Slow light and gravitational lensing

It has been proposed that the Sun's gravitational lens be used to observe distant objects, but according to relativity a spacecraft would need to be 550 AU away in order to take advantage of the effect. Would it be possible to instead slow down the light while it is being deflected using, for example, a Bose-Einstein condensate such as a large quantity of very cold sodium placed into orbit around an asteroid? An answer to this question mentioned that it is useless to do the same thing using an ordinary material with a high refractive index because of dispersion, but as I understand it the mechanism for producing slow light changes the group velocity, which is the speed at which energy is carried, instead of the phase velocity.


Hydrogen ionization energy

Proton and electron have a distance equal to the Bohr radius apart in the hydrogen atom. Knowing this, what's the ionization energy of the atom? So we know $U = -kq^2/a_B$, but this is potential energy, not ionization. How should I proceed?


What cosmic event would cause Sun rising from the west?

I'm wondering what kind of astronomical or geological events would cause the reversal of Earth's rotation. For instance, Is a meteorite passing very close to Earth able to reverse its rotation? Can the presence of Earth inside extreme magnetic field cause such event?


Phonon scattering process in raman spectroscopy

I am trying to understand diagrams that involve showing phonon scattering processes that contribute to raman spectroscopy peaks. For example, I drew the one at the bottom of this post. This is supposed to the scattering process responsible for the D peak in graphene. I'm trying to understand exactly what the arrows represent. At initial glance it looks almost as if the phonon scatters and increases energy of magnitude $E_L$ and transitions to another branch, then scatters again, losing energy of magnitude $D$, which is somehow indicative of the D peak? ![enter image description here](http://i.stack.imgur.com/dXi9j.png) Thanks!


Getting nonphysical results when solving for the index of refraction of a slab?

I'm trying to computationally find the refractive indices (real and imaginary) for a thin slab suspended in air (so the only indices to deal with are air and my material's). I've experimentally taken transmission and reflection measurements of intensity in a certain wavelength range. From a textbook chapter I read, $$r_{ij}[n_i, n_j]= (n_i - n_j)/(n_i + n_j)$$ $$t_{ij}[n_i, n_j]= (2 n_i)/(n_i + n_j)$$ $$r = (r_{ij}[n_1, n_2] + r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})$$ $$t = \sqrt{n3/n1} (t_{ij}[n_1, n_2] t_{ij}[n2, n3] e^{ 2i \pi n_2 d/\lambda})/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})$$ and $$T = |t|^2$$, $$R=|r|^2$$ for transmission and reflection of a plane wave through a 'slab' (though I may be missing a factor of 2 in a couple places. I've used Mathematica style brackets for the functions). From here, $T$ and $R$ are both functions of $n_1$ (the air, ~1) and $n_2$ my material ($n_3$ is air again, here). So mathematically I should be able to figure out $n_2$ for each $T$ and $R$ pair, at each wavelength. So I've done this. I have Mathematica (MM) find the $n_2$ that minimizes $$\sqrt{(T-T_{exp})^2+(R-R_{exp})^2}$$ at each wavelength. Then, to see how close it is, I plug $n_2$ back into the original equation and plot it against my experimental results to see how close they match up. The problem is, they're matching up incredibly well -- but the results I'm getting for $n_2$ aren't realistic (more specifically, the real part of $n_2$ is negative, and this is not a negative refractive index material...). Here is a graphical example of what I mean (the horizontal axis is wavelength, in units of microns. The vertical axis is unitless for all): ![expT](http://i.stack.imgur.com/ziXv4.png) ![ns](http://i.stack.imgur.com/VhNxv.png) And here I've plotted the values of $T$ and $R$ from my computed $n_2$'s (green and black) over the first graph. As you can see, they're so similar you can't tell, except at the leftmost tails. ![exptheoryT](http://i.stack.imgur.com/xNzzQ.png) What could be going on? One possibility is that though my solution for $n_2$ gives very close values for $T$ and $R$, there are very different values of $n_2$ that give even closer values. Someone I talked to told me that the plane wave model doesn't always apply at some scales -- which blew my mind because I have always seen it used. He said it was the far field solution to dipole radiation but that it might not apply at my length scale. Could anyone verify or refute this?


Why does a perpendicular force not change speed?

I have been taught that a perpendicular force acting on an object will only change the direction of its velocity, not its magnitude. The explanation that was provided to me is that because there no force in the direction of its velocity, there must be no work done and thus change in the object's kinetic energy. However, I have also studied the vectorial form of the equations of motion $$\vec v_f = \vec v_i + \vec a\cdot t$$ So I assume that the velocity after a constant force has acted on it for a period of time is simply the vector sum of $\vec v_i $ and $\vec a \cdot t$, which I just assume is the acceleration vector (determined by the force) scaled up by a factor of the time the force acts.Diagrammatically, I took the sum as ![Vector Sum of initial and change in velocity](http://i.stack.imgur.com/Mo9VJ.png) It is obviously clear from the diagram that the new speed (which is the length of the $\vec v_f$ vector) is bigger than the previous speed. This can be confirmed by Pythagoras's theorem, the magnitude of the $\vec v_f$ vector is $$ |\vec v_f|=\sqrt{|\vec v_i|^2 + |\vec a|^2 \cdot t^2} $$ which is very clearly not equal to $|\vec v_i| $ So, I conclude that the speed of the object has changed even though the force acting on it was perpendicular to its velocity. So my question is, what have I missed? Is my assumption that the final velocity is the vector sum of the initial velocity and the product of acceleration and time incorrect? Or have I made a mistake somewhere else? Thank you in advance for your help.


Muon production in particle accelerator

PAMELA is a particle accelerator which have two concentric rings, protons are accelerated in the inside ring. At ISIS muons are produced when a 800 MeV proton beam collides with a graphite target(carbon 12) can the same be done with this device? The accelerated protons from inside may be sent to the outside ring which will have slow carbon atoms and the collision will produce pions which will decay into muons. Do pions have any magnetic properties which can help to contain them after production?.


The Lorenz gauge in electrodynamics

What is the fundamental reason to fix the Lorenz gauge to $0$?


From this article on hindustantimes.com - Curiosity sends data about Martian surface: > The resultant flash of glowing plasma is viewed by the system’s 4.3-inch > aperture telescope, which sends the light down an optical fiber to a > spectrometer located in the body of the rover. There, the colours of light > from the flash are recorded and then sent to Earth, enabling scientists to > determine the elemental composition of the vaporized material. How do NASA's Curiosity determine the elemental composition of Mars using spectrometer? Do just examining the colors of light enabled scientist to determine the elemental composition of any object?


Having Trouble With The Principle Of Conservation Of Momentum For a Multiparticle System

I'am reading John Taylor's Classical Mechanics chapter 1 page 20 where he proves the principle of conservation of momentum which states "If the net external force $F^{ext}$ on an $N$-particle system is zero,the system's total momentum $P$ is constant. I'am not sure if i'am having trouble understanding the usage of summation notation or the entire derivation (which includes the concept) itself.Anyways here it is.A five particle system labelled by $\alpha$ or $\beta = 1,2,...,5$. The particle $\alpha$ is subject to four internal forces,shown by small arrows and denoted $F_{\alpha \beta}$ and the net external force shown by the large arrow and denoted by $F^{ext}_{\alpha}$. ![enter image description here](http://i.stack.imgur.com/5bjnf.jpg) Thus the net force on particle $\alpha$ is $$F_{\alpha} = \sum_{\beta \neq \alpha}F_{\alpha \beta}+F^{ext}_{\alpha}$$ According to Newton's second law,this is the same as the rate of change of $P_{\alpha}$:$$\dot{P}_{\alpha} = \sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$$ Then he considers the total momentum of the $N$-particle system,$$P=\sum_{\alpha}P_{\alpha}$$ Differentiating wrt time we get $$\dot{P}=\sum_{\alpha}\dot{P}_{\alpha}$$ or,substituting for $\dot{P}_{\alpha}$,$$\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$$ .The above expression is what is bothering me .I need someone to give me an intuition behind the double summation sign. How come it's $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$ and not just $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$. He eventually concludes that $$\dot{P}=\sum_{\alpha}F^{ext}_{\alpha}\equiv F^{ext}$$.


Can all the systems have a Hamiltonian description?

I have heard of mechanical systems that might not have a Hamiltonian dynamics, but I cannot figure out an example that supports it. Please help.


Why is lightning considered a plasma?

I was wondering why a lightning bolt is coined as "plasma", or a "spark" from an electrical wire/device is as well, yet flares, molten lava, and burning buildings are not(flares are pyrotechnic, exothermic, and very much like flames). I hear arguments saying that plasma is ionized gas, but how can a flare not be ionized at such high temperature(and why isn't ionized, very hot flames considered plasma)? This guy, the top answerer, says that fire is not plasma because it's not ionized: Is fire plasma? Also, some lava exceeds 3,000 Fahreheit. How is that not plasma, and what draws the fine line between plasma and fire itself(not as a state of matter, but as a relation with gas)? Fire is gas, but some say it's plasma if you can see it (check the link the poster added below in the first comment), so when does it become a full-on plasma if it's not normally one?


Effects of microwaves on optical properties of human eye

I have read a long series of paragraphs on wiki. Previously I thought that microwaves are not harmful to living beings but Wiki does not claim this explicitly. while doing an experiment in my college laboratory our teacher cautioned us not to see the face of Horn antenna. He said it will damage eyes so I am asking here to be sure about that. > What are the effects of microwaves emitted by Horn antenna upon the optical > properties of human eye(a natural optical device)?


Derivative of the product of operators

I'm asked to show that $\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$ With $\lambda$ a continuous parameter Should I use the definition $\frac{d\hat{A}}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$ applied to $\hat{A}\hat{B}$ like $\frac{d(\hat{A}\hat{B})}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$ and do some algebra to get the RHS of the first equation, or I'm missing something? Thanks for your time Another interesting derivative to pay attention to is: $\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )$?


Would it matter if the Earth rotated clockwise?

In the Futurama episode _"That Darn Katz!"_ they save the world by rotating the Earth backwards saying it shouldn't matter (which direction Earth rotates). If Earth rotated clockwise and remained in it's current counter- clockwise orbit around the Sun, would it affect anything beyond the direction the constellations move across the sky?


Can Gravity be described in terms of velocities and rotations of local Minkowski space?

