title: Why does the current stay the same in a circuit?

text: I was informed that in a circuit, the current will stay the same, and this is why the lightbulbs will light up (because in order for the current to stay the same, the drift speed of the electrons need to get faster). However, I do not understand why the current needs to stay the same from point to point. Why does the current stay the same from point to point in a circuit?


title: How can galaxies collide if everything moving outwards

text: If the universe started from Big Bang and everything is expanding outwards and actually accelerating away from each other, than " **How is it possible for two galaxies to collide as they all are moving in the same direction** ". To collide they must in opposite direction! Edit: **How is it possible for two galaxies to collide as those colliding galaxies are moving in the same direction.**


title: Thermodynamics - Sign convention

text: I use the sign convention:   * Heat absorbed by the system = $q+$ (positive)   * Heat evolved by the system = $q-$ (negative)   * Work done on the system = $w +$ (positive)   * Work done by the system = $w -$ (negative) Could anyone please tell me, that volume increased in system does positive or negative work?


title: Ionized Depletion Region, Why aren't those charged being excited?

text: Ok so I understand the PN junction, and how when 2 Semiconductor materials are placed together the Electrons will jump into the Holes near the junction creating a Negatively Ionized Atoms on the P-Side (Near the Junction) and Positively Charged Atoms near the Junction on the N-Side. Makes sense. HOWEVER, the Donor Atoms Give out Electrons at room temperature and the Acceptor Atoms move around holes at room temperature. How come these Ionized Atoms aren't Displacing Electrons to create more holes (In the Ionized P-Side) or Accepting more electrons (In the Positively Ionized Region on the N-Side)? I understand That the Electric Field is causing some resistance....but Regardless at room temperature why aren't the Electrons getting excited out of the Negatively Ionized Atoms on the P-Side near the Junction....Why are they now "fixed" to the lattice? Like why aren't these Atoms near the Junction allowing Electrons to Get Excited and move at room temperature (Like the Electrons that have filled the holes on the P-Side)? Every Book/Semiconductor Physics book i've read talks about how the Ionized Atoms are fixed to the lattice (Makes sense)....but it doesn't say why THEY themselves aren't changing due to thermal excitations at room temperature. edit: to rephrase:Why arent the acceptor atoms releasing the electrons they obtained from the N-Biased junction at room temp? Because I know That the electric field at the depletion region is caused by electrons filling holes in the P-region and Lack of Electrons in the N-Region.....im asking whats stopping those acceptor atoms near the junction in the P-region from "releasing electrons (that they just gained)" or whats stopping the donor atoms from being filled? (the atoms that are producing an electric field). I understand the electric field opposes more electrons from filling the n-side, but whats stopping the p-side negative ions from being excited again? I guess im confused how the electric fields are stable...when room temperature is what excited the electrons in the first place.


title: (How) Does A Vortex Speed Drainage?

text: When a sink drains it forms a vortex. Does that vortex speeds drainage. If yes, how does the vortex speed drainage? The only relevant thing I could find was an expensive article and a science fair project.


title: Proof of Bethe-Bloch stopping power

text: I want to prove the Bethe-Bloch stopping power formula but I don't know where to start. Any one can offer a book or paper? Related: http://en.wikipedia.org/wiki/Bethe_formula


title: Qubit, one or two complex numbers?

text: I'm currently reading up on quantum computing and it seems like I have found some contradiction about how to represent qubits. It is often stated that a qubit is represented as $a|0\rangle + b|1\rangle = (a, b)$ with both a and b being complex numbers. However, it is stated just as often, that there is only one complex number needed, namely b, since to ignore the global phase shift means, that a becomes real while b stays complex. See for example: this and this. What is it now? What don't I get here?


title: Triangle inequality Clebsch-Gordan coeffcients

text: The Clebsch-Gordan coefficients can only be non-zero if the triangle inequality holds: $$\vert j_1-j_2 \vert \le j \le j_1+j_2$$ In my syllabus they give the following proof: $$-j \le m \le j$$ $$-j_1 \le m_ \le j_1$$ and $$ -j_2 \le m_2 \le j_2$$ For maximal values: $m = j$, $m_1 = j_1$ and $m_2 = j_2$ we get: 1) $-j_1 \le j-j_2 \le j_1$ which implies $j_2-j_1 \le j \le j_1+j_2$ 2) $-j_2 \le j-j_1 \le j_2$ which implies $j_1-j_2 \le j \le j_1+j_2$ Which should prove the triangle inequality. This proof looks really simple, but I don't completely understand it though. It seems that I'm missing some essential reasoning, and I can't find where. Why for instance do they take for $m_1$, $m_2$ and $m$ all maximal values? Can't I also take $m$ maximal and $m_1$ minimal? This would give bad results though. So I really don't understand it, and I hope that someone can clarify it.


title: Why can't noncontextual ontological theories have stronger correlations than commutative theories?

text: EDIT: I found both answers to my question to be unsatisfactory. But I think this is because the question itself is unsatisfactory, so I reworded it in order to allow a good answer. * * * One take on contextuality is to develop an inequality on measurement outcomes that is satisfied for any ontological noncontextual theory, and see that it is violated by quantum mechanics. Another take would be to assume an algebraic structure and see that if one restricts the observable algebra to be commutative, the expected values of certain operators are restricted to lie in a given range, whereas if we allow non-commutativity the range is greater. These approaches coincide? I've seen plenty of works that assume that it does, but without discussing it; in particular there is this paper by Tsirelson that states (in the particular case of Bell inequalities) that both approaches are equivalent, but without proving it. Is it too obvious? At first sight, an ontological noncontextual theory is much more general than some theory embedded in a C*-algebraic framework. Why then can't it generate stronger correlations than theories with commuting algebras of observables? Can one find a more direct connection between non-commutativity and the violation of a contextual inequality?


title: Looking for description of Helstrom's measurement

text: I hope someone can help me find the page or chapter where Helstrom discusses his famous measurement for distinguishing between two mixtures in the textbook _Quantum Detection and Estimation Theory_. Thanks in advance.


title: Force on pneumatic cylinder

text: A pneumatic cylinder is kept vertically straight on a weighing scale, which is set at 0 for the setup. now a force of 1000 N is applied on the cylinder. what is the reading of scale. assume the outlet/ inlet to be blocked hence the mass of air inside the cylinder shall remain constant all the times and cylinder can move in y(vertical) direction by compressing the air inside it on application of external force. let diameter of bore of cylinder be 60 mm and stroke be 50 mm. will the reading be less than 1000N or less or more?? more info can be supplied if required.


title: What does Gribov's last paper tell about coloured states?

text: In the first days of July 1997, after a long driving effort, crossing all of Europe to come to a meeting in Peñiscola, Vladimir Gribov fell fatally sick and he passed away one month later. His paper " _QCD at large and short distances_ " was uploaded posthumously to arXiv:hep-ph/9708424 and, in edited version, to arXiv:hep-ph/9807224 one year later. Still a remaining write-up, " _The theory of quark confinement_ ", was collected, edited and uploaded later to arXiv:hep-ph/9902279. I guess that I am asking for some general review of these articles, but I am particularly intrigued by the conclusion as it is exposed in the introduction to the first one (bold emphasis is mine). > "The conditions for axial current conservation of ﬂavour non-singlet > currents (in the limit of zero bare quark masses) require that eight > Goldstone bosons (the pseudo-scalar octet) have to be regarded as > **elementary objects** with couplings deﬁned by Ward identities." > > "... the ﬂavour singlet pseudoscalar η′ is a "normal bound state of qq¯ > without a **point-like** structure" How serious is this elementary status? For instance, in order to do calculations of electroweak interaction, should the bosons in the octet be considered as point particles, say in a Lagrangian? Does it imply that the QCD string, at low energy at least, is point-like? And, does it happen the same for the colored state having two quarks? (What I mean here is the following: the color-neutral states have been produced from the decomposition $3 \times \bar 3 = 8 + 1$ of SU(3) flavour, joining a quark and an antiquark. Similarly, we could use QCD to join two quarks, and then SU(3) flavour should decompose as $3 \times 3 = 6 + 3 $) **Is the flavour sextet "pointlike"**? And the triplet then, is it still a "normal bound state"? I expect the argument does not depend of the number of flavours, so the same mechanism for SU(5) flavour should produce a point-like color-neutral 24 and, if the answer to the previous question is yes, a point-like colored 15. Is it so? Let me note that most work on diquarks concludes that only the antisymmetric flavour triplet, not the sextet, is bound by a QCD string --e.g., measured as $\sqrt 3/2$ times the meson string here in PRL 97, 222002 (2006)\--.


title: What is inner product of the vacuum state with itself?

text: If $|0 \rangle$ is the vacuum state in quantum mechanics and $\alpha$ is any complex number, what is $\langle 0 | \alpha | 0 \rangle$? I need to have that $\langle 0 | \alpha | 0 \rangle = \alpha$, but this seems to imply that $\langle 0 | 0 \rangle = 1$. Is this true? If so, why?


title: Work due to introducing a dieletric in a capacitor

text: I have a doubt about a couple of exercises tha asks to find the work done on introducing a dieletric between the plates of a capacitor. Yes, this question is in the general case, how do we procede? I really don't know how to start those questions. I know how to find the new capacitance, the new field between the plates and the new voltage, but I don't know how to find the work required to introduce the dieletric. So, given a capacitor of capacitance $C$ and charge $Q$ what's the procedure if we want to find the work on introducing a dieletric of constant $k$ inside the plates? I just want some ideas on how to start those problems. Thanks very much in advance.


title: Impedance of a circuit

text: I am trying the following problem. A resistor with $295\Omega$ and an inductor are connected in series across an AC source that has voltage amplitude of $550V$. The rate at which electrical energy is dissipated in the resistor is $224W$. What is the impedance of the circuit. I tried this: $224W=I^2R$, from which $I=.87A$. Then, we have that $V=IZ$ in an AC circuit, so $550V=.87A(Z)$, and from here I get that $Z=631.17\Omega$. I was wondering what is wrong here. Am I using the correct formulas?


title: Vasiliev Higher Spin Theory and Supersymmetry

text: Recently there is renewed interest in the ideas of Vasiliev, Fradkin and others on generalizing gravity theories on deSitter or Anti-deSitter spaces to include higher spin fields (utilizing known loopholes in the Weinberg-Witten theorem by including infinitely many higher spin fields and by working with asymptotic conditions that do not permit S-matrix to exist). There is also a conjecture for a duality for the theory as formulated in asymptotically AdS space with the O(N) vector model in the large N limit. So in this context I am curious if there are supersymmetric generalizations of the theory, and how much supersymmetry can be shown to be consistent with this set of ideas (given that the usual restriction to 32 supercharges comes from forbidding higher spin fields).


title: Selectively visible laser beam with a controllable means

text: How can parts of a laser beam be made visible while others left invisible using something electronically controllable (for example, another laser beam crossing it, or a magnetic field, or heat, etc).


