title: Why is (von Neumann) entropy maximized for an ensemble in thermal equilibrium?

text: Consider a quantum system in thermal equilibrium with a heat bath. In determining the density operator of the system, the usual procedure is to maximize the von Neumann entropy subject to the constraint that the ensemble average of the Hamiltonian has a fixed value. **What justifies this assumption?** Sakurai, in his QM text, writes > To justify this assumption would involve us in a delicate discussion of how > equilibrium is established as a result of interactions with the environment. I'd appreciate if someone could shed some light on this. References are welcome as well. I've heard the suggestion that a thermal equilibrium ensemble is simply defined by that density operator which solves the constrained optimization problem above. If this is the case, then why are real physical systems that are in weak contact with heat baths for long periods well described by such mathematical ensembles, and how would one identify the Lagrange multiplier $\beta$ that arises with inverse temperature of the heat bath?


title: Topological Insulators: is HgX a special case?

text: I got confused by reading this article: > Francois Virot, Roland Hayn, Manuel Richter, and Jeroen van den Brink. > “Engineering topological surface-states: HgS, HgSe and HgTe.” arXiv preprint > arXiv:1306.0999 (2013). The article says that topological surface states include contibutions from dangling bonds. And later, passivating the dangling bonds, they actually open a gap in Dirac cones present in surface. ![edge states for HgS and HgS+H \(passivated\)](http://i.stack.imgur.com/UxOtS.png) I get confused whether this materials are classified as TI just because they are insulators in their bulk and conductors in surface or are them TI as any other. So, my questions are:   1. Are there two definitions of TIs, one for usual TIs, and another (wider) including HgX?   2. Are these (HgX) topological insulators a result of strong spin-orbit coupling and time-reversal symmetry?   3. Weren't surface states of a TI topologically protected? Has HgS stopped being a TI by passivating its dangling bonds? (a gap has appeared) I've found these related topics, but still unclear:   1. Would HgTe be a topological insulator?   2. How Fundamental is Spin-Orbit Coupling to Topological Insulators?   3. Is edge state of topological insulator really robust？


title: Converting heat into energy

text: I'm currently building a custom desk. In this desk I will also build-in a small part with a custom electronic panel to provide power to USB devices. As my gaming pc generates a lot of heat and the panel will generate some too, I was thinking about doing something with this heat. I read another post on this physics part of stackExchange where they say it isn't efficient and you would need a alot of cold air supply. As I live in Belgium, most time of the year there is enough of cold air, just outside of my window. Another thing is, it doesn't have to be super efficient. If I could just generate enough current to give power to one or more USB connectors or store it in a battery, I would be happy. Is it possible to create enough energy for my USB, if I got a hot air flow of 50°C and a cold air reservoir of 0-10°C? Follow up question, is it practically possible? USB works at 5 Volt at a current from 0.5 to 5 Amps. Other (possible?) options I found was using thermocells and thermocouples.


title: In QFT, why does a vanishing commutator ensure causality?

text: In relativistic quantum field theories (QFT), $$[\phi(x),\phi^\dagger(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0$$ On the other hand, even for space-like separation $$\phi(x)\phi^\dagger(y)\ne0.$$ Many texts (e.g. Peskin and Schroeder) promise that this condition ensures causality. Why isn't the amplitude $\langle\psi| \phi(x)\phi^\dagger(y)|\psi\rangle$ of physical interest? What is stopping me from cooking up an experiment that can measure $|\langle\psi| \phi(x)\phi^\dagger(y)|\psi\rangle|^2$? What is wrong with interpreting $\langle\psi| \phi(x)\phi^\dagger(y)|\psi\rangle \ne 0$ as the (rather small) amplitude that I can transmit information faster than the speed of light?


title: Given Velocity formula and told to find acceleration, I need help please.

text: ![Phys Problem, ](http://i.stack.imgur.com/e6LmR.png) I tried to do this problem by taking the derivative and it did not get me the right answer. I also tried the Vdv=Ads integration and could not get the right answer. Please Help


title: Which units has the relation $E=mc^2$?

text: Fast everyone knows the relation $E = mc^2$ but I still ask me, which units the relation has. $c$ is the velocity of light. It means, a constant, isn't it? We therefore just have the relation $E = \text{constant} \times m$ Ok, nice to know the mass and the energy are directly proportional. But the constant is only a constant; with another unit system, it would be another one... What is wrong to my reflection?


title: Entropy and Crystal Growth

text: I was reading about growing single crystals and I'm a little confused about this - In most crystal growing processes, a "seed crystal" is used, and the rest of the material crystallizes _on_ the seed crystal (for example the czochrlaski process). But thinking about entropy, wouldn't it be more favourable for the vapour/liquid to not crystallize and hence maintain a larger disorder? A similar process occurs in supersaturated vapour as well, where a small liquid droplet acts as a "seed" for more condensation to occur. So I suspect I'm missing something, but I can't figure out what.


title: Can the speed of an electromagnetic wave be measured in the absence of neutrinos?

text: Let me explain better: from what I understand neutrinos are so pervasive they are literally everywhere. And since they have such a tiny electric charge they barely interact with anything and cannot be "removed" or "shielded" from an area in order to take a measurement of light speed in their absence. However, all three neutrino flavors do have some charge (see neutrino oscillation) and do interact to a very small degree. So, my question is: if we can only ever measure the speed of light in the presence of neutrinos, could it be that the limit of the speed of light is actually the "resistance" of the neutrinos to the energy passing through it?


title: Is it possible to calculate chromacity coordinates using absorbance values?

text: I have absorbance values for blood plasma. Absorbance was measured in range of 380 – 600 nm every 5 nm ending up with absorbance values for 45 wave lengths. I was browsing through literature, and I came across chromacity coordinates (XYZ) and this seems to be a very convenient way of summarizing data about color in just two data points. I have been reading some literature about different colorimetric systems, but since my background is in genetics, it didn’t make much sense for me. Therefore, I would like to ask you if it is possible to calculate these chromacity coordinates using my absorbance values and if so does anyone now some kind of software for this or exact formulas on how to do it?


title: Work done by friction

text: Suppose we have a block of mass $M$ and we are moving it up a curve, very slowly ($a=0$). The surface is not smooth, and coefficient of friction is $\mu=\mu_s=\mu_k$. To move the block we apply a force $F$ to the block tangential to the surface(on which the block is, at that time). If we take the angle of slope to be $\theta$ (of the surface), then the forces acting are friction$f$, gravity/weight $Mg$ and external force $F$. ![diagram](http://i.stack.imgur.com/38sAY.png) This is where the problem comes. If we want to find out the work done by friction, then $$W_{friction}=\int\vec{F_{friction}}.\vec{dr}=-\mu Mg\int dr cos\theta=-\mu Mg\int dx$$ But when I use the work-energy theorem, then $$W_{friction}+W_{gravity}+W_{ext}=0$$ $$W_{friction}=-W_{gravity}-W_{ext}$$ $$W_{friction}=Mgh-F\int dr$$ When I get the work done by friction using integration then it tells me it doesn't depend on the type of path and path length, but only horizontal $x$ covered. When I use the work-energy theorem, it tells me that it depends on height $h$ till which the block goes and path length $r$. What is going wrong? Could anyone hint me? Thanks in advance.


title: Intuition behind classical virial theorem

text: I am continuing to brush up my statistical physics. I just want to gain a better understanding. I have gone through the derivation of the classical virial theorem once more. I have thought about it, googled it and slept about it. The statement: $\langle x_i \frac{\partial \cal H}{\partial x_j} \rangle= kT \delta_{ij}$ is still just counter-intuitive to me. So I am at a fixed position in phase space and I'm looking at my Hamiltonian. Then I step away from my current position and watch how the Hamiltonian changes and multiply that knowledge by how far away from my initial position I have moved. I do this a lot in a random way and then I take an average. Et voilá, I have arrived at the **equilibrium temperature** of a system. Right now this is _just some math_ **to me** (which I totally get) to calculate the temperature of a system of particles in thermal equilibrium. Is there more to it? Am I not getting it? What is the intuition behind this?


title: Please explain this statement about Lorentz transformations

text: I'm reading Sternberg's Group Theory and Physics. I have a question about chapter 1.2 Homeomorphisms. * * * Background: A Lorentz Metric is defined as $||{\bf x}||^2=x_0^2-x_1^2-x_2^2-x_3^2$ And a Lorentz Transformation $B$ as one which satisfies $||B{\bf x}||^2=||{\bf x}||^2$ Identify ${\bf x}$ with $x=x_0I+x_1\sigma_1+x_2\sigma_2+x_3\sigma_3$ where $\sigma_i$ are the Pauli matrices. We have $x=x^*$ ($x$ is self-adjoint) and $\det x=||{\bf x}||^2$ Let $A$ be any 2x2 matrix and define the action of $A$ on $x$ as $x \to AxA^*$ and denote the corresponding action on the vector ${\bf x}$ as $\phi(A){\bf x}$ $AxA^*$ is also self-adjoint and $\det AxA^*=(\det A)^2\det x$ Therefore if $A \in SL(2,\mathbb{C})$ then $\phi(A)$ is a Lorentz transformation and also a homeomorphism. * * * Ok so far, here begins my question. The next bit of the text says: suppose $A \in SU(2)$, then $AIA^*=I$ so if ${\bf e}_0 = \left( \begin{array}{c} 1 \\\ 0 \\\ 0 \\\ 0 \end{array} \right)$ then $\phi(A){\bf e}_0 = {\bf e}_0$ I see why this is so. Now here's the bit I don't understand. If a Lorentz Transformation $C$ satisfies $C{\bf e}_0 = {\bf e}_0$ then $C$ also carries the space of vectors $\left( \begin{array}{c} 0 \\\ x_1 \\\ x_2 \\\ x_3 \end{array} \right)$ onto itself. I don't see why this is so. Can someone please explain the logical steps which I'm missing? Thanks.


title: What is $\gamma$ in the damping equation?

text: $x''+\gamma x'+w_0^2x=0$ That is the general equation for damped harmonic motion. What is the term or name that describes $\gamma$? Is it called the damping constant? I know its the ration between the resistive coefficient $b$ and mass of the system $m$ but what do we actually call it ?


