title: What do I see if I move quickly past a charge surrounded by iron filings?

text: This might be a straightforward exercise, in which case I apologize. Suppose I surround a charge by iron filings initially oriented in some fixed direction, and I then move past the charge at an appreciable fraction of the speed of light. > What do I see the iron filings do? More precisely, in my frame there should be a magnetic field of some kind (right?). What do I see it doing to the iron filings?


title: What is a bulk phase transition?

text: I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of what exactly a bulk phase transition is. Could someone please help me with a definition of _bulk phase transition_? I would very much appreciate it.


title: Moment of Inertia and Rotational Dynamics?

text: I'm having problems with the intuition behind the Parallel axis theorem and the Perpendicular axis theorem. I'm self studying Mechanics for the British Curriculum but, the book I've is missing the moment of inertia part. It's supposed to cover simple systems and doesn't contain any calculus. I need a small explanation of the two theorems if possible. No derivation, please. Sorry, if my question isn't appropriate for this site.


title: What is the most efficient way to destroy the universe?

text: Don't worry... I won't actually do it, just would like to know how.


title: What is the origin of the kinetic theory of temperature?

text: Count Rumford's experiments with canon barrels are often cited here, but the kinetic theory certainly precedes them. Another candidate is Daniel Bernoulli, who, in his 1738 book Hydrodynamica showed how Boyle's law follows from the kinetic model. But Bernoulli writes, "As it is well known that heat is intensified as the internal motion of the particles increases..." indicating that these ideas were already current when he wrote. Does anyone have an earlier reference?


title: Mnemonics to remember various properties of materials

text: I'm trying to figure out how to remember that   * hardness: how resistant it is to deformation   * toughness: how resistant it is to brittle failures   * stress: force on a surface area   * strength: ability to withstand stress without failure   * strain: measurement of deformation of a material Does anyone know of a mnemonic or easy way? I only know these from googling them, and I'm finding it tricky to remember them.


title: How do physicists use solutions to the Yang-Baxter Equation?

text: As a mathematician working the area of representation of Quantum groups, I am constantly thinking about solutions of the Yang-Baxter equation. In particular, trigonometric solutions. Often research grants in this area cite this as an "application" of their research. This being said, many mathematicians (definitely including myself) don't know why these solutions are important. So, I wonder: > What exactly do physicists do with solutions to the Yang-Baxter Equation > once they have them?


title: Proton therapy in cancer treatment

text: Why are protons used in cancer therapy ? Is there any advantages compared to classics X-rays treatment or electron treatment ?


title: What is an intuitive explanation of Gouy phase?

text: In laser resonators, higher order modes (i.e. TEM01, etc) accumulate phase faster than the fundamental TEM00 mode. This extra phase is called Gouy phase. What is an intuitive explanation of this effect? Gouy predicted and then experimentally verified the existence of this effect long before the existence of lasers. How did he do it, and what motivated him to think about it?


title: Do Franck-Condon oscillations have natural lineshapes?

text: I recently found a paper (for the curious, this one) that talks about observing the motion of a nuclear wavepacket in H2O, as initiated by tunnel ionization. This wavepacket should be thought of as a superposition of the different vibronic states of the molecule after ionization, and is therefore described by a Franck-Condon oscillation: the oscillation of the overlap between the ground-state nuclear wavefunction and the different eigenstates of the nuclear degrees of freedom. That paper reproduces a figure from a 1975 paper (L. Karlsson et al. Isotopic and vibronic coupling effects in the valence electron spectra of H216O, H218O, and D216O. _J. Chem. Phys._ **62** no. 12 (1975), p. 4745) that experimentally observed the Franck-Condon oscillations in energy-resolved photoelectron spectra: ![enter image description here](http://i.stack.imgur.com/kGso6.png) The measurement clearly resolves the different peaks in the spectrum, their roughly equal spacing, and even a slight displacement for the different isotopic combinations. However, it is clear to me that the shape of the lines, particularly in such an old paper, must come from inhomogeneous broadening mechanisms. (I would by default blame the thermal velocity spread of the molecules prior to ionization, but I can't tell for sure.) I would therefore like to pose the question:   * **if one were to remove all sources of inhomogeneous broadening, can one observe natural lineshapes for Franck-Condon oscillations such as these?**   * If so, **what are the physical mechanisms behind them and what theoretical tools and models exist to study them?**


title: Is quantum entanglement mediated by an interaction?

text: You can get two photons entangled, and send them off in different directions; this is what happens in EPR experiments. Is the entanglement then somehow affected if one puts a thick slab of EM shielding material between the entangled photons? Have such experiments been made? According to EPR experiments measurements of the entangled states are at odds with SR, so based on that I'd assume the answer is "no"/"don't know", but any citations would be appreciated!


title: The definition of an inertial reference frame in Einstein's relativity

text: I'm reading Sean Carroll's book on general relativity, and I have a question about the definition of an inertial reference frame. In the first chapter that's dedicated to special relativity, the author describes a way of constructing a reference frame in the following manner: > "The spatial coordinates (x, y, z) comprise a standard Cartesian system, > constructed for example by welding together rigid rods that meet at right > angels. The rods must be moving freely, unaccelerated. The time coordinate > is defined by a set of clocks, which are not moving with respect to spatial > coordinates. The clocks are synchronized in the following sense. Imagine > that we send a beam of light from point 1 in space to point 2, in a straight > line at a constant velocity c, and then immediately back to 1 (at velocity > -c). Then the time on the coordinate clock when the light beam reaches point > 2, which we label $t_2$, should be halfway between the time on the > coordinate clock when the beam left point 1 ($t_1$) and the time on the same > clock when it returned ($t^{'}_{1}$): $$t_2=\frac{1}{2}(t^{'}_{1}+t_1)$$ The > coordinate system thus constructed is an **inertial frame** ". First of all, it is not completely clear what does "the rods must be moving freely, unaccelerated" exactly mean. Unaccelerated compared to what? Secondly, and this is my main question, is the ability to synchronize clocks is unique to inertial frames? If the frame is not inertial, in the sense that Newton's second law $\vec{F}=\frac{d\vec{p}}{dt}$ does not hold, is it still possible that for a set of clocks which are not moving with respect to the spatial coordinates of this frame, that the equation $t_2=\frac{1}{2}(t^{'}_{1}+t_1)$ will always hold for any 2 points in space and a beam of light traveling between them? Can the ability to synchronize clocks be used as a criteria for inertial frames?


title: Action of the Lorentz group on scalar fields

text: The Lorentz groups act on the scalar fields as: $\phi'(x)=\phi(\Lambda^{-1} x)$ The conditions for an action of a group on a set are that the identity does nothing and that $(g_1g_2)s=g_1(g_2s)$. This second condition is not fulfilled because of the inverse on $\Lambda$. What is then the action of the Lorentz group on the scalar fields?


