title: about polarization of light

text: In the course of circuits and electronics, I remember there is an experiment to show the polarization of the wave as lissajous figures. I am wondering for polarized laser, is there any way to visualize the polarization in the similar way? I try to use a light splitter to split the (circular) polarized light into two perpendicular beams and used photo detector to receive the beam and send them into oscilloscope in X and Y channel. But the scope doesn't really show the 'lissajous' figure. So I am wondering if this is the right way to visualize the polarized light? Thanks


title: What does Friedrichs mean by "Myriotic fields"?

text: I came across K. O. Friedrichs' very old book (1953) "Mathematical Apsects of the Quantum Theory of Fields", and hardly any of it makes sense to me. One of the strange things that he refers to are "Myriotic" fields. What are these? Is there a modern account of what Friedrichs is talking about?


title: Fermi-Energy Level for Extrinsic Semiconductor

text: How does the Fermi energy of extrinsic semiconductors depend on temperature?


title: Horizon and Unruh radiation for a finite period of acceleration

text: It's a well known fact that an observer that accelerates at a constant rate from $-c$ at past infinity to $+c$ at future infinity sees a horizon in flat Minkowski spacetime. This is easy to see from a spacetime diagram once you realize that the union of past light cones on such a trajectory has a boundary that divides the spacetime into two regions - one inaccessible by the accelerating observer. This leads to the classic result of Unruh radiation when one looks at the quantum field theory for such an observer. The horizon plays a crucial role here. How does one go about determining whether a horizon is seen by a general class of worldines? In particular, is there any reason to believe that a horizon would exist for an observer that is stationary for all time except for a finite period of acceleration and deceleration? Is there any other class of worldlines other than an indefinitely accelerating observer for which a horizon is known to exist in flat spacetime?


title: Can reading glasses improve one's eyesight of objects lying in the long range?

text: Someone told me that reading glasses ( _a priori_ with a magnifying glass effect only) improve one's eyesight of objects lying in the long range distance. I am really sceptic about it since everything is blurry if I look at objects through those glasses! Is it possible (for this particular person at least)?


title: Force to stop a moving rope vs. stagnation pressure of a fluid

text: Let $\lambda$ be a linear density of a rope which is moving into a scale at velocity v. The additional force on the scale due to the collision is given as $\frac{d p}{d t} = v\frac{d m}{d t} = \lambda v^2$. For an incompressible fluid, the stagnation pressure from stopping a column of water _in excess of static pressure_ is $$\frac{1}{2}\rho v^2$$ We can easily compare the forms by, for example multiplying by the width of the column to obtain a linear density of the fluid, or consider hitting the scale with a continuum of infinitesimal ropes. It seems the $\frac{1}{2}$ factor would remain different. So what is the explanation for this relative factor of $\frac{1}{2}$? I have tossed around a few ideas but I'm curious what you may think.


title: What does "optical conductivity" mean?

text: Does it just mean "AC electric conductivity"? If so, why have a special name for it, and why mention optical specifically? The wikipedia page on it is very sparse. This (warning, PDF) document just says: > The term “optical conductivity” means the electrical conductivity in the > presence of an alternating electric field. Which sounds exactly like AC conductivity to me. Is it different? If not, then why the special name?


title: Did Maxwell invent the math to describe the ideas of electromagnetism?

text: Did he invent surface and line integrals, or did they already exist when he formulated his equations. If they did, already exist, how did they come about in pure math?


title: Is there a simulator for observing the force between two charges?

text: I basically want to observe the electrostatic force between two charges. I want to know if is there a simulator which could both help me to observe and calculate the force between two charges. I found this simulator which is pretty good I think. However, that would be great if there is a more comprehensive simulator with greater distances, more parameters etc.


title: PDE from London's Equation with Cylindrical Symmetry

text: The question is from ISSP by Kittel and as follows: (a)Find a solution of the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar coordinates, we want a solution of $$ B-\lambda \nabla^2 B = 0 $$ that is singular at the origin and for which the total flux is the flux quantum: $$ 2 \pi \int_0^\infty d\rho\rho B(\rho) = \Phi_0$$ The equation is in fact valid only outside the normal core of radius $\xi$ (b) Show that the solution has the limits $$ B(\rho) \simeq (\Phi_0/2\pi\lambda^2)\ln(\lambda/\rho) , (\xi \ll \rho \ll \lambda) $$ $$ B(\rho) \simeq (\Phi_0/2\pi\lambda^2)(\pi\lambda/2\rho)^{1/2}\exp(-\rho/\lambda) , (\rho \gg \lambda) $$ I have proceeded to solve the equation with CAS software such as Maple and Mathematica; however, they give two arbitrary constants which are to be determined. Is the problem sufficiently defined to fix these arbitrary constants? The solutions given by these programs are: $$B(r)\to c_1 J_0\left(\frac{i r}{\lambda }\right)+c_2 Y_0\left(-\frac{i r}{\lambda }\right) $$ Furthermore, how can I describe the effect of core? I found a solution manual but got confused by the method employed there (implementation of Dirac Delta) in order to describe the effect of the core. Here is a screenshot of the page. I appreciate your help and contribution. ![enter image description here](http://i.stack.imgur.com/CoStX.jpg) How can I implement 2-D Dirac delta function in cylindrical coordinates? By the way, I think one of the solutions of the homogenous equation is disregarded.


title: What Happens When A Gravitational Wave Interacts With Another One?

text: If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one.


title: Landau free energy

text: I am reading the statistical mechanics by Pathria in Chap 12. I have a question about the Landau free energy. What is the physical reasoning for that the free energy could be a functional of the order parameter? Normally, the free energy is used when we choose the canonical ensemble and we we know that the free energy is the characteristic function of $N,V,T$ of the system.


title: Equality of masses of particle and antiparticle

text: Usually we say that equality of masses of particle and antiparticle follows from CPT-theorem. But do we need it for showing this equality? The first method to show that is following.   1. The equation of free fields of an arbitrary spin $s$: all of them have Klein-Gordon or Dirac form (with some conditions of irreducibility), so the general solution is given in a form $$ \hat {\Psi}_{A}(x) = \sum_{\sigma}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2p_{0}}}\left(u^{\sigma}_{A}(\mathbf p)e^{-ipx}\hat {a}_{\sigma}(\mathbf p) + v^{\sigma}_{A}(\mathbf p)e^{ipx}\hat {b}^{\dagger}_{\sigma}(\mathbf p) \right)_{p_{0} = \sqrt{\mathbf p^{2} + m^{2}}} \qquad (1) $$   2. We must write $\hat {b}^{\dagger}_{\sigma}(\mathbf p)$, not $\hat {a}^{\dagger}_{\sigma}(\mathbf p)$, because of existence of some internal symmetries different from Poincare symmetry. If there exist some symmetry, we have conserving quantity $q$ for field, it means that corresponding operator $\hat {Q}$ commutes with hamiltonian of theory. We construct hamiltonian by having the equation of motion and its physical sense as polynomial of $\hat {\Psi}_{A}(x), \hat {\Psi}^{\dagger}_{B}(x)$ and spinor functions. It leads to the fact that $$ \hat {Q}\hat {a}^{\dagger}(\mathbf p)| \rangle = q\hat {a}^{\dagger}(\mathbf p)| \rangle, \quad \hat {Q}\hat {b}^{\dagger}(\mathbf p)| \rangle = -q\hat {b}^{\dagger}(\mathbf p)| \rangle . $$ The second method is following.   1. Suppose we don't have the equations for fields. But we know that S-operator must be Lorentz invariant operator or that our theory must be causal theory. S-operator is constructed from hamiltonian. Hamiltonian is constructed from creation and destruction operators and must be poincare scalar. So we must create invariant combination of fock space operators and some non-operator functions. Corresponding object is called creation/destruction field. By having laws of poincare transformation of creation and destruction operators we build the expression $(1)$ for field with one little difference: the first and second summands corresponds, in general, to the different masses. Then we construct the hamiltonian as polynomial of combinations of field functions and some spinor functions.    2. The statement of paragraph 1 leads us to the conclusion that $$ [\hat {\Psi}_{A}(x), \hat {\Psi}^{\dagger}_{B}(y)]_{\pm} = 0 , \quad (x - y)^{2} < 0\. $$ With a bit of derivations we can get $$ [\hat {\Psi}_{A}(x), \hat {\Psi}^{\dagger}_{B}(y)]_{\pm} = P_{AB}\left(i\frac{\partial}{\partial x}\right)\left( D^{m_{1}}_{0}(x - y)\pm (-1)^{s}D^{m_{2}}_{0}(y - x)\right), \qquad (2) $$ where $$ D^{m_{i}}_{0}(x - y) = \int \frac{d^{3}\mathbf p}{(2 \pi )^{3}2p_{0}}e^{-ip(x - y)}, \quad p^{2} = m_{i}^{2}. $$   3. We conclude that $(3)$ may be equal to sero for spacelike intervals if and only if $m_{1} = m_{2}$. Are these methods of demonstrating of the identity of particle-antiparticle masses correct?


title: Determine the number of days with North-East wind direction from the number of days with North and East wind direction?

text: I have some data like: Wind flow from north direction = “X” Numbers of days. Wind flow from east direction = “Y” Numbers of days. Then is there any formula to know numbers of days wind flows from North-East direction?


title: Why do tunneling photons outrace their non tunneling counterparts in vacuum?

text: If we describe a photons with a wave packet, moving towards a potential barrier and E smaller than V, there is a finite chance that it will tunnel to the other side. In this process it is likely that it will arrive before a photon that does not tunnel, not because it exceeds the the speed of light, but because the front of a wave packet will contribute most to tunneling. I would understand this if it was concerned with PARTICLES as follows: because each of the different waves that make a wave packet has a different momentum, the faster moving, higher energy, waves will move to the front of the wave packet when it disperses. According to the formula for tunneling probability indeed the higher energy parts will tunnel more often. BUT, why would a photon wave packet also have the higher energy elements more in front of it's wave packet, since all light-frequencies move at the same speed in vacuum right? Or does the described system indeed not work in vacuum. At page 14 of this link the mechanism I describe is presented: http://www.physics.umass.edu/sites/physics/files/admupld/Tunneling- UMass-12Feb10.pdf ![enter image description here](http://i.stack.imgur.com/2Q7Px.png)


title: What is the most efficient Organic Photovoltaic solar cell?

text: Preferably one that is researched in a paper, not necessarily in production. I have found various claims of OPVs with an efficiency of 10%, 8% and 7.3%, the only one I can substantiate is referenced in a paper in Advanced Materials (Volume 22, Issue 34 pp 3839–3856) to Solarmer but the web reference there leads to a news page where the news have been removed.


