title: Is energy exchange quantized?

text: In the photoelectric effect there is a threshold frequency that must be exceeded, to observe any electron emission, I have two questions about this. I) **Lower than threshold** : What happen with lesser frequency/energy photons? I mean there is no energy transferred? If some energy were transferred, (and it must be transfered quantized), how can be experimentally known it was quantized, if there is no electron emission? II) **Intensity dependence** : There is any known dependency with intensity? (I mean a dependency not about the number of electrons _(I am already aware of this)_ but about the energy of them) I thought energy of electron emited was independent with intensity, but then I found this link dependency with intensity (is a paper to buy that I haven't read) but relates energy with intensity.


title: Static wave's equation

text: ![enter image description here](http://i.stack.imgur.com/WNlBP.png) In my book equation for two advanced wave which created this static wave are $$y_1=asin{\frac{2\pi}{\lambda}(vt+x)} $$ $$y_2=asin{\frac{2\pi}{\lambda}(vt-x)} $$ and equation for static wave is derived by adding this two equation: $\ Y=y_1+y_2$ Q1: Though this two wave comes from opposite direction why it is added rather then subtracted? Shouldn't it be $\ Y=y_1-y_2$ Q2: Why sign of $\ x$ is opposite for two equation though it is said that it's the distance of the particular point from the reference and whose displacement for the wave is $\ y$


title: Calculating dragging frame of satellite orbiting Earth

text: Say there is a satellite polar-orbiting the Earth at 600km. How much would the satellite be dragged additionally due to dragging force? Such that $$ \Omega = \dfrac{r_s \alpha r c}{\rho^2(r^2 + \alpha^2) + r_s \alpha^2 r sin^2 \theta}$$ such that: $r_s = \dfrac{2GM}{c^2}$ is the Schwarzchild radius. $\alpha = \dfrac{J}{Mc}$ $\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta$ $\Delta = r^{2} - r_{s} r + \alpha^{2}$ I'm a bit confused how I'm supposed to use this formula as $\Omega$ denotes angular speed not distance doesn't it? Thank you so much for any help


title: Does Stephen Hawking believe that General Relativity is wrong?

text: Stephen Hawking recently said that black holes do not exist. According to Wikipedia, black holes are predicted by General Relativity. So it would seem to follow that Stephen Hawking believes that General Relativity is wrong. Am I missing something?


title: generate force accurately

text: When we like to produce known forces, we know that standard weights acting vertically on a object creates a known force. Now, what are the other methods to generate forces with accuracy. It is a DIY experiment I like to do, so any apparatus which uses simple devices like coils, motors, stepper/servo motors, or similar things would be great to try. So guys, if you have any ideas, please let me know. Cheers.


title: Instanton in sine-Gordon equation

text: This is a statement from Giamarchi's book on Quantum Physics in 1D: "For a single-particle in a cosine potential, the slightest amount of tunneling between two cosine minima leads to conduction bands, for example, and restores the translational symmetry. However, our sine-Gordon problem is a two-dimensional (one space one time) problem. In that case it is well-known that instantons with a finite action (instanton) that would connect two cosine minima cannot exist (Rajaraman, 1982). There is thus no restoring of symmetry and the field is truly locked in one of the minima. This is of course related to the Mermin-Wagner theorem stating that in two (classical) dimensions it is impossible to break a continuous symmetry but one can break a discrete one." The potential concerns us here of a scalar field $\Phi$ is: $$ g \cos(\beta \;\Phi) $$ I wonder how to **show this statement** : > "It is well-known that **instantons** with a finite action (instanton) that > would **connect two cosine minima cannot exist** (Rajaraman, 1982) ... the > field is truly **locked in one of the minima**." **Question** : (1) Are "there" any criteria or conditions when the field will be locked in one of the minima? Such as $g>g_c$ and certain values of $\beta$? (2) how to **show this statement**? How is the instanton analysis done here? ps. I read Rajaraman book and S Coleman on instantons. So please do not post an answer for recommending just the Refs. Thanks. :) NEW Edit NOTE: I suppose we are talking about this kind of 1+1D bosonic action: $$ \frac{1}{4\pi} \int_{ \mathcal{M}^2} dt \; dx \; k\, \partial_t \Phi \partial_x \Phi - v \,\partial_x \Phi \partial_x \Phi \+ g \cos(\beta_{}^{} \cdot\Phi_{}) $$ Ref: (1) Rajaraman 1982, Solitons and Instantons, Volume 15: An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland Personal Library) (2) Thierry Giamarchi, Quantum Physics in One Dimension (3) S Coleman, Aspects of Symmetry


title: Reconciling "The Big Crunch" with the 2nd Law of Thermodynamics

text: Assume "The Big Crunch" scenario (the universe will collapse to a singularity). In this case, I think of the entire universe as an isolated system; in the "Big Crunch" scenario, it seems to me gravity itself would act as the "walls" of the system, keeping all the matter and energy inside. Thus, by the second law of thermodynamics, entropy will increase. However, it seems to decrease (everything is being smashed together). I know black holes get around this through the emission of Hawking radiation, but this is the emission of radiation from black holes to the universe. In the big crunch scenario, by definition, there is no universe for the radiation to escape to because the universe itself is collapsing, taking all radiation with it.


title: Schwarzschild diagram in Einstein Cartan theory

text: I'm a very visual learner and I would like to know if the diagram representing the Schwarzschild solution is altered at all when the torsion tensor is non zero. Of particular interest - what is the graphical form of a metric with no curvature and the full range of torsion values?


title: Why is current a scalar quantity?

text: Current has both magnitude and direction. As per the definition of vector defined in encyclopedia, current should be a vector quantity. But, we know that current is a scalar quantity. What is the reason behind it?


title: Why is capacitor used?

text: I don't quite get the practical use of a capacitor. The theory is fine, but in simple words please explain how, why and where capacitors are used? Is there any other equivalent circuit for capacitors? From what I have learnt capacitors are used to store charge, but how does that make it useful in electric appliances?


title: Free Particle Path Integral Matsubara Frequency

text: I am trying to calculate $$Z = \int\limits_{\phi(\beta) = \phi(0) =0} D \phi\ e^{-\frac{1}{2} \int_0^{\beta} d\tau \dot{\phi}^2}$$ without transforming it to the Matsubara frequency space, I can show that $Z = \sqrt{\frac{1}{2\pi \beta}}$. However, I have a problem in obtaining the same result in the Matsubara frequency space: \begin{equation} \phi (\tau) = \frac{1}{\sqrt{\beta}} \left( \sum_{n} \phi_n \ e^{i\omega_n\tau} \right), \end{equation} with $\sum_n \phi_n =0, \omega_n = \frac{2\pi n}{\beta}$. And \begin{equation} Z = \int \prod_n D\phi_n\ \delta\left(\sum_n \phi_n\right)\ e^{-\frac{1}{2} \sum_n \phi_n \phi_{-n} \omega_n^2 } \end{equation} which, I think, vanishes. I guess the problem lies in the measure. Any comments? Info: I write the Schulman's derivation in imaginary time here. \begin{eqnarray} Z &=& \int\limits_{\phi(0) =\phi(\beta) = 0} D\phi(\tau) e^{-\frac{1}{2}\int_0^{\beta}d\tau\dot{\phi}^2}\\\ &=& \text{lim}_{N \rightarrow \infty} (\frac{1}{2\pi \epsilon})^{(N+1)/2} \int d\phi_1 \dots d\phi_N e^{-\frac{1}{2\epsilon} \sum_{i =0}^N (\phi_{i+1} -\phi_i)^2} \end{eqnarray} Then, we can use the identity \begin{equation} \int_{-\infty}^{\infty} du \sqrt{\frac{a}{\pi}} e^{-a(x-u)^2}\sqrt{\frac{b}{\pi}} e^{-b(u -y)^2} = \sqrt{\frac{ab}{\pi(a+b)}} e^{-\frac{ab}{a+b}(x-y)^2} \end{equation} to evaluate the sum to be \begin{equation} Z = \sqrt{\frac{1}{2\pi \beta}}. \end{equation}


title: Is the pH of antimatter the reverse of the pH for matter?

text: If it does have a pH, I mean. Though I don't see why it shouldn't have it. Both the Brønsted-Lowry and Lewis definitions of acid have to do with the ability of a substance to accept, or provide protons or pairs of electrons in certain reactions. I got interested into this when I learned that the electric charges for antimatter particles are the opposite of those for matter particles (i.e.: positively charged positrons orbiting around negatively charged antiprotons). So I was thinking... If I mixed anti-hydrogen chloride with anti-sodium hydroxide to make anti-salt, it would be the anti-HCl that would be donating negative charged ions, and the anti-NaOH that would be providing the positive charges. Does that mean that the pH of a matter substance and its antimatter counterpart are reversed in relation to each other? Would anti-LSD be a base instead of an acid? **Edit:** I understand that getting matter in contact with antimatter will cause anihilation. I know that trying to neutralize an antimatter acid with a matter base ~~would be great fun~~ would never work. While asking this question I had a "bizarro" anti-matter world in mind. I was thinking, for example, how some anti-lemmon battery would work, since the anti-ions would be going in different ways when compared to a lemon battery in our matter world.


title: Why is the absolute zero a rational number in Celcius?

text: From the question "Why is the absolute zero -273.15ºC?" I understood that 1°C is the 100th part of the difference of melting and boiling temperature of water (this is my high school physics, maybe there is a more accurate definition). So we linked the value of 1°C to a physical quantity. Then we measure another physical quantity (the absolute zero) and it turns out this is exactly 273.15°C less the melting temperature of water. So how come those two quantities have a rational ratio? If we look at other constants (gravitational, Planck or whatever) they all are irrational, aren't they? What did I miss? Is it because the melting point of water is not exactly 0°C (maybe something really close to 0 like 0.000565...)?


title: Moment of Inertia of a sector of a circle

text: I am trying to find the moment of intia about its centre of a sector of a circle of radius $a$, mass $m$ and angle $\pi/3$. I have found the answer it is $\frac{1}{2}ma^2$ but originally tried a method that was unsuccessful and don't know why this is the case. This involved finding the moment of interia of elementry rods about the centre, length $a$ and width $a\;\delta\theta$, using $p$ as mass per unit area. I then tried to integrate the moment of inertia of these rods from $\theta=0$, to $\theta=\pi/3$. Which gave me an answer that did not lead to the correct result. Please can you explain why? I have shown my workings bellow but they are not very clear: ![enter image description here](http://i.stack.imgur.com/Qs91b.jpg)


title: Is it possible to reduce the sound, when two metal objects collide (perhaps with some coating) without reducing the rigidity of the surface?

