Does friction act in the same way in different directions?

> A physics textbook is launched up a rough incline with a kinetic energy of > $X$ joules. When the book comes momentarily to rest near the top of the > incline, it has only gained ($X-20$) joules of gravitational potential > energy. About how much kinetic energy will it have when it returns to the > launch point? (Hint: Remember friction and conservation of energy). **Attempt** : So when the textbook goes up and reaches the point where it is at rest its GPE is $X$ joules. When it goes down and reaches the starting point its kinetic energy should by $X$ joules, I guess. Thats what I am not sure about. Does the friction act with the same force when the textbook goes down?


Resources for QCD colour factors

I'm looking for resources on the computation of (I think) very simple Feynman diagrams for QCD processes. The processes are q'qbar' -> qqbar and gg->qqbar (only at tree level, no loops involved) and my main problem is the computation of the colour factors. I'm not interested in the absolute value of the cross-section, but in the angular distribution (in terms of cos theta). At least for the gg->qqbar process I think I need the correct factors though, to correctly add the diagrams. I've already looked into the Quantum Field Theory book by Peskin, but there I only found the final cross-sections but not the calculation.


Can frequency be equal to 0?

Is correct to speak about frequency equal to 0 ? $$f= \frac{1}{t} $$ If $t\rightarrow\infty$ can I consider that the frequency is equal to 0 ?


Flow of freezing liquid or a melting solid

If a liquid is freezing, is equation of continuity violated? As the liquid flows, some portion of it is getting frozen. The mass of the fluid thus keeps dropping. Similarly, when a molten fluid flows over a solid, its mass may increase because the solid over which it flows also melts. In either of these flows, can we assume that div **u** = 0, where **u** is the velocity vector.


Are the Kaluza Klein Particles and Mini Black Holes associated with the ADD and Randal-Sundrum models different names for the same class of particles or are they distinct.


Should a 1D Guassian wave oscillate?

I wrote a few lines that numerically solve Maxwell's equations. The result is a moving wave that looks like a single pulse. This looks strange to me because I expect waves to move in oscillator fashion, perhaps like cos(x). Why doesn't the wave appear to oscillate? How does this kind of Gaussian wave work? Can it be broken into solutions? Do similar solutions exist in 2D? Are they still Gaussian?               %%Super basic FDTD     clear all;     gsz=100;     ez=zeros(gsz);hy=zeros(gsz);     imp=377.0;     %     steps = 100;     for i=1:steps         for j=1:(gsz-1)             hy(j)=hy(j)+(ez(j+1)-ez(j))/imp;         end         for j=2:gsz             ez(j)=ez(j)+(hy(j)-hy(j-1))*imp;         end         ez(gsz)= ez(gsz)+exp(-((i-30)^2) /100);         subplot(2,1,1);         plot(ez);         subplot(2,1,2);         plot(hy);         M(i)=getframe;     end


What properties are entangled in quantum entanglement?

When two particles are entangled, one can measure the properties of one of the particles and instantaneously know the properties of the other. This is because the two particles possess the same properties when entangled, no matter how far apart. Which properties are entangled? Is it just fundamental properties, such as spin, or can it be instantaneous/changing properties, such as the speed of the particles? It may seem very odd to suggest that speed could be entangled, but I think it's a reasonable question to ask as quantum entangles is plenty weird anyway.


Why doesn't a marble rolling on a table ever reflect back at the edge?

In light of the fact that when encountering a potential step there is a non- zero probability of a particle reflecting back, I was just wondering why it is that a marble doesn't "reflect" back from the edge of a table, but rather rolls off. Is it because the potential difference is extremely insignificant when compared to the total energy of the marble (so not just kinetic, but also internal energy), so that the reflection coefficient essentially evaluates to 0?


Atomic physics through classical resonance?

I have a rather general question regarding the theory of Quantum Mechanics. To preface this question, consider a violin string. When a violinist exposes the string to a bow, this is exposing the string to a wide range of frequencies. In response to this excitation, the string resonates at rather distinct frequencies. In other words, the energy spectrum of a violin string can be understood as a linear combination of distinct energies(frequencies). Why would it not be possible to develop an accurate atomic physics model based on resonant orbital electronic frequencies including possibly spin-orbit resonances as well?


Mass generation by Chern-Simons theory

Why the mass generation via a Higgs mechanism is different from that of Chern- Simons theory? I haven't done any formal course in Quantum field theory,so how do I understand this just having some basic knowledge in Classical field theory?


Why does Pluto's orbit cross Neptune's orbit? Is this a fault in Newton's law of gravitation?


Realistic Ocean Shears

I was wondering if anyone could point me to published data on the vertical structure of strong ocean currents, such as the Gulf Stream. I am primarily interested in the mean current profile as a function of depth (in the core of the current) near the surface, which go on to estimate vertical shears.


Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term beyond all orders of a saddle point expansion (singular terms of an asymptotic series), like in the problem of the lifetime of a bound state in 1+0 negative coupling $\phi^4$ toy model. Consider a particle with an initial (normalized) wave-function $$\psi_0(x) = e^{-(x+e^{-x})/2}$$ This specific shape defines a natural unit for $x$, note the double-exponential asymptotics of $\psi_0(x)$ as $ x \to -\infty$. Time evolution under the Hamiltonian $\mathcal{H}=-\frac{1}{2}\partial_x^2 $ transforms the wave-function to (using the textbook propagator) $$\psi(x,t) = (2 \pi i t)^{-1/2} \int e^{i (x-x')^2/(2t)} \psi_0(x') d x'$$ **My question is about the asymptotics of this integral** , especially the leading front propagating to the left. Here is where I've hit the wall: The saddle point expansion in $t^{-1}$ gives $$t |\psi(x,t)|^2 \sim e^{-e^{-x}} \left [1 + (e^{3x} -2 e^{2x}) \, t^{-1} /8 + O(t^{-2}) \right ] $$ which converges nicely (checked numerically) for $x \gtrsim 1$, but fails to capture the terms of order $e^{x/t}$ that dominate over the double exponential at negtavie $x$. For $t \to +\infty$ the solution becomes symmetric, $$|\psi(x,t \to \infty)|^2=\frac{1}{t \cosh (\pi x/t)}$$ Any ideas/hints will be appreciated.


Matrix geometry for F-strings

A stack of N D-branes has the strange property that the traverse D-brane coordinates are matrix-valued. When the matrices commute, they can be interpreted as ordinary coordinates for N indistinguishable objects. But in general they correspond to something classical geometry doesn't describe. In particular this leads to a non-perturbative description of string theory on asymptotically Minkowski spacetime: Matrix Theory. S-Duality exchanges F-strings and D1-strings. This means this odd "matrix geometry" should happen for F-strings as well. The question is, how can we see it directly, without invoking S-duality?


Tension on pulleys Physics Question

Collar A is connected to a 50lb load on a frictionless horizontal rod. Determine magnitude of P to maintain equilibrium when x=4.5. ![enter image description here](http://i.stack.imgur.com/mylsw.png) Im confused on the concept of Tension I originally got the answer 11.25 lb like the images below but my solution manual is different. ![enter image description here](http://i.stack.imgur.com/405Nx.png) The solution manual says: $$\tan \alpha= 20/4.5 = 77.3 d^\circ$$   sum of F sub X=0   $$-P + T \cos 77.3 $$   $$P= 50lb (cos 77.3)$$   $$= 10.98lb$$ (answer manual) Is this because the tension in the rope is equivalent to the mass of the hanging weight? Is there a way to solve for 10.98lb using method like the one below ...i.e. adding the vector components and solving for the unknowns? confused especially on how the solution manual is using 50lb for magnitude of T and when using vector addition getting the 11 so lbs. Thanks! but this one below is what makes sense to me. http://imageshack.us/photo/my-images/15/yci9.png/   http://imageshack.us/photo/my-images/534/y53e.png/


Why doesn't a typical beam splitter cause a photon to decohere?

In many experiments in quantum mechanics, a single photon is sent to a mirror which it passes through or bounces off with 50% probability, then the same for some more similar mirrors, and at the end we get interference between the various paths. This is fairly easy to observe in the laboratory. The interference means there is no which-path information stored anywhere in the mirrors. The mirrors are made of 10^20-something atoms, they aren't necessarily ultra-pure crystals, and they're at room temperature. Nonetheless, they act on the photons as very simple unitary operators. Why is it that the mirrors retain no or very little trace of the photon's path, so that very little decoherence occurs? In general, how do I look at a physical situation and predict when there will be enough noisy interaction with the environment for a quantum state to decohere?


No magnetic dipole moment for photon

Electrically neutral particles such as neutrinos can have nonvanishing magnetic dipole moments. Spin-1 particles, e.g., deuterium nuclei, can also have dipole moments. Googling seems to show that the Z boson has a magnetic moment. So is there an elementary argument that explains why the photon has a zero magnetic moment? By analogy with the Bohr magneton and nuclear magneton, we might actually expect that zero mass would produce _infinite_ dipole moment. Altschul 2007 has some discussion of empirical bounds and difficulties in creating a theory in which the moment doesn't vanish, but I'm having trouble translating anything in the paper into an elementary argument for why it's so much easier to have the moment vanish. > Altschul, "Astrophysical Bounds on the Photon Charge and Magnetic Moment." > _Astropart. Phys._ **29** no. 4, pp. 290–298 (2008), arXiv:0711.2038.


Why Weyl invariance is important for consistent string theory

This post is related to this link. I know there is a Weyl invariance for the Polyakov action at least in classical level. My question arises from obtaining effective action in string theory, such as section 7.3 in this lecture note > A consistent background of string theory must preserve Weyl invariance, > which now requires $\beta_{\mu\nu}(G)=\beta_{\mu\nu}(B)=\beta(\Phi)=0 $ later the lecture note tried to find effective action by requiring vanishing beta function. Also in p110 in Polchinski's string theory vol I, > We have emphasized that Weyl invariance is essential to the consistency of > string theory. Why Weyl invariance is so important for consistency of string theory?


Path of least resistance vs. short circuit

Some sources on the web claim that "electricity follows the path of least resistance" is not true, e.g. this physics SE question. However, in every explanation of "short circuits", the author says that current flows through the short because it's following the path of least resistance. How do I reconcile these two facts?


How to prove Galilean invariance?