In the paper _The River Model of Black Holes_ :   Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle   http://arxiv.org/abs/gr-qc/0411060   The authors give a way of describing the action of a rotating (and or charged) black hole through a collection of local Minkowski frames, that is, as a sort of collection of preferred reference frames, or more accurately, by the tidal effects arising from the movement from one frame to another. Each frame is defined by a "river field" $\omega_{km}$ (See around equation 74). This field is composed as follows:   $\omega_{0m} = \beta_m$ is the "velocity" of the river, while   $\mu^i = 0.5 \epsilon^{ikm}\omega_{km}$ gives the "twist" of the river. Then the motion of objects due to the black hole can be calculated from the tidal change $\delta\omega^k_m$ which is a local infinitesimal Lorentz transformation. My question is this: Can this description of a black hole be used to describe general relativity? Note that there is an obvious limitation: since this is based on a flat background metric you can't get worm holes and the like, but I mean, subject to the requirement of trivial topology, can every GR situation be described by a "river field"?


Newton's second law?

$F=ma$. A car strikes a wall at 60 mph. Its acceleration is zero at the time. The force of the car against the wall or vice versa is? To look at the car the force is not zero. Please explain.


Local minima in Ising model in a Monte Carlo simulation

Is there any way to check whether in a Monte Carlo simulation using Ising model is stuck in any (false) local minima of energy or not, particularly in 3D system ?


Where does the energy for repulsion in charges come from?

If I bring together 2 objects with the same charge, they repel. But even after the repulsion has taken place, there is no loss of charge in the objects. Where does the energy come from then?


Work Done in an Isobaric Process

I am given the information that an air parcel undergoes isobaric heating from 0° C to 20° C, and that's all I'm given. I have to determine the work done by the parcel on its surroundings. I know that $dW = pdV$, and that $pV = RT$ ($V$ is the specific volume), but I don't know how to go about solving this in terms of the change in temperature and not the change in volume.


Proof of equality of the integral and differential form of Maxwell's equation

Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in some occasions..)


Cause of sea ice freezing during an upwell event

One of the Episodes of BBC's documentary Frozen Planet describes an upwell phenomenon in the Antarctic Ocean. I am trying to understand the cause of seawater freezing when this upwell occurs. Liquid sea water can remain liquid below salt water's one-atmosphere freezing temperature when that salt water is maintained under pressure at deep ocean depths. The high pressure conditions deep in the ocean apparently prevent the salt water from freezing. According to the documentary, shifting ocean currents occasionally bring a deep mass of liquid antarctic sea water from near the bottom of the ocean all the way up to the surface very quickly. And much of that mass of liquid sea water quickly freezes due to the change in pressure. Pressure is reduces when the water moves from the bottom of the Antarctic Ocean and arrives near the surface. If temperature remains constant but the state changes from liquid to solid, I assume this state change is caused by the pressure allowing the sea water to expand and form a crystalline lattice - ice. Someone watching the BBC documentary with me believes the ice formed in such an upwell event is caused by the molecular movement of the water molecules slowing down with the reduction in pressure, and she argues the slowing of movement brought on by reduction in pressure causes the ice to form. She asserts the reduced molecular movement is the cause, not the physical expansion of the overall volume of the water as it rises. My understanding is that the extreme high-pressure water simply doesn't have enough physical space to expand into a volume which will allow the molecules to form a lattice, and this has little or nothing to do with the molecular movement. What is the correct way to understand this phenomenon?


Why is $\vert \phi \vert ^2$ infinite in QFT?

I've read here¹ that for a scalar field $\phi$, the square $\vert \phi \vert ^2$ is infinite (which gives an infinite contribution to mass), more precisely: > the square of the field – a quantity which diverges in QFT as necessary > consequence of the commutation rules of the theory and unitarity. It seems that this is related to the fact that $\phi$ is a distribution and the square would be the correlation function. > if we compute in quantum field theory a correlation function like $\langle > A(x)A(y)\rangle$ and let $x\to y$, we find a divergent quantity. Is there a simple way to see why is this true, with little knowloedge of QFT?   1. Inertial Mass and Vacuum Fluctuations in Quantum Field Theory, Giovanni Modanese, PACS: 03.20.+i; 03.50.-k; 03.65.-w; 03.70.+k 95.30 Sf


Entropy change relation to the number of lost bits

Can we use entropy change value to define the (perhaps fractional) number of bits of information that are lost by and, or, xor gates?


Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as $P^{\mu} |p,\sigma\rangle = p^{\mu} |p,\sigma\rangle $, where $\sigma$ is any other label. Then he shows that, for a Lorentz Transformation : $$P^{\mu}U(\Lambda)|p,\sigma\rangle = \Lambda^{\mu}_{\rho} p^{\rho}U(\Lambda)|p,\sigma\rangle $$ Therefore: $$U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,p)|\Lambda p,\sigma'\rangle.$$ Then he wants to find $C$ in irreducible representations of the inhomogeneous Lorentz group. For any $m$ he chooses a $k$ such that $k^{\mu}k_{\mu} = - m^2$. Then defines express $p$'s with mass m, according to $p^{\mu} = L^{\mu}_{v}(p)k^v$. Then he defines $$|p,\sigma\rangle = N(p)U(L(p))|k,\sigma\rangle$$ (where $N(p)$ are normalization constants). I didn't understand this last statement. Is $\sigma$ an eigenvalue of the corresponding operator, or just a label? I mean, if $J |k,\sigma \rangle = \sigma |k,\sigma\rangle $ then is it true, $J |p,\sigma\rangle = \sigma |p,\sigma\rangle$. If so how can we say that if $$U(\Lambda)|k,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,k)|\Lambda k=p,\sigma'\rangle$$ Thanks for any help. First pages of these notes on General Relativity from Lorentz Invariance are very similar to Weinberg's book.


How to define tensor contraction without referring to summation?

The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation convention. My question is: Can we define tensor contraction without referring to basis, components, and summation, but just the definition of tensor as a multilinear function? If components are inescapable, why bother define tensors as multilinear functions?


Is there a theory which treats particles as classical point singularities?

Is there a published theory that looks at all matter as occupying no space and only being felt because of its gravitational pull? We've been taught in school that matter has mass and occupies space. I was just wondering if anyone tried to look at matter as having no physical reality...


Fourier transform of two pulses of light

I have laser beam path that fires two pulses of light in a gaussian distribution, so the intensity graph over time is two identical gaussians separated by a distance $t_0$. In other words, a gaussian convolved with two $\delta$ functions equidistant from the origin. When I take the Fourier transform of this, I get a graph with beats, but I'm having trouble figuring out what this graph means physically. When we feed this beam through a spectrometer, we are trying to find out if we should see fringes, and if so, what their frequency is. Any help would be appreciated.


Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ (using the variational method). After having included the Darwin correction $$ H_D = D\delta(r) $$ with $D = \frac{\alpha^2\pi Z}{2}$, into the Hamiltonian $$ H_0 = -\frac12\nabla^2-\frac{Z}{r} $$ I calculated $$\langle\psi(\beta)|-\frac{\nabla^2}{2}|\psi(\beta)\rangle = \langle \psi(\beta)|T|\psi(\beta)\rangle = \frac{\beta^2}{2}$$ then I thought to use the virial theorem to calculate the part $\langle V\rangle $: $$ \langle \psi(\beta)|-\frac{Z}{r}|\psi(\beta)\rangle = \langle \psi(\beta)|V|\psi(\beta)\rangle = -2\langle \psi(\beta)|T|\psi(\beta)\rangle = -\beta^2 $$ with the Darwin part left to be easily calculated. Looking at the solutions that my professor wrote, I've just found a different result: > $$\langle V\rangle = \langle \psi(\beta)|-\frac{Z}{r}|\psi(\beta)\rangle = > \frac{Z}{\beta}\langle \psi(\beta)|-\frac{\beta}{r}|\psi(\beta)\rangle = > -2\frac{Z}{\beta}\frac{\beta^2}{2} = -\beta Z$$ Why's that? Is this somehow related to the fact that I'm going to use the variational method later? Please, help me understand.


A common definition of a scalar

Some dictionaries define a scalar as follows: > A quantity, such as mass, length, or speed, that is completely specified by > its magnitude and has no direction. -- The Free Dictionary However, it is my impression that in many contexts scalars can be signed, in which case their magnitude (their absolute value) does not specify its value. This definition is even used on a test question here. Is it true that this definition is inaccurate?


Best way to solve P-V-T Question

Typically in an exam or problem set we have a problem that's like this: Determine the specified property at the indicated state of water. Locate the state on a sketch of the T-v and P-v diagram. p= 3 bar, T= 240°C. Find v, in $m^3/kg$. I believe I can figure out $v,u,h$ when it ask to find this information, however I'm having trouble figuring out what's the best way to look at the steam table and figure out how to draw the correct information. For reference I've provided what the answer to this problem is. Here is the solution: ![Problem Set Answer](http://i.stack.imgur.com/XDuc3.png) Thanks,


EmDrive Cavity modes

The EMDrive described in patent application GB 2493361 is a simple section of a sphere surrounded by a superconducting cooling system. I have attempted to solve the oscillation modes possible and find spherical Bessel functions for radial directions and Legendre polynomials for "theta". When I apply boundary conditions, $E_r$ must be zero at the side walls so $P_n^0$(cos($\theta_w$)) = 0 can only happen for specific angles. If the wall is not at the right angle, $E_r$ must be zero every where. Same is true for $E_\phi$, but that has zeros for $P_n^1$(cos($\theta_w$)) so it would be a different mode. The claim in the patent is that putting RF into the cavity creates a force - but it does not say in what direction! The EmDrive web site talks about group velocity, but I don't see how that makes sense for a standing wave in a cavity. What am I missing? I can find $B$ from $\nabla\times E$), but is there an easy way to find energy density directly from the $E$ field solutions? Once I know energy density, finding forces should be straight forward. What I expect is the net total force to be zero, but it would be nice to prove it.


Why does $\nabla \to ik$ when you Fourier transform?

I am reading a text that describes the scattering of light by a particle with dielectric constant $\epsilon$ After a bit of maths starting from Maxwell's equations they obtain: $$\nabla (\nabla \cdot E(r)) - \nabla^2E(r)=\mu_0\omega^2\epsilon(r)\cdot E(r)$$ then say "Fourier transforming with respect to $r$ gives (replace $\nabla$ by $ik$):" $$[k^2 \hat{I}-kk]\cdot E(k) = \mu_0 \omega^2 \int \epsilon(r) \cdot E(r) \exp(-ik \cdot r) \,\mathrm{d}r$$ I don't understand why all $\nabla$ have turned into $ik$? Is there any way to visualize why this can be done?


Why can't angular momentum be used in flying vehicles?