text: I have a system, where there are ball bearings on the pistons that clamp the metal plate with special dents for ball bearings. The system should be precise, because it is used for microscopy. It also should be as noiseless as possible. It also should be fast, so the impact at high velocity is inevitable. I've thought of introducing some resin coating, but it will reduce the rigidity. Are there any solutions for this problem? Is there any strong relationship between sound and rigidity? I believe that there may be some rigid materials that somehow don't favor phonons.


title: If I had a 1cm iron sphere that I could vibrate at an rate, how fast would it have to be vibrating before it could not perceived by sense of touch?

text: Is there a vibratory rate at which my hand would just pass right through the object and it could not be perceived by touch anymore?


title: Interferometer problem about periodic fringe patterns

text: Please,kindly help me; my teacher assigned me problem set, and she never went over anything about interferometers in class. Also, there is almost no information about about how to solve interferometer problems in my book. I have no idea how to set this problem up, and I have no intuition concerning how I should solve it... Can someone help me set this problem up and explain it? I know if this example is solved, I will be able to solve the rest of the problems: Sodium D lines, of wavelength(a)=589.0 nanometers and wavelength(b)=589.6 nanometers are used on a Michelson interferometer. When the first mirror moves, these lines periodically appear and disappear. Explain, in detail, this phenomenon and write an equation that would allow you to know how much the mirror must move to make the lines reappear and disappear one time.


title: How much does a proton weigh when it is going around the LHC at CERN?

text: Considering that speed increases weight and the proton is going at almost the speed of light, I would like to know how much a speeding proton would weigh in the LHC.


title: Designing a home theater room for optimal acoustics

text: We are building an addition on our home that will allow us to build a basement home theater room with some level of flexibility regarding its size and shape. I've been reading somewhat conflicting information about room dimensions and shapes to provide optimal acoustics, and few sources seem to directly cite a scientific basis for guidelines. I kind of get the feeling that this is more art than science. My first question would be--how reliable is room modeling software (or manual calculations) in providing an accurate acoustic room response? Simulation has the allure of helping optimize room shape, but only if it's a good predictor. I've read conflicting guidelines regarding whether or not angling a wall, or multiple walls might help minimize standing wave problems, and that they certainly complicate the analysis. I've read in some places that room dimension formulas are valid and other places that they're not. It seems pretty universally accepted that dimensions shouldn't be near multiples of each other. Are there current best practices for room design that are universally accepted? For reference, the room interior width is constrained to be 11'11", length can be up to 23', and height can be as large as reasonably excavated in a home basement (6 to 10 ft).


title: Euler-Lagrange equation for continuous systems

text: I'm having a little trouble with wrapping my head around a part of a method which is fairly 'new' in some fashions to me. I imagine it should be fairly obvious, but I am not seeing something at the moment. Let's take the Lagrangian density to be $$\mathcal{ L}={1\over{2}}\left(\mu\dot{\phi}^2-Y(\partial_x\phi)^2\right),$$ where $\mu$ is mass density of rod and $Y$ is Young's modulus. One "should" be able to put this into the equation $${\partial\over \partial t}\left({\partial \mathcal{L}\over{\partial(\partial_t \phi)}}\right)+{\partial\over \partial x}\left({\partial \mathcal{L}\over{\partial(\partial_x \phi)}}\right)={\partial \mathcal{L}\over{\partial \phi}}$$ to get the equation of motion for the rod $$\mu\ddot\phi=Y\partial_{xx}\phi,$$ right? The trouble is I'm not exactly seeing where to go with the LHS of the E-L equation applied to $\mathcal L$ at the moment. I'm hoping someone would be so kind as to demonstrate that in a few lines.


title: Earphones and electric guitars

text: I remember being in my college dorm, watching a film with a friend. His computer wasn't loud enough and we couldn't hear properly. So he took his earphones and put them on a electric guitar coil. He then plugged his guitar to his amp and voilà, we could listen through the amp. Why on earth did this work?


title: Is there any interacting quantum field theory of massless spin-1 fields expressed locally entirely in terms of F, with no vector potential?

text: Is there any interacting quantum field theory of massless fields with helicity $\pm 1$ which can be expressed entirely locally in terms of the field strength Fμν with no reference to vector potentials at all? Clearly, quantum electrodynamics doesn't fall into this category. The Aharonov-Bohm effect is the reason. Classically, it's easy to come up with many such theories, but unfortunately, they happen to be nonrenormalizable. But do asymptotically safe models exist?


title: Deriving $T = F\ r = I\alpha$ for a rigid body

text: For a single point mass : $\tau=F_{t}r=ma_tr=(m r^2)\alpha = I\alpha$ For multiple point masses bound together : $\sum \tau_i = (m_ir_i^2)\alpha = I\alpha$ But how do we go from that to $I\alpha = F_tr_{app} $ (for the multiple masses case), where $r_{app}$ is the point where force is applied? That is saying that the entire sum of torques of individual masses $\sum \tau_i$ is physically equivalent to (i.e. produces the same angular acceleration of the bound rigid object as) a single torque $F_tr_{app}$ applied to the point $r_{app}$. (Or putting it another way, that applying a single force $F_t r_{app}$ will result in many $F_{t\\_i}$ forces on each mass element such that $\sum F_{t\\_i}r_{i}$ will be equal to $I\alpha$.) Every place I've looked for a derivation just jumps from $\sum \tau_i =I\alpha$, to $\tau=I\alpha$, as if they are the same thing.


title: Any resources to learn how to interpret formulae/diagrams?

text: Are there any materials exploring intermediate levels (of any sub-subject) which give plain English descriptions along side formulas/equations//diagrams/notations?


title: Sutherland's constant of argon

text: I want to compute the viscosity of argon at different temperatures. What is Sutherland's constant of argon?


title: Acceleration Force

text: A 6500-kg helicopter accelerates upward at 0.60 m/s² while lifting a 1200-kg car. (a) What is the lift force exerted by the air on the rotors?   (b) What is the tension in the cable (ignore its mass) that connects the car to the helicopter?


title: Interpretation of the Random Schrödinger Equation

text: I should preface this by admitting that my physics background is rather weak so I beg you to keep that in mind in your responses. I work in mathematics (specifically probability theory) and a paper my adviser asked me to read dealt with a discrete approximation to the random Schrödinger operator $\Delta + V(x,\omega)$ where $V$ is some random function. The physical meaning of this object is completely beyond me at this point and I was hoping that someone here could shed some light on it. The basic issues I am trying to work through now are what this actually models (one professor said something about 'disordered media') and whether there are physical reasons to believe that there should be universality of spectral statistics if one makes some kind of reasonable assumptions about the behavior of $V$. I know that proving the last statement rigorously is a major open question in mathematics, so I do want to emphasize that I am really just looking for a little hand-waving to help understand what is going on. Based on a few explicitly solvable classes of discrete approximations in the limit, I want to say that universality should hold for some broad classes of $V$, depending on the rate of decay of $V$ since the limiting spectral statistics from the discrete approximations seem to have that property if everything is taken to be independent. I have been told that this is a somewhat common phenomenon in the study of Schrödinger operators, though again I don't really understand what is going on here. Anything would help: links to good write-ups on the random (or not random) Schrödinger operator, a toy example that is illuminating, a suggestion on good papers to read--anything really. Thanks :)


title: Taylor approximation of e(v)

text: Relativistic mass $\displaystyle m(v)=\frac{m_o}{\sqrt{(1-(v/c)^2}}$ $m_o$ = mass of object measured at rest   $c$ = speed of light ($3\times 10^8\;m/s$)   $v$ = speed If the total relativistic energy of an object is given by $E=mc^2$, then find the second-order Taylor approximation of $E(v)$. The zeroth-order term is called the "rest energy." What is the second-order term usually called?


title: Terrestrial Space Elevator Construction - Plausability

text: Framework If there was a cable constructed at the equator about the circumference of the Earth, and if this cable had sufficient strength to remain intact while erect, call this tensile strength T. Question With a rocket, could you launch the cable from the surface of the Earth without exceeding the tensile strength T? ![enter image description here](http://i.stack.imgur.com/TJsxr.jpg)


title: Does a pivot exert an additional force, normal to the centripetal force, on an object rotating about it?

text: > A thin plank of mass $M$ and length $l$ is pivoted at one end. The plank is > released at 60$^{\circ}$ from the vertical. What is the magnitude and > direction of the force on the pivot when the plank is horizontal? I have a doubt: does the pivot exert an upwards force on the plank when the plank is horizontal? I think the answer to this is yes, because: The torque on the plank about the pivot, when the plank is horizontal, is $$\tau=\frac{Mgl}{2}$$ And the plank's angular acceleration at that point will be $$\alpha=\frac{\tau}{I}=\frac{3g}{2l}$$ So the vertical acceleration of the plank's CM will be $$a=\alpha R=\frac{3g}{4} $$ Therefore, the upwards force exerted by the pivot on the plank is $$F_{up}=\frac{mg}{4}$$ Is this reasoning ok for this case? (I'm doubtful of this part because there was a post on PF where this force was not mentioned.)


title: Space Warps & Frame Invariance

text: The Alcubierre Warp Drive in Higher Dimensional Spacetime (H. G. White & E. W. Davis), among other sources, proposes that the x-axis symmetry paradox for Alcubierre-style warp drives can be resolved by interpreting the warp field as producing a scalar boost on a pre-existing velocity. If that is the case, how can it obey frame invariance? In other words, how does the boost "know" which velocity, in which inertial frame, it's supposed to be multiplying?


title: Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions

text: Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT at that fixed point is indeed $3(1-\frac{2}{k})$.   * Given the above, how does it somehow obviously follow (by Zamlodchikov's c-theorem?) that if I had the potential as any homogeneous function of degree $k$ in say $n$ scalar chiral fields then the central charge at the similar fixed point will be $3n(1-\frac{2}{k})$?    * I have seen some quite terse arguments but I want to know if there is a simple intuitive way to see that the above theory does have a non-trivial IR fixed point (the claim that I initially started off believing).    * Now its also true that the CFT at this fixed point is infact a minimal model and that apprently follows from some representation theory of the ${\cal N}=2$ superconformal algebra. Can someone kindly give me a reference to that?    * Finally are there known generalizations of this construction to higher supersymmetries or higher diemensions?


title: What are some unanswered questions in electromagnetism?

text: What don't we know about how electricity and/or magnetism works?


title: Gravity and Planck scale

text: What is the connection between Planck's constant and gravity? Why is the Planck scale the natural scale for quantum gravity? I would have though the scale would be related to G, not h.


title: What does it mean that the neutral pion is a mixture of quarks?

text: The quark composition of the neutral pion ($\pi^0$) is $\frac{u\bar{u} - d\bar{d}}{\sqrt{2}}$. What does this actually mean? I think it's bizarre that a particle doesn't have a definite composition. There's a difference of 2 MeV between the quark masses and I don't understand how this can be ignored. If I were to somehow manage to make a bound state of an up and an anti-up quark, what would it be? Would it be a variation on the neutral pion or would it somehow transform into the mixture?