title: Groups acting on physics - a clarification on electrons and spin

text: My first question is fairly basic, but I would like to clarify my understanding. The second question is to turn this into something worth answering. Consider a relativistic electron, described by a spinor wave function $\psi(\vec x ,\sigma)$ and the Dirac equation. The conventional wisdom is that rotating everything by 360 degrees will map the spinor to its negative $\psi \mapsto -\psi$. However, it appears to me that this statement is "obviously false", because a rotation by 360 is, when viewed as an element of the group $SO(3)$, exactly equal to the identity map and cannot map anything to its negative. Thus, to make sense of the behavior of spin under "rotation", I have to conclude the following > The rotation group $SO(3)$ does _not_ act on the configuration (Hilbert) > space of electrons. Only its double cover $SU(2)$ acts on the space of > electrons. Is this interpretation correct? So, essentially, there is a symmetry group $SU(2)$ which acts on "physics", but its action on the spatial degrees of freedom is just that of $SO(3)$. > What other groups, even larger than $SU(2)$, are there that (could) act on > "physics" and are an extension of $SO(3)$? Is it possible to classify all > possibilities, in particular the ones that are not direct products? Of course, gauge freedoms will give rise to direct products like $SO(3) \times U(1)$ (acting on space $\times$ electromagnetic potential), but I would consider these to be trivial extensions.


title: Locality in Condensed Matter Lattice Model

text: What is a proper definition of locality in condensed matter lattice model? I emphasize "condensed matter" because there is no Lorentz symmetry or "speed of light". I think it is quite important because this constraint will rules out most nonlocal lattice Hamiltonians. First, let's say local Hilbert spaces are put both on vertices and edges of lattice, and notion of lattice may not be that regular. It could be a very random lattice.


title: What will happen to compass with north all around it?

text: If I keep a compass with bar magnets all around it, all of equal strength and a compass in a center then what will happen? Will the compass start rotating?


title: No Lagrangian description v.s. No quasi-particle description

text: This post is aimed to stimulate some discussions. We are familiar with many physical descriptions and theories of the (many-body quantum) system, with both **quasi-particle description** and **Lagrangian description**. For example: > the Landau Fermi-liquids theory. Here **quasi-particle description** are simply a way to find **effective excitations** for the (many-body quantum) system. The **effective excitations** may not be the original elementary constituents or elementary particles/spins of the system. These **effective excitations** contain **quasi-particle** , **quasi-string** , **quasi-brane** excitations, etc. $\bullet$ My question is that: >   1. what are systems with No quasi-particle description but with Yes > Lagrangian description. > >   2. what are systems with Yes quasi-particle description but with No > Lagrangian description. > >   3. what are systems with No quasi-particle description and with No > Lagrangian description. > > * * * **NOTE** : For instance, I suppose that, > 1+1-dimensional **Luttinger liquids** are **examples of 1.** No quasi- > particle description but with Yes Lagrangian description. See, e.g. T. Giamarchi, "Quantum Physics in One Dimension" Chap 2. ![enter image description here](http://i.stack.imgur.com/A9D2g.png) Fig. 2.4. _The occupation factor $n(k)$ of 1+1D Luttinger liquids. Instead of the usual discontinuity at $k_F$ for a Fermi liquid, it has a power law essential singularity. This is the signaturethat fermionic quasiparticles do not exist in one dimension. Note that the position of the singularity is still at k_F. This is a consequence of Luttinger's theorem._ On the other hand, it is likely that > **examples of 2** or **examples of 3** happens in RR fields or D-branes of > string theory, which has Yes/No quasi-particle description, with NO > Lagrangian description. Eg. see this Ref:Stability of Fermi Surfaces and K-Theory by Horava, see page 1 right column: _This implies that the RR ﬁelds are also objects in K theory, and not diﬀerential forms [9], making the low energy description of string theory on manifold Y in terms of Lagrangian (1) questionable. ... once the p-form $C_p$ are reinterpreted as K-theory objects, it is not clear how to even deﬁne Lagrangian (1). This crisis of the Lagrangian formulation of low- energy string theory is further supported by the discovery [10] of apparently non-Lagrangian phases in the partition functions of various string and M-theory vacua. Perhaps this means that the Lagrangian framework currently available is insuﬃcient for RR ﬁelds but its suitable generalization awaits to be discovered. (Important steps in this direction have been taken [11].) Alternatively, the theory may require a non-Lagrangian formulation. (This may already be suggested by the presence of a self-dual RR ﬁeld strength in Type IIB theory). Before we settle on either of these two alternatives, however, we should consider a third possibility. The subtle K-theory features of string theory could be an emergent phenomenon, with D-branes and RR ﬁelds emerging as composites of some more elementary degrees of freedom that admit a conventional Lagrangian description._ > **What else physical systems and theories are examples of 1. 2. 3.?**


title: 1D Ising Model (NN and NNN interactions) with 2 transfer matrices

text: I've tried an alternative solution for finding the partition function of this model. So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a mistake along the way):$$ $$ Starting with the Energy: $$ E = -J\sum_{i=1}^{N}S_{i}S_{i+1}-J'\sum_{i=1}^{N}S_{i}S_{i+2}-B\sum_{i=1}^{N}S_i $$ I've used the same trick used for NN but instead of two terms for the energy contribution from B I now have 3: $$ E = -J\sum_{i=1}^{N}S_{i}S_{i+1}-J'\sum_{i=1}^{N}S_{i}S_{i+2}-\frac{B}{3}\sum_{i=1}^{N}(S_i+S_{i+1}+S_{i+2}) $$ The point of doing this becomes more obvious when I write out the partition function: $$ Z=\sum_{\\{S_{i}\\}}e^{-\beta H} $$ $$ =\sum_{S_i=\pm1}...\sum_{S_N=\pm1}e^{\beta\sum_{i}[JS_{i}S_{i+1}+J'S_{i}S_{i+2}+\frac{B}{3}(S_i+S_{i+1}+S_{i+2})]} $$ Having written the energy in a way that I think is analogous to Isings method allows me to write this as the product of two exponentials and define two transfer matrices as follows: $$ Z=\sum_{S_i=\pm1}...\sum_{S_N=\pm1}e^{\beta\sum_{i}(JS_{i}S_{i+1}+\frac{B}{3}(\frac{S_i}{2}+S_{i+1}))}e^{\beta\sum_{i}(J'S_{i}S_{i+2}+\frac{B}{3}(\frac{S_i}{2}+S_{i+2}))} $$ And introducing $T_{_{NN}}= \left( \begin{array}{cc} e^{\beta(J+\frac{B}{2})} & e^{-\beta(J+\frac{B}{6})} \\\ e^{\beta(-J+\frac{B}{6})} & e^{\beta(J-\frac{B}{2})} \\\ \end{array} \right) $ and $T_{_{NNN}}= \left( \begin{array}{cc} e^{\beta(J'+\frac{B}{2})} & e^{-\beta(J'+\frac{B}{6})} \\\ e^{\beta(-J'+\frac{B}{6})} & e^{\beta(J'-\frac{B}{2})} \\\ \end{array} \right)$ Writting out the partition function in terms of these two transfer matrices and rearranging: $$Z=\sum_{S_1=\pm 1}...\sum_{S_N=\pm 1}\langle S_1| T_{_{NN}}|S_2\rangle\langle S_1| T_{_{NNN}}|S_3\rangle\langle S_2| T_{_{NN}}| S_3\rangle\langle S_2| T_{_{NNN}}|S_4\rangle...\langle S_N| T_{_{NN}}|S_1\rangle\langle S_N| T_{_{NNN}}|S_2\rangle$$ $$=\sum_{S_1=\pm 1}...\sum_{S_N=\pm 1}\langle S_2| T_{_{NN}}^{\dagger}|S_1\rangle\langle S_1| T_{_{NNN}}|S_3\rangle\langle S_3| T_{_{NN}}^{\dagger}| S_2\rangle\langle S_2| T_{_{NNN}}|S_4\rangle...\langle S_1| T_{_{NN}}^{\dagger}|S_N\rangle\langle S_N| T_{_{NNN}}|S_2\rangle$$ And using completeness : $$=\sum_{S_2=\pm1}\langle S_2|( T_{_{NN}}^{\dagger}T_{_{NNN}})^N|S_2\rangle$$ $$=Tr( T_{_{NN}}^{\dagger}T_{_{NNN}})^N$$ And voila, I have something I can plug into matlab and calculate U,F,M,$\chi$, heat capacity etc. for arbitrary J,J',B T and N (which I've done). The only problem is that it doesn't give me the same results as the NN 1D ising for J'=0 and it doesn't give me exactly the same results as the 4X4 matrix shown here: http://arxiv.org/pdf/cond- mat/9703187v1.pdf (equation 9). So I must have made some kind of mistake along the way, however, I would like to point out that the 4X4 matrix doesn't seem to reduce to the 1D ising model with NN interaction when J'=0 either. (I should specify that my matlab numerical computations have so far used values of N ranging from 20 to 500, I've also tried calculating U/N and $\frac{\chi}{N}$using the largest eigenvalues from each transfer matrix for certain parameters and obtained the same results for T>0)


title: Winter solstice, sunrise and sunset time

text: We all know the Winter Solstice comes on December the 20th or 21st, which is (by definition) the shortest day of the year. The Winter Solstice day is **_not_** the day of the year the Sun rises later (that would be one or two weeks later), and also is **_not_** the day the Sun sets earlier (that would be one or two weeks earlier)... * * * **Why is it like this? Why aren't sunset and sunrise times symmetric?**   I.e., why isn't there a "middle of the day" time valid throughout the year?   Does it have something to do with the eccentricity of the Earth around the Sun?   Does it have something to do with the fact the Earth is not a perfect Sphere? * * * I would understand a technical explanation, but I am asking for a widely- understandable, simple one. ![Analema on a globe](http://i.stack.imgur.com/Px5Zo.jpg)   _Pictured: ananalema on a globe._


title: Straw and Water

text: I drink from a glass of water with a vertical straw. What's the longest straw I can use and still drink water if the ambient pressure is 1 atm? Details and assumptions   * 1 atm is 101,325 Pa.   * The acceleration of gravity is -9.8m/s^2 .   * The density of water is 1g/cm^3. I am relatively new to physics, so I have no idea how to start. I greatly appreciate everyone's assistance. Thanks in advance.


title: Will larger balls make a Newton's cradle swing more stable?

text: I got a rather small Netwon's cradle and when I start it, the effect is not very good since all the balls start to swing where the effect we want is obvious. The balls are small and I wonder if larger balls would make the swing more ideal, like only the outer balls appear to move and the 3 inner balls should be still? I know product recommendations are off-topic, but I'm leaning towards buying a cradle with larger balls and perhaps you can advice which alternative to get that best replicates the original ideal cradle where only the outer balls move and the inner balls appear still?


title: Will Earth clock be slower than aeroplane clock when viewed from another planet?

text: I know that a clock placed on an aeroplane will have slowed with respect to a clock placed on earth because the more our velocities are, compared to the speed of light, the more the time slows down. But, the earth is moving faster than the aeroplane when viewed from outer space. Then, the clock placed on Earth should be slower than a clock placed on the plane. But the clocks are the same. How would that be different?