title: Obtaining the conserved current of the Lagrangian making the parameter depending on $x$

text: To calculate the conserved current due to an internal symmetry of the system (expressed by the Lagrangian density) we can proceed as follows: if it is invariant under $\delta \phi = \alpha \phi$, where $\alpha$ is constant we make $\alpha$ depending on the space-time $x$, and consequently the variation on the Lagrangian should be of the form $\partial_\mu \alpha * f^\mu(\phi)$ the argument for this is that when $\partial_\mu \alpha $ is zero ($\alpha$ is constant) we should recover that $\delta L=0$ my question is why it should have this form? I don't see why we couldn't have something like $\log (\partial_\mu \alpha f^\mu(\phi) + 1)$ or something like that. I will finish the argument for those who are curious, The change in the action should be zero for any change of the fields, and after integrating by parts we obtain $\partial_\mu f^\mu(\phi) =0$ and therefore $f^\mu$ is the sought current.


title: Time derivative to energy equates $0$

text: Following up on a related question. This implies that, From the reference we can write \begin{align*} E(\lambda) &= I_F [g_\lambda ] + I_K [f_\lambda , g_\lambda ] + I_V [f_\lambda ] \\\ & = \lambda^{4-d} I_F [\bar A] + \lambda^{2-d} I_K [\bar \phi,\bar A]+ \lambda^{-d} I_V [\bar \phi]. \end{align*} This must be stationary at $\lambda = 1$, therefore we must have, \begin{equation} 0= (4-d) I_F [\bar A] + (d-2) I_K [\bar \phi,\bar A]+ d I_V [\bar \phi].\tag{1} \end{equation} I think the equation (1) equates the time derivative to zero. Is it right? If this is true, then why are we taking this derivative and why does it need to be zero? * * * Source: Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43)


title: Different invariant gauge groups (IGG) on different lattices with the same form mean-filed Hamiltonian?

text: Suppose that we use the Schwinger-fermion ($\mathbf{S_i}=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$) mean-field theory to study the Heisenberg model on 2D lattices, and now we arrive at the mean-field Hamiltonian of the form $H_{MF}=\sum_{<ij>}(\psi_i^\dagger u_{ij}\psi_j+H.c.)$ with $u_{ij}=t\sigma_z$($t>0$), where $\psi_i=(f_{i\uparrow},f_{i\downarrow}^\dagger)^T$, and $\sigma_z$ is the third Pauli matrix. Now let's find the IGG of $H_{MF}$, by definition, the pure gauge transformations in IGG should satisfy $G_iu_{ij}G_j^\dagger=u_{ij}\Rightarrow G_j=\sigma_zG_i\sigma_z $ on link $<ij>$—(1) . Specifically, consider the IGGs on the following different 2D lattices: (a)Square and honeycomb lattices(unfrustrated): These two lattices can be both viewed as constituted by 2 sublattices denoted as $A$ and $B$. Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ in **the same sublattice** are _site-independent_ while those in different sublattices differ by $G_A=\sigma_zG_B\sigma_z$ and $IGG=SU(2)$. (b)Triangular and Kagome lattices(frustrated):Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ are _global_ (site-independent) and $G_i=\bigl(\begin{smallmatrix} e^{i\theta }& 0\\\ 0& e^{-i\theta } \end{smallmatrix}\bigr)$ which means that $IGG=U(1)$. So my question is: The same _form_ mean-field Hamiltonian $H_{MF}$ may has _different_ $IGGs$ on _different_ lattices?


title: How do you show from the index notation that the change of frame formula for a metric must involve the transpose?

text: Let $x^\mu$ and $x^{'\mu}$ be two coordinate systems related by $$dx^{'\mu}~=~S^\mu{}_\nu~ dx^\mu.$$ In index notation the metric in both systems are related by: $$g'_{\rho\sigma}~=~g_{\mu\nu}~(S^{-1})^\mu{}_\rho~(S^{-1})^\nu{}_\sigma.$$ In matrix form this is $$g'~=~(S^{-1})^Tg(S^{-1}).$$ How is it clear from the index notation that the matrix form must involve the transpose matrix?


title: Electric field of an infinitely long (thin) metal cylinder

text: An infinitely long thin metal cylinder of radius $a$ coaxial with the $z$ axis carries a uniform surface charge density $\sigma=\frac{\lambda}{2\pi a}$, where $\lambda$ is a constant. A thick cylindrical insulating tube coaxial with the metal cylinder carries a uniform volume charge density $\rho=\frac{\lambda}{\pi(c^2-b^2)}$ throughout its volume. The insulating cylindrical tube has an inner radius $b$ and an outer radius $c$. Find the electric when a. $$s<a$$ b.$$a<s<b$$ c.$$b<s<c$$ d.$$s>c$$ So we'd have to use Gauss's law for each case, varying the width of the Gaussian surface.


title: Distinguish between Past and Future

text: When writing the metric in Minkowski space, how can we distinguish between the past and the future? I understand the answer after drawing the light cone but I want to know how we get that by just considering the quadratic form of the metric: $$ ds^2= -dx_{1}^2+ dx_{2}^2+ dx_{3}^2 + dx_{4}^2 $$ without visualizing the light cone. Thank you.


title: process matrix - physical interpretation

text: I have a (probably) advanced question, concerning quantum process tomography. Let's say I have made a measurement with a single qubit, and calculated a $\chi$-matrix which looks like $$\begin{bmatrix}0&0&0&0\\\0&1&0&0\\\0&0&0&0\\\0&0&0&0\end{bmatrix}.$$ What's the physical interpretation for this, and why do I need a 4$\times$4 matrix for describing a 1 qubit system?


title: How do aerospace engineers choose a landing system? (Curiosity rover)

text: The Sojourner rover with the Mars Pathfinder used a entry, descent, and landing system involving airbags to land on Mars. The Spirit and Opportunity rovers each used more-or-less the same system involving airbags to land on Mars. The Curiosity rover with the Mars Science Laboratory (MSL) used a very different landing system, including "seven minutes of terror". Why did NASA engineers select such an apparently complicated landing system for MSL? How do rocket scientists calculate the impact forces for a Mars landing for a given proposed landing system, when no one has ever used that landing system on Mars ever before? How do aerospace engineers choose a landing system?


title: Can the process $u\overline{u} \rightarrow s\overline{s}$ be mediated by the EM interaction?

text: I’m working on a homework problem which asks for the dominant contribution (e.g. EM, strong, or weak) to the process $p + \overline{p} \rightarrow \Lambda + \overline{\Lambda}$). ![enter image description here](http://i.stack.imgur.com/tXSeC.png) I know that the EM interaction cannot change flavor, but I’m not sure how this rule applies to the $u\overline{u} \rightarrow s\overline{s}$ annihilation indicated in this diagram. Is it valid to say that since there is no ‘net’ flavor on either side, the photon could be the boson involved? (This almost doesn’t matter for my purposes—since the process can be strong, I think, that’ll be my answer to the question anyway. I was still curious about whether this could be an EM process though.)


title: Calculating impact force for a falling object?

text: Good evening, I'm trying to calculate what kind of impact force a falling object would have once it hit something. This is my attempt so far: Because $x= \frac{1}{2} at^2$, $t=\sqrt{2x/a}$   $v=at$, therefore $v=a \sqrt{2x/a}$   $E_k=\frac{1}{2} mv^2$, so $E_k=\frac{1}{2} m(2ax)=m \cdot a \cdot x$   Since $W=E_k=F_i s$, $F_i=E_k/s=(m \cdot a \cdot x)/s$ For an object weighing about as much as an apple, $0.182$ kg, falling $2.00$ m straight down and creating a dent of $0.00500$ m, this would result in: $$F_i=(m \cdot a \cdot x)/s$$ $$F_i=(0.182 \cdot 9.81 \cdot 2.00)/0.00500=706 \, \text{N}$$ Does this make any sense? I wouldn't be surprised at all to find out I'm missing something here. Any input would be appreciated, thanks in advance!


