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \\\ = + \sigma^z_3 \sigma^z_3 \sigma^x_4 \sigma^y_5 \sigma^z_6 \sigma^x_4 \sigma^y_5 \sigma^z_6 \\\ = + \sigma^x_4 \sigma^x_4 \sigma^y_5 \sigma^z_6 \sigma^y_5 \sigma^z_6 \\\ = - \sigma^y_5 \sigma^y_5 \sigma^z_6 \sigma^z_6 \\\ =-1 $$ Where I've pulled $\sigma_1$ through first, next the $\sigma_2$, etc. And I've used $\\{\sigma^\alpha_i , \sigma^\beta_j \\}= 2 \delta_{i,j}\delta_{\alpha,\beta} I_2 $ (so that every swapping of unequal $\sigma$'s gives a minus sign and $\sigma^\alpha_i\sigma^\alpha_i =I_2$ ). So I get $(w_p)^2=-1$, which is not what I wanted to find. All text on the subject state that $w_p$ acting on a lattice configuration yield $w_p=\pm1 $ which can be easily concluded from $(w_p)^2=1$ (the expression I didn't get). My guess is that my commutation relations are not correct, but I'm unsure. Who can help me out? A big thanks in advance! Best, L


title: Gravitational attraction of triangles

text: Suppose I have two triangles relatively close together (so they probably shouldn't really be treated as point masses). I want to calculate the gravitational force (and potentially torque?) generated between the two bodies in the 2D plane. For spheres/circles you can just treat them as point masses and go from there, but can you do that for arbitrary triangles (or tetrahedrons in 3D)? I know the answer is probably to do a spatial integral across both triangles, but it's been a long time since I knew how to do that :) The end goal is to be able to compute the gravitational force between arbitrary polygons/polyhedra. I figured decomposing it in to triangles/tetrahedrons would be a good start. ... **UPDATE:** Okay, my multidimensional calculus is a bit rusty, but I think this is a promising direction: Let: $\vec{f} = (a - c) \mu_1 + (b - c) \upsilon_1 - (x - z) \mu_2 - (y-z) \upsilon_2 - (z - c)$ be the separation vector between two points on either triangle, where $a, b, c$ are the vertices of triangle 1, and $\mu_1, \upsilon_1$ are the barycentric coordinates (corresponding to $a$ and $b$) for the point on triangle 1. Likewise for $x, y, z$ and $\mu_2, \upsilon_2$ for triangle 2. Using the barycentric coordinates let's us form the spatial integral to arrive at an answer. So the (linear, non-torque) force between them is proportional to: $\vec{F_G} = \displaystyle\int_0^1 \int_0^{1-v_2} \int_0^{1} \int_0^{1-v_1} \\! \frac{f}{||{f}||^3} \, \mathrm{d} \mu_1 \mathrm{d} \upsilon_1 \mathrm{d} \mu_2 \mathrm{d} \upsilon_2 $ I think this admits a closed form solution, though I'm still wrestling with Mathematica. I'd really be surprised if this integral hasn't been done somewhere before, though.


title: Is the physical design of the human beings the best possible?

text: The stature and design of the human body. Is it possible that the best physical structure belongs to a human being? If not. What are the flaws in it and what improvements could be done? Is there actually a perfect physical design in terms of strength and agility?


title: Maximum range of a projectile (launched from an elevation)

text: > If a projectile is launched at a speed $u$ from a height $H$ above the > horizontal axis, and air resistance is ignored, the **maximum** range of the > projectile is $R_{max}=\frac ug\sqrt{u^2+2gH}$, where $g$ is the > acceleration due to gravity. > > The angle of projection to achieve $R_{max}$ is $\theta = \arctan > \left(\frac u{\sqrt{u^2+2gH}} \right)$. Can someone help me derive $R_{max}$ as given above? I have tried substituting $y=0$ and $x=R$ into the trajectory equation $$y=H+x \tan\theta -x^2\frac g{2u^2}(1+\tan^2\theta),$$ then differentiating with respect to $\theta$ so that we can let $\frac {dR}{d\theta}=0$ (so that $R=R_{max}$), but this would eliminate the $H$, so it won't lead to the expression for $R_{max}$ that I want to derive.


title: How many measurements of a physical quantity should you do to get a representative value / distribution?

text: When we measure a physical quantity multiple times, we can calculate several values which should help us understand the whole measurement better: Standard deviation, variance, median, average, and so on. What Im interested in: How many measurements should I take to get a representative number? Clearly it depends on the variance of the measurable itself... The following equation gives a starting point, but im unsure how to use it. $\sigma$ is the standard deviation, $e$ the accuracy and $z$ a value that has different values for different wanted certainties. $$ n \geq \left(\frac{z\sigma}{e}\right)^2 $$


title: Superfluid $\Leftrightarrow$ Bose-Einstein condensate?

text: On the Wikipedia page on superfluidity one can find the sentence > not all Bose-Einstein condensates can be regarded as superfluids, and not > all superfluids are Bose–Einstein condensates. So I was wondering if someone here can give me an example of a system which is superfluid, but not a Bose-Einstein condensate.


title: Why there is a time gap between the news presenter and the field reporter?

text: This one has been in my mind for years, but I simply can not find any reason for this event. **The Situation** Imagine yourself sitting in the living room while watching television. It's 9 p.m and the news are on. Suddenly the news presenter announce of an event and contacts a field reporter, and start asking him questions or delivers us the exposition for the story. **The Problem** In these cases, most of the times, because that particular event is pretty far away from the studio, there is a time gap between the speech of the news reporter until it gets to the field reporter. Now, naturally you can think - well, thats make sence - the voice, usually travelling through some kind of electromagnetic force by using an antena or a satellite, takes time to travel long distances, thus creating this time gap between it's emerging from the studio and it's arrival to the reporter. And this is where I get to the point - When the field reporter answers back, and start talking, this time gap magically disappears, and you can see that his lips are synchronized with his voice. **The Questions**   * Why is this happening?   * Both the studio and the reporter uses the same method to tranfer their voices? If not, why? If yes, why there is a time gap only in one direction? I hope this question is physics related :)


title: Are black holes (and the universe in general) said to exhibit Ostwald's Ripening?

text: Ostwald's Ripening is a phenomenon where the surface area to volume ratio of droplets causes small particles to shrink until they disappear and for droplets above a certain volume to continuously grow. This is due to mechanisms of evaporation and condensation, which I don't fully understand in this specific case. My first thought upon hearing this is "this sounds like black holes". Hawking Radiation is a mechanism through which a black hole emits matter- energy from its event horizon. The power of this radiation is related to surface area, and in a naive sense, surface area is proportional to mass. This is as opposed to a 2/3rds power in the case of an incompressible drop held together by surface tension, but it's still monotonic and increasing which is what matters for the law to apply. Thus, all black holes are radiating, but small ones are doing so at a much greater rate relative to their mass. There is also a certain cutoff beyond which the temperature of the black hole is lower than the Cosmic Microwave Background (CMB) so the radiative balance can only permit it to grow bigger. I'm surprised to not find any real mention of Ostwald's Ripening and black holes in the same paper. **Has this terminology ever been entertained by physicists?** I think Ostwald's Ripening is a _physical_ law, although I might be trying to apply it in a cross-disciplinary sense. Perhaps it's also valid to ask if this applies for cosmology in a larger sense. If the current thinking is to accept an apparent inevitability of a big rip, would the swelling and consolidation of black holes continue indefinitely, or would a lowering of the CMB due to the universal acceleration cause all black holes to evaporate way in the future? Is this question trivial or non-trivial?