title: Electromagnetic Tensor in Cylindrical Coordinates

text: I understand that the Electromagnetic Tensor is given by $$F^{\mu\nu}\mapsto\begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\\ E_{x} & 0 & -B_{z} & B_{y}\\\ E_{y} & B_{z} & 0 & -B_{x}\\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$ where $\mu$, $\nu$ can take the values {0,1,2,3} or {$t,x,y,z$}. So, for example $$F^{01}=F^{tx}=-E_x$$ My question is, what would the following expression be? $$F^{t\rho}=?$$ or $$F^{z\rho}=?$$ where $\rho=\sqrt{x^{2}+y^{2}}$ is the radial coordinate in cylindrical coordinates? And **more generally** , how can we construct the Electromagnetic Tensor in cylindrical coordinates? Where $\mu$, $\nu$ now take the values {$t,\rho,\varphi,z$}.


title: Vectors on Different Coordinate Systems?

text: > _Consider the product of vectors coordinated relative to a given coordinate > frame, defined by_ > $$\vec{a}\square\vec{b}=((a_{1},b_{1})\square(a_{2},b_{2})):=(a_{1}b_{1},a_{2}b_{2})$$ > _Explain why that in the context of vectors this is no good._ My work: I think that it is no good because if you change the coordinate system, you may change the vector. (Like a vector in Cartesian coordinates would be a different vector in polar coordinates [Not a 100% sure about that, but seems to make sense]). If anyone could expand as to why this is true, it would be greatly appreciated. Also please note, that I am fairly new to vectors, so if the advanced physics jargon could be kept to a minimum it would be appreciated as well.


title: Meaning of generalized normal distribution

text: I asked a version of this question over on Math.SX, and never received a response… perhaps it will be more appropriate here. I'm looking at spectroscopic data (specifically a $T_2$ coherence decay curve of some NMR data). Normally, this data is fit to a single or multi-exponential decay to account for multiple components. However, I have a data set that fits best to a function with a power in the exponent near 1.4 (in between a Gaussian and single exponential decay). Is there any physical meaning for generalized normal distribution functions? To elaborate on what I mean by "physical meaning", when working with spectroscopic absorptions, an an exponential decay (n=1) indicates a system with homogeneous broadening of lifetimes, while a Gaussian decay indicates inhomogenous broadening of lifetimes. What does a power between these two values indicate? Is there a precedent for using this sort of peak shape (or decay function in this case) in spectroscopic analysis? ### \--EDIT-- To demonstrate the phenomenon, here are a couple of sample curves with some data. The depressed points at the start **may** be an experimental artifact, but I'd still be curious to know if there is any physical precedent for the exponential power between 1 and 2. $e^{-t/T_2}$ ![Exponential decay \($e^{-t/T_2}$](http://i.stack.imgur.com/XFvIb.png) $e^{-(t/T_2)^{1.6}}$ ![Modified exponential decay \($e^{-\(t/T_2\)^{1.6}}$\)](http://i.stack.imgur.com/hPjhK.png)


title: Isospin and weak Isospin

text: What is the difference between isospin and weak Isospin? What is the difference between hypercharge and weak hypercharge?


title: Why superconductors aren't used in space?

text: As we all know temperature of space is near to absolute zero.Then why super conductors aren't used there?


title: Total current of two sources in series?

text: Here is a question that a friend asked me. He had to an experiment in school and do some calculations afterwards. Those calculations require maximal current that the DC source can produce. He has measured the maximal currents for two separate sources but forgot to measure it for both connected in series. If maximal current of one battery is $a$ and maximal current of other battery is $b$, what will be the maximal current if one connects these batteries in series? I could imagine the equation system: $$I=\frac{\varepsilon_1 + \varepsilon_2}{r_1+r_2}$$ $$I_1=\frac{\varepsilon_1}{r_1}$$ $$I_2=\frac{\varepsilon_1}{r_1}$$ where $I$ is for current, $\varepsilon$ for electromotive force and $r$ for inner resistance. But it seems to have too many unknowns. And I'm not even sure if it's correct to describe the system in this way.


title: Formation of meniscus

text: If molecules at the surface of a liquid have higher energy and want to minimise the surface area, then why is a mensicus formed which of course increases the surface area?


title: How can I easily explain interference to a tour group?

text: I'm looking for unique and illustrative ways to explain the phenomenon of interference to a tour group consisting of all types of people, from elementary school kids to adults. I run into this problem from time to time at the LIGO Livingston Observatory. I often use my hands to illustrate two waves adding up constructively or destructively, but I don't think it gets across to people who aren't already familiar with the idea. I have the possibility to use small gadgets or home-made devices, but they can't be too complex or bulky.


title: Virtual Particles and Causation

text: Sometimes when people debate what type of cause a universe with a beginning may have Virtual Particles has been used as an example of a thing that can arise without a cause. So my question would be what exactly is a virtual particle and do they really pop in and out of existence without a cause?


title: (thought) experiment re: Bell's Theorem and Schrodinger's cat

text: I apologize if this question is naive. I am wondering about what would happen with the following experiment. Start with a standard Bell's Theorem setup: We have two quibits entangled in a particular way and sent in opposite directions, where they are measured by independent observers. The distribution of joint measurements is, Bell says, inconsistent with any theory of local variables. But now, before we send the quibits out, a single central person measures both quibits. That person does not communicate with either observer. My question is, **does the distribution of measurements of the observers change?** This seems like a paradox to me for the following reason. The central person actually knows what the outcomes will be of the observers' measurements. Therefore, the central person can set or calculate "local variables" that always explain what measurements will occur. Yet, to the observers, this experiment seems indistinguishable from the standard Bell's Theorem experiment (they can't even tell whether the central person is there or not!). So they should see the same distribution, which no local variable theory can explain. ... One possibility that I thought of is that the answer might depend on whether the light cones of the central person and the observers overlap, in some way, i.e. on whether the information can be transmitted at conventional speeds or not. However, we are assuming that the central person does not communicate with them in any way...(is this assumption even well-defined?). But my guess is that the observers should see the same distribution of measurements. The resolution of the paradox would be that, from the observers' perspectives, the central person and her local variables are in a superposition, just like Schrodinger's cat. Could this be right?


title: Why is Lightning referred to as High frequency source?

text: Lightning strike is categorized as DC because of its electrostatic nature. The negative flash/stroke is found to have 200KA of current with a voltage close to 300KV (cloud to ground). But why is it referred to as a High frequency source, since it does not have a frequency component? In an electric circuit diagram will lightning be represented as a voltage source or current source?


title: Is Rosehip Syrup boiling in a bottle?

text: My wife made Rose-hip syrup and the final part was boiling it in a saucepan. While still hot she placed it in a glass bottle. To keep it sterile she plugged it with a wine-saver top and sucked out some of the air - so the (inch or so) space in the bottle is at lower than atmospheric pressure. Large bubbles are spontaneously appearing in the bottle and quickly rising to the top. This has been happening for half an hour. The bottle is still quite warm . It _looks_ like it is on a slow boil: is it?


title: Does Contraction at High Speeds Happen in Any Dimension Besides Length?

text: I'm not sure if this is a duplicate. Whenever physics buffs talk about Einstein's relativity (I forget which kind) at high speeds, they always talk about "length contraction", or shortening of the object due to the high speeds. This happens even at the relatively low speeds we travel at every day, but there is so little contraction we don't notice it. However, these physics buffs only talk about **length** contraction, implying a single dimension. Why do they never talk about contraction in more than 1 dimension? Is it because the contraction happens in the direction the object is going, which is usually only a single direction? If so, are there _any_ cases which would involve 2- or 3-dimensional contraction?


title: Rotation and fictitious forces

text: A bug eats through an apple and forms a vertical, infinitesimally thin canal parallel to the vertical diameter at a distance $\frac{R}{2}$ from the center. The apple rotates at angular velocity around that center equal to $\sqrt{\frac{g}{R}}$. As the bug attains the other side of the apple it slips and falls through that canal, whose coefficient of friction is $\mu$, and I am asked to find its velocity as it passes through the bottom. Now, the length of the canal is obviously $R\sqrt{3}$. The location of the bug may be written as: $(\frac{R}{2},y)$ wrt to the center of the apple. Coriolis would have an effect in the negative $x$ direction whereas the centrifugal force, unless I am mistaken, should have an effect in both $x$ and $y$ directions. Hence, $$N = m \omega^2 \frac{R}{2} - 2m \omega \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)$$ and $$-mg + \mu N + m \omega^2 y=m\left(\frac{\mathrm{d}^2y}{\mathrm{d}t^2}\right)$$ Is this how this problem ought to be approached?


title: How to solve the Laplace equation in ellipsoidal coordinates?

text: It seems that popular textbooks on electrodynamics do not discuss how to solve the Laplace equation in ellipsoidal coordinates. I could not find any reference, but there must be references about this. Could anyone give a reference? As an example, the question can be how to calculate the charge distribution on an ellipsoid by solving the Laplace equation. (I know there is a way to solve this particular problem without solving the Laplace equation, but I want to know how the ellipsoidal coordinates works.)


title: Darcy Flow in porous material - consider porosity in cross-section area?

text: According to Darcy's Law, the volumetric flow rate Q of a fluid occuring due to a pressure difference $\Delta p$ over a distance L and through a cross- sectional area A of a porous medium (volume V) is given by: $$Q = \frac{k\cdot A}{\mu}\cdot \frac{\Delta p}{L}\tag{1}$$ When calculating the pressure increase caused by that inflow of Q over a time $\Delta t$, it is typically considered for porous media that only a fraction of its total volume can actually be filled by the fluid, where the fraction of "free space" is characterized by the porosity $n = V_{free}/V_{total}$: $$\Delta p = K\cdot Q\cdot \frac{\Delta t}{n\cdot V_{real}}\tag{2}$$ My question now is: Why is the porosity considered in the last equation, but not in the cross-sectional area of the first equation? By the very same logic I could say, that only a fraction of the cross-sectional area is "open" for fluid to flow into, couldn't I? Especially since (1) is often used for the determination of material permeabilities, but always the total cross-sectional area is used for the calculation. This is just something that seems odd to me and maybe one of you guys has some insight on that and tell me where I'm thinking wrong.


title: Stability and lifetime of soap bubbles formed with light gases like helium or hydrogen?

text: A friend asked me if it would be possible to make soap bubbles out of a gas like hydrogen and if you did, would they float higher, faster. Due to the lower mass of light gases (compared to the air) I know they will float up faster (buoyancy). I wonder how stable they would be though. **My question is, will bubbles filled with a light gas like helium or hydrogen be stable for similar time periods to bubbles filled with air?** I suspect there are complicated effects between the gas and the surface tension of the soap. Will the smaller molecules of the light gases be able to breach the soap bubble faster? I found a video that shows bubbles filled with air having helium added to them. They seem stable but I wonder if they'd be stable if they didn't have the air mixed in (pure helium).


title: Welding physics (black body and heat conductance)?

text: If I have two pieces of steel that I would like to weld with a laser (say $d$ in diameter). If I wanted to melt the two pieces to a depth of $d$ along the edge and then combine them together, how much power for the laser would I need? Is this simply $ P = A*\sigma * T_{\text{melting}}^4$? Also how fast can the laser stay on such that it wont over heat? How would you calculate upperbound on the max distance the laser can cut per second? How do you calculate whether or not molten steel will drip? Depends on viscosity and surface area right? Any help is appreciated


title: Before inflation, what sets the initial value of the inflaton field?