title: How do I calculate the time it would take for a space craft launching from earth to reach another planet?

text: I am making a matlab program the my final project for my programming class in college. I have chosen this problem, but I am bashing my head against the wall trying to figure out the equations and how they interact with each other. I have the user play with the thrust and the payload to carry to the other planet. The user also decides what planet to go to and what type of orbit to use (elliptical or parabolic). I know I have to usde the rocket equation, escape velocities have to be claculated, and time formula, but I still haven't figured out what to do.


title: event horizons are untraversable by observers far from the collapse?

text: Consider this a followup question of this one In the classical schwarszchild solution with an eternal black hole, the user falls through the event horizon in finite local time, but this event does take place for distant observers in the infinite future. As Leonard Susskind explains, measurements of objects near the horizon are subject to large uncertainties, since all the radiation used to see them is largely red-shifted and received in increasingly large intervals in the future. This is why in some still vaguely defined sense, the effective event horizon grows, even if no matter has actually yet crossed the original event horizon (the stable event horizon after the black hole collapse period) But when the event horizon radiates, the event horizon must recede faster than any infalling observer. Any infalling matter will probably be still stir-fried by the hawking radiation over the eons the evaporation takes (which in the observer frame is intensely blue shifted). The only matter that ever gets to be inside the formal event horizon (not the effective horizon) before the final evaporation is the matter that was already in the original collapsing star, around which the event horizon formed Is there still any room to say otherwise? can still be argued that matter will fall in the event horizon in finite time for distant observers (faster than the evaporation will shrink the event horizon)?


title: The Stern Gerlach Experiment Revisited

text: ## Is it possible to provide an explanation to the observations of the Stern Gerlach Experiment using the classical theories? Some Considerations: We consider the standard set-up for the Stern-Gerlach experiment. The predominant component of $\vec{B}$ is $B_{z}$ Again $B_{z}$ varies most strongly with changes in z $$\vec{F}=\nabla(\vec{\mu}.\vec{B})\approx\vec{e}_{z}\mu_{z}\frac{\partial B_{z}}{\partial z}=\vec{e}_{z}F_{z}$$ The force acting on the electrons is supposed to cause the deflection.This causes acceleration in the z direction and hence an increase in the KE in the z direction. The total KE of the electron(in consideration of the three directions) cannot change since magnetic field can only curve the path of an electron. It cannot change the magnitude of speed.Increase of speed in the z direction may be compensated by decrease of speed in the x or in the y direction Changes in the value of $B_z$ _due to the accelerated motion of the electrons_ ,is accompanied by the creation of an electric field: $$Curl{\;} \vec{E}=-\frac{\partial \vec{ B}}{\partial t} $$ Decrease of magnetic energy= increase in electrical energy, if total KE remains unchanged for each particle. When the particles pass out of the region of interaction with the magnetic field the electrical energy restores the energy of the magnetic field. Prior to this, while the interaction is going on , may write the curl B equation as: $$\int\vec{E}.\vec{dl}=-\frac{d}{dt}\int\int\vec{B}.\vec{ds}$$ The integral on LHS is a closed line integral whose plane is in the x-y direction.The electrons seem to get accelerated in the x-y direction due to the emf in action and this should tend to restore the acceleration in the z-direction. The electrical effect is just a temporary effect. Now,the greater the amount of deflecting force , due to higher value of $\mu_{z}$ ,greater the decrease in magnetic energy and greater the amount of acc in the x-y plane.The restoring effect becomes stronger for larger values of magnetic moment in the z-direction.Incidentally for each value/magnitude of $\mu_{z}$ we have to consider the two directions the ,+ve z and the -ve z directions.


title: Creating a fair 3 sided coin

text: I want to make a cylindrical three sided fair coin, with sides: heads, tails, and edge. What should the area of the edge be in relation to the area of the head of the coin? Assume it is all made of a uniform material. **Thoughts:** I was thinking that so long as the surface areas of all three sides were equal, that would be enough, but this seems to lead to tipping over and landing on one of the other faces. Another thought was that the height of the edge should be equal to the diameter of the face, but this seems much too thick. I am looking for a rigorous way of approaching the problem, as opposed to using (bad?) intuition.


title: Pauli Matrices in orthogonal space

text: In some literature there is reference to $\tau$ matrices which are the same pauli matrices in an orthogonal space. I have not seen any explicit constructions of this anywhere. Could someone tell me or point to literature on how to find the matrix elements of these $\tau$ matrices.


title: How can current be a smooth curve?

text: The following is a graph of the current across some circuit element: ![enter image description here](http://i.stack.imgur.com/9MtQH.png) Note how the current is treated like a smooth continuous function. Even in the analysis of things (i.e. analysis of the transient state of an RC circuit) current is treated like a continuous smooth function. This is how currents works: Say you have a voltage source connected to some wire which is then connected to some component. A field will be established in the wire, and electrons will move around for say $t$ seconds, creating surface charge to ensure the field is uniform and points in the correct direction. The period during which surface charge moves around to establish the field is known as the _transient state_. For a more detailed explanation, see: http://physics.stackexchange.com/a/102936/28118 This transient state lasts for a very short time - a couple of nanoseconds, but exists none the less. So there is a short time between when the voltage source is connected and a current is established. Now say the voltage source is non-constant. Say it starts of at a voltage V when you connect it to a circuit. The transient state will exist for a couple of nanoseconds and then current will be established. Now say you increase the voltage. The transient state will happen again and then a new current will be established. My point is that the current doesn't change "smoothly". Rather, it's a bunch of discrete points, with the space between the points being the transient state and therefore having an undefined current. **Q1) So why is it valid to treat current like a continuous function instead of the discrete function that it is?** **Q2) Say you have a current pulse function:** ![enter image description here](http://i.stack.imgur.com/K1jep.png) **In real life, would the function really be a pulse or would it really be an extremely steep line (but a slope of non-infinity none the less)? I would expect the slope to be non-infinity since it would take a couple of nanoseconds to establish the current instead of it being immediate as the function suggests.**


title: Why does Gauss's law work for a charge off center in a spherical surface?

text: CASE 1: Consider an enclosed spherical surface with a charge $q$ at its centre. From Gauss' law we can say that the flux through this sphere is $q/\epsilon_0$. CASE 2: The charge is inside but _off centre_. In this case the electric field will not be normal to the surface. So how can we use the formula $q/\epsilon_0$? It turned out that the answer is same for both cases. This is my confusion.


title: Why do degenerate normal modes appear as complex conjugate pairs?