title: Why does pushing a magnet inside a solenoid produce current?

text: If you push a bar magnet inside a solenoid, a current is produced. But why is that? I mean, the wire is being moved _along_ the magnetic field, so taking the cross product: $\vec{F} = I\vec{V}\times\vec{B} = I|\vec{V}||\vec{B}|\sin\theta\hat{n}$ Here, the angle between the velocity of the charge in the wires and the magnetic field is essentially zero. So there is no force on the charges in the wire and hence no current. But that does not correspond to reality. Why? ![enter image description here](http://i.stack.imgur.com/q1nly.jpg)


title: Searching books and papers with equations

text: Sometimes I may come up with an equation in mind, so I want to search for the related material. It may be the case that I learn it before but forget the name, or, there is no name for the equation yet. In this case, I may be able to recall a reference book. Searching in Internet can be a fast way though. However, there are cases that I get some idea and equations (maybe a modified one) myself. So I want to know whether there are any other people working on it before. Because of the vagueness, it is difficult to search it by keywords because I do not know the 'name' of this idea: it is either too board or no results. I can try to search google scholar, arxiv and maybe the prola, but there is no support of equation pattern matching. For example, entering ∇⋅V in google give you no useful results (it is better if you input the 'divergence of potential'). Is there any good way to search papers in this case? It is even better that I can use the combination of equations and keywords. **Edit** : Another reason is that Mathematician should have done some deep analysis on the related mathematical topics. Yet it is difficult to know their results because there is a gap between the terms used in physics and mathematics. It would be really useful if we can find and learn their results.


title: Tension in string due to surface tension

text: Suppose you take a loop of string and place it on the surface of a liquid, the string should become taut due to surface tension. How would you be able to calculate the tension in the loop of string? My main difficulty in solving this problem is the direction in which surface tension force act on the thread - perpendicular or tangential at any point on the thread?


title: What force keeps electrostatic charge from exiting a metals surface

text: I understand that in electrostatics all charges are 'static' due to equal forces acting upon them. This then leads to the fact that the arising electric field must be perpendicular to the metals surface at the surface. What i don't understand is what force is counteracting the electric field. I have read that there is a potential well but am not sure how this works. Could someone point me in the right direction? Thanks.


title: How does QFT help with defining measurement in Quantum Mechanics?

text: I learned that a quantum system has an "overall state", a vector in a Hilbert space. That Hilbert space can be decomposed in a basis of "basic states". For example if in the Universe there is only a free electron with its own spin, and I pick an axis $z$, there is an infinite basis of "position+spin", let's say $|(0,0,0);1/2>, |(0,0,0);-1/2>, |(0.1,0,0);1/2>\ldots$ I cannot obviously write all possible positions, but that was to give the idea. Some $\psi (\vec{r},s)$ contains the coefficients for all the elements of this basis and is given. This is the "overall state" of my "one-electron Universe" I was said that there is "measurement" that makes the $\psi$ collapse to one of the "basic states" and can give the observer the result (position + spin in the $z$ axis) Since a measurement involves an interaction with some sort of detector I figured out that I would need QFT (the possibility to have multiple particles and interaction) to understand measurement and add a detector into my Universe. I thought that the dynamic of the interaction between the particles of the detector and the electron could act in a way that the $\psi$ collapsed (I hoped that even in QFT there was something analogous to the original $\psi$ of the electron). Now I couldn't study QFT but from what I grasped it is much more complicated and the reason of the collapsing of the wave function is not explained by an "interaction". The wave function itself is subjective and depends on the knowledge of the observer. So I am lead to believe that QFT does not contain anything that explains the collapse of the wave function. Is it true? I know that I presented a rather subjective view. I hope that they are shared by many and that an answer that points out my misunderstandings could help others. I think that many people in the quest for a more "classical" explanation of QM resort to thinking that when they would see how the interactions work "really" so as to describe a fully fledged measurement equipment (photons, QED, QCD...) they would figure that out.


title: LIGO sensitivity in terms of minimum received power per area

text: I've been looking at LIGO figures for gravitational wave sensitivity here and it seem to be displayed in an adimensional strain ratio (which I assume that is more or less equivalent to metric perturbation $h_{\mu \nu}/g_{\mu \nu}$, correct me if I'm wrong) What I'm wondering is how can I convert this sensitivity measure into something like watts/area, as a measure of energy density flux of gravitational waves.


title: Is there physics behind the layout of a piano keyboard?

text: We have 12 different 'notes' per octave on a musical keyboard. They are set up so that every 'note' (A, B,C etc) is a second harmonic of the same 'note' in the next higher octave. With this ratio in place, it seems that everything on a keyboard should be divided up into equal groups of 2,3,4 or 6. However, the keyboard seems to be divided up into groups of 5 and 7 (one group of 2 black & 3 white keys, and another group of 3 black & 4 white keys). This does not make much sense to me. However, when I play scales, the layout is very helpful. I'm guessing that the design of the scale is related to physics and therefor the layout of the keys must be too. What is the connection between physics, scales and the layout of the piano keyboard?


title: Why do the passing objects' shadows run in the direction of the moving train?

text: While travelling in the train on the sunset, I have noticed the strange phenomenon. The shadows of the lonely trees and lampposts, that stay along the railroad, are moving on the wall in the same direction as the train. I tried my artist talents to express the situation as a picture: ![descriptive picture](http://i.stack.imgur.com/ua8gY.png) As far as I understand, the shadows should move in an opposite direction. ## Update #1 I have found the evidence video - and converted it into an animated gif: ![animated gif](http://i.stack.imgur.com/zQFqg.gif) The shadow runs on the left wall - from the the camera point of view. So, from the man's point of view, the shadow moves in the direction of the train, if it's observed on the wall behind the camera.


title: Is a communications channel based on quantum mechanics as effective as one based on any other physics?

text: What I mean by effective in the question refers to the time and space requirements for sending information over a quantum communications channel. Having read "Mike and Ike" but doing no independent research on the question, my initial impression is that a channel using an effect from quantum field theory e.g. quantum electrodynamics would be more powerful over a narrower range of algorithms than an "ordinary" or "vanilla" quantum computer i.e. one that derived information from its steady-state behavior e.g. quantum electrostatics. Are there any quantum information theorists working on proving whether this is true or false? EDIT: Allow me to quote from the portion of the book the provoked the question (p.6): > At the time of writing it is not clear whether Deutsch’s notion of a > Universal Quantum Computer is sufficient to efficiently simulate an > arbitrary physical system. Proving or refuting this conjecture is one of the > great open problems of the field of quantum computation and quantum > information. It is possible, for example, that some effect of quantum field > theory or an even more esoteric effect based in string theory, quantum > gravity or some other physical theory may take us beyond Deutsch’s Universal > Quantum Computer, giving us a still more powerful model for computation. At > this stage, we simply don’t know. I simply assume that a more powerful model of computation would give rise to a more efficient communications channel. Unless there has been a proof of the universality of a Universal Quantum Computer it seems that whoever said 'quantum is quantum' is very much getting ahead of themselves. Now, with all that being said, I reiterate my previous question: are there any quantum information theorists working on proving whether this is true or false?


title: What is the role of the color-anticolor gluons?

text: Ignoring for the moment the experimental necessity of eliminating the gluon singlet state, the nine "raw" gluon color components form two quite distinct sets. The first set consists of the six (color,anti-other-color) components: > $r\overline{g}$, $r\overline{b}$, $g\overline{b}$, $g\overline{r}$, > $b\overline{r}$, $b\overline{g}$ The second set consists of the three (color,anti-color) components: > $r\overline{r}$, $g\overline{g}$, $b\overline{b}$ Only the first six components are needed to explain color flipping in hadrons and mesons. The second smaller group cannot change quark colors since they are net neutral in color. So: For gluon exchanges between quarks in hadrons and mesons, is the role of the three gluon color-anticolor components purely attractive or repulsive? That is, are these three components more closely analogous to the photons of the electromagnetic force, which similarly carry no net electric charge? Popular explanations, even ones by very good physicists, seem usually to skip over the existence of these two distinct groups, and instead describe only the color-changing consequences of the first group of gluon color components. Finally, are there experimental implications of in the existence of these two groups? For example, would gluons that happen to carry mostly color-anticolor charge generate identifiably different jets from gluons that carry mostly color-anti-other-color charge? This is a well-studied area, of course, so I suspect I just haven't been looking in the right places.


title: Material breaking underneath its breaking point (due to repeated load)

text: I'm trying to recall some fact that I learned in my college materials science class, where we learned concepts like stress vs. strain. I remember that materials like metal have a breaking point, and if an applied stress produces a force higher than the breaking point, the metal will break. Or something like that. Now, I distinctly remember my professor showing some real-world engineering failure where a metal was used in a structure (in an airplane or bridge), and it broke even through the stress never went above the breaking point. Rather, the metal broke due to repeated use of the structure. What is this principle?


title: Could electrons be a form of antimatter?

text: I've played with this idea for years, and haven't really been able to eliminate it. So, perhaps someone here can point to simple experimental evidence that would do so. Here's the issue: Antimatter of course annihilates ordinary matter, but the more precise statement is that antiparticles annihilate _the same types_ of particles. That means that you _can_ mix antimatter with matter without it going boom, as long as you only allow one of each type in the mix. For example, positrons (anti-electrons) can pass through interior of a neutron at low speed without causing annihilations for the simple reason quarks and electrons cannot pair off for annihilation. All of which leads to this delightfully non-standard line of thought: If what we call electrons were actually anti-particles, would we even be able to tell? A more precise way to express that is this: If _all_ negatively charged particles had the internal chirality of antimatter, and all positively charged particles that of matter, would the resulting transformed theory lead to paradoxes or contradictions with the Standard Model? Or would it instead produce an experimentally isomorphic theory that describes what we see just as accurately as the Standard Model? > (The neutrinos of course have no charges, so their affiliations would be > determined by their associations with the electron types.) So my question is this: Would reclassifying the chirality of particles so that "pro" always means positive particles and "anti" always means negative particles result in a version of the Standard Model that is experimentally isomorphic to known results, or would it instead produce one or more clear contradictions or paradoxes? One final note that's too much fun not to mention: a workable pro="+"/anti="-" theory would convert the mystery of why matter dominates over antimatter into a mystery of the particle mix came out so asymmetrically.


title: How fast does the collapse travel in a multiple slit experiment?