![Animation of relationship between torque and momentum](http://i.stack.imgur.com/yIa9K.gif) If angular momentum ( **L** ) works like this animation from Wikipedia leads me to believe, why can't we put a large flywheel or several small ones on a chassis, all rotating counter-clockwise around their vertical axes and move the system upwards? It seems that there's nothing countering **L**. Am I missing something? I think the concept of such a vehicle would break Newton's first law, but right now to me that's like saying 'you can't because Newton said so'. Can anyone provide a more detailed explanation that someone with high-school- level knowledge of physics can understand?


Bloch theorem, Energy, Free electron

I'm trying to learn on my own a bit of solid physics to tackle semiconductors afterwards. I'm struggling with the Energy versus $k$ diagrams for a free electron which shows that for a single value of $k$ we can have many energies value although that there's only one band (in this example). Bloch's theorem tells us that we can label the energies the system can take with a vector $k$ and an integer $n$ (band index). And the theorem tells us also that for any vector $G$ in the reciprocal lattice we have: $E(n,k)=E(n,k+G)$ Let's apply this to the free electron. This tells us that (there's only one band index) $$\frac{\hbar^2}{2m}\lvert k\rvert^2 = \frac{\hbar^2}{2m}\lvert k +G\rvert^2 $$ for all $G$ in the reciprocal lattice$ \lvert k\rvert=\lvert k+G\rvert$ which is absurd. We find another contradiction by making the following reasoning: Since the potential is constant, we can consider that the potential is periodic with respect to a lattice with arbitrary vectors $a_1,a_2,a_3$. (In 1D, all that I'm saying is that a constant function can be seen as 1-periodic or 100-periodic or 0.0001 periodic..). So by using this fact we can prove that there is only one possible value for $E$ which is also absurd. Can someone please explain in detail what am I doing wrong?


Does Free Will Theorem imply that quantum mechanics plays crucial role in our brain’s functioning (consciousness)?

Is opposite statement of Free Will Theorem right: If elementary particles have a certain amount of free will, then so must we? Because to me elementary particles does have a bit of free will – quantum mechanics guarantees that nobody can predict what one is going to do, say in double slit experiment. So Penrose was right and origins of our consciousness lie in the laws of quantum mechanics? Is the only way our free will can come from is that of quantum mechanics?


Can any gas be liquidified by lowering the temperature? What happens with gases at absolute zero? Are there gases that remain gases at absolute zero? Do their molecules move at these temperatures? Does the gas acquire a crystallic structure at these temperatures while still remaining gaseous?


Why is powdered sugar less dense than granulated sugar?

The density of powdered sugar is about 20% lower than that of granulated sugar. There is small composition difference between the two - about 3% cornstarch and 0.5% water in powdered sugar, but that's it. It seems this is an issue of packing then. Why does powdered sugar pack worse than granulated sugar?


Are regular light bulbs better for the eyes than CFLs or "tube lights"?

I've heard that regular light bulbs with a filament are better for the eyes. Is the spectrum of one worse than the other? If so, are there any regulations for their use in industrial settings for worker safety?


Difference between Higgs and anti-Higgs Fields

I'm assuming the LHC can create a Higgs and an anti-Higgs boson. If so, would their fields be identical with respect to mass effects? How would LHC detectors distinguish between the two bosons?


Angular momentum for 3D harmonic oscillator in two different bases

I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\rangle$$ or, if solved in the spherical coordinate system: $$|N,l,m\rangle$$ The relationship between capital $N$ and the little $n_i$'s is straightforward: $N=n_1+n_2+n_3$, but this can't be said for the other quantum numbers. I want to find a way of relating the two representations, but I'm not sure how to go about doing this (my linear algebra background is pretty weak). Let's say I fix the energy to be $\frac{5\hbar \omega}{2}$, which is equivalent to saying $N=1$. There are three states corresponding to this situation in the first representation: $| 1,0,0\rangle, |0,1,0\rangle,| 0,0,1\rangle$. But what are the corresponding states in the second representation? If $N$ is fixed to be $1$, what are the allowed values of $l$ and $m$? I remember that $l=0,...,N-1$, and $-l\leq m\leq l$, but this makes no sense, since it would mean both $m$ and $l$ would have to be $0$...


What is the difference between center of mass and center of gravity? These terms seem to be used interchangeably. Is there a difference between them for non-moving object on Earth, or moving objects for that matter?


A problem concerning the force between currents or moving electrons

Concerning two identical wires carrying the same current (same direction, speed and magnitude), they will be attracted because of the Ampere force. But when I was in the frame moving with the same speed as the electron in the current, why will there be the same force between each wire (supposing that the electrons are moving at a speed far less than $c/10$). Also, when two bundles of electrons are moving in the vacuum at the same speed, will they be attracted because of the Ampere force? Will they be dis- attracted because of the Coulomb force?


Intense of liner polarized wave: with a polarizer vs. without a polarizer differences

We had this experiment in which we measured the intense of **linear polarized** wave - with and without a polarizer. I noticed that **without the polarizer** the intense was slightly lower than **with the polarizer**. How is that possible? ![enter image description here](http://i.stack.imgur.com/mTQR7.png) The wave length was about 2.8 [cm]. With polarizer I got 0.317 [Volt] . Without polarizer I got 0.2835 [Volt].


Bulk-boundary cutoffs in AdS/CFT

I'm studying the holographic entanglement entropy (HEE) in this paper (Ryu- Takayanagi, 2006). In section 6.3 they compute the HEE for a segment in a 2D CFT. To do so, they obtain the corresponding geodesic in the bulk (in the Poincaré patch) and compute its length. I understand all that process, but I'm having some trouble when they introduce the cutoff. The metric diverges when $z\to0$ so we introduce a cutoff $\epsilon>0$, I understand that. But then they say > Since $e^\rho\sim x^i/z$ near the boundary, we find $z\sim a$ Here, $\rho$ is the hiperbolic radial coordinate un the global coordinates for AdS, $$ ds^2 = R^2(-\cosh^2\rho\ d\tau^2 + d\rho^2 + \sinh^2\rho\ d\Omega^2) $$ $x^i$ and $z$ are coordinates in the Poincaré patch, $$ ds^2 = \frac{R^2}{z^2}(dz^2-dt^2+\sum_i(dx^i)^2) $$ And $a$ is the inverse of the UV cutoff of the CFT in the boundary, that is, the spacing between sites. I have two problems: 1) First, I don't see why near the boundary $e^\rho\sim x^i/z$. I made up the relations between both coordinate systems and I find more complicated relations than that (even setting $z\sim0$). 2) Even assuming the previous point, I don't understand why we obtain that relation between the CFT and the $z$ cutoff.


Pseudo-randomness of observables

There is a somewhat recent paper by Colbeck and Renner that, given the assumptions — 1) QM accurately predicts the correct statistical results in experiment (which so far has always been found to be true) and 2) the measurement can freely be chosen — that no extension to quantum mechanics can have improved predictive power. I've taken basic QM and I just started taking QFT so understanding the paper is somewhat beyond my skills; so perhaps someone can help me with this situation: Suppose you have a pseudo-random number generator (e.g. it could be any function really as long as its outputs appear random and the appropriate correlations [entanglement] between different positions exist) that is a function of position and time: $F(\vec{x}, t)$. If you knew the algorithm for this generator, you could plug in the spatial coordinates and time of your experimental measurement*, and get information that allows you to predict the exact* eigenvalue of the linear operator that corresponds to your measurement. This hypothetical PRNG would be for the whole universe, so I suppose it would count as a non-local hidden variable. My question is: why does Colbeck and Renner's paper rule out this possibility? (*Okay, you can't plug in the exact coordinate and time of your measurement because of HUP. Perhaps you just get _improved_ predictive power over QM by giving close enough coordinates and time).


Are W & Z bosons virtual or not?

W and Z bosons are observed/discovered. But as force carrying bosons they should be virtual particles, unobservable? And also they require to have mass, but if they are virtual they may be off-shell, so are they virtual or not.


What is the optical illusion called where our sun seems to disappear from view and then rise on the third day? And where can it be viewed from?

What is the optical illusion called where our sun seems to disappear for 3 days and then rise again into view on the third day? And where can it be viewed from?


Dimensional Analysis: Buckingham Pi Theorem Using a Matrix

I have the following problem - which is basically that $\omega = f(g,h,l)$ now the book claims that the two $\pi$ terms as follows $\pi_1 = \omega \sqrt{\frac{l}{g}}$ and $\pi_2 = \frac{h}{l}$. Now we know we can get different dimensional $\pi$ terms because of the nature of the theorem. The ones I get however I obtain by solving $Ax=0$ for a matrix $A$ so my solution space looks like this $x = \begin{pmatrix} a \\\ b \\\ c \\\ d \end{pmatrix} = c \begin{pmatrix} 2 \\\ -1 \\\ 1 \\\ 0 \end{pmatrix} + d \begin{pmatrix} 2 \\\ -1 \\\ 0 \\\ 1 \end{pmatrix} $ such that my $\pi$ terms are then $\pi_1 = \frac{\omega^2 h}{g}$ and $\pi_2 = \frac{\omega^2 l}{g}$ However I am uncertain as to whether or not this is the correct answer. It seems right to me given the theorem, but I don't see any way to confirm that.


Flow of water through sand

Imagine a set up with a column of sand with porosity $\phi$ and permeability $k$ contained in a cuboid container. When one pours water evenly (let's say $V_w/A$ cubic meters per square meter) on the top surface of the sand at the same time, how does the distribution of water (water content at a particular height) of the sand vary over time (until the water reaches the bottom, and after the water reaches the bottom)? The sand may not be saturated.


Helium Fusion and the Origin of Elements

In the fifth of his Messenger Lectures Feynman mentions the fact that Carbon-12 has an energy level at 7.82 MeV. He then states that this is what enables the fusion of three Helium-4 ions into Carbon and ultimately enables the production of the other elements. Moreover one can apparently calculate that an energy level of 7.82 MeV is necessary in advance. Now unfortunately I have almost no knowledge of nuclear physics and therefore do not know how fusion is modelled theoretically. And I do not know if it is hard or easy to estimate that energy level (i.e. it could simply be a matter of if energy conservation allows the process). So my question are: What goes into this kind of calculation? and more generally: What are the current methods for a quantitative (qualitative) understanding of nuclear fusion? Naively I would model the fusion process by an estimate for the transition amplitude between a $\lvert \text 3{He}\rangle$ state into a $\lvert C\rangle$ state maybe by Fermi's golden rule. This should give you approximately a delta function $\delta(E_f - E_i)$, where $E_f$ and $E_i$ are the energies of the initial and final state. Then you would need to estimate the energy levels of the $3\text{He}$ system and one of them would be approximately at 7.82 MeV. You can assume in your answers that I took graduate level courses on quantum mechanics and quantum field theory, if that helps.