title: Parity and Particle Exchange Operators

text: I would be grateful if somebody could help me brining the **parity operator** and the **particle exchange operator** together. Suppose, there is a two-proton system, where one proton is sitting at $+r$ on the $x$-axis, one is sitting at $-r$, as shown in the figure. ![Two-proton system](http://i.stack.imgur.com/tKGa9.png) In this situation, the parity operator $\hat P_{\text{pa}}$ should be the same as the particle exchange $\hat P_{\text{ex}}$ operator, right? Therefore, I will denote both operators just by $\hat P$. $$ \hat P :\equiv \hat P_{\text{pa}} = \hat P_{\text{ex}} \tag{1} $$ ## Particle Exchange Since this is a **two-fermion system** , the combined state should be anti- symmetric under the exchange of the two protons: $$ \hat P \, |\Psi_{12}\rangle = p_{12}\, |\Psi_{12}\rangle, \ \ p_{12} = -1 \tag{2} $$ ## Parity The **intrinsic parity** of each proton is $+1$. $$ \hat P \, |\Psi_i\rangle = p_i \, |\Psi_i\rangle, \ \ p_i = +1, \ \ i \in \\{1,2\\} \tag{3} $$ Since parity is multiplicative, the parity of the two-proton system should be the product of the single parities: $$ p_{12} = p_1 \cdot p_2 = (+1) \cdot (+1) = +1 \tag{4} $$ ## Why the different sign? Apparently, the parity of the two-proton system is $p_{12}=+1$, whereas the particle exchange considerations lead me to $p_{12}=-1$. I could not resolve this by considering the spin as well. Could someone give me a pointer what I am doing wrong here? Thank you!


title: Buckyballs in vacuum

text: I've read about the idea that buckyballs and other nanostructures could be used to hold drugs and things until they reach certain places in the body and then get released. So I was wondering, if you created a buckyball in air, so that some molecules that are in air (such as oxygen and nitrogen) were inside the buckyball. if you then put the buckyball in a vacuum, would the air be able to escape, or would it be trapped in the buckyball. (And vice versa, a buckyball created in a vacuum put into the atmosphere, would any molecules fit through into the buckyball?)


title: How does a photon travel through glass?

text: This was discussed in an answer to a related question but I think that it deserves a separate and, hopefully, more clear answer. Consider a single photon ($\lambda$=532 nm) traveling through a plate of perfect glass with a refractive index $n=1.5$. We know that it does not change its direction or other characteristics in any particular way and propagating 1 cm through such glass is equivalent to 1.5 cm of vacuum. Apparently, the photon interacts with glass, but what is the physical nature of this interaction? Let's not consider high-order effects like Rayleigh/Raman scattering.


title: Why CAN we see the new moon at night?

text: I understand that the Moon's phases are determined by its position in orbit relative to the Sun. (See: _Full Story on the Moon_ ). The "shadow" is not cast by the Earth (a common misconception - this is actually a lunar eclipse), but by the moon's body itself. It would appear that, in order for the moon to be a new moon, it would have to be somewhere between the Earth and the Sun. However, it would seem that, if we were looking at a new moon, we'd necessarily be looking at the Sun as well. Why, then, can we sometimes see a new moon at night? Why doesn't it vanish at night for a half-month every month, between its last/first quarters? EDIT: Here is a picture of a new moon at nighttime, for visualization purposes (and so nobody has to comb through Twilight pictures).


title: Does car have more kinetic energy when turning?

text: I asked this Phys.SE question Does car lose kinetic energy when turning? Assume a car turning without losing its speed by holding to a point by a rope. IMO, while the car is turning, its kinetic energy is constant $E=(1/2)mv^2$, but it has additional rotation energy as it changes it direction $E=(1/2)Iw^2$ (while turning) I feel this has something wrong, but I don't know. So whether or not the car has more energy when turning?


title: If the universe were compressed into a super massive black hole, how big would it be?

text: I understand only a little of general relativity, but that's why I'm here! :) Consider the hypothetical situation of some extra-terrestrial intelligence pushing all the mass in the universe, every last subatomic particle, into a giant black hole. What is the surface area of such an object? For the sake of argument, let's keep this very idealized. Dark energy, whatever it is, can be captured and thrown into the SMBH; assume there's no Hawking radiation causing it to slowly evaporate, etc.


title: Rocket engines design & materials

text: Does anyones know which materials (or type of steel) are used to make rocket engines nozzles & combustion chambers? What are the tricks to reduce temperature of the engine except trying to cool it with fuel/oxidant? Are there any good books around on rocket engines design?


title: Creating the opposite of an optical lattice

text: Is it possible to create periodic potentials that instead of creating a well for an atom to be trapped in, repulsed by that specific location? If yes, can we use this as a means to make artificial objects that we can feel, with let's say our hands, by putting lots of these repulsive points together? This is an idea that I have in order to make holograms that we can feel like the holodeck in Star Trek.


title: Break speed of light with infinite mass

text: I've heard that a spacecraft could never exceed the speed of light because it's (relativistic) mass quickly approaches infinity and therefore there could never create a big enough rocket to propel it faster and faster. I'm 99% sure this is a stupid question, but I'll ask it anyway :). Theoretically if a spacecraft had unlimited fuel (by travelling through a nebula picking up hydrogen/oxygen and whatever else it needed as a propellant), couldn't it actually break the speed of light? Yes I know it gets harder and harder to accelerate the craft because it gets more and more massive (relativistically speaking), but as the craft and propellant tanks get more and more massive so does the thrust right? 1 million pounds of propellant exploding out the back gives 1 million pounds of thrust at Newtonian speeds. As the spacecraft gets heavier and heavier, the thrust would also get more and more massive would it not? In that way more massive mass is countered by more massive thrust. If not, **_why is the reaction mass any different to the rest of the spacecraft's mass_**?


title: Mechanics + Thermodynamics: Bouncing Ball

text: In preparation for an exam, I'm revisiting old exam questions. This one seems neat, but also quite complicated: > A soccer ball with Radius $R=11cm$ is inflated at a pressure of $P =9 \times > 10^4 Pa$, then dropped from a height of $0.1m$ (distance floor to lowest > part of ball) onto a hard floor and bounces off elastically. > > **Question** : Find approximate expressions for: > >   * Surface area of ball in contact with floor >   * Amount of time the ball is in contact with the floor >   * Peak force exerted on the floor > > > if the mass is $0.42 kg$. My attempt at a solution: Assume the ball is filled with an ideal gas and that the process is adiabatic. Assume that the deformation leads to a simple spherical segment, i.e. a ball where one part is cut off flat. This gives an expression of the volume $V$ in terms of the height of the "center" of the ball, $h$ as $$V = \frac{11}{6}\pi R^3 + \frac{\pi}{6}h^3$$ The surface area then is simply $$A = \pi (R^2 - h^2)$$. Next: The ball has potential energy $mgh_0$, which is completely converted to internal energy at the peak point of the motion. The internal energy of an ideal gas is $$U = 3/2 N kT = 3/2 PV = 3/2 P_0 V_0^\gamma V^{1-\gamma}$$ where $\gamma$ is the adiabatic coefficient (1.4 for air). The _change_ in $U$ due to a changing $V$ then comes completely from the initial potential energy $E$. Some algebra and some sensible binomic approximations then yield a simple expression for the surface area of $$A = \frac{32 mgh}{9 V_0 P_0 (\gamma - 1)}$$. Using $F = PA$ then allows me to calculate the peak force. **But** what about the contact time? My initial guess was to approximate $F(t)$ as a triangle curve that goes from 0 to $F_{max}$ and back to $0$ and then use that $F$ equals change in momentum over time, $dp = Fdt$, which in this simple case would mean the total change in momentum, $\delta p$, would be equal to $1/2 F_{max} \delta t$. I can calculate the initial momentum from the initial potential energy, and since the process is elastic, the change in momentum is (minus) two times that value. Then I know everything to calculate $\delta t$. I am, however, unsure about whether I can apply this law about momentum at all, since I am not talking about a simple point of mass here, rather about a bunch of gas molecules confined to a certain volume. This is also why my standard approach to mechanics questions, i.e. Lagrangian mechanics, doesn't seem to work: A simple coordinate describing the entire process would be $h$, but what is the kinetic term in terms of $h$? **EDIT** I just realized that my formula for the spherical cap is wrong, and a bit more complicated than I wanted: If $h$ is the distance from the center of the sphere to the base of the cap, the volume becomes $$V = \frac{2}{3} R^3 - hR^2 + \frac{h^3}{3}$$ If I keep the pressure constant, the work needed for a volume change is $P \Delta V$, so we can equate: $$\Delta V = E_0 / P$$ where $E_0 = mgh$ is the initial potential energy of the ball. The only problem then is that I have a third-order equation for $h$ which, I feel, is too complicated to solve. But let's see... let $d = R - h$, then we get $V == Rd^2 - d^3/3$ and now we assume that $d \ll R$ so that we have $V \approx Rd^2$. Next, we need the surface area, $A = \pi (R^2 - h^2) = \pi (R^2 - (R-d)^2) \approx 2\pi R d$ where we again use $R \gg d$. Plugging in some numbers gives $A \approx 44 cm^2$ which amounts to a radius of the spherical cap of approx $3.7cm$. The peak force $F_{max}$ is just $AP \approx 401 N$. Assuming that the force grows linearly with time until $F_{max}$ is reached and then drops linearly to $0$, the total change in momentum over the contact time $\Delta t$ is $\Delta p = 1/2 F_{max} \delta T$. The change in momentum is two times the initial momentum, so it is $$\Delta p = 2\sqrt{2mE_0} = 1/2 * 2\pi \sqrt{E_0 R P} \Delta t,$$ which we can solve for $\Delta t$ to obtain $\Delta t \approx 9 ms$. That sounds reasonable to me.


title: Between a solid and a hollow cylinders of the same mass, which one has the greater rotational kinetic energy?

text: I know that rotational kinetic energy is $\frac{1}{2}I\omega^2$. Therefore, the rotational kinetic energy will depend on the moment of inertia. I came to the conclusion that since both have the same mass, both should have the same moment of inertia as   $$I = \sum_\text{over all mass} m r^2$$ But the answer says that since the hollow cylinder has greater moment of inertia, it has greater rotational kinetic energy. How can the hollow cylinder have greater moment of inertia even though the masses of both the bodies is same?


title: Why are there usually grids in front of sound speakers?

text: Usually, the sound speakers have this metal grid. ![enter image description here](http://i.stack.imgur.com/6PKMc.png) However, not all of them have this. Is there any purpose, maybe related to the sound quality, that justifies this? Or is it only for physically protect the speaker?


title: Radioactive half lifes

text: What is the typical half life of material released from nuclear fission? This is a question I received for yr 12 physics and I can't find a proper answer telling me what the material released is and what the half life is.


title: How would you store heat?

text: Um .. naive question perhaps but if somebody wanted to store heat, how would they go about it? Can heat be stored? I'm told that decomposing kitchen waste in a closed vessel results in a rise in temperature on the body of the vessel. I'm just wondering whether it could be stored for later use.


title: A little question about Gauss' Law

text: So I've just learned Gauss' Law a few days ago. I also worked out some applications of Gauss' Law. But I have a little confusion. In a couple of books that I referred, I found a statement that I don't quite understand. It says "Gauss' law works because the field due to a charge varies as $\frac{1}{r^2}$. Had the field not varied that way, it would have failed." What does the statement actually mean? Why would it fail? And if it did fail, would there be an equivalent law in that case? I'm confused, please help!