text: Can anyone prove this? THE DETAILS Suppose we have a system with n components, (i.e. $|~{\psi}(t)\rangle=\sum \langle x_i|~\psi(t)\rangle ~|~x_i\rangle$) where our equation of motion is described by $$\frac{d^2}{dt^2}|~\psi(t)\rangle=\hat{M}^{-1}\hat{K}~|~\psi(t)\rangle$$ And say we are looking for solutions $|~\omega\rangle$ such that $\frac{d}{dt}|~\omega\rangle=\omega|~\omega\rangle$, and thus our equation simplifies to $$\hat{M}^{-1}\hat{K}~|~\omega\rangle=\omega^2|~\omega\rangle\tag{1}$$ (Note that $\hat{M}^{-1}\hat{K}$ is Hermitian). Is there any way to prove that if $|~\omega_1\rangle,|~\omega_2\rangle$ are linearly independent and both satisfy (1), then $|~\omega_1\rangle=|~\omega_2\rangle^*$, i.e. the components of one vector are the complex conjugates of the components of the other vector? I ask this because this keeps on popping up in our class study of coupled oscillators and I want to know why this is happening Thanks!


title: Linear sigma models and integrable systems

text: I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow I have already difficulties in penetrating the literature... I'd highly appreciate any help with the following question: My aim is to relate a certain (equivariant) linear sigma model on a disc (with a non-compact target $\mathbb C$) as constructed in the exciting work of Gerasimov, Lebedev and Oblezin in Archimedean L-factors and Topological Field Theories I, to integrable systems (in the sense of Dubrovin, if you like). More precisely, I'd like to know if it's possible to express "the" correlation function of an (equivariant) linear sigma model (with non-compact target) as in the above reference in terms of a $\tau$-function of an associated integrable system? As far as I've understood from the literature, for a large class of related non-linear sigma models (or models like conformal topological field theories) such a translation can be done by translating the field theory (or at least some parts of it) into some Frobenius manifold (as in Dubrovin's approach, e.g., but other approaches are of course also welcome). Unfortunately, so far, I haven't been able to understand how to make things work in the setting of (equivariant) linear sigma models (with non-compact target). Any help or hints would be highly appreciated!


title: Large and small gauge transformations?

text: I've a questions about the difference between small and large gauge transformations (a small gauge transformation tends to the identity at spatial infinity, whereas the large transformations don't). Many sources state (without any explanation or reference) that configurations related by small gauge transformations are physically equivalent, whereas large gauge transformations relate physically distinct configurations. This seems odd to me (and some lecturers at my university even say that this is wrong), because all gauge transformations relate physically equivalent configurations. Some of the literature that mentions the difference between small and large gauge transformations: In Figueroa-O'Farrill's notes it is mentioned in section 3.1 (page 81-82) in http://www.maths.ed.ac.uk/~jmf/Teaching/EDC.html In Harvey's notes, see equation (2.13) in http://arxiv.org/abs/hep-th/9603086 In Di Vecchia's notes, see the discussion above (and below) equation (5.7) http://arxiv.org/abs/hep-th/9803026 They all say that large gauge transformation relate physically distinct configurations, but they don't explain why this is true. Does anybody know why this is true? Best regards, Hunter


title: Doesn't charging by induction violate Law of Conservation of Energy

text: Charging by induction seems very unintuitive to me. Let me explain: Consider a positively charged rod and a neutrally charged conductor. You can charge it by induction by earthing it and not lose charge on the rod. You can then use this potential difference in the spherical conductor as energy. This can be repeated many times. Does this not violate the law of conservation of energy? Also, why does charging by friction work? Consider conductors $A$ and $B$, which I charge by rubbing together. Assume that $A$ gains electrons during the charging process and $B$ loses them. But, as soon as $A$ gains an electron, that electron should move back to $B$, since we have a potential difference _and_ a path for conduction.


title: Nature of charge on conductive surfaces

text: In our introductory physics class, we were being taught about charge density ($\sigma$). Our teacher told us that a spherical conductor has a uniform charge density while conductors like a cuboidal conductor have a greater charge density at the vertices. In general, charge density is greater than average in uplifted areas. She also said that in conductors, charge was restricted only to the surface. I asked her why. She wasn't able to deliver a satisfactory answer.


title: Is there an idealist rather than realist interpretation of QM?

text: The many-worlds interpretation of QM is a realist explanation as it makes the wave function of the universe real. That is it makes the probabilities of outcomes real outcomes. One could argue that this view is biased that realist ideas of reality are better by definition. But philosophically speaking idealist ideas of realities have been mooted, for example by Berkely & the Cambridge Platonists, (or Brahman in Indian Philosophy). Is there is an idealist explanation which explains the probabilities inherent in the wave function as epistemological? That is states of knowledge of observers. Except here by observer I do not mean necessarily some one that is conscious, but simply a system that reacts to the information context around it.


title: Why does evenly heating soup with a microwave take so long?

text: My anecdotal observations (which could be incorrect, they're totally unscientific) indicate that it takes almost as long to evenly heat a big bowl of soup in a microwave as it does to heat it on the stove - 4-5 minutes in my most recent example. Yet a plate of meat or vegetables is reheated in 30 seconds or less.


title: Higgs field requires a large cosmological constant -- does the Zero Point Field balance it?

text: I just read Wolfram's blog post on the Higgs discovery. > Still, there’s another problem. To get the observed particle masses, the > background Higgs field that exists throughout the universe has to have an > incredibly high density of energy and mass. Which one might expect would > have a huge gravitational effect—in fact, enough of an effect to cause the > universe to roll up into a tiny ball. Well, to avoid this, one has to assume > that there’s a parameter (a “cosmological constant”) built right into the > fundamental equations of gravity that cancels to incredibly high precision > the effects of the energy and mass density associated with the background > Higgs field. Then I recalled that one of the great unsolved problems in physics is why the zero-point energy of the vacuum predicts a very _large_ cosmological constant which is not observed. The language used to describe these two effects confuses me, but as far as I can tell, Higgs->contraction and ZPF->expansion Any chance these two effects are in balance?


title: Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$?

text: This notation can be found on page 254 of Victor Stenger's Comprehensible Cosmos and in David Tong's Lectures on QFT (Equation 2.4 http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf), and in EDIT: on page 254 of Stenger's Comprehensible Cosmos, the Lagrangian is written with a ${\partial^i}{\partial_i\phi}$ instead of the usual ${\partial^i {\phi}}{\partial_i{\phi}}$ (that David Tong uses). Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$ in QFT ? This fact is used to calculate the Lagrange Equations of Motion (The Klein Gordon Equation) from the Lagrange Density for a Scalar Field. This clearly isn't true for elementary functions like $y^2$ because ${\partial_y}{\partial_y\ ({y^2})}$ =/= $ {\partial_y {y^2}}{\partial_y{y^2}}$


title: The history and modern understanding of spin

text: This question was inspired by Abstruse Goose :) http://abstrusegoose.com/342 It's well known that any attempt to describe the spin (of say an electron) in terms of non-internal spatial coordinates is futile since this is just orbital angular momentum and can't take half-integral values. Spin is currently understood to be simply the generator of rotations, which being Hermitian should correspond to an "observable". However, I remember hearing about attempts to model the electron as an extended object that literally spins and that in these attempts you run into problems with relativity. (A back of the envelope calculation shows that the "periphery" of the electron should be moving 35 times the speed of light)   1. My first question is whether someone knows of a serious attempt in the literature at doing something as described in the previous paragraph   2. How is spin understood (or incorporated) in the framework of (a) string theory (b) loop quantum gravity?


title: With estimates of mass constraints on magnetic monopoles, how likely is one to be found by the LHC(MoEDAL)?

