title: Would a strong diamagnetic material exhibit inductance?

text: Erm. This question got shot-down in electronics.stackexchange.com and somebody recommended I raise it on here, so ... Do metals with the property of strong diamagnetism also exhibit inductance? Would a fluctuating magnetic field induce a weaker current in a strongly diamagnetic metal as compared against a ferromagnetic/paramagnetic metal? I remember reading metal-detectors are based upon the property of inductance. So if a metal detector were to hover over a silver ingot/plate, would it succeed or fail?


title: Is a tankless water heater more or less efficient than an electric stove at heating water?

text: Consider the following scenario that I actually encounter frequently: I just finished washing a pot in the sink, and now I want to boil some water for cooking noodles. If I fill the pot with hot water from the pipes, my stove will require less energy to bring the water to boiling, but my tankless water heater will kick in and heat an equivalent volume of water to displace the water taken from the pipes. Or, I can fill the pot with cold water, which won't trip the water heater, but will instead require more energy from the stove to heat. It seems to me that if my goal is to minimize energy spent on heating my water, I need to know which is more efficient at heating water - the tankless water heater, or the stove. I have seen this question, but it is about heating water on the stove vs. in the microwave, which is a different situation. Which is the more efficient solution?


title: Bound states and scattering length

text: 1. What is the relationship between bound states and scattering length?    2. What is the relationship between scattering states and scattering length?    3. When we say, potential is 'like' repulsive for positive scattering length and viceversa, are we talking with respect to scattering states or bound states? (though the answer should be scattering states, but the literature everywhere assumes you to know it on your own.)  All these concepts are found in the theory of BCS-BEC crossover.


title: Vectors with more than 3 components

text: 1. I have some confusion over Vectors, Its components and dimensions. Does the number of vector components mean that a vector is in that many dimensions? For e.g. $A$ vector with 4 components has 4 dimensions?   2. Also, how can a Vector have a fourth dimension? How can we graphically represent vectors with more than 3 dimensions? Its hard for me to visualize such a vector, Can anyone point me to some resource where they explain this graphically and in detail?


title: How "things" radiate electromagnetic radiation?

text: How things radiate electromagnetic radiation? I don't ask why they radiate (higher temperature than 0K) but how they radiate this electromagnetic waves?


title: Does timbre consist in pitch and volume?

text: I read that the physical properties of a sound wave correspond to its audible qualities: pitch, volume, and timbre. However, an oscilloscope uses only two- dimensions to accurately depict the physical properties of a wave. Intuitively, pitch and volume seem more basic than timbre does, so I surmise that timbre must consist in those two properties.   1. Does timbre consist in those two properties?   2. How do two-dimensional waveform diagrams depict three properties of sound waves?


title: Double-slit experiment formulated in a quantum circuit

text: What would the double-slit experiment or something analogous to it look like implemented as a quantum circuit/program? Also, what about the delayed choice quantum eraser, how would that look in a quantum circuit? For example, if implemented in this quantum computing simulator: http://www.davyw.com/quantum/?example=Toffoli . I couldn't find any direct answer via Google, so even good search terms would be useful.


title: Sharp cut-off, quadratic corrections and naturalness

text: When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to which point can we trust this cut-off as the true validity limit of our theory? If we do not trust this regulator, there is no way of seeing that the corrections to the Higgs mass are quadratically divergent, right? So how can we know that, in a renormalizable theory, an elementary scalar should have his mass around the highest possible scale in the theory? Moreover, does it even make sense of talking about the cut-off of a theory when it is renormalizable? Since the notion of a cut-off, the quadratic corrections to the Higgs mass and naturalness are the driving principle to almost every BSM particle physics model I'm pretty puzzled with this. Any thoughts, hints or literature that could clarify these ideas?


title: Could dark energy be powered by force particles that obey quantum mechanics?

text: From what (little) I know about physics, I understand that the universe is expanding due to dark energy, and I understand that no one quite understands it yet. I also understand that the cosmic background radiation hints that there were quantum fluctuations in the universe's infancy. Could the background radiation fluctuations be explained by assuming dark energy consists of force carrying particles that obey quantum mechanics? That is, they generally appear to follow the path of least action, but due to their quantum-mechanical probablistic nature, may fluctuate slightly?


title: Space time curvature due to electric charge or magnetic charges

text: since we know that gravitational force is nothing but a curvature in space- time. I have a similar analogous for the electric or magnetic charges. Similarity is that both electromagnetic and gravitational forces follow the inverse square law. consequence of this can be that we can have a another interpretation of electron orbital motion causes by curvature in space due to electric charge (as mass of electron is very small so STR have no significance here). But problem is that we have two different type of charges (+ve and -ve) which gives repulsion and attraction, whereas in gravitation force mass always gives the attraction forces. Please correct me if I am wrong.


title: Heat transfer, Cold vs. Hot

text: If you boil water inside a pot the outer rim bubbles first, I imagine because its hotter. Does that same concept apply for the inside of a refrigerator for example. Is the outer rim always more extreme, or at least at first?


title: Difference between Clausius-Clapeyron and Van't Hoff equation

text: I was wondering what is the difference between the Clausius-Clapeyron equation and the Van't Hoff equation. They appear to have the exact same physical meaning and are often used interchangeably.


title: Questions from elevator ride

text: I like to play inside the lift (elevator). For instance, there are bars attached at the side of the lift, and I like to hold my body up using my two hands on the bar. I realised that I actually feel lighter when the lift is decelerating and heavier when it is accelerating. Takes the upward motion as positive. Also, I tried weighing myself inside the lift using a balance ;D As most people know, my weight increases as the lift is accelerating, and vice versa. Does this mean that when I am holding an object inside the lift, the object will actually feel heavier when the lift is accelerating? Lastly, and this question is quite stupid. My friend once told me that if you want to survive an elevator crash, just jump the moment it hit the floor. I told him that's a stupid idea, but I can't really explain why it wouldn't work! Please help me.


title: Absorption / Control / Reduction of EMF radiation using passive antennae & matched load

text: Wanted to understand the physics behind usage of passive antennae and matched load combination, to absorb, control and reduce the Electromagnetic-Field (s.a. due to microwave radiation from cellular phone towers), within a confined area (s.a. a room). Also, does the shape / size / material used for the antenna have a role to play ? If this kind of absorption does work, what might be the range / shape of the area that such a device can reduce / remove the radiation from ? I have come across EMF shielding/reduction solutions which use such combination, and also some sort of Faraday-cage effect to keep out EMF radiation, which require "grounding". Are these two based on similar / related principles ? Finally, I read in an article that not grounding the Faraday-cage (or the metallic protective shield), results in the metal starting to becoming radioactive, or maybe even emit X-rays. Please do excuse the rather layman approach to the questions, a dumbed-down but factual (i.e. can be backed by theory and empirical data, if required) explanation would be highly appreciated.


title: Does spontanous symmetry breaking affect Noethers theorem?

text: > Does spontanous symmetry breaking affect the existence of a conserved > charge? And how does depend on whether we look at a classical or a quantum field theory (e.g. the weak interacting theory)? (In the quantum case, if we don't want to speak of the Noether theorem, the question can be worded as how does the field breaking the symmetry affect the identities resulting from the Lagrangian symmetry.) * * * The question came up as I wondered if you can make a gauge theory out of every "kernel" of the process in which you compute obervables. In the sense that if $$\langle A \rangle_\psi=\int (\psi^*A\psi)\text d V,$$ the transformation $T:\phi\rightarrow \text e^{i\alpha}\phi$ is in the kernel of $\langle - \rangle_\psi$.


title: Is there a single metric for a given system?

text: Let imagine a tunnel that connect two distant places at the globe (eastern- western or north-south) There are a lot of posible "distances" or metrics, defined by maps, routes, "as the crow flies", etc.. but none of those distance can be shorter than the distance of the tunel. So if two trains travels at same speed, one inside the tunnel and other above in the surface, the one on the tunnel will reach first. ![enter image description here](http://i.stack.imgur.com/Rydj2.jpg) If this is possible, then perhaps it's possible to have differents coexisting metrics with differents dispositions or topologies, within the same system. Of course that if we describe a space-time metric surrounding a sphere, then "holes" in it would change the metric (just because it's not a sphere anymore). But it's strange for me that making a hole we could change in some way the space-time shape. In an extreme case. Could be an euclidian space _of same dimension_ be build within a non-euclidian space? I would like to have a view from people familiar with general theory of relativity, thanks


title: Curvature of spacetime: pincushion distortion?

text: This may be an elementary question, but if gravity causes a curvature in spacetime, then why isn't everything distorted when looking down on earth, or up at the moon? Shouldn't there be a pincushion effect when viewing an object that is bending spacetime? I understand that being here on the surface of the earth, everything relative appears normal, but far away, shouldn't it all look warped?


title: Walter Lewin Lectures in HD

text: I like the lectures by Walter Lewin 8.0x. However the quality of the videos is pretty bad. Is there any way (DVD, web,...) to get the lecture videos in a good quality, best in HD?


title: What is a TOE, a Theory of Everything

text: I see many learned contribution about the role of a TOE, what it might do or not do, what kind of answer it might provide, and what not. But I do not know what a TOE is, how I would recognize it if I met it in the street. Is it an axiomatic theory, a collection of equations, a sacred book, an incoherent dream, a good topic for fun discussions on a forum, or just a unification of QM and GR, which might not even tell us everything about either. It would be nice for naive users like me to know what is being talked about, and whether it is the same for all who talk. Personally, I naively took it to mean a complete description of the physical universe, which does remain very vague, and may not be meaningful.


title: Star product of two commuting spinors

text: Ok so this might be a very stupid and trivial question but I have spent a couple of hours on this little problem. I am trying to derive a simple formula in a paper. We have a real commuting spinorial variable $y_\alpha$ ($\alpha = 1,2$) ($y^\alpha = \epsilon^{\alpha\beta}y_\beta$, $\epsilon^{\alpha\beta}=-\epsilon^{\beta\alpha}$, $\epsilon^{12}=\epsilon_{12}=1$). The linear space of functions of the variables $y_\alpha$ is endowed with the structure of the algebra with the following $*$ product law: $$(f*g)(y) = \int_{\mathbb{R}^4} \\! d^2u~d^2v~ \exp \left[i u_{\alpha} \epsilon^{\alpha\beta} v_{\beta}\right] f(y+u)g(y+v).$$ I am trying to compute the $*$ commutator $$[y_\alpha,y_\beta]_{*} := y_\alpha * y_\beta - y_\beta * y_\alpha = 2i\epsilon_{\alpha\beta},$$ but when I try to use the formula for lets say computing $y_1*y_2$, I get a divergent integral $$y_1*y_2 = \int d^2u\,d^2v \,\,\, (y_1+u_1)(y_2+v_2)\exp[i (u_1v_2-u_2v_1)].$$ I feel like I am applying the formula wrong but I can't understand where. Can someone help me?