text: [This is a version of the question that I've revised based on helpful comments from Dan.] I haven't studied inflation at a technical level. My picture of the process is that we have an inflaton field $\phi$ which is a scalar and has a potential $V(\phi)$. Before inflation, the field starts at some $\phi_o$ that doesn't equal the value $\phi_m$ that minimizes $V$. It then rolls downhill to the minimum. Assuming I have this right, what I don't understand is (1) why $\phi$ initially has a _single value_ $\phi_o$ everywhere, and (2) what sets $\phi_o$. Before the onset of inflation, the universe's temperature was much higher than any scale set by $V$. When the age of the universe was on the order of the Planck time, presumably its temperature was on the Planck scale. If the temperature was that high, wouldn't thermal fluctuations cause the field to sample a wide range of values of $\phi$, contradicting #1? And if it could sample all those values, I would expect that thermodynamically, it would consist overwhelmingly of values of $\phi$ close to $\phi_m$, not some other value $\phi_o$.


title: Why is pressure gradient assumed to be constant with respect to radius in the derivation of Poiseuille's Law?

text: Poiseuille's Law relies on the fact that velocity is not constant throughout a cross-section of the pipe (it is zero at the boundary due to the no-slip condition and maximum in the center). By Bernoulli's Law, this means that pressure is maximum at the boundary and minimum at the center. But in the book I have it is assumed that the pressure gradient is independent of radius (distance from the center of the pipe), and the pressure gradient is thus extricated from a radius-integral. Can anyone justify this?


title: what is in the electrostatic field that makes the word done independent of path taken?

text: I am trying to understand how is it possible for electric force to behave like this (path independence)? I am repeatedly failing to get an intuitive meaning behind its (electric field) nature of doing the work. Does it have to do with the radial direction of electric force ? or Its inverse-square law nature?


title: Choice of color for graphs with filled area

text: I am writing an article and the central results are summarized in a graph so I want it to be very beautiful and physically intuitive. It has 3 regions in which a certain quantity is conserved, partially conserved or not conserved so I want them to be respectively green, yellow/orange and red. The problem is that I can't find a good combination of colors... Maybe someone has done something of similar before and can give some suggestion.. I post here my figure: ![enter image description here](http://i.stack.imgur.com/gdLfq.png) Maybe it is not the right place where ask this question but I can not find a more appropriate Stack Exchange site to post it...


title: Is there any quantum-gravity theory that has flat space-time and gravitons?

text: Many quantum-gravity theories are strongly interacting. It is not clear if they produce the gravity as we know it at low energies. So I wonder, is there any quantum-gravity theory that a) is a well defined quantum theory at non-perturbative level (ie can be put in a computer to simulate it, at least in principle). b) produces nearly flat space-time. c) contains gravitons as the ONLY low energy excitations. (ie the helicity $\pm 2$ modes are the ONLY low energy excitations.) We may replace c) by c') contains gravitons and photons as the ONLY low energy excitations. (ie the helicity $\pm 2$ and $\pm 1$ modes are the ONLY low energy excitations. This is the situation in our universe.)


title: What is the motivation for Mohr's circle?

text: I am very puzzled by the motivation for Mohr's circle in Wikipedia here. Please, explain why we need something called _"Mohr's circle"_. Use as little words as possible and be precise. **Helper questions** >   * To illustrate normal stresses and shear stresses? >   * Why is it used? When is it really used? >   * Some example to show why it may have some usage scenarios? >   * Is it just a tool to show two-dimensional stresses on a two-dimensional > plane? >


![enter image description here](http://i.stack.imgur.com/oar2l.jpg) How to derive the equation in my question?


title: Numerical torque calculation

text: Suppose I can compute interaction energy of two rigid bodies as a function of their coordinates of centers of masses and Euler rotation angles (total 6 + 6 degrees of freedom). Now I can numerically compute force acting on the center of mass of the body by calculating numerical derivatives e.g. $F_x = (E(x + dx) - E(x - dx)) / (2 * dx)$. But if you do the same for Euler angles this doesn't give you torques. So how do I convert numerical derivatives of energy by Euler angles to the resulting torque on a body?


title: Grain of sand attracting the sun?

text: My friend keeps telling me that according to physics... _"The sun attracts a grain of sand on the earth with the same force that the grain of sand attracts the sun"_ or _"A grain of sand on the earth attracts the sun"_ Is that true? Does in theory a grain of sand really attract the sun?


title: Modeling Syringes e.g. with the ideal gas law

text: Gentlemen I have a similar yet very practical problem that might provide further insight. I'm trying to model a moving plunger in a syringe (something like a piston in a cylinder). At time zero the plunger is at rest, the pressure (behind the plunger) is atmosphere and the volume zero. Then I open (hydrogen) gas which flows at a certain flow rate (which varies in time) and the pressure to move the plunger also varies in time. So my idea is to model using: $$\frac{\mathrm{d}}{\mathrm{d}t}(PV)=\frac{\mathrm{d}}{\mathrm{d}t}RT≡P\frac{\mathrm{d}V}{\mathrm{d}t}+V\frac{\mathrm{d}P}{\mathrm{d}t}=R\frac{\mathrm{d}T}{\mathrm{d} t}$$ If I assume constant temperature $P\frac{\mathrm{d}V}{\mathrm{d}t}+V\frac{\mathrm{d}P}{\mathrm{d} t}=RT$ Is this right? How do I go about solving this if I know the change in flow rate and pressure versus time? Note also that as the volume of gas increases so does m (the mass)....I'm puzzled...has anyone every modeled a syringe in detail?


title: Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?

text: When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics. Now when we studied QM, on the level of Griffiths, we always dealt with Hamiltonians not Lagrangians. Why is that? I know that QFT and particle physics rely instead on Lagrangian!!   1. Could someone provide some insights on why, it seems that, "only" Hamiltonians are used in undergrad QM.    2. Also it would be great if providing some insights on why "only" Lagrangians appear in high energy physics (KG Lagrangian, Dirac Lagrangian, Standard Model Lagrangian...etc)


title: Why not formulate Quantum Mechanics using Lagrangians?

text: As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some convenience to defining our systems one way over the other? Are there cases I'm not aware of where Lagrangian formalism is preferred?


title: Why is there a gravitational attraction between two objects at rest with respect to each other?

text: From my understanding of relativity, gravity is not a force, but a result of the curvature of spacetime. If Object1 moves past Object2, even though it's moving in a straight line, its direction may change due to the distortion caused by Object2's mass. However, what about the situation where Object1 is not moving? How can there be an attraction between two objects that are at rest relative to each other? i.e. what makes them move towards each other?


title: Schwarzschild solution in arbitrary dimensions

text: Is there any generalized Schwarzschild solution for an arbitrary number of dimensions? Is it necessary to calculate each individually, or is there a relationship between them?


title: When the electric field in an electrostatic charged conductor is zero, is the potential within also zero?

text: Let's say you have a point charge inside a conducting shell with an inner radius of 5cm and an outer radius of $7cm$. The point charge has a chard of $-4C$ and the shell a charge of $6C$. This means that the $-4C$ point charge attracts $4C$ worth of charge from the shell to the shell's inner surface. This leaves $2C$ of charge on the shell's outer surface. If you use the Gaussian-surface method at a radius of $6cm$, the Gaussian surface encloses both the $-4C$ point charge and the shell's $4C$ inner surface, so the net enclosed charge is $0C$, and therefore there is no electric field inside the shell (between the inner surface and outer surface). But is there a potential there? My first intuition was "no", but then I was thinking that if you put a positive point charge there, the positive charge would join the other positive charges on the shell's outer surface, which makes it seem there'd have to be some potential energy being converted to kinetic energy somehow...


title: Current Electricity and E.M.F of a cell

text: The **E.M.F** of a cell is the work done in moving a unit positive charge in a loop or from the terminal to the same terminal. The force it experiences is a **_conservative force_**. Therefore, the work-done in a loop should be zero which means the **E.M.F** of any cell is zero, which is not correct...What happens there?


title: Using Relaxation Method to Model Negative Dielectrics in an electric field?

text: How can you use the relaxation method to model negative dielectrics? The relaxation method is usually used to model electrostatics problems but negative dielectrics are only see in dynamic systems. **EDIT:** To my knowledge at frequency, $f=0$, no materials act like they have a negative dielectric properties. The university I attend offers a numerical electrodynamics course, every year as part of a homework they ask for a a negative dielectric be put into the relaxation method simulation that the students have programmed through the course. A negative dielectric slab in a parallel plate would be an example. I assume even though all of the voltages are static in time domain, that a f!=0 system is being simulated. What I am interested in is what systems can be modelled in this fashion? How do I correlate the results of the simulation to a physical system. Or a example of a physical system that can be simulated in the above fashion.


title: Distance travelled in free-fall

text: When an object is in free fall, we have: $a(t) = g - \frac{c}{m}v(t)^2$ where $g$ is acceleration due to gravity, $m$ is the mass of the object, and $c$ is the coefficient of air resistance. How does one get the distance traveled after t seconds? I tried integrating it, giving $v(t) = gt - \frac{c}{3m}s(t)^3$ $s(t) = \sqrt[3]{\frac{3m}{c}(gt - v(t))}$ Which is a function defined in terms of its derivative. How would I find a better definition of s(t)?


title: Calculating the $J$ value for atomic terms, having a lot of trouble with this. Already attempted

text: I am trying to understand this, and want to be very very clear. This is a homework question but I **already attempted to answer it** , so please don't put this question on hold. The question > What atomic terms are possible for the electronic configuration ns1nd1? > Which term is likely to lie in the lowest energy? Here is my attempt at answering this question. Some of the things I understand like calculating Multiplicity, you do that by the following relationship: > $$S = s_1 + s_2, s_1 + s_2 - 1, \ldots , |s_1 - s_2|.$$ Then apply it to the multiplicity rule: > $$ 2S + 1$$ I got the values 1, and 3 for the multiplicity and understand that the maximum multiplicity is the one that lies in the lowest energy. I also understand that the maximum $L$ value corresponds to the lowest energy, and since $L = 2$ therefore the term Symbol will be D, and the supercript for S will be either 1, or 3. What I don't get is how to calculate the $J$ values, the $J$ values the solutions manual has are 3, 2, 1 but it didn't explain how they derived those values. Do you use the Clebsch-Gordan relationship for the $J$ values as well, as such? > $$J = L + S, L + S -1, \ldots |L - S|$$ Really having trouble. Can someone please explain this to me? **Here is my attempt:** ![enter image description here](http://i.stack.imgur.com/WYEsS.png) I don't understand how they obtained the $J$ values


title: How far do atoms move in a wire when whirling it horizontally?