title: Spin up, spin down and superposition

text: I'm just starting to study quantum mechanics. Please explain the error in this thinking: You set up decay of two $\pi$ mesons and get $2\mathrm{e}^-$ on Mars and $2\mathrm{e}^+$ on Earth. On Earth you may or may not measure the spin of those positrons, with 50% probability that they are the same spin. On Mars, your buddy "immediately afterwards" takes a $\mathrm{He}^{2+}$ ion and adds the electrons. If you made the measurement, the $\mathrm{He}$ has a 50%+ chance of being in a higher-then-base energy level due to Pauli exclusion. If you didn't make the measurement, the $\mathrm{He}$ has a much lower chance of being in a higher-than-base energy level. Your buddy measures the energy state. Result: You just superluminally transmitted $\approx 0.8$ bits of information.


title: Is it possible to whirl a point mass (attacted to a string) around in a horizontal circular motion *above* my hand?

text: I'm studying circular motion and centripetal force in college currently and there is a very simple question but confuses me (our teacher doesn't know how to explain either :/), so I hope we can sort it out here >< So I draw two pictures to show what I was thinking on it. ![1](http://i.stack.imgur.com/IXKXR.jpg) In pic 1 there is a hand rotating a ball attached to a piece of string in a circular motion, by free body diagram we can easily see that the net force produced by tension and gravity is centripetal force, and it towards to the center of the circle. But in pic 2 as shown below ![2](http://i.stack.imgur.com/c5Ocp.jpg) When the hand is below the ball, the net force is actually towards downwards, not to the center of the circle. How would that circular motion happen if this free body diagram doesn't make sense? Or is there any other force acting on it?


title: Coulomb potential

text: It is known that the Coulomb potential can be obtained by Fourier transform of the propagator from E&M. Is this because one of Maxwell's equations have the form $\nabla \cdot \mathbf{E}=\rho$?


title: What could cause a 30 k g acceleration?

text: I'm reading the specs of an IC (Cypress 5LP SoC) and it says it's available in 30 k g shock resistance configuration. The fastest acceleration I heard of so far was hitting a golf ball hard, which would be around 1000 g. Does anyone have an example of a mechanical impact of 30 000 g (300 000 m$\cdot$s$^2$), applicable to electronics devices?


title: A rod of length $L$ & mass $M$ is rotating in a circle about one end then calculate tension in the rod at a distance $x$ from the support

text: A rod of length L & mass M is rotating in a circle about one end then calculate tension in the rod at a distance 'x' from the support ? For its solution why should we take mass of L-x portion of rod instead of taking mass upto x distance from support as we have the formula T = m w2 x I m in confusion....


title: calculating ballast needed to offset wobble of a 1g unbalanced centerfuge spinning at 20000G's

text: So this seemed pretty straight forward at first, but then I wasnt too sure about gravity forces... So.   * centerfuge   * 2 samples   * 1 sample is 1gram heavier than the other   * centerfuge at top speed is exerting 20000 gravitys of force on the samples What kind of weight / ballast is needed to keep the centerfuge from wobbling off the table? Id imagine this is a linear equation ... So if i could get the equation that would be TOPS! Ballast should be placed on 4 corners on the outside of the sample


title: Is it feasible to measure the energy of cosmic ray muons with a consumer Digital Single Lens Reflex camera?

text: I have read this article SIBBERNSEN, Kendra. Catching Cosmic Rays with a DSLR. _Astronomy Education Review_ , 2010, 9: 010111. and it talks about estimating the muon cosmic ray flux by means of a DSLR camera. Is it possible to measure the muon's energy with the same apparatus? If yes, how can I "calibrate" (?) the camera in order to measure the energy?


title: How much of the universe is observable at visible wavelengths?

text: Knowing that:   * The Zone of Avoidance (Looking towards the center of the Milky Way) blocks roughly 20%   * Each Milky Way star has an angular size, depending on proximity, that obscures a certain percentage of our optical view.   * Each Galaxy has an angular size, depending on proximity, that obscures a certain percentage of our optical view. And so on through Galaxy Clusters, Superstructures, etc... **What percentage of our universe can we not see at visible wavelengths?**


title: Help with understanding an Operator definition

text: The operator $\hat{F}$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a)$ Does this mean $\hat{F}=(x+a)+(x-a)$ and that $\hat{F}$ is operating on $\psi(x)$?


title: Swimming in Spacetime - apparent conserved quantity violation

text: My question is about the article Swimming in Spacetime. My gut reaction on first reading it was "this violates conservation of momentum, doesn't it?". I now realize, however, that this doesn't allow something to change its momentum; it only allows something to move (change position) without ever having a nonzero momentum. Since this is relativity, there is no simple relationship between momentum and velocity like p = mv, so this is all well and good. An object can move, with a constant momentum of zero, by changing its shape in a nontrivial cycle. However, now I'm thinking about a different conservation law and I can't see how "swimming through spacetime" is possible without violating it. The conserved quantity I'm thinking of is the Noether charge associated with Lorentz boosts, which is basically x - (p/E)t, that is, the position of the center of mass projected back to time t=0. If p = 0, then the conserved quantity is simply x, the position of the center of mass. This obviously contradicts the whole swimming idea. What's going on here? Is swimming through spacetime only possible if the spacetime is curved in some way that breaks the symmetry under Lorentz boosts? Or is there some error in my reasoning?


title: Resistance of aluminium rectangular wire

text: I am working on upgrading an electromagnet design program. There are currently two types of wire used for the coil, circular and strip (read as: rectangular). The current algorithms in this program are empirically derived with lots of undocumented constants and so one of the requirements is to move it more into the realm of accepted physics. The formula I am using to calculate resistance is: $$ R = \frac{\rho L}{A} $$ These are aluminum wires so the value of $\rho$ boils down to $2.82\times10^{-8}$ at 20°C. For circular wires this works perfectly, giving near enough the same results (enough to put it down to the increased accuracy). For strip wires this doesn't work quite so well. The problem comes from a difference in the cross-sectional area of the wire. The original program is doing $h^2$ as the area, where $h$ is the height of the wire. Now is that not wrong? The cross-sectional area should be $wh$ where $w$ is the width of the wire. For example, If the strip wire is 8.4mm by 2.8mm, the original program would do: $$ 2.8 ^ 2 = 7.84 $$ but should it not be doing: $$ 8.4\times2.8 = 23.52 $$ The entire original algorithm is: $$ R = \frac{84.09L}{h^2} $$ Is there a reason the original engineer may have done $h^2$ as apposed to the correct cross-sectional area? I don't want to just declare his calculations as wrong as the results from this program have been used for the last 30 years and none of the magnets have gone bust or underperformed. Infact if anything they would have been over performing if this calculation is wrong.


title: Classical Limit in Quantum Mechanics

text: Suppose I have a wave function $\Psi$ (which is not an eigenfunction) and a time independent Hamiltonian $\hat{\mathcal{H}}$. Now, If I take the classical limit by taking $\hbar \to 0$ what will happen to the expectation value $\langle\Psi |\hat{\mathcal{H}}|\Psi\rangle$? Will it remain the same (as $\hbar = 1.0$) or will it be different as $\hbar\to 0$? According to correspondence principle this should be equal to the classical energy in the classical limit. What do you think about this? Your answers will be highly appreciated.


title: Origin of exchage interactions

text: Can someone explain to me the origin of the exchange interaction between two electrically charged spin 1/2 fermions? Quantitative or qualitative accepted.


title: Excluded Volume Calculation in Van der Waals Equation

text: When accounting for the excluded volume for in the Van der Waals equation, it is assumed that the molecules are hard spheres and are of diameter. If we consider a cube of volume V, then we can say that the side of this cube is of length $V^{1/3}$. Consider the diameter of the molecules to be $\sigma$. Suppose that the number of molecules in this box to be $N$. If we anchor $N-1$ molecules at their positions and look at the excluded volume from the perspective of the $N^{th}$! molecule, we see that the center of this molecule can approach the walls of the cube only upto a distance of $\sigma/2$ and can approach the anchored molecules upto a distance of $\sigma$ from their centers as shown:![excluded1](http://i.stack.imgur.com/v8JE4.png). Then the excluded volume for this molecule should be $V_{ex}=(V^{1/3}-\sigma)^{3}-(N-1)(\frac{4}{3}\pi\sigma^{3})$. This follows even if we consider any other molecule and anchor the rest. But, according to wikipedia, we would be overcounting. I don't see how. The correct expression should be $V_{ex}=(V^{1/3}-\sigma)^{3}-(N/2)(\frac{4}{3}\pi\sigma^{3})$. Can anyone please explain?


There are power law of phase transition in physics and Zipf law in linguistics which are similiar to each other ,and some expert think they are in fact just the same.But the diagrams of them base on the data of physics and data of language are not that same. What is difference and linkage between power law of phase transition in physics and Zipf law in linguistics?And what is the mechanism behind them and the difference?Any reference?


title: What's the advantage of NASA's flying saucer over traditional aerodynamic models?

text: NASA has recently tested a saucer-shaped spaceship. What's the advantage of this new design over traditional aerodynamic designs? The test launch was performed from within atmosphere which would offer higher air drag. What's the positive side of this that this big thing has been ignored?


title: $SU(2)$ gauge symmetry

text: Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = \partial_\mu+i\frac{g}{2}t^aW^a_\mu$. The Lagrangian is invariant under a local $SU(2)$ transformation: $$ \psi(x) \rightarrow \exp \left[-i\theta^a(x)t^a \right]\psi(x) $$ $$W^a_\mu(x) \rightarrow W^a_\mu(x) +\frac{1}{g}\partial_\mu\theta^a(x) + \epsilon^{abc}\theta^b(x)W^c_\mu(x)$$ Often, we say that $W_\mu^a$ transforms according to the adjoint representation of $SU(2)$ but how can we say that based on the previous equation?


title: Does Hall Effect increases the resistance of a wire?

text: If a current in a wire is flowing perpendicular to a magnetic field, the Hall effect will be observed, which is caused by the forces from magnetic fields 'pushing' the electrons to one side or the wire. So, does it increases the resistance of the wire, as there is now less area for current to flow through due to one side of the wire being occupied by these (assumed)immobile electrons?


title: about MS in Theorectial Physics

text: I am an undergraduate student (Bachelors of engineering) about to complete my 1st year in the field of Information science and engineering or information technology. I am interested in pure physics especially high energy physics and some parts of quantum Physics. My question is that, is there any possibilities i can have my post-graduation study in the field of Physics(purely theoretical physics like high energy physics) ?


title: Radioactive stability of some nuclei

text: While studying radioactivity I found that even the most radioactive substances i.e substances with the shortest half lives do not completely degenerate. Suppose there is a 1 mole sample of an element **X** whose half life is **1 day**. But as the _degeneration equation_ is _exponential_ in nature even after 1 million years some amount of **X** would remain in the sample. What i want to know is how come certain **_atoms of an element possess such stability that they do not generate after a million years_** , **_while some atoms taken form the same sample degenerate after 1 day._** Is it because they have some different particles in their nucleus or something ? What is the role of **quantum tunnelling** ?


title: Where does the term "boost" come from for rotation-free transformations?