text: If we have a kilometer long million slit light source and make an observation on a slit at one of the ends, will the interference pattern fully collapse or partially collapse? is there any delay in the propagation of the collapse or is the collapse instant? Link to multiple slits. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslidi.html#c3 Additional information. I am thinking that if Alice is at one end, point A and is looking at the pattern on the wall from the million slits. Bob is at the other end a kilometer away at point B. Bob then makes an observation to detect if a photon is going through a slit. Wouln't Alice see the pattern on the wall disappear? My understanding of how this is not faster than light in the 2 slit experiment is because the change in light travels from the slit to the wall at c after the instant collapse. In the multiple slit though the collapse travels a lot further instantly and then the change in light travels at c from the slits to the wall. For Alice that is. Where is my mistake?


title: Why don't orbits expand with the Universe?

text: Consider two bodies orbiting each other. As the Universe expands would the distance between them increase? Most people say that a gravitationally bound system will not expand with the Universe. They say that such a system is not described by the FRW solution. But surely one could consider two orbiting bodies that are sufficiently far away that the spacetime between them could be approximately described by the FRW solution? In this case the recession velocity would be of the same order as the orbit velocity.


title: A fundamental equation for solitary wave and dimension analysis

text: According to the scalar Field theory we write Lagrangian as $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ What I want to do is to get a equation of solitary wave (solition) from the above Lagrangian which will lead for solving many solitary problems like life-time, dimension analysis etc. My confusion is choosing the potential because it may vary and depend on the author's selection. What are the physical constraints on choosing an appropriate potential? Another rather technical expectation is solitary wave is nonlinear equation so how will I write down the equation as 2nd order differential equation with respect to time?


title: How to convert Richter magnitude scale to approximate TNT?

text: I know the Richter magnitude scale is often used for measuring the strength of earthquakes. At the same time, explosive/destructive releases of energy are often quoted in equivalent amounts of TNT. Is there some way to relate these two scales? Or are they not equivalent? It's especially confusing since the Richter scale doesn't seem to go very high. Is a magnitude 8 earthquake really only 4 times worse than a magnitude 2? Or would converting these numbers to TNT equivalents be more revealing?


title: what's the difference between linear n-atom molecule and nonlinear n-atom molecule?

text: I am reading a material about the degree of freedom for linear n-atom molecule and nonlinear n-atom molecule. Here is my analysis for a diatomic molecule, if there are two atoms, we have to use 3 coordinates to describe the translational motion, 2 coordinates about the rotation and 1 for vibration, so total will be 6 degree of freedom for diatomic molecule. According to the equipartition theorem of the energy, when there are N molecule, each degree of freedom share $kT/2$ energy, so total $$E = 6\times \frac{NkT}{2}$$ and the heat capacitor is $$ C = \frac{dE}{dT} = 3Nk $$ But in other book, it said that the heat capacitor for diatomic molecule is $\frac{7Nk}{2}$. I am googleing the issue and someone said the vibration energy consists of kinetic energy and potential energy so there are actually 7 degree of freedom instead of 6. Is this statement correct? It is pretty confusing because in the wiki http://en.wikipedia.org/wiki/Degrees_of_freedom_%28physics_and_chemistry%29, it states that the linear N-atom molecule has 3 positions, 2 rotations and 3N-5 vibrations so total 3N degree of freedom, if N=2, then we will have 6 degree of freedom instead of 7. So which conclusion is correct? BTW, what's the main difference between the linear and nonlinear n-atom molecule?


title: Cartan equations versus Einstein equations in classical gravity

text: Are Cartan structural equations equivalent to Einstein's equations $$G_{\mu\nu}=T_{\mu\nu}$$ and why (in the case of torsionless geometries, of course)? Does it also apply with a non-null cosmological constant?


title: Charges in a molecule and rotation invariance

text: Why are the charges of atoms in a molecule not invariant under the rotations of the molecule? My intuition is that the charges should not change under the rotation of the molecule, because their relative positions do not change. But I was told that they can change.


Does the general topology of Minkowski space-time change under a Lorentz transformation? Open balls in $\mathbb{R}^{4}$ (with the standard topology) are not invariant under Lorentz transformations. Does this mean for example that observers in one reference frame would have different notions of convergence, continuity etc? Note: I'm asking from the perspective of a curious undergraduate who just finished introductory analysis and modern physics. Please answer appropriately.


title: What happens between two harmonics?

text: I know that standing waves in a simple harmonic system occur when the "echo" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because both waves have the same speed and frequency but opposite direction, they move away from eachother until they cancel eachother out, and then keep moving until they overlap again to amplify the signal. But what happens halfway between two harmonics? Would it be like a half standing wave, or would the signal cancel out?


title: Absorption of waves in a plasma

text: Suppose a plasma has characteristic frequency $\omega_p$. Since $$n = \sqrt{\left(1-\frac{\omega_p^2}{\omega^2}\right)} $$ For $\omega<\omega_p$, the refractive index will be imaginary - which corresponds to absorption of light. For $\omega > \omega_p$, the refractive index will be real. Suppose plasma frequency is $9\: \mathrm{MHz}$. Why is it that AM radio waves ($\sim 1\: \mathrm{MHz}$) are able to be transmitted further than FM radio waves ($\sim 100\: \mathrm{MHz}$)? Isn't that counter-intuitive?


title: Is redshift a reliable means to know how fast an object is moving away or towards an observer?

text: Does wavelength change with distance? How do they know that the change in wavelength can only be caused by the change of speed of the object to the observer? What about noises, distortions, and interferences as the light moves away from the source to the observer? Or perhaps it travelled through some unknown medium that changed its wavelength. I'm asking this question to understand why redshift is being used to tell how fast a galaxy is moving away from an observer. Please explain in plain English. I'm not a physics major.


title: path integral quantization of EM field derived from canonical quantization?

text: In Peskin's QFT book page 294, he formally addressed the quantization of EM field, > $$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$ Now that we have the > functional integral quantization method at our command, let us apply it to > the derivation of this expression.   >  Consider the functional integral   >  $$\int DAe^{iS[A]},$$ Actually I expected to see how he would derive the generating functional $\int DAe^{i(S[A]+J^\mu A_\mu)}$ from the canonical quantization before proceeding on to introduce the Faddeev-Popov trick. So is there a way to do this derivation, i.e., from the operator formalism using Hamiltonian, or we must presume the validity of this path integral as a starting point, which is probably what Peskin did(I guess)?


title: Noether's Theorem: Foundations

text: I'm wondering on what principles Noether's theorem foots. More precisely: _The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful considerations, however, seem to untangle them as special field variations. So what's going on here truly?_


title: Wick rotation in field theory - rigorous justification?

text: What is the rigorous justification of Wick rotation in QFT? I'm aware that it is very useful when calculating loop integrals and one can very easily justify it there. However, I haven't seen a convincing proof that it can be done at the level of path integral. How do we know for sure that Minkowski action and Euclidean action lead to the equivalent physical result? Is there an example where they differ by e.g. a contribution from a pole not taken into account while performing Wick rotation?


title: Does time expand with space? (or contract)

text: Einstein's big revelation was that time and space are inseparable components of the same fabric. Physical observation tells us that distant galaxies are moving away from us at an accelerated rate, and because of the difficulty (impossibility?) of defining a coordinate system where things have well defined coordinates while also moving away from each other without changing the metric on the space, we interpret this to mean that space itself is expanding. Because space and time are so directly intertwined is it possible that time too is expanding? Or perhaps it could be contracting?


title: Problem with Wick's theorem at first order

text: I've been struggling with a detail in Second Quantization which I really need to clear out of my head. If I expand the S-matrix of a theory with an interaction Hamiltonian $ H_I(x) $ then I have $$ S - 1= \int^{+\infty}_{-\infty} d^4x\, H_I(x) + \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} d^4 x\, d^4 y\, T[ H_I(x) \, H_I(y) ] + ... $$ where the T operator is unnecessary in the first term. Now, if I choose a $ \overline{\psi}(x)\psi(x) $ theory for example, the first term gives some contributions which I can calculate most easily by doing the expansion $ \psi(x) = \psi^+(x) + \psi^-(x) $, which is the essence of Wick's theorem. I know the contributions in this example will be trivial, but the point is Wick's theorem is not defined for the same spacetime points and everywhere I see, what everyone says is that since the T operator in the first term can be there, then when substituting $ H_I(x) $ we simply retain the operator and use Wick's theorem like the spacetime points were different and impose x=y at the end, but this doesn't make any sense since the operators T are different. Basically it is assumed implicitly that $$ \overline{\psi}(x)\,\psi(x) = T[ \overline{\psi}(x)\,\psi(x) ] $$ in the first term of $ S-1 $, but the time ordering operators aren't even the same, since this one has a minus sign in its definition. The point is, the first term will always be $$ \int^{+\infty}_{-\infty} d^4x\, H_I(x) $$ independently if I chose to put there the T operator (without the minus sign) or not, so for fermions I should have $\overline{\psi}(x)\,\psi(x)$ there and not $T[ \overline{\psi}(x)\,\psi(x) ]$, since they are not the same as far as I can see.


title: Relativistic space travel?

text: If one travels in a spaceship at speed V, the time elapsed for the traveller relative to an observer on earth is dilated by, $$ t' = t(1 - v^2/c^2)^{1/2} $$ Does this mean that a space traveller, who gradually accelerated to a speed very close to the speed of light can then travel 10 billion light years in a few minutes? Relative to the travelling observer, does this mean that effectively there is no speed limit? Even though this can be translated as a space contraction, what difference does that make - the traveller effectively travelled 10 billion light years in say 6 minutes - meaning an effective speed of 100 billion light years per hour (876 trillion times the speed of light). Does this mean that time travel into the future is possible? Would it be possible for a traveller making a round trip away and back to earth to travel to the year 3000 in 5 years?


title: Lagrangian Oscilattor

text: I want to know how to calculate the normal modes from a Lagrangian. I make the T (kinetic energy matrix) and U (potential energy matrix), and then I calculate the determinant of $|T-\omega ^2 U|$ (with omega as the angular speed). This way I find the $\omega$'s but I dont know how to find the normal modes. Are they just the eigenvectors of the $T-\omega ^2 U$?


title: Wick's theorem proof details

text: Let's have Wick's theorem in following form: for fields $$ A_{i}(x) ~=~ A_{i}^{+}(x) + A_{i}^{-}(x), $$ where first summand contains creation operator and the second contains destruction one, is a fair to say $$ \tag{1} T(A_{1}...A_{n}) ~=~ \sum_{k}(-1)^{\sigma}\overline{A_{i_{1}}A_{i_{2}}}~\overline{A_{i_{3}}A_{i_{4}}}\ldots \overline{A_{i_{2k - 1}}A_{i_{2k}}}: A_{i_{2k+1}}...A_{i_{n}}:, $$ where contractions are denoted with an overline $$ \overline{A_{i_{1}}(x)A_{i_{2}}(y)} ~=~ \langle | T(A_{i_{1}}(x)A_{i_{2}}(y))|\rangle ~=~ -iD^{c}(x - y), $$ and normal order is $$ : A_{i_{k}}(x)A_{i_{k + 1}}(y): ~=~ A^{+}_{i_{k}}(x)A^{+}_{i_{k + 1}}(y) + A^{+}_{i_{k}}(x)A^{-}_{i_{k + 1}}(y) + A^{-}_{i_{k}}(x)A^{-}_{i_{k + 1}}(y) + A^{+}_{i_{k + 1}}(x)A^{-}_{i_{k}}(y). $$ In the beginning of proof of this theorem one says that permutation of fields in the left and right side of $(1)$ doesn't change it. Why this statement is correct?