title: Photon dispersion in an optical fiber

text: What is the law of spatial dispersion of a photon in an optical fiber? Say I have a femto second photon ($\lambda$ around $700\,\text{nm}$) with gaussian shape ($\sigma = c\cdot 1\,\text{fs}\approx 3\cdot 10^{-7}\,\text m$). How will it look at the end when I send it through a 50 km long optical fiber? I guess it will still have a gaussian shape but with a much bigger $\sigma$.


title: Is it possible to impart a moment with soundwaves?

text: How can one adjust the properties of a sound wave to use it to spin an arbitrary object of shape S in a medium comprised of the same material. My intuition tells me that it would be much more difficult (if not impossible) to do this with spherical objects, so for now we can assume the object is assymetric if that makes things easier.


title: Magnetic field of a Herzian dipole antenna

text: If I am given the dipole moment of very short dipole antenna as $P = P_0 sin (\omega t)$, what will be the magnetic field and polarization of far field radiation? Do I need to consider the time variation when calculating near and far fields, as in how many half or full cycles will exists along the dipole ? Thanks in advance.


title: Air density equation in stratosphere

text: Does the density equation for air on Earth change in the outer atmospheres? I am generating a model for the Red Bull Stratos jump and am trying calculate a precise air density function.


title: Why can't missing mass be photons?

text: After a star lives and dies, I assume virtually all of it's mass would be photons. If enough stars have already lived and died, couldn't there be enough photon energy out there to account for all the "missing mass" (=dark matter) in the universe? And if there were enough photons to account for all the missing mass, what would it look like to us?


title: Energy due to polarization opposing dipoles from alignment

text: Can polarization produce heat? If so will this energy oppose dipoles aligning in the field direction?


title: Where does this derivation of the equation of motion for a simple pendulum go wrong?

text: I am attempting to derive the following equation of motion for a simple pendulum: $$\theta''(t) = - \frac{g}{l} \sin(\theta).$$ For background, see this Wikipedia article. I understand this page has multiple derivations, but I don't like any of them so I am attempting my own. My attempt does not quite work. I am likely missing something stupid. What is it? * * * The strategy is to obtain two expressions for the acceleration $a(t)$ of the pendulum bob and set them equal to one another. The first we obtain via Newton's second, the second is obtained by writing down an equation for the position $x(t)$ and deriving it twice. First, let $u(\alpha) = (\sin(\alpha), -\cos(\alpha))$ for any $\alpha$. Note that $u'(\alpha) = (\cos(\alpha), \sin(\alpha))$ is $u(\alpha)$ rotated $- \pi / 2$. The position function can be written: $$x(t) = u(\theta(t))$$ where $\theta$ is an unknown function of time describing the angle of the pendulum at each instant. Using the product rule and the chain rule, we obtain: $$x'(t) = \theta'(t) u'(\theta(t))$$ $$a(t) = x''(t) = \theta''(t) u'(\theta(t)) + \theta'(t)^2 u''(\theta(t))$$ Now, the net force on the pendulum bob is $$F = -mg \sin(\theta(t)) \ u'(\theta(t))$$ since the tension on the string and the component of gravitational force parallel to the position vector negate one another. By Newton's second, $$a(t) = -g \sin(\theta(t)) u'(\theta(t))$$ Setting the two equations equal to one another: $$\theta''(t) u'(\theta(t)) + \theta'(t)^2 u''(\theta(t)) = -g \sin(\theta(t)) u'(\theta(t))$$ If we "forget about" the second term on the left, we obtain the desired relation. Note also that $u''(\alpha) = - u(\alpha)$, so this is some component of acceleration that always normal to the arc that the pendulum travels in. Have I forgotten some reason why it shouldn't exist? What step above is incorrect?


title: How does Boscovich's argument show that force must act at a distance?

text: Apparently Boscovich's argument shows that force must act at a distance: > But how did the velocity of the faster body come to be reduced from 12 to 9, > and that of the slower body increased from 6 to 9? Clearly, the time > interval for the change in velocities cannot be zero, for then, argued > Boscovich, the instantaneous change in speed would violate the law of > continuity. Furthermore, we would have to say that at the moment of impact, > the speed of one body is simultaneously 12 and 9, which is patently absurd. > > It is therefore necessary for the change in speed to take place in a small, > yet finite, amount of time. But with this assumption, we arrive at yet > another contradiction. Suppose, for example, that after a small interval of > time, the speed of the faster body is 11, and that of the slower body is 7. > But this would mean that they are not moving at the same velocity, and the > front surface of the faster body would advance through the rear surface of > the slower body, which is impossible because we have assumed that the bodies > are impenetrable Wouldn't be possible to explain this situation by discussing how the body is compressible instead?


title: Angular diameter distance in an inhomogeneous universe?

text: Computing the angular diameter distance $D_{A}$ is a well known academic exercise in an homogeneous Universe. But now suppose that we are in an inhomogeneous Universe and that I am interested in computing the angular diameter distance of a source $S$ viewed by an observer $O$. Consider furthermore that I know :   * the trajectory of a photon from $S$ to $O$ (ie null geodesics)   * the redshift $z$ along the trajectory   * the metric $g_{\mu\nu}$ along the trajectory Question: how to compute (numerically) $D_{A}$ using these information ?


title: Explain polarization in RF in which the conductor is stationary

text: Consider a metal rod parallel to $x$-axis moving with velocity $\vec v =(0,v,0)$ perpendicular to magnetic field $\vec B=(0,0,B)$. ![](http://i.stack.imgur.com/jZeu5.jpg) Lorentz force will give rise to the electric field $\vec E = - \vec v \times \vec B$, so the rod becomes polarized. Now, how to explain this force in a reference frame, in which the rod is stationary? Is such a reference frame, the field should be $\vec E = 0$, no? And then, how to explain the polarization in this RF? Thanks!


title: Why do we still get sharp scattering spots with quasi-crystal?

text: In a quasi-crystal, there is no translational invariance. This means there is no delta-function in the Fourier transform. But to get a sharp scattering spot, we need a delta function. Physically, in a crystal, we get sharp scattering spots because it is possible to get all the scattering waves from each cell in phase. But for a quasi- crystal, apparently, this is impossible.


title: How to find the force of the compression at the core of a planet?

text: For example the core of the planet Jupiter, how to find out the magnitude of the force that compresses its core? and can that be thought of as the total force that holds the planet together?


title: The proof that Dirac's hamiltonian commutes with inversion operator

text: I tried to check the statement that Dirac free Hamiltonian commutes with inversion operator. For $$ \hat {P}\Psi(\mathbf r , t) = i\hat {\gamma}_{0}\Psi (-\mathbf r , t), \quad \hat {H} = (\hat {\alpha} \cdot \hat {\mathbf p}) + \hat {\gamma}_{0}m $$ I got # $$ [\hat {H}, \hat {P}]\Psi (\mathbf r, t) = i\hat {H}\hat {\gamma}_{0} \Psi (-\mathbf r , t ) - \hat {P} \hat {H}\Psi (\mathbf r, t) = $$ $$ i ((\hat {\alpha} \cdot \hat {\mathbf p }) + \hat {\gamma}_{0} m)\hat {\gamma}_{0}\Psi (-\mathbf r , t) - i (-(\hat {\alpha} \cdot \hat {\mathbf p }) + \hat {\gamma}_{0} m)\hat {\gamma}_{0}\Psi (-\mathbf r , t) = $$ $$ =2i(\hat {\alpha} \cdot \hat {\mathbf p} )\hat {\gamma}_{0}\Psi (-\mathbf r , t). $$ Where is the mistake? Maybe, my mistake is in the following: $$ \hat {P} \hat {H}(\mathbf p) \Psi (\mathbf r , t) \neq i\hat {H} (-\mathbf p )\hat {\gamma}_{0}\Psi (-\mathbf r , t), $$ the correct one is $$ \hat {P} \hat {H}(\mathbf p) \Psi (\mathbf r , t) = i\hat {\gamma}_{0}\hat {H} (-\mathbf p )\Psi (-\mathbf r , t). $$ But I don't understand, why. For example, when I acting by $\hat {P}$ on energy expression $$ \hat {P} E = \hat {P} \int \Psi^{+}((\hat {p} \cdot \hat {\alpha}) + \hat {\gamma}_{0} m)\Psi d^{3}\mathbf r = \int \hat {P}(\Psi^{+}) (-(\hat {p} \cdot \hat {\alpha}) + \hat {\gamma}_{0} m)\hat {P} (\Psi )d^{3}\mathbf r, $$ the $\hat {P}$ only changes the sign in $\hat {\mathbf p}$ summand of hamiltonian, while the $\hat {\gamma}_{0}$ doesn't act on it.


title: Using mass of the observable Universe to estimate an energy equivalent

text: For quite some time now, physicists have been able to estimate the mass of the observable universe. Reportedly it's around $10^{50} \:\mathrm{kg}$. There is also general relativity, which states that $E=mc^2$. **If we can calculate the energy equivalent of the observable Universe, can we extend the same logic to the rest of it?** Let's leave electromagnetic radiation out of the picture for starters, since I know nothing about neither average photon density nor spectral distribution. Although if anyone does, please feel free to elaborate.


title: Trace of the number operator in second quantization

text: I would like to find the helmholtz free energy of a system $F=-T ln(Tr[e^{\frac{-H}{T}}])$ namely a bcs superconductor (using Annett's notation) $H=\sum E_k (b_{k\uparrow}^\dagger b_{k\uparrow}+b_{-k\downarrow}^\dagger b_{-k\downarrow})$ I've managed to get my expression for F down to $F=-T \sum ln(Tr(e^{-\frac{E_k}{T}b_{k\uparrow}^\dagger b_{k\uparrow}})Tr(e^{-\frac{E_k}{T}b_{-k\downarrow}^\dagger b_{-k\downarrow}})) $ ie $F=-T \sum ln(Tr(e^{\frac{-E_k}{T}n_{k\uparrow}})Tr(e^{\frac{-E_k}{T}n_{-k\downarrow}})) $ But I do not know how to evaluate the trace of operators in the second quantization formalism. My hunch tells me that $Tr[e^{\frac{-E_k}{T}n_k}]$ is $1+e^{\frac{-E_k}{T}}$, because these are fermions, so the eigenstates are $|0\rangle$ and $|1\rangle$. Is this correct? If not, how should I proceed from this point? Have I attempted to take a dead end approach?


title: Two equations for $y$ in projectile motions

text: I have two equations on my equation sheet: $$y=v_o t \sin \theta - \frac{1}{2} g t^2.$$ And then another: $$y_{\max} = \frac{v_o^2 \sin 2 \theta }{2g}.$$ I understand that the first one is an equation for the height of the projectile (as a function of time $t$) but what is the second equation for?