title: Electric shock from a frayed wire

text: Let's say it's a laptop charger not connected to a laptop, and it is frayed near the tip. I am guessing about 20Volts runs through that area. Now if a person touches a bit of plastic below the frayed area, and they receive a painful shock in the finger but not enough to throw them back, about how much percentage of that 20V would they receive? Assume the person was standing barefooted and was reasonably dry.


title: Negative and positive energy and Hawking

text: I don't have any physics background (except the material we did in high school-long time ago).   I was watching a documentary with Stephen Hawking about whether God created the Universe and I could not follow one of the arguments.   **Note:** This is **not** a theological question. I am not trying to determine if the conclusion about God is right or wrong, I am trying to understand the argument from the physics perspective. One of the arguments (at least as much as I was able to understand it) was that: > For each positive energy in the universe (plannets etc) there is an equal > negative energy. So the sum of both adds up to nothing. As a result there is > no need for a God. To make the argument more clear, the following analogy was presented:   A man on a flat area of sand or dirt wants to make a hill. So the man starts to dig and pile up the sand (or dirt) which is increasing and piling up as a hill.   In the end a hill has been created BUT at the same time a hole has also been created taking equal space as the hill in the oposite direction.   In this example it is clear that both do add up to nothing since if you reverse the process then you return to the original flat area. What I can not understand are the following: How can in the analogy and in reality things add up to nothing?I mean in the analogy with the pile of sand(dirt) the hill and the hole don't add up to nothing since you always have the original flat area with sand you started with. So how is this starting state ignored and it is stated that the 2 opposite forms add up to nothing? Why is the original flat area of sand which is a prerequisite for the existence of both the hole and the hill being ignored? If someone could help me understand these, but explain it in layman's terms (as I have no physics background) it would be much appreciated.


title: Symmetries of a Uniform Magnetic Field

text: Simple question. A system with a uniform electric field everywhere in space has translational invariance in the directions perpendicular to the electric field but no translational invariance parallel to it. This system also has rotational invariance in the plane perpendicular to the electric field. What about a uniform magnetic field? Judging from the Hamiltonian $(\vec{p}-q\vec{A})^2/2m$, it would seem that the symmetries depend on our choice of gauge. Choosing a different $\vec{A}$ breaks different symmetries. Is there a most symmetric choice for $\vec{A}$?


title: 'Hole in the clouds' in the Sarychev Peak eruption

text: One of my favourite ever pictures taken from space is a picture of the 2009 eruption of Sarychev Peak (Ostrov Matua island, Japan) taken by an ISS astronaut during a lucky fly-over. ![enter image description here](http://i.stack.imgur.com/O1FfD.jpg) Image Source: Earth Observatory Image of the Day. I heartily recommend clicking through and seeing the animation. I always assumed that the hole in the cloud cover around the island was caused by some sort of shockwave originating in the eruption plume, most probably as a result of increased temperatures evaporating the cloud, or some such. However, upon revisiting the IOTD page, I was surprised to learn that this is not necessarily the case and that the origin of this hole is a matter of some controversy: > **Editor’s note: Following the publication of this photograph, the > atmospheric and volcanic features it captured generated debate among > meteorologists, geoscientists, and volcanologists who viewed it. Post- > publication, scientists have proposed—and disagreed about—three possible > explanations for the hole in the cloud deck above the volcano. > > One explanation is that the hole in the clouds has nothing to do with the > eruption at all. In places where islands are surrounded by oceans with cool > surface temperatures, it is common for a sheet of clouds to form and drift > with the low-level winds. When the cloud layer encounters an island, the > moist air closer to the surface is forced upward. Because the air above the > marine layer is dry, the clouds evaporate, leaving a hole in the cloud deck. > These openings, or wakes, in the clouds can extend far downwind of the > island, sometimes wrapping into swirling eddies called von Karman vortices. > > The other two possibilities that scientists have offered appeared in the > original caption. One is that the shockwave from the eruption shoved up the > overlying atmosphere and disturbed the cloud deck, either making a hole or > widening an existing opening. The final possibility is that as the plume > rises, air flows down around the sides like water flowing off the back of a > surfacing dolphin. As air sinks, it tends to warm; clouds in the air > evaporate. **Has this controversy been settled?** Is there a convincing, accepted explanation for the origin of the hole in the cloud cover?


title: Natural macroscopic amplification of quantum randomness

text: What are some good examples? For example, chaotic systems where you can show that quantum nondeterminism sets the initial conditions.


title: Will the heat flow of Joule heat be different, if the Joule heat is dissipated in a material that has a temperature gradient beforehand?

text: Let us assume one dimensional heat transfer, for example a finite length wire starting at point $0$ and ending at point $\ell$. If the current passes the wire, the Joule heat $I^{2}R$ will be generated and dissipated into the wire and its thermal surroundings. Had the wire had a constant temperature $T$, the half of the power $I^{2}R / 2$ will be passing the left end, the other half will be passing the right end. Will the situation change if the non-zero temperature gradient $\nabla T $ is present before the Joule heating starts? I cannot grasp, which principle has "higher priority" in this case - be it either principle of dissipation of heat which should be considered "a random walk" or the second thermodynamic principle which states that on average more heat will flow from colder to hotter parts. Motivation for this question are heat transfer equations in thermoelectricity. Thank you in advance for any answer of insightful comments!


title: The equivalent electric field of a magnetic field

text: I know that Lorentz force for a charge $q$, with velocity $\vec{v}$ in magnetic field $\vec{B}$ is given by $$\vec{F} =q \vec{v} \times \vec{B}$$ but there will exist a frame of reference where observer move at same velocity with that of charge $q$, so according to him $v=0$. hence he will see no magnetic force is exerted on charge $q$. I have work on this problem for a while and found that the special relativity predicts equivalent electric force will acting upon charge instead. I want to know the relationship between this equivalent electric force and magnetic force. Thanks in advance


title: Can two distinct spatially separated many-body systems in the ground state contain entangled particles?

text: In particular, I am asking if two distinct many-body systems (e.g. system A and system B) separated at some arbitrary distance will necessarily be found to contain entangled particles (such that particle A1 is entangled with particle B1...etc) **_simply as a consequence of them both being forced into a ground state_**? Specifically, can two experimenters go off to separate labs with no preparations made and then force their systems to the ground state and expect to **_later_** find through conventional testing of entanglement and sharing of statistical results that in fact they did share entangled particles? Or will any entanglement only be coincidental? The reason I ask is because I'm trying to find a way in which entanglement can be acheived by two separated labs remotely (without having to meet up..though classical comms would be allowed) since entangled states are so fragile. It is impractical, it seems, to meet up to prepare entanglement in one lab and then try to keep it from falling apart as you migrate to separate labs to take advantage of its utility or to test its sensitivity to distance (I'm assuming there is none, of course). Has this been achieved and if so, how is it achieved? Also, I have read a paper that documents that two distinct **_subsystems_** of a many-body system are entangled when the many-body system as a whole is in the ground state. What I am asking here is that if that can be possible because the particles making up the many-body system do not have identity, why can't it be true of the particles comprising two separate many-body systems? Why are they not equally indistinguishable, if in fact they are not? Thanks


title: How does beta+ decay actually occur?

text: I'm having difficulty in understanding beta plus decay. How can a proton which has slightly less mass than neutron transform into a neutron, positron and neutrino? Form where does the extra mass for neutron,electron and neutrino come? Is it that during the decay some of the binding energy gets converted into mass?


title: Why is the voltage drop across an ideal wire zero?