text: I had never seen rotation free transformations called "boosts" (I think I have it right) before reading some questions here. I'm too old perhaps. I have not found the etymology after some searching, though it sounds like something V.I. Arnold would think up, or jargon from inertial navigation. Anyone know where/how it started or was popularized? (If it is in MTW or Ohanian (old edition) or Weinberg, I promise I'll facepalm)


title: Why does an airplane have more lift near the ground?

text: I've noticed that an airplane appears to have more lift when it's almost touching the ground then it has 100 feet or more in the air. What causes this to occur?


title: Duhamel formula for propagators

text: Let $\dot{z} = A(t)z + b(t)$ with $ z(t) \in \mathbb{R}^n$ and $A(t)$ be a linear map from $\mathbb{R}^n \rightarrow \mathbb{R}^n$. A propagator is also a linear map $P(t,s):$ $\mathbb{R}^n \rightarrow \mathbb{R}^n:$ $z(s) \rightarrow z(t)$ and has the following properties:   1. $P(t,r)P(r,s) = P(t,s)$   2. $P(s,s) = id$   3. $\frac{\partial}{\partial t}P(t,s) = A(t)P(t,s)$ The Duhamel formula is the following (a general solution to: $z(t) = P(t,s)z(s) + \int_s^t d\tau P(t,\tau)b(\tau)$ I'm trying to prove it by simply deriving it: $\dot{z} = \frac{\partial}{\partial t}P(t,s)z(s) + \frac{\partial}{\partial t}\int_s^t d\tau P(t,\tau)b(\tau) = A(t)P(t,s)z(s) + P(t,t)b(t) + \int_s^t A(t)P(t,\tau)b(\tau) = A(t)z(t) +b(t) + \int_s^t A(t)P(t,\tau)b(\tau)$ This is nearly the form that I want (I used equation 2 and 3 and standard properties of the integral and dervative, for those wondering), but I can't see right now how the intgeral equals to zero. Anyone can tell me? Cheers!


title: Cell connections

text: I have an UPS with two 6V batteries connected in series. Is it possible to use the same system (UPS) to charge another two pieces of 6 volts battery if all the batteries are connected in parallel to maintain a 12 volts circuit? Will this affect the charging system of the ups if the connection is made?


title: An atomic bomb explodes inside of an "unbreakable" container which is on a scale. Does the "weight" of the container change?

text: This may or may not be an incredibly stupid thought experiment, but a short time ago I read that most of the "mass" in the proton was actually energy from the quarks and gluons, as opposed to the actual mass which was coupled to the Higgs field. This made me start thinking about objects on macroscopic scales and the effects of energy and mass in a closed macroscopic system. I have only a superficial highschool level understanding of relativistic mass, and I have read extremely vigorous debates and arguments about the significance of the mass-energy equivalence, and what it means in terms of the "weight" of an object. So my question is that if said atomic bomb were to explode inside of a completely impenetrable container of some sort, would the measured weight of the container change, assuming no radiation was able to escape the container itself?


title: Why does my refrigerator door resist opening?

text: After closing my refrigerator's door I noticed that it's much harder to reopen it immediately, as if there's an underpressure. Only after a minute or so it opens normally. How can this be explained?


title: Hyperfine structure vs Lamb shift in the hydrogen atom

text: The hyperfine structure of the energy levels of the hydrogen atom refers to the shifts in the evergy levels due to the magnetic moments of the nucleus and of the electron. This is an effect of non-relativistic QM, if I understand correctly. On the other hand, the Lamb shift in the energy levels is due to completely different reasons: it has nothing to do with the spin of nucleus, and can be explained only by the relativistic theory, i.e. QED. How to compare these two effects? Which one is stronger? Is there any literature about this? I do know references to literature about the Lamb shift. It would be helpful to have a reference to a detailed discussion of the hyperfine structure, and to a comparison with the lamb shift.


title: Amount of thermal energy in the Earth?

text: Does anyone know the amount of thermal energy that the Earth's mantle and core possess? I don't mean the maximum limit of electrical power we could generate with geothermal plants, but rather: if you took the Earth and magically cooled it down so that its temperature became homogeneously 0K, what would the change in the Earth's internal energy be? (ignoring the Sun) If we don't have that data, does anyone have an idea how to perform an order of magnitude estimate?


title: Is the color gauge group spontaneously broken in QCD by the Savvidy vacuum?

text: One model for confinement in quantum chromodynamics is the Savvidy vacuum. This is a spontaneous symmetry breaking of color gauge symmetry by the gauge fields. The vacuum is divided into Savvidy domains. Is this process hidden by confinement? L.Motl: I erased one "d" from Savvidy - everywhere. Link to Savvidy vacuum added.


title: Uncertainty relation and Energy-Position interference

text: How would you prove that the simultaneous measurements of position and energy are not subject to interference? I was thinking in calculate the commutation relation between $x$ and $H$ (Because $\Delta E=\Delta H$), but I realized that $[H,x]\neq0$, so I tried to use a more general expression for the Uncertainty Principle that says that if $H_1$ and $H_2$ are Hermitian operators then $\Delta H_1 \Delta H_2\geq\frac{1}{2}|\langle [H_1,H_2]\rangle|$, but again, $[H,x]\neq0$. Can you suggest me a way to do this? Thanks.


If you look at most published articles, or those on the preprint server, you find the vast majority of articles have more than one authors. Are collaborations needed to be productive in physics? In other fields like math, single author articles are the norm.


title: Which subjects in physics should I choose if I want to help tackling today's energy and environment related problems?

text: I was wondering what subjects a freshman in mathematics ought to choose in the future if s/he wanted to help working on energy and environment-related issues we are currently facing, and will very likely be even more profound in the future. I am currently day-dreaming about using the skills I will acquire as a mathematician to create more efficient solar cells or wind-mills, but I guess a little understanding of the physics involved wouldn't hurt. Which subjects should I choose to enhance my knowledge on this subject? Could you perhaps suggest a rough pathway by means of which I could deepen my understanding of the subject matter?


title: What is the density of virtual photons around a unit charge?

text: It seems that virtual photons also exist in vacuum. So the precise question is: What is the additional virtual photon density due to the electric field of a unit charge? Or: How many virtual photons per volume are found around a unit charge? The answer will depend on distance, but what are the exact numbers?


title: A wobbly pan on an induction cooker, is it less effective?

text: According to the Wikipedia, one of the limitations of the induction cooker is that the bottom surface of the pot should be flat. Accordingly, I commented on a question on Seasoned Advise, but I'd like to know whether this is actually true (My username is BaffledCook there). For a wok, it seems reasonable to believe the induction cooker will not be effective, but how about a wobbly pan? How wobbly should it be before it really affects efficiency?


title: Calculating size/weight of the base of a banner to withstand wind load

text: I want to build what is essentially a banner - a rectangular piece of fabric that will be mounted to a stand and displayed vertically. This object will be outdoors, so the most important factor in it staying upright is that it be able to withstand wind load. I know how to calculate wind load on an object, and I understand that the force of that wind load can be considered to act on the center of mass of the banner, but I'm not sure how to figure the size or weight of the base into the system to determine whether the object will fall over under the wind load. For example, if I were to make a banner 3 feet wide and 6 feet tall, With a base also 3 feet wide, and 3 feet deep, how much would the base need to weigh to withstand a 30 mph gust of wind? (Of course, the ideal answer would show me how to calculate this for myself, to try different scenarios)


title: How DC and AC relays works?

text: I was told long time ago that DC relay had a coil. There was a switch (2 wires, one is stable, the other one is flexible) inside the coil. The switch was parallel to the axial direction of the coil. Today, I am thinking how AC relay works. I go back to think about DC relay. If what I was told is right, the magnetic field direction is parallel to the switch direction. Then how the magnetic field makes two wires touch? Also, how does AC relay work?


title: Boiling Pasta resulting in a Torus like shape?

text: I noticed when I was boiling pasta the other day that the pasta uniformly spread out and formed a donut like torus. Why does this happen? Does it have to do with the shape of the pot? I tried to take a picture when it was boiling, but steam got all over my lens. So, this is a more 2d version of what the pasta looked like with boiling (while boiling, the pasta strands were uniform and seemed to be repelled out uniformly from the center of the pot.![I tried to take a picture when it was boiling, but steam got all over my lens. So, this is a more 2d version of what the pasta looked like with boiling \(while boiling, the pasta strands were uniform and seemed to be repelled out uniformly from the center of the pot.](http://i.stack.imgur.com/B9RD4.jpg)


title: Tape adhesive force

text: So I was bored and had some clear Scotch tape nearby. Decided to mess around, and now I have questions.   * What is the physics behind the force of tape adhesive?    * Why does the tape hold so much better if it is pulled parallel to the surface plane it is on? (Excuse my poor drawing skills)    * Is there a rating for adhesion? Can I found out how "sticky" the tape is by measuring the area of tape on a table and the force at which it breaks loose from the table in a similar situation below? ![Pulling on tape](http://i.stack.imgur.com/diduF.png) Is this even an appropriate question? Or should I close it? It's definitely not homework.


title: Universality in Weak Interactions

text: I'm currently preparing for an examination of course in introductory (experimental) particle physics. One topic that we covered and that I'm currently revising is the universality in weak interactions. However I don't really understand the point my professor wants to make here. Let me show you his exposure to the topic: We stark by looking at three different weak decays:   * $\beta$-decay in a nuclei: $^{14}O \rightarrow ^{14}N^{*} + e^{+} + \nu_{e}$ (Lifetime $\tau$=103sec)    * $\mu$-decay: $\mu^{+}\rightarrow e^{+} + \nu_{e} + \overline{\nu_{\mu}}$ ($\tau$=2.2$\mu$sec)   * $\tau-decay$: $\tau^{+}\rightarrow$ $\mu^{+} + \nu_\mu + \overline{\nu_{\tau}}$ or $\tau^{+}\rightarrow$ $e^{+} + \nu_e + \overline{\nu_{\tau}}$ ($\tau = 2 \cdot 10^{13}$ sec). Now he points out that the lifetimes are indeed quiet different. Never the less the way the reactions behave is the same? (Why? Okay we always have a lepton decaying into another leptons and neutrinos? But what's the deal?) Then he writes down Fermi's golden rule given by: $W=\frac{2\pi}{\hbar} |M_{fi}|^{2} \rho(E')$. Now he says that universality means that the matrix element $|M_{fi}|$ is the same in all interactions. The phase space however is the same (Why? First of all I have often read on the internet that universality means that that certain groups of particles carry the same "weak charge"? And secondly: What do particle physicists mean when they talk about greater phase-space? Three dimensional momentum space? But how do you see or measure that this space is bigger? And bigger in what respect? More momentum states?) Now he says that the different phase spaces come from the different lifetimes. He calculates $\rho(E')=\frac{dn}{dE}=\int \frac{d^{3}p_{\nu} d^{3}p_{e}}{dE} = p_{max}^{5} \cdot \int \frac{d^{3}(p_{\nu}/p_{max}) d^{3}(p_{e}/p_{max})}{d(E/p_{max})}$. Now the last integral is supposed to be identical for all decays. And $p_{max}$is suppose to be different in all decays . But why? And what is the definition of $p_{max}$? Now he has $\tau = \frac{1}{M}$. So he gets $\tau = const \cdot \frac{1}{|M_{fi}|^{2} p_{max}^5}$. Hence he gets in the ln($\tau$)-ln($p_{max}$) diagram a line with slope -5. Now he claims that this "proofs" that $|M_{fi}|$ is constant in all processes. Again why? Can someone please give me some overview and explain to me why he is doing all that stuff? I din't really have much background when it comes to particle physics. So can someone explain it to me in a clear and easy way?