title: Operator that describes particle detector

text: In non-relativistic QM, the position of a particle is an observable. In QFT, fields are the observables. However, particles must have some sort of position, otherwise we wouldn't see pictures like the ones below. What is the (linear, Hermitian) operator that describes what we are observing? I've tried considering making a "local" version of $N = \int dk\, a^+ a^- = \int dx \, \phi^+(x) \phi^-(x)$ by replacing this with $N_R =\int dx \, f(x) \phi^+ \phi^-$ where $f$ is some function that is concentrated in the localized region $R$. However, this is not Hermitian! ![enter image description here](http://i.stack.imgur.com/b3YlF.jpg)


title: Does Archimedes law depend of lack of matter?

text: I would like to know if Archimedes law take in equation the lack of matter. Fluid density at 1. An object with density of 0 (or near 0) is put in liquid, it's fixed. The external sphere where liquid is inside is fixed too. The lack of matter change the attraction in all the sphere itself, no ? Density change with depth, no ? How forces can be calculate ? ![e](http://i.stack.imgur.com/Jn1KB.jpg)


title: Speed of sound in non-newtonian fluids

text: Recently I saw some videos of non-newtonian (shear-thickening) fluids like corn starch mixed with water(sometimes known as oobleck), where the fluid is placed on top of a speaker cone and starts to create odd shapes as it becomes thicker in some places. Now I wondered what would happen in the following situation: Suppose a room with reflective walls is fully filled with such a fluid. On one side of the room a sound is produced by a speaker. On the other side of the room is a microphone. How will a sound wave propagate through such a substance? Does the fluid increase its viscosity before, during or after a wave moves through it? What will the speed-of-sound be in such a case? I've read that the speed of sound is much faster in solids than in liquids, but how is that for this kind of substance?


title: Is the Normal force dependent on the density of the object the body is resting on?

text: If a body is resting on the ground, the Normal force equals its weight. If I keep putting another body on top of it, eventually the body will start moving into the ground until the new Normal equals the total weight of the body. I can add more weight on top and the body will move further into the ground. So it appears that the denser the object gets, the more the Normal force. So my question is 3 folds:   1. Is it accurate to say that the normal force depends on the density of the object that the reference object is sitting on?    2. It appears that when the body starts moving into the ground, the equation at any given time t will be Mg - N(t) = Ma where M is the total mass which essentially says that the force that the ground feels on it at time t is N which will keep changing with time and become Mg at some point where acceleration becomes 0?    3. If I draw a free-body diagram of an object moving horizontally forward(just to distinguish from backward) because of force applied on it, I will have two forces - The external force applied on it and the force of friction resisting its motion and the equation will be   F - $\mu$mg = ma or   F = m($\mu$g + a)   When I draw the free body diagram of the person pushing the object, I will have three forces - the reaction from the body that is being pushed, the force of friction in the forward direction and the force applied by the person in the opposite direction to his/her movement (a kick backwards). The equation for the person will be   $\mu$Mg -R -F$_{b}$ = Ma or   R = M($\mu$g -a) - F$_{b}$   It would seem that perhaps the reaction to Force F will not be the same as F. Shouldn't action and reaction be the same?


title: Kinematics of scratching a rolling tire

text: Recently a "U" shaped scratch appeared on my car's front tire. It looks like someone rolled past a curb and scraped it. This has led me to the following problem: Generate a plot (ideally polar) of a scratch pattern on a tire given tire radius and the ground height of a protrusion. Assume the protrusion is a single point and remains stationary while the tire rolls past. Also assume that the tire rolls straight past the protrusion, which scratches the side of the tire (i.e. hubcap, trim, etc). If I wasn't so rusty on this material I would have taken a stab at it myself. So, any help in setting up and solving this problem would be greatly appreciated.


title: How big should a lens of a telescope be so we can see the American flag on the Moon from the Earth's surface?

text: How big in diameter should the lens of the imaginary telescope to be so we can see the American flag from e.g. some observatory in Hawaii?


title: How does a comet form?

text: As the title explains,   1. How does a comet form?   2. What are the elements, what is a comet composed of?    3. Why didn't they become part of planets, moons or asteroids?


title: Is there a way to fill Tank 2 from Tank 1 through Gravity alone?

text: I am a newbie in water system design but I am currently faced with the exact situation below on my land, and I need to know whether gravity alone is sufficient in order to fill Tank 2 from Tank 1, as I already experienced backflow. Please have a look at this image: ![enter image description here](http://img819.imageshack.us/img819/184/qhzf.png) _Note: let's not worry about how Tank 1 is being filled as it probably does not matter - I just made sure I never used a pipe diameter smaller than 0.5" between Tank 1 and Tank 2._ As you can see:   * My water source is the overflow of Tank 1   * I have put a 1" diameter pipe in the first 'downhill' section of my path   * The second, longest section of the path is a pipe of 0.5" diameter   * The system is 'powered' through gravity alone **Questions**   * Is there any missing variable (eg path length?) on my drawing in order to resolve this system? (What are the important variables?)   * The flow at the source (the overflow of Tank 1) can be nearly 0. How does this variable influence the system? (Can a larger flow at the source help going up the 'uphill' sections between the tanks?)   * Is the 'head' variable important at all?   * I used a bigger pipe for the first section in an attempt to 'make more weight' and create sufficient pressure to go up the small uphill that follows. Should I limit the use of that bigger pipe strictly to the downhill section, and does it actually make sense to have used a bigger pipe at all?


title: Second law of Newton for variable mass systems

text: Frequently I see the expression $$F = \frac{dp}{dt} = \frac{d}{dt}(mv) = \frac{dm}{dt}v + ma,$$ which can be applied to variable mass systems. But I'm wondering if this derivation is correct, because the second law of Newton is formulated for point masses. Furthermore if i change the inertial frame of reference, only $v$ on the right side of the formula $F = \frac{dm}{dt}v+ma$ will change, meaning that $F$ would be dependent of the frame of reference, which (according to me) can't be true. I realize there exists a formula for varying mass systems, that looks quite familiar to this one, but isn't exactly the same, because the $v$ on the right side is there the relative velocity of mass expulsed/accreted. The derivation of that formula is also rather different from this one. So my question is: is this formula, that I frequently encounter in syllabi and books, correct? Where lies my mistake.


title: Why does angular velocity lies in the axis passing through the center of the circumference?

text: I understand that it can't be placed anywhere on the radius because it doesn't vary with it ( and so of course it doens't make sense to place it anywhere else on the plane), but why do we place it there? I undertand all the advantages that came _after_ $$\vec v = \vec\omega\times\vec r$$ $$\vec a = \vec\alpha\times\vec r + \vec\omega\times\vec v$$ but I can't see what's before. PS. I know there is another question opened: Direction of angular velocity but I don't understand the answers given there (it's stated that angular velocity _points in every direction of the circumference's plane_!?) and I can't comment yet so please don't close this one. Thank you.


title: How to evaluate spin operators in second quantization for spin symmetry-broken Slater determinants?

text: Suppose we have the following Slater determinant: \begin{equation} | \Psi \rangle = \prod \limits_{i,i'} a^+_{i\alpha} a^+_{i'\beta} | \rangle \end{equation} where $a^+_{i\alpha}$ creates an electron in state $i$ with spin $\alpha$ and, in general, $i \neq i'$. I want to evaluate $\langle \Psi | S^2 | \Psi \rangle$ using second quantization. We can express the $S^2$ operator as \begin{equation} S^2 = S_- S_+ + S_z (S_z +1) \end{equation} with \begin{equation} S_- = \sum_p a^+_{p\beta} a_{p \alpha} \hspace{1.5cm} S_+ = \sum_p a^+_{p\alpha} a_{p \beta}. \end{equation} Since $|\Psi \rangle$ is an eigenfunction of $S_z$, evaluating $\langle \Psi | S_z | \Psi \rangle$ terms becomes trivial and the problem reduces to the evaluation of $\langle \Psi | S_- S_+ | \Psi \rangle$. In the standard restricted Hartree-Fock method, $i = i'$ and it is easy to show that $\langle \Psi | S_- S_+ | \Psi \rangle = 0$ using the canonical anticommutation relations. When $i \neq i'$ (unrestricted Hartree-Fock) we must have that $\langle \Psi | S_- S_+ | \Psi \rangle = N_\beta - \text{Tr}(PQ)$ (this I verified in books) where $N_\beta$ is the number of $\beta$ electrons and $P_{ij} = \langle \Psi | a^+_{j\alpha} a_{i\alpha} | \Psi \rangle$ and $Q_{ij} = \langle \Psi | a^+_{j\beta} a_{i\beta} | \Psi \rangle$. My question is then, specifically, how do we obtain $\langle \Psi | S_- S_+ | \Psi \rangle = N_\beta - \text{Tr}(PQ)$ using the anticommutation relations of second quantization? If I try to use these anticommutation relations just in the way they are written in textbooks I do not get the right answer. Clearly, I am getting something wrong or some special consideration needs to be taken into account when $i \neq i'$. If someone can show me how to correctly use second quantization to get the right answer here, I would appreciate it a lot.


title: Component of force field tangent to a curve

text: I recently had to solve the following problem: > _A force $F=10\hat x- 8\hat y$ is applied to an object that is constrained > to travel towards increasing values of x along the path defined by $y=x^2$ > and $z=0$. Find the component of $F$ that is tangent to this path at the > point $(2,4,0)$._ Attempt: I used Pythagorean's theorem to $ F=12.81 N$ since the $x$ and $y$ components are given. I just wasn't sure if this is what the questions is asking for, or is there something else I have do do? **Edit.** More generally, suppose that one is given a curve and a force field on the plane. How would one go about computing the component of such a force field in the direction tangent to the curve at a given point?


title: Why doesn't my pinhole camera work?

text: We all know that light travels in straight a line, which can be proved by pinhole imaging as in the picture shown : ![pinhole camera](http://i.stack.imgur.com/lNIN3.png) But when I'm doing this little experiment with an apple, no matter how I change the distance between the object and the pinhole, an image can never be observed on the cardboard behind. So what's going wrong? Please help! ![pinhole camera experiment](http://i.stack.imgur.com/Rp92x.jpg)


title: Absorption cross section for direct dissociation and predissociation - Photoelectric absorption

text: Why the absorption cross section in direct dissociation process is wide and structureless while the absorption cross section in the predissociation process is structured and containing lines which are normally Lorentzian in shape?


title: Can the solar system really fit in a thimble?

text: Almost every time somebody talks about atoms, at some point they mention something like this: > If we remove the spaces between the atoms and atomic components, we can fit > the solar system in a thimble. Or > If we remove the spaces between the electrons and the nucleus, we can fit > the universe in a baseball. I know that atoms are mostly empty, but I've always thought that those statements are exaggerating. Can we really fit the solar system in a thimble (if we remove all those spaces)?


title: What is the purpose of differential form of Gauss Law?

text: I am learning the differential form of Gauss Law derived from the divergence theorem. $${\rm div}~ \vec{E} =\frac{\rho}{\epsilon_0}.$$ So far in my study of math and physics, the word "differential" has been associated with a infinitesimal change in a value (e.g. dv, dr...etc) I do not really understand what the physical interpretation of the Gauss Law in differential form is. In general, what is the purpose of putting an equation into differential form?


title: Adiabatic evolution for initial Hamiltonian on Hadamard basis and problem Hamiltonian as diagonal

text: This is spawned from a comment at the answer to one of my previous questions. Someone suggested to me that claiming the following statement might be NP- hard. Could anyone please help me to figure out whether it is true? > If the eigenstate of the initial Hamiltonian are superposition of all > possible states we cannot perform the adiabatic evolution with a diagonal > Hamiltonian as the problem Hamiltonian. Thanks in advance!