text: I'm having trouble conceptualizing why the voltage drop between two points of an ideal wire (i.e. no resistance) is $0~V$. Using Ohm's Law, the equation is such: $$ V = IR \\\ V = I(0~\Omega) \\\ V = 0$$ However, conceptually I can't see how there is no change in energy between these two points. It is my understanding that the _electrical field_ of this circuit produces a **force** running counterclockwise and parallel to the wire which **acts continuously on the electrons** as they move through the wire. As such, I expect there to be a _change in energy_ equal to the **work**. Voltage drop is the difference in electric potential energy per coulomb, so it should be greater than $0~V$: $$ \Delta V = \frac{\Delta J}C \\\ \Delta J > 0 \\\ \therefore \Delta V > 0 $$ For example, suppose I have a simple circuit consisting of a $9~V$ battery in series with a $3~k\Omega$ resistor: ![Simple electric circuit of a 3 kilo-ohm resistor in series with a 9 volt battery](http://i.stack.imgur.com/OANVv.png) If the length from point `4` to point `3` is $5~m$, I would expect the following: $$ W = F \cdot d \\\ W = \Delta E \\\ F > 0 \\\ d = 5 > 0 \\\ \therefore W > 0 \\\ \therefore \Delta E > 0$$ Since **work** is positive for any given charge, the _change in energy_ for any given charge is _positive_ \-- therefore the **voltage drop** must be _positive_. Yet, according to Ohm's Law it is $0~V$ since the wire has negligible resistance. Where is the fault in my logic?


title: Sign wrong in angular momentum (Quantum Mechanics)

text: For small angles $\theta$ the rotation along a particular axis $n$ is given by $R(n,\theta)(r)=Id+ \theta (n \times r)+ o(\epsilon)$. Now, the rotation operator in Quantum Mechanics is given by $R(n,\theta)(r)=r-\frac{i}{\hbar} \theta \langle n , L \rangle r + o(\epsilon)$ But if I check this for $n=e_z$ we have: $R(n,\theta)(r)=r+ \theta (e_z \times r)+ o(\epsilon)=r+ \theta (x e_y - y e_x)+ o(\epsilon)$ and $R(n,\theta)(r)=r-\frac{i}{\hbar} \theta l_z r + o(\epsilon)=r-\theta (x \partial_y - y \partial_x)r + o(\epsilon)=r-\theta (x e_y - y e_x) + o(\epsilon)$ so obviously the last two expressions of the last two rows differ in a sign and I do not see why.


title: The orbit of 2010 TK7

text: Having just heard about the asteroid 2010 TK7 in _Trojan asteroid seen in Earth's orbit by Wise telescope_ , I want to know more about its orbit. The BBC article says it moves above and below the ecliptic. I was under the impression that everything orbits in ellipses around the Sun, so what causes this motion? Also the Wikipedia article about it states that it shuttles between Lagrangian points 3 and 4, but just above that that it orbits 60 degrees in front of Earth which seems contradictory? And that its path oscillates? What does that mean and what exactly is a Lagrangian point? You can see animations in _Earth Finds Dance Partner in a Trojan Asteroid_ of 2010 TK7. It is circling the Sun because of the Sun's gravity, but why is it spiraling in its orbit? Does that mean there is an object in the center, around which it spirals? Does it disobey Newton's first law? > A body in motion will continue its motion unless it is opposed by another > force. Also, is there some 3D model or program that can demonstrate its orbit?


title: Why is a new moon not the same as a solar eclipse?

text: Forgive the elementary nature of this question: Because a new moon occurs when the moon is positioned between the earth and sun, doesn't this also mean that somewhere on the Earth, a solar eclipse (or partial eclipse) is happening? What, then, is the difference between a solar eclipse and a new moon?


title: The skin effect and the reflectivity of gold

text: I am simulating a waveguide in COMSOL, a FEM solver. My model looks like this (it is similar to a standard Quantum Cascade Laser geometry): ![geometry](http://i.stack.imgur.com/Ca9Rj.png) Therefore there is a very thin (30nm) layer of gold sandwiched by two regions consisting of Gallium Arsenide ($ \epsilon_r \simeq 13$). When I perform a mode analysis around 3THz (the frequency of my laser, $\simeq 100 \mu m$), the mode most likely to propagate (lowest losses, matching effective mode index) is the one below (E field norm is plotted here): ![mode](http://i.stack.imgur.com/fprKN.png) Now, I would expect this behaviour **if** the gold layer in between was thick enough, because the plasma frequency of the gold, according to the Drude model, is well above 3THz. The absorption coefficient is equal to $$\alpha = \frac{2 \kappa \omega}{c}$$ and the skin depth, defined as $\delta = 2/ \alpha$, is around 40-50nm at this frequency. Therefore, due to the skin effect, the reflectivity of the thin gold layer would be very low and the wave would leak out into the bottom layer. The E field does penetrate the gold, but decays very rapidly. This is the norm of E in the gold layer (0 to 0.03 on the y-axis), zoomed in a lot and the color range adjusted (notice $|E_{max}|>300$ on the original plot): ![gold zoom](http://i.stack.imgur.com/hR5lV.png) This is what I got from the support: > The reflection appears to be the result of the normal flux conservation > boundary condition $Dy_1=Dy_2$. Since in the gold Er is much larger than in > the material, $Ey_1<<Ey_2$. However, the tangential components of the > electric field (which should indeed decay after one skin depth) are almost > continuous thorough the gold thickness. But when I look at the displacement current (y component), I get a clearly discontinuous plot: ![Dy](http://i.stack.imgur.com/CX4tS.png) I got another reply, commenting on this issue: > This discontinuity is attributed to transition between a conductor and a > dielectric. The D1-D2=pho_s condition is derived from the charge > conservation law, J1-J2=dt phos_s, where J1 and J2 are the sum of > displacement and conduction currents. In case that the conductivity in both > domains is zero, you get the electric flux conservation. When passing > through a conductor-dielectric boundary, the induction current becomes > discontinuous and therefor, the displacement current will too be > discontinuous. The displacement current is directly related to the flux > density. Thus you see the jump in the normal flux component. If you plot > emw.Ex or emw.Jy you will see that they are continuous. The different > current components (emw.Jiy, emw.Jdy) and emw.Dy will be discountinuous. If I understand correctly, this just means that there will be charge accumulation on the Au/GaAs interface due to the existence of a Schottky junction. **Am I wrong in my assumptions, am I misunderstanding the skin effect, am I plotting the wrong thing? If not, why am I getting the wrong result?** From the technical point of view, the mesh in the software is small enough to resolve the thin layer, so it can't be the problem.


title: Is the CMB rest frame special? where does it come from?

text: It seems that we are moving relative to the universe at the speed of ~ 600 km/s. This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is "where does the CMB rest frame come from?".


title: Why does a dielectric have a frequency dependent resistivity?