**Thinking this would be obvious, I was trying to prove the Galilean invariance of Newton's second law of motion, but I failed. This is what I've got so far:** If we define a world line in Galilean space-time $\mathbb{R}^{4}$ as the following curve $$\bar{w}\colon I\subset\mathbb{R}\to \mathbb{R}^{4}\colon t\mapsto (t,\bar{x}(t))$$ $$\bar{x}\colon I\subset\mathbb{R}\to \mathbb{R}^{3}\colon t\mapsto (x(t),y(t),z(t))$$ where $\mathbb{R}^{3}\subset\mathbb{R}^{4}$ Euclidean, then the acceleration is given by $$\bar{a}\colon I\subset\mathbb{R}\to \mathbb{R}^{4}\colon t\mapsto \frac{d^{2}\bar{w}(t)}{dt^{2}}=(0,\frac{d^{2}\bar{x}(t)}{dt^{2}})\equiv(0,\tilde{a}(t))$$ where $\tilde{a}$ the classical acceleration and the force field that causes the acceleration $$\bar{F}\colon\mathbb{R}^{4}\to\mathbb{R}^{4}\colon\bar{w}(t)\mapsto m\bar{a}(t)=(0,m\tilde{a}(t))$$ So if Newton's second law of motion is written as $\bar{F}(\bar{w}(t))=m\bar{a}(t)$ then a Galilean transformation causes $\bar{F}=G\bar{F}'$ and $\bar{a}=G\bar{a}'$ since they both live in $\mathbb{R}^{4}$. Therefore $$\bar{F}(\bar{w}(t))=m\bar{a}(t)$$ $$\Leftrightarrow G\bar{F}'(\bar{w}(t))=mG\bar{a}'(t)$$ $$\Leftrightarrow \bar{F}'(\bar{w}(t))=m\bar{a}'(t)$$ which is what we needed to prove (if F=ma in the unprimed frame then F'=ma' in the primed frame). Note that this is analogue to how Lorentz invariance is shown in special relativity. **However $\bar{a}$ doesn't transform like $\bar{a}=G\bar{a}'$** A general Galilean transformation $G$ in $\mathbb{R}^{4}$ is given by $$t=t'+t_{t}$$ $$\bar{x}=R\bar{x}'+\bar{u}t+\bar{t}_{\bar{x}}$$ The relation between velocity and acceleration before and after a Galilean transformation is given by $$\bar{v}(t)=\frac{d\bar{w}(t)}{dt}=\frac{d\bar{w}(t)}{dt'}\frac{dt'}{dt}=\frac{d\bar{w}(t)}{dt'}= (1,R\frac{d\bar{x}'}{dt'}+\bar{u})$$ $$\bar{a}(t)=(0,\tilde{a}(t))=\frac{d\bar{v}(t)}{dt}=(0,R\frac{d^{2}\bar{x}'}{dt'^{2}})=(0,R\tilde{a}'(t'))$$ $$\Leftrightarrow\tilde{a}(t)=R\tilde{a}'(t')$$ This is not the same as $\bar{a}=G\bar{a}'$ because $$\bar{a}(t)=G\bar{a}'(t')$$ $$\Leftrightarrow\begin{pmatrix} 0 \\\ \tilde{a}(t) \\\ 1\end{pmatrix} = \begin{pmatrix}1&0&t_{t}\\\ \bar{u}&R&\bar{t}_{\bar{x}}\\\ 0&0&1 \end{pmatrix}\cdot\begin{pmatrix} 0 \\\ \tilde{a}'(t') \\\ 1\end{pmatrix}$$ So the acceleration of a world line transforms not with $G$ but with the linear part of $G$, meaning that the translation part must be zero: $(t_{t},\bar{t}_{\bar{x}})=\bar{0}$. Can someone help me out of this mess? **Edit:** Let me try again, this time forgetting that we're talking about forces and just consider a 3D vector field. Of course Galilean space-time is still 4-dimensional and a general Galilean transformation $G$ in $\mathbb{R}^{4}$ is still given by $$t=t'+t_{t}$$ $$\bar{x}=R\bar{x}'+\bar{u}t+\bar{t}_{\bar{x}}$$ and the relation between the classical acceleration in inertial frames (primed and unprimed) related by a Galilean transformation $G$ is still given by $$\tilde{a}(t)=R\tilde{a}'(t')$$ One could say that the acceleration transforms with $R$ if the frame transforms with $G$ because $\tilde{a}$ lives in the associated vector space of Galilean space time (therefore the affine translation $(t_{t},\bar{t}_{\bar{x}})$ doesn't apply) and moreover lives in the 3D Euclidean subspace of this vector space $\mathbb{R}^{3}\subset\mathbb{R}^{4}$ (therefore the boost $\bar{u}$ doesn't apply). Suppose now that we define a 3D vector field on a world line as $$\bar{F}\colon C\subset\mathbb{R}^{4}\to\mathbb{R}^{3}\colon\bar{w}(t)\mapsto m\tilde{a}(t)$$ In this case, by the same reasoning as for the acceleration, we can say that the vector field (defined in the unprimed frame) transforms with $R$ if the frame transforms with $G$ $$\bar{F}(\bar{w}(t))=R\bar{F}'(\bar{w}'(t'))$$ If we use this together with $$\bar{F}(\bar{w}(t))=m\tilde{a}(t)=mR\tilde{a}'(t')$$ it follows that $$\bar{F}'(\bar{w}'(t'))=m\tilde{a}'(t')$$ So if we define a vector field as $\bar{F}(\bar{w}(t))=m\tilde{a}(t)$ (forget that we're talking about force) and since both sides live in the same space ($\mathbb{R}^{3}$), then they also transform in the same way under a Galilean transformation (whatever this way is, in this case $R$). Therefore it doesn't matter in which frame we define the vector field on a world line as the acceleration multiplied by the mass, it will have the same form in all frames. Therefore we could say that the definition of the vector field is Galilean invariant. The problem I'm having is that one says that "Newton's second law of motion is Galilean invariant". This implies that it is invariant, regardless the nature of the force. So if we make the nature of the force abstract, we can just forget that we're talking about force and consider a 3D vector field. Then everything boils down to showing that the definition has the same form in all inertial frames, which it has as shown above. Is this a valid point of view?


Problem evaluating $C^{-1}M^\dagger C$

How can I show the following? $$\overline{\psi_L}M^\dagger (\psi_L)^c=\overline{\psi_L}CM^\dagger\overline{\psi_L}^T$$ where $\psi^c=C\overline{\psi}^T$ and $C=i\gamma^2\gamma^0=-C^T=-C^\dagger=-C^{-1}$. In particular, what is $C^{-1}M^\dagger C$? I'm stuck at this point. If I can show, $$C^{-1}M^\dagger C=M^\dagger$$, the job is done. EDIT: http://arxiv.org/abs/hep-ph/0001264 see equation 24 of this review


Lagrangian approach to spinning thread reel

I am trying to better understand Lagrangian dynamics and am struggling to complete the following question: > A reel of thread of mass $m$ and radius $r$ is allowed to unwind under > gravity, the upper end of the thread being fixed. Find the initial > acceleration of the reel.                                  ![Diagram of spool and thread](http://i.stack.imgur.com/EWSJq.png) I believe there are three generalised co-ordinates here $(x,y,\theta)$, as shown in the diagram and the constraint that $x=0$ (as there is no acceleration in the $x$-direction). We therefore have kinetic energy: $$T=\frac{1}{2}m\dot{y}^{2}+\frac{1}{2}m r^{2}\dot{\theta}^{2}y \cos(\theta)$$ And potential energy given by: $$U=-mgy$$ We therefore can define the Lagrangian: $$\mathcal{L}(y,\theta,t)=\frac{1}{2}m\dot{y}^{2}+\frac{1}{2}m r^{2}\dot{\theta}^{2}y \cos(\theta) + mgy$$ We therefore have the Euler-Lagrange equations: \begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial \mathcal{L}}{\partial \dot{y}}\right) &= \frac{\partial \mathcal{L}}{\partial y} \\\ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial \mathcal{L}}{\partial \dot{\theta}}\right) &= \frac{\partial \mathcal{L}}{\partial \theta} \end{align} Expanding these we get: $$m\ddot{y}=\frac{1}{2}mr^{2}\dot{\theta}^{2}\cos(\theta) + mg$$ And: $$mr^{2}\ddot{\theta}y\cos(\theta)-mr^{2}\dot{\theta}^{2}\sin(\theta)+mr^{2}\dot{\theta}\dot{y}\cos(\theta)=0$$ Solving both of these simultaneously gives: $$\ddot{\theta}=\frac{\dot{\theta}^{2}\tan(\theta)-\dot{y}\dot{\theta}}{y},\quad \ddot{y}=\frac{1}{2}\left(2g + r^{2}\dot{\theta}^{2}\cos(\theta)\right)$$ Which I am unable to solve to yield anything useful; and therefore am assuming I am on the wrong track. I would be grateful for any pointers as to where I have misunderstood things. * * * Just to clarify what the issue is, given the expression above for $\ddot{y}$ and the boundary conditions $y_{0}=0$, $\dot{y}_{0}=0$, $\theta=0$ and $\dot{\theta}=0$ we get: $$\ddot{y}_{0}=g$$ Which is what I expected, however the answer to the question states that $\ddot{y}_{0}=\frac{2g}{3}$.


about orthogonal catastrophe

I am reading Wen's book, QFT of many-body systems ( @Xiao-Gang Wen ). I am a little confused about the orthogonal catastrophe introduced in Chap.5. Below Eq.(5.1.6), it is stated that ``the influence of the core electron on the _conduction electron_ Green's function is not important...''. If I understand correctly, it is exactly the (almost) vanishing overlap between the initial and final state wavefunctions of the _conduction electrons_ that causes the orthogonal catastrophe. Why does the asumption that ``the influence of the core electron on the _conduction electron_ Green's function is not important'' works? Thanks in advance for any help.


Vector decomposition validity

**Is force or field decomposition into component vectors always valid?** Lets say a constant electric field $\vec{F}$ is acting in space such that it makes an angle $\phi$ with respect to the horizontal direction. The component along x axis (horizontal direction) is $F\cos(\phi)$ and along the vertical $F\sin(\phi)$. Is there any assumption that the space is homogeneous or any other such (uniform space) condition while applying decomposition? OR is it like decomposition of vectors always refers to a single point in space, and doesn't matter upon the nature of medium/space?


What does a supernova look like at its peak luminosity?

I know that in some types of supernovae, the cause of the increased luminosity is the radioactive decay of certain elements ejected during the explosion, so a question came to my mind. If the ejected material carrying the isotopes that decay to give the electromagnetic radiation is expelled at velocity of say 5% the speed of light, and given the fact that some supernovae stay extremely luminous for more than 4 weeks, then by that time the radioactive isotopes will have traveled more than 30 billion kilometers from the exploded star. So, does that mean a supernova at 4 weeks can be expected to look like a star with a radius of 30 billion kilometers and luminosity of $10^8$-$10^9$ times the solar luminosity ? Or I am getting the idea of the radioactive decay as the source of the supernova luminosity wrong ?


Is it possible to create an audible sound source in mid air by intersecting ultrasonic sound beams?