text: so the question is: > I whirl a mass m attached to a wire with length L and diameter d around my > head in the horizontal plane. The mass takes t seconds to move around a > circle. How far do the atoms in the wire move apart, compared to their > spacing at rest? Young's modulus is given as Y. I know the value of Young's Modulus(Y). So: Y = Stress/Strain For me to find the Strain I need to find the Stress first, which requires force(F) Stress = Force/A , A = cross-sectional area = pi*(d/2)^2 => Y = (F/A)/strain = (F/(pi*(d/2)^2))/strain So, to find the strain, I need to find the Force and from the answer I can find how far the atoms move. My question is how to I go about to find the force? Thanks


title: How would Hamiltonian for several fermions with spin look?

text: All discussions of Pauli exclusion principle I read usually talked about antisymmetric wavefunctions, from which the princinple appears. But I would like to see a Hamiltonian for multiple fermions, which would have Pauli exclusion principle already incorporated into its eigenstates. I've looked into Pauli equation, but it seems to only describe one fermion. The same would hold for the more complete Dirac equation. So, my question is: what equation describes _multiple_ (more than two) fermions _with spin_ in non-relativistic regime, so that its eigenstates would automatically obey Pauli exclusion principle? Do I need the full machinery of QFT for this?


title: Why p-wave superconductors are rare in nature?

text: I have the basic question that why so many superconducting materials are s-wave and d-wave pairing, but the p-wave superconductors are so rare in nature? An equivalent question may be that why topological materials are so rare in nature, but the answer to the this question may be more complicated then my first question.


title: Learning the stress-energy tensor

text: I am learning dynamics in **special relativity** and come across the stress- energy tensor. I have real trouble understanding it. I would love answers on   1. How to motivate the definition of this tensor.    2. How to deduce the equation of motion, i.e. the divergence of the stress-energy tensor is 0. A toy model would be best for a demonstration. (You may use the language of Riemannian Geometry freely.)


title: Andromeda/Milky Way collision: How, and how accurately, can a galaxy's lateral velocity be measured?

text: Some sources suggest that the Andromeda Galaxy is likely to collide with our own in approximately 3 to 5 billion years. We can estimate the distance to the Andromeda Galaxy using various techniques, including measuring the apparent brightness of Cepheid variable stars; its distance is currently estimated to be about 2.5 million light-years. We can measure its radial velocity (i.e., the rate at which it's either approaching or receding from us) using Doppler shift. One source, the same Wikipedia article I linked to above, indicates that its radial velocity with respect to the Sun is about 300 km/s in our direction; another article says the radial velocity relative to our galaxy is about 120 km/sec, also in our direction. (Presumably the difference is due to the Sun's orbital motion around the core of the Milky Way.) But that's just the radial component of the velocity. Taking the 120 km/sec figure, it could be moving directly toward the Milky Way (more precisely, its core could be moving directly toward the core of the Milky Way) at 120 km/sec, or it could be moving at a 45° angle at about 170 km/sec, or any of a number of other possibilities. Without an estimate of the lateral component of the velocity, there's no way to be sure whether the collision will occur or not. I'm reasonably sure we can't measure the lateral velocity directly; 120 km/sec over a century would cause Andromeda to move only about 0.04 light-year (if my calculations are correct). And yet this Wikipedia article says: > The best indirect estimates of the transverse velocity indicate that it is > less than 100 km/s. with a reference to "Abraham Loeb, Mark J. Reid, Andreas Brunthaler and Heino Falcke The Astrophysical Journal, 633:894–898, 10 November 2005", but the link is invalid. So how can a galaxy's lateral velocity be measured, or at least estimated? How accurate can such an estimate be with current technology? Can we expect improvements in the near future?


title: Form factor and Guinier regime

text: sorry for reposting. I dont get why it is such a good idea to plot the logarithm of the form factor vs $Q^2$ in Guinier plots. It seems arbitrary to me.


title: Heat engines and "Angular momentum" engines?

text: We know that the theory of heat engines is that, if you accept the second law of thermodynamics, $\Delta S > 0$ then you can define temperature using $\frac{1}{T} = \frac{\partial S}{\partial E}$ And you would arrive at the conclusion that heat can only go from higher temperature to lower temperature reservoirs. And the best efficiency we can get is to use a reversible engine. Now my question is "What's so special about energy?" If we replace energy with any conservative, for example, one component of angular momentum $L$. Then is it possible to define a "temperature" as $\frac{1}{T_L} = \frac{\partial S}{\partial L}$ And we would conclude that "in order to get useful angular momentum out of a reservoir, there is a maximum efficiency achieved by a reversible angular momentum engine" ?


title: Are there planets that do not rotate on their axis?

text: I was reading a thread about how a pendulum would be affected if the Earth did not rotate and Larian's answer made me wonder if all planets rotate necessarily due to physics. So that's the question: is it possible for there to exist a planet* that doesn't rotate at all and why? _* In any solar system or galaxy._


title: Baryonic density in collision experiments

text: Does anyone know any way of estimating the net baryon density in collision experiments, e.g. in LHC, RHIC, or the upcoming ones at GSI-FAIR? I have comes across many hand-waving arguments, sample -   1. Electron-positron colliders start with $\rho_B = 0$, and since conservation laws have to respected, matter and anti-matter must be produced in equal amounts. Hence, barring inhomogeneities, LHC is essentially in the $\rho_B \approx 0$ regime.   2. Antikaon-proton interaction is known to be attractive, through various model-dependent calculations, so if we have a ${\bar K}-p$ collision experiment, it shall involve the formation of quasi-matter having a larger density than usual, and hence, via these kind of reactions, GSI-FAIR plans to study QCD matter at high densities (especially in the Compressed Baryonic Matter experiment), while it remains inaccessible at LHC. (Please correct me if the hand waving is inappropriate.) I am wondering, if there is any way of making this a little more calculation oriented, though I understand that a fully rigorous calculation would be very difficult. (I don't want that, too.) Also, is it possible to have any model-independent answers, since model- dependent answers can be found with a bit of googling, e.g. Phys. Rev. C 72, 034613 (2005), which even has the evolution of these densities in lead-lead collisions. Though these calculations are the order of the day, model- independent calculations, if possible, teach you much more.


title: Why does a string connected between a D0-brane and an anti-D0-brane turn into a tachyon upon their annihilation?

text: Consider a string stretched between a D0-brane and an anti-D0-brane. In this case as the stretching energy is greater than the quantum zero point energy the string will have a positive mass. But, as the D0-brane and the anti-D0-brane come closer to annihilate the string becomes a tachyon as now the major is from quantum zero point energy as the string is no longer stretched. But will the sudden "coming closer" of the brane and anti-brane give the string extra vibrations? If so,is it that the energy due to these vibrations is not enough to overcome that of zero point energy finally resulting in a tachyon?


title: Why does quantum zero point energy contribute negative mass to strings?

text: A string which doesn't have any kind of vibrations will have mass whose square is negative due to quantum zero point energy. But why does it contribute negative rather than positive mass to strings?


title: What is "hydrostatic uplift pressure"?

text: I was reading Wikipedia about the St. Francis Dam and came across this sentence. > Water that collected in the drainage pipes under the dam to relieve the > hydrostatic uplift pressure was carried off in this manner as well. What is hydrostatic uplift pressure? The word "uplift" in particular confuses me.


title: Why am I experiencing some type of odd motion?

text: So I noticed for a few years ago while I'm riding on the bus and as we stop at a corner i tend to just look at the road while I'm listening to music, I noticed that as the bus driver had stopped i would see the ground shift forwards as if the bus is going backwards, but we are stopped and not in reverse. Also I have never been in an earthquake and I didnt feel any vibrations so that's excluded. So could my eyes are playing tricks on me or am I actually seeing some sort of movement. Also the shift isn't big; it stops after a second or two and it shifts at an angle rotating CCW.


title: Understanding Momentum

text: I'm trying to learn more about momentum and I'm a little confused. Based on my understanding, in an isolated system, total momentum is conserved in a collision. Today in class the professor went over an example of a car on a ferry driving from one end to the other, in the opposite direction of the ferry. According to him, the total momentum for the system was zero in this case. I understand why total momentum is zero in a collision: because the objects come at rest. But in the example both the car and the ferry are moving (the goal is to find the ferry's new velocity as the car takes off). So my question is: Is total momentum always zero if two objects are in touch with each other and are applying the same amount of force (but in the opposite direction) to each other? Also, the water applies the same amount of force to the ferry (again in the opposite direction) but how come it's not considered in our "system"?


title: How do internal forces affect motion?

text: As per newton's first law, only an external force can bring any change in the acceleration of a body, internal forces cannot. So, when we apply brakes to an accelerating car, aren't those brakes(opposing force) part of internal forces? How can they put the car to a halt?


title: Maxwell's Stress Tensor

text: What really _is_ the Maxwell Stress Tensor? I understand that it's derived from $$\mathbf {F} = \int _V ( \mathbf E + \mathbf v \times \mathbf B )\rho \ d \tau$$ Griffiths describes this as "total EM force on the charges in the volume $\mathcal V$". $$T_{i j} = \epsilon_0 \left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0} \left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right)$$ This leads us the the stress tensor, but there's something that I don't understand. The description given is > Physically, $T$ is the force per unit area acting on the surface. What surface are we speaking about here? An arbitrary surface? In the case of example 8.2 (net force on a uniformly charge sphere's upper hemisphere), the surface in question is clearly the boundary of the upper hemisphere and it's "disk" separating the two hemispheres. In other cases, such as problem 8.4, where we have two point charges separated by a distance, we must integrate over a particular surface. For such a problem, we must "integrate the stress tensor over the plane equidistant from the two charges", but _why_? How would summing up the force on the plane separating the two point charges equal the force on each charge? How could there be "force per unit area" across an empty plane?


title: How does power consumption vary with the processor frequency in a typical computer?

text: I am looking for an estimate on the relationship between the rate of increase of power usage as the frequency of the processor is increased. Any references to findings on this would be helpful.


title: Battery fully loaded, pull out the chord?

text: I recently bought a new phone (Samsung Galaxy 3) and when the battery is fully loaded, it says like "Battery fully loaded, pull out the chord". Is this a typo from Samsung, or would there theoretically be any reason (for battery saving purposes) to always have it going from the battery, and not from the chord. _(Ftr my phone is set to Swedish language)_


title: what breakthrough Physics needs to make quantum computers work?

text: I read some posts on this forum and some articles which repeatedly state that it is not impossible to build q-comps but to make it successful, physics needs a great breakthrough. I tried finding but most of the material talks about how it works and what are current limitations, but how physics is going to help solving it? Can somebody please explain precisely 'what is that breakthrough?'.