Difference between horizon and flatness problems & how inflation solves flatness (w/out math)

Layman here so I'm hoping for an answer for my query that doesn't involve math. I'm reading about inflation and how it solves the flatness and horizon problems. I get that the horizon problem deals with the uniformity of matter/energy in the universe. Flatness deals with the curvature. But isn't the universe flat because of the uniformity of matter/energy? Or is it the uniformity plus also the density (i.e. how far apart everything is)? In this way, is it correct to say that the flatness problem encapsulates the horizon problem? Finally, is there any way to explain why inflation would have given this particular density without using math?


What does it mean to "solve an equation"?

I don't understand what is meant by there being a "solution" to an equation. For example, what does a solution to the wave or heat equation represent, and _what are we solving for_? Of course, we can use such equations to perform computations, but what can the "solutions" tell us, and how can we use them? (Please excuse my knowing almost nothing about differential equations, but I am curious)


What is a "lateral fringe displacement"?

I encountered the term "lateral fridge displacement" in my optics homework for a problem about inserting a thin plate of glass over one of Young's double splits. So what does "lateral fringe displacement" mean in this context? I Googled but found nothing quite useful. Most of the results are about how this phenomenon emerges, but not specific about the exact definition.


Problem counting spin states

I can't figure out how many different spin states I can create with a four- electron system. I think I can create a spin-zero state, three spin-one states, and five spin-two states. That gives me nine possible states altogether. My problem is that I can specify a maximum of eight (complex) numbers to completely describe the spin states of the four electrons. But the nine spin- states I can seemingly create correspond to the spherical harmonic functions and I know for sure that they are linearly independent. It seems very wrong. This discrepancy in the counting doesn't appear until you get to four electrons. It gets worse as you add more electrons. Does anyone else have a problem with this? EDIT: Thanks for the excellent answers, especially from Lagerbeer. It turns out I was even more messed up than I thought I was. I don't run into trouble with _four_ electrons....I'm already in trouble with _three_. I didn't realize it because I was counting 2n electron-parameters instead of n^2, so I had 2, 4, 6, 8... (parameters to describe electron spin) as opposed to 2, 4, 8, 16... as people have pointed out. And this has to relate to (l,m) spin states of 2, 4, 6, 9, 12, 16... So there's already a problem with three electrons and it boils down to this: you have two different states with z-axis spin 1/2: the are (3/2, 1/2) and (1/2, 1/2). To make them up from electrons you have at your disposal these three states: {A} = duu {B} = udu {C} = uud The obvious thing to do is add the three together (and of course normalize); and I believe when you do, you get the (3/2, 1/2) state. The question is: how do you create the (1/2, 1/2) state? I think I know the answer and I've posted it on my blog. Anyone want to take a stab at it?


Quantum entanglement and spooky action at a distance

When quantum entanglement is explained in "layman's terms", it seems (to me) that the first premise, that we have to accept on faith, is that a particle doesn't _have_ a certain property (the particle is not IN one state or another) until that property is measured. For example, I have read that a particle may have "spin up" or "spin down" but -- here's where I get lost -- it's not that we don't KNOW which spin the particle has until we measure it, but actually the particle has NEITHER spin until we measure it. Now, if we accept that premise, then we can see how spooky it is, when two particles are created with correlated spins -- if one has spin up, the other has spin down, but neither particle "has" either spin until one of them is measured. At that instant, the other particle "assumes" the other spin. At least that's how I interpret things I have read. Certainly, if we accept that a particle doesn't actually HAVE the property until it is measured, but rather the particle exists in a superposition of states until the measurement is made, then we "regular citizens" can understand what is meant by "spooky action at a distance". (As soon as one particle is measured, the other particle assumes the "other" spin, no matter how far away it is. How did the second particle know the first particle was measured? That, I think, is the spooky action at a distance (and the information transfer can be faster than light).) However, from a layman's perspective, I want to cry out "but the particle does have a definite spin, we just don't KNOW what it is, until it is measured! Duh!" Now, I can't believe that all of the physics community hasn't thought of that objection -- but -- here's my point -- although the spooky action at a distance can be explained to us, once we accept the premise of superposition -- Why isn't the fact of superposition explainable also, in terms mortals can understand? I have looked, but I haven't found a layman's explanation of why we should accept the fact that a particle doesn't have a particular spin, or other property, until it's measured. It sure seems that the particles "have" the properties, even though we haven't measured them yet. I know that much of quantum mechanics is not "intuitive". If anyone can explain why particles don't have a definite property, even before we measure them, I would be grateful. Thanks.


Does Galileo's Tower of Pisa argument contradict quantum mechanics?

(My questions are at the end, but they may not mean much without explanation below.) Galileo argued that because the mass of a falling object can always be redistributed in ways that asymptotically approximate a set of smaller falling objects, e.g. linked by light strings, then the equations of motion for an object falling in a vacuum cannot depend on its mass. That is, because dependency on mass would lead to paradoxical motion predictions if in one case you used the total mass $m_\Sigma$ and in another one of the asymptotically free submasses $m_i$ were used as the arguments to the equation of motion. Thus for example, the equation for the instantaneous velocity of any object falling in a vacuum does not include mass as a significant parameter: > $v_i=\sqrt{2gd}$ Galileo's argument was insightful but largely tautological. The problem is that the equations of motion for objects falling though fluids, such as terminal velocity, provide an existence proof that self-consistent equations of motion with mass as a significant parameter _can_ be constructed. However, such self-consistency becomes possible only if the distribution of object mass in space also becomes a non-trivial parameter. This additional information allows paradoxes to be resolved by converting seemingly inconsistent state predictions into asymptotic limits of a larger continuum of predictions. For example, in fluid mechanics a mass $m_{\Sigma}$ is correctly predicted to have a higher terminal velocity than the same mass configured as two smaller spheres $m_1$ and $m_2$ linked by a thin string. The latter form is an example of how the predicted behavior of a single falling sphere can be made to approach asymptotically the prediction for two smaller spheres by slowing "morphing" the original object into one that closely approximates the case of two smaller spheres. What Galileo really proved, then, was that since experimental evidence showed that mass was not a relevant parameter for the equations of motion of objects falling in a vacuum, then the specific distribution of mass in any one object must also be irrelevant to those equations. What Galileo could never have realized was that exists a domain of physics for which his argument is very relevant: quantum physics. The de Broglie wavelength $\lambda=h/p $ of an object falling at modest speeds in a vacuum is a function of its instantaneous momentum $p_i=mv_i$, so: > $\lambda_i=\frac{h}{mv_i}=\frac{h}{m > \sqrt{2gd}}=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ While not an equation of motion, this equation violates Galileo's argument by predicting that the de Broglie frequency of an object falling in a vacuum _is_ a function of its mass. As in the case of fluid dynamics, the inclusion of mass as a significant parameter results in paradoxical predictions unless and until the configuration of the mass in the object is also taken into account. In other words, the de Broglie wavelength of an object must necessarily depend upon its shape and how mass is distributed within that object. The resulting more closely resembles solid state acoustic resonances in complexity than it does the simple singular mass parameter of pointlike particles isolated in space. For example, if $m_1=m_2=m$ and $m_\Sigma=m_1+m_2=2m$, three state equations are relevant: > $\lambda_{\Sigma}=\lambda_{2m}=\frac{h}{2m}\sqrt{\frac{1}{2gd}}$ > > $\lambda_1= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ > > $\lambda_2= \lambda_m=\frac{h}{m}\sqrt{\frac{1}{2gd}}$ Self-consistency alone argues that if these equations are even approximately correct, then $\lambda_{2m}$ should dominate when the $2m$ mass takes the form of a compact spherical object. Conversely, reconfiguring the $2m$ sphere into two smaller $m$ spheres linked by a slender thread should asymptotically approach the case of two separate $m$ spheres, and so should be dominated by the $\lambda_m$ wavelength. However, any single wavelength cannot be accurate in any of the cases. There should instead be a spectrum of wavelengths and intensities that is highly dependent on the particulars of how the mass is distributed and connected in the object. That is why I mentioned earlier that an accurate equation model for predicting de Broglie wavelengths for non-point object will necessarily be comparable in complexity and subtly to resonance models in solid state physics. I do not readily see a way out of this. Galileo's argument, while largely specious in the case of the equations of motion for objects falling in a vacuum, is extraordinarily relevant to quantum mechanics, where it appears to say that the entire concept of "mass" requires revisiting if it is to be used accurately and meaningfully. A correct model should instead have more kinship with the equations of motion for objects falling though a fluid, in the sense that both must define very carefully how the distribution and "connection" of mass impacts the predictions. * * * So, my two main questions are:   1. Do there currently exist mathematical models of de Broglie wavelengths that pass the Galilean self-consistency test? For example, are they perhaps hidden away in the mathematical details of phenomena such as quantum models of molecular vibration? (My impression is "no", but I could certainly be wrong.)   2. What are the particle physics implications, if any, of quantum mass being dependent on form?


Relative motion question

I'm sitting on my seat in a train and the train is moving at a very high speed, let's say 600 mph. If I throw a ball vertically up in the air (while still sitting on my seat), will the ball fall back in my hands or will it fall in the seat behind me since my position would have changed from the time I threw the ball, but the balls vertical trajectory will remain the same?


Ball thrown in a moving train

> A ball is thrown upward in a train moving with a constant velocity. Where > will it land? My intuition tells me that the ball will fall at my back. But my book says that it will return back to the thrower.


What is $k$ in Newtons Law of Cooling?

I am writing the definition of quantities for Newton's Law of Cooling and am not sure what to actually call k$,$ or its units. This is what I have so far $$\frac{dT}{dt}=-k(T-T_s)$$ Where $\frac{dT}{dt}$ is the rate of cooling measured in degrees Celsius per second $\left[°C\cdot{}s^{-1}\right]$ $T$ is the temperature of the object measured in degrees Celsius $\left[°C\right]$ $T_s$ is the ambient temperature of the objects surrounding measured in degrees Celsius $\left[°C\right]$ $k$ is a constant and has no units? I don't think that's a good enough description


Practical personal aircraft today?

Is it possible to build, today, a personal aircraft that not use an impracticable amount of fuel for everyday use? What are the physical concepts that could be used to build it?


Momentum and indirect bandgap

When an electron is excited from valence band to conduction band, it has to have a finite momentum in the case of indirect bandgap. Does that mean that the electron cannot be created at rest ? does it have a non zero "speed" ?


