Conservation Laws and Symmetries

Usually, in Quantum Mechanics, an observable is an operator on the space of the possible quantum states (labelled as $|\psi\rangle$). If this quantity is conserved, in the meaning that the associated operator $\hat D$ is constant: $$\frac{d\hat D}{dt}=0$$ it has to commute with the Hamiltonian operator $\hat H$. To prove this, in the Heisenberg picture of Classical QM, we use the Schroedinger Equation $$i\hslash\frac{\partial}{\partial t}|\psi\rangle =\hat H|\psi\rangle $$ On the other hand, the Noether theorem states that if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. And this is true either for classical, relativistic and quantum systems. Therefore, first question:   * Is it true, even in the relativistic case, when the Schroedinger eq. doesn't apply, that a conserved quantity has to commute with the hamiltonian to be conserved during the motion?  Then, one usually studies the transformations generated by $\hat D$, for instance if $\hat D$ is the angular momentum, the transformation $U(\theta)=e^{-i\hat D \theta}$ (a rotation of $|\psi\rangle$ with angle $\theta$). And assuming the angular momentum conserved, or verifying this by means of the commutation properties, we can construct the space of the eigenstates of $\hat H$ knowing that they are also eigenstates of $\hat D$. For instance we construct the eigenstates of a spin-1/2 particle using $\hat D = \hat S$, and knowing that $[S^2, S_z]=0$. Since it exists only two states: spin up and spin down, $|\psi\rangle$ will be a two-components spinor, and $\hat S$ will be a 2x2 matrix. We see that the Lie Group of the $\hat U$ transformation is generated by the Pauli's matrices throught the expressions $$\hat S_i=\hslash\sigma_i/s$$. So, I understand that, wanting to impose the conservation of $S$, we get into the $SU(2)$ Lie-group, because of the commutation relations satisfied by the $S_i$, and the infinitesimal decomposition of each $\hat U$. Final questions:   * If we want to conserve something else than the spin, as it happens in the strong interaction study, when we have the Isospin and the Strangeness to be conserved, we use $SU(3)$. Why?    * Who is the corresponding Casimir operator? (the analogous of $S^2$)   * Why in this case the generators should be 8, and not 3?   * What is the relationship between a normal rotation in the physical space, for which the generator is $\hat L$, and the rotation in the total space $\hat J$?    * Why in the case of the strong interaction we don't consider any spin or orbital angular momentum in the labeling and classification of the eigenstates?    * At last, is there any good textbook you would suggest to have a global view of this subject? Because I've been studying this problems in several courses during my undergrad courses, in QM classes, RQM classes, Nuclear Physics, and so on, but every time they've just mentioned some concepts without getting deeper, and this causes me a lot of confusion...


Cramer's rule, Origin of Quarks Fractional electric charge?

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns. 2u+1d=1 1u+2d=0 $$a_1d+b_1u=c_1$$ $$a_2d+b_2u=c_2$$ $$u=\frac {c_1b_2-c_2b_1}{a_1b_2-a_2b_1}$$ $$d=\frac {a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$$ u=+2/3 d=-1/3 there ain't no experiment that could be done, nor is there any observation that could be made, that would say, "You guys are wrong." The theory is safe, permanently safe. Is that a theory of physics or a philosophy? I ask you.


Mathematical Formulation for Accelerating Charges Generating Light?

I haven't been formally taught the fact that accelerating charges generate light, and I have seen the picture of an accelerating charge sending ripples down its field lines, but is there a mathematical formulation someone can provide that show that an acceleration on a charge produces E&M waves?


Density as a function of Temperature?

I know that volume = mass/density....and density is the function of temperature. Is there any standard equation that describes density as a function of temperature? The system I am interested in is like a liquid tank or reservoir


Actions at a distance vs. contact interaction

The ancestors could not imagine an action at a distance (in German: "Fernwirkung"). Today physicists don't take serious its _opposite_ anymore (in German: "Kontaktwechselwirkung"). So my first question is: > What is/was the official _english_ name of the opposite of an "action at a > distance"?   >  Something like "contact interaction"? Next to terminology there is the matter itself: The prototypical _repelling_ "contact interaction" has always been collision. > But has there ever been a prototypical _attracting_ "contact interaction"?   > Something like agglutination? What have been prototypical examples, eventually?


Why does the monitor make a cracking noise?

After switching off the monitor, there's a single crack after a while. I wonder where exactly it comes from. I know that this is normal and not an indicator of being defect. This sound is also not related to a specific type of monitor. I've heard this sound from my CRT as well as from my TFT. I guess it possibly has to do with cooling down. I found a lot of statements that second this thought but there's no reliable source confirming this theory though. However, an explanation that only says that it has to do with the cooling process is not sufficient at all. I'm interested in what exactly makes this sound? Is it, for example, a capacitor which has fully unloaded (if this is the proper term). I really haven't the slightest inkling what exactly can make this crack noise.


How intense a magnetic field would it take to keep an hypothetical iron-made moon orbiting around it?

The intention of the question is to provide an example of the weakness of gravity. I imagine a horseshoe magnet located at the Earth's centre (remove the Earth), and a ferromagnetic moon. How intense would the magnetic field need to be to keep such a moon in orbit at the same distance that the moon is from the Earth?


Axion related questions

I have several question regarding axion. Could anyone give me some brief introduction to what is a axion string, axion field and how is this related to fermion zero mode and chiral zero mode?


How irreversible processes are possible?

Susskind says that all laws of mechanics are reversible and any valid mechanic law most be reversible: you can always determine the previous state of any physically valid system. However, the simplest operations in computation, _reset_ as well as the binary AND and OR operators, are irreversible. You may say that "this is only a virtual simulation" and everything is reversible in reality. However, because of the dissipation, almost all processes in the world are irreversible. So, despite irreversibility is completely forbidden, everything seems to be irreversible and entropy only increases. I want to understand how this contradiction is resolved (at least in one case). I also do not understand why do you represent a system configuration space by a box. You say that the state is a point in that box and its trajectory represents the state evolution. However, reversibility requirement demands that the trajectories do not intersect. This means there must be only one trajectory. So, you should rearrange all the dots of the configuration box into a line (it might be a segmented into unconnected circles). However, that might be another question.


How is this classical "paradox" resolved in electromagnetism?

A magnet and a coil move relative to each other. In the frame of reference of the magnet, there is a magnetic field and consequently a force acting on the charges in the coil according to the Lorentz force $F=qv\times B$ but there is no net electric field. In the frame of reference of the coil, there is a magnetic field and also an electric field, induced by the magnet, $E'$ that moves the charges in the coil, producing a current. But, in the first case no work is done on the charges, since the force is perpendicular to the velocity. In the second case, the force $qE'$ does work on the charges. How is this "paradox" resolved in classical electromagnetism?


Why point mass leaves it's own gravitational well?

I suppose that point mass has its own gravitational well. Why this point mass is still attracted by other gravitational waves. I expect that this point mass is surrounded by "gravitational hills". Why it is not blind to other weaker gravitational waves? If we think about gravity in terms of shape of spacetime. I mean that point mass (left chart) creates "sharp" gravity potential. Lowest gravity potential is exactly where point mass is located. Point stays in place (?). When mass is spread across spacetime (right chart) this is different. Lowest gravity potential is moved from the centre of the object. Object accelerates. Does point mass exist in real world? ![enter image description here](http://i.stack.imgur.com/A5RFM.png)


Compute and plot satellite position on 2D Earth map

I would like to create a simple satellite tracking software, that will compute the position of a satellite at any time, and show its location on a 2D map of Earth. Let's assume it is possible to find online position of a satellite in the Earth-centered inertial (ECI) coordinate frame ( in form of TLE for example ) and use some method to propagate its state to a moment in future. Then if I want to plot the position on Earth map, I will need to know orientation of Earth at a given moment in relation to the original ECI frame, probably with the help of ECEF (Earth-Centered, Earth-Fixed). And here come the questions, how do I :   * transform from ECI to ECEF    * take in account Earth precession and nutation   * do the proper time conversion, to also involve leap years, leap seconds, added seconds...  **EDIT :** During my online searches, I came across this article. It seems to be describing what I am looking for. However it uses a lot of terminology that I am unfamiliar with. It also doesn't tell a lot about precision of this technique and where the equations come from ( I need to be confident that I am implementing the right procedures ). I would still like to address all of the issues I listed in the question above.


What properties does the conductor making the plates for Casimir effect have?

Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM field vanish on its boundaries. This looks much like unphysical model of rigid body in relativistic mechanics. Still, zeta regularization appears to somehow correctly remove the divergence and give sensible results. This must mean that effectively, it replaces our "too ideal" conductor with a "moderately ideal" one. So, the boundary conditions for EM field are no longer as simple as zero. So, my question is now: what boundary conditions does zeta-regularization implicitly impose on the EM field instead of the original ones? Does such change of boundary conditions in fact mean that the force will depend on which conductor the plates are made of?


Is the electromagnetic spectrum discrete?

I'm just starting to learn physics and I have a question (that is probably stupid.) I learned that energy levels that the bound electron can have are discrete. I also learned that when an electron transitions from one level to another, a photon with a specific wavelength (energy) is created and released into the wild. My question is this: are there holes in the electromagnetic spectrum, values of frequency that the created photon can't possibly have?


How can such a high exponent arise in this physics equation?

This site http://what-if.xkcd.com/14/ states that during a helium flash, "the reaction rate is proportional to the 40th power of the temperature". Taking for granted that this is true, how can such a large exponent arise in a physic equation? It just makes no sense to me that some physical process could give rise to this phenomenon. An explanation of how it can arise for a helium flash would be satisfactory, but the question is really more general. How can a physical process accumulate enough factors to get up to x^40? Thermodynamic processes tend to be in the x^2 or x^3's and geometry can only add a few more x^3's. Note that we are not talking about the scale of things (obviously the 40th power is not a big deal in the length or time scales), we are talking about a physics /equation/.


Why can you "suck in" cooked spaghetti?

We all know that there is no "sucking", only pushing. So how are cooked spaghetti pushed into your mouth? The air pressure applies orthogonal on the spaghetti surface. Where does the component directed into your mouth come from?