title: Loop corrections to propagator (QFT of Srednicki)

text: Perhaps this is a very basic question. In chapter 14 of Srednicki's QFT textbook (2007), $O(g^2)$ loop corrections to the propagator of $\phi^3$ theory is discussed. However, I don't know how to derive the Eq. 14.34 on page 103 (I can't present the full context as it would be too long). From \begin{equation} \Pi(k^2)=\frac{1}{2}\alpha\Gamma(-1+\epsilon/2)\int_0^1\,dxD(4\pi\mu^2/D)^{\epsilon/2}-Ak^2-Bm^2+O(\alpha^2), (14.32) \end{equation} and take the $\epsilon \rightarrow 0$ limit using \begin{equation} A^{\epsilon/2}=1+\frac{\epsilon}{2}ln(A)+O(\epsilon^2) \end{equation} and the following property of Gamma functions \begin{equation} \Gamma(-n+x)=\frac{(-1)^n}{n!}\left[\frac{1}{x}-\gamma+\sum_{k=1}^nk^{-1}+O(x)\right], \end{equation} he got \begin{equation} \Pi(k^2)=-\frac{1}{2}\alpha\left[\left(\frac{2}{\epsilon}+1\right)\left(\frac{1}{6}k^2+m^2\right)+\int_0^1\,dxDln\left(\frac{4\pi\mu^2}{e^\gamma D}\right)\right]-Ak^2-Bm^2+O(\alpha^2), \qquad (14.34) \end{equation} where $\alpha\equiv\frac{g^2}{(4\pi)^3}$. I don't know how to do the integral correctly (although it seems elementary). Can anyone familiar with this part help me out?


title: How do I predict volume loss due to evaporation when boiling water?

text: Suppose I have a pot with diameter $D$ containing a volume of water $V$, being heated by a flame under it. If the ambient air temperature is $T$ and relative humidity is $R$, how can I calculate the expected rate of loss due to evaporation over a period of time, $t$? Assume the period of time begins once the water begins boiling. **Edit:** For simplicity's sake, also assume the flame is just hot enough to get the water boiling.


title: Is the spin-rotation symmetry of Kitaev model $D_2$ or $Q_8$?

text: It is known that the Kitaev Hamiltonian and its spin-liquid ground state both break the $SU(2)$ spin-rotation symmetry. So what's the spin-rotation-symmetry group for the Kitaev model? It's obvious that the Kitaev Hamiltonian is invariant under $\pi$ rotation about the three spin axes, and in some recent papers, the authors give the "group"(see the **Comments** in the end) $G=\left \\{1,e^{i\pi S_x}, e^{i\pi S_y},e^{i\pi S_z} \right \\}$, where $(e^{i\pi S_x}, e^{i\pi S_y},e^{i\pi S_z})=(i\sigma_x,i\sigma_y,i\sigma_z )$, with $\mathbf{S}=\frac{1}{2}\mathbf{\sigma}$ and $\mathbf{\sigma}$ being the Pauli matrices. But how about the quaternion group $Q_8=\left \\{1,-1,e^{i\pi S_x}, e^{-i\pi S_x},e^{i\pi S_y},e^{-i\pi S_y},e^{i\pi S_z}, e^{-i\pi S_z}\right \\}$, with $-1$ representing the $2\pi$ spin-rotation operator. On the other hand, consider the dihedral group $D_2=\left \\{ \begin{pmatrix}1 & 0 &0 \\\ 0& 1 & 0\\\ 0&0 &1 \end{pmatrix},\begin{pmatrix}1 & 0 &0 \\\ 0& -1 & 0\\\ 0&0 &-1 \end{pmatrix},\begin{pmatrix}-1 & 0 &0 \\\ 0& 1 & 0\\\ 0&0 &-1 \end{pmatrix},\begin{pmatrix}-1 & 0 &0 \\\ 0& -1 & 0\\\ 0&0 &1 \end{pmatrix} \right \\}$, and these $SO(3)$ matrices can also implement the $\pi$ spin rotation. So, which one you choose, $G,Q_8$, or $D_2$ ? Notice that $Q_8$ is a subgroup of $SU(2)$, while $D_2$ is a subgroup of $SO(3)$. Furthermore, $D_2\cong Q_8/Z_2$, just like $SO(3)\cong SU(2)/Z_2$, where $Z_2=\left \\{ \begin{pmatrix}1 & 0 \\\ 0 &1\end{pmatrix} ,\begin{pmatrix}-1 & 0 \\\ 0 &-1 \end{pmatrix} \right \\}$. **Comments:** The $G$ defined above is **even not a group** , since, e.g., $(e^{i\pi S_z})^2=-1\notin G$. **Remarks:** Notice here that $D_2$ can _not_ be viewed as a _subgroup_ of $Q_8$, just like $SO(3)$ can _not_ be viewed as a _subgroup_ of $SU(2)$. **Supplementary:** As an example, consider a two spin-1/2 system. We want to gain some insights that what kinds of wavefunctions preserves the $Q_8$ spin- rotation symmetry from this simplest model. For convenience, let $R_\alpha =e^{\pm i\pi S_\alpha}=-4S_1^\alpha S_2^\alpha$ represent the $\pi$ spin- rotation operators around spin axes $\alpha=x,y,z$, where $S_\alpha=S_1^\alpha+ S_2^\alpha$. Therefore, by saying a wavefunction $\psi$ has $Q_8$ spin-rotation symmetry, we mean $R_\alpha\psi=\lambda_ \alpha \psi$, with $\left |\lambda_ \alpha \right |^2=1$. After a simple calculation, we find that a $Q_8$ spin-rotation symmetric wavefunction $\psi$ could **only** take **one** of the following 4 possible forms: $(1) \left | \uparrow \downarrow \right \rangle-\left | \downarrow \uparrow \right \rangle$, with $(\lambda_x,\lambda_y,\lambda_z)=(1,1,1)$ (Singlet state with _full_ $SU(2)$ spin-rotation symmetry), which is annihilated by $S_x,S_y,$ and $S_z$, $(2) \left | \uparrow \downarrow \right \rangle+\left | \downarrow \uparrow \right \rangle$, with $(\lambda_x,\lambda_y,\lambda_z)=(-1,-1,1)$, which is annihilated by $S_z$, $(3) \left | \uparrow \uparrow \right \rangle-\left | \downarrow \downarrow \right \rangle$, with $(\lambda_x,\lambda_y,\lambda_z)=(1,-1,-1)$, which is annihilated by $S_x$, $(4) \left | \uparrow \uparrow \right \rangle+\left | \downarrow \downarrow \right \rangle$, with $(\lambda_x,\lambda_y,\lambda_z)=(-1,1,-1)$, which is annihilated by $S_y$. Note that _any_ kind of superposition of the above states would _no longer be_ an eigenfunction of $R_\alpha$ and hence would break the $Q_8$ spin-rotation symmetry.


title: How do I know if an object comes from the Solar System or not?

text: I'm told that if the total energy of an object near the Sun is negative, then it comes from the Solar System; if positive, it is extrasolar. I don't understand why. Can someone please explain this?


title: Can a wave be two dimensional?

text: I am having a hard time picturing waves, the image that comes to mind is a bobbing device submerged in still water which generates pulses in all directions (similarly in air). Then how can a wave be two dimensional? The classic image of an electromagnetic wave is a 3D wave function, x-y represent the electrical component, z-x represents the magnetic component. But this is a wave function, so its just a plot, and that is not how I should picture it (or is it?). So in real life, if I have a source of EMR in space, I can imagine concentric spheres of Electrical fields, but what about the magnetic fields? How will they be perpendicular to a sphere of electric fields? Can someone help me picture EMR? Thanks! (apologies in advance for the noobish language, I am an absolute beginner to physics)


title: Ringing sound when you flip a coin?

text: When you flip a coin, you hear a ringing sound. I know that the source of the sound is the thumbnail hitting the coin, but it seems to be filtered by the spinning of the coin. Specifically, the faster the coin spins, the smoother the tone and it seems the spectrum becomes more uniform. So my question is, physically, what is the exact phenomenon that is causing the filtering? It seems to be some form of frequency modulation (or amplitude modulation? I suspect it's the former but I could be wrong). my other question is, what are the exact characteristics of this filter? I know that the filter will be a function of the angular frequency of the coin, which I'll call $\omega_m$, but other than that I don't know where to start on how to characterize it.


title: Power radiated by a star, black body model

text: let's say I want to model a star of radius $R$ at a distance $r$ from the Earth. I need to show that the apparent luminosity for frequency $\nu$ is equal to $$\ell(\nu)=\frac{2\pi h}{c^2}\left( \frac{R}{r}\right)^2\frac{\nu^3}{\exp\displaystyle\left(\frac{h\nu}{kT}\right)-1} $$ Which is Planck's law multiplied by $$\pi\left(\frac{R}{r}\right)^2$$ Given the nature of the factor (adimensional function of $R,r$) I would say that the result follows from geometrical considerations, however I fail to see which ones. Thanks.


title: Solitonic nature of RR sources

text: In the famous paper by Polchinski where he shows that D-branes are sourcing RR fields, he says (before we known the result) that RR sources must be objects with tension going like $1/g_s$ (page two of the paper). How do we know that? I understand that it must be non-perturbative in $g_s$ (essentially because strings aren't charged under RR fields), but why it has to be precisely $1/g_s$ ? Thanks.


title: Is speed an intensive property?

text: I remember being taught in elementary physics that while it makes sense to add volumes, masses, or heat, it makes no sense to add temperatures. As I wanted to use that to illustate some other issue, I checked it on wikipedia, and discovered the concept of intensive and extensive properties of materials and systems, which is apparently a century old, though I do not remember it being taught to me (it was not as old then :-). The concept seems to be particularly useful in material and systems thermodynamics. But I was wondering how far it extends, and I was keeping in mind my original problem of determining when adding values in a given unit (not necessarily a physical one) made sense or did not. I noticed the fact that the ratio of two extensive variables is intensive. So I started looking, a bit ramdomly I confess, at ratios, and the first that came to mind was speed. And I wondered whether adding speeds made sense. We do that all the time, so it should. But then, my initial problem was not about adding two quantities, but a long list of them (it was a database question). While adding a few speeds (or velocities) makes sense when I move in the bus, or analyse the motion of the Moon in the solar system, I cannot imagine it would ever make sense to add a list of a hundred or a thousand speeds (though computing the average would make sense, as it would for temperature). But I makes perfect sense to sum the masses of a thousand objects. So there is apparently something special, not quite right, about adding speeds. Of course, I know that slightly older results state that adding speeds is not done with simple addition, but I feel that my problem is elsewhere (though there may possibly be a connection). I am aware that temperature in some materials is related to speed of motions inside it, so I am not too surprised. But speed in general seems to go beyond that (sorry for the vagueness). Also speed is not listed by Wikipedia as an intensive property. So, my question is : why does it seem improper to add many speeds (or velocities)? I guess this must also be true of other physical quantities, and I am wondering what is the right way to look at this, and understand it. Is there a more general notion than intensive and extensive?