title: Would a distant enough electromagnetic force be non-continuous?

text: To explain, with a source of 'normal' photons, as the source becomes more and more distant the luminosity fades until finally the stream of photons is no longer continuous, giving the receiver individual peaks of energy with none in between. As recession continues the gaps between energy peaks (photon impacts) becomes longer. If the electromagnetic force is carried by virtual photons, and these photons are still bound by the speed of light, then unless an infinite number are created there should be some distance at which the observed electromagnetic force applied should no longer be continuous, but instead would break up into individual force applications. I doubt we have anything like the level of sensitivity to be able to detect such tiny fluctuation, especially with all the background noise being generated by all sources of electromagnetic force that are much closer, but in theory would the force break up at sufficient distance into discrete instances of force instead of a continuous force?


title: Is the orbit of earth around the sun chaotic?

text: The orbit of the earth seems to be very predictable. But as it is a many-body problem having sun, earth, moon, jupiter and so on, is it really that stable or will it start making strange movements sooner or later? Are there asteroids or moons in our solar system with obviously chaotic orbits?


title: Why are no CMB experiments able to measure the whole range of multipole moments?

text: Have a look at the following diagram: ![CMB measurements](http://i.stack.imgur.com/zB8KY.png) This shows measurements of the CMB by various experiments, with multipole moment $l$ on the x-axis and the temperature of the corresponding moment on the y-axis. (Please correct me if I have that wrong.) Looking at the plot, I can see that WMAP's results extend to about $l=1000$, and they become poor as we go to higher values of $l$. Meanwhile, the other experiments can't measure the CMB at lower values of $l$ around $l < 500$. I'm guessing that a ground-based approach with a large collector is required to get the high resolutions needed to measure the power of the component at high values of $l$. Is this correct? What I don't understand is, why can't the other experiments get measurements in the lower part of the plot?


title: How to find a hyperbolic escape orbit with a starting point and a desired velocity after sphere exit

text: How would you find an escape orbit which includes a given starting point (not assuming a circular starting orbit) and escapes the sphere of influence of the current body with a given remaining velocity? A problem perhaps similar to Lambert's Problem, but instead of an arrival location and time, it is a final velocity. I have a solution that someone suggested to me, that takes the final velocity as a hyperbolic excess velocity, uses that to calculate specific energy and semi major axis of the departure orbit. However they then assume that you're departure burn will occur at the periapsis of the new orbit and derive the eccentricity, and determine other elements of the departure orbit using that. The approach would work, though you have to use various guesses as the starting position until you find one that results in an orbit departing in the correct direction and the start is always at the periapsis of the departure orbit. I am hoping that there is an answer to this problem that doesn't require starting at the periapsis of the escape orbit, as I think that would broaden the range of potential departure directions from a given point. However I don't know how I would begin to calculate that.


title: Why is the change in height for mass be $2h$ in this question?

text: Could anyone tell me why the height is $2h$?, and is the total gpe in a system where the masses are balanced 0? ![enter image description here](http://i.stack.imgur.com/OcQae.png)


title: Kac-Moody algebras in 5 dimensional Kaluza-Klein theory

text: I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein we start with pure (5 dimensional) gravity. Then we assume a $M_4×S^1$ ground state where we have Minkowski space with a circle. The topology of this ground state breaks the 5D diffeomorphism invariance because now only periodic transformations are allowed. So considering a coordinate change $$x\to{}x'=x+\xi,$$ we can Fourier expand $$\xi(x,\theta)=\sum{}\xi(x)e^{in\theta}.$$ It is known that if we truncate at $n=0$ we get a theory of gravitation in 4 dimensional space-time, electromagnetism and a scalar field (sometimes called the Dilaton). So, the remnants of the initial 5 dimensional diffeo invariance are 4 dimensinal diffeo invariance, $U(1)$ gauge and a scale invariance for the dilaton. In the non truncated theory we will keep all the modes in the previous Fourier expansion and we will have more symmetries. The 4d diffeo and $U(1)$ gauge do survive in the non truncated version but not the scale invariance of the dilaton. So far so good. But, how to know the symmetries of the full theory? well, with Kac-Moody generalization of Poincaré algebras. For example in http://arxiv.org/abs/hep- th/9410046 says this. But HOW exactly? I would like a clear explanation of how this Kac-Moody algebra encodes the remnant symmetries of the original diffeo 5 and also how the zero modes of this algebra correspond to diffeo4 and $U(1)$ EDIT:: this is the algebra I am talking about $[P_{\mu}^{(n)},P_{\nu}^{(m)}]=0$ $[M_{\mu\nu}^{(m)},P_{\lambda}^{(n)}]=i(\eta_{\lambda\nu}P_{\mu}^{(m+n)}-\eta_{\lambda\mu}P_{\nu}^{(m+n)})$ $[M_{\mu\nu}^{(n)},M_{\rho\sigma}^{(m)}]=i(\eta_{\nu\rho}M_{\mu\sigma}^{(m+n)}+\eta_{\mu\sigma}M_{\nu\rho}^{(m+n)}-\eta_{\mu\rho}M_{\nu\sigma}^{(m+n)}-\eta_{\nu\sigma}M_{\mu\rho}^{(m+n)})$ $[Q^{(n)},Q^{(m)}]=(n-m)Q^{(n+m)}$ $[Q^{(n)},P^{(m)}_{\mu}]=-mP^{(n+m)}_{\mu}$ $[Q^{(n)},M^{(m)}_{\mu\nu}]=-mM^{(n+m)}_{\mu\nu}$


title: Problem with shell model and magnetic moment of Lithium-6

text: I have a problem with the calculus of magnetic moment of Li-6. The configuration of protons is $1p_{3/2}$, and the neutrons' one is the same. I have to add the magnetic moment of uncoupled proton and uncoupled neutron. I use the following formula for $J=l+\frac{1}{2}$ (J is the particle spin): $$ \frac{\mu}{\mu_N}=g_lJ+\frac{g_s-g_l}{2}$$ For the proton I have: $g_l=1; g_s=5.58 \rightarrow \frac{\mu}{\mu_N}=J+2.29=3.79$ For the neutron I have: $g_l=0; g_s=-3.82 \rightarrow \frac{\mu}{\mu_N}=-1.91$ So the total $\frac{\mu}{\mu_N}=3.79-1.91=1.88$, exactly 1 more than the correct value, 0.88! What's wrong?


title: Poles for a particle scattered in a delta potential

text: I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also like to go ahead and apologize for not knowing how to format correctly. I was given that a particle is scattered with the given Hamiltonian: $$ H = P^2 - g\delta(x) $$ Where $\delta(x)$ is the Dirac delta function. I was able to find the states in momentum representation, and was supposed to Fourier transform them to get the position representation states. Doing this leads to an integral with simple poles on the real axis, and can be solved by moving the poles above or below the real axis by some constant, applying the residue theorem, and taking the constant to zero. This leads to three different solutions, depending on if I move one pole up and one down (two possibilities), or both poles into a contour. While I was talking to my professor, he mentioned that the solutions only work for certain values of $x$, and that the range of $x$ is given by what makes the arc in the contour go to zero. I'm actually having trouble seeing this, since that introduces an ambiguity in the solutions. If I shift the left pole up and the right pole down and close the contour in the upper plane, this means that x has to be positive, in order to get a decaying exponential in the integral. However there is nothing stopping me from moving the poles in the opposite way, and closing above to obtain a different solution for $x>0$. Is there something wrong in the math, or is the way I move the poles governed by the physical situation I'm interested in? (Which would be no plane waves moving the left for $x>0$ d.) I should mention that I am assuming: $$ |\psi> = |p> \+ |\psi_{sc}> $$ Where p is the incoming momentum and $|\psi_{sc}>$ is the scattered portion of the wave function. Working in momentum representation, I obtain: $$ \psi_{sc}(k) = \frac{g - \frac{g^2 i}{2p+ g i}}{2\pi(k^2-p^2)} $$ Where k is the momentum variable, and p is the fixed momentum of the incoming particle. The transform I obtain is: $$ \psi_{sc}(x) = \eta \int_{-\infty}^{\infty} \frac{e^{i k x}}{k^2-p^2} dk $$ This is where I run into the problem with the poles. I know there shouldn't be any waves traveling to the left for $x>0$. My final goal is to check my states by checking the reflection and transmission coefficients and confirming they add up to 1.


title: How does a fabric containing 10% stretch material make it stretchy?

text: Why should adding a small amount of a stretchy material make an otherwise non- stretchy fabric stretch? Shouldn't the non-stretch fibres still constrain the maximum stretch of the fabric?


title: Energy required to transport fluid through tube

text: We have a tube (rigid or compliant) filled with a fluid (Newtonian or non- Newtonian). We apply a pressure drop and mobilize the fluid. What is the amount of energy required to transport a certain amount of fluid. I am looking for a general answer which takes into account both the type of fluid and type of tube otherwise it is straightforward to get an answer for the special cases.


title: black hole no-hair theorems vs. entropy and surface area

text: I was revisiting some old popular science books a while ago and two statements struck me as incompatible.   * No-hair theorems: a black hole is fully-described by just a few numbers (mass, spin etc) irrespective of the type or configuration of the matter/energy within the event horizon.   * Surface area measures entropy: you can't reduce the total entropy of the universe by throwing a box of hot gas into a black hole, its size/entropy will increase by the necessary amount. Suppose I have two black holes, A and B. They are identical and interchangeable, same mass, spin, charge... I have two boxes, BoxA and BoxB. BoxA contains a kilogram of salt in the form of a single crystal and BoxB contains a kilogram of salt as loose powder. **BoxA has less entropy than BoxB** but is otherwise identical. I throw BoxA into A and BoxB into B. The no-hair theorems seem to imply that A and B will increase in size by the same amount. The entropy theorems seem to imply that B will end up larger than A. What am I missing here?


title: Particles for all forces: how do they know where to go, and what to avoid?

text: Here's an intuitive problem which I can't get around, can someone please explain it? Consider a proton P and an electron E moving through the electromagnetic field (or other particles for other forces, same argument). They exert a force upon one another. In classical mechanics this is expressed as their contributing to the field and the field exerts a force back upon them in turn. In quantum mechanics the model is the exchange of a particle. Let's say one such particle X is emitted from P and heads towards E. In the basic scenario, E absorbs it and changes its momentum accordingly. Fine. How does X know where E is going to be by the time it arrives? What's to stop E dodging it, or having some other particle intercept X en route? Are P and E emitting a constant stream of force-carrying particles towards every other non-force-carrying particle in the universe? Doesn't this imply a vast amount of radiation all over the place? I am tempted to shrug of the entire particle exchange as a mere numerical convenience; a discretization of the Maxwell equations perhaps. I am reluctant to say "virtual particle" because I suspect that term means something different to what I think it means. Or is it a kind of observer effect: E "observes" X in the act of absorbing it, all non-intercepting paths have zero probability when the waveform collapses? Or have I missed the point entirely?