Relation between complex dielectric constant and resistance

I am having a real hard time understanding **how** I reach the expression that relates resistance and $\varepsilon''$ as the complex part of the dielectric constant. I was trying to follow a clear and eloquent way of describing this. I understand how the idealized concept of a capacitor requires a $\pi/2$ phase shift between polarization and electric field which is related by the dielectric constant. If it is not $\pi/2$ , then expressing the the dielectric constant as a complex number will allow us to express that phase shift. I know that not having a phase shift equal to $\pi/2$ means that I do not have a ideal capacitor, but also a related resistance. My professors keep telling me that the following expression is a _definition_ , not a deduction: $$\varepsilon''=\sigma/{\varepsilon_0\omega} $$ i.e. $$\varepsilon''=d/{A\varepsilon_0\omega R} $$ where _d_ and _A_ are thickness and contact area of the resistance _R_ But it seems to me that I should be able to explain it better... How?


Why is the direction of a flame is always upward?

why a candle's flame direction is always upward? What is the coposition of fire? Don't say it's energy. There is something else.


Why is general relativity considered to be a gauge theory?

I have studied the first five chapters of Carroll's book (up to the Schwarzschild solution). I see similarities to the Yang-Mill theories such as the covariant derivative to account for curvature in GR which is similar to the covariant derivative to account for local gauge transformations in Yang- Mill theories. I have heard that some consider GR to be a gauge theory, but I do not understand what the redundant degrees of freedom are required to make it gauge theory.


Is Lagrangian a scalar?

I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ (some number). $$L=\frac12m\left(\frac{dz}{dt}\right)^2-mgz\to L=\frac12m\left(\frac{dz}{dt}\right)^2-mgz-mga$$ thus the Lagrangian changed under this coordinate transformation. However Euler-Lagrange equations ARE invariant under coordinate transformations. **So some scalars do vary under coordinate transformation!** Thus components of vectors are scalars, thus time is. Again, try to say the following sentence: > the vector $a$ is the first component of the vector $b$. Can you please clarify?


Time travel in the past

The past weeks I watched some episodes from "Through the Wormhole" with Morgan Freeman. When talking about traveling back in time, they said you can only travel back up to the date when the first time travel occured. So, if for instance time travel will be discovered in 2045, all time travelers(from any given time) won't be able to travel in time earlier than 2045. Why is this? Can anyone explain this please? I know this is just theoretical but still I would like to know the concept behind it.


Gas kinetic representation of trans-critical conditions

From a molecular point of view, can we think of the super-critical conditions as conditions where T and p are large enough that the collisions of gas molecules are frequent and powerful enough to prevent the molecular bonds that would form a liquid cannot be sustained. So if we consider a single fluid in a constant pressure system and with a temperature gradient from sub-critical temperatures to super-critical temperatures, how accurate is this diagram? ![Trans-critical event from a molecular dynamics point of view](http://i.stack.imgur.com/NgSSC.png) How can we represent the disappearance of surface tension on the right side? Are molecules being stripped from the surface by collisions? Because the molecules inside the compressed liquid are still bonded right? What would you change in this crude diagram to make it more representative of what really happens? To give a bit more context, I'm more of a fluid mechanics guy modeling inject. So I might not be using the exact terms that I should. The terminology I use is within the context of this diagram: ![O2 phase diagram](http://i.stack.imgur.com/W1wRj.png) Phase diagram of oxygen in the $p-T$ space. The black dot represents the critical point at the intersection of the long-dash lines for the critical temperature and pressure. The short-dash line is the pseudo-boiling line (Oschwald_2006a) or Widom line (Simeoni_2010a) and corresponds approximately to the region of maximum specific heat at constant pressure. The density is represented by the colored contours in the background.


Does performing a measurement on a system change its internal energy?

I'm studying Quantum Mechanics in my spare time from a general point of view (no technical details) so some fundamental question came into my mind: How is it possible to detect a single photon without making any change to it? How does observation is related to the total energy of a system and does the act of performing measurement alter the internal energy of the observed system as a general rule?


Observations in the cathode ray tube experiement

![enter image description here](http://i.stack.imgur.com/6drQ1.png) 1.One of the observations I learned was that the glass tube begins to glow with a brilliant green light. Many websites I read through refer to a fluorescent material. However, as shown in the above diagram there was no fluorescent material in the experiment carried out first on the cathode ray tube. **So where does the green glow come from. Is this the color of the radiation itself?** 2." _Cathode rays travel in straight lines. That is why, cathode rays cast shadow of any solid object placed in their path. The path cathode rays travel is not affected by the position of the anode._ " I just can't seem to understand this explanation of the one of the observations.Also, different websites analyses this observation differently. For example, " _The cathode rays consist of material particles because they produced shadow of objects placed in the way_ " 3.Two of the conditions of the experiment were air at very low pressure and secondly a very high potential difference. Could someone please tell me why these conditions were necessary? I know the questions are very silly but because different websites refer to different things, I am becoming confused with something that should be simple to understand.


Characteristic waves in a plasma

What does "characteristic wave" mean in the context of plasma waves? For example, when propagating parallel to the plasma magnetic field, the characteristic wave is circularly polarized. Does this mean that only a circularly polarized wave can propagate in this direction? Could an elliptically polarized wave travel in this direction instead?


Critical dimension in quantization of p-branes

So I have what might be a fairly basic question, but my understanding that in the quantization of the the string, or the 1-brane, there are conditions on the number of spacetime dimensions to ensure that the Lorentz algebra still holds. For the bosonic string we have D=26 and for the superstring we have D=10. Now I know that an equivalent condition on the critical dimension comes up by requiring that the conformal anomaly vanish. This is the prelude to my question: Are there any similar or equivalent restrictions on the number of spacetime dimensions are come up in the quantizing more general branes? Is it only the string because only the worldsheet theory has conformal invariance that we want to ensure holds in the quantized theory? If this statement is equivalent to the Lorentz algebra closing, why for a general p-brane do we not need to impose any conditions on the dimensions for this to hold?


Is it possible that universe might not be speeding up expansion?

I'm not sure but I was thinking of galaxies shrinking with time while still moving apart from each other at almost a constant speed or less (i.e: uniform/slightly decelerating expansion). This may need further evidence such as tracking the distance between stars inside our galaxy to check if they are getting closer. If so, would this make an alternative explanation for visualizing the universe as if it were expanding in an accelerating rate?


What causes the collapse of a Magnetic Field?

It's theorized that Mars once had a magnetic field which collapsed at some point. There's also lots of speculation that a similar process will eventually happen to Earth. More recently has been the information that the earth's magnetic field is slowly dropping, presumably preparing for its 'flip' that happens every few hundred thousand years. But what causes a magnetic field to eventually collapse to the point that it ceases to be?


What is the physical meaning of charges at light-like infinity in asymptotically flat space-times?

In the case of charges defined at space-like infinity, I can understand the physical meaning of them because they can be related to measurements made by a physical observer (that is an observer whose wordline is time-like). For example in four dimensions, for the Schwarschild solution, the ADM mass coincide with the mass measured by an observer that is motionless (in Schwarschild coordinates) and far from the center of the solution. My puzzle is then the following : since no physical observer can reach light- like infinity, how are physically interpreted the charges defined at light- like infinity? How are they related to any measurement? Many thanks for your help.


Statistical Mechanic

One can define entropy as $$S=k\log{\omega(E)},$$ where $\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined as$$Z_N=\sum_{\phi}e^{-\beta E[\phi]}=e^{-\beta F(\beta,N)},$$ where the sum run on states $\phi$ and the free energy is defined as $F(\beta,N)=U-TS.$ The mean value of the internal energy and the entropy i learned that would be $$\langle U\rangle=\frac{\partial(\beta F)}{\partial\beta}$$ $$\langle S\rangle=\beta^2\frac{\partial(F)}{\partial\beta}.$$ For definition, the mean value of any physical observable is $$\langle O\rangle=Z_N^{-1}\sum_{\phi}O[\phi]e^{-\beta E[\phi]}.$$ I'm quite sure that there will be a problem of amount $k$, if one verify the definition of $\langle S\rangle$. Am i wrong?


Circular Polarization

As one observes an clockwise (cw) circular polarized electromagnetic wave which is reflected off a denser medium or metal interface it is changed to a counter-clockwise (CCW) polarized em wave plus a phase of PI. A counter clockwise polarized wave which is reflected off a lower dense medium interface only get an additional phase of PI.   1. This means the reflected wave always has the opposite polarization?   2. But **ONLY** because the reference coordinate system is changed (since by definition of polarization the wave always travels away from the observer) and NOT because of some phase which is added only to one or the other linear polarization axis?


In Noether's theorem, what is a "classical solution of the equations of motion"?

I'm reading a book which states that: > for each generator of a global symmetry transformation, there is a current > $j^{\mu}_{a}$ which, _when evaluated on a classical solution of the > equations of motion $\phi_{cl}$_ , is conserved. I.e. > $\partial_{\mu}j^{\mu}_{a}\vert_{\phi=\phi_{cl}} =0 $ I get the general principle, but I'm uncertain about the bit I have made _italics_. Can anyone shed some light on this?


Rabi oscillations and two level dynamics

I'm currently looking at Rabi Oscillations, and not I have a look at the following equations: $$W = \sqrt{\Omega^2+\delta^2}.$$ The amplitude: $$\frac{|\Omega|^{2}}{\delta^{2}+|\Omega|^{2}}$$ Now, I see that if the detuning $\delta = (\omega - \omega_{0})$ is large, the two level approximation is bad, since the ratio between level 2 and 3 is getting closer to 1. But why does the two level system not work for high $\Omega$?


How does concave bowl amplifies the sound?

I recently came across an article / answer in Quora see here that mentioned that if we place the iPhone in a bowl and play music, the volume is amplified. What is the physics principle behind this?


Why is $c$ considered as the speed of the photons?

Maxwell equations brought $\ c_{o}=\frac{1}{\sqrt{\mu_{o}\varepsilon_{o}}}\ $. Since this is a constant, it made all physicists at that time wonder where was the frame of reference? They ended up with an _aether_ answer until the Michelson–Morley experiment denied it. Now there was no frame of reference, but the genius of Einstein resolved it. However, although Einstein's theory is mathematically good, it does not imply there can not be other ones mathematically good. I just read an article written by someone not knowledgeable about physics who had this interesting idea: There is no need to consider c as the speed of the photons unless natural evidence tell us that, we could consider c as the difference between the speed of the photon that just appeared (c value) and the one at (0 value) that is going to speed up and the result is a constant speed as well, c, for all references. That is what he says, if I understood it correctly. Here is the article: http://jesusgonzalezllorente.com/ http://vixra.org/abs/1202.0013 The pictures are horrible though. So, is it actually possible to consider light as he states or is there any logical, mathematical physical law that does not allow it?