text: Fermilab seems to have ruled out monopoles with mass less than 850 GeV, but I have seen some estimates of the mass thought to be in the order of up to $10^{18}$ GeV, which, of course, would make them undetectable in any accelerators. By 2013, the LHC is scheduled to reach up to 14 TeV. The only disputed sighting of a cosmic ray produced magnetic monopole was in 1982 when Blas Cabrera reported discovering one (Valentine event monopole). This has never been duplicated. CERN has set up the Monopole and Exotics Detector MOEDAL. Other experiments set up to detect them are the Antarctic Impulsive Transient Antenna--ANITA, and the Antarctic Muon And Neutrino Detection Array, aka AMANDA. While both of these detected neutrinos, neither detected magnetic monopoles (looking for Cerenkov radiation produced by products of monopole interaction). Another method of detection is to look for induced current in a superconducting ring when a monopole passes through. Joseph Polchinski called the existence of monopoles “one of the safest bets that one can make about physics not yet seen.” This shows some optimism on his part that one will be seen. Is there any reason or way to limit the mass of the monopole on the upside to 14 TEV or less?


title: Presence of planets in Milky Way and other galaxies

text: Are there only planets in the Milky Way galaxy, or are there other planets in other galaxies? If planets are only in the Milky Way, why aren't there planets in others?


title: Pearlite or spheroidite?

text: If steel alloy having either pearlitic or bainitic microstructures is heated to, and left at, a temperature below the eutectoid for a sufficiently long period of time (e.g.24hr), spheroidite will form. However from the phase diagram we know for eutectoid composition liquid, if we give it enough time pearlite will form (all phases in the diagram assume long transformation time). So what will we get, spheroidite or pearlite, if we cool very slowly?


title: Has anyone actually produced any braided anyons (quasi-particles) yet? (quantum computer)

text: Quasiparticles theoretically can form braids on a 2D surface. The braids apparently are quite stable for reasonably long periods, allowing a superposition state more time before it decoheres through environmental interaction. This would be a big step in quantum computing. Has this actually been achieved or is it theory only at this point?


title: Thermodynamics for Dummies: Entropy and temperature

text: I do not study physics and I have never had a course in thermodynamics. I have no idea what it is about, but I am currently taking a course where we had something about entropy. Would be great if you could help me in easy words.. Wikipedia and other Websites did not help. So: We are looking at a box full of gas with N particles in it (see Maxwell's demon). The question is now to look at two different states. The first state ist when the whole box is used for the gas. The second state is when only one half is used. Now we consider the Transition. 1) -> 2): The second state has less states, hence less entropy. So we lose entropy. This Forces us to do work, to invest energy, otherwise this would not happen (or only with small probability), as the 2nd law of thermodynamics states. So what does this mean for the temperature in this box? Are we considering the temperature in this box or around this box, or both? Does it get warmer or colder? Where? 2) -> 1): This is more probable, and energy is set free. We have more entropy then. But also, this has to do something with temperature, so what are the consequences? I know how to calculate the energy that has to be invested to enforce 1)->2), but what I do not know is the consequences for the temperatures. So we talk about $\Delta S \geq - N k_B ln 2$ (why $\geq$ and not $=$), the difference in entropy, so the temperature difference is $\Delta Q = T \Delta S$.. Sometimes I have read that more entropy means hotter. As if particles are hotter, they move faster, so there's more uncertainty. But then, I also read that if we go 2) -> 1) that the temperature gets colder. But the entropy grows. So I have no idea. My guess is, that sometimes one talks about the temperature in the gas, and sometimes one talks about the temperature outside this box. Thank you for your help. Hope I am right here.


title: Why is it important that Hamilton's equations have the four symplectic properties and what do they mean?

text: The symplectic properties are:   1. time invariance   2. conservation of energy   3. the element of phase space volume is invariant to coordinate transformations   4. the volume the phase space element is invariant with respect to time I'm most inerested in what 3 and 4 mean and why they are important.


title: When/Where will the planets colide?

text: Given N bodies of known position, mass, speed and direction, if they were to collide, where it will happen and when? PS: this is not a homework... im working on a physics simulation engine... **EDIT1:** The bodies interact gravitationally with each other


title: How do you find the uncertainty of an weighted average?

text: The following is taken from a practice GRE question: > Two experimental techniques determine the mass of an object to be $11\pm 1\, > \mathrm{kg}$ and $10\pm 2\, \mathrm{kg}$. These two measurements can be > combined to give a weighted average. What is the uncertainty of the weighted > average? What's the correct procedure to find the uncertainty of the average? I know what the correct answer is (because of the answer key), but I do not know how to obtain this answer.


title: Flow behind a plate causes pressure drop?

text: I've done an experiment where we direct an air jet from a fan onto a set of electronic scales. By measuring the velocity of the air jet at the exit of the fan nozzle, and applying the conservation of linear momentum, I've found the force the jet exerts on the scales, and found this agrees pretty well with the reading on the scales. Next, I put a large wooden plate about 10cm behind the scales, normal to the top plate of the scales. This causes the reading on the scales to increase. My understanding is that this is due to a lower pressure in the area inbetween the back of the scales and the added wooden plate, which then exerts an extra pressure force on the scales. My question is, why/how? This is one of those (in my opinion) awful "you don't know this yet, but make something up" questions that some professors seem to like, so I'm more interested in an idea of the phenomena at play than great detail. An additional minor confusion; I used a thin metal rod with a piece of light string to hold in the flow in order to see the direction of flow. Without the additional plate, the flow direction was in the same plane as the scales. With the extra plate added, the flow direction now starts in the same plane at the edge of the top plate of the scales, but then starts to curve in towards the extra plate. How can, seemingly, a plate placed in a place where there is no very little flow affect the direction of flow upstream?


title: Electron photon interaction potential in old fashioned perturbation theory (OFPT)

text: In this PDF on old fashioned perturbation theory (OFPT) we find from equation (14) the potential describing the interaction between the electron and photon: $$ V = \frac{1}{2}e \int \mathrm{d}^3x\, \psi_e(x)\phi(x)\psi_e(x) \tag{14}$$ where $\phi=photon. $ My question is why $V$ looks as it does and where exactly does it come from?


title: What is a nonlinear field?

text: I have read two possible definitions. A nonlinear field is   1. A field taking values on a manifold.   2. A field whose equation is nonlinear. What do you understand by a nonlinear field or a nonlinear theory?


title: Macroscopic Forces from QED

text: In QED the carrier for electromagnetic interaction is a photon, while macroscopic forces are due to electromagnetic interaction (by macroscopic forces I mean: normal force, object collision, friction etc..). But for example, in the collision of two billiard balls, if 1 kgm/s of momentum is exchanged between the balls, the amount of Energy required in the form of photon virtual particles would be enormous (3E8 J). And while they do not necessarily obey the mass-shell relation, still the farther off shell interactions are, the less they should contribute to the actual phenomena. For least action to be satisfied it seems like some other type of exchange interaction would occur. Is there some kind of larger scale decoherence type interaction going on?