title: Time Dependent HydroHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? gen Wave Function

text: 1) How vwoulHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? go about writing the time dependent wave function given the wavefunction at $t=0$? Suppose How would I go about writing the time dependent wave function given the wavefunction at $t=0$? an electron is in a superposition state in the hydrogen atom at $t=0$ with normalized wavefunction: How would How would I go about writing the time dependent wave function given the wavefunction at $t=0$? I go about writing the time dependent wave function given the wavefunction at $t=0$? $$\psi(r,\theta,\phi) ~=~ A(2R_{10}Y_{00}+4R_{21}Y_{1,-1}).$$ 2) What woHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? uld the tiHow wouHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? ld I go about writing the time dependent wave function given the wavefunction at $t=0$? me dependent wave function look like? 3) Also how couHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? ld I tell if the probabilities of measuring a certain energy, or angular momentum arHow would I go about writing the time dependent wave function given the wavefunction at $t=0$? e time dependent?


title: How much more effective is it to stir in both directions?

text: I have been told that industrial mixing machines (say, for cake batter) switch directions periodically, first stirring in one direction, then the other, because this mixes the material more thoroughly. I imagine (but don't know for sure) that stirring in only one direction will tend to create helical structures in the mixed material, where each helix is more or less uniform but two helices might be quite different from one another; and that switching directions tends to break up and mingle these helices. Is this at all correct? Is there a way to quantify the effectiveness of different methods of stirring? If so, how much better is it to stir in alternating directions, and how often should one switch directions?


title: Is there a maximum amount of photons that can exist in a certain amount of space?

text: If you have a set amount of space, lets say 10 cubic centimeters, and you would be able to trap photons in there.   If you would then add more and more photons to that space, could you then go on infinitely or would you eventually run into a maximum amount of photons that can be in that space?


title: How do I find the magnitude and angle of the velocity?

text: I'm confused about the following problem. > A ball is shot from the ground into the air. At a height of $9.1\text{ m}$, > its velocity is $v = (7.6\hat{\imath}+ 6.1\hat{\jmath})\text{m/s}$, with > $\hat{\imath}$ horizontal and $\hat{\jmath}$ upward. > >   1. To what maximum height does the ball rise? >   2. What total horizontal distance does the ball travel? >   3. What is the magnitude and angle (below the horizontal) of the ball’s > velocity just before it hits the ground? > The given answers are: (1) 11m, (2) 22.76m, (3) 16.57m/s, 62.69 We know $S=ut + 1/2 at^2$. So, $$Y = Y_0 + V_{0y}t - 1/2gt^2$$ and, $$X=X_0 + V_{0x}t$$ and $v=u+at$, so $$V_x = V_{0x} - gt$$ and $$V_y = V_{0y} - gt$$ and we're given that $V_x=7.6$ and $V_y=6.1$ I can realize that here $Y = 9.1\text{ m}$ but I cant find the $V_0$ and also the initial angle. How can I do that?


title: Electrophorus: Why can't we use a magnet in substituion of a charged body to induce the polarization?

text: Since a magnetic field can induce a current in a coil, moving electrons from one side to another. Why isn't possible to use the same principle in a electrophorus using one magnet instead of charged body? What's the difference between induction with magnetics and a negatively charged body? P.S: I'm not asking for clarification about the difference about a magnetic field and a electric filed. Maybe it's related, but my point is that if both can induce charges, why it cannot act as a substitution on a electrophorus?


title: Correct expression for D'Alembert operator in $c=1$ units

text: In QFT texts with $c=1$ units (most of them), D'Alembert operator is written as: $$\Box ={\partial^2 \over \partial t^2} - \nabla^2$$ For pedagogical purposes, however, some texts don't set $c=1$, and then $$\Box =\frac{1}{c^2} {\partial^2 \over \partial t^2} - \nabla^2$$ Is that right? I am a bit confused because Wikipedia uses the second form here despite the fact that, in the very next line it is stated > We have assumed units such that the speed of light $c=1$ Am I missing something very basic, or is that page in Wikipedia simply wrong? * * * EDIT: In other words, when $c=1$ is assumed, $c$ NEVER appears in the equations. Is that correct in QFT, just as it happens in GR?


title: Supersymmetric Chern-Simons theories in $d=3$

text: I am reading up on Chern-Simons matter theories in $d=3$. Here is the quote (from http://thesis.library.caltech.edu/7111 page 15) that I am having trouble with: _One could also add a supersymmetric Chern-Simons term for the gauge multiplet. This restricts one to at most N = 3 supersymmetry. However, in the presence of a Chern-Simons term alone (i.e., no Yang-Mills term), the gauge field is nonpropagating, and with a clever choice of matter content and superpotential, one can obtain very large amounts of supersymmetry_ Context: The discussion is about possible supersymmetric constructions using vector multiplets. There are two things I do not understand:   1. Is it true that in the presence of a CS term alone (No Yang-Mills, no matter), the supersymmetry is limited to ${\cal N}=3$. If so, why?   2. I understand that when we add matter in a clever way one can obtain extended supersymmetry (as in the ABJM paper http://arxiv.org/abs/0806.1218). What I don't understand is that what does have to do with the gauge field being non-propagating?? I guess that the answer to my second question lies in the answer to the first. Any help?


title: Importance of Ampere's Law

text: Consider a hollow cylinder carrying a current $I$ and a wire outside the cylinder carrying a current $I'$. Let's say the cylinder is symmetrical with even current distribution etc.. so the $\mathbf{B}$ field at any point (due to current in cylinder) within the cylinder is zero by Amperes Law. However, this doesn't mean the $\mathbf{B}$ field is zero within the cylinder entirely - there is a $\mathbf{B}$ field contribution from the wire. So my question is: What is the usefulness of Amperes Law? Does Ampere's Law only tell me something about the $\mathbf{B}$ field from a particular source? Also say we have a solid cylinder inside a hollow cylinder with radii $a$ and $b$ respectively. They have opposite current directions. Then by Ampere, the $\mathbf{B}$ field at some point $P$ where $a < P < b$ is given as $B = \frac{\mu I}{2\pi r}, I $ the current in the solid cylinder. Is it really? The $\mathbf{B}$ field from the hollow cylinder will be in the opposite direction at $P$ and so acts to cancel the $\mathbf{B}$ field at $P$ from the solid cylinder thus resulting in zero net $\mathbf{B}$ field, no? Yet the $\mathbf{B}$ field at $P$ is in fact nonzero? I understand how the non zero $\mathbf{B}$ field was obtained using Ampere's Law, but the Amperian loop which coincides with $P$ does not simply shield the $\mathbf{B}$ field from the hollow cylinder. So I am struggling to see why the $\mathbf{B}$ field would be nonzero. Many thanks.


title: Do black holes have charges?

text: Do black holes have charges? If so, how would they be measured? Also, does electricity behave the same way? Black holes affect photons, which are carriers of EM radiation, so do black holes have any effect on the electric force?


title: Ferromagnetism with mobile spins

text: How can electron spins in Iron at room temperature have ferromagnetic order even though they are travelling at very high speeds? One could argue that spin and motion are completely uncorrelated and hence you can have superfast electrons that still somehow manage to orient their spins - but then how do you explain domains?


title: Observer effect, do this mean literally someone or just any interaction with other matter?

text: I am a layman and was wondering, the quantum observer effect. The regular notion to laymen seems to be literally "if you look at it", but as I am coming to understand the world I live in better I feel it means just coming in contact with something. Is this an ongoing question? Whether a particle that is not interacting with anything undergoes changes in state? Though now that confuses me too, If it is in two states at once. What denote say spin to the left from spin to the right, what denotes the origins of the calculations we make?


title: Electro Magnetic Waves can cause matter displacement?

text: After watching this video: http://www.youtube.com/watch?v=j1jIjx0XF_U The experience is made with a speaker that generates a sound wave or mechanic wave. Can you use this to establish a link to HAARP which I believe uses ELF EM radiation? Thanks


title: Mathematical equivalence between Liénard-Wiechert potential and 4-potential in Rindler coordinates

text: I'm studying the problem of the radiation of an uniformly accelerated point charge: $$x^{\mu}(\lambda)\to(g^{-1}\sinh g\lambda,0,0,g^{-1}\cosh g\lambda)$$ I found that when a point charge is moving along the $z$ axis with a constant acceleration $g$, the components of the 4-potential can be found using Rindler coordinates: $$z=Z\cosh g\tau, \qquad t=Z\sinh g\tau \qquad \mathrm{I} $$ $$z=Z\sinh g\tau, \qquad t=Z\cosh g\tau \qquad \mathrm{II} $$ $$z=-Z\cosh g\tau, \qquad t=-Z\sinh g\tau \qquad \mathrm{III} $$ $$z=-Z\sinh g\tau, \qquad t=-Z\cosh g\tau \qquad \mathrm{IV} $$ with the metric $$ds^{2}=\epsilon(-g^{2}Z{}^{2}\, d\tau^{2}+dZ^{2})+dx^{2}+dy^{2}$$ where $\epsilon=+1$ in regions I and III, and $\epsilon=-1$ in regions II and IV. And the numbers indicate the space-time region: ![enter image description here](http://i.stack.imgur.com/LBqaG.png) Now, I understand that the potentials obtained with the Rindler coordinates must be equivalent to the Liénard-Wiechert potential, because the change of coordinates (Minkowski->Rindler) is equivalent to change from an inertial frame of reference to an accelerated one. The problem is, the article > Radiation from a uniformly accelerated charge. D G Boulware. _Ann. Phys._ > **124** no.1 (1980), pp. 169–188. Available on CiteSeerX finds that the components of the 4-potential (Rindler coordinates) are: $$A_{\tau}=-\frac{eg}{4\pi}\frac{\epsilon Z^{2}+\rho^{2}+g^{-2}}{\left[\left(\epsilon Z^{2}+\rho^{2}+g^{-2}\right)^{2}-4 \epsilon Z^{2}g^{-2}\right]^{1/2}}$$ where $\rho^{2}:=x^{2}+y^{2}$. $A_x=A_y=0$, and $$A_{Z}=-\frac{e}{4\pi Z}$$ In regions I and II. My **question** is: How can I conclude that this components are actually equivalent to the components of the Liénard-Wiechert potential?