Pushing, inertia of a cart system

I have a cart with another cart on top which gets pulled down by another cart on the side. There is no friction. The question is: > How strongly do I have to push with $F$ to keep the cart $m_1$ stable? ![](http://wstaw.org/m/2012/02/04/m62.png) I work in the system of the cart $m_3$. This system accelerates with some $a$. The force of inertia pulls back on $m_1$ with $m_1 a$. That is opposed by the gravitational force of $m_2$, which is simply $m_2 g$. The acceleration that I need for this to be stable is: $$ a = \frac{m_2}{m_1} g $$ Here comes the point I am not too sure about: The force is on $m_3$ and on $m_2$, so the driving force would be this: $$ F = (m_2 + m_3) \frac{m_2}{m_1} g $$ On second thought, I think also need to accelerate $m_1$ some way or another, so that my total force would be a little higher: $$ F = (m_1 + m_2 + m_3) \frac{m_2}{m_1} g $$ I prefer the latter, but what is the right solution?


Relationship between mass preserving four-fources and proper acceleration

**Background** I studied Rindler's book on Relativity. Relevant information from this book is available on-line here: > http://www.scholarpedia.org/article/Special_relativity:_mechanics Given that we work with four-vectors $\mathbf{R} = \left(\vec{r},ct\right)$ where the four-velocity is defined as $$ \tag{1} \mathbf{U}=\frac{d\mathbf{R}}{d\tau}=\gamma\frac{d\mathbf{R}}{dt}=\gamma\left(u\right)\left(\vec{u},c\right), $$ $\vec{u}$ is the spacial "three-velocity" and the four-acceleration is defined as $$ \tag{2} \mathbf{A}=\gamma\frac{d\mathbf{U}}{d\tau}=\gamma\frac{d}{dt}\left(\gamma\vec{u},\gamma c\right)=\gamma\left(\dot{\gamma}\vec{u}+\gamma\vec{a},\dot{\gamma}c\right), $$ we can define the four-force as $$ \tag{3} \mathbf{F}=\frac{d}{ {d\tau } } \mathbf{P}=\frac{d}{ {d\tau}}(m_{0}\mathbf{U})=m_{0}\mathbf{A}+\frac{ {dm}_{0} }{ {d\tau} }\mathbf{U}. $$ The four-force $\mathbf{F}$ is said to be **mass-preserving** if $dm_0/d\tau$ equals zero, i.e. the force leaves the resting mass of the particle unchanged. Moreover $$ \tag{4} \mathbf{F}=\frac{d}{d\tau}\mathbf{P}=\gamma(u)\frac{d}{ {dt} }( \vec{p},mc)=\gamma (u)\left(\vec{f},\frac{1}{c}\frac{dE}{dt}\right). $$ We note that $\mathbf{A}\cdot\mathbf{U}=0$ and therefore for a mass preserving four-force $\mathbf{F}\cdot\mathbf{U}=0$ from Eq. (3) or from Eq. (4) $$ \tag{5} \mathbf{F}\cdot\mathbf{U}=\gamma^2(u)\left(\frac{dE}{dt}-\vec{f}\cdot\vec{u}\right)=0 $$ which means that $$ \frac{dE}{dt}=\vec{f}\cdot\vec{u} $$ and $$ \tag{6} \mathbf{F}=\gamma(u)\left(\vec{f},\vec{f}\cdot\vec{u}/c\right). $$ We also have the relationship $$ \tag{7} \gamma(u)\vec{f}=m_0\frac{d^2\vec{r}}{d\tau^2}. $$ Moreover we have that the **proper acceleration** $\alpha$ is defined as $$ \mathbf{A}\cdot\mathbf{A}=\mathbf{A}^2=-\alpha^2. $$ I studied the questions formulated in > Why proper acceleration is $du/dt$ and not $du/d\tau$? > > relativistic factor between coordinate acceleration and proper acceleration and realized that the spatial vectors can be broken down into a parallel component and an orthogonal component, i.e. $\vec{v}=\vec{v}_\parallel+\vec{v}_\perp$ or $\vec{a}=\vec{a}_\parallel+\vec{a}_\perp$ where one makes a distinction between the component of the velocity that is parallel with the acceleration and vice versa. **Question** Now it can be shown that for a force $\vec{f}$ acting upon a particle traveling with a velocity vector $\vec{u}$ where there is an angle $\theta$ between $\vec{f}$ and $\vec{u}$ the relationship between the proper acceleration of the particle and the force $\vec{f}$ is $$ \tag{8} \vec{f}^2=m_0^2\frac{\gamma_\parallel^2}{\gamma^2}\alpha^2 $$ where $$ \tag{9} \gamma_\parallel=\frac{1}{\sqrt{1-u_\parallel^2/c^2}}\;\;\;\text{and}\;\;\;\gamma=\frac{1}{\sqrt{1-u^2/c^2}}. $$ Now how can I derive or prove equation (8)? I have tried every feasible combination of Eq. (1) to (7) without success. I've also tried the more basic variants of the Lorentz transformation > https://en.wikipedia.org/wiki/Lorentz_transformation but it gets very confusing when there are two components to consider when assessing the relativistic effects. Any ideas?


Determine orbital position after change in velocity

I am working on a satellite simulator for a project/game and I am stuck on this one bit of physics. So far I have a satellite that revolves around earth on a 2D plane following Keplerian motion using Kepler's equation. Everything is fine and the satellite will orbit many times without problem. I can also change the velocity of the satellite to manipulate apo/peri points as well as augment of periapsis. However, the problem occurs if I change the velocity once the satellite passes the `PI/2` mark, the entire orbit gets flipped (augment of periapsis is offset by `PI`) and the true anomaly resets to 0. Here is how it is currently implemented: With the assumption that the satellite starts at the periapsis at t0, I find the eccentric and true anomaly given a time since periapsis $w$. Then I find the radius at that angle to get the final position. $$t_0 = 0$$ $$E_0 = 0$$ $$M(0) = E_0 - e \sin E_0$$ $$n = \sqrt{ \frac{GM }{ a^3}}$$ $$E = {\rm kepler}(e, M)$$ $$v = 2 \tan^{-1}{(\sqrt{\frac{(1+e)}{(1-e)}}\tan\frac{E}{2})}$$ $$r = a\frac{1-e^2}{1 + e\cos{v}}$$ When I update the velocity of the satellite (flight path angle $\phi$ is recomputed after change in velocity), I recalculate a new true anomaly using $\phi$, I then find the difference between the old true anomaly and the new true anomaly, and add that to the augment of periapsis $w$. After that I recompute time since periapsis $t$ using the new true anomaly and recompute the orbital parameters as well. $$vel = \sqrt{GM\frac{2}{r} - \frac{1}{a}}$$ $$\phi = \tan^{-1} ( \frac{e\sin{v}}{1 + e \cos{v}})$$ $$v_2 = \tan^{-1}{\frac{r v g m \cos{\phi} * \sin{\phi}}{r v g m \cos^2{\phi} - 1}}$$ $$w \mathrel{{+}{=}} v_2 - v$$ $$E = \cos^{-1}{ \frac{e + \cos{v}}{1 + e \cos{v}} }$$ $$M = E - e \sin{E}$$ $$n = \sqrt{ \frac{GM}{a^3} }$$ $$t = \frac{M}{n}$$ $${\rm calculateOrbit}()$$ I noticed if I were to $v = v + \pi$ if ever $v_2 < 0$ and $v > 0$, then the weird orbit/position flip won't occur until the $v = \pi$ Otherwise I do not know what the problem is. I would really appreciate it if someone can point me the way, as I have been scratching my head over this for a long time now.


What is spontaneous symmetry breaking in QUANTUM systems?

Most descriptions of spontaneous symmetry breaking, even for spontaneous symmetry breaking in quantum systems, actually only give a classical picture. According to the classical picture, spontaneous symmetry breaking can only happen for non-linear systems. Classical linear systems, such as Harmonic oscillators, can never have spontaneous symmetry breaking. (Here "linear" means that the equation of motion is linear.) But the real QUANTUM systems are always linear, since the Schroedinger equation is alway linear. But how can a linear quantum system have spontaneous symmetry breaking? Do we have a simple intuitive understanding spontaneous symmetry breaking WITHIN QUANTUM mechnics? (ie without using the classical picture, such a Mexican hat -- the logo of physics.stackexchange) The Mexican hat does give us a intuitive and pictorial understanding of spontaneous symmetry breaking in classical systems. Do we have a intuitive and pictorial understanding of spontaneous symmetry breaking in quantum systems.


How is energy produced in Alpha emission (radioactivity)?

![enter image description here](http://i.stack.imgur.com/pW1vD.gif) In the above Alpha decay reaction, we get thorium by the alpha decay of uranium with an alpha particle. But we also get energy of amount Q by this process. And according to my course books, this Q energy comes from E = mc2 i.e. some of the mass is converted into energy. My question is that which mass converted into energy? I mean if you add the nucleons of alpha particle (helium nucleus), you will get the total nucleons present in Uranium. But, where did the energy come from then?


What is mass of free up and down Quark?

1. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation.   2. Color confinement, often simply called confinement, is the physics phenomenon that color charged particles (such as quarks) cannot be isolated singularly, and therefore cannot be directly observed.   3. In theoretical physics, quantum chromodynamics (QCD) is a theory of the strong interaction (color force), a fundamental force describing the interactions between quarks and gluons which make up hadrons (such as the proton, neutron or pion).   4. Yang–Mills theory List of unsolved problems in physics Yang–Mills theory in the non- perturbative regime: The equations of Yang–Mills remain unsolved at energy scales relevant for describing atomic nuclei. How does Yang– Mills theory give rise to the physics of nuclei and nuclear constituents? When you read about quarks, All folders of Quark model have not been defined. I just want to know what is mass of free Quark? But in my calculations seem free quark have no mass. while Quark become free is like to melt ice.


Computing the efficiency of a heat engine

With the heat engine ![enter image description here](http://i.stack.imgur.com/ulLxt.png) we have an ideal gas working the substance. I want to show that the efficiency of this heat engine is $$\eta = 1 - \frac{1}{\gamma}\frac{\bigg(\frac{V_3}{V_1}\bigg)^\gamma - \bigg(\frac{V_2}{V_1}\bigg)^\gamma}{\bigg(\frac{V_3}{V_1} - \frac{V_2}{V_1}\bigg)} $$ where $\gamma = \dfrac{C_p}{C_v}$ is the adiabetic index. So we have that $Q_1$ enters during the expansion $B \to C$ and $Q_2$ leaves during the compression $D \to A$. Therefore, $$\eta = \frac{Q_1 - Q_2}{Q_1} = 1 - \frac{Q_2}{Q_1}$$ I'm kind of stuck at this point. Any further help would be appreciated.


Can we observe diffraction even if the slit size is tending to zero?