title: Special relativity and massless particles

text: I met an approval that massless particle moves with fundamental speed c and this is the consequence of special relativity. Some authors (such as L. Okun) like to prove this approval by the next reasoning. > Let's have $$ \mathbf p = m\gamma \mathbf v ,\quad E = mc^{2}\gamma \quad > \Rightarrow \quad \mathbf p = \frac{E}{c^{2}}\mathbf v \qquad (.1) $$ and $$ > E^{2} = p^{2}c^{2} + m^{2}c^{4}. \qquad (.2) $$ For the massless case $(.2)$ > gives $p = \frac{E}{c}$. By using $(.1)$ one can see, that $|\mathbf v | = > c$. But as for me, this is non-physical reasoning. Relation $(.1)$ is derived from the expressions of impulse and energy for the massive particle, so it's scope is limited to massive cases. We can show, that massless particle moves with speed of light by introducing hamiltonian formalism: for the free particle $$ H = E = \sqrt{p^{2}c^{2} + m^{2}c^{4}}, $$ for massless particle $$ H = pc, $$ and by using Hamilton equation it's easy to show, that $$ \dot {|r|} = \frac{\partial H}{\partial p} = c. $$ But if I don't like to introduce hamiltonian formalism, what can I do for proving an approval about speed of massless particle? Maybe, expression $\mathbf p = \frac{E}{c^{2}}\mathbf v$ can be derived without usage the expressions for massive case? But I don't imagine, how to do it by using only SRT.


title: ratio between work and heat

text: I am really stuck on a problem in my textbook: Water is heated in an open pan where the air pressure is one atmosphere. The water remains a liquid, which expands by a small amount as it is heated. Determine the ratio of the work done by the water to the heat absorbed by the water. MY ATTEMPT: We are given that: $P = 1.013 \cdot 10^5 Pa$ We then have: $$\frac{W}{Q} = \frac{P \Delta V}{cm \Delta T} = \frac{P \beta V_0 \Delta T}{cm \Delta T} = \frac{P \beta m \Delta T}{ cm \rho \Delta T} = \frac{P \beta}{c \rho} = \frac{1.013 \cdot 10^5 \cdot 207 \cdot 10^{-6}}{4186 \cdot 1} = 5 \cdot 10^{-3}$$ But according to the textbook, the solution should be $4.99 \cdot 10^{-6}$. If anyone can help me by pointing out what I'm doing wrong here, I would be extremely grateful!


title: Explanation for self-rupture glass is needed

text: I witnessed a phenomenon that I couldn't conclude its cause. Please bear with me for the length of the recall, for I merely want to include any details that might help us to investigate. I had a **cooking glass lid** sat on a wooden shelf that is **away from** the stove and oven and other heating objects. The shelf is nailed on the wall and is situated just above my eye level, and a counter top is also on the same side of the wall where the shelf is installed. Now here comes the surprise. In a winter afternoon 2011, my room had almost the same temperature as an autumn morning, and while I was cutting my lettuce on that counter top which I pointed out in above passage, **a pounding sound** , as if a heavy car door slam or a tree trump falling on top of the roof, knocked its introduction from the shelf that was just above my eye level. First, I thought I may had knocked something around me off(which I didn't believe that for there wasn't anything around me to knock off); then I thought it may be my neighbor next door dropping a heavy box; last, I suspected somewhere my roof top collapsed. But it was my third suspicion directed me to meet that glass lid I mentioned above, and **I found it had ruptured completely like glacier creaked BUT still having all broken pieces bounded without any pieces scattering toward random direction! Only the nob of the lid popped out partially.** Before this happened, I hadn't used that lid for cooking for years, and I didn't removed it from any heating object nor there was something on top of the lid that day, and I believe what the lid had maybe just an invisible layer of dust. I was glad my face hadn't been stung by any glass residues, but ponder what really happen to that glass lid and why it ruptured without collapsed. Below, I attached 2 pictures of the scene from that day. If you have any similar experience or know the theory behind it, may you please drop me an explanation to this incidence? Thank you in advance. I'm new and need reputation to post pictures. I definitely post that 2 pictures once I earn enough reputation point.


title: Why do some beverages stay cold longer than others?

text: For my daughter's science experiment, she placed six beverages (cola, diet cola, milk, chocolate milk, apple juice, and water) in the exact same amount in the exact same type and size of plastic cups, and placed all of the cups in a refrigerator at the same time and allowed them to remain in the refrigerator for the same period of time (4 hours). When the beverages were taken out and set on the same counter, in a room at approx. 70 degrees, a cooking thermometer was then used to measure the temperature of each of the beverages at different time intervals (the intervals were the same for all beverages)---i.e., the temperature of each was measured at 15 minutes, 45 minutes, 1 hour, and 90 minutes. The results were that all but two of the beverages held the same temperatures throughout the measuring period. The two beverages with different (colder) temperatures were diet cola and (white) milk. Of course, my daughter is supposed to cite to references that support (and explain) the basis of the test results/conclusions, and we are finding nothing that addresses this. "Viscosity" is the closest principal that we have found, but we cannot find anything that provides the viscosity levels for the separate beverages (hence, no conclusion can be drawn). Any ideas on where to find research references related to this? thanks,


title: Spin half for the value of $|1 0\rangle$?

text: Spin-1/2 The eigenspinor , $X=aX_++bX_-$ $$X_+=\left( \begin{array}{cc} 1\\\ 0\end{array} \right) $$$$X_-=\left( \begin{array}{cc} 0\\\ 1\end{array} \right)$$ They are define like this because they work well in the following? $S_zX_+={\hbar}/2X_+$ and $S^2X_+={\frac{3}{4}}{\hbar}X_+$. But for $|s m \rangle$, I don't understand why do we need to put $|1 0\rangle = \frac{1}{\sqrt2} | \uparrow \downarrow + \downarrow \uparrow\rangle$ Because without $\frac{1}{\sqrt2}$ we can prove the eigenvalue of $S^2$ is 2har. Why $\frac{1}{\sqrt2}$?


title: What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics?

text: I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics? The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful. I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.


title: Duration of Satellite orbit in the shadow of the Earth

text: I want to determine how many minutes a satellite is in a circular orbit around the Earth at about $1000 km$ altitude. I assumed that the Sun-Earth vector lies exactly in the orbital plane of the satellite. Also, in this case, the Sun can be seen as a point light source and the distance to Earth is infinite. Is it possible to make an approximation of the duration that the satellite is on the 'dark' side of the Earth? Or do I need more information, like the speed of the satellite? The radius of the Earth is $6378 km$. And one orbital period is ofcourse $24$ $hours$.


title: Need to use lasers,mirrors and prisms for a project.How to make the light visible and presentable at the same time?

text: Alright so I really want to put some art into science for my project.project. I'm thinking of creating a beautiful image,preferably in air.I read about the visible lime-green lasers but i still want to be able to use other colors,and I need to make it visible.How can i make "fog" for it so that a pretty image is seen yet at the same time the inspectors won't be suffocating.Any ideas on how should I make it and what other things I can use to create pretty images?Please explain in details as I might not know some terms in optics.Thanks in advance!


title: What happens to the amplituhedron in a non-peturbative context?

text: The **Amplituhedron** has recently been popular; it supposedly encodes **perturbative scattering amplitudes** in a simple, **geometric** fashion. What happens to it in a **non-perturbative** context? Is there still some sort of amplituhedron, somehjow? If so, a side question: Can the amplitudihedron be able to solve the problem of 5-Brane scattering amplitudes in M-Theory?


title: Why don't we include $\rho g h$ in a hydraulic jack below?

text: Consider a hydraulic jack with massless pistons as follows. ![enter image description here](http://i.stack.imgur.com/MVJaU.png) The famous equation for this system is $$ \frac{F_1}{A_1}=\frac{F_2}{A_2} $$ My question is why isn't the equation as follows? $$ \frac{F_1}{A_1}=\frac{F_2}{A_2} + \rho g h $$ It is based on my understanding that any points in the same horizontal line have the same pressure. Could you spot my misconception?


title: Helmholtz Free Energy, Partition Function

text: I'm trying to develop some basic intuition here, so this comes mostly as a jumble of commentary/questions. Hope its acceptable. Helmholtz Free Energy: $A = -{\beta ^{-1}}lnZ$. I find this statement to be incredible profound. Granted, I found it yesterday. Suppose my system has one energy state with no degeneracy. $Z = e^{-\beta E_1}$, then $A = E_1$, which I suppose says if the system consists of one particle, all its internal energy is available for work. That's nice. Now, if we introduce some degeneracy $\gamma$, we get $Z = \gamma e^{-\beta E_1}$, and so $A = E_1 - \beta ^{-1}ln \gamma$, and we have clearly lost some of our free energy to the degeneracy (ie. to the fact that there are multiple microstates for our given macrostate, and so we have limited information about the actual configuration of the system, which is free to explore its micro states, limiting the energy we can get from it). So that's nice too. We can go further by introducing more energies, so $Z = \Sigma \gamma_i e^{-\beta E_i}$, but nice analysis is confounded by my inability to deal coherently with sums in a logarithm. Though I managed to show that $A$ for such a multi-state system is strictly less than $\Sigma [E_i-\beta ^{-1}ln\gamma _i]$, ie. less than the sum of the free energies for independent systems of a given energy $E_i$ and degeneracy $\gamma_i$ . This result, however, requires $E_i > 0$, which I take for granted, but makes plenty sense. Now, what does it mean for A to be negative? Perhaps more importantly, how does one simply go about obtaining work from a system with some A (a practical question)? Or, perhaps even more importantly, is it this requirement that there be a second final state, seemingly of lower free energy, that makes $A$ itself not so significant, but rather $\Delta A$? And if so, what happens to the intuition about a system with only one state having exactly its energy as free-energy? Your insights on these and related matters pertaining to legendary $Z$ and its relation to $A$, as well as pointers on where my thinking may be flawed or enlightened, are much appreciated.


title: Constraint on a metric

text: Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$ why does it follow that $a=b$? I can't quite spot a constraint condition...