title: Curved spacetime as a coherent state in string theory

text: I have a question about Polchinski's string theory book, volume I, p 108. When we write the Polyakov action in curved spacetime, it is said > $$ S_{\sigma} = \frac{1}{4\pi\alpha'} \int_M d^2 \sigma g^{1/2} g^{ab} > G_{\mu\nu}(X) \partial_a X^{\mu} \partial_b X^{\nu} \tag{3.7.2} $$ _[...] A > curved spacetime is, roughly speaking, a coherent background of gravitons, > and therefore in string theory it is a coherent state of strings._ My question is, how to see a coherent state here? Is coherent state defined as   $$ \hat{a} |\alpha \rangle = \alpha | \alpha \rangle $$ as in wiki?


title: Amount of energy required to hover.

text: I've noticed a motionless kingfisher over a lake looking for prey and wondered what amount of energy does a bird, weighing 0.15kg, require to hover for 15s?


title: Is metabolic rate the amount of heat given off?

text: The Basal Metabolic Rate is intuitively expressed as "the amount of energy you consume just sitting around." In some places it seems to be equated with the amount of heat that you give off. Is this true? It seems to me that energy could be used to repair cells etc. which would not necessarily all convert to heat energy.


title: Ground state degeneracy of a variation of Toric Code model

text: We know that the ground state degeneracy of Toric Code model is 4. An easy way of seeing this is the following: Consider a 2D spin model where all the spins live on the links. The Hamiltonian is $H=-\sum_v{A_v}-\sum_p{B_p}$, where $A_v=\sigma_{v1}^z\sigma_{v2}^z\sigma_{v3}^z\sigma_{v4}^z$ is the product of 4 spins around a vertex $v$, and $B_p=\sigma_{p1}^x\sigma_{p2}^x\sigma_{p3}^x\sigma_{p4}^x$ is the product of 4 spins around a plaquette. All the $A_v$ and $B_p$'s commute with each other and they all square to one; thus each $A_v$ and $B_p$'s can take values $\pm1$. To see the ground state degeneracy, we notice that $\prod_vA_v=1$ and $\prod_pB_p=1$, being two constraints on the $A_v$ and $B_p$'s. Thus different $A_v$ and $B_p$'s can now only label $2^{2N}/4$ different configurations ($N$ being the total vertex number) because of the 2 constraints. But the total Hilbert space is still $2^{2N}$-dimensional, thus on average each configuration has a 4-fold degeneracy. Now suppose we add a third term in Toric Code model, making it $H=-\sum_v{A_v}-\sum_p{B_p}-\sum_p{C_p}$, where $C_p=\sigma_{p1}^y\sigma_{p2}^y\sigma_{p3}^y\sigma_{p4}^y$ being the product of 4 $\sigma_y$'s around a plaquette. What happens to the ground state degeneracy? Note that after adding the third term, the Hamiltonian is still a sum of commuting projectors, thus the counting method should still work; I'm just having trouble counting the number of constraints. My guess is that after adding the $C$ term, the ground state degeneracy reduces to 2. Anyone can prove/disprove me?


title: Rabi oscillation

text: Let's consider simple two-level system with frequency gap of $\omega_0$ between ground and excited state. Now, when we turn on external electromagnetic field with frequency $\omega < \omega_0$, there is non-zero probability that starting from ground state system will become excited. Energy of exicted state is $\hbar \omega_0$ but electromagnetic wave contributes only $\hbar \omega < \hbar \omega_0$. So my question is - what's with the missing $\hbar (\omega_0 - \omega)$ ?   I heard the explanation that it is measuring equipment that fills the energy gap, but I'm not quite convinced. Could you help?


title: Do we hear sounds differently on the highest mountains?

text: Some searching gives that above 6,000 meter altitude the air density is less than half of that at sea level. Speed of sound is about 15-20% slower and "acoustic impedance" seems to change too. Do humans notice these differences in sound? Does the pitch of tones change noticeable?


title: When will the Hubble volume coincide with the volume of the observable Universe?

text: The Hubble volume is the volume that corresponds to objects so far from the Earth that the space between us and them is expanding faster than the speed of light. (I.e. objects outside this volume could never again be visible to us, even in principle.) The volume of the observable Universe extends from us to the maximum distance light could have travelled since the Universe became transparent; when It was roughly 380,000 years old. Since c1998 we have known the expansion of the Universe is accelerating, implying that the number of galaxies within the Hubble volume is decreasing. Since the Big Bang (well infinitesimally close to it at least) we know that time has been going _forwards_ , and thus that the observable Universe is expanding. **When do these two volumes coincide with one another and what will the corresponding maximum volume of the observable Universe be at that time?** The associated calculation or link to a suitable reference would also be very much appreciated. Supplementarily/ I have often wondered about this; ever since reading Professor Sir Roger Penrose's _Cycle's of Time_ a year or two ago. I thought about it again today after reading a somewhat unassociated article about how recent results had shown the accelerating expansion of the Universe cannot be explained by the "Hubble Bubble" hypothesis. Before asking this question I did of course search the site for an existing answer: this question is extremely similar but does not appear to include an exact answer (in fact the answers appear to somewhat contradict each other. Moreover, it does not appear to address the subtleties involved in making the estimate; such as the variation in inter-galactic recession speeds, due to the change in gravitational force between them, since the Big Bang.


title: Gravitational lens to illuminate a planet?

text: I've been involved in a Internet debate: **Is it possible to have a special cosmic configuration consisted of multiple bodies such that a planet's surface can be lit up by light focused by gravitational lens effect caused by some giant mass bodies inside the system?** Somebody say it is a silly idea because they think even such configuration exist it certainly won't last long before the huge mass bodies drags the planet into themselves and the stable system can never be correctly set up by all means. I am not a physics major myself. I wonder if anyone can clarify the matter using mathematical terms here. (Source: http://jandan.net/2014/08/01/remote-lens-galaxy.html in Chinese)


title: How should two particles accelerate to have their separation Lorentz-contract?

text: Suppose we have two particles with initial separation of $|x_2^\text{init}-x_1^\text{init}|=d$. At time $t=0$ they start accelerating with proper acceleration of $w$. Then position of the $i$th particle in laboratory frame will be: $$x_i(t)=\frac{c^2}w\left(\sqrt{1+\frac{w^2t^2}{c^2}}-1\right)+x_i^\text{init}.$$ But then we can see that $$x_2(t)-x_1(t)=d=\text{const}.$$ So, if both particles uniformly accelerate starting from some equal moments of time, distance between them doesn't Lorentz-contract. Why is it so? How should they instead accelerate to have the distance Lorentz-contract? Should they continuously synchronize their velocities according to _laboratory_ frame times instead of their proper times?


title: Could the singularity that gave rise to our universe be past-eternal?

text: Are there any compelling models that include a past eternal singularity that ultimately gave rise to our universe? Does the "no-boundry" hypothesis that utilizes imaginary time have past eternal imaginary time and real time being created at the big bang? What are the underlying aspects of physics that allow or disallow different past eternal models of the universe? Are these questions even really answerable without a complete theory of quantum gravity? Your answers to my hopefully somewhat comprehensible questions are extremely appreciated.


title: Velocity of electron in electrostatic field. Does radiation matter?

text: There's a voltage difference of 1000 Volts between two points 2 meters apart. An electron starts at the point of lower potential and is left to travel alone in a straight line until it reaches the other point. **Question** : What speed does the electron have when it reaches the second point? My concern is whether the retardation effects of the radiation of the electron are important here. I tried doing it using the relativistic formulae for the kinetic energy and got $$v=c\sqrt{1-\left(\frac{m_ec^2}{m_ec^2+V}\right){}^2}$$ where $m_e$ is the electron mass, $c$ the speed of light, $V$ the voltage difference between the two points, and $v$ the final speed of the electron. This is not a homework exercise. I am just curious as to the effects of radiation. **Edit** : What if the potential difference was 10 000 Volts in 2 meters? What if it was 1 000 000 Volts in 2 meters?


title: In a composite Higgs, should the Z0 decay at the same rate that the neutral pion?

text: I am sorry to ask a question that obviously is a sum of separate questions about a process: Should the decay rate of Z0 be related to the decay of the "higgs" composite field it is eating? And, should this composite field, being a pseudoscalar, decay at the same rate of the neutral pion? But I hope that people will be able then to give at least partial answers. It is a fact, surely so trivial that it is rarely mentioned, that the pseudoscalar neutrals have about the same electroweak decay rate, simply scaled by the cube of its mass. Just as the charged mesons scale by the quintic of its mass, a case which is more popular in textbook exercises. Time ago the particle data group provided the table of decay rates and masses for all the particles, and then I drew this plot. Surely some point has moved slightly by now, but you can see the blue line, cubic, where all the neutrals align to scale, and the green quintic power for the charged particles, whose decay is ruled by the Fermi mass, and then a lot slower than the neutrals. ![decay widths against mass, for electroweak decays](http://i.stack.imgur.com/i1la3.jpg) You can see how the blue line touches the neutral pion in the left part of the plot and the Z0 in the right part. Yes, the muon and tau are also there, as they also decay via the electroweak interaction. (EDIT: current pdg numbers are off two sigma. More precisely we have for Z0: 2495.2/91187.6^3 = 3.291*10^-12 and for the neutral pion, using only indirect measurement: (6.58211928*10^-22/(8.30*10^-17))/134.9766^3 = 3.225*10^-12. Given that the pion lifetime has a huge error (it is 8.30 \pm 0.19), the discrepancy is still within one-sigma, but it goes up to almost 3 sigmas if we include the only direct measurement of the decay, ATHERTON 1985) So, can this be interpreted as slight evidence of, or arguments for, technicolour?


title: Relative Velocity and Momentum

text: The question is, "A $45.5~kg$ girl is standing on a $140~kg$ plank. Both originally at rest on a frozen lake that constitutes a friction-less, flat surface. The girl begins to walk along the plank at a constant velocity of $1.47~m/s$ relative to the plank." (a) What is the velocity of the plank relative to the ice surface? (b) What is the girl's velocity relative to the ice surface? In a commentary given on this problem, they are essentially that, when the velocity of the girl relative to the plank is $\vec{v}_{gp}$, and the velocity of the plank relative to the ice is $\vec{v}_{gp}$, the velocity of the girl relative to the ice is simply $\vec{v}_{gi}= \vec{v}_{gp} + \vec{v}_{gp}$. I just can't conceive why finding the velocity of the girl relative to the earth is a simply sum. Could someone please elucidate this for me? Also, they treat this situation as if momentum is conserved, that is, $\Delta \vec{P}=0$; but isn't it necessary that friction be present, between the plank and the girl's shoes, for there to be any motion? Wouldn't this force cause a change in momentum?


title: Is 2.5x speed of light possible between two objects?