In the book _Daemon_ by _Daniel Suarez_ a technique gets described, that enables a device to create an acoustical illusion of a moving person inside a house. > Just then a voice called out clearly from the end of the hallway upstairs. > ‘Who’s there?’ […] No reply. But they heard walking again. The footsteps > came down the marble stairs to their right, some distance away from them. > They could clearly see the staircase, but no one was there. They could hear > the sound of a hand sliding down the metal railing. […] The footsteps were > moving across the floor to them now. […] Then, **in midair not five feet in > front of them** , a man’s voice shouted, ‘You don’t belong here!’ _(page > 94)_ As I searched the internet for more information on the mind-shaking technology described by Suarez, I just encountered papers (and videos) on the use of a single hypersonic sound beam, that creates audible sound where it hits a surface. This leads to an effect that most people describe as a 'god voice' similar to the experience of the use of bone conducting speakers. _However what I am searching for is an answer to the question, if it is possible to create sound in mid air (at a chosen position in 3D space) as the author is shaping the picture of the technology, cited below._ > A HyperSonic Sound system – or HSS – does not use physical speakers. HSS > pulsates quartz crystals at a frequency thousands of times faster than the > vibrations in a normal speaker – creating ultrasonic waves at frequencies > far beyond human hearing. Unlike lower-frequency sound, these waves travel > in a tight path – a beam. Two beams can be focused to intersect each other, > and where they interact they produce a third sonic wave whose frequency is > exactly the difference between the two original sounds. In HSS that > difference will fall within the range of human hearing – **and will appear > to come from thin air**. This is known as a **Tartini Tone** – in honor of > Giuseppe Tartini, the eighteenth-century Italian composer who first > discovered this principle. * * * PS: The reason why I even bother to search for information about a technology mentioned in a novel, is the author's concern, to just describe technology, that is in development in some lab and not placed in science fiction. He has a website for the description of some technology used in his book.


What's a good reference for this classical picture Feynman's talking about?

I have a mathematics background but am trying to educate myself a little about physics. At the beginning of Feynman's QED book (not the popular one) is the following: > Suppose all of the atoms in the universe are in a box. Classically the box > may be treated as having natural modes describable in terms of a > distribution of harmonic oscillators with coupling between the oscillators > and matter. I guess this is something that physicists learn, but I have never heard of it. What is Feynman talking about and where can I learn more about it? The Wikipedia article on harmonic oscillators gives no indication that physicists do this.


Hamiltonian of oscillators quantized proof

https://docs.google.com/open?id=0BxrBcN1-BZWUOXNxR1l4S0l2MjQ http://www.2shared.com/complete/Qjy1_uzp/Quantum_Mechanics_in_Simple_Ma.html (I uploaded a pdf file that contains the parts of the textbook that I am having some difficulty understanding. Seven pages.) (two links point to the same pdf.) A matrix $N$ has all entries except diagnoal ones zero, and first row, first column also zero, other diagonal entries depending on its column, row place; for example, second row, second column has one as its entry, third row, third column has two as its entry and so on. It is known that $H = \hbar\omega (N + \frac{1}{2}) $. The question is, my textbook says that if there are a continuous range of possible values, then for some states, $\langle (N - \langle N \rangle )^2 \rangle \leq [\frac{1}{2} (n - \langle N \rangle)]^2$ , where $n = 0,1,2,3,...,$ then progresses to say that since $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$ then $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2$. Then it says that this proves that there cannot be a continuous range of possible values, as $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$. I am not sure how one gets $\langle (N - \langle N \rangle )^2 \rangle \geq (n - \langle N \rangle)^2 (\langle I_0 \rangle + \langle I_1\rangle + ...)$. Can anyone show me the proof? Also, how does one get the part - $(0-\langle N \rangle)^2$, $(1 - \langle N \rangle)^2$ and so on cannot be smaller than $(n - \langle N \rangle)^2$?


Continuum model of the n-body problem?

Consider a collection of $n$ point masses evolving under the influence of gravity. For very large $n$, one might be interested in making a continuum approximation of this system (for cosmology, say). However, my naïve attempt to derive a continuum model directly from particle-particle interactions fails. In particular, consider a function $\rho: \Omega \rightarrow \mathbb{R}$ that describes the mass density over some domain $\Omega$. To compute the gravitational force at a point $x \in \Omega$, I might try to simply integrate the force contributed by every other point $y$: $$ f(x) = \int_\Omega \frac{\rho(x)\rho(y)}{|x-y|^2}\frac{y-x}{|y-x|} $$ Even if we exclude the point $x$ (i.e., we integrate over $\Omega \backslash x$ instead of all of $\Omega$) this integral diverges -- we effectively have particles arbitrarily close to $x$. So the question is: what's the right way to compute the gravitational force in this situation? It would also be nice to have some pointers to continuum models used in cosmology **_and their derivation from the particle model_** (many references simply start with a continuum model and do not explain its origin). By the way, I really only care about classical models -- let's forget about relativity, etc., for now. Thanks!


Do higher frequency EM waves travel longer than lower EM frequency waves?

In radio communication, the distance traveling seems to be an important parameter before we choose the frequency band of communication. There is also a path-loss model that describes a distance $\leftrightarrow$ frequency dependence of the attenuation with an isotropic antenna. However, I fail to find any specific physical law that demonstrates the frequency dependence of the attenuation. Some even say that higher the frequency, higher the energy content. What are the factors for propagation distance?


The appearance of volume $V$ in the Fourier series representation of a periodic cubic system

In the textbook _Understanding Molecular Simulation_ by Frenkel and Smit (Second Edition), the authors represent a function $f(\textbf{r})$ (which depends on the coordinates of a periodic system) as a Fourier series. I quote from page 295 of the text: > Let us consider a periodic system with a cubic box of length $L$ and volume > $V$. Any function $f(\textbf{r})$ that depends on the coordinates of our > system can be represented by a Fourier series: > > $$f(\textbf{r}) = \frac{1}{V} \sum_{\boldsymbol{\ell} = -\infty}^{\infty} > \tilde{f}(\textbf{k}) e^{i \textbf{k} \cdot \textbf{r}} \; \; \; \; > \textbf{(12.1.6)}$$ > > where $\textbf{k} = \frac{2\pi}{L}\boldsymbol{\ell}$ with $\boldsymbol{\ell} > = (\ell_x, \ell_y, \ell_z)$ are the lattice vectors in Fourier space. The > Fourier coefficients $\tilde{f}(\textbf{k})$ are calculated using > > $$\tilde{f}(\textbf{k}) = \int_V d\textbf{r} \; f(\textbf{r}) > e^{-i\textbf{k} \cdot \textbf{r}} \; \; \; \; \textbf{(12.1.7)}$$ Now, the authors use equation **(12.1.6)** to write the electric potential $\phi(\textbf{r})$ in Fourier space: $$\phi(\textbf{r}) = \frac{1}{V} \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}}$$ The authors write: > In Fourier space, Poisson's equation has a much simpler form. We can write > for the Poisson equation: > > $$-\nabla^2 \phi(\textbf{r}) = -\nabla^2 \left( \frac{1}{V} > \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}} > \right) = \frac{1}{V} \sum_{\textbf{k}} k^2 \tilde{\phi}(\textbf{k}) > e^{i\textbf{k} \cdot \textbf{r}} \; \; \; \; \textbf{(12.1.8)}$$ My question is, why is the $\frac{1}{V}$ factor present in equations **(12.1.6)** and **(12.1.8)**? What is the significance of the $\frac{1}{V}$ factor in $\phi(\textbf{r}) = \frac{1}{V} \sum_{\textbf{k}} \tilde{\phi}(\textbf{k}) e^{i\textbf{k} \cdot \textbf{r}}$? In contrast, the article on Wikipedia does not include this prefactor. I realize that that article is dealing with the general case, whereas here we are considering a system with a cubic box of volume $V$. But shouldn't the units of $\phi(\textbf{r})$ be the same as those of $\tilde{\phi}(\textbf{k})$? The $\frac{1}{V}$ seems to preclude $\phi(\textbf{r})$ and $\tilde{\phi}(\textbf{k})$ having the same units. Do you have any advice? Thanks.


What does imaginary number maps to physically?

I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, Circuits, Control Theory and now Quantum Mechanics, and it seems that I always understand the math, but fail to grasp the concept in terms of its physical mapping. Hence my question, what does imaginary number maps to physically? Any help would be much appreciated


Why does aperture size affect depth of field in photography?

Why do small and big holes (the aperture) in a lens change the depth of field in the image? (if you have big hole you have smaller depth of field). I wondered if I should ask this on photo.stackexchange.com, but then realize that I want optic based answer which I think will get here.


Higgs field questions

The Wikipedia article about Higgs field poses some questions to me.   * The article says that the Higgs field is a "nonthermal" field, a field whose energy does not decrease as the universe expands, yet it claims that in the early universe Higgs field was stronger which led to inflation. How the field's strength could decrease then?   * If the Higgs field was stronger in early universe, why the article claims that elementary particles had no mass at early stages of the universe?   * The article claims that Higgs field is repulsive and is behind the inflation. Yet it does not interact with photons. Would not that make the photons not to follow the universe's expansion so that we would see distant stars not actually in the direction where they should be?   * The article claims that all electrons in neutron beta-decay are left-handed only for a stationary observer while for a moving observer this changes. Does it mean that one can measure his speed relative to the Higgs field by counting the right-handed electrons in neutron's decay?   * The article claims that in absence of Higgs field electrons would have no rest mass. But I cannot imagine a charged particle without rest mass because the electromagnetic field has potential energy. Does it mean that in absence of Higgs field electrons are chargeless?


Observation of cosmological topological defects

Are there any projects/experiments running for the observation of topological defects (domain walls, monopoles, etc)? Are there any _past/canceled_ or _future_ such projects?


Trying to rhyme Peskin and Schroeder with Weinberg

This is a follow up question of this one. In the Vol 1, Weinberg derives how a unitary operator $U(\Lambda)$ acts on one-particle states, which is given by equation (2.5.2): \begin{equation} U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,p)|\Lambda p, \sigma'\rangle \end{equation} Am I correct in thinking that Peskin and Schroeder use this result to derive the Lorentz transformation on a annihilation operator of the quantized Dirac field (given by equation (3.106)): \begin{equation} U(\Lambda) a^s_{\mathbf{p}}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a_{\Lambda \mathbf{p}}^s \end{equation} However, Peskin and Schroeder are _only_ considering Lorentz transformations that do not mix states with different spin indices! In other words, the first equation I wrote becomes: \begin{equation} U(\Lambda)|p,\sigma\rangle = C|\Lambda p, \sigma\rangle \end{equation} where $C$ is normalization constant which can be determined to be $C=1$. If this is not correct, can you please let me know how they (Peskin and Schroeder) have derived equation (3.109)? I hope this question is not too confusing. Edit: for you to see my reasoning. We can now write (for $C=1$): \begin{equation} U(\Lambda)|\mathbf{p},s\rangle = \sqrt{2 E_{\mathbf{p}}} U(\Lambda)a^{s\dagger}_\mathbf{p}U^{-1}(\Lambda)|0\rangle=|\Lambda \mathbf{p},s\rangle = \sqrt{2 E_{\Lambda \mathbf{p}}} a^{s\dagger}_{\Lambda \mathbf{p}}|0\rangle \end{equation} which implies: \begin{equation} U(\Lambda)a^{s\dagger}_\mathbf{p}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a^{s\dagger}_{\Lambda \mathbf{p}} \end{equation} Subsequently, using the unitarity of $U(\Lambda)$, we can take the adjoint of the above equation: \begin{equation} U(\Lambda)a^{s}_\mathbf{p}U^{-1}(\Lambda) = \sqrt{\frac{E_{\Lambda\mathbf{p}}}{E_{\mathbf{p}}}} a^{s}_{\Lambda \mathbf{p}} \end{equation} which is equation (3.109).