According to Huygens principle, if we consider any point of the primary wave fronts, each point is the source of secondary disturbance. If the point size we consider tends to zero, the number of points on that wave front which enters the slit tends to infinity. Thus, if we consider the slit how much ever small, we must see the diffraction pattern. Because we can consider many points on the wave front that enter slit, whose size is very very small-thus producing interference of waves emanating from them which can give diffraction pattern. But in reality as I know we don't see any diffraction pattern if we have the slit of very small in width compared to wavelength. Is the point size that we consider in diffraction is restricted to some value? If not as explained above, we can have many points of very very small size on the wave front that enter the slit, and we can expect diffraction even if slit size is tending to zero. I don't know whether I have misunderstood anywhere, if yes pardon me and explain. If not, is it that (according to huygens principle) we can observe diffraction pattern even if slit size is tending to zero? Sometimes I might not be communicating with you better with my problem, if so please comment on where there is problem.


What is the difference between 'running' and 'current' quark mass?

When looking at the PDG, there is a difference between the 'running' and the 'current' quark masses. Does anyone know which is the difference between these two?


Exercise QFT and CFT

Consider the action functional $S[z;t_1,t_2]=\int_{t_1}^{t^2}[g(z,\bar{z})\dot{z}\dot{\bar{z}}]^{\frac{1}{2}}dt$ with $z(t)$ a complex path with end points $z_i=z(t_i),\; i=1,2$. $g(z,\bar{z})$ is a positive real function on $H=\\{z\in C: Im(z)>0\\}$. After some easy questions the exercise asks me to determine (up to a multiplicative constant) the function $g(z,\bar{z})$ such that the action is symmetric under the transformation $z\rightarrow \gamma(z)= \frac{az+b}{cz+d}$, with $a,b,c,d \in R$. Can somebody help me or give me some hints?


Question regarding Schrödinger's equation

Here is my understanding of Schrödinger's equation and wave functions: (I'm considering the time independent equation.) > The equation takes in values of energy as an input and, being a differential > equation, gives functions as outputs. These functions correspond to the > different orbitals. > > When we input coordinate values we get an answer that we can use to plot the > probability distribution of an electron around the nucleus. Can someone tell me if my understanding is correct, and correct me if I'm wrong?


Isotropy and Noise

If you have a field which value is just Gaussian noise plus a constant do you call it isotropic?   * there is no preferred direction   * however it is not "the same" in all directions if "the same" means "constant" The question is about a definition of a word, thus obviously conflicting answers can arise. However what definition do **you** use or have seen used: "no preferred direction" or "constant wrt translation"?


Free fall from space

If you leave a ball weighing 10 kg at a height of 500 km above sea level (neglecting air friction). How can calculate how long the ball hits the ground and what will be its speed? ![enter image description here](http://i.stack.imgur.com/5Emdd.png) I know that: On earth I can calculate the time taken to the object to fall by this equivalent:               x = x0 + v0t + (at^2)/2     x = 0     x0 = 500000     v0 = 0     a = -g = 10 m/s^2     0 = 500000 - 5t^2     t = 316.227sec      Also I know how to calculate the force between objects:               g = GMm/r^2


Instability of a thrown tennis racquet

Someone once mentioned to me that it's impossible to throw a tennis racquet (or similarly shaped object) into the air, perpendicularly to the string plane, in such a way that it won't turn. What is this effect, or was he talking rubbish?


Does perpetual motion machine really exist?

I recently saw a video on Youtube about a perpetual motion machine. Here is the link: http://www.youtube.com/watch?v=OS1KXMsE2qk It's on Robert Boyle's flask. Where does the energy come from?


Young's Double Slit Experiment : What would happen if the "first slit" was too wide?

![A picture showing Young's Double Slit experiment](http://i.stack.imgur.com/QTGeQ.gif) I was wondering what would happen to the fringe pattern displayed on the screen if the first slit (as shown in the picture), which is also known as "single slit", was made a bit wider. I read it in my book. I don't understand it. Anyway, it quotes > _If the single slit is too wide, each part of it produces a fringe pattern > which is displaced slightly from the pattern due to adjacent parts of the > single slit. As a result, the dark fringes of double slit pattern become > narrower than the bright fringes, and contrast is lost between the dark and > the bright fringes._ Please answer the question in your own words and try to explain to me what the quote is trying to say. Also, when it says "parts", what does it mean?


Explanation for this type of (magic-trick) suspension?

Well let's start off with that I'm not a physicist but I'd like some thoughts on something I came across in my hometown. This guy: ![enter image description here](http://i.stack.imgur.com/G3U6R.jpg) Is it possible that due to the electrical charge of magnets this guy can make the illusion that he can float ? Or is this probably a cheap trick that fools the eye ? I was standing there for quite some time watching the guy and he keep moving his feet. The resistance that he appeared to have was from a magnet force keeping him afloat. So after I passed this guy I did some physics searches on the web and the first thing that caught my eye was the electrical charge of magnets. So the question is : Is this related to the electrical charge of a magnet or a cheap trick ?


Tree level and loop level

I'm trying to read through a paper which explains the following about Universal Extra Dimensions (UED) vs ADD models: > The new feature of the UED scenario compared to the brane world is that > since there is no brane to violate translation invariance along the extra > dimensions, momentum is conserved at tree level leading to degenerate KK > mode masses at each level and conservation of KK number in the interactions > of the four dimensional effective theory. This statement is broken at the > loop level, where the fact that the extra dimensions are compact leads to > (calculable) violations of the full Lorentz symmetry, and as a result shifts > the masses of the KK modes away from their tree level values. I'm unfamiliar with the concepts of tree-level and loop-level. Could someone perhaps explain this in context? If I would have to guess, I'm thinking that tree-level correspond to the scattering correlation functions of the classical theory and loop-level are a kind of correction?


Flux if there were only one type of charge in the universe

There was this question that i saw in a book and it also had an answer given. The Question was: > If there were only one type of charge in the universe, then: > >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} \neq 0 $ on any > surface. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} = 0$ is the > charge is outside a surface. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} $ will not be > defined. >   * $\phi = \oint \boldsymbol{E}\cdot \partial\boldsymbol{A} = > \frac{q}{\epsilon_o}$ if the charge is inside the surface. > The answers were given as option second and the last one. (Gauss' law statements) How?


Do ~1THz oscillators output "light"?

Upon looking at some radiation levels higher than Microwaves, I have come across "Terahertz radiation". According to the article, there are numerous ways to generate such radiation, and that brings questions to me. My confusion lays in how it is different (if it is!) from a heating fan emitting infrared radiation (simple), or an LED emitting visible green light, or a flashlight emitting various spectral frequencies. Are those examples emitting Terahertz radiation? I would hypothesize that is, but can see no reference to light being called "Terahertz radiation" even though it is radiating. Although I am unsure if the technologies listed in the article can go above infrared in to visible light, but assuming so, would it emit "light" (in the respective frequencies)? How would this be different than an ordinary visible light emitter? At that point is it no different from an "ideal" colour changing LED?


Partial derivative potential energy of 'free' vibration

I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial x_i\partial x_j} x_ix_j.$$ Or if we use $b_{ij}$ for the Hessian : $$U(x_i)=\frac{1}{2}\sum_{i,j}b_{ij}x_ix_j.$$ I want to calculate the force: $k_i=\frac{-\partial U}{\partial x_i}$. This should be $-\sum_{j} b_{ji}x_j$. My questions:   1. Why is the summation over the index $i$ gone?   2. Why are the indices of the hessian $b$ switched? I hope someone can give me an answer. I put this in the physics section because it's a physics problem, but my question is actually purely mathematical.


What is the friction between cylinder and wall (ground)?

A hollow cylinder (radius $R$) is rolling against the wall at angular speed $\omega$. The coefficient of friction between the cylinder and the wall(ground) is $\mu$. After how many rotations the cylinder will stop rotating? So I figured I need to find the time taken for cylinder to stop moving, and that would be $$ \beta=-\omega/t => t = -\omega/\beta $$ Where $\beta$ is angular acceleration, which is known from torques: $$ 2*F_f*R = I*\beta $$ That's where I got stuck... How do I know the friction? I'm familiar with such equation: $$ F_f = \mu*F_n $$ How do I find the normal force? Does it have anything to do with centripetal force?


Which experiments prove atomic theory?

Which experiments prove atomic theory? Sub-atomic theories:   1. atoms have: nuclei; electrons; protons; and neutrons.   2. That the number of electrons atoms have determines their relationship with other atoms.    3. That the atom is the smallest elemental unit of matter - that we can't continue to divide atoms into anything smaller and have them retain the characteristics of the parent element.   4. That everything is made of atoms.  These sub-theories might spur more thoughts of individual experiments that prove individual sub-atomic theories (my guess is more was able to be proven after more experiments followed).


Transformation of $t=0$ line in moving frame of reference

How does $t=0$ transform into $t - vx/c^2 = 0$ if a frame of reference is moving as given in here? It seems that the relativistic transformation is given by $$ \begin{bmatrix} x' \\\ ct' \end{bmatrix}= A\begin{bmatrix} x \\\ ct \end{bmatrix} = \left( \begin{array}{cc} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} & -\frac{v}{c \sqrt{1-\frac{v^2}{c^2}}} \\\ -\frac{v}{c \sqrt{1-\frac{v^2}{c^2}}} & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\\ \end{array} \right).\begin{bmatrix} x \\\ ct \end{bmatrix} = \begin{bmatrix} \frac{x-t v}{\sqrt{1-\frac{v^2}{c^2}}} \\\ \frac{c^2 t-v x}{c \sqrt{1-\frac{v^2}{c^2}}} \end{bmatrix} $$ I was wondering if the transformation is equivalent to measuring in new basis $A\begin{bmatrix} 1 \\\ 0 \end{bmatrix}$ and $A\begin{bmatrix} 0 \\\ 1 \end{bmatrix}$ for an event?


Could there be a correlation between the Island of Stability and dark matter?

I can only boast of my passion for science and not of any credentials. I ask because I am curious, and it is not the type of question one can easily Google! What if the key to discovering and unlocking the secrets of dark matter and energy lie in the Island of Stability? Am I missing a vital reason that outright contradicts an attempted connection between the two?


Definition: Coupling

What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?


Wormholes and the illusion of time?