Returning Light

What happen if someone is just close enough to a black hole, and he emits light tangentially out, and the light got bent and circles around, what will he see, can he see the light source from his back?                            +---------->                 /            \        light   /              \               ^                \     observer  +   O black hole /     sees light\               /                \             /                  <----------- light path


Newton's second law, gravity and buoyancy

> A body of mass $m$ and volume $V$ is immersed into the sea. The body moves > under the action of two forces: gravity and buoyancy (Archimedes). > > Gravity force has a magnitude $mg$, where $g$ is a constant acceleration due > to gravity. Buoyancy force has a magnitude $\rho V g$, where $\rho$ is the > water density, and acts in the direction opposite to the gravity force. > > Part 1. > > Assume that the water density is a constant, $\rho V >m$ always holds, and > at the initial time moment the body is released from rest at the depth of > $h>0$ metres below the sea surface. Assume that the origin is at the sea > surface and the positive direction of motion is towards the centre of the > Earth. > > (a) Complete the diagram by showing all forces acting on the body, and the > initial displacement of the body. > > (b) Write down Newton's second law of motion. Solve this equation, i.e. > write down the expressions for the velocity and the displacement as > functions of time. > > (c) Show that the body will move towards the sea surface, and when it > reaches the surface it has the velocity $v= -\sqrt{\frac{2gh(\rho > V-m)}{m}}$. 1a) I can do 1b) I know that Newton's second law is $mx=F$ with $F$ being the forces. I have written the formula as $mx=\rho Vg - mg$ but I am not sure if this is correct. Then I've divided both sides by $m$ to get $x$ on its own and integrated with respect to t (time) to get $x$ as a function of time. Giving me $x=-gt+\frac{\rho Vgt}{m} + c$ 1c) I'm not sure where the $h$ has come from, or how I could input it into my previous equation


relationship between kinematics and area under curve

My teacher is giving us an honors pre-calc/calculus introduction to physics and the kinematics learning target he has given us is to understand the "relationship between area under a curve and the equation that calculates motion". I understand that the area under a curve (from the curve to the x-axis) will calculate the slope of the curve (which makes absolutely no mathematical sense to me), but I don't know how that relates to calculating motion. Please understand that my teacher is a very strict, high-standards, superfluous teacher who works at a university speed (he says so himself, and I'm only in HS) so there may be gaps in my understanding. I know what $\Delta$ means and I have a very, very loose understanding of the relationships between acceleration, velocity, and distance.


How to get a function for the voltage across a capacitor connected to an AC voltage source?

I am looking for the way of obtaining a solution for $V_{c}$ ,as a function of $t$ depending of $\omega$, of the following differential equation related to an electrical circuit involving a low-pass filter : $ \frac{ d V_{c}(t)}{d{t}} + \frac{V_{c}(t)}{\tau} = \frac{V_{s}(t)}{\tau} $. where, $V_{c}$ is the voltage across the capacitance, $V_{s}$ is the voltage given by the AC voltage source, $\tau$ is the time constant, considering that, $\tau = R C$ , $V_{s} = V_{in} \sin{( \omega t )}$ , $I_{R} = I_{C} = \frac{V_{s}(t) - V_{c}(t)}{R} = C \frac{dV_{c}(t)}{dt}$. I approached the problem by first solving the homogeneous part ( $ \frac{ d V_{c}(t)}{d{t}} + \frac{V_{c}(t)}{\tau} = 0$ ) and for which I get the following solution : $V_{c}(t) = K e^{\frac{-t}{\tau}}$ (where $K$ is a constant). I now need find the particular solution (to get the general solution : $S_{general} = S_{homogeneous} + S_{particular}$ ). I think the particular solution might be of the type : $V_{c}(t) = A \cos{(\omega t + \phi)}$. Edit : Once I replace the particular solution in the equation, I come to something depending on $\omega$ , $\phi$ and $\frac{A}{V_{in}}$ , but I don't see how to continue using the sum function sum formulas.


Single-mode fibers and ray-theory of light

I have read ray-theory cannot be applied to single-mode fibers. Why? I know rays are kind of an approximation to describe the real behavior of light, and electromagnetic theory and photonics are closer to nature, but are there no "rays" in single-mode fibers? Thank you.


How is a vacuum formed and measured by this pneumatics attachment?

Please excuse me if this question is misplaced and also for my narrow understanding of physics. Firstly, I'm asking simply because the robot I've now been set to work with uses an air pneumatics system. By way of a small vacuum it can move about objects from point A to point B. This vacuum is somehow created from a small attachment that is connected to a steady supply of pressurised air. This crude illustration is made on MS Paint shows how the vacuum is made. I'm still puzzled as to why the input of air doesn't simply flow out of **both** of the other openings. ![crude illustration](http://i.stack.imgur.com/CwyCl.png) **EDIT** I can intuit that the volume of air from the exhaust is the sum of the input and the additional air taken from the vacuum but **what is the force of the vacuum and how is it measured?** (Side note) > Does the surrounding environment (atmospheric presure) have any impact on > the vacuum? Any response is much appreciated.


Why are Topological Superconductors hard to make?

Topological insulators (TI) have already been made in lab. Topological superconductors (TSC), being close cousins of TI, seem harder to make. Why is that? It seems that materials in connection with majorana zero modes all have bad fate in lab. Are there any deep reasons behind?


What is a boundary condition for capacitors/dielectrics?

I am extremely confused about what boundary conditions are. One minute ago I was solving easy capacitor questions and the next minute I am being asked boundary condition questions and there is no such section in my textbook on boundary condition. I have some questions below if you could please explain from scratch/beginner level what boundary condition is and what the goal is. I have some questions below you could use to explain what these conditions are… I have looked all over google and search various videos and I still cant find any information on them. * * * ![enter image description here](http://i.stack.imgur.com/enZ48.png) * * * ![enter image description here](http://i.stack.imgur.com/qNbaO.png)


How can I subtract Spectra?

I am writing an application that should allow to view and manipulate spectra ( **Raman** ). I am only given one normal spectrum consisting of 1024 coordinate points. Plus I have a background spectrum that is supposed to be subtracted from the first spectrum. f.e.: The first point in spectrum_1 is: **-55.988** , **9373** The according point in background spectrum: **-55.988** , **9382**. After subtracting the two in a third-party software, the first point of the resulting spectrum is: **-55.988** , **-641.968** So I asked myself how this is calculated. According to the Documentation of this software, it is using the following formula: > result spectrum(n) = multiple component spectrum(n) > > – Factor1 * single component spectrum (n) > > – Factor2 * single component spectrum (n) Unfortunately I can't do anything with it. What is meant by "component spectrum(n)"? Is it the Spectrum equation? Also what could the both factors be and what does the (n) mean? Here is an image of all three spectra that are involved in the calculation. Here is an image of all three spectra that are involved in the calculation. http://imageshack.us/photo/my-images/546/spectrumsubtractionresu.jpg/ From Top to bottom:   1. First spectrum   2. Background spectrum   3. The resulting spectrum after the subtraction I simply can't imagine or reproduce the calculation that leads to this result. Any help and/or advice is welcome. I am highly willing to learn and understand so please don't hesitate to share some links or other references that could help. Thanks in advance, BC++


Photonic. vs Plasmonic

I hear a lot about photonic and plasmonic structures being the future in many different applications, but what are the differences between them? Why would one want a material with both as opposed to just one?


How is it possible the universe expanded faster than the speed of light during inflation?

In a documentary written in collaboration with Stephen Hawking, the narrator (supposedly Stephen Hawkings) says that by the time the cosmos was 10 minutes old, it had already expanded thousands of light years in diameter. How is it possible that matter was traveling faster than light?


Quantum Mechanics in terms of *-algebras

I'm currently trying to find my way into the geometric description of Quantum Mechanics. I therefor started reading: Geometry of state spaces. In: Entanglement and Decoherence (A. Buchleitner et al., eds.). Lecture Notes in Physics 768, Springer Verlag, Berlin, New York, 2009, 1-60. A document that can also be found as a manuscript via: http://www.physik.uni- leipzig.de/~uhlmann/PDF/UC07.pdf Even though I thought that I have a solid background in abstract algebra I somewhat got lost in Chapter 2 when he's trying to classify all the *-algebras that represent actual physical systems (starting at page 24 in the document). Do you have some recommendations for texts that introduce the *-algebra language in Quantum Mechanics in a more 'detailed' way. Because I kind of have the feeling that at a certain point Uhlmann just keeps skipping steps and I also lack some of the physical intuition concerning partial traces, canonical traces, purification and all that. From time to time I'd also be happy to see a concrete example. I'm looking forward to your responses. Best regards.


Dirac equation in QFT vs relativistic QM

How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The question was not clear. So I ask again. Why is the Dirac equation not meaningful as a single particle wave equation? Does many-particle interpretation of Dirac equation make it meaningful? If yes, how?


Toroidal Planets

I have read online in numerous places about the possible existence of toroidal planets, and I most people seem to believe that they could exist, but they also have no evidence to support this claim. I recently did a project in which I calculated the gravitational field around one-such planet, and I would like to continue working on it to either prove or disprove the possibility of these planets. Before I continue, I would like to know if any mathematically rigorous investigations into this question have already been conducted. I have searched, but I haven't found anything on planets, only black holes.


Lorentz force from velocity-dependent potential and Lagrangian

There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ is a scalar potential and $\mathbf{A}(t,x,y,z)$ is a vector potential. The Lagrangian $L=T-U$ Where $T$ is the potential energy. I'm trying to solve this with Lagrange equations for the x coordinate, and for the term $$\frac{d}{dt}\big(\frac{\partial{L}}{\partial\dot{x}}\big)$$ I'll show you how I'm trying to compute this with detail: $$\frac{d}{dt}\big(\frac{\partial{L}}{\partial\dot{x}}\big)=m\ddot{x}-q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)+q\frac{d}{dt}(\frac{\partial\mathbf{A}}{\partial{\dot{x}}}\cdot\mathbf{v}+\frac{\partial\mathbf{V}}{\partial{\dot{x}}}\cdot\mathbf{A})$$ Now, here's my big problem, I'll may have some trouble understanding the condition in wich i can interchange the differentiation with respect to $\dot{x}$ and $t$, i.e for example, $$q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)=q\frac{\partial{}}{\partial{\dot{x}}}(\frac{d}{dt}\phi)$$ This is done some pages before when Goldstein is deriving the lagrange equations, and I solved a problem in this chapter usig this. So now I have: $$q\frac{\partial{}}{\partial{\dot{x}}}(\frac{\partial{}}{\partial{x}}\phi\frac{dx}{dt}+\frac{\partial{}}{\partial{y}}\phi\frac{dy}{dt}+\frac{\partial{}}{\partial{z}}\phi\frac{dz}{dt}+\frac{\partial{}}{\partial{t}}\phi)$$ Now I have looked now for many hours for many ways in wich this derivation is made and the the expression above vanishes, but i don't understand why, is it because I'm assuming x,y,z in the scalar potential are functions of time themselves? If not i can see why it would vanish. Also this vanishes because i can't exchange the order of differentiation? * * * I totally see what you mean, I was applying it wrong, after you answered me I encountered that final term $\frac{\partial \phi}{\partial q_i}$, I had to substract it because it was actually: $$q\frac{d}{dt}(\frac{\partial{}}{\partial{\dot{x}}}\phi)=q[\frac{\partial{}}{\partial{\dot{x}}}(\frac{d}{dt}\phi)-\frac{\partial f}{\partial q_i}]$$ Thanks a lot you're awesome!


Is light red shifted in optical tweezers?

This is a question I put to my supervisor during my PhD many years ago, and never really got a satisfactory answer to. In an optical tweezers, assume that a beam of light is used to move a glass bead. My question is whether the outgoing light is red-shifted. If it is not I cannot reconcile how energy is conserved in the system, as work has been done to move the bead? Answers appreciated as always...


Moving "virtual" singularity?

Imagine two close, really big black holes rapidly spinning around each other. That setup would emit a terrible amount of gravitational waves. My question is, could those gravitational waves, if big enough, temporarily collapse space around them into a singularity? That is, not the black holes themselves forming a singularity, but the ripples they form on space-time, constructively interfering to form a singularity in a place where there is no actual mass.