text: This question has come about because of my discussion with Steve B in the link below. Related: Why is glass much more transparent than water? For conductors, I can clearly see how resistivity $\rho\,\,(=1/\sigma)$ can depend on frequency from Ohm’s law, $\mathbf{J}=\sigma\mathbf{E}$. So if the E-field is an electromagnetic wave impinging on a conductor, clearly the resistivity is frequency dependent. In a similar fashion, the frequency dependence of the electric permittivity $\epsilon=\epsilon_0n^2(\omega)$ can be derived through the frequency dependence of the electric polarization and impinging electromagnetic wave (see How Does $\epsilon$ Relate to the Dampened Harmonic Motion of Electrons?).   1. What does it mean physically for a dielectric to have a frequency dependent resistivity from (i) classical and (ii) quantum viewpoints? I am especially interested in the optical frequency range.   2. Can a simple mathematical relationship be derived similar to the frequency dependent resistivity (for conductors) and electric permittivity (for dielectrics)? Thank you in advance for any help on this question


title: Question on the uncertainty principle

text: **The problem statement:** > Measurement detects a position of a proton with accuracy of $\pm10pm$. How > much is the position uncertainty $1s$ later? Assume the speed of a proton > $v\ll c$. **What i understand:** I know that in general it holds that: \begin{align} \Delta x \Delta p \geq \frac{\hbar}{2} \end{align} This means i can calculate momentum uncertainty for the first measurement: \begin{align} \Delta x_1\Delta p_1 &\geq \frac{\hbar}{2}\\\ \Delta p_1 &\geq \frac{\hbar}{2\Delta x_1}\\\ \Delta p_1 &\geq \frac{1.055\times10^{-34}Js}{2\cdot 10\times10^{-12}m \rlap{~\longleftarrow \substack{\text{should I insert 20pm}\\\\\text{instead of 10pm?}}}}\\\ \Delta p_1 &\geq 5.275\times10^{-24} \frac{kg m}{s}\\\ \Delta p_1 &\geq 9.845 keV/c \end{align} **Question:** Using the position uncertainty $\Delta x_1$ I calculated the momentum uncertainty in first measurement $\Delta p_1$. How do I calculate the position uncertainty $\Delta x_2$ after $1s$? I am not sure what happens $1s$ later but is a momentum uncertainty conserved so that it holds $\Delta p_1 = \Delta p_2$? I know that the problem expects me to use the clasic relation $p=m_ev$ somehow but how?


title: Would connecting p-type and n-type semiconductors work as a diode?

text: If we placed `p-type` and `n-type` semiconductors close enough to be touching (see fig. 1), would this arrangement work as a diode? Please explain. ![connecting p-type and n-type semiconductors](http://i.stack.imgur.com/n5UzB.jpg) Fig. 1 - Connecting p-type and n-type semiconductors


title: Water flushed down, water pumped up (in buildings)

text: I live in a tall building (20 floors) on a mountain. Because the water pressure from the water company is not enough, there is a water pump at the last floor which is activated each time someone is using fresh water in his apartment. To me, that's a big waste of energy, and I think that in theory it would be possible to reuse the used water which is going down to pump some fresh water up. I wonder if such device already exist on the market, and I would like to know how such device would be called. Would there be a problem to operate with such a height difference?


title: How can one describe electron motion around hydrogen atom?

text: I remember from introductory Quantum Mechanics, that hydrogen atom is one of those systems that we can solve without too much ( embarrassing ) approximations. After a number of postulates, QM succeeds at giving right numbers about energy levels, which is very good news. We got rid of the orbit that electron was supposed to follow in a classical way ( Rutherford-Bohr ), and we got orbitals, that are the probability distribution of finding electron in space. So this tiny charged particle doesn't emit radiation, notwithstanding its "accelerated motion" ( Larmor ), which is precisely what happens in real world. I know that certain "classic questions" are pointless in the realm of QM but giving no answers it makes people asking the same questions over and over.   * If the electron doesn't follow a classic orbit, what kind of alternative "motion" we can imagine?   * Is it logical that while the electron is around nucleus it has to move in some way or another?   * Is it correct to describe electron motion as being in different places around nucleus at different instants, in a random way?


title: Why do people rule out zeta regularization for renormalization?

text: Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. However, I have not seen anywhere. For example, I do not know why in physics you can use zeta regularization to define infinite products $ \prod _{n=0}^{\infty}a_{n} $, but they do not use zeta regularization for the regularization of divergent integrals like $ \int_{a}^{\infty}x^{m-s}dx $ for any $m$ even though there is a known formula $$ \begin{array}{l} \int\nolimits_{a}^{\infty }x^{m-s} \text dx =&\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} \text dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s} +a^{m-s} \\\ &-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)} (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} \text dx, \end{array} $$ which I think is better than Pauli-Villars or dimensional regularization. Why do people ignore it? Is there a plot against zeta regularization?


title: Off-diagonal terms of the Husimi $Q$ function?

text: The Husimi $Q$ function of a quantum state $\rho $ is defined as $ Q (\alpha)=\langle \alpha \vert \rho \vert \alpha \rangle $, where $\alpha = (x, p) $ is a phase space coordinate and $\vert \alpha \rangle$ is a coherent state. Is the off-diagonal generalization $ Q (\alpha, \beta)=\langle \alpha \vert \rho \vert \beta \rangle $ used for anything? Does it have a name? This is an interesting object because it essentially measures coherence (or decoherence) in the overcomplete basis of wavepackets .


title: How can we calculate heat transfer through hot gas to moving thermic fluid?

text: Heat is to be transfer from $CO_2$ gas to fluid. $CO_2$ is being heated by solar energy. How can we calculate heat transfer from the hot $CO_2$ gas to the moving fluid?


title: Do image charges radiate?

text: Suppose I have a charge moving back and forth above an infinite, grounded, conducting plane. Can I calculate the total radiated power by using image charges? That is, are the scalar and vector potentials the same in the upper- half space for all time for both the image charge "picture" and the standard picture?


title: Nuclear reactions conservation laws

text: I'd want to know the basic rules to apply the conservation laws in nuclear reactions (nuclear fusion, nuclear fission, radioactive decays, etc.) to determine parity and angular momentum of the products. I know that these principles lay deep in quantum mechanics, but I don't need a very thorough explanation, just the rules. I have searched the Internet a lot but I didn't find simple and clear information. I know that this question would probably require a quite long answer, so if you'd post just a exhaustive link about this it would be enough. Thank you


title: A nonintegrable quantum system whose classical limit is integrable?

text: In this discussion: http://chat.stackexchange.com/rooms/4243/discussion- between-arnold-neumaier-and-ron-maimon Arnold Neumaier suggested that there might be a close link between classical and quantum integrability, while I think there are many more classically integrable systems than quantum integrable ones. The reason is that classically integrable systems are easy to make up--- you make up an infinite number of action and angle variables, and change canonical coordinates in some complicated way to x,p pairs, and say this x-p version is your system of interest. But quantum systems don't admit the same canonical transformation structure as classical systems, so there might be systems which have an integrable classical limit, but no real sign of integrability outside of the classical limit. But I don't know any examples! Most of the 1+1d integrable stuff is for cases where the classical and quantum integrability are linked up, for the obvious reason that people are interested in finding integrable systems, not examples where they are not. The reason I think finding an example is not trivial is because the classical integrability guarantees that the motion is not classically chaotic, and that the asymptotic quantum energy states are pretty regular. So I don't think one can look for a counterexample in finite dimensions, where all high enough energy states are permanently semi- classical. But consider a field theory on a lattice in 2+1 dimensions (continuous time). The lattice is so that the dynamics can be arbitrary, no continuum limit, no renormalization. Even if you have an integrable classical dynamics for the field theory, the energy can still dissipate over larger volumes (this isn't 1+1 d), and eventually the classical field will be weak enough that the classical limit is no longer valid, and you see the quanta. This allows the possibility that every finite energy state to eventually leave the semi- classical domain, and turns quantum, and then the integrability is lost. So is there a 2+1 (or 3+1) dimensional lattice scalar field theory where the classical dynamics is integrable, but the quantum mechanical system is not? By saying that the quantum system is not integrable, I mean:   * the many particle S-matrix doesn't factorize or simplify in any significant way (aside from the weak asymptotic relations implied by having a classical integrable limit)   * there are only a finite number of quantum conserved currents (but an infinite number of conserved currents in the classical limit).


title: Is there any significance to the negative sign on this speed?