title: Working out the mean velocity of particles in a gas

text: I'm trying to answer the following question: > Air consists of molecules Oxygen (Molecular mass = 32$amu$) and Nitrogen > (Molecular mass = 28$amu$). Calculate the two mean translational kinetic > energies of Oxygen and Nitrogen at 20($^\circ C$) To solve it I have done: Use $E = \frac{3}{2}kT$ Energy = $\frac{3}{2} \times (1.38 \times 10^{-23}) \times (20+273) = 6.07 \times 10^{-21}$ Use $KE = \frac{1}{2}mv^2$: For oxygen: $\sqrt{\frac{6.07 \times 10^{-21}}{2 \times (32 \div 6.02\times 10^{23})}} = 15.11$ However, 15.11 isn't the answer in the textbook (the answer is 480m/s) For nitrogen: $\sqrt{\frac{6.07 \times 10^{-21}}{2 \times (28 \div6.02\times 10^{23}) \div 32}} = 16.12$ 16.12 isn't the answer either (it's 510m/s) I know that my answers are wrong (gas molecules don't move as slow as I calculated at room temperature) but I can't see why my method doesn't work. Any help?


title: Why is spectral sensivity of a photodiode expressed in A/W

text: Can someone explain me the meaning of the A/W unit of the photosensivity when reading a spectral response function of the wavelength characteristic of a photodiode?


title: Increase in velocity by loss of mass?

text: > _A trolley of mass 300kg carrying a sand bag of 25kg is moving uniformly > with speed of 27km/h on a frictionless track. After a while, sand starts > leaking out of a hole on the floor of the trolley at the rate of 0.05 kg/s._ > > _What is the speed of the trolley after the entire sand bag is empty?_ I was so surprised when I read this question. It doesn't make sense to me. I can't comprehend how the loss of sand creates an **external** unbalanced force on the trolley such that it affects its velocity. Maybe I haven't analyzed the question enough but I find this a bit conceptually challenging for me. Maybe I have to consider how the sand particles affect the back wheels of the trolley or maybe consider the sand to be a propellant?


![enter image description here](http://i.stack.imgur.com/1YRyD.png) The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1,1/2,1/3,… form the (musical) harmonic series. How can Hilbert spaces be used to study the harmonics of vibrating strings?


title: Electrostatics/ magnetostatics: why is $\int_\text{all space} d\vec r \; \nabla \cdot(\vec A \times \vec B)$ equal to 0?

text: I'm reading electrodynamics notes and come across that: $$\int_\text{all space} d\vec r \; \nabla \cdot(\vec A \times \vec B)=0$$ in case of magnetostatics and: $$\int_\text{all space} d\vec r \; \nabla \cdot(\phi \vec E)=0$$ in case of electrostatics. ($A$ is the magnetostatic potential, $B$ the magnetic field and $\phi$ the electrostatic potential) My question: is why are they equal to $0$? * * * The above mentioned formulas are used to show that: $$W=\frac{\epsilon_0}{2 \ } \int_\text{all space} d \vec r\vec E ^2$$ $$W=\frac{1}{2 \mu_0} \int_\text{all space} d \vec r\vec B ^2$$ starting from $$W=\frac{1}{2 \ } \int_\text{all space} \phi(\vec r) \rho(\vec r) $$ $$W=\frac{1}{2 } \int_\text{all space} d \vec r \vec j \cdot \vec A$$ with $\rho$ the charge density, and $\vec j$ the current density * * * I've tried using Gauss's Theorem: $$\int_{\partial V} d\vec r \; (\vec A \times \vec B)\cdot d\vec S=0$$ and: $$\int_{\partial V} d\vec S\; \cdot (\phi \vec E)=0$$ but this doesn't bring me any further to solving my problem.


title: Fock state and Slater determinant

text: Let's have Fock state for fermions: $$ | \mathbf p_{1} , \mathbf p_{2}\rangle = \frac{1}{\sqrt{2}}\hat {a}^{+}(\mathbf p_{1})\hat {a}^{+}(\mathbf p_{2})| \rangle , \quad | \mathbf p_{2} , \mathbf p_{1}\rangle = -| \mathbf p_{1} , \mathbf p_{2}\rangle . $$ How to get the expression for Slater determinant by starting from definition above?


title: Momentum-Representations in Quantum Mechanics

text: Why do we get information about position and momentum when we go to different representations. Why is momentum, which was related to time derivative of position in classical physics, now in QM just a different representation brought about by some unitary transformation. Is Ehrenfest's theorem the only link? I just started studying QM. So please suggest some references explaining the structural aspects and different connections.I don't want to start with noncommutative geometry. I would like something of an introductory nature and motivating.


title: How can you solve this "paradox"? Central potential

text: A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. The equation of motion can be obtained also from the Lagrangian. if we substitute, however, the conserved angular momentum herein then the centrifugal potential arises with the opposite sign. So if we naively apply the Euler-Lagrange equation then the centrifugal force appears with the wrong sign in the equations of motion. I don't know how to resolve this "paradox".


title: What's the physical significance of the inner product of two wave functions in quantum region?

text: I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the significance of that product? I Googled that and someone call that probability amplitude, but that product could be complex, so does it tell any phyical significance? In terms of quantum interference, shall we add the wavefunctions or the probability amplitudes before taking the square of modulus? Sorry I am just starting on learning the quantum mechanics and many concepts are pretty confusing to me.


title: Quantum field theories with asymptotic freedom

text: QCD is the best-known example of theories with negtive beta function, i.e., coupling constant decreases when increasing energy scale. I have two questions about it: (1) Are there other theories with this property? (non-Abelian gauge theory, principal chiral field, non-linear sigma model, Kondo effect, and ???) (2) Are there any simple (maybe deep) reason why these theories are different from others? It seems that the non-linear constraint of the non-linear sigma model (and principal chiral model) is important, but I have no idea how to generalize this argument to other theories.


title: phase transformation of liquid under motion

text: I was testing in my lab with water and found that it starts to solidify when it is stationery at 0 degree centigrade.but when I move the liquid with some velocity it dosent change its phase.my interpretation of this happening because when I move liquid it has kinetic energy which dosent allow it to crystallize.but at certain velocity and further lowered temperature it should start crystallising again.is there any graph which shows at what velocity and temperature it crystallize with atmospheric pressure?as in phase transition of water under temperature and pressure?


title: Strange light polarization effect?

text: I spent a while working with MgF2-windowed xenon flash / discharge lamps. Primarily, I characterized their spectra with two normal-incidence spectrometers against a calibrated Deuterium lamp. In this particular case, it included a Czerny-Turner-type design. As side investigation, I rotated one of the xenon flash lamps by 360° in about 45° steps and took time-resolved spectra in the range of 200 to 500 nm for each rotation step. Time-resolved means, that I took measurements / spectral scans at an interval of 0.1 µs with an exposure time of 0.168 µs. Because the exposure time is longer that the interval, I took only one spectral scan per one flash, for each flash delayed by another 0.1 µs. I assume this is ok because the intensity of the light emitted by the discharge over its entire spectrum is stable with $$ \sigma = 0.0294 $$ So for each rotation step and each interval, I integrate over an entire calibrated spectrum. If I plot the result, it looks somewhat like this: ![some plot](http://i.stack.imgur.com/zyb88.png) (The actual flash is "visible" for about 1.5 µs, while there is an "afterglow" mostly in longer wavelengths with less than 5% of the maximum intensity for about another 40 µs.) My understanding is, that normal-incidence spectrometers behave like polarization filters. So I would expect perfect ellipses - point symmetric - if the the light is (partially) polarized. What I see looks different, unsymmetrical. If have also done another type of plot. The first one shows how it theoretically should look like: ![some plot](http://i.stack.imgur.com/C8roc.png) And this is what I get: ![another figure](http://i.stack.imgur.com/icVSv.png) I assume, that I see ellipses. The semi-major-axis / maximal radius is indicated red (1), the semi-minor-axis / minimum radius is indicated blue (-1). My questions ... is there any physical context, that could explain the unsymmetrical behaviour (other than uncertainties in measurements?). If there is not, would it be appropriate to fit ellipses into the data? **EDIT (1):** ![another one](http://i.stack.imgur.com/X2sSK.png) Based on the reply by @akhmeteli, I looked into different wavelengths. Top left: 230 nm; top right: 260 nm; bottom left: 362 nm; bottom right: 461 nm. The first three represent spectral lines, the fourth one is a random choice without any specific feature. My data has a resolution of 0.2nm. Here, I integrated from lambda-0.2 to lamda+0.2 nm, kind of as narrow as possible. **My impression is, that the observed effect does not (very much) depend on the wavelength.**


title: Curved spacetime point particle Lagrangian density

text: This is probably trivially related to the question: Action for a point particle in a curved spacetime , but am a bit unsure how to write it as a Lagrangian density. In curved spacetime the action is related to the Lagrangian density by: $$ S = \int \sqrt{-g} \, \mathrm{d}^4 x \ \mathcal{L}(x^\mu)$$ The simpilest way I can think of describing a point mass taking a path through space time would be something like: $$ \mathcal{L}(x^\mu) = -m\int d\alpha \ \delta^4(x^\mu-s^\mu(\alpha))$$ Where $\alpha$ is parameterizing a path $s^\mu(\alpha)$ through spacetime. Is that the correct Lagrangian density? Can someone show me how to manipulate these mathematical structures to check that this simplifies somehow to the known action for a free particle: $$ S = - m \int d\tau $$ or how it relates to what was written in the other question? $$\mathcal S =- m \int\sqrt{g_{\mu\nu}(x)\frac{dx^\mu(\xi)}{d\xi} \frac{dx^\nu(\xi)}{d\xi}} \ \ d\xi$$


title: How are atomic bonds created?

text: From what I have learned in my chemistry course, Electrons with similar quantum numbers but with opposite spin are attracted to each other. What does this mean when there is a covalent bond being formed between lets say hydrogen and fluorine? I can think of four different results: A) A bond is not formed until an unpaired fluorine electron meets an unpaired hydrogen electron with the same energy shift (in this case, 0). B) The fluorine or hydrogen electron forces the other electron to have the same energy shift. C) The electrons don't change, but pair up regardless of quantum numbers. D) The electrons from both atoms enter into a different state that I have not learned yet. Thanks in advance for any help received!


title: Can a solid object float on a frictionless liquid?

text: Assuming the solid is less dense than the liquid, will a solid object float on a frictionless liquid? I can imagine that due to the pressure gradient the object will float, but I can also imagine that without friction there would be no upwards force on the object and it would sink. What would happen?


title: Gravastars: Are they observationally distinguishable from Black-Holes?