text: These news are in Finnish: http://www.hs.fi/ulkomaat/Kaukaisin+havaittu+galaksi+et%C3%A4%C3%A4ntyy+maasta+valoa+nopeammin/a1305618680897 The main excerpt is: "Distant galaxy moves away from us as much as 2.5 times the speed of light" I'm not a physicist, I can somehow comprehend the idea that 2x speed of light can be reached, like in the case of 2 light beams shot to opposite directions. But 2.5x? Would 1000000x speed of light possible in theory between 2 objects?


title: Can you put a magnetic ball into a hollow magnetic sphere?

text: if all magnets have to have two poles(one north one south), is it possible to construct a hollow sphere where the inside face of the sphere was one pole, and the outside face another pole? is it also possible to make a magnetic sphere (not hollow) where the outside face is one pole? If both of these ideas are possible, what would happen if we put a small sphere (non hollow) inside of a hollow sphere, where the polarity of the hollower sphere's inside face was the same as the polarity of the smaller ball? the ball would levitate theoretically, but what if the magnets were so strong that they pushed the ball more and more into the center until it had nowhere else to move?


title: Does quantum mechanics predict instantaneous action at a distance even without entanglement?

text: The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a common origin. My question is: does quantum mechanics predict that this happens even when there is no entanglement? In particular, does this happen when two quantum systems interact in such a way that the wave function of at least one of them changes drastically? Take, for example, an interaction between a hydrogen atom and a hydrogen-oxide radical: H + HO = H2O. While the hydrogen atom is free, the wave function of its electron allows it to be anywhere in the universe. Or rather, it is everywhere in the universe. Instantly after the interaction has completed, the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a part. In the rest of the universe, the wave function has taken on other values, if not instantaneously – the reaction could take some time – then at least far quicker than light. One could object that the electron is never really at more than one point at any one time, only we don’t know which until we make a measurement. But that amounts to a refusal to accept quantum mechanics. The effect of a wave on its environment is determined by the totality of the wave, including even those places where its amplitude is partly or entirely imaginary. For example, the hydrogen atoms in a water molecule are positioned at an angle of 108 degrees from each other, as seen from the oxygen nucleus, due to some very particular properties of the wave functions which define the hydrogen-oxygen bonds. That results in water having some very peculiar properties, including the property that its solid form – ice – is less dense than the liquid form and therefore floats on it. Even the mere stability of water molecules is due to the wave function of the electrons which the hydrogen atoms share with the oxygen atom: in this wave function the electrons are as much as near the oxygen atom as the hydrogen atom.


title: First order phase transition in a classical system

text: I've never liked discontinuous quantities in classical physics, so I find the discontinuity in heat capacity weird. My question is, do first order phase transitions ever really exist? Or are our discontinuous experimental $C_v$ vs. $T$ graphs just really steep curves? Discontinuous for all practical and theoretical purposes. I understand that the theoretical model of a pure component system has a first order phase transition - and therefore a discontinuity in $C_p$. But its theoretically impossible to have a pure component (the chemical potential for any impurity in $A$ is infinite for a pure $A$). So effectively we only ever have multicomponent systems.


title: Barrier in an infinite double well

text: I am stuck on a QM homework problem. The setup is this: ![enter image description here](http://i.stack.imgur.com/uRIu3.png) (To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction vanishes when $|x|>b+a/2$.) I'm asked to write the Schrödinger equation for each region, find its solution, set up the BCs, and obtain the transcendental equations for the eigenvalues. **Where I'm at** : I understand the infinite potential well easily and I have done a free particle going over a finite barrier before (which I understood less well, but I can deal with it it).   * The problem asks me to make use of "a symmetry" in the problem, which is a vague hint. Are they trying to get me to make $\psi$ an even function?   * I am supposed the condition for there to be one and only one bound state for $E<V_0$. How do I go about that?


title: Problem with convergent geodesics at 2D sphere

text: There is a chapter on general relativity in the book _Spacetime Physics Introduction To Special Relativity_ by Taylor and Wheeler, which qualitatively explains how attractive gravitational force can be understood in terms of space(time) curvature. They give (chapter 9.5, pp. 281-283 in 2nd edition) an example of two travelers on a 2d surface of sphere who start their motion some distance from each other at the equator straight north and move _along geodesic_ (great circle). Because of the surface curvature they _accelerate_ toward each other, and finally meet at the north pole (Fig. 9-4, attached). ![enter image description here](http://i.stack.imgur.com/x2xDm.jpg) My problem begins once I (probably wrongly) extend the idea and imagine that the surface of the sphere represents the entire closed universe (in which case it would be a 3D sphere). In the case of flat Euclidean universe the two observers moving along straight lines are in inertial motion (they are in free fall though isotropic, homogeneous space) and their distance changes linearly with time, that is without accelaration. When the universe is closed and its curvature is positive and constant, they would observe acceleration although the universe is still homogeneous. What exactly is wrong in my speculation (I assume that in reality such two observers in Einstein 1917 spherical universe would not meet at a single point) and what makes the difference? The fact that real space has 3 dimensions, the fact that one has to consider 3D+1 space*time* (with Minkowski rather than Euclidean geometry in the flat case), the fact that Einstein universe is unstable, or something else?


title: Why exactly does a boomerang return back to the thrower?

text: I was always intrigued by the phenomena that govern the returning back of a boomerang to the thrower. Even if it is dependent on various factors such as velocity, aerodynamics, wind resistance and so on I was wondering, there must be some concise explanation to it?


title: Is coherent light required for interference in Young's double slit experiment?

text: In this Veritasium video, a home experiment is presented which appears to produce a very good double-slit interference pattern with normal sunlight. The experiment is an empty cardboard box with a visor and a placeholder for a microscope slide with two slits on one side. This is arranged with the slits and visor facing the Sun, so the interference forms on the bottom of the box. ![enter image description here](http://i.stack.imgur.com/x2ssa.png) They claim to observe a good interference pattern from the two slits: ![enter image description here](http://i.stack.imgur.com/4ZNX6.png) Discussions of interference in optics textbooks often stress that coherent light is needed to produce such patterns, and that sunlight and other thermal sources of light do not have such coherence. How, then, is this possible?


title: Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$

text: What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?


title: Can special relativity be deduced from $E=mc^2$?

text: So instead of assuming that the velocity $c$ is a maximal velocity, proving that while assuming $E=mc^2$.


title: Feynman Rules for massive vector boson interactions

text: I am stuck at the beginning of a problem where I am given an interaction term that modifies the regular QED Lagrangian. It involves the interaction between a photon field and a massive vector boson: $\mathcal{L}_{int_1} = \frac{1}{4}g_1G_{\mu \nu}F^{\mu \nu}$. Here, $F^{\mu \nu}$ is the electromagnetic field tensor. Similarly, $G_{\mu \nu} = \partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ where $B_{\mu}$ is the massive vector field. I am trying to derive the Feynman rule for this vertex. I am not sure what to do. Usually, I just bring down a 4-momentum when I have a field with derivatives in the Lagrangian, but in this case, it looks like two of the 4-momenta would end up forming a dot product together, while the other two would not, and I would be left with a term without any indices and two terms with indices, which doesn't make sense. Does anyone know what the Feynman rule for this vertex is and why it is so? EDIT: I think the Feynman rule might be $i\mathcal{M} = -i g_1 (g_{\mu \nu}-\frac{k_\mu a_\nu}{k \cdot q})$. I obtained this by an analogous computation for finding the photon propagator. Is doing this allowed? The main problem that I am trying to solve involves the diagram $f\bar{f} \to \phi\phi^*$. From comments, I realize that I have two options: 1) I can diagonalize the mass matrix and use the new mass as the mass in the propagator between the two end vertices or 2) I can consider the infinite series of $A$ propagator > $B$ propagator > ... > $A$ propagator > $B$ propagator and obtain the same answer. The problem I am having with diagonalizing the mass matrix is that in previous problems where I have diagonalized a mass matrix, my interaction term did not contain derivative terms. So my attempt at a solution (which I am trying now) is to just bring down a 4-momentum and go from there. But then wouldn't that make my mass momentum dependent? With the second method, I think I would need to still know the feynman rule for the interaction vertex, which circles back to my original problem.


title: Does the wave function need to be zero at the boundaries?

text: Recently I had a quiz in my physics class and I feel like the professor made a mistake on the solution for it. Yes, I already have the answer to this question. I am not trying to get people's answers for homework or anything. But I would like to hear people responses to this. The question was: Which of the following conditions are necessary restrictions imposed on the solutions of the one dimensional Schrodinger equation? (Choose all the apply)   1\. The wave function must be continuous.   2\. The wave function needs to be normalized.   3\. The wave function must converge to zero as x tends to positive and negative infinity.   4\. The wave is zero at the boundary conditions I know that the first three are correct because they are right out of the book. But my problem lies with the last one. So far we have only dealt with a particle in a rigid box and in the derivation, the wave function was zero at the boundary conditions. That's why I said 4 was correct too. This was before we studied finite potential wells where the wave function exponentially decays at the boundaries. So with knowing our limits to the material for the quiz, do you guys think it is fair that he take points off for saying the wave function zero at the boundary condition when it really is zero at the boundary with what we were studying?


title: A moving charge in an uniform magnetic field

text: An electron is in a region where there is an uniform magnetic field created by a large disc-shaped magnet. As is well known, if the electron is moving with velocity $\vec v$ parallel to the flat surface of the magnet (ie, perpendicular to the magnetic field) experiment the Lorentz force, $\vec F=q(\vec v\times\vec B)$. But what would happen if the magnet is moving with velocity $-\vec v$ while the electron is at rest? How we can explain the origin of the force? (Just make a qualitative analysis.) Do not use the Lorentz transformation, we want find the physic origin of the force.


title: Neglecting mass at asymptotic spacelike momenta

text: What is the rational/reason for neglecting masses at asymptotic non- exceptional space-like momenta. I have come across this as a first fix for being able to extract information from the Callan-symanzik and Gell-Mann-Low methods. Is it purely physical, or is it both physical and mathematically sound reason? All responses are welcome.