Question on inflation

I have two particular questions regarding the inflationary scenario. They are: 1.) What is the physical origin of the inflaton field? 2.) Why has the potential of the inflation field its particular form?


How to find the distance of the source of sound in this problem?

On a cloudy day, the sound of thunder was heard 4.5 s after the flash of light was seen. How far was the cloud? Given that speed of sound = 340 m/s **This is what I've tried:** speed of light(c) = 10^6 * speed of sound( _v_ ) therefore, speed of light=340*10^6 m/s If _x_ is the distance of the cloud, then               x/(340*10^6)  -  x/340 = 4.5          =>(340*10^6  -  340)/x =  1/4.5     => x=1529998470`      but the correct answer is 1530m . Where am I going wrong?


What is the maximum mass of planet HD 40307 g

Recently discovered planet HD 40307g around HD40307 Star system is claiming that its inside a habitual zone of a solar system. Besides It has water where life could be evolved. I tried to calculate `g` of that planet. But failed! This is because the true mass of the planet could not be found. What we have is Minimum Mass which is `7.1` times Earth mass. My question is how do I know the Maximum Mass?


Force on a bar magnet

A bar magnet is freely falling vertically inside a conducting ring placed in horizontal plane. Will it fall with an acceleration equal to g, less than g or greater than g? ![magnet](http://i.stack.imgur.com/KsRlC.png)


Toroidal magnet in Tonomura et al. demonstration of Aharonov-Bohm effect

What does the magnetic field look like in Tonomura's toroidal magnet coated with a superconductor? ![Tonomura's toroidal magnet coated with a superconductor](http://i.stack.imgur.com/Q0e6x.jpg) See this paper to find results showing conclusive proof of the A-B effect and flux quantization. Context: For the A-B effect to work, there needs to be a magnetic flux in the area between the alternate paths which will produce the interference fringe patterns (but not necessarily along the paths themselves). The paper shows fringe patterns due to interference between the center of the toroid and outside the toroid, which means that there needs to be a flux in the plane of the toroid for A-B effect to be observed. Doesn't the superconductor coating quantize the flux penetrating the center of the toroid rather than the in- plane flux?


How to find the electric field due to a point charge in 3 dimensions?

A point charge with charge $+q$ is situated at $(x,y,z)$. How do I find the electric field at $(p,q,r)$? $E=k\frac{q}{r^2}$, right? So why isn't the electric field, $E=k\frac{q}{(x-p)^2+(y-q)^2+(z-r)^2}$ ??? Question: A charge $q=1\,\mu C$ is placed at a point (1m, 2m,4m). Find the electric field at point P (0m,-4m,3m). I end up with $E$ = 2.38x10$^2$ N/C; the given answer is 1.46x10$^3$ N/C


Experiments which show the phenomenon of closed electric field lines

In german physics introductory texts often the following experiment is cited which "shows" that a changing magnetic field can induce an electric field with has closed field lines. Here is a picture of the setup: ![](http://i.stack.imgur.com/iUV4v.jpg) A high frequency ac-voltage is applied to a coil around a glass sphere which is filled with a gas. Then a circular lighting effekt occurs as shown on the image. I have some questions about this:   1. How is this experiment called in english? Do you have references of textbooks or papers which discuss this in detail?   2. Why is it reasonable to say that this experiment "shows" that there "must" be a circular electric field? Let general epistomology aside, I think that the rough theory about this is, that the electrons wiggles back and forth because of the changing induced electric field. Since the magnetic field changes in sign the mean lorentz force due to the magnetic field is zero, so the lorentz force doesn't matter here. But if you don't know that the field lines are circular, especially that the field vectors are tangential to the circle, how conculde this from the experiment? For example it could be conceivable, that the field changes from radially invards to radially outwards and we get a radial wiggling of the charges which produce the lighting!? What (and what not) can one reasonable "conclude" from this experiment about the nature of the induced electric field and why?   3. How can one describe this experiment mathematically in detail?   4. Are there other, perhaps better or clearer experiments which demonstrate the existance of closed electric field lines produced by induction?


Modified Maxwell's equations

In 1985, Harmuth wrote that Maxwell's equations are incompatible with causality, and overcame the problem by adding a term for magnetic dipole currents, and as a consequence the problem of infinite zero-point energy and renormalization disappears. At least according to Harmuth's book: Calculus of finite differences in quantum electrodynamics By Henning F. Harmuth, Beate Meffert The foreword is readable at http://books.google.com/books?id=EQFENquY6fwC Are Harmuth's modifications generally accepted by the physics community as a more accurate description of reality than the unmodified equations ?


advantage of string theory over other theory-of-everything candidates

I am getting curious over why string theory, especially M-theory, is the most popular candidate for the theory of everything. It seems that all candidates of the theory of everything lack substantial experimental supports, so why is string theory so special? Thanks.


Can neutrino detectors tell what direction the neutrinos came from?

I was reading this question and got to thinking. Can neutrino detectors give us any clue where the neutrinos came from or when a supernova may occur? I was unsure and decided to ask that here.


Electrons Orbiting Sphere Magnet in Crooke's Tube

Were I to assemble a Crooke's Tube and insert a spherical magnet, with the poles of the magnet perpendicular to the electron stream, would the electrons begin to orbit the poles of the magnet as in the images of Terrella experiments? I am pretty sure yes. If so, are there equations to deduce the ideal distance from the electron emitter that the spherical magnet should sit? To determine how strong my magnet should be? What should I be reading to become more fluent in these sorts of experiments? Thank you for any assistance.


Converse of the Lagrangian form-invariance

The form-invariance of the Lagrange equations implies the existence of a function $\ A( q_k, t)$ so that $\ \begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \frac d {dt} A( q_k, t) \end{equation}$ Is there any way of proving the converse? That is, asuming the existence of the funcion $A$ prove the form-invariance of the Lagrangre equations.


Can virtual image be photographed

Is there any instrument that can photograph a virtal image? How would such a device work?


Can acceleration feel like constant gravity for indefinitely long?

So here's the setup: I'm in a spaceship, without windows as always, and the ship is accelerating upwards at a constant rate of $1\,\text{g}$. So inside the spaceship it feels like I'm being pulled down by gravity as normal. But this constant acceleration can't last forever: I have to approach $c$ asymptotically. I believe the equation for my speed is: $$v/c = \tanh(at/c).$$ That's what an outsider would observe. Correct? Not really my question though. What I want to know is, as my speed approaches $c$ and the outsider observes my acceleration slowing down, do I feel that? Or does time dilation and length contraction mean that I would feel the exact same acceleration/gravity inside, forever?


Simple harmonic oscillator system and changes in its total energy

Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to point $B$ and $CB=d$. Its motion is harmonic. The total energy of such a system is simply $\frac{1}{2} k \left(\frac{d}{2} \right)^2=\frac{1}{8} k d^2$ (because the equilibrium point is at $d/2$ and $d/2$ is the amplitude). Now, suppose we drop vertically some plasticine with the same mass $M$ from some height $h$. After it hits the oscillating object, it just sticks to it. My question is - why the total energy of the system doesn't change if the plasticine hits the object at point $C$ but it changes when it hits the object in the middle of $CB$ (it equals there to $\frac{1}{16} k d^2$)? Intuitively, I do understand that such a plastic collision contributes to the loss of energy, but I'm not sure how exactly it fits here and how the energy is lost. And even so, I do not understand how the position of the body influences the change in energy. * * * **Proposed solution:** The total mechanical energy at point $d/2$ is $E_{k,max}=P_{total}=\frac{1}{8} k d^2$: $E_k=\frac{M}{2}V^2\\\ V=\sqrt{\frac{2E_k}{M}}=\sqrt{\frac{2P_{total}}{M}} $ Due to the conservation of the horizontal momentum we get: $MV=2MV'\\\ \frac{2E_k}{M}=4V'^2\\\ V'=\sqrt{\frac{E_k}{2M}}=\sqrt{\frac{\frac{1}{8}kd^2}{2M}}=\sqrt{\frac{kd^2}{16M}}$ Therefore, the momentary kinetic energy (which is the total energy) after the hit is: $$E_{k, after}=\frac{2MV'^2}{2}=M \frac{kd^2}{16M}=\frac{kd^2}{16}$$


Coarse-graining on a second channel decreases mutual information?

Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$. This can be thought of as a bound on the capacity of a quantum channel called 2 (for example, you either perfectly correlate input $B_2$ with output $X_2$ (measurement result), or input $A_2$ with output $Y_2$, depending on the value of a bit $C_2$). Do we have: $I(X_1:B_1|C_1=0) \geq I(X_1\oplus X_2:B_1\oplus B_2|C_1=0, C_2=0)+ I(X_1\oplus Y_2:B_1\oplus A_2|C_1=0, C_2=1)$ ? $I$ is the mutual information and $\oplus$ the sum modulo 2. If this inequality is true, how would you show it?


Hyperkahler manifolds and their use in theoretical physics

Just as the title says:   1. What is the easiest definition of a Hyperkahler Manifold?   2. Could you give some examples of Hyperkahler manifolds, and manifolds which fail to be hyperkahler?   3. Why are such manifolds considered to be interesting in physics, and how do they arise in the study of supersymmetric gauge theories?   4. Apart from SUSY, are there any other branches of physics in which Hyperkahler manifolds appear?


Normal force and reaction

It is said that the normal force comes into play any time two bodies are in direct contact with one another, and always acts perpendicular to the body that applies the force. This force is a reactionary force because its value depends on the force the body from which the Normal force is emanating experiences and changes with change in the force applied. When I apply force on a body at an angle to the normal, even then I am getting a reaction equal to my force. This, too, is emanating from the body that is reacting. So why is the normal special?? When we discuss the free body diagrams of forces applied on bodies, we do not always show this normal - harness attached to a horse, rope on a pulley, wire attached to picture etc.?


Is light's path a wave?

In a lot of textbooks I see a schematic of light drawn as a squiggly line. I have even heard that some things are too small to be seen because they are smaller than the wavelength of light (and therefore light wiggles around them.) But now I think that light actually travels in a perfectly straight line and that no particles will feel any force from this light when they are not on this line. My friend says I'm wrong, and that if you increase the amplitude of the light, the ray will wiggle into the particle. To back it up, he linked me to a video about the single photon double slit experiment, where a single photon seems to interfere with itself. I'm not sure that this experiment proves him right, but it does seem to say that light wiggles around.


Absorption of Water

1. I was just watching the videos Chris Hadfield put on youtube from space, and was wondering why water doesn't get absorbed as well onto his toothbrush in space? And what characteristic of toothbrushes and wash clothes causes them to absorb water?   2. What exactly is water absorption?    3. Also, when he put water on his eye, it spread out over the surface. Why would water still want to spread out over his eye's surface without the force of gravity pulling on it?


Function to determine Earth-Moon distance vs. time

I had no luck trying to find a function that determines the distance between Earth and Moon versus time.   (Moon ground level to Earth sea level) _Accuracy:_   * the acceptable time range would be for years in [ 1900, 2100 ]   * 1 km error margin (or better)


Symmetry Breaking And Phase transition

1. Is every phase transition associated with a symmetry breaking? If yes, what is the symmetry that a gaseous phase have but the liquid phase does not?   2. What is the extra symmetry that normal $\bf He$ has but superfluid $\bf He$ does not? Is the symmetry breaking in this case a gauge symmetry breaking?


how to determine the direction of a wave propagation?