text: Are observations of Hawking radiation at the acoustic event horizon in Bose- Einstein condensates consistent with Gravastars? To reconcile the second law of thermodynamics with the existence of a black hole event horizon, black-holes are necessarily said to contain high entropy while Gravastars not at all. An Event-Horizon forming out of a collapsing star's intense gravity sufficient enough to force the matter to phase change transforming into Bose-Einstein condensate would be such that nearby matter would be re-emitted as another form of energy, and all matter coming into contact with the Event-Horizon itself would become incorporated. So, it seems reasonable to wonder if Black-Holes are distinguishable from Gravastars since Gravastars appear to be better emitters, and Black-holes better entropy sinks. What do observations of Hawking radiation from acoustic black holes from Bose-Einstein condensate seem to suggest?


title: What is the difference between a battery and a "polarized" charged capacitor?

text: I know that the main difference between the two is that a battery can provide a constant voltage whereas a capacitor's voltage decreases as the charge stored decreases. But what about the internal structure ? A battery also has chemical reactions going inside of it while a **polarized** capacitor also has an electrolytic structure. So is there or is there not any difference between the internal workings of a **polarized** capacitor and a battery?


title: Hilbert space of a free particle: Countable or Uncountable?

text: This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but the Hilbert space of a free electron is uncountable. But the arguments about smoothness and delta functions in the answers to the previous question convince me otherwise. Why is the Hilbert space of a free particle not also countable?


title: Hilbert space of harmonic oscillator: Countable vs uncountable?

text: Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of possible basis states be the set of $\delta$-functions $\psi_x = \delta(x)$, and that indicates that the size of my Hilbert space is that of $\mathbb{R}$. On the other hand, we all know that we can diagonalize $H$ by going to the occupation number states, so the Hilbert space would be $|n\rangle, n \in \mathbb{N}_0$, so now the size of my Hilbert space is that of $\mathbb{N}$ instead. Clearly they can't both be right, so where is the flaw in my logic?


title: Ambiguity in number of basis vectors

text: The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\\{|n\rangle\\}$ in the problem of particle in a box which can be used to expand a general state $|\psi\rangle$ as: $$|\psi\rangle=\sum\limits_{n=0}^{\infty} C_n |n\rangle$$ Here, the basis implies that the Hilbert space has **countably infinite** number of basis vectors. But one could well expand $|\psi\rangle$ in any other basis as well, say the continuous position basis. Then that superposition, $$|\psi\rangle=\int dx \psi(x)|x\rangle$$ is also possible. But this basis $\\{|x\rangle\\}$ is continuous and **uncountably infinite**. Such sets cannot be equal because they have different cardinality. Then my questions is how many independent basis vectors are really there in this Hilbert space?


title: Is the universe a quantum computer - is light speed barrier a computational constraint

text: There is currently a debate ongoing on leading maths blog Gödel’s Lost Letter, between Gil Kalai and Aram Harrow, with the former arguing that building a quantum computer may not be possible due to noise propagation, and the latter arguing to the contrary. I am wondering if there is any argument to show that building a quantum computer is possible, by virtue of showing that quantum computation is evident in the physical world. So the question is: (A) Are there any known examples of physical interactions where macro level state transitions could be determined to only be in correspondence with an underlying quantum computation? I.e. similarly to Shor's algorithm being exponentially faster than any known classical factoring algorithm, are there any examples of known physical processes, for example perturbation stabilization in a very large particle cluster, that could be shown, assuming P<>NP, to only be efficiently solved by a quantum computation. Some, I admit highly speculative, additional questions would then be: (B) Is the speed of light barrier possibly a natural computational limit of our particular universe, so that for the computational complexity class of quantum mechanics, working on an underlying relational network-like spacetime structure, this is the maximum speed that the computational rules can move a particle/wave representation through a network region of the lowest energy/complexity (i.e. a vacuum)? (C) Is quantum mechanics an actual necessity for the universe to follow classical physical laws at the macro level? The informal argument being that in many-to-many particle quantum level interactions, only the capability of each particle to compute in parallel an infinite or quantum-quasi-infinite number of paths is what allows the universe to resolve a real-time solution at the macro level. Requesting references to research along these lines, or any arguments to support or contradict these speculations.


title: What's a good reference for the electrodynamics of moving media?

text: The answer to a previous question suggests that a moving, permanently magnetized material has an effective electric polarization $\vec{v}\times\vec{M}$. This is easy to check in the case of straight-line motion, using a Lorentz boost. I suspect this formula is still correct for motion that is not in a straight line, but I'm not interested in reinventing the wheel. Does anyone know of a textbook or journal article that derives this $\vec{v}\times\vec{M}$ term? Even better, does anyone know of experimental observation of this effect? EDIT:   Followup question: What is the electric field generated by a spinning magnet?


title: Is causality synonymous with continuity?

text: In general relativity, we use the term "time-like" to state that two events can influence one another. In fact, in order for an event to physically interact with another one, they have to be inifnitely close both in time and space. As far as I know (correct me if I'm wrong) this principal of "near action/causality" is conserved in all branches of modern physics and that is one of the reasons people are looking for "force carriers". If this is the case, then would it be accurate to say that causality is simply a measure of continuity in all dimensions - and not only the time dimension? (I don't know anything about continuum mechanics other than its name, but it may have something to do with what I'm asking)


title: What is the difference between gravitation and magnetism?

text: If you compress a large mass, on the order of a star or the Earth, into a very small space, you get a black hole. Even for very large masses, it is possible in principle for it to occupy a very small size, like that of a golf ball. I started to think, how would matter react around this golf ball sized Earth? If I let go of a coffee mug next to it, it would go tumbling down toward the "golf ball". Isn't that exactly how magnets work, with paperclips for example? Magnets are cool because they seem to defy the laws of gravity, on a scale that we can casually see. Clearly, the force carrier particles that produce electromagnetic attraction are stronger than gravity on this scale (or are at least on par: gravity plays some role in the paperclips path, but so does electromagnetism). My question is, why do we try to consider gravity as anything different than magnetism? Perhaps "great mass" equates to a positively (or negatively) charged object. Pull so much matter in close and somewhere you've crossed the line between what we call electromagnetic force and gravity force. They are one in the same, no?


title: How to understand order emerging from many-body system?

text: ![enter image description here](http://i.stack.imgur.com/5bscW.jpg) Like the order behavior shown in the image, is it due to the universality of some fundamental mathematic theory? Is there some general physics explanation for it? \- edit: This question comes after I read Terence Tao's talk on universality (pdf).


title: How would a semi-classical particle react to the double slit experiment?

text: As far as I know, there is a smooth transition between quantum and classical regimes, so that even classical particle like a massive object has a wavefunction associated with it. However, the double slit experiment can either show quantum character, where the particle supposedly passes through both slits and interferes with itself, or classical character, where the particle passes through one slit with no interference. What would a semi- classical particle do when faced with this situation? In addition, I have a problem imagining a semi-classical particle in general. What is a good example of one?


title: Why is the (free) neutron lifetime so long?

text: A neutron outside the nucleus lives for about 15 minutes and decays mainly through weak decays (beta decay). Many other weakly decaying particles decay with lifetimes between $10^{-10}$ and $10^{-12}$ seconds, which is consistent with $\alpha_W \simeq 10^{-6}$. Why does the neutron lives so much longer than the others?


title: How can I calculate the flow rate necessary for a given cross sectional area, angle, and length

text: ![enter image description here](http://i.stack.imgur.com/z8fR1.png) I'm making a toy for my kids and this problem came up. I have a channel on a slight angle (angle is between ground and length of channel) and I'm pouring water into it. I want to know how quickly I need to pour water in to make it flow continuously at a given height. So the water is (meant to be) .5m wide, .2m high, and 1m long, angled at 5 degrees. How fast is that water going to fall out, in other words, how fast do I have to pour it in? I'm particularly looking for the technique to do this, not just an answer. The part I find very confusing is that the water at the top of the box will accelerate a little, and have a low velocity, while the water at the bottom of the box will have accelerated for a while, and have a higher velocity. But intuition tells me the water will stay cohesive, for lack of a better word. Can someone explain what's going on here and how this can be calculated?


title: How do we end up with a gravity-dominant macroscopic universe from a quantum world having weakest gravity?

text: At quantum scale, gravity is the weakest force. Its even negligible in front of weak force, electromagnetic force, strong force. At macroscopic scale, we see gravity everywhere. Its actually ruling the universe. Electromagnetic force is also everywhere, but its at rank 2 when it comes to controlling motion of macroscopic bodies. And, there's no luck finding strong force and weak force. How can that be? Is that because gravity only adds up but others cancelled out too? I am unable to understand how resultant of weakest force can be so big. Can you please show it with calculation?


title: How to prove that the ground state of the Hubbard model is not a Slater determinant?

text: Of course it is expected. But how to prove it analytically? Slater determinant is mentioned in almost every quantum mechanics textbook. But it is necessary to warn the undergraduate students that not every fermionic wave function is a Slater determinant. So is there any model whose ground state can be easily and rigorously shown to be not a Slater determinant? I had this problem since my paper: http://arxiv.org/abs/1309.1848, in which we considered the best Slater approximation of a fermionic wave function.


title: Does an interaction of entangled particles with each-other cause decoherence?

text: I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question. Given a system of entangled particles (eg, 2 or more electrons), possibly in a superposition state: if the particles interact with each-other, what effect does this have on their quantum state? Is their state now determined (but perhaps unknown until observed)?