text: A girl is riding a bicycle along a straight road at constant speed, and passes a friend standing at a bus stop (event #$1$). At a time of $60$ s later the friend catches a bus (event #$2$) If the distance separating the events is 126 m in the frame of the girl on the bicycle, what is the bicycle's speed? $$u = u' + v$$ can be written as: $$Δx/t = Δx'/t + v$$ $$v = Δx/t - Δx'/t$$ $$v = 0m/60s - 126m /60s$$ $$v = -126 m/ 60s$$ $$v = -2.1 m/s$$ Just wondering if the negative holds any significance? I know we're talking about speed which is scalar but how come the calculation gives a negative? Sorry I am just beginning to learn about relativity.


title: Why are cosmic conspiracies considered undesirable in quantum interpretations?

text: Quantum interpretations like superdeterminism, nonlocal hidden variables, etc. are regularly dismissed by the vast majority of physicists because they require "cosmic conspiracies" which can potentially involve anything and everything within the universe, including distant quasars billions of light years away (e.g. from Bell inequality violations), events arbitrarily long ago in the past or far away in the future (e.g. from delayed choice experiments), the "will" of experimenters and their brain states (from complementarity), etc. . This conspiracy has also got to be good enough to escape detection from within. Then, proponents are accused of "paranoia" and as such, not worth debating with any further. Other than premature accusations of "paranoia", are there any good objections against "cosmic conspiracies"? Let me put it this way. Other than "gut feeling" metaphysical _pre_ -judices, what other good objections against cosmic conspiracies are there?


title: How can you test what color different people perceive?

text: If I would show someone a yellow object and ask them, "is this object yellow?"   That person would say "yes". But I could never know if my perception of the color yellow is the same as that other person's.   Because he or she could actually be seeing, what I know to be the color green.   But then tells me that its the color yellow because that has been taught to him or her from young age. So how can you test if people are really seeing the same color? Question closed and can now be found @ http://biology.stackexchange.com/questions/5728/how-can-you-test-what-color- different-people-perceive


title: Is there any evidence that everyone sees colours the same way?

text: We all know that the sky is blue. Grass is green. Blood is red. But, how do we know that everyone sees these colours more or less the same? How does one know that my `green` isn't your `red` and vice versa? In other words: if I would see what someone else sees, would I see the same colours, or could it all be completely different? Note that I'm talking about the perception of colours, not the distinction of colours themselves.


title: When was it established light travels forever?

text: The range of electromagnetic radiation is indefinite. When was that established? Doesn't Hubble's limit have an effect?


title: How to relate second variation to quadratic terms?

text: I am working on a physics problem, but my issue is math-related. My professor skips some steps based on 'intuition' that I lack: In a conservative system, to find out the nature of equilibrium points, we are looking at the potential energy functional, in general form given by $P[f(s)] = \int F(f(s)) ds$. (The problem is about deflection of beams, $s$ being the coordinate of the deflected elastic line.) To me, it makes sense to perturb the function $f(s)$ to $f(s)+\epsilon g(s)$, $\epsilon$ being a small number. Then, $P[f + \epsilon g]$ is an ordinary function of $\epsilon$, so Taylor expansion works: $$ \Delta P = P[f+\epsilon g] - P[f] = \epsilon \frac{dP[f+\epsilon g]}{d\epsilon}\big|_{\epsilon=0} + \frac{1}{2}\epsilon^2 \frac{d^2P[f+\epsilon g]}{d\epsilon^2}\big|_{\epsilon=0} +O(\epsilon^3). $$ Equilibrium requires that the first term on the RHS is zero, and for a stable equilibrium, the potential energy should be minimal, so it is necessary that $$ \frac{1}{2}\epsilon^2 \frac{d^2P[f+\epsilon g]}{d\epsilon^2}\big|_{\epsilon=0} \geq 0 \qquad (1) $$ for all $g$. My problem arises when my professor says the above is more formal and involved, and that it would suffice to just collect powers of $f(s)$ and write the functional as $$ P[f(s)] = P_0 + P_1 + P_2 + \cdots, $$ where $P_i$ has $i$th-order terms in $f$. Then, apparently, equilibrium dictates that $P_1=0$ and a minimum requires that $$ P_2 \geq 0. \qquad (2) $$ Could anyone please show me how to go from (1) to (2)? I have tried a second Taylor expansion, now in $F$, but it confuses me, especially since it is not a variable but a function.


title: Why does a black hole have a finite mass?

text: I mean besides the obvious "it has to have finite mass or it would suck up the universe." A singularity is a dimensionless point in space with infinite density, if I'm not mistaken. If something is infinitely dense, must it not also be infinitely massive? How does a black hole grow if everything that falls into it merges into the same singularity, which is already infinitely dense?


title: Calculating the most probable radius for an electron of a hydrogen atom in the ground state

text: This link describes a method for determining the most probable radius of an electron for a Hydrogen atom in the ground state. It states that : > _The radial probability density for the hydrogen ground state is obtained by > multiplying the square of the wavefunction by a spherical shell volume > element._ When I went to solve this problem myself I multiplied the square of the given wavefunction by the volume of a sphere, which gave me the wrong answer as I know it should be the Bohr radius. When I thought about this problem it seemed reasonable to multiply it by the volume of a sphere rather than the surface area of a sphere (my gut feeling). The link doesn't really explain to me **_why_** it uses a surface area and not a volume of a sphere and I would like some help in getting an intuition for why the surface area is used and gives the right answer and the volume of a sphere does not.


title: The metric expansion of space and Hawking

text: In a Wikipedia article I read that the "Metric Expansion of Space" exceeds the speed of light. If this is true then we must be being disconnected from very remote parts of the universe since gravity can only travel at the speed of light. In his book "A Brief History of Time", Hawking seems to still believe in the big crunch. Well, it does appear that way, as he devotes a lot of space about the vector of time reversing due to the big crunch. But how can the big crunch still be a valid theory if the above statement is true, or is it not true? This question is not about dark energy,although i do appreciate the replies.


title: Why doesn't the no-cloning theorem make lasers impossible?

text: As I understand lasers, you start off with a few photons that are in an identical state, and other photons that are created later tend to have the same quantum numbers due to Einstein-Bose statistics. Isn't each photon that "joins" the group of preexisting ones a clone of the previous ones? Why doesn't this violate the no-cloning theorem?


title: What are some problems with Hugyen's wave model of light?

text: In terms of the different light phenomenon (reflection, diffraction, rectilinear propagation and dispersion), what are some problems with Huygen's wave model of light?


title: Einstein's relation and osmotic pressure

text: How can I derive the Einstein's relation $D=k_{b}TB$, where $D$ is the diffusion coefficient and B is the mobility coefficient, from the concept of osmotic pressure?


title: M-Theory and computer simulation

text: I have a question. Is it possible to perform computer simulations based on the M-Theory? I was looking for such simulators or source codes but I have not found anything. However, M-theory must have a mathematical model that can be saved as a code to be executed by the computer. Something like this might work? Leaving aside the problem of computational complexity.


title: Calculate work done in an inclined plane

text: How can you calculate the work done by a force (of unknown quantity) exerted on a 10kg block on an inclined plane. The force is pointing upwards and parallel to the incline (which is inclined 30 degrees with respect to the horizontal). a. frctionless plane b. coefficient of friction = 0.12 So the forces acting the block are the normal force, its weight, the friction force (for letter (b)), and the force exerted upwards the incline. All are given or can be solved almost instantly except for the force upwards denoted by F. How do I solve this problem? I am not sure what value of acceleration to use in the axis of the incline for F=ma. Sorry I could not provide a diagram for this.


title: Getting first and second time derivative of mean motion of spacecraft - without TLE?