Some hints for special case of metric tensor in GR

Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + \partial_{y}^{2}f = 0$. The straightforward method is obvious, but does some method especially for this metric exist? Edit. As I see, first is replacing the variables as $$ u = (t - z), \quad v = (t + z). $$


Dirac paper quoted at Wikipedia

From Wikipedia's http://en.wikipedia.org/wiki/Uncertainty_principle: > In 1936 Dirac offered a precise definition and derivation of the time-energy > uncertainty relation in a relativistic quantum theory of "events". The citation to the paper is missing. Can anyone provide the title of this paper and where can I find it? Thanks.


Do strong and weak interactions have classical force fields as their limits?

Electromagnetic interaction has classical electromagnetism as its classical limit. Is it possible to similarly describe strong and weak interactions classically?


Mixed state after measurement

I'm looking at Section 2.4.1 of Nielsen and Chuang's _Quantum Computation and Quantum Information_ were they derive the density operator versions of the evolution and measurement postulates of quantum mechanics and something is bugging me. Let $$\\{(p(i), | x_i \rangle) \colon i=1,2,...,n\\}$$ be an ensemble and suppose you perform a measurement on this ensemble that results in outcome $m$. Then the post-measurement ensemble is $$\\{(p(i|m), | x'_i \rangle) \colon i=1,2,...,n\\}$$ where $p(i|m)$ is the probability that the state of the system was originally $| x_i \rangle$ given that the measurement outcome was $m$. It seems natural enough to use $p(i|m)$ for the probabilities of the post-measurement ensemble. However, mathematically, I do not see why that should be the case. My question then is this: Given only the definition of what an ensemble is and the state vector version of the postulates of QM, is there a way to derive the rules to compute the post-measurement probabilities of an ensemble?


How does Hubble's constant affect the Earth's orbit

If Hubble's constant is $2.33 \times 10^{-18} \text{ s}^{-1}$ and the earth orbits the sun with average distance of 150 million kilometers; Does that mean the earth's orbital radius increases approximately $11\text{ m}/\text{year}$? Does the earth's angular momentum change? If so, where does the torque come from? If the angular momentum doesn't change, does the earth's orbital velocity (length of a year) change? If so, where does the lost kinetic energy go? * * * Aside: the 11 meters per year figure comes from Hubble expansion of space the distance of the earth's orbital radius integrated over an entire year. $$(2.33 \times 10^{-18}\text{ s}^{-1}) (1.5 \times 10^{11} \text{ m}) (3.15 \times 10^7 \text{ s}/\text{year}) = 11 \text{ m}/\text{year}$$


Young's double-slit experiment with detectors

_Related:Accuracy of various optical instruments_ In many books, it's written that knowing which slit a photon passes through (by placing a detector before the slit) in a Young's double-slit setup beforehand destroys the interference (producing two bright fringes). My question is, does this require there to be only one photon in the double slit experiment at a time? The one-photon interference is explained by there being "half a photon" coming from each slit, and these interfere. If we know which slit it comes through, then it becomes a full photon in one slit and it can't interfere with itself. But can _other_ photons interfere with it? I'm quite sure that light can be either a particle or a wave in such experiments, never both. Aside from this, how was the photon detected without absorbing it? If I keep a detector behind a slit, yes, it will detect all the photons, but it won't let any past it to the main apparatus. Unless the detector was something else.. UPDATE: A clearer version of this question:   1. If you know which slit a photon went through, its wavefunction is collapsed. One can say that it ceases to be wave like. This collapsed photon cannot interfere with itself   2. Can it interfere with others? It is, after all, not really a wave anymore (it is a wave, but collapsed to a point).   3. In a normal double slit experiment (beam of light, no doodads), are photons interfering with themselves, each other, or both? (See comment: Is Time Significant in the Double Slit Experiment)


Parity of a decay

If a particle of unknown intrinsic parity decays into 2 particles each with negative intrinsic parity, does that necessarily imply that the original particle also has negative parity?


A foundational question about entanglement

The recent Nature article http://www.nature.com/news/data-teleportation-the- quantum-space-race-1.11958 prompts me to ask this question, which is of the same tenor as that asked at the recent Perimeter Institute meeting attended by Zeilinger et al., i.e. "Do the pairs somehow communicate though some still- unknown information channel?" It seems to me that entanglement could be described as behaving as if in some sense the coordinate separation is described by an extra degree of freedom (dimension) in which either a) the two entangled components are co-located in the extra dimension, allowing instantaneous communication; or b) in the extra dimension, the speed of light is no longer a limitation, and so co-location is not a requirement. ... or some mixture of the two. This could all be rephrased in terms of the concepts of "brane" and "bulk", of course. Are there many other such "unconventional" interpretations of the nature of entanglement out there?


What is the difference between a vortex ring and a circular line vortex?

Can someone explain the difference between a vortex ring and a circular line vortex? Or are they the same?


How do I find the initial velocity in this problem?

> An X-ray tube gives electrons constant acceleration over a distance of > $20\text{ cm}$. If their final speed is $2.0\times 10^7\text{ m/s}$, what > are the electrons' acceleration? I know this equation is relevant: $$v_\text{final}^2 = v_\text{initial}^2 + 2a(x-x_\text{initial})$$ Also I have that $x = .2\text{ m}$ (after conversion from cm), and $v_\text{final}$ is given. How do I find $v_\text{initial}$? I need that so I can find acceleration.


Dipping a Dyson Ring below the event horizon

The basic assumption about inescapability beyond the Event Horizon is that the necessary escape speed (orbital speed) would exceed speed of light, therefore no object can achieve it. Now, would it be possible to escape it by other means, say, by mechanical push against the gravity, through application of force without increase of speed? * * * First, short introduction: What is a Dyson Ring? (feel free to skip or skim this section if you know it): Dyson Ring is a hypothetical, artificial, ring-shaped structure surrounding a massive body (e.g. a star), rotating - essentially orbiting it - rotating at speed slightly higher or lower than orbital speed necessary to maintain orbital equilibrium - state of freefall near its surface. This way, its rotation creates centripetal force simulating gravity for inhabitants. It would create a habitat much larger than any planet, given some "rim" it could hold atmosphere, and essentially is a neat sci-fi alternative for a planet - similar conditions, vastly more surface. The one property of it is that tensile strength of its construction allows it to exist at distance from the central body (star) that is off from normal orbit matter moving at this speed would take. Most theoretized Dyson Rings move slightly faster than needed, stretched out, with the inner side inhabited, and the Sun in zenith for all of their surface. It is possible though (if not recommended due to buckling risks) to make one that moves slower, the star's gravity not fully overcome by rotation, compressive force applied to the construction. * * * Now - technological problems aside - of building a Dyson Ring (not habitable and much smaller; a scientific device of ~30km radius.) and bringing it close enough to a Black Hole. Let's imagine a Dyson Ring that can change its circumference (say, built of segments connected with expanding/contracting actuators) - and, as effect, radius within certain range - specifically between outside and inside of Schwarzschild radius of a selected Black Hole. It is also supplied with power sources that allow to vastly increase its rotary speed. Now, with the actuators expanded the ring is placed around the black hole, outside the even horizon, rotating so fast that its segment orbit the black hole. Its angular velocity so high the linear velocity of its surface is near to speed of light. Normally the centrifugal force would tear it apart, but it remains in equilibrium with centripetal force of gravity of the black hole. Now, actuators contract. The angular velocity rises a little, with accordance of law of conservation of momentum, but since we're at relativistic speeds, the increase of angular speed is minimal; it's mostly the ring gaining some mass. Now, the ring is below the Event Horizon. Since at this altitude orbital speed would exceed the speed of light, any object there would quickly spiral into the black hole. Still, the ring does not, due to its own tensile strength preventing further decrease of its own radius. It moves slower than the orbital speed, but the mechanical strain overcomes the gravity. (note, while below the Event Horizon, we're still pretty far from the singularity, so the forces exerted are not yet extreme enough to cause collapse of matter, and the rapid rotation overcomes great most of the compressing gravitational force) Expand the actuators, increase the radius of the ring - and it is forced back outside the Event Horizon, able to broadcast its findings. Possible physically, or am I missing something?


Intuitive understanding of the entropy equation

In thermodynamics, entropy is defined as $ d S = \dfrac{\delta q_{\rm }}{T}$. This definition guarantees that heat will transfer from hot to cold, which is the second law of thermodynamics. But, why do we denote entropy as$\dfrac{\delta q_{\rm }}{T}$ other than $\dfrac{\delta q_{\rm }}{T^2}$,$\dfrac{\delta q_{\rm }}{e^T}$,or something else? Is there an intuitive explanation for this $\dfrac{\delta q_{\rm }}{T}$?


quantum optics of a polarizing beam splitter

I would like to use the Heisenberg picture in quantum field theory to model a polarizing beam splitter. Is there an easy way for someone to show me how the field operators ($a^\dagger_{input1}$, $a^\dagger_{input2}$etc) transform across a polarizing beam-splitter?


What's the relationship between glow and arc discharge?

I'm trying to understand the differences between an arc and glow discharge. Most texts I have found on the Internet only describes the observed differences but doesn't really explain what happens on the atomic level in the two types of the discharge. So what I know:   * Glow discharge happens on higher voltage in the mA current range. It has recognizable structure (eg. the Faraday dark space and the Crookes dark space). It's faint and cool.   * Arc discharge happens on lower voltage, and it's in the Amp current range and higher. It runs from electrode to electrode without dark spaces. It's often blindingly bright and hot.  A characteristic of the normal glow discharge is that the voltage is stable, only the current changes, hence it's used as a voltage stabilizer. As you increase the current through the glow, it tends to cover more area of the cathode. When it covered all of it, the voltage starts to increase to supply more and more energetic electrons, this is the abnormal glow. **What happens at point`I` in the graph below that causes the voltage drop again?** ![enter image description here](http://i.stack.imgur.com/6CYxA.png) Glow discharge is often demonstrated with discharge tubes. Countless of youtube videos show amateur experimenters pumping down their tubes, causing initially an arc discharge which turns into a glow discharge as the tube pumped down, **what causes this mode transition?** Whenever you turn on a compact fluorescent bulb it glows for a second then the discharge turns into an arc as the cathodes heat up and the bulb brightens up. I know hot electrodes emit more electrons, but **why more electrons cause transition into the arc mode?** I suppose it's due to the same reason for all cases. Can anybody help me put the things together, so I can get the point?


Quantum mechanical proof of Conway's SPIN and TWIN axioms?

I think I understand correlations of EPR-state and GHZ-state which deal with spin-1/2 particles and (-1, 1) measured values. Conway's state is spin-1 particle state with (-1, 0, 1) measured values. Which quantum mechanical state vector correspond to Conway's state? And how to prove SPIN and TWIN axioms of Conway's free will theorem from quantum mechanics?


What is limiting line in series spectrum?