In the textbook, it said a wave in the form $y(x, t) = A\cos(\omega t + \beta x + \varphi)$ propagates along negative $x$ direction and $y(x, t) = A\cos(\omega t - \beta x + \varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\pm$ x direction, to my understand, we can drop the time term and ignore the initial phase $\varphi$ while analyzing the direction, i.e.$y(x, 0) = A\cos(\pm\beta x)$, however, because of the symmetry of the cosine function, $\cos(\beta x)\equiv \cos(-\beta x)$, so how come we determine the direction of propagation from that? I know my reasoning must be something incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\beta x$ corresponding to the propagation on positive x direction but not the opposite? Thanks.


What is $p$ in this candidate solution for the motion of N masses?

In this problem: > Consider the system of $N$ masses shown in the figure (see picture 1). > > a) Using the approximation for small angles, write the equation of > transverse movement for the n-th particle. > > b) Propose a solution of the form: > ${ψ_n}^{(p)}(t)=A^{(p)}\cos(nk^{(p)}+α^{(p)})\cos(w^{(p)}t+\phi^{(p)})$. > Find the dispersion relation and decide whether it depends on the contour > conditions. > > **Figure 1:** > > ![picture 1](http://i.stack.imgur.com/MuQu5.jpg) > > **Figure 2:** > > ![picture 2](http://i.stack.imgur.com/NLmaK.jpg) With regard to part b), I don't understand the notation of the proposed solution, to be more specific, I don't understand what $p$ is, is it an index? Could someone explain me what $\ddot{y_n}$ would be just to see what $p$ stands by?


What formulae for calculating fluid pressure change involves these parameters?

Okay... this is a bit desperate... I am reverse engineering an excel program that is used for calculating outputs for a pressure vessel used for industrial process fluid heating. There is a part in the excel sheet where a pressure drop is being calculated. But there are too many unit conversions and other magic constants in intermediate steps i cannot regonise. I have some physics background but not enough in fluid mechanics to decipher. For starters, I have intermediate step formulas like:               (fluid flow rate[lbs/h]]) * (1545) * ((inlet plus outlet temp[deg c] / 2)*9/5+460) /          ((molecular weight[??])*((pressure[psig])+14.5)*144)      ## The formula looks like this: $\frac{\dot{m} \times 1545 \underbrace{\left[\frac{(T_\text{in} + T_\text{out})}{2}\times(\frac{9}{5}) + 460 \right]}_\text{Degrees C to Deg F to Rankine}}{M_w \times (P_g + 14.5) \times 144}$ I am hoping someone who deals with fluid mechanics will be familiar with many formulae to tell what it might be. The inputs to the entire calculation steps consists of flow rate, inlet and outlet temperatures, molecular weight, viscosity and thermal conductivity; and also dimensions of the vessel. Greatly appreciate everyones help! Thanks. **EDIT** : The actual problem is here http://www.jakesee.com/se/physics_se_16403.png and the birthplace of the image is here http://stackoverflow.com/questions/7647478/generate-a-flat-list-of- all-excel-cell-formulas


Entropic force in rubber bands

The formulation for entropic force of stretching a rubber band is now known. Are there any other such daily life examples that also have mathematical formulations of entropic forces associated with them? Or can the idea and theory be extrapolated in such a way that more physical phenomena can be explained by it?


Distance increase between Sun and Venus

Is the distance between the Sun and Venus increasing?   What is the measurement?   May I request for a formula?


Satellites and gravitation

A satellite with mass $m$ orbits a planet of mass $M$ in a circular path with radius $r$ and velocity $v$. Due to some internal technical failure, the satellite breaks into two, similar parts with mass $m/2$ each. In the satellite's frame of reference, it appears the two parts move radially, in opposite directions, along the line connecting the original satellite and the planet's center, each with velocity $v_0/2$. I am expected to show that right after the technical failure, each of the two parts has a total energy equal to $-3GM/(16r)$ and angular momentum equal to $(m/2)√(GMr)$, wrt the planet's center. The total energy of each of the two parts should be, I believe: $E_{tot} = mv_0^2/16 - GmM/(2r)$. Now, isn't angular momentum preserved despite the failure? However, why isn't the angular momentum zero if the two parts are moving in opposite directions?


Can you use a laser to measure the speed of light with a rotating mirror?

I have learned that the classical measurement of the speed of light with a rotating mirror does not work with a laser (as opposed to, say, a mercury- vapor lamp). Can you tell me if and how coherency of light is necessary/contraindicated for the success of this experiment?


Two-body problem questions

I am self studying the two body problem and I'm stuck on the following: I have given $$\ddot{\vec{x}}_1= - G m_2 \frac{\vec{x}_1-\vec{x}_2}{|\vec{x}_1-\vec{x}_2|^3}$$ and $$\ddot{\vec{x}}_2= - G m_1 \frac{\vec{x}_2-\vec{x}_1}{|\vec{x}_1-\vec{x}_2|^3}$$ with masses $m_1,m_2$ at the position $x_1,x_2$ and gravitational constant $G$. How can you show only with these two equations that total momentum $m_1\dot{\vec x_1}+m_2\dot{\vec x_2}$ is constant in time? And why is the total mechanical energy $$\frac12m_1|\dot{\vec x_1}|^2+\frac12m_2|\dot{\vec x_2}|^2-\frac{Gm_1m_2}{|\vec{x}_1-\vec{x}_2|}$$ conserved?


Why does a laser beam diverge?

I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on photons rather than wave physics.


Minimal Kinetic energy for particle in a box

This is driving me crazy! The question goes as follows: A proton is enclosed in a zone of length 2pm along the x-axis. The minimal kinetic energy of the proton lies closest to:   * 5000eV   * 0.5eV   * 50eV   * 500eV   * **5eV** (this is supposed to be the correct answer) So first I calculate the energy level for a particle in a box which is given by the equation $E_n = p_n^2/(2 m) = (n^2 h^2)/(8 m L^2) $ Here $L$ is the length of the zone. So I find $E_1=94007.53 eV$.. So.. How do I relate this to the minimal kinetic energy? \-- EDIT 1 -- After the first hint from someone, I changed my calculations (I used the mass of an electron instead of a proton..) NOW I get $E_1 = 51eV$, which is still wrong..


Special relativity, acceleration and being in two places at once (How to resolve this paradox?):

Imagine I am on Earth I have a clock which measures time $t$. When my Earth clock reads time $t$, then when I look at a moving spacecraft's clock I see the time $t'$. Let us assume time is measured in years. Let us suppose the clocks were initially synchronized so that when $t=0$, $t'= 0$. Now let us suppose that the speed of the rocket continues to increase in the following way (Note that I have increased the speed in discrete time steps for convenience, but I'm sure a continuous speed function could be obtained but just mathematically tedious): At $t = 0$, $v = c\sqrt{3/4}$. So when $t = 1$, $\delta t'= \frac{\delta t}{\gamma} = 0.5.$   At $t = 1$, $v = c\sqrt{15/16}$. So when $t = 2$, $\delta t'= \frac{\delta t}{\gamma} = 0.25.$   At $t = 2$, $v = c\sqrt{63/64}$. So when $t = 2$, $\delta t'= \frac{\delta t}{\gamma} = 0.125.$ and so on... If $v$ continues to increase over time in this way, then $t'$ will approach 1, but will never reach 1. It appears from the perspective of the Earthling, that the person in the rocket never ages by over 1 year. From the person in the rocket however, he will experience time as normal. Now let us assume that after the person in the rocket has aged 10 years, he leaves his rocket ship, and travels back to Earth, leaving his rocket to continue accelerating in the same pattern. Now when the man reaches Earth, will he not see himself in the rocket, where he sees himself not having aged 1 year? (Ignore optical effects for the purpose of this question.)


Air velocity in a double-skin facade

I was wondering how the air flows in a double skin facade? Say I have a glass facade with a depth of 1m, a height of 200m and a width of 10m. It is a single vertical shaft all the way up with openings at the bottom to let air in and at the top to let air out: ![](http://i.stack.imgur.com/w4Xrc.png) There is a solar radiation of 100W/m2 hitting the outside of the facade and the air velocity at ground level is 1m/s. Can I expect the velocity of the air to increase as it travels through the shaft? and to what degree?


size and distance of mirror used in projected "smart" whiteboard

Apologies in advance for complete ignorance of optics (beyond conic sections) How is the size, distance, and angle of the mirror used here calculated? The mirror is I believe the open flap in front of the projector. ![enter image description here](http://i.stack.imgur.com/BNEiv.jpg)


What are the rainbow and ladder approximations in a solid state physics context?

All references I find talk about quarks and gluons, where I have only very limited knowledge about. From it's name (rainbow) I guess it applies to fermions coupled to bosons and we're interested in the single particle Green function for the fermion, and we approximate the sum over all diagrams by only summing those diagrams where no phonon lines cross (so that Feynman diagrams look like rainbows). Is that correct? I also noted that all search results always talk simultaneously about the rainbow and the ladder approximation, as was also pointed out in a comment below.


Flow rate of a syringe

Suppose a syringe (placed horizontally) contains a liquid with the density of water, composed of a barrel and a needle component. The barrel of the syringe has a cross-sectional area of $\alpha~m^2$, and the pressure everywhere is $\beta$ atm, when no force is applied. The needle has a pressure which remains equal to $\beta$ atm (regardless of force applied). If we push on the needle, applying a force of magnitude $\mu~N$, is it possible to determine the medicine's flow speed through the needle?


Optical Theorem ,how can experiment distinguish the unscattered wave from the forward scattered wave?

How can experiment distinguish the unscattered wave from the forward scattered wave? The Optical Theorem says the imaginary part of the forward wave determines the cross section for an initial two-particle state $|i\rangle$ in the center of mass frame through the equation $$2E_Tp_{in}\sigma=Im{\cal M}_{ii}$$


Simularity transformation of Heisenberg XXZ Hamiltonian

I am considering the Heisenberg XXZ Hamiltonian: $$ H(\Delta, J) = J\sum_{i=1}^L\left(\sigma^x_i\sigma^x_{i+1} + \sigma^y_i\sigma^y_{i+1} + \Delta \sigma^z_i\sigma^z_{i+1} \right) $$ Apparently, one can show: $$-H(-\Delta) = UH(\Delta)U^{-1}$$, where $U=\Pi_{m=1}^{M/2}\sigma_z^{2m}$ is a unitary transformation. I am having some trouble showing this. Of course the transformation commutes with the z-Pauli matrices and using $(\sigma^z)^{\dagger} = \sigma_z$ and $(\sigma_z)^2 = \textrm{id}$ this term is unchanged. However, I am left with mix terms for the first and second term. Furthermore, my reference claims that because of this simularity transformation, $J$ can be omitted if $-\infty<\Delta<\infty$. I don't really see how this follows from the simularity transform. Any help is much appreciated!