title: Majorana equation in two forms

text: Let's have two forms of Majorana equation. First form (standart or spinor representations of gamma-matrices). $$ i\gamma^{\mu} \partial_{\mu}\Psi - m\Psi = 0, \quad \Psi = \Psi_{c} = \hat {C} \bar {\Psi}^{T} = \begin{pmatrix} \Psi_{a} \\\ \bar {\Psi}^{\dot {a}}\end{pmatrix}, \quad \hat {C} = diag (\varepsilon_{\alpha \beta}, \varepsilon^{\dot {\alpha} \dot {\beta}}) = diag (-i\sigma_{y}, i\sigma_{y}). $$ I only say that the spinor is equal to its charge-conjugated, so the corresponding particle doesn't have an electric charge. The second form (Majorana representation of gamma-matrices). $$ i\tilde {\gamma}^{\mu} \partial_{\mu}\Psi - m\Psi = 0, $$ where all coefficients are real, so we can take $\Psi$ as the real function (for the Majorana fermions we must take $\Psi$ as the real function). So, the question: how these forms are connected? First, of course, I must get the unitary transformation $\Psi ' = \hat {U} \Psi$, which leads to $\tilde {\gamma}_{\mu} = \hat {U}^{+}\gamma_{\mu} \hat {U}$. But what to do with the definition of charge conjugation? Do I need to transform $C$-matrix? If I think correctly, will transformation $\hat {U}^{+} \hat {C} \hat {U}$ lead to the new charge conjugation operation, which consists only complex conjugation of the spinor? **An edit.** I decided to check my assumptions about charge conjugating. It is determined as $$ \Psi^{c} = \hat {C} \gamma_{0}^{T} \Psi^{*}, $$ where $\hat {C}$ refers to the charge conjugation operator. I started from spinor basis: $$ \gamma_{0} = \begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix}, \quad \hat {C} = \begin{pmatrix} -i\sigma_{y} & 0 \\\ 0 & i\sigma_{y}\end{pmatrix}. $$ Standart (or Dirac) basis: $$ U_{spinor\to standart} = U_{1} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\\ 1 & -1\end{pmatrix} \Rightarrow $$ $$ \gamma_{0}^{Dirac} = U_{1}^{+}\gamma_{0}U_{1} =\begin{pmatrix} 1 & 0 \\\ 0 & -1\end{pmatrix}, \quad \hat {C}^{Dirac} = -\begin{pmatrix} 0 & i\sigma_{y} \\\ i\sigma_{y} & 0\end{pmatrix}. $$ Finally, Majorana basis: $$ U_{standart \to Majorana} = U_{2} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & \sigma_{y} \\\ \sigma_{y} & -1\end{pmatrix} \Rightarrow $$ $$ \gamma_{0}^{Majorana} = U_{2}^{+}\gamma_{0}^{Dirac}U_{2} =\begin{pmatrix} 0 & \sigma_{y} \\\ \sigma_{y} & 0\end{pmatrix}, \quad \hat {C}^{Majorana} = \begin{pmatrix} -i & 0 \\\ 0 & i\end{pmatrix}. $$ So $$ \Psi^{c} = \begin{pmatrix} 0 & i\sigma_{y} \\\ -i\sigma_{y} & 0\end{pmatrix}\Psi^{*} \neq \Psi^{*}. $$ It is very strange, because Majorana fermion is real in Majorana representation, so charge conjugation must be equal to complex conjugation. Where is the mistake? **One more edit**. The answer is found.


title: Why isn't quantum entanglement just a lack of information?

text: From this question and answer from joshphysics, I didn't understand one thing, even after reading the comments: Why should assume that entangled electrons will only "decide" their state after observation? Isn't it just a lack of information? For instance, knowing that we have a black and white marble and picking one afterwards with your eyes closed, doesn't mean that the marble "decides" to be white or black after the observation, i.e. there is no entanglement. Therefore, I do not know what makes the states of the electrons so special.


title: Kinetic energy loss when a rigid body falls into water

text: Assume you have a rigid body falling into the ocean at terminal velocity. Also assume that the rigid body does not break on impact. How could you figure out how much kinetic energy would be lost in the resulting splash? I promise this isn't a homework question, but an engineering thought experiment. I began to reason that water is incompressible so the body has to displace the water. I know the volume of my object so I could say I know how much water gets displaced. I also have a figure that the splash (radial jet) leaves at 20-30 times the impact speed. From there I can calculate a KE, but I don't believe it because this doesn't account for the surface tension. Any guidance would really be appreciated.


title: Is it possible that Cauchy stress be asymmetric?

text: According to conservation of linear momentum and angular momentum, one can derive that Cauchy stress tensor is symmetric and hence has only 6 independent components. Is it possible that, when breaking one or both of conditions, Cauchy stress tensor is not symmetric?


title: Why must the Dirac equation multiplied by its complex conjugate give the KG equation?

text: This may be a simple question. I can show this is the case mathematically but cannot explain why it happens. It was only when asked why this happens when I realised I couldn't explain it intuitively/physically. So I suppose there must be some way in which the Klein-Gordan equation can be recovered from the Dirac equation, just like a lot of things in Physics. But why does this $\textit{have}$ to be done by multiplying by the complex conjugate and not some other operation?


title: Same partition functions, different theories

text: I am reading the book "Basic Concepts of String Theory" by Blumenhagen, Lust and Theisen and in page 290 they say: "It follows that the $E8\times E8$ and the $SO(32)$ heterotic string theories have the same number of states at every mass level which are however differently organized under the internal gauge symmetries. So, even though the partition functions are identical, the theories are nevertheless different. The differences show up in correlation functions." I am trying to understand how these differences would show up in correlation functions, ie what would be an example of two correlation functions that are different in these two theories. Any references where this is done/explained would be greatly appreciated!


title: Contribution to angular momentum $ L_z$ - due to rotation of probability fluid?

text: I'm doing a course on QM and this concept is entirely new to me: "The eigenvalue $m\hbar$ of $L_z$ can be understood as the result from the rotational motion of probability fluid around the z-axis. " Then I thought, suppose we have a volume element $dV$ at some point $r$ away, and I'm trying to find the contribution to $L_z$. I think this is related to $J_{\phi} = \frac{m\hbar}{\mu r sin\theta}|\psi|^2 $. Is the contribution $$J_{\phi} \space dA = J_{\phi} \space r^2 sin\theta \space d\theta \space d\phi= \frac{m\hbar}{\mu} r \space d\theta \space d\phi $$. Am I right? Also, how do I obtain $m\hbar$?


title: Number of Parameters Required to Specify n-Qubit States and Quantum Operations

text: How many parameters are required to specify the density matrix of a $n$-qubit system, and how many parameters are required to specify a quantum operation (completely positive maps between states) on an $n$-qubit system? For example, when $n=1$, there are 3 parameters required to specify the density matrix and 12 parameters are required to specify a completely positive map.


title: Influence of a Capacitance Matrix of a voltage measurment

text: I am having a problem in understanding how in a multiconductor system the Capacitance Matrix influences the measure of voltage between two electrodes. In the case that interest me we have 4 electrodes, 2 emitting ones and 2 receiving ones. We transmit a known sinusoidal current into the the emitting electrodes and we measure a voltage on the receiving ones. We are looking for the phase and amplitude difference between the current emitted and the measured voltage. It is my understanding that the measured voltage is influenced by any conductor present in the medium close-by, therefore we have to determine the Capacitance Matrix to obtain the real voltage measured. $$ V_{real} = V_{measured} * C_{ij} $$ What I don't understand is the relation between the measured voltage ($V_{measured}$), the capacitance matrix ($C_{ij}$) and the real voltage ($V_{real}$). If any of you could help understand this I would be very grateful.


title: How does the choice of a particular vacuum in a field theory problem decide the number of Goldstone bosons?

text: How does the field expansion method (by this I mean expanding your fields about a chosen VEV and plugging into a given potential so that the masses of the fields are given by the coefficients in front) corrolate with the vacuum you choose. In SU(3), choosing v = (0,0,v) gives 5 NGB, while v = (v,0,0) gives 6 NGB (using the Gell-Mann matrices as a basis). However, when you do a field expansion and plug it into the potential, for either of the two vacua above, you get 5 massless fields and 1 massive field. How does this match up and where have I gone wrong?


title: The Reeh-Schlieder theorem and quantum geometry

text: There have been some very nice discussions recently centered around the question of whether gravity and the geometry and topology of the classical world we see about us, could be phenomena which emerge in the low-energy limits of a more fundamental microscopic theory. Among these, @Tim Van Beek's reply to the question on "How the topology of space [time] arises from more fundamental notions" contains the following description of the Reeh-Schlieder theorem: > It describes "action at a distance" in a mathematically precise way. > According to the Reeh-Schlieder theorem there are correlations in the vacuum > state between measurements at an arbitrary distance. The point is: The proof > of the Reeh-Schlieder theorem is independent of any axiom describing > causality, showing that quantum entanglement effects do not violate Einstein > causality, and don't depend on the precise notion of causality. Therefore a > change in spacetime topology in order to explain quantum entanglement > effects won't work. which is also preceded with an appropriate note of caution, saying that the above paragraph: > ... describes an aspect of axiomatic quantum field theory which may become > obsolete in the future with the development of a more complete theory. I had a bias against AQFT as being too abstract an obtuse branch of study to be of any practical use. However, in light of the possibility (recently discussed on physics.SE) that classical geometry arises due to the entanglement between the degrees of freedom of some quantum many-body system (see Swingle's paper on Entanglement Renormalization and Holography) the content of the Rees-Schilder theorem begins to seem quite profound and far- sighted. The question therefore is: Does the Rees-Schlieder theorem provide support for the idea of building space-time from quantum entanglement? or am I jumping the gun in presuming their is some connection between what the theorem says and the work of Vidal, Evenbly, Swingle and others on "holographic entanglement"?


title: Laws of fluid flow in porous medium

text: What are the equations that describe the flow of a fluid in a porous medium? Is there a variation of the Navier-Stokes equations? I would like to model the flow of air through a sponge-like structure. Instead of modelling the structure of the sponge, I would like to treat it like a homogenous area, where I solve appropriate equations for conservation of momentum, mass and energy.


title: is gravity always the weakest force

text: > **Possible Duplicate:**   >  What does it mean to say "Gravity is the weakest of the forces"?   >  Strongest force in nature It is usually taught that the gravitational force is the weakest but I wanted to know if it will be always so. For example, On earth gravity is weaker than electromagnetic force or weak nuclear force, so is it that on black hole also if gravity is strongest than other forces shall be more stronger than it or it's that gravity becomes most strong suppressing others. Because if other forces increase in same proportion than it will have serious implications.


title: What does it mean to say "Gravity is the weakest of the forces"?

text: I can understand that on small scales (within an atom/molecule), the other forces are much stronger, but on larger scales, it seems that gravity is a far stronger force; e.g. planets are held to the sun by gravity. So what does it mean to say that "gravity is the weakest of the forces" when in some cases, it seems far stronger?


title: Why do we say that the electric forces are stronger than gravitational forces?

text: May be I am asking a stupid question but I am very confused...I understand both forces are inversely proportional to the distances squared, and that the gravitational constant is roughly $10^{-20}$ times the Coulomb's constant. But one involves charges, while the other involves mass. To me this makes as much sense as a saying a second is larger than a meter. Is it related to the energy densities of these fields? If yes, how? I think that it is not the duplicate of the mentioned one because the point asked in that question is why do we consider gravitational fields weaker even if they are more dominant in universe at macroscopic levels but I want to ask that on what criteria we define stronger or weaker fields....What do we exactly mean to say by declaring any one field stronger than the other ? Is it about their energy densities?