> The wavelength of first line in the Balmer Series is 'whatever(in nm)' . > Calculate the wavelength of the second line and the 'limiting line' in the > Balmer Series. I found this question in an ancient question paper in the library. Now, I have solved the first part by calculating the atomic number from the first relation and then applying it while calculating the wavelengths of the second line in the Balmer series which must mean the line after Balmer (which is paschen). **But for the second part 'what is limiting line', and that's exactly what my question is. Also since the given wavelength is in 'nanometer' what whould be the unit of energy if found.** I certainly don't need any solution for the above noted question( from the question paper), so this question cannot be declared as an 'homework question' due to some weird rules, (ofcourse they are meant for a mass good, but the tag name 'homework' hurts, anyway don't want that to be a topic of discussion here, but please consider it.)


Microscopic definition/expression of/for the heat current

Often I see the following microscopic definition/expression of/for a heat current due to an external field $$ {\bf j}_Q = 2 \int \frac{\text{d}{\bf k}}{(2\pi)^3} \frac{\hbar {\bf k}}{m} (\epsilon_{{\bf k}} -\mu) f(\epsilon_{{\bf k}}), $$ where $f$ is the solution the the appropriate Boltzmann equation. Note that the factor of two is there for the spin of electrons. I do not really understand where this expression comes from. Especially the factor $(\epsilon_{{\bf k}} -\mu) $ bothers me. Does it tell us that only electrons near the Fermi level contribute to the heat current? Or does it have another meaning?


Propagator of Chern-Simons Abelian gauge theory

I need to compute the "topologically massive photon" propagator. I've started with : $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\mu}{4}\epsilon^{\mu\nu\lambda}A_\mu\partial_\nu A_\lambda $$ $$ =A_\mu\underbrace{[\frac{1}{2}g^{\mu\lambda}\partial^2+\frac{\mu}{2}\epsilon^{\mu\nu\lambda}\partial_\nu]}_{(\Delta^{-1})^{\mu\lambda}}A_{\lambda} $$ So how can I invert The under braced part which will yield the topologically massive photon propagator?


Inner product of particle-anti-particle spinor components

Suppose I have four-component spinors $\Psi$ and $\bar \Psi$ satisfying the Dirac equation with $$\Psi(\vec x) = \int \frac{\textrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\vec p}}} \sum_{s = \pm \frac{1}{2}} \left[ a^s_p u^s_p e^{i \vec p \cdot \vec x} + \tilde b^s_p v^s_p e^{-i \vec p \cdot \vec x} \right] $$ $$ \bar \Psi(\vec x) = \int \frac{\textrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\vec p}}} \sum_{s = \pm \frac{1}{2}} \left[ \tilde a^s_p \tilde u^s_p e^{-i \vec p \cdot \vec x} + b^s_p \tilde v^s_p e^{-i \vec p \cdot \vec x} \right] \gamma^0$$ with these definitions: $$ \tilde a \equiv a^\dagger \quad ; \quad \gamma^0 = \begin{pmatrix} 0 & 1_2 \\\ 1_2 & 0 \end{pmatrix} \quad ; \quad u^s_p = \begin{pmatrix} \sqrt{p \cdot \sigma} \xi^s \\\ \sqrt{p \cdot \bar \sigma} \xi^s \end{pmatrix} \quad ; \quad v^s_p = \begin{pmatrix} \sqrt{p \cdot \sigma} \eta^s \\\ -\sqrt{p \cdot \bar \sigma} \eta^s \end{pmatrix}$$ $$ \sigma = (1_2, \vec \sigma) \quad ; \quad \bar \sigma = (1_2 , - \vec \sigma) \quad ; \quad p = (p_0 , \vec p)$$ where $\vec \sigma$ is the usual vector of Pauli matrices and $\xi, \eta$ are two-component spinors. The lecturer then goes on to choose an appropriate basis for $\xi$ $$ \xi^1 = \begin{pmatrix}1 \\\ 0\end{pmatrix} \quad ; \quad \xi^2 = \begin{pmatrix} 0 \\\ 1\end{pmatrix}$$ and similarly for $\eta$ such that $\tilde \xi^r \xi^s = \delta^{rs}$ and $\tilde \eta^r \eta^s = \delta^{rs}$. This appears sensible and using this, we can compute various inner products of $u$ and $v$. In the examples discussed in the notes, mixed terms, that is, those containing products of $\xi^r$ and $\eta^s$ never occur (multiplied by 0, hence irrelevant) or cancel out. In an example assignment, however, we are meant to calculate $\bar \psi(-i \gamma^i \partial_i + m) \psi$ which then leads me to terms such as $$ \sum_{s,t} 2 m \tilde a^t_p \tilde b^s_{-p} (p_i \sigma^i) \tilde \xi^t \eta^s$$ which also cancel out in the end, but pose the question in the meantime how $\tilde \xi^t \eta^s$ is actually defined/can be calculated. The identities gained from an appropriate choice of basis for $\eta$ and $\xi$ ‘feel’ wrong here, since, after all, $\xi$ comes from a particle-spinor and $\eta$ comes from an anti-particle spinor. I take it my question could be rephrased as to whether $\xi$ and $\eta$ live in the same space or belong to two different spaces (which would make their inner product an even more thrilling exercise).


Gravitational lensing or cloud refraction?

**My current understanding of gravitational lensing follows** When a star or other massive body passes between us and another star, the phenomenon generally labeled 'gravitational lensing' occurs. The term 'gravitational lensing' refers to the bending of light rays due to the gravitational influence of the massive body. Since the bends concentrate light for the observer, they act like a magnifying lens. **OTOH** But since I was young I have pondered other explanations for such observations. One thought about what (else) might cause the observed effect is a cloud of gas around the massive body itself. The assumption being that any change in density between the thin gas of interstellar space and the thicker gas surrounding a massive object should form a refractive boundary. In the case of an active star, that gas & might be the heliosphere of the star, and for any massive object that is _not_ creating an heliosphere, it could be due to clouds of gas that the massive object is sucking in. I raised this idea with a physics teacher in high school & he assured me that any possible refraction from clouds of gas had been ruled out in observations of gravitational lensing. I never got to see papers on the subject, so I cannot confirm that (and am not convinced). My questions are:   1. Does anybody know of papers on gravitational lensing that discuss the refraction from clouds of gas around the lensing body?   2. How could an observer/experimenter go about quantifying the likely refractive index of such a gas cloud.


What's the average position of oscillating particles in a box with periodic boundary conditions?

Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is "teleported" on the opposite boundary (typical periodic boundary position in 3D). Now put a hundred of oscillating particles inside. Each particle position is identified in a uniq fashion through time. **I want to get the average position of each of these particles.** **EDIT:** Thanks for your input. But I think I did not explain myself correctly. Please forgive me for that. That's why I removed most of my wordy description and created the following picture as a 1D reduction of my problem instead. ![1D simplification](http://i.stack.imgur.com/McuWL.png) _**Both pictures:_**   * This is a 1D projection of the 3D problem.   * The axis is infinite.   * The `blue` line represents the _1D unit cell_ repeated by translation along the axis without overlapping.   * The `red` dot represents the position of one unique and identified particle.    * The `green` dots represents the equivalent particles to the `red` one obtained by translation. These are really equivalent to the `red` one: same nature, velocity, force field, etc.   * As an input, I only have particle positions inside of the _1D unit cell_ (the `blue` line). _**Top picture:_**   * The `red` particle is present in the unit cell for each time step.   * I take the particle position $x(t)$ and average it over $t$ to get $\bar{x} = \frac{1}{4}$ of the unit cell in this case (`black arrow`). _**Bottom picture:_**   * The `red` particle is **crossing** the boundary of the _unit cell_.   * However, $x(t)$ is only defined in the _unit cell_ , therefore in this case at:      * $t=0$, the position of the `green` particle (on the right) is taken into account ;     * $t=1$, the position of the `red` particle (on the left) is taken into account     * _and so on_ ;   * If the average $\bar{x}$ is calculated without much care, it is false since this average is done between two different particles (`red arrow` instead of `black arrow`). I developed a small algorithm that tries to correct this assuming that $x(t+\Delta t) - x(t) < \frac{\Delta x}{2}$, with $\Delta x$, the size of the _unit cell_. However this algorithm is quite slow. I know I can improve the algorithm itself and my implementation of it. But I also know that this is a trivial (and inefficient) way to do it. I also hate to reinvent the wheel again and again... Do you know any smart implementation or algorithm able to calculate this kind of "periodic" average?


FWHM in resonance amplitude square derivation

Consider a linear harmonic oscillator subject to a periodic force: $$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$ The solution tends to: $$A \cos (\omega t - \delta)$$ where: $$A^2=\dfrac{f _0 ^2}{(\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2}$$ The maximum of this function is reached when: $$\omega ^2 = \omega_0^2-2\beta ^2$$ which is: $$A^2 _{max}=\dfrac{f_0 ^2}{4\beta ^2\omega_0 ^2-4\beta^4}\approx \dfrac{f_0 ^2}{4 \beta ^2\omega_0 ^2}$$ if $\beta$ is small compared to $\omega _0$. **I need to find theFWHM of $A^2$**. I put: $$\dfrac {A_{max}^2}{2}=\dfrac {f_0 ^2}{8 \beta ^2\omega_0 ^2}=^!\dfrac{f _0 ^2}{(\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2}$$ $$\iff (\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2=8\beta^2\omega_0^2$$ My book gives the hint "Is it ok to consider $\omega + \omega _0 \approx 2\omega _0$ or $\omega \approx \omega _0$, it is _not_ ok to consider $\omega - \omega _0 \approx 0$". With this hint it's not hard to find the solution: $$( \omega - \omega _0)^2(\omega + \omega_0)^2+4\beta^2\omega^2\approx 4\omega _0 ^2[(\omega-\omega_0)^2+\beta^2]$$ $$\therefore (\omega-\omega_0)^2\approx\beta^2\iff\omega=\omega_0 \pm\beta $$ solved. Now my question is: it's pretty clear that the approximation $\omega -\omega _0 \approx 0$ wouldn't work. It is also clear to me that it is not contradictory to assume $\omega \approx \omega _0$ , for we use this approximation for the other term $\beta ^2 \omega ^2 $. _Is there a more precise way to see this (i.e. that the approximation $\Delta \omega \approx 0 $ is not correct)??_


relating spinor and fundamental representation for $E_8$

While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is $Tr(F^6)=\frac{1}{48}TrF^2TrF^4-\frac{1}{14400}(TrF^2)^3$, where trace is in adjoint representation. I am able to prove this relation but while doing so, I have some identities which relates the spinor representation $128$ of $SO(16)$ to fundamental representation of $SO(16)$ which I must show but this is not working out. The simplest one being $TrF^2=16trF^2$, where $Tr$ is in spinor representation $128$ and $tr$ is in fundamental representation. There are other relations showing the equality between $TrF^4$ and $tr(F^2)^2$ and $trF^4$. I am aware that spinor representation would be $\sigma_{ij}$ which is $128$ dimensional. While trying to prove these identities, I have noticed that if $F^2$ in the fundamental representation is diagonal with only two elements -1 and -1 and if $\sigma_{ij}^2=\frac{-I}{4}$ where $I$ is $128$ dimensional identity matrix then I can get the result. But I can not convince myself why it should be true. Any details would be appreciated of how to prove it. The identities can be found in GSW chapter 13 last section (VOL.2).


Lenses (refractor) or mirrors (reflector) telescope?