title: Trying to understand "recursion at the lowest level of matter"

text: I'm reading the book Gödel, Escher, Bach: an Eternal Golden Braid and on Chapter V, Hofstadter talks about different examples of recursive structures and processes. By page 142 of the 20th anniversary edition, he starts talking about recursion at the lowest level of matter, that is, according to him, related to the structure of elementary particles. To explain his point, he draws a little example I'm unable to understand due to, most likely, my poor knowledge of physics in general. He starts by limiting himself to two kinds of particles: electrons and photons: "Imagine first a dull world where a bare electron wishes to propagate from point A to point B. A physicist would draw a picture like this:" ![Initial diagram](http://i.stack.imgur.com/4FmjZ.png) Based on this, my initial set of questions are:   * He says in such a world there is an easy mathematical expression to represent this line, does anybody know which expression is that?   * And how exactly is an electron simply traveling from point A to B? Next, in a world where electrons and photons interact, the electron is now capable of emitting and then reabsorbing "virtual photons - photons which flicker in and out of existence before they can be seen.": Electron and photon interaction: ![enter image description here](http://i.stack.imgur.com/HqEQ9.png) "Now as our electron propagates, it may emit and reabsorb one photon after another, or it may even nest them, as shown below:" ![Nested interactions](http://i.stack.imgur.com/NHzTF.png) Also here he says that "the mathematical expressions corresponding to these diagrams - called Feynman diagrams - are easy to write down." So my last set of questions are:   * How do these diagrams relate to Feynman ones?   * I'm not quite sure I actually understand them, could someone help me with a gentle introduction to the topic?   * More importantly to the book reading at hand, where is the recursion in all this? I fail to see it. For those interested, I found all of this part of chapter V online. Edit: I accepted the answer given by Gugg below. For completion, follow the interesting discussion we had on chat as well.


title: Rosenberg-Coleman effect

text: I know only that it states some variations of the Sun's magnetic fields. What is the Rosenberg-Coleman effect specifically?


title: Rutherford's Gold Foil Experiment

text: Can anybody explain how Rutherford bombarded a 0.0004 cm thick gold foil? How did he put it in a photographic sheet? Wasn't the foil too thin to be held? How did he know that the atoms were deflected at various angles? Did he calculate every alpha particle's angle of deflection?


title: How to convert a hologram into an image?

text: Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the boundary condition for the wave equation, you should in principle be able to solve for the phase and intensity of light at every point in space (up to some global phase factor that changes in time). Now, how do you translate this into an image? Say, if the hologram is given by a complex field $\phi(x,y)$ on the $XY$ plane, the light has wavelength $k$ and I have a pinhole camera located at position $\vec{p}$ pointing in direction $\vec{r}$, is there a simple formula to translate this field into an image intensity? In more detail: think of the light at each point in space as being given by a complex number, where the intensity is the magnitude squared and the phase is, well, the phase. I'm leaving out the $z$ coordinate of the field because it shouldn't matter for this problem. The Green's function for the wave equation (basically, the field emitted by a point source located at $\vec{s}$) is given by $G(\vec{x},\vec{s},t) \sim e^{i(k||\vec{x}-\vec{s}|| - \omega t)}$ where $\omega$ is the frequency of the light, and I've dropped a constant term that doesn't really affect the answer. Also, the $\omega t$ term in the solution shouldn't matter either, since it will be the same globally and doesn't change the magnitude of the field. We can solve for the field $\psi(\vec{s},t)$ at any point in space by taking the integral $\int \int dx dy \phi(x,y)G(\vec{s},(x,y,0)^T,t)$ but how do we go from this field to an image? Basically, how is the light transformed on passing through the pinhole?


title: Pion decay in particle physics

text: I'm taking a particle physics course and we're using Perkins Introduction to High Energy Physics as the text. I am looking at problem 1.7. It asks whether $$\pi^0\rightarrow e^- + e^+$$ is allowed or forbidden by the Standard Model based off of conservation laws. I feel it doesn't violate any of them. Energy and momentum can clearly be conserved, charge is conserved, lepton number is conserved, etc. However, I know Dalitz decay is a very similar process, but that the pion decays into $$\pi^0 \rightarrow e^- + e^+ + \gamma$$ so feel as though I must have overlooked something. I would be greatly appreciative if someone could point me in the right direction.


title: Variance of Nested Experimental Uncertainty

text: I have to find the uncertainty of a quantity $Q$ doing two mean values. For example for a set of parameters I measure ten times $Q$, I obtain a mean value $Q_1$ and variance ${\rm Var}(Q_1)$. Then for a different set of parameters I measure ten times $Q$ and obtain $Q_2$ and ${\rm Var}(Q_2)$ etc. At the end I compute the mean value which is the sum of $Q_i$ but these 10 from one set of parameter to the other are correlated so I don't know how to compute the variance. Put differently, I don't know how to compute the variance if I have two averages.


title: Gauge invariance and form of the vacuum polarization tensor

text: In quantum field theory or condensed matter physics, the fermionic one-loop diagram gives rise to the polarization tensor $$ Π^{µν} = Tr[ γ^µ G γ^ν G ] $$ If we couple the electrons to an electromagnetic field, we can integrate out the electrons again and obtain an effective action for the electromagnetic field $A^µ$ as $$ S \propto \frac14 F^{µν}F_{µν} + A_µ Π^{µν} A_ν + \dots $$ where the dots indicate corrections of higher order in $A$. (Apologies for my prefactors, they are probably all wrong.) Of course, the effective action for the e.m. field has to be be gauge invariant, which poses certain restrictions on the polarization tensor $Π^{µν}$. > What is the most general form of the polarization tensor $Π^{µν}$, i.e. the > form dictated by gauge invariance alone? According to my calculations, gauge invariance imposes the restriction $Π^{µν}q_µ = 0$, but I'm not entirely sure whether this is correct. Also, I have heard that one possible form is $$ Π_{ij} = \tilde Π(q)(\delta_{ij} - \frac{q_iq_j}{q^2}) $$ at least for the spatial components $i,j$, but I don't think that this is the most general form. > What is the general form of the polarization tensor in a condensed matter > context? Are there other restrictions aside from gauge invariance?


title: How does $F = \frac{ \Delta (mv)}{ \Delta t}$ equal $( m \frac { \Delta v}{ \Delta t} ) + ( v \frac { \Delta m}{ \Delta t} )$?

text: That's how it's framed in my Physics school-book. The question (or rather, the explanation) is that of the thrust of rockets and how the impulse is equal (with opposite signs) on the thrust-gases and the rocket itself. $( m \frac { \Delta v}{ \Delta t} ) = -(v \frac {\Delta m}{ \Delta t} ) = F_i$ I suppose it's a problem with how I see the transfer of impulse and exactly which part of the equation relates to which part of the physical world (gases, rocket). So we can start from there. _**Title equation:_** $F = \frac{ \Delta (mv)}{ \Delta t} = \left ( m \frac { \Delta v}{ \Delta t} \right) + \left ( v \frac { \Delta m}{ \Delta t} \right)$ _Grade:_ The equivalent of G-10 in the US.


title: Efficiency of parallel and sequential heat pumps

text: Consider two identical heat pumps, for example, split-system air conditioners. There're two ways to make them work - in parallel or sequential. Parallel means that "hot" radiators of the machines are put outside the room, while the "cold" radiators are put inside the room. Sequential means that the cold radiator of the first machine is inside the room, while the hot radiator is put inside, say, a large isolated box, where it is cooled by the cold radiator of the second machine, and the hot radiator of the second machine is outside the room. In which case - parallel or sequential - the efficiency of the cooling would be better?


title: Which new insight did $E=mc^2$ give us?

text: I had a special relativity course at university. Now I'm trying to extract what new insight $E=mc^2$ did give us. I mean that moving mass has/is energy (kinetic) not new. The energy merely changed from classical kinetic energy to some relativistic form. And the offset with rest mass doesn't matter for energy. I'm hoping for some in-depth answer. Given a few hours I could probably recall physics myself, but I suppose someone will have a nicer, more fundamental answer :)


title: Is there an overlap between quantum dynamics and math of brownian motion?

text: Suppose you have dynamics of a coherent state. The state presents a normal distribution of finding the particle. Does anyone know of any attempts to connect modern advances in the probability theory to quantum dynamics of a coherent state.


title: What is the mathematical reason for topological edge states?

text: There are many free fermion systems that possess topological edge/boundary states. Examples include quantum Hall insulators and topological insulators. No matter chiral or non-chiral, 2D or 3D, symmetry protected or not, their microscopic origins are similar. Explicitly speaking, when placing such a system on a geometry with open boundary in one spacial dimension (say the $x$-axis), and closed boundary in other spacial dimensions, the bulk model Hamiltonian is always reduced to one or several copies of the following 1D Hamiltonian along the open-boundary $x$-axis direction (see B. Zhou et al.,PRL 101, 246807) $$H_\text{1D}=-i\partial_x\sigma_1+k_\perp\sigma_2+(m-\partial_x^2+k_\perp^2)\sigma_3,$$ where $\sigma_{1,2,3}$ are the three Pauli matrices, and $k_\perp$ denotes the momentum perpendicular to $x$-axis (and could be extended to a matrix in higher dimensions). The existence of the topological edge state is equivalent to the existence of edge modes of $H_\text{1D}$ on an open chain. It was claimed that the edge modes exist when $m<0$. After discretize and diagonalize $H_\text{1D}$, I was able to check the above statement. But my question is that whether there is a simple mathematical argument that allows one to judge the existence of the edge mode by looking at the differential operator $H_\text{1D}$ without really solving it? I believe there should be a reason if the edge mode is robust. PS: I am aware of but not satisfied with the topological argument that the bulk band has non-trivial topology, which can not be altered without closing the bulk gap, thus there must be edge states on the boundary. Is it possible to argue from the property of $H_\text{1D}$ without directly referring to the bulk topology?


title: Double- well potential and Mexican potential

text: Is double well potential related to Maxican hat potential? I have found on Quantum Field Theory in a Nutshell by A. Zee He wrote the double well potential as : $V (φ) = (λ/4)(φ^ 2 − v^2)^2$. Can anyone give me an argument for writing the equation like that.


title: Second baryon octet

text: Let's temporarily ignore spin. If 3 denotes the standard representation of SU(3), 1 the trivial rep, 8 the adjoint rep and 10 the symmetric cube then it's well-known that 3 x 3 x 3 = 1 + 8 + 8 + 10 Interpreting the 3's as the space of up/down/strange flavour states for a quark, the tensor cube is interpreted as the space of baryon states that can be obtained by combining three light quarks. Obviously spin matters, but at least this should give a classification of baryons modulo spin into SU(3)-multiplets. **There are two octets here, but in the literature I have only seen one of them described. What is the second octet?** I appreciate that the answer may be "it's more complicated than that".


title: Is speed of electrons in atoms a constant during explosions?

text: E.g. in exploding nuclear bomb or some other big explosions. I mean if the speed of electrons as waves/particles is a constant or changes according to other "forces" involved?


title: Does the black body emit more than any other type of body?

text: I found this on Wikipedia article on black bodies: > A black body in thermal equilibrium (that is, at a constant temperature) > emits electromagnetic radiation called black-body radiation. The radiation > is emitted according to Planck's law, meaning that it has a spectrum that is > determined by the temperature alone (see figure at right), not by the body's > shape or composition. Now I'm a bit confused. Isn't the black body radiation given by $\frac{P}{A}=\sigma T^4$ So it doesn't depend on body color, composition or anything else than its size. Shouldn't the radiation be the same for all bodies, but we use the black body because its a convenient model because all of its radiation comes from emission and not reflection or anything else?


title: Connections between classical and quantum mechanics?