Does Gravity / curved space cause rotation? Meaning, if a spaceship is heading not directly toward Earth, but slightly off to one side, and when finally being close to the Earth it falls into earth orbit, does the spaceship continue to point in the same directions as it was when approaching the earth, or does it now rotate at a frequency that is equal to its orbital frequency? I would expect the spaceship to be pointing in the opposite direction after completing one half of an earth orbit.


title: Ground State Degeneracy in ferromagnetic Heisenberg model

text: I am reading the book "Lecture notes on Electron Correlation and Magnetism" by Patrik Fazekas. It says, "The ground state (of Heisenberg FM model) is not unique. We have just found that the system has the maximum value of the total spin $S_{tot}= LS$. $S^{z}_{tot} = LS $ state which is maximally polarized in the z-direction. However the Hamiltonian is spin-rotationally invariant , hence turning the total spin in another direction does not change the energy: the ground state must be (2LS+1)-fold degenerate." where, L is the no. of lattice sites. I don't understand why the ground state should be (2LS+1)-fold degenerate and not infinitely degenerate. Are we not considering a global rotation symmetry of the system? I understand it in the following way: All the spins in the system have the same quantization axis which is along z-direction. Now, if every spin is rotated by the same angle,it produces a new state but the energy of the system remains unchanged because it depends on the scalar product of spins in the Heisenberg Hamiltonian. Therefore, a global rotation chosen from any of the infinite no. of possible rotations should produce infinitely many possible states and leave the energy of the Hamiltonian invariant (equal to the ground state energy). " Does the system has a single quantization axis(i.e. same quantization works for each spin)? or Do we need to consider a unique quantization axis for each spin? What happens to the quantization axis/axes of the system/spins as we consider a different state which has been globally rotated by some angle? If the quantization axis (z-axis) remains fixed, then the system no longer has the maximum value of $S^z$ in the new state obtained after a global rotation. But, if the quantization axis rotates with the system, then, why are there only, (2LS+1) possible rotations? In other words, how does L get into the degeneracy? Thanks


title: kadanoff and cohomology

text: for those that combine Homology group and some form of Kadanoff scheme for coarse graining on a lattice, am I having a good argument when saying this: (practical thinking now) 1\. I obtain the Homology groups for a lattice space, I do some coarse graining on it, everything looks fine 2\. I go to the cohomology. Cohomology induces the linking product, I have a new structure involving possible linked chains. 3\. I do coarse graining While keeping the points of the lattice resp. the duals (the links) and coarse graining isn't it possible that I miss the consistency of the linking product? I mean, coarse graining should destroy the consistency of the linking product so I may end up with some different space (note that linking product may generate spaces not visible by homology alone) Shouldn't there be some sort of beta function in cohomology that should take care of this?


title: What does "sites" mean in the lattice language?

text: I acknowledge that this question is quite trivial. But in the lattice jargon, what does a $N$-sites lattice mean?   * it's a lattice $N\times N$ or   * it's a lattice with $N$ vertices?    * another option perhaps. From the context I cannot infer it. I just know that $N$ is quite large, i.e. in the thermodynamic limit $N\to \infty$, one also has $N^2\to\infty$ and $N^{1/2}\to\infty$. * * * _Edit:_ The lattice is in 2D.


title: Seeing inside an atom

text: ![Hydrogen  atom](http://i.stack.imgur.com/DDuwM.png) After seeing this photograph and reading this article : The First Image Ever of a Hydrogen Atom's Orbital Structure I was really happy to come to know that orbitals have actually been seen now, but it also made me wonder that how was the orbitals illuminated ? I know that when we send light through a colloidal solution its path is illuminated, but how could the experimental physicists who performed this experiment illuminate an atomic orbital. As I understand it is only a probability density rich region of finding the electron. How is it possible to see/detect a probability density wave ? Addendum : I understand from the article that the nodal structure is inferred from the interference pattern observed, but how does the interference pattern account for nodal structure, also is this an experimental prove of the results of schrodinger wave equations ? Addendum 2 : after struggling through this paper http://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.110.213001 I understood most of the experiment, and got stuck here, I understoot the electrons are ionised and then there interference patterns are observed, but how do the interference pattern represent the shapes of atomic orbitals ? Can the expirement be taken as a solid proof for the establishment of shapes of atomic orbitals ?


title: Solving straight-line motion question for time

text: I apologise in advance if this question doesn't appeal to the advanced questions being asked in this Physics forum, but I'm a great fan of the Stack Exchange software and would trust the answers provided here to be more correct than that of Yahoo! Answers etc. A car is travelling with a constant speed of 80km/h and passes a stationary motorcycle policeman. The policeman sets off in pursuit, accelerating to 80km/h in 10 seconds reaching a constant speed of 100 km/h after a further 5 seconds. At what time will the policeman catch up with the car? The answer in the back of the book is 32.5 seconds. The steps/logic I completed/used to solve the equation were:   \- If you let x equal each other, the displacement will be the same, and the time can be solved algebraically. Therefore:   $$x=vt$$ As the car is moving at 80km/h, we want to convert to m/s. 80/3.6 = 22.22m/s   $$x=22.22t$$ As for the policeman, he reaches 22.22m/s in 10 seconds.   $$\begin{aligned} x &= \frac12 (u+v) t \\\ &= \frac12 \times 22.22 \times 10 \\\ &= 111.11 \mathrm m \end{aligned}$$ The policeman progresses to travel a further 5 seconds and increases his speed to 100km/h.   100km/h -> m/s = 100 / 3.6 = 27.78m/s. $$\begin{aligned} x &= \frac12 (u+v) t \\\ x &= \frac12 \times (22.22 + 27.78) \times 5 \\\ x &= \frac12 \times 50 \times 5 \\\ x &= 250 / 2 \\\ x &= 125 \mathrm m \end{aligned}$$ By adding these two distances together we get 236.1m. So the equation I have is:   $$22.22t = 27.78t - 236.1 $$ Which solves to let t = 42.47s which is really wrong.


title: How can I prevent drinking water from freezing in cold weather?

text: I'd like to keep a container of potable water in my car for when I get stuck, but in the winter it gets cold enough for bottled water to freeze. I can't put salt in it, because then it won't be drinkable anymore (or at least, will make my condition worse if I do drink it), and I can't mix it with alcohol in case I get the car unstuck and have to drive. What else could be added to water to sufficiently lower its freezing point without making it undrinkable? (Or to keep it in line with physics rather than chemistry, how do I calculate the amount of pressure the water need to be under to prevent it from freezing at a given temperature, say 15-20o F?)


title: Why does a slight drip of water protect pipes from freezing?

text: I've heard that turning on faucets to a slight drip will prevent pipes from freezing, but I've never understood why this is the case. Can anyone out there help me to understand? Thanks!


title: How does watering your plants help protect against freezing?

text: I've always heard that watering plants if the temperature goes a few degrees below freezing will help prevent them from freezing, but I've never quite understood the physics behind it. Can you guys help me to understand this a bit better? Thanks!


title: What is the role spin-networks play in Loop Quantum Gravity?

text: This is a second question (in what will probably become a series) in my attempt to understand LQG a little. In the previous question I asked about the general concepts behind LQG to which **space_cadet** provided a very nice answer. Let me summarize that answer so that I can check that I've understood it properly. In LQG, one works with connections instead of a metric because this greatly simplifies the equations (space_cadet makes an analogy to "taking a square root" of K-G equation to obtain Dirac equation). The connections should determine the geometry of a given 3D manifold which is a space-like slice of our 4D space-time. Then, as is usual in quantizing a system, one proceeds to define "wave-functions" on the configuration space of connections and the space of these functionals on connections should form a Hilbert space. * * * _Note:_ I guess there is more than one Hilbert space present, depending on precisely what space of connections we work with. This will probably have to do with enforcing the usual Einstein constraints and also diffeomorphism constraints. But I'll postpone these technicalities to a later question. * * * So that's one picture we have about LQG. But when people actually talk about LQG, one always hears about spin-networks and area and volume operators. So how does these objects connect with the space_cadet's answer? Let's start slowly >   1. What is a spin-network exactly? >   2. What are the main mathematical properties? > Just a reference to the literature will suffice because I realize that the questions (especially the second one) might be quite broad. Although, for once, wikipedia article does a decent job in hinting at answers of both 1. and 2. but it leaves me greatly dissatisfied. In particular, I have no idea what happens at the vertices. Wikipedia says that they should carry intertwining operators. Intertwinors always work on two representations so presumably there is an intertwinor for every pair of edges joining at the vertex? Also, by Schur's lemma, intertwinors of inequivalent irreps are zero, so that usually this notion would be pretty trivial. As you can see, I am really confused about this, so I'd like to hear >  3\. What is the significance of vertices and intertwinors for spin- > networks? Okay, having the definitions out of the way, spin-networks should presumably also form a basis of the aforementioned Hilbert space (I am not sure which one Hilbert space it is and what conditions this puts on the spin-networks; if possible, let's postpone this discussion until some later time) so there must be some correspondence between connection functionals and spin-networks. >  4\. How does this correspondence look precisely? If I have some spin- > network, how do I obtain a functional on connections from that?


title: Two magnets with Iron truncated cone or pyramid. How low inclination can the truncated cone or pyramid be made?