title: Derivation of Brillouin-Wigner theory for coupled subpaces

text: I recall faintly from my quantum theory lecture that there was a really neat way to derive Brillouin-Wigner perturbation theory for the special case of two coupled subspaces that involved a geometric series in reverse. I know that the beginning was to have the Hilbert space split into subspaces 0 and 1 so that the Hamiltonian reads $$H = H_{00} + H_{01} + H_{01}^\dagger + H_{11}$$ where the two indices indicate what subspace goes "in" and what subspaces comes "out" of each of the components. If $P$ is a projector onto subspace $0$ and $Q$ a projector onto subspace $1$, this means, for example, that $$H_{00} = PH_{00}P, H_{01} = PH_{01}Q$$ and so on. In matrix notation $$\begin{pmatrix} H_{00} & H_{01} \\\ H_{01}^\dagger & H_{11}\end{pmatrix} \begin{pmatrix} \psi_{0} \\\ \psi_1\end{pmatrix} = E \begin{pmatrix} \psi_0 \\\ \psi_1\end{pmatrix}$$ Now we can formally solve the Schrödinger equation for $\psi_1$ only: $$|\psi_1\rangle = \frac{1}{E - H_{11}} H_{01}^\dagger |\psi_0\rangle$$ and insert that back into the SG for $\psi_0$ to obtain $$H_{00} |\psi_0\rangle + H_{01} \frac{1}{E-H_{11}} H_{01}^\dagger |\psi_0\rangle = E|\psi_0\rangle$$ Now I know that some cool trick with geometric series and an inverted matrix was going on to immediately write down the expansion of $|\psi\rangle$. Usually in the literature one first shows the iterative formula and THEN notes that this is a geometric series, but here it was done in the reverse, somehow an operator of the type $1/(A-B)$ with $A$ and $B$ matrices, was found and then expanded in the geometric series, but no matter how I massage the equations, I cannot seem to make it work, because there are so many different points where one could substitute one form into the other etc. Any ideas on how this works? I'm also relatively sure that the work was done using operators, not for "individual" matrix elements. Personally, I find this more elegant because it makes it clearer what's going on on a more abstract level instead of dealing with the low-level matrix elements. It could also be that at this point already the approximation $E \approx E_0$ was made where $E_0$ is a typical energy of subspace $0$, assuming that this subspace is near-degenerate and well-separated from subspace 1.


title: generation of arbitrary potentials

text: Suppose you have as many electrically charged particles as needed (even countably many) and consider the open unit ball centered at some point in space. For every continuous real valued function on the unit ball, is there a configuration of the particles (outside of the ball) that would generate an electric potential corresponding to said function on the ball? Does an answer somehow trivially follow from Gauss' law?


title: What's an ideal wire?

text: I'm not talking about an ideal wire in a circuit (a wire with infinite conductance). I'm talking about an ideal wire in the case of the magnetic field of an infinite current carrying wire. What dimensions must an ideal wire have to better approximate an infinite current carrying wire in terms of its magnetic field? For example, for a parallel plate capacitor, the field better approximates two infinite plates as $$A >> d$$ where $A$ is the area and $d$ is the length. Or for a solenoid, the field better approximates an infinite solenoid as $$L>> d$$ where $L$ is the length and $d$ is the diameter. For a current carrying wire, what dimensions must it have to better approximate an infinite wire?


title: What distance has the soldier travelled?

text: > A troop $5$ meters long starts marching. A soldier at the end of the file > steps out and starts marching forward at a higher speed. On reaching the > head of the column, he immediately turns around and marches back at the same > speed. As soon as he reaches the end of the file, the troop stops marching, > and it is found that the troop has moved by exactly $5$ meters. What > distance has the soldier travelled? I thought that some info is lacking until my friend showed it in a book. How to get the answer?


title: What would be walking speed in low gravity?

text: In $1g$ the average adult human walks 4-5 km in an hour. How fast would such a human walk in a low gravity environment such as on the Moon $(0.17g)$ or Titan $(0.14g)$? Let's ignore the effects of uneven terrain (regolith or ice/snow/sooth); suppose our human walks on hardened pavement.


title: Scalar Field of a long thin wire

text: Given the situation that $B=-\nabla A$ where B is magnetic field and A is some Scalar Field. How can I calculate the scalar field, A. We are dealing with current free region, here. I know we can calculate B for a infinitely long thin wire given $I$. But I am not sure how we can calculate $A$.


title: Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

text: I actually posted this to math.stackexchange.com a few months ago but never got any answers. I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film simplification of the Navier-Stokes equation, similar to the Reynolds or Lubrication Equation, but including inertial terms as well. To walk through the major steps to the point where I have questions:   1. Start with basic N-S equation (eq 1 in the paper): $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot(\mathbf{u u}) \right) = -\nabla p + \rho \mathbf{g} + \nabla \cdot \mathbf{\underline{T}} $$ Here $ \mathbf{\underline{T}}$ is the deviatoric stress tensor, and I have left off the final body force term since it isn't used in the rest of the paper.   2. Use the thin-film assumption that $u_z=0$ and define (eq 4 in the paper) $$\bar{\mathbf{u}}=\frac{1}{h}\int_0^h \mathbf{u}\, dz$$   3. Integrate the N-S equation with respect to $z$ from $0$ to $h$ (equation 8 in the paper). $$ \rho \frac{\partial}{\partial t} (h \bar{\mathbf{u}}) + \rho \int_0^h \nabla \cdot (\mathbf{u u})dz = -h \nabla p -\left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=0} $$ Obviously there are a few steps being skipped over here. For the time-derivative term, I use the Leibniz formula to derive the following: $$ \int_0^h \rho \frac{\partial \mathbf u}{\partial t} dz = \rho \frac{\partial}{\partial t} \int_0^h \mathbf u\,dz - \rho \frac{\partial h}{\partial t} \mathbf u (x,y,h,t) + \rho \frac{\partial 0}{\partial t} \mathbf u (x,y,0,t)$$ Obviously the last term is $0$ and can be dropped. Also $ \rho \frac{\partial}{\partial t} \int_0^h \mathbf u\,dz = \rho \frac{\partial}{\partial t} (h \bar{\mathbf{u}}) $, giving the form seen in the equation. However, I don't see how $ \color{blue}{\rho \frac{\partial h}{\partial t} \mathbf u (x,y,h,t)} $ can be taken to be $0$. The height of the film is certainly changing with time, and the top surface has a von Neumann boundary condition, not a Dirichlet no-slip boundary. Any insight here? Also, the deviatoric stress has to be integrated as well. I think the divergence theorem can be used here: $$ \int_V \nabla \cdot \mathbf{\underline{T}}\, dV = \int_S \mathbf{n} \cdot \mathbf{\underline{T}} \, dS$$ In this case that should come out to be $$ \int_0^h \nabla \cdot \mathbf{\underline{T}}\, dz =\left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=h} - \left.\mu \frac{\partial \mathbf{u}}{\partial z}\right|_{z=0} $$ The top surface stress is $0$, leaving the bottom stress term as is found in the derived equation, right?   4. Now we get to my main question, the integration of the $\nabla \cdot(\mathbf{u u})$ term. The author is able to evaluate this term by using a combination of Pohlhausen's method of assuming a cubic profile for the liquid flow, and the Reynold's Averaged Navier-Stokes method of splitting the velocity into an average velocity and deviation from that average. For the cubic profile he defines: $$ \mathbf{u}(x,y,z) = u(x,y,\xi) \text{, where} $$ $$ u(x,y,\xi) = a_0 + a_1\xi + a_2\xi^2 + a_3\xi^3,\quad \xi \in \langle 0,1 \rangle,\; z=h\xi $$ Then he applies the boundary conditions and integral relation to obtain $$ u(x,y,\xi) = \mathbf{u}_{disk} + (\bar{\mathbf{u}}-\mathbf{u}_{disk})\left( \frac{12}{5}\xi - \frac{4}{5}\xi^3 \right) $$ This step is fine, I had no problems figuring it out. Then the author defines the velocity fluctuation (with respect to the vertical direction) $\mathbf{\tilde u}$ as $$ \mathbf{u} = \mathbf{\bar u} + \mathbf{\tilde u} \text{. This makes:} $$ $$ \int_0^h \mathbf{\bar u}\, dz = h\,\mathbf{\bar u}\text{, and } \int_0^h \mathbf{\tilde u}\, dz = 0 \text{.}$$ Anyway, so the author does this to integrate the advection term: $$ \int_0^h \nabla \cdot ( \mathbf{uu} )\,dz = \nabla \cdot \left( \int_0^h \left[ \mathbf{\bar u} \mathbf{\bar u}+\mathbf{\bar u} \mathbf{\tilde u} + \mathbf{\tilde u} \mathbf{\bar u} + \mathbf{\tilde u} \mathbf{\tilde u} \right]\,dz \right)$$ So on the RHS the 1st term is just a constant, the 2nd and 3rd terms become $0$, and then he uses the derived polynomial form of $u$ to evaluate the last term. However, how was he able to pull the divergence operator out of the integral? He didn't use the divergence theorem, and I don't know if you can use the Leibniz formula on a divergence operator. If you could do that though, wouldn't you have a term that's something like $\nabla \cdot h \left.(\mathbf{u}\mathbf{u})\right|_{z=h}$ leftover as well?


title: Geometriclength and effective length

text: > Why is effective length of a bar magnet shorter than its geometric length? > Blockquote in a textbook, author wrote that, effective length is equal to 0.85 times geometric length. But didn't mention the mathematics behind it. I don't understand which keyfactor can determine the differences between geometric length and effective elgth?


title: Completing the square for Grassmann variables

text: When working with path integrals of both bosonic and fermionic field variables, I'm a bit unsure of how to do the usual complete the square trick when an interaction between the two is concerned. Say you have a generic partition function like \begin{equation}Z=\int D\phi D\bar\psi D\psi \,e^{iS[\phi,\bar\psi,\psi]}\end{equation} where the action has your standard quadratic part, but also an interaction term like $\mathcal{L}_{int}\sim \bar\psi \phi\psi$. If you wanted to integrate out the Bose fields first, you would complete the square but that would involve making a substitution like $\phi\rightarrow \phi'= \phi-\bar\psi\psi $ and integrating over $\phi'$ (I may be wrong here). I'm confused on what it means to:   1. Subtract a product of Grassmann numbers from an ordinary number (they aren't the same kind of number)   2. Integrate over the new Bose field, which is technically a function of the Grassmann variables (are there any subtleties with that process?) Any help would be greatly appreciated.


title: Would synchronized dancing be a good way to describe entangled atoms to a laymen?

text: I was talking my professor about entanglement swapping between light and matter and it is briefly described here: You start out with a crystal capable of doing parametric down conversion of incoming photons. When they go in, they undergo a physical process which produces two entangled photons that come out. At the same time, you get two atoms of the same type, say two Hydrogen atoms that are trapped in a harmonic oscillator potential in the, let's say x direction. Then, you send each entangled photon pair to each of the atoms along the x direction and have them interact. You keep on sending entangled photon pairs until the two atoms come to a steady state where they oscillate in a synchronous fashion. You essentially transfer the momentum of the photon to the atoms and do the opposite of laser cooling. The atoms are entangled in position and momenta. When one is measured at x1 = 1, the other is at x2 = -1. Their momenta are equal. p1 = p2. So, if we are to imagine two people. These two people would be the Hydrogen atoms. The photons that bounce off each of the two people are like the entangled photons. With their vision, absorption of photons, the two people can come into sync. When two people dance mirror images of each other, this can be viewed as if they are in "opposite" positions but since they are moving at the same velocity, they share equal momenta. You can't take the analogy too far, but would this be good?