What differentiates, in terms of practical quality, not technical implementation, a refractor from a reflector telescope? Why would one prefer a refractor over a reflector, when reflectors come with such large diameters at a smaller price?


Does water have surface tension in a vacuum?

I could be totally wrong here but I was thinking about water surface and what creates that. My thought is it is the thin mixture of water and air separating the two. This mixture creates the boundary between water and air that has the property of surface tension. So does the surface of water in a vacuum act in the same way? Is there a surface with surface tension when there is no air to make the mixture?


Spatial light modulation for mode multiplexing, how does it work?

If a laser light becomes spatial phase modulated (by phase only spatial light modulator) what happens to the intensity distribution and frequency spectrum of the laser light ? I know that this technology is used to perform mode multiplexing. But how affects the spatial phase modification the mode field distribution (e.g. transform a SMF mode to a MMF mode)? I suppose the Fourier Transformation can help here? But how exactly ?


Interpretation of decay associated difference spetrum

We measured transient absorption spectra and we got $\Delta A(\lambda,t)=A(\lambda,t)-A(\lambda,0)$. If we use DAS (decay associated spectra) for interpretation, we assume that our data have this dependance: $\sum_{i}A_i(\lambda)e^{-t/\tau_i}$, where $\tau_i$ is a lifetime of a component $i$. Decay associated spectra are then $A_i(\lambda)$. If we focus on one of the $i$ spectra ($A(\lambda)$), is there any easy way how to interpretate it? I am confused with mixing decacy associated spectra together with transient absorption.


Speed of light in different parts of the universe

Do we have enough evidence to claim that speed of light remains the same at 'c' even at the most distant parts of the visible universe?


Why is the second order perturbative correction to the ground state energy always down?

What is the physical/deeper reason for the second order shift of the ground state energy in time independent perturbation theory to be always down? I know that it follows from the formula quite straightforwardly, but I could not find a deeper reason for it even if if I took up examples like the linear Stark effect. Would you please tell me why?


Flaw in the radius of a spherical electron model (classical electron radius)

My textbook asks me to derive an equation for the potential energy ($U$) of sphere ($r_0$) filled with an electric charge of uniform density ($\rho$), expressed in terms of the total charge $Q$. The equation can be derived by treating the potential energy ($U$) as the total work it would require to build the sphere and computing an integral [which came from the standard charge through a sphere formula (if you feel the need for a diagram or intermediate mathematical steps let me know)]: ($Work$) = $\int_0^r ((4\pi\rho)^2r^4\cfrac13)~dr$ which means that: ($Q$) = $4\pi\cfrac13r_0^3\rho$ , and from the relationship between potential energy and Work, we obtain: $$U = \frac{3Q^2}{5r_0}$$ The next question (based off of the equation we just derived) asks us to set our answer equal to $mc^2$, and attempt to calculate the proper radius of an electron (the value we receive is, I now know is the "classical radius" of the electron, or the "Lorentz radius" $\approx$($2.8 *10^{-15}\;\rm m$). Finally we arrive at our problem : It asks us to explain why this theory does not adequately analyze the proper radius of an electron, or to find out what the flaw in this theory is. I really do not know, I wouldn't have used this method in the first place. Any thoughts here? Is looking at an electron as a sphere inncorrect? I actually attempted to research this matter on my own, and could not ascertain the reason that the classical electron's radius is a poor theory, only that it does not work well in quantum mechanics.


Could symmetric molecular stretching cause a physical object to become amorphous?

I understand very little of this, but my google-foo has yielded me next to nothing, I thought I might then just ask. My real question here is if it might be possible through the use of vibration to cause a solid object (rock) to become amorphous/polymorphous by causing the bonds between particles within a molecule to weaken. If the bonds that made up each molecule were weakened and the particles that it were made up of were somehow dissipated or transferred temporarily, it makes sense to me that the object might become polymorphic and manipulatable with ease. Possible? Yes/no would be helpful and then how/why. Thanks in advance! _Edited to ask a single more specific question_


How can a point-like particle "feel" gravity, if locally the curvature of spacetime is always flat?

I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that > _Locally, as expressed in the equivalence principle, spacetime is > Minkowskian, and the laws of physics exhibit local Lorentz invariance._ > (Wikipedia:General relativity) But if this is the case how does a point-like particle know, which direction it should follow in a gravitational field? Can this be a seen as a hint that point-like particles cannot be used as underlying concept for a field theory like general relativity?


Video documentaries about physics?

Does anyone know about good video documentaries about physics, possibly on YouTube? I've seen a couple of them about string theory, but I could not find many others.


Is there a four dimensional form of Born's Rule -redub

Generalizing Born's Rule for 4-dimensions $x_4$, write $$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$   1. Is this consistent with quantum mechanics?    2. Is this a generalized form of the Born's Rule for the phase space of the wave function? @DavidZas I was thinking of a four dimensional space where Time is not a parameter. There been recent work on Time Reversed light pulses. http://prl.aps.org/abstract/PRL/v106/i19/e193902 and the PHYSORG description http://www.physorg.com/news/2011-05-physicists-time-reversed-pulses.html And this leads me to wonder about abstract phase space, and I mean phase both in the epoch angle of a wave function and the phase space of a Hamiltonian (which yes, is very confusing). The way I see it, the wave function implictly has the Action of phase space built into it. $\psi = e^{i(\omega t - k x)} = e^{\frac{i}{\hbar }(\mathbb{E} t - \mathbf{p} x)}$ Where the phase and the action are connected by $\frac{\mathbb{E} t - \mathbf{p} x}{\hbar }$ Imagine dividing the phases by the action of the particle It's a bit like there's a blob (note the very precise language) of phase space wobbling about like the liquid drop model of the nuclear physics. Only this is a more abstract phase space that's oscillating. So, again, would Born's Rule apply to such a system? Actually after rereading what I just written, I'll be surprised if anyone replies this time! :)


Where can I get latest full updated list of supernovae candidates from the SDSS-II Supernova Survey?

I am doing a project where I need the list of supernovae candidates from the Sloan Digital Sky Survey II (SDSS-II) supernova survey. **One list** is available as part of the data-release-7 (DR7) website. The list contains 902 confirmed candidates. On the other hand, **there is a list** on another website for the supernova- survey project, where the list is supposedly confirmed _and_ updated. This list has 1030 candidates. Not only is is the number of candidates on each list different, but each list has candidates that are not on the other list. **The question at first, then, was which list is correct one?** There is no information on the SDSS DR7 website on when the list was last updated, but on the main site for the external project site, the footer says that the page was last modified in 2009, which is after DR7. I am not sure if this date is the same for the page with the actual list, though. When I inspected the overlapping entries (those sharing the same internal SDSS id), I noticed that the larger list at least for the ones that I looked at, in some places had information in a field that was blank in the smaller list. This was usually the value of the redshift for the given supernova candidate. I took this as an indication that the larger list, i.e. the one that is not on sdss.org, is actually the newer of the two lists. **But what about the confirmed candidates that are still listed in the older (smaller) list? Are they no longer considered supernovae candidates?** I do not want to have to guess which list is better one to use, and I am not sure that merging the candidates from the older confirmed list that are not on the newer into the newer list, since there must be a reason why they are not included in the latter in the first place. But this rationale for this decision is not given alongside the list or anywhere else on the project webpage that I looked. _What I really want is the latest complete and updated list of supernovae candidates available._ **So the question is, where do I obtain such this list? And when was it made?**


How should I be thinking about tides?

I am working on a project for physics that involves tides. This is my current mind set when thinking about tides: The earth applies a gravitational force on some mass `M`. The moon & sun apply a gravitational forces on the mass `M` away from the earth. The bodies of water on the earth have some acceleration towards the moon & sun due to their forces (F=ma). The bodies of water accelerate till they reach a sort of equilibrium between forces. I would like you to poke holes in my understanding of tides and try and answer my naive questions bellow. How would you describe tides effected by the moon using forces? How can that be used to calculate the height of tides?


Speed of the Moon

Why the motion of the Moon looks very slow in the sky? Doesn't it need the high speed in order to escape the earth's gravity?


Boltzmann Distribution of Electrons in Confining Potential

I have a particle simulation wherein many non-interacting electrons are trapped in a electric potential well. I would expect, and therefore I initialize according to this, that the electrons would take the shape of a Boltzmann distribution. However, when the simulation is allowed to run for a long time, the charge density converges to a slightly different shape: ![enter image description here](http://i.stack.imgur.com/kAjFj.jpg) In this chart, the horizontal axis denotes position. The red line is the negative of the confining potential. I flipped it just for readability. This is on one vertical axis. On the other vertical axis is both the charge density as the system is initialized (green), and the charge density that the system settles into (blue). The colors are somewhat hard to see, so to clarify, the initialized (green) state is the one that is more negative in the region from around z=20cm to z=60cm Any ideas as to why the Boltzmann distribution is not the correct initialization? UPDATE: This is the graph with the electron-electron interaction turned on. It has pretty much settled into equilibrium by now ![enter image description here](http://i.stack.imgur.com/6icd6.jpg)


Calculate water flow rate through orifice

I'm not very good with fluid physics, and need some help. Imagine the following setup with water contained in-front of a wall with an opening on the bottom: ![Fluid image](http://i.stack.imgur.com/GjPyZ.jpg) How do I calculate the water flow $Q$?. I have made some re-search and found I need to (partially) calculate the pressure across the opening (orifice). But I don't know the pressure on the back side of the orifice. Can this be solved in any way? **Note:** I'm not saying _"please give me the solution, I'm lazy"_. I want to figure it out myself. But since, in this case, I only found formulas involving calculating pressure drop, I canno't use them to solve the problem. Therefore I'm turning my face to you, to see if there's another way to solve this problem. **Update:** The "tank" holding the water is actually `a big lake`, and the opening is how much the water gate have opened. I need to very precisely calculate how much water flows through the opening.


Geophysics with earth science

I am a communications engineering student. I am interested in doing a project on geophysics. The concept is for a hole-digging robot with a camera and sensors and wireless transmitter that is made of high mp alloys. It is intended to make a hole down to 3000 km and perform a sample-return for any interesting things that could be found at that depth. This could help scientists learn more about the interior of Earth. Is this something that could be done?


Categorizing solutions to Hierarchy problem

We know that no gauge symmetry can prevent a term $m_\phi^2|\phi|^2$ for a scalar field, and that, given the quadratic loop corrections, the natural scale is $m_\phi \sim M_P$. This is related to the hierarchy problem. Now, I was thinking, is it fair to categorize the solutions to the hierachy problem into two sets:   1. **Solutions which _transfer_ a symmetry to the scalar:**     * Supersymmetry, in which $\phi$ is protected via a symmetry transferred from a fermion.     * Little Higgs, in which $\phi$ is a Goldstone-boson protected via a shift symmetry linearly realized from the original scalar that precipitated SSB.     * Composite Higgs, in which the fundamental degrees of freedom at high energies are not scalar, but fermionic, and thus $m_\phi$ is protected.   2. **Solutions which lower the true $M_P$**. i.e. the true $M_P \sim M_W$, but on the brane $M_P^\textrm{eff}\sim 10^{19}$ GeV, e.g. ADD and extra-dimensions. Is this division of solutions accurate? Are there any solutions that don't fall into these categories? And are there any other ways to transfer symmetries to scalars?