title: How can a detector distinguish between a photon and a gluon

text: 1. Considering that both gluons and photons have no mass, no charge and spin 1, I was wondering how one can tell the difference, if they hit a detector after a collision at the LHC.   2. I know that gluons are supposed to have a color, but I have never heard, that someone measured the color. How can you measure the color of a gluon, which would certainly provide a way of telling the difference between a gluon and a photon?


title: How were noble gases discovered?

text: Noble gases are chemically neutral. They don't react with anything. So, how were they discovered?


title: Can I survive a free fall using a ramp and a rope?

text: Can I survive a free fall by carrying a _very light and resistant_ ramp using a rope? ![enter image description here](http://i.stack.imgur.com/yXc8t.png) **Note:** lets assume the ramp is a little bit heavier at the bottom and I am very skilled at making it always land correctly, also I am wearing a ultra- oiled suit.


title: Can a nearly-extremal black hole be stable against Schwinger vacuum breakdown?

text: I was doing some basic algebra to estimate the range of possible masses $M$ and electric charge $Q$ for a nearly extremal Reissner-Noström black hole. I want to see if the logic is correct the electric field near the event horizon of such a black hole would be $$ E= \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0} \frac{q c^4}{G^2 M^2} = 10^{66} \frac{\text{Volt Kg}^2}{\text{m Coul}} \frac{q}{M^2}$$ since Schwinger-pair vacuum breakdown occurs at $10^{18} \frac{\text{Volt}}{\text{m}}$, it implies that $$ \frac{q}{M^2} \lt 10^{-48} \frac{\text{Coul}}{\text{Kg}^{2}}$$ Would be a bound that guarantees electromagnetic vacuum stability **Question:** can a charged black hole that is stable against this breakdown still be nearly extremal? well, it seems so, because the extremality condition: $$ r_Q = \frac{r_s}{2}$$ Implies that $$ Q \sqrt{ \frac{G}{4 \pi \epsilon_0 c^4}} = \frac{GM}{c^2}$$ $$ \frac{Q}{M} \approx 10^{-16} \frac{Coul}{Kg}$$ So the above two relations would imply that a black hole with a mass $$ M \gt 10^{32} Kg \approx 100 \text{ Solar masses}$$ Can be both stable (against the specific Schwinger-pair decay) and extremal. In particular, if the mass is $10^{32}$ kilograms, a charge around $10^{16}$ Coulomb would make the black hole arbitrarily close to extremal as one might want, and is not at all clear that it would decay from the extremality state But probably i'm overlooking other ways than the extremality of the black hole can decay (in particular, i'm ignorant about how Hawking radiation will contribute in here, specially if the temperature is below 511 KeV). Any insights into this would be greatly appreciated


title: Determining the moment of force

text: A force $F = 3i + 2j$ passes through through a point $P$ with respect to an origin $O$. How do I determine the moment of the force at the origin.


title: name of experiment

text: I'm seeking the name of or reference for an experiment I once saw in a college physics class. At the beginning of one class the instructor repeatedly wound a wiper that spread a blot of some type of ink all over the interior of a glass jar. Then during the lecture (which I admittedly don't remember very well) he must have explained something about the second law of thermodynamics or entropy and that once a large system gets all mixed up, there's really no chance for it to return to its original state. Then he concluded the class by winding the wiper in the opposite direction, and clearly to his delight our jaws all dropped---the film of ink, which had been spread all over the interior of the jar, reappeared in the original blot. Surely, there are folks on this site who demonstrate this every semester. What on earth is this experiment?


title: Would it be possible to have an electron-less solid?

text: We can create plasmas quite easily, indeed you can buy a plasma cutter and generate it all day long for less than $500. Would it be possible to trap a plasma, say magnetically, and cool it so much that it became a solid/liquid/gas without allowing it to receive electrons? Or would the energy requirement exceed what is possible? I'm mostly curious about how the nuclei would interact with each other in the absence of electrons as a solid or liquid. Is there some principle that makes such an endeavor impossible?


title: Finding an equation for velocity and acceleration

text: I'm trying to derive an equation for the velocity and acceleration of an object undergoing simple harmonic motion. I have the equation for displacement: $x = A\sin (2 \pi ft)$ If I differentiate the equation with respect to $t$ then I should get an equation for the velocity and if I differentiate again, I should get an equation for the acceleration, right? So if I differentiate once, I get: $v = 2 \pi f\cos (2 \pi ft)$ If I differentiate again, I get: $a = -2 \pi fsin (2 \pi ft)$ However, I don't think these equations are correct. Am I making a silly mistake? N.b. This isn't homework, I'm making extra revision notes :P


title: How does Newton's 2-prism experiment help to explain why light does not get dispersed into 7-colors in a parallel glass slab?

text: ![enter image description here](http://i.stack.imgur.com/Tze2p.png) In a real parallel glass slide(with two prisms imagined to be touching each other to form a parallel glass slide),   The ray of light should pass through the Z in between without any dispersion or change in direction because the density in both of the prisms is the same(and they are touching each other). So Newton's experiment won't help in explaining this because in it, there is a vaccum(or air) in between and hence the change in direction of light. I hope that you are getting me, if not I'm ready to elaborate more. _[In case anyone is wondering that I did post a Q about dispersion earlier too, then that is true - but I did not get the answer of the question that I really meant to ask and hence this, I hope that you guys will help me better understand this.]_


title: What's jet multiplicity?

text: That's a basic thing, but, surprisingly, it is very difficult to find concise explanation of:   * What is the definition of jet multiplicity?   * Why is it interesting?


What does it mean for a phase space trajectory to be "long" and "stable"? I understand the concept of a trajectory in phase space but not how these adjectives can be applied to one. Thanks.


title: Distribution of Dark Matter around galaxies

text: It is well-known that measurements of the velocity profile in galaxies are not compatible with Newtonian laws. A way to circumvent the problem is to assume that galaxies are surrounded by a spherical halo of Dark Matter. The density of mass of this Dark Matter can then be computed from the velocity profile with Newton's laws. My question is the following: the formation of a galaxy involves essentially only the gravitational interaction. According to Newton (and General Relativity too), the trajectory of a particle does not depend on its mass. As a consequence, I do not understand _how usual matter and Dark Matter could have followed different trajectories leading to such different mass distributions_ nowadays. Can somebody explain this (apparent) paradox ?


title: In QFT, how does matter field excitations create Macroscopic world having different behavior?

text: Is the answer similar to that of classical Quantum Mechanics: Decohrence? If yes, how does that work? I am unable to picture that with QFT. Is there any analogy between Matter Field (of QFT) and Matter Wave (of QM)?


title: Spacetime assumptions at OPERA

text: I would like to know what are the assumptions about the spacetime metric between the neutrinos emiter and detector, I mean, perhaps (just a stab in the dark) could be a mistake to asume the metric of the middle spacetime, because we don't really know how clocks behave at every centimeter of the beam path. I've asked about this (before knowing the noisy news about OPERA results) Is there a single metric for a given system? What are the assumptions about the spacetime metric? Could those be the problem?