title: Escape Velocity question

text: I would like some help in understanding the derivation of the formula for escape velocity. Why does $t$ approach infinity? And why is $v(t)$ approaching zero? I know that $r(t)$ is infinite because the gravitational field is infinite. But I don't understand why $t$ (time) and velocity would be zero. I don't think I understand what $v(t)$ is. Is it velocity at a given point in time? And when time increases, you are farther away from the body producing the gravitational field and thus velocity needed for escape is lower? .... Here is a photo of the derivation: ![](http://i.stack.imgur.com/KIH3X.png) From the wikipedia page: http://en.wikipedia.org/wiki/Escape_velocity


title: Who works professionally on reformulation of QFT?

text: P. Dirac was worried with the infinities and their discarding in QED. He wanted us to reformulate the theory in order to eliminate infinities and renormalizations from the very beginning. Is there anybody who works professionally on this subject? I mean, who earns his salary for that. I know several amateurs but no one professional.


title: Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

text: My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many instances where they use the former. They cannot be the same as the don't have the same transformation laws. What I think is that since $\delta_{j}^{i}$ = ($\delta_{j}^{i}$)$'$, but $\delta_{ij}$ doesn't, so the latter cannot be a tensor. But the problem is that both have the same value :- ($1,0$) depending upon the indices. So it makes me think that $\delta_{ij}$ is just the identity matrix $I$ and not a tensor, and $\delta^{i}_{j}$ is a _function_. But since $\delta_{j}^{i}$ also has the same output as $\delta_{ij}$, WHAT IS THE DIFFERENCE? I think it could be matrix representations. IN GENERAL, _is there a difference between matrix representations of $\delta_{ij}$, $\delta^{ij}$ and $\delta_{j}^{i}$_ (or any other tensor for that matter). Please answer _these_ (the difference between mixed indices AND matrix representations).


title: Do Baryons have Quantized Vibrational and Rotational States?

text: Given that baryons are not point particles but are composed of three quarks, are there quantized vibrational and rotational states analogous to those of molecules? If not why not, and if so are there spectroscopically observable transitions between such states?


title: Is it possible to have $\Delta T\neq 0$ with no heat exchange $Q=0$, no work $W=0$ done, and no change in internal energy $\Delta E=0$?

text: Is it possible to have $Q=0$, $\Delta E=0$ and $W=0$, but $\Delta T\neq 0$? In particular, if there is no change in internal energy, doesn't that imply it is an isothermal process, and therefore that $\Delta T=0$?


title: Kinetic energy relationships with momentum and mass

text: If I have some arbitrary particle, we will call particle 1 and another particle 2 that have the same momentum, yet particle 1 has half the mass of particle 2, how does the kinetic energy compare between the two particles? I guess I am looking for a ratio K2/K1. I can't seem to think it through.


title: Tension in a curved charged wire (electrostatic force) - does wire thickness matter?

text: Consider a conducting wire bent in a circle (alternatively, a perfectly smooth metal ring) with a positive (or negative) electric charge on it. Technically, this shape constitutes a torus. Assume that the radius of the wire is much smaller than the radius of the circle. **Find the tension in the wire due to the repulsive electrostatic force** (in Newtons). For this problem, here are example physical values, (may or may not be necessary to solve the problem) charge, $Q = 1 C$, circle radius, $R = 1 m$, wire radius, $r = 1 mm$. Use the most straightforward notations possible for Coulomb's law, $F=\frac{k q_1 q_2}{r^2}$. This is a question about the mathematics of a set of inverse-square problems, not about theoretical E&M. Other notations that will be of useful to this discussion:   * $\lambda = \frac{Q }{ 2 \pi R } $ = linear charge density    * $\sigma = \frac{\lambda }{ 2 \pi r }$ = surface charge density **What happens to the tension in the wire as the wire radius goes to zero** with a constant circle radius, assuming:   * A constant charge on the entire thing (also constant linear charge density)    * A constant surface charge density    * A constant voltage    * If it really bothers you, consider the tensile stress for the above as opposed to the wire tension, although my question is for the latter The following are some helping equations. Any of them could be wrong. Electric field from an infinite straight line charge $$F(r) = \frac{ 2 \lambda k }{ r }$$ Electric field face-on for a ring of charge, R is radius of ring and d is distance away $$F(d) = \frac{2 \pi \lambda R k d }{ (d^2+r^2)^{3/2} }$$ Electric potential in vicinity of ring of charge, $\rho$ is distance from z-axis with torus in xy-plane, centered at origin. Electric field is simple knowing this.   * $l = \sqrt{ (\rho+R)^2 + z^2 }$ = distance to furthest point on ring    * $K(x) =$ Elliptic integral of the 2nd kind $$E(\rho,z) = \frac{ 4 R k \lambda }{l} K\left( \frac{ 2 \sqrt{ R \rho } }{ l } \right) $$ Now, I think that a valid approach would be to find the net effective field that a differential part of the ring experiences. _If_ you found this, I believe the tension could be written as follows, with T being tension, R the circle radius, and F the electric field. $$T = \lambda R F$$ I have my own thoughts on the answer (and more equations than you would ever want), and I'll return to offer them later, but I certainly don't have a complete picture, and I want to see what approaches people here use. Maybe this could be answered very simply and maybe some people will know intuitively which cases limit to infinity.


title: How many boats does it take to find an acoustic buoy by Doppler shift?

text: Inspired by this question on the Doppler shift, suppose there is buoy somewhere on the surface of the ocean emitting a pure frequency. You get to place some boats wherever you want on the surface of the ocean, moving whatever direction you want with whatever speed you want. The boats listen to the pure frequency, which will in general be Doppler shifted due to the relative motion between buoy and boat. If you do not know ahead of time the frequency the buoy is emitting, is it possible to deduce the buoy's location simply by observing the frequency measured by the boats? What is the minimum number of boats necessary, how should they be positioned, and what velocities should they have? This is a toy problem with the mathematics of the Doppler effect, so let's leave out attenuation of sound over distance and assume the surface of the ocean is a plane. Also, if the boats could continuously monitor the Doppler shift, they could collect extra information based on their own changes in position and velocity, so imagine locating the buoy based solely on a single frequency measurement from each boat.


title: Distance traveled by a projectile into a block

text: This problem is part of one of the free response questions on the 2011 AP Physics C Mechanics exam [PDF of free-response questions] that was administered last week. (Sorry the problem statement is so long; I wanted to make sure I'm not inadvertently changing the meaning by shortening it.) > A projectile with mass $m$ and velocity $v_x$ is fired horizontally into a > block of wood that is clamped to a tabletop so that it cannot move. The > projectile travels a distance $d$ into the block before it stops. > > (d) Derive an expression for the average force $F_b$ exerted on the > projectile as it comes to rest in the block. > > Now a new projectile and block are used, identical to the first ones, but > the block is _not_ clamped to the table. The projectile is again fired into > the block of wood and travels a new distance $d_n$ into the block while the > block slides across the table a short distance $D$. Assume the following: > the projectile enters the block with speed $v_x$, the average force $F_b$ > between the projectile and the block has the same value as determined > earlier, the average force of friction between the table and the block is > $f_T$, and the collision is instantaneous so the frictional force is > negligible during the collision. > > (e) Derive an expression for $d_n$ in terms of $d$, $D$, $f_T$, and $F_b$, > as appropriate.   >  (f) Derive an expression for $d_n$ in terms of $d$, the mass $m$ of the > projectile, and the mass $M$ of the block. For part (d), I simply used energy to find $F_b$ using that $E = \frac12 m v_x^2 = F_b d$. For part (e), I used energy again but with another term for friction: $\frac12 m v_x^2 = F_b d_n + f_T D$, giving $$d_n = \frac{ m v_x^2 - 2 f_T D }{ 2 F_b } = \frac{ F_b d - f_T D }{ F_b }.$$ I'm pretty sure I'm right up to this point, but part (f) is quite confusing – it seems to me that there's absolutely no way that $d_n$ can be expressed only in terms of those three quantities because it needs to depend on the frictional force between the block and the table. Can anyone explain if I'm missing something? If I'm not missing anything, what is the most likely original intention of the question-writers? Thanks!


title: Three-Dimensional Gravity

text: Does anyone have any references that discuss gravity in three-dimensions? I'm trying to make my way through some papers by Witten relating $SL(2,\mathbb{C})$ Chern-Simons theory and gravity in three spacetime dimensions and would like some explanation about the latter of the two topics. Thanks.


title: Complex Fourier Particular Solution

text: I have found the complex Fourier series for my desired force. I now need to find the steady-state forced vibration of my oscillator as a Fourier Series. (The particular solution to the inhomogeneous differential equation) How can I go about finding this.


title: Charged plane in an electric field acceleration

text: A perpendicular plane to an electric field's lines of force has more electric flux than a plane that is in parallel with the lines of force, right? Does this mean that a **_charged_** plate would experience less **_acceleration_** if parallel to rather than perpendicular to the lines of force? **EDIT** Another example of this question would be a charged object inside a charged sphere... would any force act upon it?


title: Is position fundamentally different from other observables?

text: I apologize in advance for what might be a very naive question, and for its science-fictionesque flavor. It's still, I think, a real physics question. Suppose I have a (quantum) particle, whose state is represented by some element $x$ of some Hilbert space $H$. Given an observable $O$, I can express $x$ as a sum of eigenstates of $O$. For example, I could take $O$ to represent the particle's position --- but so far, at least, "position" --- as opposed to any other observable --- seems to play no special role. On the other hand, at the macro level, our brains seem to assign a very special role to position --- namely, when we're deciding whether to think of some collection of particles as an "object", one of our main criteria is (I think) whether these particles occupy nearby positions, and whether they continue to do so over time. Moreover, we ourselves are objects by this criterion. One can (perhaps barely) imagine some other sort of creature that perceives a collection of particles as an object when (and only when) those particles have, say, similar momenta (regardless of their positions) and continue to do so over time --- or similar values of some linear combination of position and momentum, or similar values of some other observable $O$ for which we don't even have a name. Presumably these creatures would themselves be collections of particles that are $O$-localized but not necessarily position-localized. But for us humans, position seems to play this unique role. **Question 1.** Is the unique role of position a matter of physics or of biology? In other words, is there anything in fundamental physics that rules out (or renders unlikely) the sort of $O$-localized (but not position- localized) creatures I'm imagining? **Question 1A.** Let me reword that. It seems clear that (spacetime) position plays a special role in relativity, which quite plausibly rules out my $O$-localized creatures. So let me ask more specifically whether there is anything in **quantum mechanics** (as opposed to all of physics) that explains these creatures' absence. **Question 2.** If the answer to Question 1A is yes, then how does one reconcile this with the fact that any basis of the Hilbert Space $H$ can be transformed via an isomorphism to any other basis, which seems to say, in essence, that there can be nothing special about the position basis, and hence nothing special about the position operator? **Question 3.** If relativity grants a special role to position and quantum mechanics does not, does that all by itself constitute a fundamental incompatibility between relativity and quantum mechanics? And can it be viewed as a significant failure of quantum mechanics that it cannot account for this aspect of the world? **Edited to add:** I'm glad for the answers I've gotten so far, but I infer from some of them that I need to be clearer about what I'm asking. The question is not just "Why can't there be $O$-creatures?". Instead, it is: "If we have good reasons for thinking that $O$-creatures are unlikely, while knowing that position-creatures like us exist, how can we reconcile this with the fact that quantum mechanics sees no essential difference between $O$ and position? (Or am I wrong about the "no essential difference" part?)