Particle accelerators by focused laser beam?

My (layman) estimate is that such accelerators could relatively easily and cheap override the current high-end, multimilliard-$ accelerators. Some years ago I've read some interesting breakthrough in the news, but practically nothing since that. What is the current focus of the research/development? Is there any obstacle found? What is the current estimate to the practical energy limit of such accelerators?


Proof the superstring action is Weyl invariant

The superstring action is: $$ S = k\int \mathrm d\sigma \sqrt{-h} \left [ h^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X_\mu + 2i {\bar{\psi}} ^\mu \rho^\alpha \partial_\alpha \psi_\mu - i {\bar \chi}_\alpha \rho^\beta \rho^\alpha \psi^\mu \left (\partial_\beta X_\mu -\frac{i}{4}{\bar \chi}_\beta \psi_\mu \right)\right ]$$ This is Weyl invariant, but the symmetry isn't manifest. All the books and lecture notes I've seen just state this outright, without proving it. Is there a reference where I can see the proof done in detail? Even when they give the action in the superconformal gauge ($ h_{\alpha \beta} = \eta_{\alpha \beta}$ and $\chi_\alpha = 0$), they don't bother to prove it.


Resistance of Light Dependent Resistor (LDR)

Is there a mathematical expression relating the resistance of Light Dependent Resistor (LDR) with light intensity?


How does a ducted fan suck in air?

I am thinking of a simple ducted fan with a single set of rotating blades, no stator vanes behind. If the effect of the fan is to raise the pressure of the air, how is there flow into the fan against this pressure gradient? I understand that the rotating fan blades have the effect of increasing the stagnation enthalpy of the flow. For an incompressible flow, this is equivalent to the stagnation pressure increasing via the fan blades imparting kinetic energy into the flow. I know that static pressure and velocity increase across the fan, so how can there be flow against this pressure gradient? I think the explanation will have something to do with the different frames of reference chosen to analyse the problem.


Where and how exactly does string theory and Q.E.D. use zeta function regularization?

In the video they mention it being used in many fields of physics inclusing String and QED theory. https://www.youtube.com/watch?v=w-I6XTVZXww But I remember reading somewhere that 1+2+3..=-1/12 is obviously a "mathematical trick" (something about stupidly equating incompatible sets), and if so, how does this turn out to be true for things that are real (like QED)?


Spring Damper System for a Vibrating Motor

Good day people of SE I have a friend that has a final year project and is stuck. He has a motor with a small weight at the end of the shaft that causes vibrations. This motor is on a thin plate. Beneath the plate, in each corner, is a spring and damper. He has to formulate a model that describes the power absorbed by the spring damper system. The aim is to minimize or negate the vibrations to the platform under the spring-damper system. The weight, spring and damper can be adjusted. The level of the student is about 3rd year engineering. The standard of the mechanical engineering department is quite low compared to the other engineering practices, so a purely mathematical solution will be sufficient. I have no information regarding the position of the weight on the shaft. I assume it will be a rectangular weight positioned slightly to one side to allow some serious vibrations, but this is a guess. Where do you suggest we start researching this? Thank you EDIT : Thank you Tobias for the comment. I changed the question as instructed.


When a wire is stretched by a load, is there any change in its volume?

When a wire, or a rod(whose diameter is not negligible) is subjected to a tensile stress, is there any change in its volume? If yes, does the volume increase or decrease? Take into consideration both the longitudinal elongation and the lateral contraction. If no, then what is wrong with the following derivation? $V=\pi d^2 l/4$$\ \ \ \ \ \ \ \ \ \ \ (V$ is the volume of the wire) $\implies\Delta V/V=2\Delta d/d +\Delta l/l$ $\implies\Delta V/V=(1-2\sigma)\ \Delta l/l$ $\ \ \ \ \ \ \ (\Delta d/d=-\sigma \Delta l/l)$ where $\sigma$ is Poisson's ratio.


Quantum Mechanics: Show that the expectation value of angular momentum does not change with time

The potential is given by $V\left(\left\|(x,y,z)\right\|\right)$, so $[\hat{L}, \hat{H}] = 0$. Using the definition of $\langle \hat{L} \rangle$ and the time-dependent Schrödinger equation, show that the expectation value of angular momentum does not change with time. Expectation value of angular momentum, $\langle \hat{L} \rangle$ = $\langle \psi \mid \hat{L} \mid \psi \rangle $ where $\psi$ is a generic wavefunction.


The chiral ring of N=2 WZW models

Does anyone knows if the chiral ring of the Kazama-Suzuki models have been found? If yes, please send me the reference. Thanks


QED photon propagator to one-loop order gets different answers

I'm a self-studying 14-year-old who has a passion for particle physics. I'm currently trying to calculate the QED photon propagator to one loop. However, in all the places I've looked, even with the same renormalization scheme, the results are different. I don't get it. Also, I need a step-by-step tutorial on how to calculate that trace in the numerator, specifically, the one with those gamma matrices.


Why does it take so long to make a nuclear bomb?

So as I know nuclear bombs are derived from fission reactions: By providing the nucleus with enough power to trigger a chain reaction. If uranium was present why does it take so much to make a nuclear bomb.


Is it possible to clearly receive weak signal from space?

Imagine there is communication satellite on geostationary orbite. It transmits signal in a defocused beam, "illuminating" whole Earth hemisphere it's floating over. For example, signal is amplitude-modulated with 10GHz carrier frequency. What if satellite transmitter power is very, very low, like 25mW, is it possible to receive this signal on Earth with temporal resolution high enough for high-speed data transfers by just using sufficiently large or huge satellite antenna dish; or will noise get amplified too and this approach will be fruitless? What if two identical satellites are transferring signals of their own, and floating close enough (say, 1° away) to each other to be visible at once from surface, is this possible to receive separate signals from both satellites, or interfence will cause total mess? Are there ways to overcome this?


Can we measure the "Tilt-A-Whirl" effect of the Earth+Sun orbiting the galactic hub?

You've probably ridden on the fairground ride called the "Tilt-A-Whirl", or-- as Disneyland calls it--the "Spinning Teacups", as well as other fairground rides that employ epicycles. You can really feel the centrifugal force strongly when your car spins complementary to the spin of the main rotor. Now, can that extra force be measured on Earth as our orbit around the Sun complements the Sun's orbit around the galactic hub? Has anyone done it?


Transformers rate equation

Going back to transformers after some years I realize they hide a possible missconception I'm not sure I really understand. It is not easy to find a full derivation of the well known relation $\frac{\Delta V_2}{\Delta V_1}\\!=\\!\frac{n_2}{n_1}$, as if this question were being circumvented, and that's why in an aim to give an intuitive idea of it I fall in a nonsense. ![Typical transformer arrangements...](http://i.stack.imgur.com/UfNmA.png) I start thinking in the secondary winding: attending to Maxwell-Faraday-Lenz's Law, $\varepsilon=\Delta V_2=-\frac{d\phi}{dt}$, and given a certain primary winding signal frequency, the higher the maximum flux $\phi$ is, the higher the induced e.m.f. $\varepsilon$ will be. Then we want a secondary coil plenty of loops (high $n_2$), as this increases its effective surface, and it is _pierced_ by the same field $\vec{B}$ lines $n_1$ times. Ok, so $\Delta V_2\varpropto n_2$. On the other hand, to increase that flux, we need a strong field due to primary solenoid. And this, **for a given intensity $I_1$** (let $R_1$ be $n_i$-independent), is fulfilled if we have high $n_2$. Then we would say that, at the end, $\Delta V_2\varpropto n_1 n_2 \Delta V_1$, which is not true, and which would produce some symmetry problems. What physics am I missing? Something related to induction in the primary too? Is due to not considering that $I_i=\frac{\Delta V_i}{n_i R_o}$? What makes, physically, that by decreasing $n_1$ one increases $\Delta V_2$? How to understand that? Usual derivations, as in Wikipedia, apply that $\Delta V_1=-\frac{d\phi}{dt}$ should be verified, but I find it a bit misleading, as the _cause_ and _effect_ roles symmetry is not clear. Given the true relation, in fact, what is the reason not to set always just $n_1=1$? Just to have a more precise ratio control and/or to have a more efficient system with $\vec{B}$ lines better guided inside the iron or is there any other reason?


Why batteries in serial don't discharge each other?

A question that I've asked myself for long time about why batteries in series work as they do. I'll try to explain: If you take two batteries, let's say A and B, and you put A's anode touching the B's cathode (only that, without closing the circuit). It turns out that not only A and B poles don't discharge each other but the resulting device sums the voltages of both batteries. When I think of it my idea is that A and B poles should get "lost" (they have cancelled) but the poles that remain unconnected still have a different potential so it shoud work at that voltage, not the sum. After all you are connecting a + pole to a - one and it's supposed that one pole has a different charge that goes to the other until they are balanced. So, why batteries discharge when closing the circuit but not when it's open if poles of different batteries are in touch? Why the charge from one pole doesn't flow to the other? And why the voltages sum? I'm afraid that the explanation is quite simple but...which is? Thanks in advance! **NOTE: I edit my own question.** As Murphy states, once published I've found there can be a possible explanation here: Why two connected 1.5 volt battery become 3 volt I'll see if it tells me what I want. Nevertheless, feel free to give me your own answer. **NOTE 2:** Ok, guys, thank you very much for your answers. Finally I think I got it! The point is in the electrons that cause the potential difference. The accumulation of electrons in one pole makes impossible that the electrons of the other battery enter the possitive pole. They are rejected electrostatically. Allowing it you would have the two poles charged but the battery doesn't work that way. So, when you close the circuit, both accumulations of electrons sum because one battery would help expelling the other one's charge pushing with it's electrons. Am I right?


How to calculate velocities after collision?

I'm currently writing a program for a particle simulator. One of the requirements is that the particles collide in a realistic way. However, I don't know how to calculate the final velocities. For each collision, I have the $x$-component and $y$-component of each velocity, as well as the displacement and mass of each particle. Is it possible to calculate the direction and magnitude of their velocities after the collision? If so, how?


Why is the objective function used in Nudged Elastic Band method reasonable?

In Nudged Elastic Band (NEB) method, which is used to find reaction pathways when both initial and final states are know, an objective function is first constructed and then minimized to find reaction pathways. NEB creates a string of replicas (or 'images') of the system between initial and final states, and connects them with springs to represent a path connecting initial and final states. Initially, the images may be interpolated between reactant and product linearly. Then an optimization algorithm is applied to relax the images down towards the Minimum Energy Path (MEP). The objective function is constructed as a combination of potential energy of each image and "spring energy" between each image as equation below shows. $$ S(\vec R_0,...,\vec R_N)=\sum_{i=1}^{N-1}E(\vec R_i)+\sum_{i=1}^N\frac{k}{2}(\vec R_i-\vec R_{i-1})^2 $$ Where $\vec R_i$ is the coordinates of i-th image. NEB solved "corner-cutting" and "sliding-down" problems in this model by considering vertical and parallel components of potential force and string force. I will not introduce here, you can look at this article for more details. My question is why the objective function above is reasonable. Does minimization of this objective function sufficiently leads to the MEP? If yes, how to prove it? If no, why is this "chain-of-states" model widely used? Thank you very much for answering.