text: The Celestrak website provides information on reading the TLE ( Two-Line Element Set ) format. In `Line 1, Column 34-43 & 45-52` give information about `First Time Derivative of the Mean Motion` and `Second Time Derivative of Mean Motion`. Is there any way to calculate or estimate these two parameters for a spacecraft, if you don't have its TLE at disposal to simply read them from there ? And what do they actually mean ?


title: Bogoliubov quasiparticle current

text: How to calculate the quasiparticle current density starting from bogoliubov - de gennes equation: $$\left( \begin{array}{cc}H_{0} - E_{F} & -i\sigma_{y}\Delta \\\ i\sigma_{y}\Delta^{*} & E_{F} - H_{0}^{*}\end{array}\right)\Psi = \mathcal{E}\Psi$$ I am not assuming that the single particle hamiltonian $H_{0}$ is $x,y,z,$ dependent and has electromagnetic field dependence. How to start? What is the definition?


title: Calculate Mass based on Velocity and Acceleration

text: I am attempting to design a obstacle avoidance system with the Arduino. My position (for now) is going to be stationary. I will be detecting an incoming object and I want to use the below known variables to predict if it will hit me or not. Unfortunately this requires physics and I am weak. I plan on doing the following calculations: Object detected at point 1 = Known Distance and azimuth. Object detected at point 2 = Known Distance and azimuth and speed. Object detected at point 3 = Known Distance and azimuth and speed and acceleration. Now that I have the above values I think I can plot a trajectory but I will need to account for gravity. I can't account for gravity until I know my objects mass. So, given that I have Velocity and acceleration, is there anyway to calculate, taking into account for gravity, a graph of its trajectory in a 2D and 3D space. (Please use 2D for the example, I'll figure 3D out later.)


title: Convective and Diffusive terms in Navier Stokes Equations

text: My question has 2 parts:   1. I just followed the derivation of Navier Stokes (for Control Volume CFD analysis) and was able to understand most parts. However, the book I use (by Versteeg) does not derive it in its entirety. He pulls a lot of results directly from Schlichtling and continues his analysis. I want to understand the derivation in its full form. **Is there any resource other than Schlichtling (My library doesn't have it) for deriving the NV equations in their full form? I would prefer an online (free) PDF or similar.** (I'm not sure my library would have many books on this including the ones discussed here)   2. After the derivation, most books follow it up with : $$\underbrace{\frac{\partial (\rho\phi) }{\partial t}}_{\text{Rate of increase of }\phi} +\underbrace{\text{div}(\rho\phi \vec u)}_{\text{Convective Term}} = \underbrace{\text{div}(\Gamma \text{ grad}(\phi))}_{\text{Diffusive Term}} + \underbrace{S_\phi}_{\text{Source Term}}$$ where $\phi$ is property per unit mass, $\vec u$ is velocity vector and $\Gamma$ is diffusive term (Like viscosity or thermal conductivity). **What I can't understand is the use of the terms "convective" and "diffusive"? What do they mean? What is the physical interpretation of these terms?** Their dictionary meanings seem to exacerbate the situation: `(convection) the transfer of heat through a fluid (liquid or gas) caused by molecular motion.` `(diffusion) The spreading of something more widely or the intermingling of substances by the natural movement of their particles`


title: Potential in the middle of a square

text: > Find the electric potential in the middle of a square with side length $a$ > and charge $Q$. If I put the origin in the middle of the square, for the potential I get: $$4\frac{1}{4\pi\epsilon_0}\int_0^\frac{a}{2}\int_0^\frac{a}{2}\frac{\sigma}{\sqrt{x^2 + y^2}}dxdy$$ ,where $\sigma$ is surface charge density. Then I'd proceed to change to polar coordinates, so I get something like: $$\frac{\sigma}{\pi\epsilon_0}\int_0^{\frac{\pi}{2}} \int_0^{\frac{a}{2\cos\varphi}} dr d\varphi=\frac{\sigma}{\pi\epsilon_0}\frac{a}{2}\int_0^{\frac{\pi}{2}}\frac{d\varphi}{\cos \varphi}$$ which is a divergent integral. What am I doing wrong? I think I can imagine changing the approach in such a way that the limits of integration are different, but what's wrong with this one?


title: How are contact forces electromagnetic forces?

text: How are contact forces electromagnetic? What are they: Coulombic forces between polarized molecules in contact? Or they are the magnetic forces due to the motion of subatomic particles? I also would like to know what the feeling of 'Mechanical Contact' means? First I thought that it only means normal reaction force, but I think that if a sufficiently strong laser beam falls on our hand we will also feel the 'Mechanical Contact' with that beam. Will this be due to the normal reaction force exerted on our hand? If it is so, are electromagnetic waves capable of exerting normal reactions on another body? If yes then owing to which mechanism? In short, what are contact forces? I also would like to know to what extent these forces act (can they act between two protons in contact)?


title: Significance of mc/h constant in Klein-Gordon equaiton

text: The are several ways, in which one can write the Klein-Gordon equation, the most straightforward being probably the following: $$ \hbar^2 \partial_t^2 \psi(x) = (\hbar^2 c^2 \Delta + m^2c^4) \psi(x) $$ However it is possible to use the d'Alembert operator $\square$ and write KG equation like this: $$ (\square+\mu^2)\psi = 0 $$ where $\mu := \frac{mc}{\hbar}$. the $\mu$ in natural units ($c = \hbar = 1$) is obviously just the mass of the particle, however I wonder if is it possible to interpret it somehow in non-natural units (e.g.: $[\mu] = m^{-1}$). I mean something along the lines of $mc^2$, which we can interpret as energy and therefore easily analyze particle collisions in terms of possible masses that appear in such events. Just a soft question, because I don't have a lot of experience with relativistic quantum mechanics.


title: Where does tidal energy come from?

text: Kind of an odd, random question that popped into my head. Tidal energy - earth's ocean movement, volcanism on some of Jupiter's moons, etc. - obviously comes from the gravitational interaction between large bodies. On earth the interactions with the moon are pulling water around the surface, creating some amount of heat due to friction, etc. My question is, where does that energy come from exactly? More specifically, what potential energy source is getting depleted to do that work? Is the earth minutely slowing down in its spin - or are the orbits of earth and the moon subtly altered over time by the counteractive movement and friction of liquids and gasses?


title: How do I tackle this problem WITHOUT focusing the whole system as a "single mass"?

text: **Problem/Solution** ![](http://img215.imageshack.us/img215/6696/problem2f.jpg)! ![](http://img59.imageshack.us/img59/4281/sol2pp.jpg)! In the third FBD, they made the whole system as one object. What happens if I _don't_ want to do that? What if I jsut apply Newton's second Law to the mass M? I tried it out but it didn't work. Could someone point out my mistake? Note that my other force diagram are like the ones in the solutions. Note that $n_1'$ and $n_2'$ are the reaction forces for mass $m_1$ and $m_2$ respectively. ![](http://img528.imageshack.us/img528/799/progk.jpg)! Writing out Newton's Second law for all the other massses For M $\begin{cases} \sum F_x = F - n'_2 = Ma \\\ \sum F_y = n = n_1' \+ Mg + m_2g \end{cases}$ For $m_1$ $\begin{cases} \sum F_x = T = m_1 a \\\ \sum F_y = n = m_1g \end{cases}$ For $m_2$ $\begin{cases} \sum F_x = n_2 = m_2 a \\\ \sum F_y = T = m_2g \end{cases}$ If I eliminate the system of equations, I would get something absurd like $F = (M + m_2)a$ Could someoene please help me nail down this concept? I really appreciate it.


title: Quasicrystals - Projections from higher dimensional regular crystal lattices

text: ## Why are quasicrystals projections from higher dimensional regular crystal lattices? See for example wikipedia: > »Mathematically, quasicrystals have been shown to be derivable from a > general method, which treats them as projections of a higher-dimensional > lattice.« Mathematics aside, is there a physical reason, why this has to be the case?