title: Question about simple permutation of covariant derivatives

text: I must to compute value $$ [[D_{\mu}, D_{\nu}],D_{\lambda}]A^{\rho}. $$ It is equal to $$ [D_{\mu}, D_{\nu}]D_{\lambda}A^{\rho} - D_{\lambda} ([D_{\mu}, D_{\nu}]])A^{\rho} - [D_{\mu}, D_{\nu}]D_{\lambda}A^{\rho} = -D_{\lambda} ([D_{\mu}, D_{\nu}])A^{\rho}. $$ So, the question: can I formally take $A^{\rho}$ under the sign of the derivative for using the identity $[D_{\mu}, D_{\nu}]A^{\rho} = R^{\rho}_{\quad \sigma \mu \nu}A^{\sigma}$ and, afer that, take $A^{\sigma}$ outside the derivative? I'm afraid that no, but I hope that it is possible.


title: Experiments looking for monopoles

text: **Background:** (skip it if you know it) In the easiest formulation of classical electromagnetism magnetic monopoles do not exist. In fact, the Maxwell's equation $\nabla \cdot \vec{B}=0$ implies (using Gauss' Theorem) that the surface integral of the flux of $\vec{B}$ over the bounday of any finite surface is zero. Therefore no isolated magnetic charges (i.e monopoles do exist). However Dirac discovered that even if a single monopole existed in the universe, we could then explain in a rather easy way why the electric charge is quantized. Note that since the charge is not an observable its quantization is completely different than the quantization of energy or momentum in QM, for example. Furthermore, in the recent developings of QFT, the theoretical model we usually assume implies that every time a gauge symmetry is broken monopoles (and other kind of topological defects such as solitons) arise. Since in the hot big bang model it is usually belived that many gauge symmetries were broken in a primordial of the universe, monopoles of various kinds (not just magnetic monopoles but also Yang-Mills ones) could (at least this is what teorists say) have been produced. Up to date, not a single monopole has been found by experiments. With this in mind I ask the following **Question:** > Which experiments are currently being carried over to look for magnetic, and > other types of, monopoles?


title: Why Don't the Ladder Operators Commute?

text: I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very strange to me. Second, I don't understand why applying the raising operator and then the lowering operator is different from applying the lowering operator and then the raising operator. The two are explained in terms of moving up and down energy levels, so shouldn't going up then down, or down then up, be the same operation? What is asymmetric about harmonic oscillators that causes this asymmetry?


title: Where do the equations for resistive force come from?

text: I've been watching MIT's Walter Lewin physics lectures. In lecture 12, Lewin pulls, out of nowhere, a couple of equations regarding resistive/drag forces in fluids: $$\begin{align} \vec{F_{res}} &= -(k_1v+k_2v^2)\hat{v}\\\ \lvert F_{res}\rvert &= c_1rv + c_2r^2v^2\text{ (for spheres)} \end{align}$$ The equations are introduced immediately and it is not necessary to watch past 5-10 minutes or so to see what I'm talking about. I'm usually OK with light handwaving, but something is really bugging me about these equations - I can't seem to find them outside of this lecture. I own a couple of textbooks, neither of which feature them, the Wikipedia articles on resistive forces and drag forces don't include them, and Googling "viscous term" or "pressure term" doesn't turn up anything similar similar to what was presented in the lecture. The apparent idiosyncrasy of these equations is very troubling to me. Where do they come from?


title: Does small objects still move "smoothly" or are there a measurable incrementation?

text: First off all: I'm no physicist so I am sorry if this question is badly written or stupid even. But it's so exiting to read and try to learn about this kind of stuff! Lets say you have a quark (or a smaller particle) traveling at the velocity `V`, and you (the observer) would have been shrunken down to the same scale as the quark (you would be really small). Would you see the particle moving `at a constant, smooth speed` or would the quark seem to be `jumping through space` like a stop-motion movie? I thought of this when I read about the Plack-time and -length.


title: How can perturbativity survive renormalization?

text: The most usual way to renormalize quantum field theories is by re-writing the Lagrangian in terms of physical (finite) parameters plus counter-terms. Take $\lambda \phi^4$ theory for instance: $$ {\cal L} = {1\over2}(\partial_{\mu}\phi)^2-{m^2 \over 2} \phi^2 - {\lambda\over 4!}\phi^4 + {\cal L}_{CT} $$ $$ {\cal L}_{CT} = {\delta Z\over2}(\partial_{\mu}\phi)^2-{\delta m \over 2} \phi^2 - {\delta\lambda\over 4!}\phi^4 $$ All parameters with $\delta$ in ${\cal L}_{CT}$ are divergent quantities. Then what we do is to treat everything in ${\cal L}_{CT}$ as interactions and calculate it perturbatively My question is: how can we do that? Considering that the "couplings" in this case ($\delta Z$, $\delta m$ and $\delta \lambda$) are huge numbers?


title: Can we measure temperature of a object just by the sound it makes?

text: I been thinking if temperature is a basic property of macroscopic objects rather than of quantum or microscopic objects and it is as a result of average kinetic energy of particles residing in the object either through movement of vibration. That being said, its also described similarly in Wikipedia thus I'm sure it is a good description of temperature however this is the only description I'll use for this question but there are way more descriptions of temperature. That however is similar to sound which is also as a result of vibration in an medium which will result in transfer due to vibrations and movement of particles being oscillated through the medium which will transfer the energy across. That being said cant we detect a temperature just by hearing it using a detector but obviously we have to know its vital information like density of the medium and density of the material and such but not the temperature of the object Can this be possible? I'm in middle school so please excuse my lack of scientific knowledge therefore if any descriptions are mathematical please explain it.


title: Simple 2D Vehicle collision physics

text: I'm trying to create a simplified _GTA 2 clone_ to learn. I'm onto vehicle collisions physics. The basic idea I would say is, To apply force F determined by vehicle A's position and velocity onto point P on vehicle B. But what should that do exactly? My vehicles have a center position and an angle, but I'm not sure what something like that applied force should do? How could I determine the new center and angle of the vehicle after the hit? It doesn't have to be realistic, just something that would get the job done would be appreciable...


title: The Hendriks-Teller Model

text: So I am working on understanding the Hendriks-Teller model of 1D disorder. So the way I understand it is the following. You have a random smattering of particles. Each mass is separated by some unit distance with probability _p_ or some larger distance 1 + (rho) with probability 1- _p_. I want to derive the structure factor for said model. Further information can be found in Principles of Condensed Matter Physics By P. M. Chaikin, T. C. Lubensky (question 2.1) and Hendriks and Teller's original paper in the Journal of Chemical Physics, 1942. The final structure factor should be: $$S(q) = \frac{p(1-p)[1-\cos{\rho q}]}{1-p(1-p)-p\cos{q}-(1-p)\cos{[(1+\rho)q]}+p(1-p)\cos{\rho q}}$$ I figure I should start with something like this, the structure factor for a discrete system: $$S(q)=\frac{1}{N}\langle \sum_{jk} e^{i \vec{q} (\vec{R_{j}}-\vec{R_{k}}) } \rangle$$ Thanks all.


title: Quantum of Time

text: I am not a String Theorist, but I just want to ask if there is anything which specifies a quantum of time, as per M-theory? Can a quantity such as $$ t_p = \left(\frac{\ell_p}{c}\right),$$ where $\ell_p$ is Planck length and $c$ speed of light in vacuum, be called as time quantum?


title: Why do mirages only appear on hot days?