Could we use the coriolis effect to travel

Given a platform that is floating independent of Earth's gravity, would it be possible to put a platform over the equator and over the day, would the Earth rotate from under it allowing you to travel around the world?


Gauss's Law - Electric Field outside a Shell?

I'm somewhat confused as to why the electric field outside a spherical shell is $\frac{Q}{4\pi r^{2}\epsilon_{0}}$ Going through the work: $$ Q = 4 \pi r^{2} \sigma ; \frac{4\pi r^{2}\sigma}{ε_{0}} = \int \int_{s} \vec{E} \cdot d \vec{A} $$ Which, doing the integral leaves = $$\frac{4\pi r^{2}\sigma}{ε_{0}}= 4\pi R^{2}E$$ which means that $$E = \frac{r^{2} \sigma}{ε_{0}R^{2}}$$ Is this correct, or am I making some silly mistake?


Will the absorption of high energy gamma rays end up heating the absorping material?

By high energy I mean 100+ MeV gamma rays. I think that at this energy of photons, pair production is the dominating absorption process. So, as known, in pair production an electron and a positron are produced and any excess energy of the photon over the rest mass of the electron and the positron will be in the form of the kinetic energy of the new electron and positron. So now, how will the electron and positron end up in a dense material ? like will the electron come to a stop releasing its in energy in the material and thus heating it up ? or will it escape the material ? I am assuming using the attenuation length of Lead at this gamma ray energy as the absorbing material.


Is there a thermodynamic law or theorem that expresses how systems "break" or "change" when enough energy is added?

I have a simple question about thermodynamic laws, and I am hoping you can help me. Let's say that I have a sphere container with some pressurized gas in it. I can slowly increase the pressure over time, and eventually the container will have to rupture (no matter what it is made of). More generally, if energy is slowly added to a system in a consistent manner, there will be some point in time when the system either changes or breaks as it can no longer contain the energy. Which of the thermodynamic laws would explain this thought experiment? Could you point me in the right direction? Thanks


Can magnetic energy be used to create a magnetic motor which can power vehicles and machines?


Thermodynamics: Are pressure and temperature enough to calculate enthalpy?

![enter image description here](http://i.stack.imgur.com/VG8Km.gif) From the chart above, it looks like it is possible to calculate enthalpy from just pressure and temperature. After a few hours of Googling my question, I'm not able to find anything helpful. My company is interested in finding the effectiveness of an air conditioning system by measuring the change in enthalpy from one side to the other. As a software person, I need a formula that I can use to calculate the enthalpy. I know this formula would need to include different properties of the gas, can you elaborate on what those constants are? Thank you very much.


Finding potential difference (PD) between two points

How do I find the potential difference between two points in a circuit without using Kirchhoff laws?


Time to travel distance given acceleration/deceleration

> A body accelerates at 0.2 m/s$^2$ & retards at 0.3 m/s$^2$. Maximum velocity > obtained by it is 30 m/s. Find the least time in which it covers 6 km Well, if I do not assume that the body starts from 0 m/s and stops at 0 m/s. I get stuck with six equations, three from each condition. And if I do assume it, the condition of body crossing 6 km doesn't meet. All I wish to know is how to begin with this problem.


First-order wave equation: Why is its presence not common?

The (one-dimensional) **wave equation** is the second-order linear partial differential equation $$\frac{\partial^2 f}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}\tag{second order PDE}$$ that admits as its solutions functions $f$ of the form $$f=f(x\pm vt),\tag{solution}$$ as can be verified in a straight-forward manner. These solutions have a convenient interpretation that justifies the phrase _wave equation._ I noticed there are **first-order** partial differential equations which have as solutions functions of the form $f(x\pm vt)$: $$\frac{\partial f}{\partial x}=\pm\frac{1}{v}\frac{\partial f}{\partial t}\tag{first order PDE}.$$ A quick Google search shows this is indeed called the first-order wave equation, but it usually shows up in the context of math classes. **So now the question:** Why is the usual second-order PDE favored over these first-order ones if both admit the same solutions? Is there a physical reason? Are this first-order equations useful in their own right? Perhaps there other solutions that one admits that isn't desired, or maybe it just looks cleaner since one doesn't have to carry around the $\pm$ symbol in the differential equation.


From Minkowski to Euclidean Time in Path Integrals

I'm trying to prove the following equality: $$ <x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge x\left(t_{i}\right)=x_{i}\right\\} }\mathcal{D}x\exp\left\\{ -\frac{1}{\hbar}\int_{t_{i}}^{t_{f}}dt\left\\{ \frac{1}{2}m\left[x'\right]^{2}-\left(-V\left[x\right]\right)\right\\} \right\\} $$ where the definition of $ <x_f, t_f|x_i, t_i>$ is given by: $$ <x_{f},\, t_{f}|x_{i},\, t_{i}>\equiv\mathcal{N}\int_{\left\\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge x\left(t_{i}\right)=x_{i}\right\\} }\mathcal{D}x\exp\left\\{ \frac{i}{\hbar}\int_{t_{i}}^{t_{f}}dt\left\\{ \frac{1}{2}m\left[x'\right]^{2}-\left(V\left[x\right]\right)\right\\}\right\\} $$ What I have done so far:   * Assume that the domain of integration, that is, $\left\\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge x\left(t_{i}\right)=x_{i}\right\\}$ is such that all the functions in this set can be analytically continued to a new domain of integration $ \left\\{ x\in\mathbb{C}^{\mathbb{C}}:\, x\left(it_{f}\right)=x_{f}\wedge x\left(it_{i}\right)=x_{i}\right\\} $. Is this valid?   * Plug in the definitions: $<x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\\{ x\in\mathbb{C}^{\mathbb{C}}:\, x\left(it_{f}\right)=x_{f}\wedge x\left(it_{i}\right)=x_{i}\right\\} }\mathcal{D}x\exp\left\\{ \frac{i}{\hbar}\int_{it_{i}}^{it_{f}}dt\left\\{ \frac{1}{2}m\left[x'\right]^{2}-\left(V\left[x\right]\right)\right\\}\right\\}$   * Now to compute $ \int_{it_{i}}^{it_{f}}dt\left\\{ \frac{1}{2}m\left[x'\right]^{2}-\left(V\left[x\right]\right)\right\\} $ make a change of variable (is this valid?? Don't you need Cauchy's theorem and also to assume that the time boundaries go to infinity?) $t \mapsto -it$ to get: $i \int_{t_{i}}^{t_{f}}dt\left\\{ \frac{1}{2}m\left[x'\right]^{2}+V\left[x\right]\right\\} $ so you get the correct exponent.    * But how do you prove that $\mathcal{N}\int_{\left\\{ x\in\mathbb{C}^{\mathbb{C}}:\, x\left(it_{f}\right)=x_{f}\wedge x\left(it_{i}\right)=x_{i}\right\\} }\mathcal{D}x = \mathcal{N}\int_{\left\\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge x\left(t_{i}\right)=x_{i}\right\\} }\mathcal{D}x$? Is it even the same $\mathcal{N}$?


Is There an Equation To Calculate Water Evaporation Taking Into Account Environmental Variables

I am starting to learn more in my area of science...Meteorology and the fact that I own a pool, I have the perfect testing environment. I just so happened to have a water level drop from an actual leak that I learned how piping systems work, and will continue to learn more there. The leak was plugged yesterday. But, it got me thinking about my good old weather schooling and physics days. So, here is my question guys: Is there an equation for the following: Calculation of water level evaporation with following variables (X):   water volume, water flow/speed, ambient air temperature, dew point/humidity, chemical composition of water (pool), temperature of water, wind speed. Constants and Assumptions: No leaking or structural damage that holds the water, contained to a finite amount of gallons. I understand that there may be other environmental variables to control, but this is a good start. If there is a standard deviation to follow in all of the above variables, that can be measured, can or is there an equation for this? What I am trying to learn: Is there correlation in any of the variables to the measurement of the change in the water level (Y). Then I would like to apply the formula and change some of the constants to measure on larger scales. Any answers related to the subject at hand is most helpful. Thank you.


I'm trying to reconcile these facts,   * If I had paints of the primary colours and I mixed them together, I would produce gray.    * White light contains every colour.  My most convincing attempt to answer the question: The colours of objects that don't produce light are the colours those objects don't absorb. A white object doesn't absorb any colour, a blue object absorbs all colours but blue, a green object absorbs all colours but green, etc. So by mixing paints of the primary colours I mix influences that collectively absorb all colours and reflect all colours; whereas, if I had paint that only absorbed all colours I would have black paint, and if I had paint that only reflected all colours I would have had white paint. Gray is a combination of white and gray, so I found this answer intuitively satisfying. However, I don't know if it's an accurate explanation of the observation. Why does mixing every paint colour produce gray instead of white? Thank you.


Force applied to an inclined plane

Below is a picture of the problem. Any guidance would be helpful. This problem isn't actually from any assignment, per se. I'm hoping that, by understanding this, it'll help me to understand a more complicated system where one inclined plane is resting on another inclined plane, and there's a mass on the top of the second plane, and the bottom incline is being accelerated by an applied force--but since I don't understand how this much simpler system works, I don't seem to have a chance at the more complicated system. ![Diagrams](http://i.stack.imgur.com/gHLO3.jpg) Forgot to include what the question is asking for. Since this isn't really a question I've been asked, I guess what I want to know is the acceleration of the system.


Why does a spurion analysis work independently of the UV physics?

In short, my question is why does a spurion analysis work to produce the correct symmetry breaking terms regardless of the high energy physics? The context that this question arose is from an Effective Field Theory course (for more context, see here, Eq. 5.50). Consider the QCD Lagrangian, \begin{equation} {\cal L} _{ QCD} = \bar{\psi} \left( i \gamma^\mu D_\mu - m \right) \psi \end{equation} The kinetic part is invariant under a chiral transformation: \begin{equation} \psi \rightarrow \left( \begin{array}{cc} L & 0 \\\ 0 & R \end{array} \right) \psi \end{equation} however, the mass term is not. Now the claim I don't understand is as follows. Suppose the mass transformed as, \begin{equation} m \rightarrow L m R ^\dagger \end{equation} In that case the mass term would be invariant under such a transformation. To write down the correct chiral symmetry breaking terms in our Lagrangian we find the terms invariant given this transformation for $ m $ and then make $ m $ a constant again. The way I understand this physically is that the breaking arises from a high energy spurion field, $ X $, which gets a VEV, $ m $. When we write down all possible chiral symmetry preserving terms using the transforming $m$, we are writing down all the terms that the spurion couples to. The VEV is then inserted and is equal to $ m $. But this procedure assumes that the spurion obeys the chiral symmetry, $ SU(2) _L \times SU(2) _R $, and transforms as, $ X \rightarrow L X R ^\dagger $. How do we know this assumption is true? In fact it seems to fail for the case of QCD since the ``spurion field'' is really the Higgs field, which is a singlet under $ SU(2) _R $.