I was watching a video on Youtube by Brian Greene, "the illusion of time."http://www.youtube.com/watch?v=j-u1aaltiq4 In that video, he introduces to me the idea of a "brane," or a slice of the universe at specific points in time. And then, he talks about an alien that is tens of billions of lightyears away from Earth. At that distance, he says that even the slowest movements can angle his brane, or slice of time, such that he is angling it towards the Earth's past (if the alien is moving away from us) or future (if the alien moves towards us). So I got to wondering. What if you had a wormhole here on Earth, and walked through it to appear somewhere tens of billions of lightyears away from Earth. Depending on your speed of crossing through it, would you appear further into their past/future (depending on which direction you cross)? And, if you had another wormhole there connecting back to the Earth, wouldn't you be able to travel back into Earth's past? Which would violate causality? I know lots of people don't like the idea of wormholes, but it doesn't seem to have been scientifically ruled out. So, my question is, despite the fact that wormholes seem to possibly violate causality, why can't it be ruled out? Why is it still a theoretical possibility? As a side question, if the past, present, and future exist at the same time -- what does that even mean? (If anyone could edit my post, that would be great. My internet is giving me a horrible screen to work with. I have totally no ability to see my own tags -- look at this interface: https://dl.dropbox.com/u/106018191/phystackex.png) (The image above is going to take a while to upload because of the slow internet, possibly a few hours at worst.)


Where and how is the entropy of a black hole stored? Is it around the horizon? Most of the entanglement entropy across the event horizon lies within Planck distances of it and are short lived. Is it stored near the singularity? How can you pack so much information in such a small region? Can you please help me?


Overlap integral and probability

I have a question regarding how to extract probability from an overlap integral. Specifically, I am calculating the probability of a particle in a bound state in a delta potential $V=-\alpha \delta(x)$ ($\alpha$ is positive) remaining in a bound state after $V$ changes strength to $V=-\beta \delta(x)$. I am able to calculate the overlap integral of the bound state wavefunctions of these two $V$'s just fine; I am just wondering if I need to square the value of the overlap integral to find the probability of remaining in a bound state or if we just keep the value of the overlap integral as our probability. And if we do have to square the value of the overlap integral, why is that?


In Copenhagen, can this idea preserve locality for Bell inequalities?

Generate an entangled pair of qubits. Send to Alice and Bob far away from each other. Both measure along basis in one of two possible orientations. The result is sent to Charlie at some later time, who compares the corellations and concludes the Bell inequalities have been violated. But in Copenhagen, Charlie can say, the pair of results didn't become real until I collected both observations from Alice and Bob. They weren't real when either of them measured them. They only became real once I observed them. Then, both results, when they materialized, materialized at the same place. So, no nonlocality?


On the theoretical aspects of the development of the first nuclear bombs

I've just read that 68 years ago _Little Boy_ was dropped on Hiroshima, which made me wonder about some rather historical facts about the development of the first nuclear bombs; they seem to be several questions, but they boil down to the same thing: the theoretical aspects of the development of the bombs. As far as I know, nuclear fission was already understood at the time, so was all the _competition_ between _the american team_ and _the german team_ to develop the bomb first, a mere matter of engineering? Or what was the role of theoretical physicists such as Richard Feynman? Also, I've heard things like that the _team_ led by Werner Heisenberg had _misconceptions_ about nuclear fission, so that his _team_ could not develop the bomb first. Can anyone put in perspective the theoretical aspects of the development of the bombs taking care of this historical issues?


Can entanglement be explained as a consequence of conservation laws?

This article at NewScientist magazine (subscription required) describes entangling photons by passing them through a half silvered mirror. http://www.newscientist.com/article/mg21929282.100-quantum-weirdness-the- battle-for-the-basis-of-reality.html It talks about the "weirdness" of entanglement and how it means that either information is traveling faster than the speed of light or other fundamental assumptions about reality are wrong because the spins of the emerging photons are always correlated when measured. My question is just what's so strange about it? I don't get it... It seems to me to be just another variation of conservation laws. It would be like saying that it's "weird" for a photon in pair production to always produce particles of opposite charge. That's a given... Is there something to the timing of it that ruffles everybody's feathers? I think I'm missing something.


Bounded and Unbounded (Scattering) States in Quantum Mechanics

I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state implies that $E > V_0$. But I do not understand why for bounded states, the complete set of wave- function solutions to the time-dependent Schrodinger equation is constructed from a **discrete** linear combination of solutions, whereas for scattering states, the complete set, or general solution, is instead written as an integral over the **continuous** value of $k$ (or more appropriately, as a Fourier-type integral). Could anyone clarify why this difference is true for bound and scattering (unbound) states?


Are there reasons for the discrepancies in absolute temp units - Kelvin vs. kelvins vs. degrees Kelvin?

Before 1968, the units for absolute temperature were described as "degrees Kelvin" or "degrees absolute." After that, the SI system got rid of the idea of "degree" for absolute temperature, so the new unit should apparently be expressed as a "kelvin" (with lowercase k) and abbreviated simply "K" (without the degree sign). Also, official SI conventions suggest that not only should the unit name be lowercase, but it should be pluralized as other units would be: "Il en résulte que la température thermodynamique du point triple de l’eau est égale à 273,16 kelvins exactement, Ttpw = 273,16 K." Or, in English: "It follows that the thermodynamic temperature of the triple point of water is exactly 273.16 kelvins, Ttpw = 273.16 K." Despite the official SI usage, however, it seems that there are still a variety of conventions in use. Many of the questions on this forum, for example, use Kelvin (with a capital) instead of kelvin in referencing the unit. Also, it appears that the plural usage is somewhat mixed in the physics literature: something like "200 kelvins" occurs, but more rarely than "200 kelvin" or even "200 Kelvin." The NIST guidelines do not list the kelvin as an exception to the normal pluralization rules: "the following plurals are irregular: Singular — lux, hertz, siemens; Plural — lux, hertz, siemens." On the other hand CERN's writing guidelines suggest that there is an exception: "And note that it is always kelvin, even when plural (not kelvins or degrees kelvin)." Given all of this, here is my question: Is the SI standard actually to pluralize kelvins, as would be suggested from the quotations from the official SI guides above? Is this officially stated anywhere in some standards organization's guidelines? Or, is there some rationale given somewhere for the continued use of plural "kelvin" (as in the CERN guidelines) or even "Kelvin" (with an apparently anomalous capital)? Or, is it -- as I suspect -- just a failure to treat the kelvin as an actual SI unit, despite the redefinition from "degrees Kelvin" to "kelvins" that happened decades ago? (Perhaps we just dropped the "degree" but effectively still treat it the same way as Celsius or Fahrenheit?) **EDIT:** Just to be clear -- I did NOT intend for this to be only a question about linguistic convention. The SI redefinition of the absolute temperature scale and units was apparently meant to refine or change some physical conception about temperature. Resistance to this change may indicate some other elements about the underlying physics (perhaps including the fact that temperature is an intensive property, as suggested in one answer below). While I'd certainly be interested in other official standards and usage recommendations, I also wanted to know if there were other PHYSICAL rationales for the inconsistent units.


Pulse Reflection

I wanted to ask why a pulse that reaches to a free end will reflect off and return with the same direction of displacement that it had before reflection. I am looking for explanation with forces.


Form of the Classical EM Lagrangian

So I know that for an electromagnetic field in a vacuum the Lagrangian is $\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu}$, the standard model tells me this. What I want to know is if there is an elementary argument (based on symmetry perhaps) as to why it has this form. I have done some searching/reading on this, but have only ever found authors jumping straight to the expression and sometimes saying something to the effect that it is the "simplest possible". Thanks for the great answers, I accepted the one that was exactly what I was looking for, but the other two long ones were equally interesting and useful.


Confused about relating spring force to maximum distance

I keep working through this problem and getting the wrong answer: > A spring with spring constant k is positioned vertically and then compressed > from its equilibrium length by a distance Δy. A ball of mass m is placed on > top of the spring, and is launched into the air when the spring is released. > The ball travels to a maximum height hmax. Using the Work-Kinetic Energy > theorem, determine hmax in terms of the given variables and any other > constants you deem necessary. The thought process was ∆K = $K_f - K_i$ = W --> $\frac{mv^2}{2}-\frac{mv_0^2}{2}$ = W $W$ = $W_s - W_g$ where $W_s$ is a spring and $w_g$ is gravity The final velocity is also zero so $K_f$ = 0. Which gives us $\frac{mv_0^2}{2} = -k∆y^2 - mgh_m$ (solve for $h_m$) $h_m = \frac{-k∆y^2}{2mg}+\frac{v_0^2}{2g}$ I might be completely off but I can't figure out what else to do. I would be very grateful is someone walked me through it ![enter image description here](http://i.stack.imgur.com/19aGI.png)


Modelling a matter dominated universe collapsing into a black hole

With the FLRW equations we can get solutions for a matter dominated closed universe in which the finale is an ultimate collapse, but this is only in terms of $a$ (the scale factor) and $t$ (time) and some $q_0$ and $\theta$: $a = {\frac {q_0}{2 q_0 -1 } } (cosh (\theta) -1) $, (1) $t = {\frac {q_0}{2 q_0 -1 } } (sinh(\theta) - \theta)$ (2) where $dt=a d\theta$ Unfortunately, the FLRW does not show that the model should collapse into a black hole. So it only tells you when things might end depending on $q_0$. So I will appreciate if someone can give me an opinion of whether the following solution is appropriate for a universe that has collapsed into a black hole. Please feel free to rip it apart if necessary. And secondly, how can the final black hole equation be factored into the FLRW solution to get a complete model? Here it goes. First the energy of the matter is simply: $E_{m}= M c^2$ (3) The gravitational potential energy $U$ of the universe is (see http://arxiv.org/abs/1004.1035): $U = - \frac {3 G M^2} {5 R}$ (4) For a particle away from the event horizon the Newtonian gravity is simply: $g r^2 = G M $ (5) where $r$ is the radial coordinate of the observer. If I assume that $U = E$ once the universe has completely collapsed then I get, $Mc^2 = - \frac {3 G M^2} {5 R}$ (6) Substituting (5), I get: $c^2 = - \frac {3 r^2 g}{5R}$ (7) And finally, $R = - \frac{3}{5} \frac {r^2 g}{c^2}$ (8) Now it turns out that the formula for the Schwarzschild radius in Newtonian gravitational fields is remarkably similar http://en.wikipedia.org/wiki/Schwarzschild_radius): $R_s = 2 \frac {r^2 g} {c^2}$ (9) The obvious issue here is that I got the constant $-3/5$ instead $2$. So the real question now is, are these computations correct and there is a reason for the difference? (for example, a supermassive BH Universe is not the same as your conventional BH). Or is it this a completely wrong approach that yields a completely wrong answer? If wrong, what is the right answer? Note that I tried to avoid solutions with $M$ and $G$ because that results in absurd answers such as those arrived in If the universe were compressed into a super massive black hole, how big would it be? (i.e., the radius of the BH would turn out larger than the current radius of the Universe, etc, etc).