title: Self-adjoint and nonpositive differential operators

text: I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as > $$ \frac{1}{v_0}\frac{\partial^2}{\partial t^2} \begin{pmatrix}p \\\ > r\end{pmatrix} = \begin{pmatrix} > 1+2\epsilon&\sqrt{1+2\delta}\\\\\sqrt{1+2\delta} &1\end{pmatrix} > \begin{pmatrix}G_{\bar x \bar x}+G_{\bar y \bar y}&0\\\0&G_{\bar z \bar z} > \end{pmatrix} \begin{pmatrix} p\\\r \end{pmatrix}\tag{20} $$ and they claim that > To achieve stability, the rotated differential operators $G_{\bar x \bar > x}$, $G_{\bar y \bar y}$, and $G_{\bar z \bar z}$ should be self-adjoint and > nonpositive definite as are the second-order derivative operators > ($\tfrac{\partial^2}{\partial x^2}$, $\tfrac{\partial^2}{\partial y^2}$ and > $\tfrac{\partial^2}{\partial z^2}$). (see eq. 14 and statement under eq. 20). They also claim that self-adjointness and nonpositiveness of the differential operators of the wave equation are necessary to conserve the energy in this system, and that if they were not self-adjoint, numerical instabilities occur. > We have solved the problem by introducing the self-adjointness to the > operator matrices in equation 20 to make sure that energy is conserved > during the wave propagation to avoid amplitude blowup in the modeling. In this case the wave equation consists of two coupled elliptical PDEs. What happens in general, when some operators are not bounded and linear? Unfortunately I don't have the mathematical background to understand this statement.


title: How do we know superposition exists?

text: How do we know superposition exists? Has it been observed, or has it been deduced, and how certain are we? The Copenhagen Interpretation seems to imply that superposition collapses into one state once measurement has occurred, so I don't understand how we can observe it. The reason I ask is I'm totally bewildered by the mainstream interpretations of Quantum Mechanics. I understand its probabilistic; I'm very accepting of the idea of there existing fundamentally stochastic scenarios where the initial state does not entirely determine the next state. What I don't understand is why superposition is so necessary to the theory. The two mainstream explanations are the Copenhagen Interpretation and the Many Worlds Theory, which both seem so incredibly farfetched and counterintuitive. Thanks for any help or insight.


title: What are the limits of applicability of Coulomb's Law?

text: Coulomb's law is formally parallel to Newton's Law of Universal Gravitation, which is known to give way to General Relativity for very large masses. Does Coulomb's Law have any similar limits of applicability? What physics takes over then? Does the law hold for very large and very small charges?


title: Calculating how a polygon bounces off a plane

text: ![enter image description here](http://i.stack.imgur.com/9zeX7.png) I'd like to calculate how polygons bounce off a plane. In this picture, the square doesn't bounce straight up, but instead it bounces somewhat to the right and starts spinning. But I have no idea what angle it bounces and how much energy is converted into rotational kinetic energy.


title: Why do we care about black hole interiors' physics?

text: Whatever happens in there is not falsifiable nor provable to the outside. If for (amusing) example the interior consisted of 10^100 Beatles clones playing "Number Nine" backwards, do we know how to unscramble the Hawking radiation to divine this? The same question applies to this new firewall furor. So of what use is a description of the interior to our physics on the outside? The only possibility of usefulness I can see is if our own universe can be described as an interior up to the cosmic horizon in de Sitter space. But that's only an "if".


title: System with no entanglement but consuming quantum discord

text: I have come across an article which talks about quantum discord (Observing the operational significance of discord consumption. M. Gu _et al. Nature Physics_ **8** , 671–675 (2012) doi:10.1038/nphys2376), but I am struggling with the abstract, which says: > The amount of extra information recovered by coherent interaction is > quantified and directly linked with the discord consumed during encoding. > **No entanglement exists at any point of this experiment**. Thus we > introduce and demonstrate an operational method to use discord as a physical > resource. So how can this happen? Usually quantum computations are thought to consume entanglement to perform calculations, and only pure states are involved. However, when quantum discord is discussed, the mixed state description makes it hard to understand where 'discordance' comes from. Are there explicit (mathematical) examples that demonstrate whether it is possible to have   * (a) zero entanglement, non-zero discord;   * (b) non-zero entanglement, zero discord; and   * (c) non-zero entanglement, non-zero discord? * * * In classical information theory, the (Shannon) entropy is defined by $ H(A) = -\sum p_i \log(p_i) $ for a given probability distribution $p_i$. The mutual information can be defined as $$\mathcal{I}(A:B) = H(A)+H(B)-H(A,B)$$ or $$\mathcal{J}(A:B) = H(A)+H(B|A),$$ where $H(A,B)$ and $H(B|A)$ are the entropy of joint system and the conditional entropy, respectively. These two expressions are equivalent in the classical case, and the second one can be derived from the first one using Bayes' rule. Similar expressions can be defined in the quantum system, with the entropy replaced by the Von Neumann entropy, $$H(\rho)=-Tr(\rho \ln \rho).$$ However, the expressions $\mathcal{I}(A:B)$ and $\mathcal{J}(A:B)$ are not always equivalent in quantum systems and the difference is called the quantum discord: $$ \mathcal{D}(A:B) = \mathcal{I}(A:B) - \mathcal{J}(A:B).$$ Therefore, it can be treated as a measure for the non-classical correlations in a quantum system.


title: Electromagnetic Energy and Momentum

text: I'm trying to show that the electromagnetic energy density and momentum density satisfy the inequality: $ u \geq c |\vec{\wp}| $, where $\wp$ is the momentum density and $u$ is the energy density. I can at least show that $$|\wp|^2 = \epsilon_0 \mu_0 \vec{S}\cdot\vec{S} = \epsilon_0 ^2 (\vec{E}\times\vec{B})\cdot(\vec{E}\times\vec{B})$$ and therefore $$|\wp|^2 = \epsilon_0^2 (E^2+B^2-2 \vec{E}\cdot\vec{B})$$ but I do not know how to proceed. How do I even get the $\mu_0$ back into the momentum density equation, given that $$u= \frac{1}{2}(\epsilon_0 E^2+\frac{1}{\mu_0}B^2)$$


title: Impact of LHC on other science and technologies, in particular on mathematics?

text: The Large Hadron Collider (LHC) "remains one of the largest and most complex experimental facilities ever built" (Wikipedia); it may even be the most complex project in humankind's history (?). Such projects usually have impact beyond their original target and boost science and technology in a non-trivial way. I wonder, then, what kind of impact it has or might have on other science and technology fields, in particular on mathematics (if any), and what specific impact has mathematics had (or might have) on the LHC project (if any)? Cross-post on mathoverflow: Impact of LHC on math.


title: why boundary condition in steady electric current?

text: when we electric field between two conductors in certain direction the current density should pass in its direction why current density direction change at boundary although the direction of electric field is the same for both conductors ![enter image description here](http://i.stack.imgur.com/qOEiH.png)


title: How long will it take to cool?

text: Is it possible to cool down a system in milliseconds? And if so how can I? If I have a red hot metal at 773.15K can I dope it to 673.15K super fast?


title: How to derive the Bethe stopping power formula

text: I need the derivation of Bethe formula for stopping power, but I can't see the corresponding paper to this matter. > Application of Ordinary Space-Time Concepts in Collision Problems and > Relation of Classical Theory to Born's Approximation. E. J. Williams. _Rev. > Mod. Phys._ **17** no. 2-3 (1945) pp. 217-226. Can anyone help me about this paper please?


title: Can thick-film reflection holograms be used to create true mirrors?

text: _Experimental alert:_ Someone may be able to answer this question experimentally simply by going to a shopping mall and finding the right piece of holographic jewelry. My question is whether the type of front-view "thick film" hologram often seen in jewelry can be used to create a true mirror, one in which you can see an image of yourself reflected in the hologram. I'm afraid I messed up the terminology quite badly in my first try at this question! The type of hologram I was trying to ask about is correctly called a _reflection_ (or _thick film_ or _volume_ ) hologram because of its ability to reflect light back in the direction from which it came. It does that using only photographic emulsion, via a wave exclusion effect similar to the one that gives peacock feathers and opals their bright colors. (For the record, I was originally thinking, quite incorrectly, that "reflection holograms" meant the ones that use holes in a smooth metallic surface, such as those seen on almost any credit card. Now I'm unsure what those ones are called.) At least one non-SE site correctly points out that you cannot create a holographic mirror by using a _transmission_ hologram, which is the kind where the light source passes through a film and you view it from the other side. (Again, I was using this term incorrectly in my first version of this question.) That's not a very deep answer, however, since light from the viewer plays no role at all in what is being seen in a transmission hologram. You can't reflect something whose light plays no role in the image being shown! So, my question is this: Can the wave exclusion effect that is used in volume holograms, which lack any true metallic surfaces despite their shiny appearance, be configured in a way that makes it possible to reflect true images of objects in front of the hologram? I suspect it's possible for two reasons: (1) Somewhere I still have a thick- film hologram of the insides of a watch. I clearly recall watching a bright spot move left and right on the image of a curved metal part within the watch as I moved a background light move left and right in front of the hologram. While a moving bright spot is hardly a complete image, it does indicate that a hologram is capable of a visible response to an object not in the original hologram. (2) I'm not aware of anything from diffraction theory that says you _cannot_ use multilayer diffraction patterns to create simple, mirror-like reflecting surfaces. (Nor am I aware of anything that says you _can_ for sure, either.) So, experimenters: Does anyone out there have a hologram in hand that seems to reflect non-trivial light patterns? And theorists: Regardless of how one would create it in the lab, is it mathematically possible to create multilayer diffraction patterns that, like metallic mirrors, would reflect light in a way that depends on the incident angle of the light?


title: Specific Internal Energy Decreases with Increasing Pressure?

text: I was looking at "Properties of Compressed Liquid Water" (Table A-5) in Fundamentals of Engineering Thermodynamics by Moran. Table A-5 shows that the specific internal energy $u$ of compressed liquid water decreases as as the pressure increases at a fixed temperature. For instance, $u=418.24$ kJ/kg at $T = 100$ F and $P = 2.5$ MPa. While $u=412.08$ kJ/kg at $T = 100$ F and $P = 25$ MPa. Also, this same trend in specific internal energy is shown in Table 3. Compressed Water and Superheated Steam. This trend that the specific internal energy decreases with increases pressure goes completely against my intuition and the numerous examples of compressing a gas or two phase mixture in a cylinder with a piston. Any thoughts on how the internal energy is decreasing?


title: Building a LED lamp, which is the brightest? 3 LEDs of 2500 mcd or 1 LED of 6000 mcd?

text: I want to build the brightest lamp, for it to be seen at daylight, should I use 3 2500 mcd LEDs or 1 6000 mcd? And also Why?


title: What makes an equation an 'equation of motion'?

text: Every now and then, I find myself reading papers/text talking about how _this_ equation is a constraint but _that_ equation is an equation of motion which satisfies _this_ constraint. For example, in the Hamiltonian formulation of Maxwell's theory, Gauss' law $\nabla\cdot\mathbf{E}=0$ is a constraint, whereas $\partial_\mu F^{\mu\nu}=0$ is an equation of motion. But why then isn't $\partial_\mu j^\mu=0$, the charge-conservation/continuity equation, called an equation of motion. Instead it is just a 'conservation law'. Maybe first-order differentials aren't allowed to be equations of motion? Then what of the Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi=0$? This is a first-order differential, isn't it? Or perhaps when there is an $i$, all bets are off... So, what counts as an equation of motion, and what doesn't? How can I tell if I am looking at a constraint? or some conservation law?