text: Two magnets with Iron truncated cone or pyramid: Frustrum **How much low inclination can the truncated cone or pyramid: Frustrum be made, without non homogeneous lines of magnetic field coming out from the sides ?????????** ![Cone_Inclination](http://I.stack.imgur.com/IqOxV.png) Two magnets with Iron truncated cone and Yoke ![Two magnets with Iron truncated cone and Yoke](http://I.stack.imgur.com/9Ak7B.png) Link_to_Example_1 Two magnets with Iron truncated pyramid and Yoke in a different set up. ![Two magnets with Iron truncated pyramid and Yoke](http://I.stack.imgur.com/w9927.jpg) Link_to_Example_2 Knowing the magnetic flux at the larger base of the truncated cone or Frustrum, the area of the smaller base will be the one in which you get the magnetic saturation of the Iron. But, ... **How much low inclination can the truncated cone or Frustrum be made, without non homogeneous lines of magnetic field coming out from the sides ?????????** ![Cone_Inclination](http://I.stack.imgur.com/IqOxV.png) Also in the image of Link_to_Example_2 the gray rectangular number 2 is a block between the iron pieces to keep the distance between the two parts. It must also be a non ferromagnetic material like brass, wood, stainless steel, etc. **But what's the best material to hold strong compression and not catch magnetic field drawing it away from the main gap (number 4) ???** Help please. Thank you in advance.


title: Gauge invariance and Feynman path-integrals

text: Let me look at the Hamiltonian of a charged particle in a plane in a constant magnetic field ($\vec{B}$) pointing upwards - then in usual notation it is, $$\hat{H} = \frac{1}{2m}\biggl(\hat{p} + \frac{e}{c}\hat{A}(\hat{r})\biggr)^2$$ To convert this in a Feynman-path-integral language, I pick say a gauge $\vec{A}=(-\frac{B}{2}y,\frac{B}{2}x)$and then in this gauge $\hat{p}$ and $\hat{A}$ commute and that makes rewriting in the path-integral language much easier. If I put this through the usual process of "deriving" a Feynman path-integral then I would get the expression, $$\int \bigl[\mathcal{D}\vec{r}\bigr]\bigl[\mathcal{D}\vec{p}\bigr] \exp\biggl[i \int dt \biggl(\vec{p}\cdot\dot{\vec{r}} - \frac{1}{2m}\Bigl(\vec{p}+\frac{e}{c}\vec{A}\Bigr)^2\biggr)\biggr]$$   * Is it obvious (or true ) that the above expression is independent of the gauge I chose to the calculation? Is there a way to write the system in the path-integral language without explicitly choosing a gauge? (I have worked through calculations in Yang-Mills theory in the Fadeev-Popov gauge which does exactly that in those cases but I can't see a way out here...)    * Can't I have written down the above path-integral without even going through the usual process of finding infinitesimal transition amplitudes and then collecting them together? I mean how often is it safe to say that for a Hamiltonian $H(p,q)$ the path integral representation of the transition amplitude will be $\int [\mathcal{D}p][\mathcal{D}q]e^{i\int dt (p\dot{q}-H(q,p)) }$?  Now the Heisenberg's equation of motion will tell that, $$\frac{d\vec{r}}{dt} = \frac{1}{m} \biggl(\vec{p} + \frac{e}{c}\vec{A}\biggr)^2$$ In the Feynman path-integral the position and the momentum vectors are treated to be independent variables and hence it would be wrong to substitute the above expression into the path-integral but outside one can I guess do this substitution and one would get for the action (whatever sits in the exponent), $$S = \int dt \biggl(\frac{p^2}{2m} - \frac{e^2}{2mc^2}A^2\biggr)$$   * But the above expression doesn't look right!? The integrand isn't what the Lagrangian should be. right?  Now in the derivation of the Feynman-path-integral if one integrates out the momentum for every infinitesimal transition amplitude and then reconstitutes the path-integral then one would get the expression, $$\int [\mathcal{D}\vec{r}] \exp\biggl[i\int dt \biggl(\frac{m}{2}\dot{\vec{r}}^2 - \frac{e}{c}\dot{\vec{r}}\cdot\vec{A}\biggr)\biggr]$$   * Now what seems to sit in the exponent is what is the "correct" Lagrangian - I would think. Why did the answer differ in the two different ways of looking at it?    * I wonder if given a Hamiltonian its corresponding Lagrangian can be "defined" as whatever pops out in the exponent if that system is put through this Feynman re-writing. In this case keeping to just classical physics I am not sure how to argue that the $\frac{m}{2} \dot{\vec{r}}^2 - \frac{e}{c} \dot{\vec{r}}\cdot\vec{A}$ is the Lagrangian for the system with the Hamiltonian $\frac{1}{2m}\bigl(\vec{p} + \frac{e}{c}\vec{A}(\vec{r})\bigr)^2$  I think I have seen examples on curved space-time where the "classical" Lagrangian differs from what pops out in the exponent when the system is path- integrated.


title: Is mass-energy conversion in chemical reactions experimentally observable

text: This is a common point of argument on internet forums. I think it is fairly well established theoretically that there is a very small amount of mass converted to energy in an exothermic chemical reaction. Has this ever been observed experimentally? If not, how far off is our current experimental state of the art from being able to detect this change? Is it something that could conceivably be accomplished in the next few decades, or are we unlikely ever to be able to observe it?


title: Hypercharge for $U(1)$ in $SU(2)\times U(1)$ model

text: I understand that the fundamental representation of $U(1)$ amounts to a multiplication by a phase factor, e.g. EM. I thought that when it is extended to higher dimensional representations, it would just become a phase factor times the identity matrix. Can someone explain where the hypercharge comes into the $U(1)$ generator matrices in $SU(2)\times U(1)$ model, e.g. $Y = -(1/2) I$ in $(2, -1/2)$ representation? I don't quite understand where the "$-1/2$" comes from. Where do all these hypercharges come from? What is the logic behind choosing a particular value like $-1/2$? Hope someone can clarify.


title: why is time order invariant in timelike interval?

text: why do two observers measure the same order of events if we are inside the light cone? (e.g. if ds^2 > 0 time order is preserved according to the class mech book I am reading, but it doesn't give any proof of this) I assume there is some simple geometrical argument I am missing and analgously why do two observers measure possible different order of events if we are outside the light cone?


title: Mass converted to energy in a common fire?

text: In a common wood fire such as a campfire, is matter converted to energy or is it simply an exothermic chemical reaction and all the mass can be accounted for in the ash and soot?


title: When is energy conserved in a collision and not momentum?

text: Consider the following example: > _A bullet of mass 45g is fired at a speed of 220 m/s into a 5.0 kg sandbag > hanging from a string from the ceiling. The sandbag absorbs the bullet and > begins to swing. To what maximum vertical height will it rise? (Assuming > negligent air resistance and heat loss due to friction)_ The way to solve this would be to calculate the kinetic energy of the bullet and then convert it to gravitational potential energy of the bullet + sandbag. $$E_{k} = \frac12 * 0.045\,{\rm kg} * (220\,{\rm m/s})^2=1089\,{\rm J}$$ We could then find the gravitational potential energy here by substituting in $E_{p} = E_{k}$. But, what I'm curious about here is, what would be the speed immediately after the bullet impacts the sandbag? Either the kinetic energy, or the momentum could be conserved, but not both. $$P = 0.045\,{\rm kg} * 220 \,{\rm m/s} = 9.9 \,{\rm kg} \cdot \,{\rm m/s}$$ Then we find the speed after the collision of the sandbag + bullet to be: $$9.9 \,{\rm kg} \cdot \,{\rm m/s} = 5.0\,{\rm kg} + 0.045\,{\rm kg} * X \,{\rm m/s}\to 1.96234 \,{\rm m/s} = X$$ So if momentum is conserved then the kinetic energy would be: $$E_{k} = \frac12(5.0\,{\rm kg} + 0.045\,{\rm kg}) * (1.96324 \,{\rm m/s})^2=9.72\,{\rm J}$$ Should this question specifically state whether momentum or energy was conserved? How can you tell which one will be conserved?


title: What is the Weyl algebra of a confined bosonic particle?

text: The abstract Weyl Algebra $W_n$ is the *-algebra generated by a family of elements $U(u),V(v)$ with $u,v\in\mathbb{R}^n$ such that (Weyl relations) $$U(u)V(v)=V(v)U(u)e^{i u\cdot v}\ \ Commutation\ Relations$$ $$U(u+u')=U(u)U(u'),\qquad V(v+v')=V(v)V(v')$$ $$U(u)^*=U(-u),\qquad V(v)^*=V(-v)$$ It is a well known fact (Stone- von Neumann theorem) that all irreducible representations $\pi:W_n\rightarrow B(H)$ of this algebra such that $$s-\lim_{u\rightarrow 0}\pi(U(u))=U(0),\qquad s-\lim_{v\rightarrow 0}\pi(V(v))=V(0)$$ are unitary equivalent to the Schroedinger one: $$H\equiv L^2(\mathbb{R}^n)$$ $$\pi(U(u))\equiv e^{i\overline{u\cdot X}},\qquad \pi(V(v))\equiv e^{i\overline{v\cdot P}}$$ where $X,P$ are the usual position and momentum operators defined (and essentially self-adjoint) on the Schwartz space $S(\mathbb{R}^n)$. For this reason, this algebra is usually associated to the quantization of a spinless non relativistic particle in $\mathbb{R}^n$. But what happen if one considers a spinless particle confined in a subregion $K\subset\mathbb{R}^n$, for example a circle $S^1\subset\mathbb{R}^2$, a sphere $S^2\subset\mathbb{R}^n$ or a generic open subset? The state space could still be chosen as $L^2(K)$, but what about the associated Weyl Algebra? I think it can still be defined abstractly but, while the commutation relations must be conserved in some way, the parameters domain $(u,v)$ cannot still be taken as $\mathbb{R}^n\times\mathbb{R}^n$ otherwise we fall in the previous context. Is there some prescription for such a general situation? To be more precise: Given a bosonic particle confined in a sub region K, is it possibile to associate a family of abstract elements $U(u),V(v)$, $u,v$ belonging for example to some additive groups, such that the above Weyl properties are satisfied?


title: Possible spin states?