title: What is the sensation by man carrying a water bucker with an object floating in it?

text: A man is carrying a bucket with water. If a object is kept in it which floats in water, man will feel 1) heavier 2) lighter 3) none I think when the body floats its weight is balanced by upthrust so no further be felt. Is it right or wrong?


title: Do "shape" and "volume" in "space" have any meaning for an electron?

text: The Standard Model (SM) predicts an "aspherical" electron due to its possesing a non-zero electron electric dipole moment (EEDM). An experiment by Hinds (2011) placed an upper bound on the asphericity of the electron without absolutely being able to say the EEDM was zero. Most sources I've read somewhat authoritatively say that an electron is a point and thus possesses no volume. If the SM calls something this small "aspheric", does that logically mean that it predicts the electron possesses what we would call a shape in space and thus a volume? (Even if this shape/volume is a probabality distribution function like atomic orbitals shapes) Does "shape" even have a meaning at these length scales due to QM? (If quantum foam theory is right, wouldn't the electron's shape be continually buffeted and deformed by the foam thus rendering any shape/volume as "indefinite"?) Does the fact that we have nothing smaller than an electron to probe the electron mean that we can't ever know its shape even if it has one? I know my questions probably indicate I'm helplessly stuck in the classical world, but any insights to help are appreciated.


title: Is heat energy from electronics wasted in the winter?

text: In the winter, are not all electronics heating the house - thus no energy is wasted? **Considering the energy I'm paying for:**   If I leave my computer running while I'm out, how much energy is actually wasted in the winter? Is all of the energy used in the computer expelled as heat into my home? To be even more specific:   **If my computer ran at a constant 500 watts and I had a 500 watt space heater, would they heat my home the same (thus cost the same)?** If so, wouldn't it make sense to leave the computer on in the winter, since the heat produced is not really wasted? That way I always have instant access. (Ignore variables like computer lifespans, gas furnaces, and cost- effectiveness for the sake of the question.)


title: Is everything 100% efficient at heating?

text: Everyone always talks about the efficiency of their appliances. I was wondering if everything was 100% efficient at heating its surroundings ?


title: Are all means of heating equally efficient?

text: In terms of cost of electricity, does it matter if I heat my room using a space heater, an open oven, a flood light, an ASIC? Assume no energy escapes the room, that the heat quickly becomes uniformly distributed, and ignore the fact that these produce heat at different rates. Consider only the amount of heat produced per unit of electricity.


title: How many photons does it take to measure a linear polarization?

text: A star emits perfectly (100%) linearly polarized light at an arbitrary angle.   How many photons must you detect to measure this angle to a precision of n binary digits? (with greater than 50% probability of being correct.) What is the best strategy for performing this measurement?


title: Overcharging a black hole

text: Hubeny's 1998 paper got a lot of people interested in determining whether cosmic censorship can be violated by dropping too much charge onto a black hole. It suggested that you might be able to get a naked singularity from generic initial conditions rather than requiring infinitely fine-tuned initial conditions as in the 1997 example by Choptuik. The only more recent information I've seen about this was a video of a talk by Eric Poisson in 2011. In that talk, Poisson generalized Hubeny's work to spinning black holes, and claimed that his group had made progress on technical issues involved in computing the self-force on a test charge. Hubeny and others in the field had recognized the self-force issue and had realized that it made it impossible to reach firm conclusions about cosmic censorship. In 2011, Poisson seemed to be optimistic that he'd be able to prove fairly soon that Hubeny's scenario wouldn't actually go. My question is, what is the current state of research on this topic? Hubeny, "Overcharging a Black Hole and Cosmic Censorship," 1998, http://arxiv.org/abs/gr-qc/9808043 Related but different: What prevents the accumulation of charge in a black hole? That question asked for a general explanation of what the mechanism was that would prevent overcharging. Mine is asking for specific information on a recent research program.


title: Looking for modern results in semiclassical physics and relevant references

text: What are some important approximations, especially those that are state-of- the-art, used to approximate the many-body dynamics of atoms and molecules in the semiclassical regime? To be clear, I'm not really looking for semiclassical approximations for time-independent Hamiltonian systems. Are there any modern reference-level texts/monographs or review articles that develop the mathematical theory and its applications for said models?


title: Semi-major axis and ellipticity of a binary system?

text: In the image below (source at bottom), it seems to be suggesting that \begin{equation} a = a_{1} + a_{2}, \hspace{8cm}(1) \end{equation} where $a_{1}$ and $a_{2}$ are the semi-major axis of the two orbits, and $a$ is the semi-major axis of the trajectory for the equivalent reduced-mass star in the one-body problem. It also says that \begin{equation} e_{1} = e_{2} = e, \hspace{8cm} (2) \end{equation} which is to say that both orbits have the same ellipticity, which is also equal to the ellipticity again of the trajectory for the equivalent reduced-mass star in the one-body problem. **What I'd like to know is: why are (1) and (2) true? Specifically, I have a binary system and I'm working in the centre of mass frame. I'd like go to the reduced-mass particle system and so I want to know how the properties of this new system relate to those of the first system.** ![enter image description here](http://i.stack.imgur.com/RW1zM.png) Image courtesy of: http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Gravity/BinaryStars.html


title: How much refraction occurs as a fraction of all reflection and refraction?

text: ![enter image description here](http://i.stack.imgur.com/Ec4jC.png) When light reaches a boundary between materials below the critical angle, some of it refracts and some of it reflects. For example, glass acts as a partial mirror with a dark background. Assuming no light is lost in the process, how much of total light in the incident ray is refracted as a function of $\theta_1$ (but of course $\theta_c ,n_1,n_2$ and other properties of the materials will probably matter too (whether the reflection was hard or soft, for instance)). Intuitively, the amount refracted seems to be maximum when $\theta=0$, so perhaps $A\cos(\pi\frac{\theta}{\theta_c})$ is a viable function for the fraction refracted when $\theta > \theta_c$. I hope there's an explanation that doesn't explain this away with a single experimental constant associated with each material (that is, is theoretically derived).


title: Spectral properties of CFT

text: What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac equations, Klein-Gordon equations or even the Maxwell equations...But, how do we work with conformal invariance, either in 2d, where the conformal group is infinite-dimensional or in d>2 where the conformal group is "finite-dimensional"? Note: I think It is related to correlators of conformal fields but I need a more precise and intuitive(if possible) physical/mathematical idea of how we face CFT.


title: Quantum graph theory: complex spectra

text: In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely developed branch for spectra of "infinite graphs". Some useful reference would be welcome (for spectral graph theory of infinite graphs).


title: What happens at the interface between two universes with opposite thermodynamic arrows of time?

text: I was trying to think but cannot figure it out. For instance, if the interaction is small, for instance limited to a windows, the observers in each universe will see that the other goes in reverse. But what if they can start a more meaningful interaction? (for instance, if one observer crosses the window and moves to the other universe, will anything change? why and how fast? (at least at the beginning his thinking will move in the same time arrow than in his universe, but the physical processes around him will still violate the second law?


title: In what sense is a quantum field an infinite set of harmonic oscillators?

text: In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point?   When is it useful to think of a quantum field this way? The book I'm reading now, QFT by Klauber, claims its not true, which is it? I would like to understand this analogy a little better.


We went to a planetarium last night last night and watched "Black Holes" narrated by Liam Neeson. If I recall correctly, he said that scientists believe that they have identified a number of Black Holes, and that they think there are White Holes at the bottom of Black Holes and instead of drawing matter/light in, the White Holes spew matter out. I didn't get to ask this question after the session was over: Have scientists been able to find an example of a possible White Hole?


title: Radiance equation

text: I am trying to understand the equation of the radiance, but there is one thing i don't understand: $L_r = \frac{d^2\Phi}{d\omega \space dA\cos{\theta}}$ Why is that second exponent there in the numerator? Thanks ahead!


title: What sort of propulsion would we require for interstellar travel?

text: Further to this question I asked recently, lgritz makes a very astute observation about the massive fuel requirements to travel 36 lt yrs with known fuel technology today. So, if conventional rocket fuel is completely impractical, then what are the alternatives for interstellar travel ? And how big would the space craft need to be, assuming that we could make a structurally sound spacecraft of the required size and then what would be the estimated fuel requirements to travel 36 lt yrs taking into account acceleration and deceleration ?


title: Is water necessary for forming the sedimentary rocks found on Mars' surface?

text: So, I've just been reading a recent ScienceDaily article, and came across this interesting quote > "Clays form in more neutral, less acidic conditions than the sulfate-rich > sandstones we've been looking at," Arvidson says. "Our hypothesis is that if > there are clay minerals, the water was less acidic and therefore more > conducive to life. The presence of zinc, bromine, phosphorus, chlorine, and > sulfur in the Tisdale rock is exciting in that it points toward water moving > through and altering the rock, although we have no evidence for clay > minerals in Tisdale. Further, the Tisdale surfaces were very rough and we > were unable to make use of the Rock Abrasion Tool (RAT) to clean off the > surfaces to expose fresh materials. > > "Thus, we moved on and are now in the thick of the hunt to find these > elusive clay minerals by traversing to other rock targets and making > detailed measurements, including making chemical and mineralogical > observations of natural, brushed, and ground surfaces. The rock Opportunity > is sitting over currently is called Chester Lake and will be our first > target in which we will use the RAT to progressively expose deeper and > deeper surfaces," Arvidson says. Here's the thing: we know that liquid water once existed on the Martian surface - that's for sure. But could we tell right away once we found sedimentary rocks on the Martian surface, or could Martian sedimentary rocks form through some other process? (like maybe Martian's dust storms)


title: Coupling oscillator

text: I am currently doing the following problem: > Two identical undamped oscillators, A and B, each of mass m and natural > (angular) frequency $\omega_0$, are coupled in such a way that the coupling > force exerted on B is $$\alpha m(\frac{d^2 x_A}{dt^2}), $$ > > and the coupling force exerted on A is > > $$\alpha m(\frac{d^2 x_B}{dt^2}), $$ > > where $\alpha$ is a coupling constant of magnitude less than 1. Describe the > normal modes of the coupled system and find their frequencies. Could someone explain to me the concept of "Coupling force" ?