text: I've done basic or introductory mechanics at the level of Resnick and Halliday. I'm currently studying calculus of variations and the Lagrangian formulation of mechanics on my own. I read somewhere that an understanding of classical mechanics is necessary for appreciating quantum mechanics. Can someone please give me the big picture, where I can see the connections between classical and quantum mechanics? (I am fairly new to quantum mechanics). Is it the case that the Lagrangian and Hamiltonian formulations are what one uses, primarily, in quantum mechanics?


title: Coordinate Transformation of Scalar Fields in QFT

text: By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu $. Correct? Now when I look in the book "Introduction to QFT" by Peskin and Schroeder they state (in an example) that the scalar field $\psi[x]$ under an infinitesimal coordinate transformation $x^\mu \to x^\mu -a^\mu$ transforms as $\psi[x] \to \psi[x+a] = \psi[x] +a^\mu\partial_\mu\psi[x]$. One possible solution I thought was that a scalar field $f:M\to\mathcal{R}$ from a variety (in this case spacetime) is invariant under coordinate transformations, but that the coordinate representation $\psi[x]$ does change under coordinate transformations (obviously). Is this anywhere near correct?


title: Calculating Integrated Precipitable Water using GPS

text: I'm trying to calculate the Integrated Water Vapor (IWV) using GPS data. I'm following the equations in the following publication: > GPS Meteorology: Remote Sensing of Atmospheric Water Vapor Using the Global > Positioning System, MICHAEL BEVIS et al. 1992. link which are the same equations of this website. The problem that I'm having is that I'm obtaing a IWV of an order of magnitude different than I should. Where's the gps data that I'm using:               latitude=-3.09589*pi/180; % degrees to radians     height=85.8/1000; % converting to km     temperature=28.7 + 273.15; % celsius to Kelvin     ztd=2583.3; % mm     pressure=995.9; % mbar      The constants that I'm using: $Rv=461.5$ $k_2'=17$ $k_3=3.776 \mbox{ x } 10^5$ And the calculations that I'm doing: $zhd=(2.2768pressure) / (1-0.00265cos(2latitude)-0.000285height)$ $zwd=ztd-zhd$ $tm=0.72temperature + 70.2$ $\pi=1/(10^{-6}(k_3/tm+k_2')Rv)$ $iwv=\pi .zwd$ I'm obtaing as result a $iwv$ of **504.5616** $kg/m^{2}$ when accordingly with (BEVIS,92) results (looking at fig. 1) I should obtain something like **50.45616** $kg/m^{2}$. Can you help me to find where this problem of magnitude comes from? P.S. If you need more information feel free to ask me.


title: When studying electrodynamics do we assume Maxwell's Equations or derive them?

text: This question is because something made me confused. I always thought that the idea behind electrodynamics was to postulate some things, like Coulomb's law in electrostatics and so on, and then manipulate those things to derive Maxwell's Equations. After that we could then use Maxwell's Equations at will. Now, I'm confused because I've seen people assuming the equations. Namely saying that the electromagnetic field is a pair of vector fields $(E,B)$ such that $E$ and $B$ satisfies Maxwell's equations. Then with this the consequences are derived. But this seems strange to me: if you start by Maxwell's Equations, is because you already know them, but to know them, you needed to start by assuming other things, instead of the equations themselves! My thought was: historically we start really with Coulomb's law and so on, and then we find out that there's a system of $4$ differential equations that completely specify the magnetic and electric field. Then since we found this out, practically speaking, we simply forget about the rest and restart from these equations. Now, why do we do this? Why do we start from the equations rather than deriving them?


title: How to calculate the density of a polycrystalline sample?

text: I'm trying to figure out the density (g/cm^3) of La2CuO4. I know that the mass is 405.355 g/mol; what do I have to do to calculate the volume? Thanks!


title: Does an electomagnetic field "modulate" an electric field?

text: Lets imagine an electromagnetic wave is vertically polarized. If you ignore the magnetic component, the electric field lines would point upwards from the ground or downwards in a cyclical way and of course vary in field strength. Now lets say you attach a battery to two metal plates and you produce an electric field having an horizontal polarization/direction. Does the presence of an electromagnetic wave "modulate" the horizontally orientated electric field, in terms of it's strength and direction?


title: Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

text: I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: http://www.staff.science.uu.nl/~wit00103/ftip/Ch12.pdf. Why is this? I don't think this is necessary from a mathematical point of view. For instance, for $\mathrm{SU(2)}$ we can consider the $2 \times 2$ generators: \begin{equation} \begin{array}{ccc} \displaystyle J_{1/2}^1=\frac{1}{2}\begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix} \; ,& \displaystyle J_{1/2}^2=\frac{1}{2}\begin{pmatrix} 0 & -i \\\ i & 0 \end{pmatrix} \; ,& \displaystyle J_{1/2}^3=\frac{1}{2}\begin{pmatrix} 1 & 0 \\\ 0 & -1 \end{pmatrix} \end{array} \end{equation} However, also the following $3 \times 3$ generators satisfy the Lie algebra: \begin{equation} \begin{array}{ccc} \displaystyle J_{1}^1=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0\\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{pmatrix} \; ,& \displaystyle J_{1}^2=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\\ i & 0 & -i \\\ 0 & i & 0 \end{pmatrix} \; ,& \displaystyle J_{1}^3=\begin{pmatrix} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & -1 \end{pmatrix} \end{array} \end{equation}


title: What's a good book for an advanced undergraduate/early graduate student to learn about symmetry, conservation and Noether's theorems?

text: What's a good book (or other resource) for an advanced undergraduate/early graduate student to learn about symmetry, conservation laws and Noether's theorems? Neuenschwander's book has a scary review that makes me wary of it, but something like it would be great.


title: Gentle slope vs steep slope

text: Take for example a slide of 3m tall. Would an object (starting from rest) sliding down a gentle slope have a lower speed than a steep slope? (Note: Height of slide is the same,disregard friction.) Why is this so? What if the slide was a spiral one?


title: The workings of the Hall effect?

text: I want to ask about the workings of the Hall effect. Why do the electrons come to rest on the edge of the wire? The magnetic field pushes them up, and the electric field pushes them forward. Shouldn't they reach the top of of the conductor and then stop moving up but continue moving forward? Why does the magnetic field stop them from continuing to flow? The magnetic force is up so why would it effect their velocity forwards?


title: What low-level process drives a frequency-doubling crystal?

text: I was reading about second-harmonic generation (SHG) crystals (or frequency- doubling crystals) used to produce green laser light from IR. What low-level process in the crystal is actually driving the combination of two photons into one? I'm thinking of a process at the QED level (or at least motivating what happens at that level). The wikipedia entry on SHG doesn't really address this (it just mentions it's a non-linear phenomena..). Or put another way - what's the simplest system possible (theoretically) that can show the phenomena even if at a very low efficiency? Edit: A supplement to the question, is what entropy-related constraints are there on the incoming photons in these processes, in order to avoid for example turning heat into nicer higher-energy photons?


title: Do I need to take weight of the rocket into account when calculating escape velocity?

text: Here there is the old problem. I know from the old problem that the work $W_v$ that I need to make a rocket fast enough to reach the escape velocity is $$W_v= G \frac{mM}{r}$$ therefore because $$W_v=F\cdot S = G \frac{mM}{r} \rightarrow F_v=\frac{W}{S}=G \frac{mM}{rS} $$ that is the force I need to make a rocket fast enough to reach the escape velocity **BUT**   _Do I also have to count the weight of the rocket?_ If yes then the equation will be like this: $$F_f=F_g - F_v= G \frac{mM}{r^2}-G \frac{mM}{rS}=G \frac{mM}{r}\biggl(\frac{1}{r} \cdot \frac{1}{S}\biggr) = G \frac{mM}{r}(rS)^{-1} $$


title: How is wireless communication possible?

text: Like all my questions, I fear this will be very naive, because my physics background is very limited. Please bear with me. I think of the electromagnetic field as a section of a vector bundle over spacetime, but I think nothing will be lost if we just treat it as a function $f:{\mathbb R}^4\rightarrow{\mathbb R}^6$. Now suppose I want to send you a wireless message (by radio, cellphone, TV, whatever). I encode my message by jigglinig some electrons around so as to change the electromagnetic field from $f$ to a new function $f+g$. You are able to observe some values of this function $f+g$, or at least to infer something about some values of that function from the way the electrons in your receiver are jiggling around. So at the very most, you are able to observe the function $f+g$. How on earth, then, can you infer anything about the function $g$? I understand that you can decompose the field $f+g$ into Fourier components, but given that the ambient field $f$ can be anything at all, I don't see how that helps. The ambient field $f$ is affected by all sorts of things, from other people's communications to cosmic rays, to the fact that I happen to have just carried a 9-volt battery across my living room, much of which you have no knowledge of. Now I add a function $g$ to this completely unknown function $f$, and you're supposed to recover $g$ by observing the sum. Clearly, this can't work. Clearly it goes ahead and works anyway. What am I missing?


title: Finding the Center of pressure of a body immersed in liquid

text: I came across some problems related to finding the center of pressure. Please guide me how to go about them. Posting them all so that I can find out the intricacies involved when there are so many different geometrical shapes involved.   1. A quadrant of the ellipse $$x^2+4y^2 = 4$$ is just immersed vertically in a homogeneous liquid with the major axis in the surface. Find the center of pressure.   2. Find the depth of the center of pressure of a triangular lamina with vertex in the surface of the liquid and the other two vertices at depths $b$ and $c$ from the surface.   3. A circular area of radius $a$ is immersed with its plane vertical and its center at a depth $c$. Find the position of its center of pressure. Thanks in advance!!


title: Topological quantum computation : Anyon model

text: Could someone tell me about Frobenius-Schur indicator and the associated cups and caps notation in context of anyon model. One possible reference could be Parsa Bonderson thesis which is freely accessible here: http://thesis.library.caltech.edu/2447/2/thesis.pdf


title: How to calculate fluid's temperature change in a pump?

text: To calculate the pressure at the outlet of a pump we use pump performance characteristics i.e. charts giving pump head as a function of volumetric flow. When the fluid flows through a pump, it's temperature slightly rises. > Is there a formula or other method to calculate the temperature rise in a > pump?


title: What will happen if we will be able to produce a quark-gluon plasma and the we cool it down?

text: Lets assume that we can produce a quark-gluon plasma and then we try to cool it down, what will happen to it ?


title: Why will choice of coordinates impose functional relations on the metric?

text: I am reading Steven Weinberg's _Gravitation and Cosmology_. On page 10 he says: > In $D$ dimensions there will be $D(D+1)/2$ independent metric functions > $g_{ij}$, and our freedom to choose the $D$ coordinates at will allows us to > impose $D$ arbitrary functional relations on the $g_{ij}$... Can anyone tell me why the choice of coordinates will impose functional relations on the metric?


title: Variational calculus problem

text: I've been wondering lately about a problem that comes from, among other places, an old video game "Lunar Lander". In the game there is a spaceship that has a small tank of fuel, and you're supposed to pilot the ship from an initial height down to the surface of the moon. You win if the landing is sufficiently soft. Your only control (at least in a one-dimensional version of the game) is a dial to control the fuel for thrust during the descent. Finding the optimal strategy is seemingly a problem in the calculus of variations---out of all possible ways to spend the thrust during the descent, you want the one that minimizes the speed when the lander reaches the surface. Besides the game, given that we've landed real ships on the moon or Mars, I'm guessing this optimization problem is well known. Could someone point me to results in textbooks or the literature?