title: Physical significance of negative temperature

text: I read some answers regarding negative temperatures but I think my question is new. I want to know that what is the physical significance of negative temperature. Suppose I say a body has temperature -2 K. Can I interpret it physically?


title: Geomagnetic reversal?

text: I have always been told that the geomagnetic field acts "as if" there were a bar magnet inside the Earth. I now this is not true, but I feel that knowing what is actually going on would help me understand the answer to the question that I'm about to ask: Why does geomagnetic reversal happen? Is it something gradual or relatively fast? (What is the origin of the geomagnetic field? Can we answer my questions starting from this mechanism?


title: Why is a hard sphere gas correlated?

text: In stat mech we calculated the radial distribution function (a.k.a. pair correlation function) for a classical gas by using perturbation theory for the BBGKY hierarchy. (I could post more details of the calculation if you want, but it is a rather long winded but standard perturbation theory type calculation.) The result we got was $$ g_2 (r) = \mathrm{e}^{-u(r)/T} \left[ 1 + n_0 \int\mathrm{d}^3r'\ f(r')f(|\vec{r}-\vec{r}'|)\right],$$ where $u(r)$ is the interaction potential, $T$ is the temprature, $n_0$ is the density and $f(r)=\mathrm{e}^{-u(r)/T}-1$ is the Mayer function. $g_2$ roughly measures the probability of finding two particles seperated by a distance $r$. $n_0$ is the small parameter of the perturbation theory. If you then apply this result to hard sphere (infinite repulsive potential of diameter $a$), you get this: ![enter image description here](http://i.stack.imgur.com/95oKU.png) Now it makes perfect sense that $g_2$ is zero for $r<a$. Also the asymptote to one at large $r$ is part of the definition of $g_2$, meaning that particles are uncorrelated at large distances. The problem is the peak at $r\sim a$ which implies that you are more likely to find particles clustered together, despite the complete absence of any attractive forces! _Why is that_? Our lecturer seems to think it is because when two particles collide they stop, then bounce, hence spending more time in the vicinity of each other than for an ideal gas. But this seems dubious because perfect hard sphere collisions are instantaneous. I can imagine three possibilities:   * This argument could be formalized as a limit of soft sphere scattering and is the correct explanation of the correlation,   * there is some other (presumably entropic) explanation,   * the correlation doesn't exist - the perturbation theory gives a qualitatively wrong picture (seems unlikely in this case). So what is it?


title: How does a fixed amount of transmitted radio energy supply an unknown number of destinations?

text: I did some maths and physics up to the age of 18, and hold an amateur radio licence. This thing has puzzled me for a while - does reception of an electromagnetic wave imply an interaction with the transmitter? Does it drain some of the transmitter's energy?


title: Degenerate electron gas state

text: What is degenerate in degenerate electron gas state? Why is it called degenerate?


title: What is the equation for the tension on the ends of a cable suspended at different heights?

text: I'm looking for an equation to find the tension on the ends of a cable suspended between two poles (one higher than the other) with no load but the cable itself. ![catenary curve section](http://i.stack.imgur.com/1rkOa.png) I determined that the tension would be different on each end, that the shape of the suspended cable would be a catenary curve truncated at one end, and that the following would be the variables necessary in the equation:   * cable weight   * cable length   * vertical distance between ends   * horizontal distance between ends Some links that could be helpful:   * http://www.scribd.com/doc/12736614/Sag-Tension-Calculation   * http://mysite.du.edu/~jcalvert/math/catenary.htm


title: Heisenberg Uncertainty Principle: Which formula is correct?

text: Some websites and textbooks refer to $\Delta x \Delta p \geq \frac{\hbar}{2}$ as the correct formula for the uncertainty principle whereas other sources use the formula $\Delta x \Delta p \geq \hbar$ . The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error. This is the mentioned derivation: > $\sin \theta = \frac{\lambda}{\Delta y}$ > > _assuming $\theta$ is small:_ > > $\tan \theta = \frac{\lambda}{\Delta y}$ > > _de Broglie equation:_ > > $\lambda = \frac{h}{p_x}$ > > $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$ > > _but also:_ > > $\tan \theta = \frac{\Delta p_y}{p_x}$ > > _equalize $\tan \theta$:_ > > $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$ > > $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y > \Delta y \approx h$ > > $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$ > > $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$ So: Which one is correct and why?


title: Heisenberg's uncertainty principle - Planck's (reduced) constant divided by two or not?

text: The most common form of Heisenberg's uncertainty principle I've seen online is $$ \Delta x \Delta p ~\geq~ \dfrac{\hbar}{2}.$$ However, I also regularly see $$\Delta x \Delta p ~\geq~ \hbar. $$ Sadly, I used the latter one in a project recently and I'm afraid it's incorrect. Obviously the upper one one is true if the latter one is, but why are these two versions used instead of only one of them?


title: Planet in which satellite(moon) and star(sun) appear together once a year

text: With a hypothetical system, where the moon would be always on the opposite side of the planet than the sun, in a way that the moon would only be visible at night on the planet. I don't know if this is possible, but if it was possible how they would behave. For example, I think that if this being possible, maybe at dusk/dawn both moon and sun would appear on the horizon at opposite sides. And the last thing, could this system have a periodic event where both star and satellite would appear entirety above the horizon line. Another thing I am not sure, is if the place at the planet would affect this, but if it doesn't matter I can work with both scenarios. Old question: Is there any way to calculate how a planetary system would behave, where the satellite and the star appear simultaneously in the sky only once a year? I'm writing a novel where this day would be of great importance, but would like to have some facts as realistic as possible, for example the duration of one day and one year for this scenario to be possible.