title: Do we move at the speed of light relative to light?

text: My understanding of relativity isn't very sophisticated, but it seems to me that relative to a photon moving at the speed of light, we are moving at the speed of light. Is this the case?


title: What color would a proton be if it were visible to the human eye?

text: If a photon hits a proton, would it have a color? What color would it be?


title: With what probability does nuclear fusion occur at energies far below the Coulomb barrier?

text: Even at the core of the sun, the temperature of $\sim 10^7$ K only results in $kT\sim1$ keV, which is about a thousand times less than the electrical potential energy of $\sim1$ MeV needed in order to bring two hydrogen nuclei to within the ~1 fm range of the strong nuclear force. Therefore nuclear fusion reactions can only occur inside the sun, or in any other normal star, through the process of quantum-mechanical tunneling. The low probability of this tunneling, along with the need for a weak interaction in order to fuse two protons into a deuterium nucleus, are the two factors that make stars have lifetimes billions of years long. How is the tunneling probability calculated?


title: Newtonian gravity from the holographic principle?

text: Can one understand Newton's law of gravitation using the holographic principle (or does such reasoning just amount to dimensional analysis)? Following an argument similar to one given by Erik Verlinde, consider a mass $M$ inside a spherical volume of space of radius $R$. The holographic principle says that the mass $M$ must be completely described on the surface of the sphere in terms of the energy in elementary entities or strings. By the equipartition theorem, the total energy, $E=Mc^2$, on the surface is given by the number of strings, $N$, times the number of degrees of freedom per string, times $1/2\ k_BT$ per degree of freedom: $$E = N \times d_f \times \frac{1}{2} k_B T.\ \ \ \ \ \ \ \ \ \ (1)$$ The number of strings on the surface is given by the Bekenstein-Hawking formula: $$N = \frac{A}{4},$$ where $A$ is the sphere surface area in units of the Planck area $G\hbar/c^3$. In terms of the radius $R$ the number $N$ is given by: $$N = \frac{\pi c^3 R^2}{G \hbar}.$$ By substituting into equation (1) we obtain an expression for temperature $T$: $$M c^2 = \frac{\pi c^3 R^2}{G \hbar} \times d_f \times \frac{1}{2}k_BT$$ $$T = \frac{4}{d_f}\frac{\hbar}{2\pi ck_B}\frac{GM}{R^2}.$$ If there are $d_f=4$ degrees of freedom per string then the above formula gives the Unruh temperture for an object falling through the surface with acceleration $g$: $$g = \frac{GM}{R^2}.$$ This of course is the acceleration given by Newton's law of gravity due to the presence of a mass $M$. P.S. Could $d_f=4$ come from the 4 dimensions of spacetime?


title: Error analysis and how values in references are determined

text: Question 1:Most science textbooks have appendixes that have a value for some physical property of some object. This includes diameter of electrons, viscosity of fluids, boiling points, etc. My question is, are the values presented in such appendixes (or other data bases) averages? Question 2: Also, suppose I try to calculate the boiling point of water using equations. I then compare my result with data obtained from the CRC Handbook. How would I determine how accurate my result is? Would I use percentages or is there another way to calculate the error? Note: by percentages I mean $$\vert\frac{result-reference\,\,value}{reference\,\,value}\vert\cdot100$$


title: Explain Heat Transfer

text: I would like to know what are these formulas used for. There is no intro about it in my book at all, and I am reading Heat Transfer book. ![enter image description here](http://i.stack.imgur.com/nHffh.jpg) If needed Q. can be edited.


title: Classical regime for Fermi-Dirac and Bose-Einstein gases

text: I'm studying statistical mechanics, in particular classical regime for Fermi Dirac and Bose Einstein gases. Time average value for occupation numbers in FDBE statistics: $$ \langle n_\epsilon\rangle_{FB} = \frac{1}{e^{(\epsilon-\mu)\beta}\pm1} $$ For Boltzmann Statistics: $$ \langle n_\epsilon \rangle_B = e^{(\mu-\epsilon)\beta} $$ How can one work out a nice condition of classical regime in which $ \langle n_\epsilon\rangle_{FB} \rightarrow \langle n_\epsilon\rangle_B $ ? An obvious option is $e^{\frac{(\epsilon-\mu)}{kT}}\gg1$. However, I don't really like it, since it implies convergence at low temperature. Moreover I'm expecting an $\epsilon$-free asymptotic expression in terms of temperature and density. * * * @Adam : i've read your comment again and things are much more clear now :)! Here's what i've got: I'll assume $ \beta|\mu|>>1 $ and $\mu<0 $ or $z \rightarrow 0 $. In terms of z: $$ \langle n_\epsilon\rangle_{FB} = \frac{1}{\frac{e^{\epsilon\beta}}{z}\pm1} \, \,\underrightarrow{z\rightarrow0} \, \,\langle n_\epsilon\rangle_{B}$$ Being $z=\lambda^3_t \rho$, i can say FDBE gases behaviour like classical one when the particle's thermal wavelenght is small if compared to the typical particle distance. Almost the "low density, high temperature" condition i was looking for. At low temperature Boltzmann statistics lose physical mean (for example it's easy to recover the classical Sackur–Tetrode entropy from his thermodynamic) . Approximating in this scenary, although it may look mathematically legitimate, is conceptually wrong. Quantum statistcs have to be handled carefully on their own. Am i doing it right :)? Sorry for the poor english. Thanks you so much


title: Comparing effect of electric and magnetic dipoles on their fields

text: So at the end of one of my prof's lectures he gives us something to think about: > Both electric and magnetic dipoles tend to line up with their respective > fields. > > Materials made out of electric dipoles cause the electric field that turns > the dipoles to be **reduced**. > > Materials made out of magnetic dipoles cause the magnetic field that turns > the dipoles to be **increased**. > > Why are the two types of dipoles different in this regard? Now, this isn't a homework assignment or anything. It is just something to ponder on. I'm curious why this happens.


title: Do the states forming an orthonormal basis have the same amount of entanglement?

text: If $\\{|\psi_{i}\rangle\\}$ is an orthonormal basis for a bipartite system, will $E(|\psi_i\rangle) = E(|\psi_j\rangle)$ for all $i, j$, where $E$ is some entanglement measure?


title: intuition behind Coulomb's friction force

text: I would like to know why dry frictional force does not depend on speed(assuming that the heat generated by friction is small so it does not alter the coefficient of friction). Let me explain how I think about dry friction. When I slide one object on another than their small irregularities on surfaces are bashing to each another and this causes the friction. If I higher the sliding speed than those irregularities will bash to each other more frequently so it seams to me that the friction should rise. But Coulomb's friction force says that the force does not depend on speed. So can anybody give me intuitive explanation why is that?


title: Energy conservation and quantum measurement

text: Consider a particle in a potential well. Let’s assume it’s a simple harmonic oscillator potential and the particle is in its ground state with energy E0 = (1/2) ℏω0. We **measure** its position (measurement-1) with a high degree of accuracy which localises the particle, corresponding to a superposition of momentum (and therefore energy) states. Now we **measure** the particle’s energy (measurement-2) and happen to find that it’s E10 = (21/2) ℏω0. Where did the extra energy come from? In the textbooks it’s claimed that the extra energy comes from the act of observation but I wonder how that could work. Measurement-1 which probed the position of the particle can’t have delivered to it a precise amount of energy, while measurement-2 might just have been passive. No doubt there is entanglement here between the particle state and the measuring device but where, and which measurement?


title: Weak measurement and Hardy's paradox

text: How the notion of _weak measurement_ resolves Hardy's paradox?


title: Lorentz transformation of the vacuum state

text: In general, the Hamiltonian $H$ has non-zero vacuum expectation value (VEV): $$ H \left.| \Omega \right> = E_0 \left.|\Omega \right>, $$ where $\left.|\Omega\right>$ is the vacuum state. The zero-point energy of the quantum oscillator is a good example to have non-vanishing VEV. The problem strikes me is that **how is this vacuum state transformed under Lorentz transformation?** On one hand, we can think of this vacuum state is the trivial representation of the Poincare group, which is a Lorentz invariant. ( In Wigner's classification, vacuum state is in the $(0,0,0,0)$ class, if I understand correctly.) So $$ U(\Lambda) \left.| \Omega \right> = e^{-i \theta} \left.| \Omega \right> $$ On the other hand, the vacuum state has non-zero 4-momentum: $p = (E_0, \vec{0})$. So it seems it should transform like other irreps: $$ U(\Lambda) \left.|p, \sigma\right> = \sum_{\sigma'}C_{\sigma'\sigma} \left.| \Lambda p, \sigma' \right>, $$ Now, the important thing is, $\Lambda p$ could have non- zero 3-momentum. But the vacuum state should have zero 3-momentum. Bang! In textbooks, a common procedure to reconcile this contradiction is to shift the Hamiltonian $H \to H - E_0$; but, the Poincare algebra would be affected: $$ [ K^i, P^j ] = \delta^{ij} H \to [ K^i, P^j ] = \delta^{ij} H + \delta^{ij} E_0 $$ Thank you.


title: Power and speed

text: I'm asked to calculate how much POWER a 1210kg car needs to drive with a 85 km/s speed up a 655 meter long slope of 4.5°. I can find how much energy and work is required to do this, but isn't POWER=WORK over TIME? I don't have any time in my problem. So I ask: If I have a 1000kg car and I drive 500 meters...Could I go 100 km/h with even 1 horse power as long as I got enough time? Is horse power only an indicator of torque (i.e. work over time)?


title: Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?

text: The Feynman Checkerboard Wikipedia article states: "There has been no consensus on an optimal extension of the Chessboard model to a fully four- dimensional space-time." Why is it hard to extend it to more than 1+1 dimensions?


title: Meaning of the anti-commutator term in the uncertainty principle

text: What is the meaning, mathematical or physical, of the anti-commutator term? $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle [ A,B ] \rangle \vert^{2} + \dfrac{1}{4} \vert \langle \\{ \Delta A, \Delta B \\} \rangle \vert^{2}$, where $\Delta A, \Delta B, A$ and $ B$ are operators. The inequality is still true, and the anti- commutator term "strengthens" the inequality, but why does it appear?


title: Mysterious spectra?

text: In my blog post _Why riemannium?_ , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hygrogen-like) problem have energy spectra, respectively, equal to 1) $$ E\sim n^2$$ 2) $$ E\sim n$$ 3) $$ E\sim \dfrac{1}{n^2}$$ Do you know quantum systems with general spectra/eigenvalues given by $$ E(n;s)\sim n^{-s}$$ and energy splitting $$ \Delta E(n,m;s)\sim \left( \dfrac{1}{n^s}-\dfrac{1}{m^s}\right)$$ for all $s\neq -2,-1,2$?