title: To write work capacity of heart by symbols?

text: I know that the work capacity of the heart is               Work capacity of the heart = pumping capacity x aortic impedance.      I would mark work by W and impedance by Z. However, I am not sure how to mark the pumping capacity. I think the units of W are joules and of impedance Ns / m. So pumping capacity m / s^2. I think something wrong, since does not feel right. I think the units of pumping capacity should be m^3. So aortic impedance is N/m^2, which is the same as Pressure. Aortic impedance should not be the same as pressure. My problem is to understand the word capacity. Normally, I relate energy per temperature unit with it but here I cannot. **How can you write this sentence with correct units?**


title: Do electrons in multi-electron atoms really have definite angular momenta?

text: Since the mutual repulsion term between electrons orbiting the same nucleus does not commute with either electron's angular momentum operator (but only with their sum), I'd assume that the electrons don't really have a well- defined angular momentum (i.e., they do not occupy a pure $\left|lm\right>$ state). I would assume that the actual wavefunction is dominated by one such state compared to others, so it is approximately pure, but is there really a points in enumerating electrons according to their momenta, like the s, p, d, f and so on sub-shells?


title: The field of uniformly charged ball (without Gauss theorem)

text: The solution of Poisson equation is given by $$ \mathbf E = \int \frac{\rho (\mathbf r )(\mathbf r_{0} - \mathbf r )}{|\mathbf r_{0} - \mathbf r|^{3}}d^{3}\mathbf r. $$ I tried to use this term for a field of uniformly charged ball and got incorrect result: for the field inside the ball integration gives $$ \int \frac{\rho (\mathbf r )(\mathbf r_{0} - \mathbf r )}{|\mathbf r_{0} - \mathbf r|^{3}}d^{3}\mathbf r = \rho \int \frac{\mathbf r_{0}}{|\mathbf r_{0} - \mathbf r|^{3}}d^{3}\mathbf r - \rho \int \frac{\mathbf r}{|\mathbf r_{0} - \mathbf r|^{3}}d^{3}\mathbf r = $$ $$ = |\mathbf r_{0} = r_{0}\mathbf e_{z}| = \rho r_{0}\mathbf e_{z} \int \limits_{0}^{R} \int \limits_{0}^{\pi} \int \limits_{0}^{2 \pi} \frac{r^{2}dr sin(\theta )d\theta d \varphi}{(r^{2} + r_{0}^2)^{\frac{3}{2}}} + 0 = 4 \pi \rho r_{0}\mathbf e_{z} \left( \frac{1}{\sqrt{2}} - ln(\sqrt{2} + 1)\right), $$ which isn't correct. Where did I make the mistake?


title: Pattern in the Michelson Interferometer

text: I have a question about the Michelson Interferometer with spherical waves: I know that the pattern produced by this kind of waves is a pattern of concentric circles. But what I'm not sure is how the pattern changes when I move one mirror. I believe that If I move the mirror away from the beam spliter, the circles in the pattern will move from the center of the screen to the sides of the screen, i.e. the radius will increase. Am I right?


title: When should I take wine out of the fridge - transient heat transfer problem

text: I am hosting a dinner tonight for which I'll be serving white wine (Riesling to be more specific). Generally white wine is best served chilled (not COLD!) at around `50 F` or `10 C`. Just for kicks, I thought I'd treat this as a problem of transient conduction. I assume that (forced)convection is negligible since I will leave my wine bottle in my kitchen which has _still air_. **The following assumptions are made:**   * The wine bottle is assumed to be cylindrical with an outside to inside radii, $r_o/r_i$ ratio of `1.10`   * The only mode of heat transfer for this bottle is conduction (perhaps a poor assumption?). The kitchen air is considered to be still and at `25 C`   * The un-open bottle of wine is a **closed thermodynamic system**. The glass material has a conductivity $k$ of `1.0 W/m-K` and the wine itself has a specific heat at constant volume, $C_v$ of `2.75 kJ/kg-k` as per this   * The volume of the bottle of wine is `750 mL` or $750 \times 10^{-6} m^3$   * The wine is at a temperature of `5 C` and hence needs to be warmed for a while. The entire wine is assumed to have a **lumped capacitance** (all the wine is at the same temperature with little variation with radius).   * The temperature difference between the wine and the bottle wall is assumed to be $\sim 10 C$ and so is the temperature difference between the bottle wall and the room (just a rough order of magnitude). The first law of thermodynamics (transient) is applied to this closed system bottle of wine: $$\frac{\mathrm{d}{E}}{\mathrm{d}t} = \delta\dot{Q} - \delta\dot{W}$$ The $\delta\dot{W}$ term is zero for this closed system as only heat is exchanged with the kitchen atmosphere. $$\frac{m C_v \Delta T_\text{wine-bottle}}{\Delta t} = \frac{2 \pi k \Delta T_\text{bottle-kitchen}}{ln(r_o/r_i)}$$ This gives me the time the bottle of wine needs to be placed in my kitchen **outside** the fridge as: $$\Delta t \approx 0.025 \frac{\Delta T_\text{bottle-air}}{\Delta T_\text{wine-bottle}} C_v \approx 68 \text{ seconds}$$ This seems to be a rather small amount of time!!! Are my assumptions wrong? Should I improve this with convective heat transfer between the bottle and the kitchen air? Will my guests be disappointed? `:P` **EDIT::Including convective heat transport:** $$\underbrace{\frac{m C_v \Delta T_\text{wine-bottle}}{\Delta t}}_\text{Change in total/internal energy w.r.t time} = \underbrace{\frac{2 \pi k \Delta T_\text{bottle-kitchen}}{ln(r_o/r_i)}}_\text{conduction} + \underbrace{h A \Delta T_\text{bottle-kitchen}}_\text{convection}$$. Here $h$ is the heat transfer coefficient $\sim 1 W/m-K$, $A$ is the surface area of the cylinder. Based on the volume of the cylinder being $70 mL = \pi r_i^2 h$. The height of the bottle is about $1 foot$ or $0.3048 m$ and the generally close assumption that $r_o \approx 1.1 r_i$, I have (all $\Delta T$'s are close and cancel out): $$\Delta t = \frac{m C_v ln(r_o/r_i)}{\left[ 2 \pi k + 2\pi r_o(r_o + h) ln(r_o/r_i)\right]} \\\ \Delta t \approx 260.76 \text{ seconds} \approx 4 \text{ minutes} $$ This seems more plausible..... But I start doubting myself again.


title: How does this argument imply that entropy does not change in a quasi-static adiabatic process?

text: I am working through some notes of Gould and Tabochnik here and I am confused by their argument showing that entropy does not change in a quasi-static adiabatic process. First, entropy is defined to be an additive function of the state which is not allowed to decrease in an isolated system. Then a bit later there is the following paragraph > For pure heating or cooling the increase in the entropy is given by $$dS = > \left(\frac{\partial S}{\partial E}\right) dE.\qquad (2.68)$$ In this case > $dE = dQ$ because no work is done. If we express the partial derivative in > (1.68) in terms of $T$, we can rewrite (1.68) as $$dS = dQ/T \qquad > \text{(pure heating)}. (2.69)$$ We emphasize that the relation (1.69) holds > only for quasistatic changes. Note that (1.69) implies that the entropy does > not change in a quasistatic, adiabatic process. But (2.69) (mis-labelled as 1.69) only applies when no work is done. But certainly work might be done in a quasi-static adiabatic process, so how can (2.69) possibly apply? Instead, is the following argument valid? A quasi-static process is one which passes through a sequence of equilibrium states. In each equilibrium state, the entropy has to be at its maximum value. Thus, the entropy cannot increase in passing from one equilibrium state to another. So overall the entropy can't change in a quasi-static process either. But I don't think this argument can be right either, because then the same argument would apply to any quasi-static process, so why do we need "adiabatic" as well?


title: Can the quantum angular momentum operator be derived from its commutation relations with position and momentum?

text: Exercise 12.2.2 in Shankar's Principles of Quantum Mechanics asks to derive the expression for the angular momentum operator $L_z$ \begin{equation} L_z = XP_y-YP_x \end{equation} using its commutation relations with the coordinate and momentum operators \begin{align} [X,L_{z}]&=-i\hbar Y\\\ [Y,L_{z}]&=i\hbar X \\\ [P_{x},L_{z}]&=-i\hbar P_{y}\\\ [P_{y},L_{z}]&=i\hbar P_{x} \end{align} I have seen a solution which uses the coordinate representations of the momenta. **However, I would like to find one that relies purely on the abstract relations.** All I have been able to prove so far is that $L_z$ and $XP_y-YP_x$ commute. Multiply the first relation on the right by $P_{y}$ and the third relation \begin{align} [X,L_{z}]P_{y}&=-i\hbar YP_{y}\\\ Y[P_{x},L_{z}]&=-i\hbar YP_{y} \end{align} Equating the LHS's we obtain \begin{align} (XL_{z}-L_{z}X)P_{y}&=Y(P_{x}L_{z}-L_{z}P_{x})\\\ XL_{z}P_{y}-L_{z}XP_{y}&=YP_{x}L_{z}-YL_{z}P_{x} \end{align} Now subtract $XP_{y}L_{z}$ from both sides \begin{align} XL_{z}P_{y}-XP_{y}L_{z}-L_{z}XP_{y}&=YP_{x}L_{z}-XP_{y}L_{z}-YL_{z}P_{x}\\\ X[L_{z},P_{y}]-L_{z}XP_{y}&=(YP_{x}-XP_{y})L_{z}-YL_{z}P_{x}\\\ \end{align} Add $L_{z}YP_{x}$ to both sides \begin{align} X[L_{z},P_{y}]+L_{z}YP_{x}-L_{z}XP_{y}&=(YP_{x}-XP_{y})L_{z}+L_{z}YP_{x}-YL_{z}P_{x}\\\ X[L_{z},P_{y}]+L_{z}(YP_{x}-XP_{y})&=(YP_{x}-XP_{y})L_{z}+[L_{z},Y]P_{x}\\\ [L_{z},XP_{y}-YP_{x}]&=X[P_{y},L_{z}]-[Y,L_{z}]P_{x} \end{align} However, using the commutation relations we deduce that $X[P_{y},L_{z}]=[Y,L_{z}]P_{x}=i\hbar XP_{x}$, therefore \begin{equation} [L_{z},XP_{y}-YP_{x}]=0 \end{equation} Any other manipulations left me spinning in logical circles. I would appreciate any help in completing the argument.


title: Water, how do you make ripples

text: How do I make a ripple effect in a bowl of water. I have tried throwing small pebble sin but it just splashes and sinks. I have read that I need to make the water a thicker liquid ie add glycerine, will this work and why?


title: Absorbance by using Poynting vector

text: How should I compute the amount of energy of an EM wave absorbed by a material? Can I just use the divergence of the Poynting vector?