title: Other Gross-Neveu like theories?

text: By "Gross-Neveu like" I mean non-supersymmetric QFTs whose partition function/beta-function (or any n-point function) is somehow exactly solvable in the large $N_c$ or $N_f$ or 't Hooft limit. (..supersymmetric examples would also be helpful to know of if in case there are no other theories like the above..)


title: Vertical component of moving weight at a 45 degree angle

text: Here's an easier one. I use the leg press machine at the gym so I don't have to worrying about hurting myself while lifting heavier weight. The weight glides on a track that looks to be 45 degrees. What's the equation to figure out how much weight I would be able to squat normally. IE the vertical component of moving 400lbs at a 45 degree angle.


title: Why does a picture of a person seem to be looking in the same direction irrespective of the angle of observation?

text: If you observe a picture of a person hanging on a wall who seems to be looking directly towards you always seems to be looking at you even though you change your angle of observation to the extremes. The same can be observed in a television. If a television is watched by many people from different angles all observe that a person on the screen is looking at them. Why does it happen like that? Update: I First thought that it may be because of some data being lost due to conversion of 3D to 2D. But same is observed in a theater while watching a 3D movie.


title: How to express $ds$?(when we know expression for $ds^{2}$)

text: We know that $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ Can you say how to calculate $ds$?


title: Density of states (treating states in continuum)

text: If we have a particle in a 3D infinite square well box, with length $L$, e.g. an electron in a conduction metal. By solving the Time independent Schrodinger equation, we can get the formula of $$k^2=(n_x^2+n_y^2+n_z^2)\pi^2 / L ^2$$ Now each state occupies a volume of $(\pi/L)^3$ I don't get why we need to use the density of states, considering that at the bottom of the infinite square well, the energy levels are discrete, and in higher energy levels, there are more states than the bottom. Does that mean we shouldn't use density of states for the lower energy levels region? My professor split the the whole system into ($e^j$) levels, where each level ($e^j$) was individual energy level was smeared by ($g^j$) states i.e. there are $g^j$ number of them, where $j$ is the superscript denoting the number of level it is in. E.g. 10 energy levels At $j=1$ there are $g^1=10$ levels where $g^1 = 10$ I still couldn't get why they can treat it using a continuum?


title: How to judge whether a symmetry will be spontaneously broken while only given a Hamiltonian preserving this symmety

text: As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking? Any examples?


title: Kingston corners

text: It is approximately 275km from WVWHS to Kingston corners. If your car gets 30.0 miles per gallon and gas costs 3.85 per gallon, how many cents of gas is burned from going from school to Kingston corners?


title: Gravitational lensing image from merging black holes

text: What kind of image would a transcient toroidial black hole merger produce from it's gravitalional lensing effect? http://arxiv.org/abs/gr-qc/9905039 What does a torus shaped lens actually look like? Are there any other hypothesized transcient shapes for gravitational lenses that could arise from a merger or collapse?


title: numerical investigation of stability

text: I am trying to find the required specifications of an paul RF trap, in which a proton can be confined.(trap dimensions,voltage frequency and amplitude used, etc). I have to solve the equations of motion numerically because the potential doesn't have a closed form for this specific geometry and equations of motion can not be solved analytically (indeed , only a series solution can be obtained for potential using Legendre polynomials and this series is present in the system of equations of motion) . for this purpose, I solve this system of ODEs numerically (Runge-Kutta) and see if the ion leaves the trap. you know, I can simulate the motion for a **limited time span** (typically about some milliseconds at most). So if the ion happen to leave the trap after a longer time span , I can't notice that. **The question is that how and when I can claim that the ion is confined in the trap?(using numerical methods)**


title: How to determine which one would not be the resultant?

text: i had a physics exam yesterday, which was all in all pretty good except for this one question which i just don't get.: ` 9) Two forces 4N and 6N act at a point. All of the following could be the magnitude of their resultant except. A) 1N B) 6N C)10N D)4N E) 8N ` Could you please help me to understand the logic behind this question?


title: Inverse of the D’Alembert wave operator

text: Let $f(x)$ a solution of the wave equation $\square f(x) = 0$. I don't understand why the following operator $$ \mathcal{D} \equiv \frac{1}{2}(\nabla^2)^{-1} (x_0\partial_0 - C) $$ for arbitrary constant C, works as an inverse of the D'Alembert wave operator, namely: $$ \square \mathcal{D} \ast f(x) = f(x) $$


title: How to calculate the energy required to generate a vortex?

text: I am looking for the formula to calculate how much energy it requires to create a vortex. Can anyone assist please?


title: Problems book recommendation on supersymmetry, supergravity and superstring theory

text: I'm learning supersymmetry, supergravity and superstring. I want some problems books to have some idea in this area. Is there this kind of books? Or are there some papers that have many solved model?


title: Pulley problem with two masses

text: Assume a pulley and a massless rope. On one end there is a 10.0kg mass. On the other there is a 34.5kg mass. What is the acceleration of the 34.5kg mass? I created a free-body diagram with $m_1g_1$ pointing up and $m_2g_2$ pointing down, so the total tension is $m_1g_1 - m_2g_2$. This results in 240 N downwards, so I divide by 34.5kg to get $6.96 m/s^2$. However that is incorrect. What is my mistake?


title: Magnetic field of a solenoid at the poles?

text: It's known that at the center of an electromagnet/solenoid the magnetic field $B$ is strongest there. At the pole/edge of the solenoid/ or electromagnet is the magnetic field $B$ strong? Let's assume $B$ = $1$ $Tesla$ at the center, how different is $B$ at the edges? ![](http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/barsol.gif) In general, is it simple to create a $1$ $Tesla$ magnetic field or higher? MRI's use field's above $4$ $Tesla$ yet it seems very complex and requires massive power & cooling.


title: The Ozma Problem

text: The "Ozma problem" was coined by Martin Gardner in his book "The Ambidextrous Universe", based on Project Ozma. Gardner claims that the problem of explaining the humans left-right convention would arise if we enter into communication (by radio waves, no images allowed) with life on another planet. We can ask the aliens to perform any experiment they want. It is claimed that the classical experiments with magnets, electrical currents, light polarization, gyroscopes etc. can't solve the problem. The only simple experiment that solves the problem is the beta decay in which the parity is not conserved and this can be used to distinguish humans left-right spatial relations. But all this are quite old, the parity violation in weak interactions was experimentally proved in 50s by Chien-Shiung Wu. I was wondering if something has changed since then and if there are some other experiments (maybe some thought experiments) and theories that can solve the Ozma Problem.


title: Why is it that when driving in a car, and a lightning bolt strikes, my AM radio gets cut off for a while, but FM stays on?

text: I noticed this one day, a lightning/thunder occurred and my Fabulosa Spanish music died for a second. But not FM?


title: Mirror "roughness"

text: In optics one of "diffraction-limited" criteria is wavefront tolerance: a textbook example is is optical system with 1/4 wavelength limits producing image of a point source with 68% of the energy contained in the airy disk. Some optics books, however introduce a "roughness" idea, implying that certain mirror surface pattern would imply light scattering, and supporting this idea with naive ray-trace diagram. Does it make any sense, I would think one can made quite strong assertion, namely that _any mirror surface shape within 1/8 wavelength tolerance of paraboloid would produce an of a point source with 68% of the energy contained in the airy disk area_. Any proofs/refutations?


title: Double Pendulum

text: The equations of motions for the double pendulum is given by $$\dot{\theta_1} = \frac{6}{ml^2}\frac{2p_{\theta1} - 3\cos(\theta_1 - \theta_2)p_{\theta2}}{16 - 9\cos^2(\theta_1 - \theta_2)}$$ and similarly for the other pendulum. In respect to what does the change in angle for the first pendulum refer to? Is it with respect to time? So that $\dot{\theta_1} = \frac{d\theta}{dt}$?


title: Problem in average height

text: Let, diameter of cylinder hole of height 30m and diameter 2m filled with water. In a book, when author tried to find out the power for lifting all of the water then he used average height. why is that. I mean he wrote average height $ h = \frac{0+30}{2}$. why is that?


title: Faster than the speed of light and future travel

text: I had read somewhere that if a person attains twice the speed of light, he can actually reach the future.   1. But since the world belongs to one time and space, how will it change for one particular person?    2. And if a person actually attains that speed wouldnt his body burn out because of such high velocity?


title: How to derive the expression for the electric field in terms of the potential?

text: How can I derive that $$\vec{E}=-\vec{\nabla}\phi-\frac{\partial \vec{A}}{\partial t}$$ where $\phi$ is the scalar potential and $\vec{A}$ the vector potential?


title: Laws of addition of Vectors

text: How **Triangle Law and Parallelogram law of addition** of Vectors are different? _Ain't they._ Please don't tell me the things written in book......give me the appropriate reason.And how do i distinguish between the two while adding vectors,what am i trying to say is i get really confused where to use parallelogram law and where triangle law P.s:my basics are really weak though!May be the question might not be right


title: Historical background of wave function collapse

text: I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse paradigm.) Are there any instances where cases once thought to be examples of collapse have since been explained as the normal time-evolution of the wave function? EDIT: I'm going to have to make an objection to Ron Maimon's very excellent answer about particle tracks as evidence of collapse. I've been waiting for someone to suggest what I personally have always considered the prototype of the wave function collapse, namely the appearance of flecks of silver on a photographic plate when exposed to the light of a distant star. This has the essential elements of collapse in a way that ordinary photographic exposures do not. The mere appearance of dots on a photographic plate does not signal the collapse of anything: it is readily explainable as a consequence of the rate of silver-bromide reduction being proportional to light intensity. It is only when the intensity becomes so very low that the time taken to accumulate enough energy for a single conversion becomes unreasonable that we must consider the explanation of wave function collapse. The tracks in the cloud chamber do not demonstrate this phenomenon since the energy needed for the creation of the tracks is already available in the supersaturated gas. It is not necessary for the incoming particle to supply energy for the creation of the track, so there is no need to collapse its wave function. The straightness of the tracks is explained by Mott as an ordinary consequence of time-evolution of the wave function. There is no experimental proof that a single "particle" cannot be responsible for multiple tracks in the cloud chamber, because the tracks are not tagged according to which particle created them.


title: Why does an object when filled with water sink, but without water inside float (in a body of water)?

text: Why does an object sink when filled with water, even if the same object would float without water inside? For example, put an empty glass cup into water, and it floats. But if you put a plastic container filled with water in water, it'll sink. Why is that?


title: How to calculate the projected area at different angles/vectors?

text: Please help me with the following. I want to know if there is an equation/set of equations to find out the projected area of a (3-D) cube when it is oriented at different angles of attack to the fluid flow, rendering the velocity vector of the impinging fluid on the cube surface to change with each angle of attack. In particular, what is the projected area when the cube is oriented at such an angle that the flow velocity vector passes through one of the vertex and the geometric centre of the cube? Thanks.