title: Most sophisticated experiment you could perform in your garage?

text: What are some of the most elegant/complex/sophisticated physics experiments one could perform in his garage, if he has enough spare time and technical/theoretical know-how, but a relatively limited budget? Say, a retired emeritus physics professor? These could be either novel or reproductions of known experiments, either qualitative or quantitative demonstrations.


title: Why do power lines buzz more when it is low-humidity outside?

text: I have observed that the power lines buzz louder when there is less moisture in the air. Why is this? If it will help the lines are located on the foot hills of a nearby mountain.


title: Energy Conservation Law validity

text: I will probably be laughed out of town for saying this, but why can't the Energy Conservation Law be broken? Everybody thought electricity to motion was impossible until Faraday made his motor, nobody believed Tesla would make a brushless motor, and now everyone ignores the potential existence of energy creation. I see that no device has been made yet that destroys the Conservation Laws, but that does not mean it is impossible. Is there anything in our understanding of physics that would break down if this were the case? **P.S :** In the physics videos I have seen, the professor demonstrates that energy cannot be created. His method was hardly scientific. I agreed with him that his setup would not create energy. In other words, saying a brushless motor is impossible while demonstrating a brushed motor proves nothing. Please note: I mean no offense... just trying to get to the details


title: Charge on capacitor plates in series combination?

text: Suppose 2 capacitors are connected in series, the plates connected to the battery terminals receive charges $+q$ and $-q$, and the isolated plates in the combination receive equal and opposite charges through induction. Now my question is, why is that the induced opposite charge on the plates isolated (i.e. not connected to the rest of the circuit through wires as in any ordinary series combination of capacitors) from the other two plates equal to the inducing charge on the plates connected to the battery?


title: Why does the milk frother on my coffee machine make so much noise?

text: I have a Sunbeam home espresso machine with a steam wand. The steam roars out straight from the end of the wand. When it's first placed in the cold milk it really screams! Once the milk has a bit of a whirlpool action going it's much quieter, so I guess that the noise is because of fast-moving steam hitting stationary cold milk. Something something fluid dynamics?


title: Does a Peltier Thermoelectric layer make a good regulator for LN2 heat-sink interfacing

text: I understand that when cooling water with LN2 directly (through pipes for example), that it can be too effective, freezing the water (and blocking the pipe). From what I read, regulation of the heat transfer is considered complicated to control. Often employing at least a fixed insulation to regulate the rate of heat transfer. If I immersed a fixed copper heat sink into LN2, the copper interface would be below $0C$ and still risk freezing the water. My question is directed at those familiar particularly with Peltier cooling devices. Considering an aparatus, where a Peltier is:   * A) interfaced directly to a flat interface on a heatsink; or   * B) interfaced with a fixed insulation The following sub-questions apply:   1. Does a cooler heatsink on the heated side, improve the max. cooling rate?   2. If turned off, would the peltier device exhibit any insulating effect?   3. Would there be a minimum operating temperature, in the case where the Peltier was turned off and cooled toward LN2 temperatures? Given favourable answers to the sub-questions, and any additional comments, the title question would be answered. Some helpful additional comments may relate to industry standard heat-exchanger/regulator apparatus and their performance/cost. **Update** To reiterate the specifics of my theoretical apparatus. The "heatsink" is in the LN2 canister, the Peltier device outside with the hot interface on the "heatsink". When the peltier voltage drive is 0, the peltier would have an insulation effect R1, resulting in a Rate of heat transfer (ROHT) on the coldside of C1, changing the voltage to full on the Peltier changes the insulation effect to a negative value R2, and increases the ROHT to C2. For intermediatary voltages, the insulation effect is between R1 and R2, and the RoHT is between C1 and C2.   1. What is R1 (the insulation effect) of a non-specific Peltier device with no voltage applied?   2. What would be a guestimate for C1 given an LN2 heatsink?


title: Is force a contravariant vector or a covariant vector (or either)?

text: I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that displacements are contravariant vectors and gradients of scalar fields are covariant vectors, but in _A Student's Guide to Vectors and Tensors_ by Fleisch I found this statement: > [I]t's not the vector itself that is contravariant or covariant, it's the > set of components that you form through its parallel or perpendicular > projections. (p.121) If physical concepts can be represented as either type of mathematical object, I don't understand how a displacement could be represented as a covector. Wouldn't its components transform the wrong way if the coordinates were changed? If covariantness/contravariantness is part of the definition of a physical concept, I don't understand how force is classified. The gradient of a potential would have dimensions of length in the denominator, making it a covariant vector. Mass times acceleration has dimensions of length in the numerator, making it contravariant. **Edit:** I read through the accepted answer to Forces as One-Forms and Magnetism. One thing I don't understand is whether in relativistic spacetime any vector quantity can also be represented as a 1 form (because there is a metric) or whether its classification as a 1 form or vector depends on how its components change under a coordinate transformation. Doesn't a displacement have to be a vector and not a 1 form?


title: Why Landau Level quantization is observed only in low temperature and strong magnetic field in real experiment?

text: I know that Quantum Hall Effect and Fractional Quantum Hall Effect origin from Landau Level quantization. In magnetic field, the energy of in-plane(plane perpendicular to magnetic field) degree of motion is quantized, which is $E=(n+1/2)\hbar\omega$, $n$ is integer. In experiment, both QHE and FQHE are observed in low temperature and strong magnetic field, suggesting that landau level quantization is observable under the condition $k_B T<<\hbar\omega$, where $k_B$ is Boltzman constant. I am not sure why this condition is important in experiment.


title: What is the optimal burn direction to lower periapsis of hyperbolic orbit?

text: I am an engineering student who is interested in orbital mechanics. I am doing some self study before taking some orbital mechanics courses next year. I was learning about various orbit types (elliptical, parabolic, hyperbolic, etc.) and the effects of burning in various directions. I have found a lot of good information about about how to manipulate elliptical orbits (raising/lowering apo/periapsis, changing inclinations, etc.). However, I haven't found a lot of information on manipulating a hyperbolic trajectory. I have found a lot of good information like this in calculating various parameters (impact parameter, turning angle, etc.) but little specifics on how to change one. For example, say you were on a hyperbolic flyby like in this example. However, you wanted to lower your radius at periapsis by a couple hundred km for some reason (take some measurements, increase turning angle etc.). What would be the most efficient direction to burn? I could see doing it 2 different ways but unsure of which would be more efficient. You could burn retrograde lowering your velocity magnitude which would pull you closer to the planet. Or you could burn perpendicular to your current velocity vector in the direction of the planet changing your approach angle? Perhaps some combination of the 2? Anyone know how to determine what would be the most optimal?


title: Two particles state of a 1D massive scalar field

text: Perfectly localized states are not normalized so do not belong to the Fock space (they belong to the rigged version). Suppose we approximate localized states with gaussians, what is the mathematical expression for two "approximately" localized particles at points $x_1$ and $x_2$ that takes into account correlations due to vacuum entanglement? how do you build it from the original creation/annihilation operators eigenstates? The field is supposed to be a 1D massive scalar one. Thanks.


title: Kirchhoff law of thermal radiation

text: Kirchhoff's law of thermal radiation states that for thermal equilibrium for a particular surface the monochromatic emissivity $\epsilon_{\lambda}$ equals the monochromatic absorptivity $\alpha_{\lambda}$: $$\alpha_{\lambda} = \epsilon_{\lambda}$$ I wonder how this can be only valid for thermal equilibrium, because $\epsilon_{\lambda}$ and $\alpha_{\lambda}$ seem to depend only on the temperature of the surface under consideration, and it seems to me that they don't depend on the other bodies around the surface. So according to this, it should follow that they are equal always, when they are equal in thermal equilibrium, which isn't correct. So I wonder how the lack of thermal equilibrium can change the values of $\alpha$ and $\epsilon$?


title: The mathematics of entanglement

text: I've finally managed to get a grasp on the Bell test experiments and all that they imply about our reality. Now I'm curious about the mathematical derivation which allowed Schrodinger to predict the existence of entangled entities. Can someone please provide an explanation of these equations or direct me towards good sources of information on the subject? Thanks.


title: Does the global financial system violate laws of thermodynamics and energy conservation?

text: For the past years I am thinking about relating concepts from physics to concepts from economics. Especially since the financial crisis made obvious the instability of the financial system I ask myself whether this could be explained by physics, e.g. by lack of inertia and dissipation during transactions, conservation of energy as the total amount of money available continuously increases, or whatever. Unfortunately I am indeed not firm in physics and economics myself. However, I am yet very interested whether these ideas make any sense, or whether there is any actual research in this direction.


title: How would nucleosynthesis be different if the neutron were stable?

text: If the strong nuclear force were just 2% stronger, the neutron would be a stable particle instead of having a half life of about 13 minutes. What difference would that have made to Big Bang nucleosynthesis, to the growth of structure, to the formation of stars, nucleosynthesis in stars?


title: Renormalization Group: Different fixed points

text: Extending the Gaussian model by introducing a second field and coupling it to the other field, I consider the Hamiltonian $$\beta H = \frac{1}{(2\pi)^d} \int_0^\Lambda d^d q \frac{t + Kq^2}{2} |m(q)|^2 + \frac{L}{2} q^4 |\phi|^2 + v q^2 m(q) \phi^*(q)$$ Doing a Renormalization Group treatment, I integrate out the high wave-numbers above $\Lambda/b$ and obtain the following recursion relations for the parameters: $$\begin{aligned}t' &= b^{-d} z^2 t & K' &= b^{-d-2}z^2 K & L' &= b^{-d-4}y^2 L \\\ v' &= b^{-d-2}yz v & h' &= zh \end{aligned}$$ where $z$ is the scaling of field $m$ and $y$ is the scaling of field $\phi$. One way to obtain the scaling factors $z$ and $y$ is to demand that $K' = K$ and $L' = L$, i.e., we demand that fluctuations are scale invariant. But apparently, there is another fixed point if we demand that $t' = t$ and $L' = L$ which gives rise to different scaling behavior, and I wonder a) why I can apparently choose which parameters should be fixed regardless of their value ($K$ and $L$ in one case, $t$ and $L$ in the other case) b) what the physical meaning of these two different fixed points is... (My exposure to field theory/RG is from a statistical physics approach, so if answers could be phrased in that language as opposed to QFT that'd be much appreciated)


title: Energy and momentum as partial derivatives of on-shell action in field theory

text: According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... $$E = -\frac{\partial S}{\partial t}$$ $$p_i = \frac{\partial S}{\partial q_i}$$ Is there a straightforward generalization of this to field theory? Something that would give the energy and momentum densities by differentiating the on- shell action (with respect to... something)?