What is the interaction with Higgs field(s) that give the quarks so much different masses?

The masses of quarks are:                mu 2∼3 MeV    md 4∼6 MeV      mc 1.3 GeV    ms 80∼130 MeV      mt 173 GeV    mb 4∼5 GeV


Quantum XOR: How do you generalize it?

Consider the classical XOR Gate: Given a 2 bit system: $G = [u_1, u_2]$ $$XOR(G) = (u_1 + u_2) \ mod \ 2$$ Is the following a good generalizaiton of a Quantum XOR Gate: Given a 2-qubit system: $Q = a_1|00] + a_2|01] + a_3|10] + a_4|11]$ We now write: $$XOR(Q) = a_1|0] + a_2|1] + a_3|1] + a_4|0] = (a_1 + a_4)|0] + (a_2 + a_3)|1]$$ Notice that if we set $a_i = 1$ and the rest to $0$ that the quantum XOR behaves exactly like a classical XOR but it also has the ability to generalize to fractional and complex $a_i$


Lightbulbs in circuits. Why does the current go UP?

> **Possible Duplicate:**   >  Why does the light bulb's brightness decrease? ![enter image description here](http://i.stack.imgur.com/uZeX6.jpg) So I have a circuit in the picture with three resistors. If I were to remove/disconnect $R_2$ from the circuit, what would happen to the overall brightness of $R_3$ and $R_1$? Now the answer I already have is that $R_1$'s brightness will go down and $R_3$ will go up. I understand that $R_1$'s brightness goes down because by removing the $R_2$, the net resistance goes up (destroying a parallel combination) But I don't understand why $R_3$ would go up? Conceptually this makes no sense to me. How do I know that the new current would always be larger than when the current splits at the junction? I would like a conceptual **physics** answer and not equations comparing inequalities.


Reissner-Nordström Black Holes

The Reissner-Nordström black holes are described by the metric, \begin{align} ds^2 = -\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)dt^2 + \frac{1}{1-\frac{2M}{r}+\frac{Q^2}{r^2}}+r^2d\Omega^2 \end{align} where $M$ is the mass of the black-hole and $Q$ is the charge of the black hole. I am using the convention of $(-,+,+,+)$ here. The minimum mass horizon, which one can calculate, is given by $M_{min} = Q^2$. Part of the derivation is at: http://aesop.phys.utk.edu/ads-cft/L3.pdf. (Notes on AdS/CFT Correspondence from University of Tennessee, notes 3). Suppose we approach the horizon, the metric begins to approach the $AdS_2 \times S^2$ metric, given by, \begin{align} ds^2 = R_1^2\frac{-dt^2}{z^2}+R_1^2 \frac{dz^2}{z^2} +R_2^2 z^2 d\Omega^2. \end{align} Note that in the link I provided, there is a sign switch (since the note writes for a Euclidean AdS). I was wondering, what would be the overall generators of the group? Is it just the generators of $AdS_2$ and the generators of $S^2$ (which are from $SO(3)$)? Are there not some more complicated generators that mix the two spaces?


Particle-hole symmetry of Bogoliubov - de Gennes equation

Bogoliubov - de Gennes equation reads, $$\left( \begin{array}{cc} H_{0} - E_{F} & -i\sigma_{y}\Delta \\\ i\sigma_{y}\Delta^{*} & E_{F} - H_{0}^{*} \end{array}\right) \left( \begin{array}{c} \psi_{e}^{\uparrow} \\\ \psi_{e}^{\downarrow} \\\ \psi_{h}^{\uparrow} \\\ \psi_{h}^{\downarrow} \end{array}\right) = \mathcal{E} \left( \begin{array}{c} \psi_{e}^{\uparrow} \\\ \psi_{e}^{\downarrow} \\\ \psi_{h}^{\uparrow} \\\ \psi_{h}^{\downarrow} \end{array}\right) $$ The above Hamiltonian which is $4$x$4$ obeys particle-hole symmetry, $$H = -\mathcal{C}H\mathcal{C}^{-1}$$ where $\mathcal{C} = \tau_{x}\mathcal{K}$, with $\tau_{x} = \sigma_{x} \oplus \sigma_{x}$ and complex conjugation operator $\mathcal{K}$. This symmetry implies that if You have solution $\Psi$ with energy $\mathcal{E}$, then You also have solution $\mathcal{C}\Psi$ with energy $-\mathcal{E}$. I always thought that if Hamiltonian posseses some symmetry then You have solution with the same energy. I am confused, cause I also read that holes are time-reversed electrons, $$\psi_{h} = \mathcal{T}\psi_{e}$$ with $\mathcal{T} = i\sigma_{y}\mathcal{K}$, but I can't show this from above BdG equation.


Cosmic bubble collision

I'm reading this review right now. The claim seems to be that when you have an expanding "false vacuum", finite size bubbles form due to phase transition (cause by finite action instantons) (which, in itself might then evolve, etc). My questions is, if I imagine these bubbles to be moving away from each other due to exponential expansion of the false vacuum, then how will they come near each other and collide?


on quantum steering

I have become interested in quantum steering after listening a talk and tried to read more about it. I think I am more confused now. My understanding is as follows: > Sharing a (entangled) state, Bob wants to make a series of measurements and > Alice wants to make a guess of the measurement outcome. After which I am totally at loss, as a few papers which I have seen refers to a steering inequality. However it is heavily state dependent, and not like Bell (or Bell type) inequalities. Can someone point out some basic references or clear up my confusion. I am sorry for not writing it more clearly. May be by the questions, I shall be able to understand. Advanced thanks for any help.


Difficult to visualize Franck-Condon overlap

The franc condon principle says about the vertical transitions, means the coordinates of the nuclei remains unchanged during an electronic transition. But, what is the physical meaning or how to visualize 1) Franck Condon overlap or Franck Condon Factor? 2) I specifically came across the vibrationally mediated photodissociation (VMP). Here it says VMP is effective when there is a favorable Franck Condon overlap between vibrationally excited states of the ground electronic states and those of the electronically excited dissociative or predissociative states. Does this mean the potential energy surfaces of the said states really overlap? if anyone can give me a pictorial explanation, that would be nice.


Pendulum (highschool)

I have a question concerning pendulum. On our physics lessons we built simple pendulum consisting of weight and a thread ($\ell=0.31$ m). We tried to calculate $g$ from the well known formula $$T_m = 2\pi\sqrt{\frac{\ell}{g}}$$ which leads to $$g = \frac{4\pi^2 \ell}{T_m^2}$$ Of course, this formula comes from conclusion that $\sin(\alpha)-\tan(\alpha)$ is neglectable for "small" $\alpha$. Hence our target was to find formula, which works for larger $\alpha$ as well. According to this two sources, this should do the trick: $$T_m = 2\pi\sqrt{\frac{\ell}{g}}(1+\frac{1}{4}\sin^2(\frac{\alpha}{2}))$$ from which I came to this conclusion: $$g=\frac{\ell\pi^2(\cos(\alpha)-9)^2}{16T_m^2}$$ Here is my problem: After using this formula and our measured data, I got $\overline g = 8.9085$, which is obviously not very accurate (for ~48°N). Any ideas where I've made a mistake?


Can you gauge a $U(1)_L$ symmetry?

I recently calculating the one loop correction for the propagator of a gauge boson, $\hspace{5cm}$![enter image description here](http://i.stack.imgur.com/gk0dJ.png) I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop correction was, \begin{equation} = \frac{ - 4i }{ ( 4\pi ) ^{ d / 2}} \int dx \frac{ \Gamma ( 2 - d / 2 ) }{ \Delta ^{ 2 - d /2 }} \left[ ( g _L ^2 + g _R ^2 ) x ( 1 - x ) \left( g _{ \mu \nu } - \frac{ p _\mu p _\nu }{ p ^2 } \right) + g _{ \mu \nu } ( g _L ^2 - g _R ^2 ) m ^2 \right] \end{equation} Now we also know that for a $ U(1) $ invariant theory we should have the Ward identity and hence this one loop correction would be of the form, \begin{equation} \Pi _{ \mu \nu } = \Pi ( p ^2 ) \left( g _{ \mu \nu } - \frac{ p _\mu p _\nu }{ p ^2 } \right) \end{equation} So it appears that in order for this propagator to arise from a $U(1)$ invariant theory, we must have $ g _L = g _R $ (unless I made a mistake). I found this very strange. Is it not possible to gauge a $ U(1) _L $ symmetry and if so why? As a working example I invented the theory below which seems like it could be gauged: \begin{equation} {\cal L} = i \psi ^\dagger \bar{\sigma} ^\mu \partial _\mu \psi + i \chi \sigma ^\mu \partial _\mu \chi + \phi ^0 \chi \chi + \phi ^{+ + } \psi \psi + \phi \mbox{ terms} \end{equation} where $ \psi $ is a left chiral spinor and $ \chi $ is a right chiral spinor. The particles transform under the symmetry as, \begin{equation} \psi \rightarrow e ^{ i \alpha } \quad \chi \rightarrow \chi \quad \phi ^0 \rightarrow \phi ^0 \quad \phi ^{ + + } \rightarrow e ^{ - 2i \alpha } \phi ^{ + + } \end{equation} This seems like a perfectly fine theory, whose symmetry can be gauged but if my conclusion above is correct for some reason I shouldn't be able to gauge this $ U(1 ) _L $ symmetry. Did I make a mistake or is there some deep reason this can't be done?


Order of Monte Carlo integration and frequency summation

I am currently trying to calculate an integration formula of a linear response function by Monte Carlo method. It is a multiple integration over three 3D vectors, i.e., nine dimensions in all. And because of green function, there will also be an infinite discete frequency summation. This can be done either inside the integral, i.e., when evaluating the integrand, or outside the integral, i.e., summing up the integral result for each discete frequency point. My question is that whether this order issue matters or not.


Electric field near surface of a conductor?

> If the electric field on a positive charge is non-zero, then the charge > accelerates in the direction of the field. The field at the surface of a conductor is perpendicular to the surface. Why don't the surface charges accelerate away from the surface of the conductor then?


A question about a complex integration in Peskin's QFT textbook

In page 27 (2.52), the integration is $$\int_{-\infty}^{\infty}dp \frac{p e^{ipr}}{\sqrt{p^2+m^2}}$$ He says that there are two branch cuts starting from $\pm im$ ![enter image description here](http://i.stack.imgur.com/pyO89.png) But I learn in complex analysis that $\sqrt{z^2+m^2}$ has only one branch cut from $-im$ to $im$, because point going around $im$ or $-im$ only will gets a minus, but point going around $\infty$ only will keep the sign. Therefore $\infty$ point is not the branch point and the branch cut is from $-im$ to $im$. So who is wrong?


Information loss in a black hole

How does the Holographic Principle help to establish the fact that all the information is not lost in a black hole?


Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to sublattice $B$, which comes from the fact that there are two atoms per unit cell. $a$ is the annihilator on sublattice $A$ and $b$ on sublattice $B$. This operator is Hermitean. I can use its Fourier representation $$ H \sim \sum \limits_k H(k) e^{ikr} $$ Where $r$ denotes lattice vectors and $k$ the corresponding lattice momentum. The amplitudes $H(k)$ are not Hermitean but they satisfy $H(k)^\dagger = H(-k)$ obviously, so the above sum remains Hermitean if I take out pairs of $(k,-k)$. I cannot figure if by discretizing the elementary cell in $k$-space, thus effectively taking out some of the $k$'s in the above sum, the Hermiticity of this truncated Hamiltonian would get lost.