text: Given a system of two particles with spin up and down, I have troubles to understand the possible states of this system. I would have normally thought, that the possible states are the tensor products of the states of each particle with one of the other. This would have given us the states that are listed in Wikipedia under: substituting in the four basis states. So why do we define these new states(three with total spin $1$ and one with total spin $0$) that are given by triplet and singlet representations? So maybe you don't know what I am asking so I repeat my question: What is the meaning of these triplet and singlet states, because I would have thought that the four states listed above are the possible states of a system? The reference to what I am talking about: wikipedia reference


title: Origin of Space-time fabric

text: I have seen on this site some questions regarding the nature of gravitational- force, and the ways in which it could be interpreted. Most of them mention that the space-time fabric came into existence due to the big-bang. But I cannot understand why it came into existence during the big-bang, and had no earlier existence. Also, how should I develop my notion regarding the form of a gravitational-singularity? My current concept holds it that a gravitational singularity, is a compact-clump of spatial and temporal curves, where laws of Classical Physics are rarely(?) valid. But more often, I "like" to compare a garvitational-singularity with a huge go-down of energy, that is longing to burst open, and thus, decrease its entropy, as well as energy. If in-any way my current-notion is to a-certain degree acceptable, would it then be correct for me to guess that "minute"(?)-fabrics of space-time come into existence whenever there is a conversion of matter into an equivalent amount of energy, according to Einstein's equation E=m.(c^2) ? * * * Please take care of the fact that I am a high-school student-with a corresponding limited-knowledge on Physics, and especially, Quantum-Mechanics.


title: The Lagrangian as a metric

text: My question is, _can the (classical) Lagrangian be thought of as a metric_? That is, is there a meaningful sense in which we can think of the _least- action_ path from the initial to the final configuration as being the _shortest_ one? Then the equations of motion would be geodesics - not in physical space, but in phase space, with the metric defined by the Lagrangian. Below is a slightly expanded outline of this idea, outlining why I think it might make sense. (Classical) Lagrangian mechanics is derived by assuming the system takes the path of minimum action from its initial state to its final one. In the formalism of classical mechanics we normally write time $t$ as a special type of coordinate, quite distinct from the other (generalised) coordinates $\mathbf{q}$. However, if we want we can lump these together into a single set of "generalised space-time coordinates" $\mathbf{r} = (t, q_0, q_1, \dots)$, to be considered a vector space of dimension $1+n$, where $n$ is the number of degrees of freedom. Then the action is just a function of curves in $\mathbf{r}$-space, and the equations of motion are determined by choosing the least-action path between two points, $\mathbf{r_0}$ and $\mathbf{r_1}$. In this picture, $L$ depends upon the position and direction of the curve at each point along it, and the action is obtained by a line integral along the curve. This has a pleasingly geometrical feel, and so my question is, is there a meaningful sense in which I can think of the least-action path as being the _shortest_ path, with the Lagrangian playing the role of a metric on $\mathbf{r}$-space? Or is there some technical reason why $L$ can't be thought of as a metric in this sense? Note that this isn't in general the same as a metric defined on space-time. For an $m$-particle system in $3+1$ space-time, $\mathbf{r}$ will have $3m+1$ dimensions, rather than 4. I'm interested in the notion of a metric in $\mathbf{r}$-space defined by $L$, rather than the more usual picture of $L$ being defined in terms of a metric on space-time.


title: Electric field near a conducting surface vs. sheet of charge

text: I know perfectly well how to derive the magnitude of the electric field near a conductor, $$E = \frac{\sigma}{\varepsilon_0}$$ and near a sheet of charge, $$ E = \frac{\sigma}{2\varepsilon_0} .$$ In fact, I can explain with clarity each step of the derivation and I understand why is one two times larger than the other. But here's what bothers me... When I try to think about it _purely_ intuitively (whatever the heck that actually means), I find it difficult to accept that a planar charge distribution with the same surface density can produce a different field. Why should it _care_ whether there's a conductor behind it or not... ? I repeat, I understand Gauss' law and everything formally required, but I want to understand where my intuition went wrong. **EDIT:** _Thanks to you people, I developed my own intuition to deal with this problem, and I'm happy with it, you can see it posted as an answer!_


title: Theoretical need for Newtonian Gravity

text: I've been wondering: Are there, still, some advantages, for current research, to study Newtonian gravity? I mean, not experimentally, where Newton gravity is a very good approximation to everyday phenomenon or, even, to the solar system (aside some perihelion problems, gravitational lensing, etc...) but to some study of General Relativity or some approximation to the Einstein field equations where knowing the Newton potential is needed.


title: Constructing Tauc Plots

text: How can we construct a tauc plot using data from the transmission spectra? How does one calculate the absorption coefficient for plotting tauc plots?


title: What is path of light in the accelerating elevator?

text: 1. Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator?   2. What is the difference between an ordinary derivative and covariant derivative (which is used in curved geodesic)?


title: Launching a unpropelled rocket

text: I have read about the application of railguns and other ground powered launch, where there is no need to carry any propellant in the rocket. I am wondering, to **what velocity should the vehicle achieve to enter into the orbit of earth?** I know the orbital velocity and escape velocity calculations and their values too. I want to know the practical velocity needed, **_by accounting for the atmospheric resistance_**. Will the rocket enter into an orbit when launched at any angle? Or _is there a range of angle of launch_ to achieve orbit?


title: Sign in front of QFT kinetic terms

text: I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} \partial _\mu \phi \partial ^\mu \phi \end{equation} Based on the answer given here, this makes perfect sense since we want to have positive kinetic energy $\propto \dot{\phi}^2$. So would the Hamiltonian with a negative in front of the kinetic term be unbounded? Does this logic extend to the Dirac Lagrangian typically given by, \begin{equation} \bar{\psi} i \partial _\mu \gamma ^\mu \psi \quad ? \end{equation} i.e., would having a negative in front of the Dirac Lagrangian make the Hamiltonian unbounded?


title: What happens when work function = hf

text: What happens when the photon which hits a metal surface has energy equal to the work function of that surface? $$\phi = hf$$ I realise the emitted electron will have no kinetic energy after escape, but then what does it do? Hover above the surface? Or does it have momentum from escaping? I also don't understand what the work function is caused by - is it the electrostatic attraction between nuclei and the electrons?


title: Annihilation and Creation operators not hermitian

text: The annihilation and creation operators are given below $$ \hat a|n\rangle=\sqrt{n}|n-1\rangle\qquad\text{and}\qquad\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle $$ Now I know the operator a^ is not Hermitian but as to why I'm a little confused. Is it because they 'create' and 'annihilate' photons hence they're not hermitian? I don't think that explanation would be good for an exam if this question was asked though...


title: Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook

text: When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value because it's not a multivalued function. I'm very confused. And when I learned the complex analysis, I've not watched this formula, can anybody tell me where I can find this formula's proof. 2) $$\frac{d}{dx}\ln(x+i\epsilon)=P\frac{1}{x}-i\pi\delta(x)$$ 3) And I also find this formula. Seemingly $f(x)$ has a branch cut, then $$f(z)=\frac{1}{\pi}\int_Z^{\infty}dz^{\prime}\frac{{\rm Im} f(z^{\prime})}{z^{\prime}-z}$$ Can anyone can tell the whole theorem and its proof, and what it wants to express. ![enter image description here](http://i.stack.imgur.com/oK0o1.png) Now I am very confused by these formula, because I haven't read it in any complex analysis book and never been taught how to handle an integral with branch cut. Can anyone give me the whole proof and where I can consult.


title: 4-acceleration of rotating frame

text: Consider the 3-dimensional Minkowski space $$ ds^2=dt'^2-dr'^2-r'^2d\phi'^2 $$ Now we transform it into a rotating frame: $$ t'=t,r'=r,\phi'=\phi+\omega t $$ Then the metric becomes $$ ds^2=(1-r^2\omega^2)dt^2-dr^2-2\omega r^2d\phi dt-r^2 d\phi^2 $$ Consider a static observer in the rotating frame with coordinate $(r,\phi)=(R,0)$. Could anyone show me how to compute the 4-acceleration of this observer? I am expecting the answer to be $(0,-\omega^2 R,0)$ based on my knowledge of Newtonian mechanics. But my calculation does not give this. I have calculated the 4-velocity to be $$ u=\left(\frac{1}{\sqrt{1-R^2\omega^2}},0,0\right) $$ Then I use the formula $$ a^\mu=u^\nu\nabla_\nu u^\mu=u^t\nabla_tu^\mu=u^t(\partial_tu^\mu+\Gamma^\mu_{t\alpha}u^\alpha)=u^tu^\alpha\Gamma^\mu_{t\alpha} $$ Then it turns out only $\alpha=t,\mu=r$ is meaningful, and $$ a^r=-\frac{2R\omega^2}{1-R^2\omega^2}R^2(1-2\omega^2R^2) $$ which I think is definitely wrong...


title: Accelerometer validation trace

text: I am attempting to validate an accelerometer. I have a triaxial accelerometer and placed it on a wheel which goes at various frequencies from 1 to 3 Hz. Am I right in thinking that I would expect the acceleration in _z_ to be 1g? Then in the _x_ and _y_ plane I would expect a read out of acceleration in $X(t) = -2\pi fr\cos{(2\pi f t)}$ and acceleration in $Y(t) = -2\pi fr\sin{(2\pi f t)}$ where $r$ is the radius of the wheel and $f$ is frequency. How can I compare these with my readout from the sensor, and what is the best way to report accuracy? I'm confused as surely I will need to quite accurately time align the two traces.


title: $\nabla ^2\psi$ equals $\psi -$ average value of $\psi$ at neighboring points

text: Let $\psi (x,y,z)$ be a scalar field. I found the following statement in Morse & Feshbach Methods of Theoretical Physics: > The limiting value of the difference between $\psi$ at a point and the > average value of $\psi$ at neighboring points is > $-\frac{1}{6}(dxdydz)^2\nabla ^2\psi$ By taking a small sphere centered at the point of radius $r$, I was able to show that the difference between $\psi$ at a point and the average value of $\psi$ at neighboring points equals $-\frac{1}{6}r^2\nabla ^2\psi$. But I have not been able to get the expression given in the book. Thanks. Edit: To obtain $-\frac{1}{6}r^2\nabla ^2\psi$ I first expanded $\psi$ in a Taylor series up to quadratic terms about the point in question. Then I integrated $\psi$ (rather, its Taylor expansion up to quadratic terms) over the surface of a small sphere of radius $r$ centered at the given point and divided this by the surface are of the sphere (this gives the average value of $\psi$ in the sphere). Subtracting this average from the value of $\psi$ at the center of the sphere gives the result $-\frac{1}{6}r^2\nabla ^2\psi$.


title: Can entropy be explained in terms of a cleaning/keeping your room clean?

text: I'm trying to relate the concept of entropy to keeping my room clean as suggested by my high school teacher ~1993... Comparing the two scenarios:   1. Every day I come home and throw an empty can on the floor and every night before bed I pick it up and throw it in the trash.   2. Every day I come home and throw an empty can on the floor and every six months I bag them all up at once. So to me it's obviously more efficient to let entropy run its course and then undo it in a "batch" as opposed to constantly undoing it. Is that right though? I'm thinking since entropy is related to energy, then is has to be conserved and the same amount of effort should be required to keep a room clean than to clean it after its become horribly messy.