title: Argument for proving Earth-Centered Earth-Fixed (ECEF) reference frame is non-inertial

text: Today I heard an argument to prove that the Earth-Centered Earth-Fixed (ECEF) reference frame is non-inertial. It seriously doesn't make any sense to me but I also heard that same argument was made by Einstein, so I am asking it here. Please explain how it proves that Earth frame is non-inertial. The argument goes like this: Suppose the Earth frame is inertial. Now, all the heavenly bodies will have to go around the Earth once a day. But, this implies a very fast speed. Hence, the Earth frame can't be inertial.


title: Is it possible to create a nuclear reactor the size of a melon?

text: > **Possible Duplicate:**   >  What is the minimum amount of fissile mass required to acheive criticality? I want to make a red laser the only problem is that the power is not sufficient so I had to resort to a nuclear reactor.


title: Frictional Forces

text: In the figure, blocks A and B have weights of 45 N and 23 N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if μs between A and the table is 0.21. (b) Block C suddenly is lifted off A. What is the acceleration of block A if $\mu_k$ between A and the table is 0.14? ![enter image description here](http://i.stack.imgur.com/9e0Ks.png) I found the weight of block C to be $64.5 N$ , and I was told that was correct. For part b, here is my work. > $\mu_k = 0.14$ > > $F_k = \mu_k F_N$ $\text{ }$ $F_k = \text{frictional force due to } \mu_k$ > > $F_k = 0.14 * 45N = 6.3N$ > > $F_{net} = F_B - F_k$ > > $F_{net} = 23N - 6.3N = 16.7N$ > > $F = ma$ > > $16.7 = \frac{45N}{9.81 {m\over s^2}}*a$ > > **$a = 3.64 {m\over s^2}$** However, I am told that this answer is wrong. Did I make a mistake anywhere? Any help would be appreciated!


title: Can I measure the Reynolds number or the heat transfer coefficient

text: Suppose I have a flow of hot air around a cold and unevenly shaped object with holes and tunnels (think about it as a bed packed with some objects). I would like to know the Reynolds number of this flow and its convective heat transfer coefficient. The definition of the Reynolds number contains a "characteristic length" that is somehow mysterious to me and that I do not have at hand. And I am a bit reluctant to use the formula for the packed bed Reynolds number. Is it possible to measure it? How would I design an experiment for this? I would like to avoid temperature readings of the object. In this article I found a relation between the Reynolds number of a packed bed and the heat transfer coefficient, so measuring only the Reynolds number would be a good start, even though I do not have a real packed bed. Could this be done by simple pressure drop readings?


title: Riemann curvature of a unit sphere

text: The Riemann curvature of a unit sphere is shown in many textbooks to be sine- squared theta where theta is the azimuthal angle of spherical co-ordinates. But what is the significance of the angle and how would a value be assigned to it?


title: How to prepare a desired quantum state?

text: Given a quantum state function, we can Fourier expand it in terms of stationary states of the Hamiltonian. So if we want to build that same quantum state approximately all we need to do is to superpose stationary states with proper amplitudes. Assuming that we can prepare such stationary states individually, how is their superposition done experimentally? As a specific example, how to prepare experimentally, an ensemble described by the quantum state that is a superposition of 4 stationary states of the particle in a box (or the hydrogen atom) that corresponds to n = 1, 2, 3, and 4 say ?


title: Macroscopic quantum gravity phenomena

text: A theory of quantum gravity is said to be needed when quantum and gravitational effects are strong at the same time i.e. at black hole singularities and at the big bang. This also makes it difficult to test quantum gravity. But what about testing macroscopic quantum phenomena in different gravity regimes like flying a superconductor or liquid helium into Earth orbit and back again - would you expect gravitational time dilation or high-g accelerations to alter macroscopic quantum behaviour in a way that could test quantum gravity theories ?


title: New physics at high energies, cosmic rays, particle-detectors in space

text: New physics is expected at high energies and cosmic rays have high energies, so have there been or are there any plans to put particle detectors in space to study cosmic rays for new physics ?


title: Experimental evidence for non-abelian anyons?

text: Since non-abelian anyons have become quite fashionable from the point of view of theory. I would like to know, whether there has actually been experimental confirmation of such objects. If you could cite original literature, that would be great!


title: Flow from the bottle

text: I made some calculations about flow from the bottle, but problems occured when the experimental data came back. Need to refine calculations, but problems occur when trying to link the angle to the flow, because volume changes constantly, and volume that has left in the bottle is governed by the force sensor. And the force sensor is governed by the algorithm that links volume with weight, and weight and volume change with the change of the flow, which again is a function of the change of the angle. As the bottle is oppened, angle of the fluid will always be parallel to the ground, because of the free surface and $p_{a}$. But problem here is that not every bottle is the same shape, so the height and the angle will not be the same for every bottle, and pouring process will not start for at the same angle for different bottles. With some bottle necks being convex effects of the surface tension will be much grater, and the poured amount of the liquid will be different than the deisred one. Then also, expansion waves will cause turbulance within the flow, and the flow should at least remain laminar, for easier pouring and calculations, but in real time pouring this is not that easy to get, well offcourse on higher pouring speed, with lower pouring speed this is not that big of a problem, but surface tensions tends to splurge liquid under the bottle. ![Pouring positions](http://i.stack.imgur.com/zVbTj.jpg) So for the diagram, I can put this picture, it's selfexplanatory. As I stated angle should correlate with the flow, and weight should be governed by force sensor, with this the force sensors senses the change of the weight (or volume) in the bottle, and this tells the framework on how much fluid has been poured and how much is remaining. Angles that I got from calculations and experiments differ, some of this I can contribute to surface tension. But the flow and the angle of the bottle to correlate to the change of the weight has proved different in a bigger manner. For the geometry, different bottle types were used, so there is no specific geometry, but formulations were made for a generic bottle, so there could be some referencing points to the problem at hand. ![Bottle dimensions](http://i.stack.imgur.com/hjMR9.png) Any suggestions on how to model this mathematically and to do it correctly.


title: Why would a rotating charged sphere not have time varying electric field?

text: I have seen Gauss Law being used for a uniformly charged hollow sphere rotating with $\omega$. How is that valid to use Gauss law since it is an electrostatic law and if it is valid, why do we get a net constant electric field outside the sphere inspite of the accelerating charges - does the net time dependence cancel out ? How do I see this cancellation intuitively ? Also, can I see intuitively why the net radiation emitted is zero instead of integrating the Poynting flux ?


title: Temperatures at extreme densities

text: Cosmology (and astrophysics) talk about the "initial singularity" (IS, became the big bang) and "black hole singularities" (BS, inside black holes), and these appear to be quite different: The IS is described as having **zero volume and infinitely high temperature and density** , whereas the BS is likewise zero volume and infinite density, but with a **temperature close to 0 K**. (Personally, I suspect that both the IS and the BS equations suffer from a division be zero error (density = mass/volume, and in both cases the volume is zero (well so they say...)). My question is: **If there is infinite density, how can there be any room for thermal activity (temperature)** , let alone **infinitely high** temperature? Shouldn't the IS be stone cold, just like the BS? The BS case seems more reasonable, other than the difficulty I have trying to grasp "infinite mass in zero volume".


title: Reduced charge for attracting electrons?

text: The reduced mass in the two body problem is $\mu= \cfrac{m_1 m_2}{m_1 + m_2}$. Is there any analog to this with interacting charged particles (or at least that is of use somewhere in physics)? I have not seen anything like this anywhere, but was curious if someone else has. I would imagine that this is not possible for the repulsive case. Edit: To clarify, I understand that the same equations hold for the attractive force since they are both $\frac{1}{r^2}$ force laws. I was curious if anyone had used this for charges before.


title: What is the buoyant and viscous force acting on a ball in free fall?

text: I was thinking about a situation were I have a long cylindrical jar with some viscous liquid in it. I also have a spherical ball with me. I drop the ball into the liquid inside the jar with some initial velocity. Just after that(before the ball acquires terminal velocity) I drop the system(the jar with the liquid and ball) from certain height. (A) What would be the buoyant and viscous force in free fall compared to those that were acting before free fall? (B) Will the answer change if I drop the ball(with some initial velocity) into the jar and drop the jar simultaneously? (C)I somehow got the courage to jump along with the jar(without the ball in it). Now, I put the ball inside the cylinder with some initial velocity.Will the answer change now? (Remember that the whole system along with me is in free fall). **I tried to answer my own question like this:** In all these cases I think the buoyant force will be zero. By Archimedes Principle it is equal to weight of the liquid displaced.But, as something in free fall has zero weight there will be no weight for the liquid displaced and hence no buoyant force on the ball. From Stoke's law we know that the viscous force is directly proportional to the velocity of the ball in the liquid. Consider a frame of reference R to be moving at an acceleration g downwards. Thus, in all these situations the acceleration of the ball and the cylinder due to free fall is zero relative to R. Hence, from R's perspective the viscous force would the same to that when the system was at rest. I'm able to explain about the viscous force only in this way. What will be the viscous force for a man standing on the ground? I'm not very good at answering my own questions so, please help me with your ideas about the situation.


title: How can I calculate the initial velocities of a set of objects with a consistent force?

text: I'm trying to figure out which object would have the largest time of flight when launched from a slingshot. To do this, I need the initial velocity. How can I calculate this if I know the mass and diameter of the object? The slingshot always has the same force. I've seen a number of formulas, but none of them take into account the mass and size, which would matter in this case (a 10kg, 10cm rock will not go as fast or as long as a 1kg, 1cm rock).


title: Difference between inorganic and organic semiconductors: electronic structure or configuration, or?

text: Organic semiconductors differ from inorganic semiconductors. In organic semiconductors the molecules are held together by weak van der Waals interactions and in inorganic semiconductors by covalent bonds. So the bonds are different. How do you express the main difference between the twos? Is it the electronic structure or rather the electronic configuration, or something else?


title: Spin Control and Entanglement

text: I have a thought-experiment sort of question and I don't know where to start. Suppose you have an entangled pair, e1 and e2, and you split them. Then BEFORE reading them, you spin control e1 to +, then e2 would be -, correct? Can you then, AFTER, reapply spin control to e1 and make it -. Would e2 then flip back to + also?