Young's Double slit experiment (Finding the value for slit separation)

The question illustrates mono-chromatic red light passing through 2 slits and forming an interference pattern on the screen. The question then asks to suggest a suitable value for the slit separation. How do I find a value for this or what basis should I approximate my answer?


Super High Frequency Electromagnetic Radiation - String Theory

I am a serious high school student with one year of physics class experience, so please point out if there are any flaws in my question or reasoning. Thanks! Gamma ray radiation possesses a wavelength below 10 picometers ($10^{-11}\: \mathrm{m}$) and is capable of ionizing atoms. If a photon was endowed with sufficient energy ($~1.24 \times 10^{10}\: \mathrm{J}$) to yield a wavelength on the scale of a Planck length ($~1.62 \times 10^{-35}\: \mathrm{m}$), the theorized length of strings, would it be capable of producing interference or some other influence on strings, if they exist? From what I have learned, string theory postulates that all matter and energy is composed of tiny vibrating strings that require 11 dimensions (please correct me if I am wrong). Would EM radiation with a wavelength on the scale of a string's length then be able to cause some sort of interference or influence on the string's properties or vibrational pattern? Considering that an electron is so much larger than a string, such an experiment seems like trying to cause a ripple through a floating one inch piece of string by rapidly shifting an adjacent mountain one inch back and forth. At such high energy levels, would an electron even be able to remain stable, or would it dissociate into its constituent particles? If so, would there be some amount of sufficient energy that would dissociate even those particles? Here is the equation that I used to approximate the energy required to endow an electron with a wavelength of $~1.6 \times 10^{-35}\: \mathrm{m}$: $$E=\frac{hc}{\lambda} = \frac{6.63 \times 10^{-34}\: \mathrm{Js} \cdot 3.00 \times 10^8\: \mathrm{m/s}}{1.62 \times 10^{-35}\: \mathrm{m}}= 1.23 \times 10^{10}\: \mathrm{J}$$


How to reconcile angular momentum measurements?

Imagine a particle tracing a counter-clockwise circular path on a flat table with a certain speed. The particle is tied with a massless string of length $R$ to a point $P$ at the center of the circular path. Will the particle rotate about $P$ forever at constant speed in the absence of any external force? Consider using different origins to measure the physical quantities. **Measurement Case 1 (origin $O$ at $P$):** If I choose the origin $O$ of a Cartesian coordinate system to be at $P$, the flat table is the xy-plane and the particle rotates about the point $O$ as described. Specifically, the angular velocity $\vec{\omega}$ of the particle points in the direction of the positive z-axis, the position of the particle at any time is specified by the position vector $\vec{r}$, the angle between $\vec{\omega}$ and $\vec{r}$ is always $\frac{\pi}{2}$, and the angular momentum $\vec{L}$ is as follows: $\vec{L} = \vec{r}\times\vec{p}\;\ldots\text{ definition} \\\ \hphantom{\vec{L}} = m\,(\vec{r}\times\vec{v})\;\ldots\text{ definition of linear momentum }\vec{p} \\\ \hphantom{\vec{L}} = m\,(\vec{r}\times(\vec{\omega}\times\vec{r}))\;\ldots\text{ definition of linear velocity in terms of angular velocity} \\\ \hphantom{\vec{L}} = m\,(\vec{\omega}\,(\vec{r}\cdot\vec{r}) + \vec{r}\,(\vec{r}\cdot\vec{\omega}))\;\ldots\text{ scalar triple product}\\\ \hphantom{\vec{L}} = m\,(\vec{\omega}\,(r^2) + \vec{r}\,(0))\;\ldots\;\vec{r} \perp\vec{\omega}\\\ \hphantom{\vec{L}} = (m\,r^2)\,\vec{\omega}$ That is, $\vec{L}$ and $\vec{\omega}$ have the same direction, and $\vec{L}$ does not change direction and magnitude while the particle is rotation about $P$. Therefore, the particle will rotate about $P$ forever at constant speed in the absence of any external force because the angular momentum $\vec{L}$ is conserved (the presence of a centripetal force does not affect the angular momentum $\vec{L}$ in any way because a centripetal force is always parallel to the moment arm, and therefore, does not give rise to a torque $\vec{\tau}$). **Measurement Case 2 (origin $O'$ vertically directly beneath $P$):** Now if I choose to describe the same problem by choosing an origin $O'$ of a Cartesian coordinate system to be vertically directly beneath $P$, the particle's position vector $\vec{r}'$ makes an angle $\phi$ with the z-axis and the particle's angular momentum $\vec{L}'$ also makes a $\phi$ angle with the z-axis. As the particle rotates, however, $\vec{r}'$ also rotates about the z-axis, and therefore, the angular momentum $\vec{L}'$ keeps changing direction as shown below: $\vec{L}' = \vec{r}'\times\vec{p}'\;\ldots\text{ definition} \\\ \hphantom{\vec{L}'} = m\,(\vec{r}'\times\vec{v}')\;\ldots\text{ definition of linear momentum }\vec{p}' \\\ \hphantom{\vec{L}'} = m\,(\vec{r}'\times(\vec{\omega}\times\vec{r}'))\;\ldots\text{ definition of linear velocity in terms of angular velocity} \\\ \hphantom{\vec{L}'} = m\,(\vec{\omega}\,(\vec{r}'\cdot\vec{r}') + \vec{r}'\,(\vec{r}'\cdot\vec{\omega}))\;\ldots\text{ scalar triple product}\\\ \hphantom{\vec{L}'} = m\,(\vec{\omega}\,(r'\,^2) + \vec{r}'\,(r'\,\omega\,\cos(\angle(\vec{r}', \vec{\omega})))) \\\ \hphantom{\vec{L}'} = (m\,r'\,^2)\,\vec{\omega} + (r'\,\omega\,\cos\phi)\,\vec{r}'$ Because the angular momentum $\vec{L}'$ is not conserved due to changing direction, and a changing $\vec{L}'$ requires the presence of an external force to give rise to a net torque $\vec{\tau}'$ about the point that is used to measure $\vec{L}'$, which is the origin $O'$, the particle will not rotate about $P$ forever in the absence of any external force. But then, a contradiction arises: **the same phenomenon has a different outcome depending on the choice of origin $O$ or $O'$!** That is _unacceptable_ in Physics! How to resolve this contradiction?


Notation of plane waves

Consider a monochromatic plane wave (I am using bold to represent vectors) $$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, $$ $$ \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}. $$ There are a few ways to simplify this notation. We can use the _complex field_ $$ \tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} $$ to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the _complex amplitude_ ($\tilde{\mathbf{E}}_0$) when adding or manipulating fields. However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity: $$ \tilde{\mathbf{E}}_0 e^{i\mathbf{k} \cdot \mathbf{r}} $$ My question is, what is this quantity called? I've been thinking _time- averaged complex field_ , but then again, it's not really time-averaged, is it? _Time-independent_? Also, what is its notation? $\langle\tilde{\mathbf{E}}\rangle$?


Dimensional Analysis to Determine a Formula

The kinetic energy of a particle confined to a spherical region with a uniform internal potential depends on its mass, the radius of the sphere, and the Planck constant. An electron, confined to such a region of radius $1$ nm, has kinetic energy $0.38$ eV. Use dimensional analysis to find the kinetic energy of a proton confined to a region the size of an atomic nucleus (radius ≈ $5 × 10^{−15}$ m). I cannot produce a formula which results in $0.38$ eV as in the example.


Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?


Find the points where potential is null

Let's say we have two charges called $q_1$ and $q_2$, respectively $20 \, C$ and $-40\,C$, at a distance $d=1\,m$ We want to find all the points where electric potential is null. I solved the equation $$\frac{q_1}{4\pi\epsilon_0r_1} + \frac{q_2}{4\pi\epsilon_0(d-r_1)}=0$$ For $r_1$ (distance from $q_1$), and found $r_1=\frac13\,m$ However this is not the only solution: there is another point not in-between the charges, but $1\,m$ left from $q_1$ ($r_1=-1\,m$) How can I set up an equation giving me both the solutions?


Maximising entropy when energy is shared between systems

![enter image description here](http://i.stack.imgur.com/9kYAR.png) This is a problem to do with statistical physics, and the exchange of energy when we have two microcanonical ensemble. I don't understand why there should be a minus sign in the middle, if Energy* is the energy of system 1 , Energy total - E* is the energy of another system, Two system were put together and interact, I was told this equation can maximise entropy, but I can't really see why? I thought S = (S1(E*))+S(E total - E*) is the total entropy. therefore it should be a positive sign when you differentiate.. Just to clarify, S(E*) and S(E total - E*) , they are just labels, not function of E* and E total - E*


Angular momentum and EM wave

Is there any sense in saying that circularly polarized EM waves have angular momentum?


Some questions about spacetime topology, causality structures and other GR businesses

1) What are the exact conditions required for the canonical transformation? Most papers just assume away with global hyperbolicity, but is there a more general condition for it? "Quantum gravity in 2+1 dimensions" does contain an ADM description of the 2+1 D equivalent of the Kerr black hole, which also contains CTCs (but no event horizon since vacuum solutions in 2+1D have a vanishing Riemann tensor). And some spacetimes with CTCs (or non-time orientable) do have a well-enough behaved Cauchy problem, so I am wondering how far the causality can be weakened until you cannot define a hamiltonian. 2) Why is De Sitter space always assumed to have the topology RxS^3? I know it cannot be S^4, which would be the most natural (just a Wick rotation of the sphere metric), since S^4 does not admit a Lorentz structure, and I guess that it cannot be R^4 because I think it would suffer from geodesic incompleteness (missing pole and all), but why RxS^3, and is there any other manifold that would admit that metric? I guess that SxS^3 would work, and whatever other terrible quotient manifold you could do with S^3.


Is the sun's solar radiance spectrum matching up with water's absorption spectrum just coincidence?

People frequently point out that water has a pretty narrow range in which it isn't very absorbing of light, reaching a minimum at a wavelength of about 500nm: ![h2o](http://i.stack.imgur.com/973K4.png) And that our eyes have made good use of this tiny available range, through evolution: ![eye](http://i.stack.imgur.com/s9Sfa.jpg) That part makes sense to me, because the eye changed to match the physical parameters available. But if you look at the solar radiance spectrum: ![solar](http://i.stack.imgur.com/o4zaF.jpg) You'll see that its peak matches VERY closely with the range water lets light transmit through. But the sun's radiance is just roughly a black body, and the absorption of water is just something inherent to the chemical structure of water. So is it coincidence that they match up? Thanks!


The common major difference in thermal conducting properties between a random rock sample and metal?

what are some common major differences in thermal conducting properties between a random rock sample(non-metal) and metal?


NPT simulations: what is the inertia of volume fluctuations (using periodic boundary conditions)?