title: Rigid body/moment of inertia problem

text: I have a homework assignment about rigid body dynamics. Take a disc of radius $r=2m$ with uniform mass density $\rho=1$ $kg/m^2$ in the x-y plane, resting in an inertial frame. At some instant, a force of $F = (0,0,1)kN$ is applied at the point $A=(0,r,0)$. What's the acceleration of the point $A$ at that instant? This should be straightforward computation, but this is new to me and I think I'm making a fundamental mistake somewhere. Could you help me find it? In particular I'm not sure I'm treating the angular velocity $\omega$ correctly. I can assume the equations of motion for a rigid body, as well as the equation for the acceleration of an arbitrary point of the body given the acceleration of the center of mass and the angular acceleration. The disc is just a circle of homogenous mass $\rho = 1$, so $$M = \rho \int_{R=0}^r\int_{\theta=0}^{2\pi} R d\theta dR = \left.\frac{\rho 2\pi R^2}{r}\right|_{R=0}^2 = \left.\pi R^2\right|_{R=0}^r = 4\pi~\mbox{kg}$$ where though the the problem gives $r=2$ I'm just going to leave it as symbols. Without integration, I suppose we might have noticed that $M = \rho \pi r^2$. Now we compute the tensor of inertia, $$\begin{pmatrix} \int_V \rho x_2^2 +x_3^2 dv & -\int_V \rho x_1 x_2 dv& -\int_V \rho x_1x_3dv \\\ -\int_V \rho x_1 x_2dv & \int_V \rho x_1^2 +x_3^2dv & -\int_V \rho x_2 x_3dv \\\ -\int_V \rho x_1 x_3 dv& -\int_V \rho x_2 x_3dv & \int_V \rho x_1^2 +x_2^2dv \end{pmatrix}$$ Since our object is symmetric about the x-y axes, we can eliminate the cross- moments. Since our object has no mass in the z direction, we can eliminate $x_3$ in the tensor, so \begin{align*} I &= \begin{pmatrix} \int_V \rho x_2^2 dv & 0& 0 \\\ 0 & \int_V \rho x_1^2 dv & 0 \\\ 0 & 0 & \int_V \rho x_1^2 +x_2^2dv \end{pmatrix} \\\ &= \begin{pmatrix} \rho \int r^3 sin^2\theta drd\theta & 0& 0 \\\ 0 & \rho \int r^3 cos^2\theta drd\theta & 0 \\\ 0 & 0 & \rho \int r^3 dr d\theta \end{pmatrix} \\\ &= \begin{pmatrix} \frac{\rho}{2} \int r^3 (1-\cos(2\theta)) drd\theta & 0& 0 \\\ 0 & \rho \int r^3 (1+\cos(2\theta)) drd\theta & 0 \\\ 0 & 0 & \frac{\pi \rho r^4}{2} \end{pmatrix} \\\ &= \begin{pmatrix} \frac{\rho}{2}\left[ \frac{2\pi r^4}{4} - \frac{1}{2} \sin(4\pi)\right] & 0& 0 \\\ 0 & \frac{\rho}{2}\left[ \frac{2\pi r^4}{4} + \frac{1}{2} \sin(4\pi)\right] & 0 \\\ 0 & 0 & \frac{\pi \rho r^4}{2} \end{pmatrix} \\\ &= \begin{pmatrix} \frac{\rho \pi r^4}{4} & 0& 0 \\\ 0 & \frac{\rho \pi r^4}{4} & 0 \\\ 0 & 0 & \frac{\pi \rho r^4}{2} \end{pmatrix} \end{align*} The angular momentum vector should be $\begin{pmatrix} \omega_x,& 0,& 0 \end{pmatrix}^T$, taking the cross product of the force vector and the vector from the center of mass to $A$. Now we plug this in to the equations of motion to solve for the linear and angular acceleration, $a_c$ and $\alpha$. They are, \begin{align*} (Mv_c)' &= S \implies & a_c = \frac{1}{M}\begin{pmatrix} 0,& 0,& F \end{pmatrix}^T \\\ (I\omega)' &= M_c = \tilde r \times F \implies & \dot \omega = \frac{r}{I_{11}} e_y\times |F|e_z = \frac{r|F|}{I_{11}} e_x \end{align*} We should now be able to plug this into the equation for angular acceleration (where the angular velocity at the instant of impulse is $0$ so we ignore it), \begin{align*} a_A &= a_c + \alpha \times e_y \\\ &= \frac{|F|}{M}e_z + \dot \omega \times(r e_y) \\\ &= |F|(\frac{1}{M} + \frac{r^2}{I_{11}} )e_z \\\ &= |F|(\frac{1}{\rho \pi r^2} + \frac{4}{\rho \pi r^2} )e_z \end{align*} Which I guess seems natural enough. But by newton's formula we would have \begin{equation*} M a_A = 5 |F| \end{equation*} But does it make sense that the force should be $5F$? Not really...


title: Graviton polarization in higher dimensions

text: It's not difficult to see that the graviton in $D$ spacetime dimensions has $(D-3)D/2$ polarizations. In $D=4$ there are two $\epsilon^{\pm}_{\mu\nu}$. What I find curious is that in $D=4$ I can actually pick $\epsilon^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\mu}$ where $\epsilon^{\pm}_{\mu}$ are the two polarizations (of definite elicity $\pm1$) for a massless spin-1 particle like the photon. In higher dimension this doesn't seem possible since the photon has $D-2$ polarizations so that the number $(D-2)(D-1)/2$ of $\epsilon^{\lambda}_{\mu}\epsilon^{\lambda^\prime}_{\mu}$ pairs doesn't match the number $(D-3)D/2$ of graviton polarization. Well, unless somehow I consider only a smaller subset of them, say adding a constraint or removing one of them $$(D-2)(D-1)/2-1=(D-3)D/2$$ as in $D=4$ where $\epsilon^{+}_{\mu}\epsilon^{-}_{\mu}$ is discarded having zero elicity. Is there an analogous constructions for $\epsilon^{\lambda}_{\mu\nu}$ in terms of the spin-1 polarizations $\epsilon^\sigma_\mu$ in higher dimensions? I suspect that something similar may happen only if I involve polarizations of higher elicity as well.


title: Clarification on "central charge equals number of degrees of freedom"

text: It's often stated that the central charge c of a CFT counts the degrees of freedom: it adds up when stacking different fields, decreases as you integrate out UV dof from one fixed point to another, etc... But now I am puzzled by the fact that certain fields have negative central charge, for example a b/c system has $c=-26$. How can they be seen has negatives degrees of freedom? Is it because they are fictional dof, remnant of a gauge symmetry? On their own, they would describe a non unitary theory, incoherent at the quantum level.


title: Proof of Gauss' Law

text: How would you prove Gauss' law for an asymmetrical closed surface? I can find it for symmetrical surface but couldn't for Asymmetrical surfaces.


title: How to prove Gauss's law div(E) = rho/epsilon from Coulomb's law?

text: As we know from coulomb's law that: $$\vec{E} = \frac{q}{4\pi\epsilon R^2} \hat{R}$$ using the above equation, how can I verify that: $$\vec{\nabla}\cdot \vec{E}=\frac{q}{\epsilon}$$ I have tried to do it a home, but couldn't figure it out. Explicitly showing all steps would be very much appreciated.


title: Gauss law from Gauss divergence theorem

text: Apply Gauss divergence theorem to the gravitational field due to a spherical object of mass M and uniform density located at origin. Obtain Gauss law for gravitation in integral and differential forms.


title: Generic features, semi-heuristic derivations

text: Einstein was able to make some predictions before GR was fully formulated. e.g. gravitational time dilation. Such predictions before the full theory are referred to as "semi-heuristic derivations" here: http://motls.blogspot.cz/2012/09/albert- einstein-1911-12-1922-23.html and have also been called "generic features" by Smolin I think about loop quantum gravity although I can't find the right reference right now. What such phenomena are predicted today about theories that we don't yet have in full form ?


title: Is it true to say *Space time curvature* $\Leftrightarrow$ *Matter*

text: Is it true to say Space time curvature and Matter are just the same thing, part of the same coin and that therefore _Space time curvature_ $\Leftrightarrow$ _Matter_? In other words is _Space time curvature_ just _Matter_ and _Matter_ just _Space time curvature_?


title: how do they rip one single atom from something?

text: this is simple. what i actually want to ask is, when they do the subatomic particle collision experiments, how do they produce one single subatomic particle, e.g proton, neutron? how do they rip one single electron or photon off something? where do they store the single particle? still, how do they keep the subatomic particle on track, on the designed trajectory? even further, how do they make sure 2 such small particles to have a head-on collision? it's really a long shot, for they are incredibly small. how do they aim?


title: Understanding a paper: What is the meaning of $b_0$?

text: I am looking at this paper (Multicoated gratings, J. Opt. Soc. Am., 1981) and I am getting confused around equation 22. I do not completely understand where he comes up with the equation $$\xi^j_q=b_q^j(R^{-1}V_q^j)\qquad \text{(22)}$$ And then what is the meaning of the $b^{j+1}$. I initially thought they were the eigenvalues of the T matrix he defines, but all he says is that it is a vector of components $b^q_j$. This includes the $\bf b^0$ and $\bf b^{Q+1}$ which I do not see how they are vectors?


title: Does current flow back to the source through earth?

text: We know that if Single Line to Fault occurs, then fault current flows to the earth. I want to know whether the current will return to the source or not. For the current to flow we need a closed path. How it is possible for current to flow in a open circuit if you say that current does not return to the source.