NPT molecular dynamics (MD) are most often implemented by adding some kind of piston that regulates the pressure. The inertia of this piston determines the dynamics of volume fluctuations in the system, i.e. how fast can the volume increase/decrease. When periodic boundary conditions are imposed, the system is not in contact with such pistons. What would be the intrinsic (not chosen by the user) inertia of the volume fluctuations? If I imagine an infinite periodic system, any volume change of the periodic box seems impossible, as it would make the unit cells at large distance sweep back and forth over very large distances, which would require an enormous kinetic energy. So, should I conclude that the unit cell can not change its shape/volume by itself under periodic boundary conditions? The application I have in mind is an extension of the "normal mode analysis" which is used for computing vibrational frequencies, by diagonalizing the mass-weighted Hessian matrix. The Hessian matrix $H$ contains the second derivatives of the energy with respect to atomic positions, while the mass matrix $M$ is a diagonal matrix with the masses on its diagonal. The mass- weighted Hessian is then given by $\tilde{H}=M^{-1/2}HM^{-1/2}$. Now I want to extend this principle to the volume degrees of freedom. The second derivatives of the energy with respect to shape and volume deformations of the unit cell are given by the elastic tensor $C$. So what I am now wondering about, is what is the corresponding "mass" matrix $M_C$? If I then diagonalize $\tilde{C}=M_C^{-1/2}CM_C^{-1/2}$, I will find the frequency of the volume fluctuations. Normal mode analysis can also be derived by applying the Lagrangian formalism on the total energy (kinetic plus potential) of the system after first approximating it by a Taylor expansion up to second order in the displacements. That is why I wonder about the "correct" MD equations of motion (without pistons attached to the system, but rather the system by itself). I hope to recognize the inertia $M_C$ in these equations.


The Difference between Thomas-Fermi Screening and Lindhard Screening

Assuming the general theory of screening related to electron-electron interactions, I was wondering if anyone could provide a clear, yet _conceptually_ complete explanation of the differences between the Thomas- Fermi and the Lindhard theories? Following the treatment by Ashcroft and Mermin (1976),I get the impression that the main difference is that the Thomas-Fermi model assumes that the electrostatic potential is slowly-varying in $\vec{r}$: $$E_i(\vec{k}) = \frac{\hbar^2k^2}{2m}-e\phi(\vec{r})$$, but I'm having trouble grasping the full physical significance of this statement.


Why are do neutral atoms shrink as their valence shells approach 8 electrons?

Why do neutral, unbonded atoms shrink in size as they approach having 8 electrons in their valence shells? A good example is elements 3 through 10 in this table, that is, lithium (1 valence electron) through neon (8 valence electrons). * * * For posterity, below is my earlier _incorrect_ version of my question. It's incorrect because it based on a table that used the atomic radii of _bonded_ atoms to describe their sizes. That is a very different measurement from the sizes of neutral atoms, since for bonded atoms you are tossing in additional electrons via the bonds. Very small atomic radii for halogens are easy to explain in that case, since the incomplete shells must fight very hard over the shared electrons to complete their own octets, making their bonding radii anomalously short. This question had an interesting back-and-forth, and I thank the two contributors both who both caught my chart interpretation error and answered the real and interesting question that lurked beneath my incorrect one. * * * If you look here at the relative sizes of charge-neutral atoms, you can see something I never noticed or even thought about until I tried to think through this question in the new Chemistry beta. Why are nearly-complete atomic shells, such as the seven valence electrons of fluorine, so compact in comparison to the bloated-but-very-stable situation created by adding one more electron to make neon? I thought I could answer that in terms of Pauli exclusion and wound up completely baffled. Can anyone help on this one? Why does "almost complete" equal "very small" in a charge-neutral atom?


Rotational kinetic energy during vertical circular motion of a particle

Why is it not necessary to take into account rotational kinetic energy when using the Law of Conservation of Mechanical Energy to solve vertical circular motion problems? After all, the particle is rotating about the centre of the circle and does have rotational KE, doesn't it? All the examples I have seen just use KE= 1/2 $mv^2$, e.g. here: http://www.physicsforums.com/showpost.php?p=2312566&postcount=4


Why is there no time dilation for frequency of a wave?

Since frequency of a wave is a function of time, then for a particular ray of the wave, why would the frequency remain the same when observed from a moving reference frame? The frequency should change according to time dilation. No?


Assumptions in Bell's Theorem

It is often Stated that Bell's Theorem is equivalent to the statement: No theory of Local Hidden Variables can reproduce all of the predictions of quantum mechanics. I see nowhere in Bell's Theorem the assumption of hidden variables. As far as I can tell, there is one assumption: locality, which is shown to conflict with quantum mechanics. If I am wrong on this, could someone demonstrate exactly how Bell failed to demonstrate that quantum mechanics is non-local by showing how the assumption of hidden variables is used in his theorem? To me -- and by the way Bell -- the problem is locality.


About vector form of friction

I read a text on mechanics and in the chapter on friction, there written that the kinetic friction is in the form $$f_k = \mu_k F_N$$ where $f_k$ is the kinetic friction, $\mu_k$ is the kinetic friction coefficient, $F_N=mg$ is the normal force due to the weight of the object. In another chapter, there mentioned the vector form of the force. But it is confusing that if $f_k = \mu_k F_N$, should the vector form written as $\vec{f}_k = \mu_k\vec{F}_N$? It doesn't look right to me since $\vec{F}_N$ is downward but $\vec{f}_k$ is along horizontal. So what's the right way to write the vector for, of kinetic friction? How does people know that $f_k = \mu_k F_N$? From experiment? I am guessing that if the object is moving at the direction $\hat{\mathcal{d}}$, so the vector form of kinetic friction should be $$ \vec{f}_k = -\mu_k(\vec{F}_n\cdot\hat{\mathcal{d}})\hat{\mathcal{d}} $$ Is that correct?


How the Lorentz transformation affects the metric tensor?

After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure: ![enter image description here](http://i.stack.imgur.com/5NiiK.gif) and in such coordinate system, according to this Wikipedia article, the metric will have off-diagonal non-zero elements: $$g_{ij}=\mathbf{e}_i.\mathbf{e}_j$$ which is not that of a flat spacetime: $$ g = \begin{bmatrix} 1 & 0 & 0 & 0\\\ 0 & -1 & 0 & 0 \\\ 0 & 0 & -1 & 0 \\\ 0 & 0 & 0 & -1 \end{bmatrix}$$ What's the problem?


is centrifugal release equal to explosive release

**Scenario # 1** I put gunpowder and then a ball bearing in an old musket and fire the bullet. **Scenario # 2** Lets imagine I had a motor with a disk on it and there was the same ball bearing stuck on the edge of the disk somewhere. If I spin the motor up fast and then release the ball bearing somehow, it will fire off perpendicular to the tangent line of the release point. Lets imagine for the purposes of comparison, I spin up the motor with enough rpm such that the ball bearing’s velocity equaled the velocity of the musket shot in scenario 1. Newton's 3rd law of motions states that "for every action there is an opposite and equal reaction". With that in mind, it shouldn’t really matter how I expelled the hunk of lead, would both systems react in the opposite direction the same at the moment of release? (I’m aware that the spinning system would have a torque while it was spinning, but lets ignore that) The reason I’m asking that is; intuitively the ball bearing is desperate to escape the faster the disk is being spun. It would seem weird that it would suddenly kick the system backwards when it was already straining to leave the system the whole time. Also, there seems to be an intuitive difference because the expelling force in the spinning system is 90 degrees to the applied force.


A question about the Weyl transformation for the vertex operator of the closed-string tachyon

I met a problem of derving the Weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103. Given the vertax operator of the closed-string tachyon > $$V_0 = 2g_c \int d^2 \sigma g^{1/2} e^{ik \cdot X} \tag{3.6.1}$$ It is said the Weyl transformation of Eq. (3.6.1) is > $$\delta_W V_0 = 2g_c \int d^2 \sigma g^{1/2} \left( 2 \delta \omega( > \sigma) - \frac{k^2}{2} \delta_W \Delta (\sigma, \sigma) \right) [ e^{ik > \cdot X(\sigma)} ]_r \tag{3.6.8}$$ here renormalized operator $[]_r$ and $\Delta (\sigma, \sigma)$ are defined as > $$ [ \mathcal{F}]_r = \exp \left( \frac{1}{2} \int d^2 \sigma d^2 \sigma' > \Delta(\sigma, \sigma') \frac{ \delta}{\delta X^{\mu}(\sigma)} \frac{ > \delta}{\delta X_{\mu}(\sigma')} \right) \mathcal{F} \tag{3.6.5} $$ $$ > \Delta(\sigma,\sigma') = \frac{ \alpha'}{2} \ln d^2(\sigma,\sigma') > \tag{3.6.6} $$ where $d(\sigma,\sigma')$ is the geodesic distance between > points $\sigma$ and $\sigma'$. I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \delta g^{1/2} = \frac{1}{2} \frac{1}{g^{1/2}} \delta g= \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} \delta g_{ab} = \frac{1}{2} \frac{1}{g^{1/2}} g g^{ab} 2 \delta \omega g_{ab} = 2 g^{1/2} \delta \omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \delta e^{i k \cdot X} = e^{i k \cdot X} i k^{ab} X^{ab} \delta g_{ab} = e^{i k \cdot X} i k^{ab} X^{ab} 2 \delta \omega g_{ab} = 2 i k \cdot X e^{ik \cdot X} \delta \omega,$$ but where do the $\Delta(\sigma,\sigma)$ and renormalized opeator $ [ e^{ik \cdot X(\sigma)} ]_r $ in (3.6.8) come from?


What if the universe is rotating as a whole?

Suppose in the milliseconds after the big bang the cosmic egg had aquired some **large** angular momentum. As it expanded, keeping the momentum constant (not external forces) the rate of rotation would have slowed down, but it would never reach zero. What implications would that have to measurments to distant supernova and CMB radiation? Do we have any experimental data that definitely rules out such as scenario? And to what confidence level? **Edit** A Recent Article suggests that the universe might indeed be spinning as a whole. Anyone care to poke holes at it?


What restores balance to a repulsive electric force on an electroscope?

I am an amateur physics enthusiast (during the day I am a police officer), and recently, I learned how to build an electroscope. I regret that I am learning how wonderful our universe is so late in my life, and I suppose it is hard to teach an old dog new tricks ( I do not know how to properly research a question, or really how to locate a correct answer surrounded by a lot of other information). My question is this: On a simple electroscope, when one applies a charged object, the thin pieces of foil will separate, due to a separation of charges. However, I am curious as what factor balances the electric force of repulsion, so that the leaves do not continue to separate any further.


Free vs bound vectors and torque

When considering basic Newtonian mechanics, we can treat vector as free and move their point of application at will. This is consistent with the affine nature of Euclidean space. However, when calculating torque on a body, we must treat forces as bound to their point of application. What is the mathematical reason for this? What does it imply for affine structure of Euclidean space*


Determine the normalisation constant of a piecewise wavefunction

I'm trying to find the normalisation constant $N$ for the following wavefunction: $$ \psi\left(x\right) = \left\\{ \begin{array}{lr} N \left(x^2 - l^2\right)^2 &\: \left|x\right| \le l \\\ 0 &\: otherwise \end{array} \right. $$ Using: $$ \int_{-\infty}^{\infty} \left|\psi\left(x\right)\right|^2 \, dx = 1 $$ The answer should be: $$ N = \sqrt{\frac{315}{256}} \frac{e^{i \phi}}{\sqrt{l}} $$ However I get: $$ \int_{-\infty}^{\infty} \left|\psi\left(x\right)\right|^2 \, dx \ = \int_{-l}^{l} N^2 \left(x^2 - l^2 \right)^4 \, dx \ = \ \frac{N^2}{10} \left[\frac{\left(x^2 - l^2\right)^5}{x}\right]_{\,-l}^{\,l} = 0 $$ Which is clearly wrong, and I do not understand where the phase could have come from. Am I approaching this completely the wrong way? * * * I have now corrected the integration, however I get (double checked with Mathematica): $$ N = \sqrt{\frac{315}{256}} \frac{1}{l^\frac{9}{2}} $$ Which is the wrong power of $l$ (it should be $\frac{1}{2}$). Substituting either the answer or my answer into the original integral does not yield $1$ either. (I am working though these quantum physics notes: http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture- notes/MIT8_04S13_Lec04.pdf)


Recovering 4-vector Lorentz transformation from spinor formalism

I'm trying to recover the 4-vector transformation laws using spinors. I have defined $$v^{\dot{a}b} = v^{\nu} \sigma_{\nu}^{\dot{a}b}$$ as usual, with $\sigma_0=1$. Now with the rules for dotted and undotted spinor indices i get the transformed v for a boost in the z-direction $$v^{\dot{a}b} \rightarrow v^{\dot{a'}b'}= \left({\mathrm{e }}^{ - \theta \frac{\sigma_3}{2}} \right)^{\dot{a'}}_{\dot{a}} \left({\mathrm{e }}^{ \theta \frac{\sigma_3}{2}} \right)^{b'}_{b} v^{\dot{a}b} = \begin{pmatrix} {\mathrm{e }}^{- \frac{\theta}{2}}&0 \\\ 0&{\mathrm{e }}^{ \frac{\theta}{2}} \end{pmatrix} \begin{pmatrix} v_0+v_3&v_1-iv_2\\\v_1+iv_2&v_0-v_3 \end{pmatrix} \begin{pmatrix} {\mathrm{e }}^{ \frac{\theta}{2}}&0 \\\ 0&{\mathrm{e }}^{ - \frac{\theta}{2}} \end{pmatrix}$$ where i have used the fact that $\sigma_3$ is diagonal and that ${\mathrm{e }}^A= \begin{pmatrix} {\mathrm{e }}^{A_{11}}&0\\\0&{\mathrm{e }}^{A_{22}} \end{pmatrix}$ holds for every diagonal matrix A. This gives me the wrong transformation! It would give me the correct transformation if I had $$v^{\dot{a}b} \rightarrow v^{\dot{a'}b'}= \left({\mathrm{e }}^{ - \theta \frac{\sigma_3}{2}} \right)^{\dot{a'}}_{\dot{a}} \left({\mathrm{e }}^{ \theta \frac{\sigma_3}{2}} \right)^{b'}_{b} v^{\dot{a}b} = \begin{pmatrix} {\mathrm{e }}^{- \frac{\theta}{2}}&0 \\\ 0&{\mathrm{e }}^{ \frac{\theta}{2}} \end{pmatrix} \begin{pmatrix} v_0+v_3&v_1-iv_2\\\v_1+iv_2&v_0-v_3 \end{pmatrix} \begin{pmatrix} {\mathrm{e }}^{ - \frac{\theta}{2}}&0 \\\ 0&{\mathrm{e }}^{ \frac{\theta}{2}} \end{pmatrix}$$ But I can't figure out why this should be the case. The only possibility I can think about would be if a relation like $\ A'_{\nu\mu} = M_{\mu}^{\ \rho}(M_{\nu}^{\ \theta})^{-1}A_{\rho\theta }$ $\rightarrow$ $A'=MAM $ would hold, but i can't find a formula like this. Doing the summation by hand i get the same result as with normal matrix multiplication without using the inverse matrix on the right hand side. Any tip or help would be much much appreciated!


Why do stars have absorption spectra?

Absorption spectra are a result of light of a certain wavelength exciting an atom from a lower energy level to a higher one and at the same time being absorbed. However, the atom should eventually go back down to its lower energy state, and at the same time emit a photon of same frequency it absorbed earlier. Overall, no changes to the star's spectrum should occur.


Supermarket refrigerator - why is it noticebable colder in this shop area?

As we know fridge can't cool room in which it is (according to Second Law of Thermodynamics, heat emitted by fridge is greater than heat absorped). However, when we go next to the fridge in supermarket we can feel coldness. Why? Thanks!


Force and torque on rigid body in space

![enter image description here](http://i.stack.imgur.com/kAclJ.jpg) Assume that we have two disks in space. If we act same force F on each disk like above figure, how they react? I first thought that since $F_{ext}=\frac{dP_{cm}}{dt}$, the CM(center of mass) momentum will become same. But what about angular momentum thus energy? In first case, there will be no torque, so no angular momentum. But for second case, there exists external torque about CM, so the ball will rotate. Then, it will have spin motion, so there will be additional kinetic energy. Is this correct? How same force can give different total energy change?


Time to immerse in a fluid

Whenever you drop an object in water, it takes some time to get fully immersed. I was wondering if this depends upon the buoyant force of the water on the object, slowing it down. However, I was not able to come up with a general formula for this. I wanted to know, what would be the time taken for an object of density $d_1$, dropped from a height $h$ to be fully immersed in a liquid of density $d_2$. Take gravity as $g$. Part of the reason I was not able to do this was because I thought it might involve calculus (since the volume of water displaced changes over time and so does the buoyant force). I'm not competent enough with calculus to be able to attempt this sort of problem. I know friction will affect the time, but what if there were $no$ friction? Can this be further generalized to gases as well?


Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are $[D,P_{\mu}] = iP_{\mu}$ and $[D,K_{\mu}] = -iK_{\mu}$, so that $P_{\mu}$ and $K_{\mu}$ are raising and lowering operators, respectively, for the operator $D$. This is analogous to the operators $\hat a$ and $\hat a^{\dagger}$ being creation and annihilation operators for $\hat H$ when discussing the energy spectra of the $n$ dimensional harmonic oscillator. My question is, while $\hat a$ and $\hat a^{\dagger}$ raise and lower the energy by one unit $( \pm \hbar \omega)$ for each application of the operator onto eigenstates of $\hat H$, what is being raised and lowered when we apply $P_{\mu}$ and $K_{\mu}$ onto the eigenvectors of $D$? Secondly, what exactly do we mean by the eigenvectors of $D$? Are they fields in space-time? Using the notation of Di Francesco in his book 'Conformal Field Theory', the fields transform under a dilation like $F(\Phi(x)) = \lambda^{-\Delta}\Phi(x)$, where $\lambda$ is the scale of the coordinates and $\Delta$ is the scaling dimension of the fields. Can I write $F(\Phi(x)) = D\Phi(x) = \lambda^{-\Delta}\Phi(x)$ to make the eigenvalue equation manifest? Thanks for clarity.


Formation of a black hole and Hawking radiation

From the perspective of an outside observer it takes infinitely long for the black hole to form. But if the black hole is no extremal black hole, it emits Hawking radiation. So the outside observer can detect the Hawking radiation of a black hole that hasn't yet formed from his point of view. How can this be?


Falling off a chair, how best to save yourself

If I consider a man sitting on an office chair that reclines backwards iff you lean backwards. What could be done to prevent hin from falling? a) raising his legs till they are parallel to ground. b) bring the feet closer to himself(as close as possible) and press them down on the ground. Please don't suggest any other way. I want to compare these two. Ie, which is better. Argument in favor of a) The center of mass moves away from the rear end of the chair towards the forward end of the chair so torque of weight will restore the chair. (This works if you are in an armchair, and are rocking to and fro). However, intuition says contrary to this. I feel scared of falling If I do this. Argument in favour of b) Intuition favours this. Normally when we talk about balancing our body, what "feels right" is better for the safety. So, I can not say what is right. Please explain what your views are in this case. Again, the purpose of this question is not to ask what should be done, only what is better to do out of these two choices.


A question from Hilbert and Courant's Vol II of Methods of Mathematical Physics (I might have spotted an error)

In page 751 (I hope some folks have a copy of it, legal or otherwise, I have a legal one :-D), I am attaching scans of pages 750-751. ![enter image description here](http://i.stack.imgur.com/9S24K.jpg) ![enter image description here](http://i.stack.imgur.com/1TQ97.jpg) Anyway, I don'tunderstand two things, the equation in page 751, the RHS of $u(x_0,0,0,t_0)$, in $\psi$, shouldn't the x-component be $\alpha t_0/2 + x_0$? and in the second term shouldn't it be just $\alpha t_0 \psi_x$? Here is my reasoning, the x-component should be calculate according to Lorentz transformation in page 750: $$x = \frac{t_0}{\sqrt{t_0^2-x_0^2}} (\frac{\alpha}{2}\sqrt{t_0^2-x_0^2})+\frac{x_0}{\sqrt{t_0^2-x_0^2}}\sqrt{t_0^2-x_0^2}=t_0 \alpha/2 +x_0$$ and as for the second term there, I might be wrong but: $$\chi_{x'} = \psi_{x'} = \psi_x \frac{\partial x}{\partial x'} = \frac{t_0}{\sqrt{t_0^2-x_0^2}} \psi_x $$ So shouldn't it be $\alpha t_0 \psi_x$, instead of $(\alpha t_0+x_0) \psi_x$. I guess I missed something. Thanks in advance...


Name for thermodynamic derivative $dP/dT|_V$?

While trying to express the isoentropic sound speed as partial derivatives of $V$ and $T$ only I end up, as part of the longer expression with $dP/dT|_V$ (which according to a Maxwell relation is the same as $dS/dV|_T$). Does such quantity $dP/dT|_V$ have a given name? or it is simply expressible as a combination to other quantities (like $V dP/dT|_V$) with a given name? (By "given name" I mean, for example that $-V dP/dV|_T$ is the Isothermal Bulk Modulus)


Chandrasekhar Limit

A white dwarf is essentially a degenerate electron gas, in which pressure of degenerate electrons supports gravitational pressure. As a simplified model of such an object, consider a spherical star of ideal gas consisting of electrons and completely ionized hydrogen atoms, and of constant density. Radius of star is $R$, Mass of star is $M$, $N$ = number of electrons. A) Consider the electrons to be ultra relativistic. Density of star is given to be ${10}^{12}\, \mathrm{kg \cdot m}^{-3}$ . What is the critical mass for which the dwarf can be in equilibrium? I am trying to find the gravitational pressure and the fermi pressure due to the electrons, and then I will equate them. I have successfully calculated the gravitational pressure, but to calculate Fermi pressure, I need total energy to Pauli Exclusion principle applied to electrons. The total energy is coming to be a function of N, and due to some mistake (which I do not know , where am I making) , the radius term is not cancelling on both sides (which should). How to calculate total energy due to pauli exclusion principle?