text: A previous question asked why the road sometimes appears wet on hot days. The reason is that when there's a temperature gradient in the air, it causes a gradient in the index of refraction, causing the light to bend. This diagram (from Lagerbaer's answer to the previous question) is ![enter image description here](http://i.stack.imgur.com/8eZCm.png) I have two questions about this. First, Wikipedia says that in order for the effect to appear, there must be a temperature gradient on the order of a few degrees per hundred meters above the asphalt. To investigate this claim, I assumed the air near the surface of the asphalt is at constant pressure, so the density is inverse-proportional to the temperature (i.e. I assumed the density variation due to the weight of the air is small compared to the density variation induced by temperature gradient. Otherwise we would see mirages all the time.) Then I assumed the difference of index of refraction of air from one is proportional to the density, i.e. $n = 1 + a\frac{\rho}{\rho_0}$ with $n$ the index of refraction of air, $a$ some dimensionless constant, and $\rho_0$ some reference density. Looking up the index of refraction of air online, I guessed $a = 0.0003$. Then I assumed the temperature above the Earth is modeled by $T = T_0 + gy$ with $y$ the height and $g$ a temperature gradient in ${}^{\circ}\mathrm{C}/\mathrm{m}$. This completes the model. Fermat's principle gives a variational problem to solve for the path of the light. However, the resulting differential equation was hard to work with, so to first order I approximated that the light starts at a horizontal distance $x = -L$ at height $y =h$, slopes down towards the ground with some slope $m$ until it gets to $x=0$, then slopes back up at the same slope $m$ until it gets to $x = L$ and $y = h$ again. Then I chose $m$ to minimize the travel time of this path. To first order for small $h$, I found $$m = \frac{agL}{2T_0}$$ The problem is that when I plug in $g = 5^{\circ}\mathrm{C}/100\mathrm{m}$, $a = 0.0003$, $T_0 = 300 \mathrm{K}$, and $L = 100 \mathrm{m}$ I get $m = 2.5*10^{-6}$, which is too small to explain the mirage. It would only allow light to dip a quarter millimeter in the path from a car $200 \mathrm{m}$ away to me. In the Wikipedia image of a mirage, the light clearly dips down at least 1000 times that much (check out the blue car). ![enter image description here](http://i.stack.imgur.com/eb2iN.jpg) So my first question is: what's wrong? Why are there mirages when this model predicts that there are not? My second is: why do we see them only when it's hot? This model depends only weakly on absolute temperature, and hot absolute temperatures actually decrease the effect. Evidently, there are only high temperature gradients on hot days, by why should that be? A temperature gradient over the asphalt comes from sun heating the asphalt directly more than it heats the air. Shouldn't that happen on cold days as well as on hot days?


title: Is a given volume of sparkling water lighter or heavier than the same volume of still water?

text: I can see it two ways - if all the $CO_2$ has turned into bubbles I would imagine it is lighter. However dissolved $CO_2$ atoms are probably heaver than the $H_2O$ atoms they replace, or fit in between.


title: Conservation of quantum Noether current

text: The Noether current for a set of scalar fields $\varphi_a$ can classically be written as: $$j^\mu(x)=\frac{\delta \mathcal L(x)}{\partial(\partial_{\mu}\varphi_a(x))}\delta \varphi_a(x)$$ The divergence of this current can then be written as: $$\partial_\mu j^\mu(x)=\delta \mathcal L(x)-\frac{\delta S}{\delta \varphi_a(x)}\delta \varphi_a(x)$$ If the classical field equations are satisfied the second term on the right hand side vanishes. However in quantum theory the classical field equations are not satisfied. Why is the current still conserved for a symmetry in this case?


title: Proofs that every professional physicist should know

text: There's been a lot of questions about books that every physicist should read, or what are the most important papers in physics. I would say there are also proofs in physics that have that wow factor about them; proofs which suddenly switch the light on. An elementary example is Euler's laws of motion that are proven from Newton's laws of motion. What other proofs should every professional physicist know?


title: SO(3) x SU(2) Symmetry of the Hamiltonian

text: I have a question converning multiplets when describing atoms. Let $H=\sum\limits_{k=1}^{N} (p_{k}^2 - \frac{Z}{|x_{k}|} + \sum\limits_{i<k}^{1..N} \frac{1}{|x_{i} - x_{k}|}$ be a (self-adjoint) operator on the Hilbertspace of totally antisymmetrized wave functions of N particles H^(N)_{a}. In my notes it is written that SO(3) x SU(2) is a symmetry group of this hamiltonian. The group SO(3) x SU(2) consists of pairs (R,A)$\in SO(3) x SU(2)$ and has a unitary representation U on H^(N)_{a} given by: $(U(R,A)\phi)(x_{1} s_{1} ,...., x_{N} s_{n}) = As_{1}s'_{N} ... A s_{N} s'_{N}\phi(R^{-1}x_{1} s_{1} s'_{1} ... R^{-1} x_{N} s'_{N}) $ Here comes my problem. I don't really understand this definition. It's not given in my notes what the s' are and I don't understand what $As_{1}s'_{N}$ means...A vague guess might be that you multiply the vector $s_{1}$ in $C^{2}$ is multiplied by the matrix A in SU(2) and then scalar mulitplied by $s'_{N}$. Butit's not clear to me..Can anyone help me to understand this definition better? In addition to that I thought that if a group G is a symmetry group of a Hamiltonian H then it means that [H,U]=0 for ANY representation U of the group G. But in the above example one only gives one particular representation. I'd be really happy if someone could help me. Thanks in advance.


title: What are the mathematical problems in introducing Spin 3/2 fermions?

text: Can the physics complications of introducing spin 3/2 Rarita-Schwinger matter be put in geometric (or other) terms readily accessible to a mathematician?


title: Understand audio signals

text: Most audio signal are described by a curve having two parameters: The amplitude and the frequency (amount of time from positive to negative amplitude value). What I want to understand is how the sound level and absolute pitch are encoded to these two values. How would amplitude and frequency change if I'd change volume or pitch? Thank you in advance!


title: st. venant shallow water equations for pipe flow

text: As far as I know from the fluid dynamics class, St. Venant equations (the shallow water equations) are derived by depth-integrating the Navier-Stokes equations. This depth-integrating is done with the assumption that the horizontal scale is much larger than the vertical scale, hence it would be used for overland flow or surface runoff modeling. But I was wondering if the St. Venant equations can also be used in open channel pipe flow, say to model the flow inside the drainage pipe. I imagine the flow would more or less be similar as that in a river channel, but in this case the horizontal dan vertical scale would be the same at some point of water depth, given the geometry of the pipe. So my question is, can St. Venant equations be applied correctly for modeling open channel pipe flow? If they can, will there be any constraint from the water depth condition? For example, if the water depth is 5cm in a 1m diameter pipe then perhaps SWE can be applied, but when the water depth is more than half of the pipe then wouldn't the horizontal to vertical scale ratio be the same?


title: James Joule's Conservation of Energy

text: Is there a manifest or documentation of the original Conservation of Energy as stated by James Joule word for word? **Edit 1.1:** I found Joule's Memoir on google, skimming through to find any conclusive evidence.


title: Gravitational force due to asymmetric Earth shell

text: http://en.wikipedia.org/wiki/Shell_theorem According to above link the gravitational force inside a symmetrical shell due to itself is zero. Is it also true for an asymmetrical shell?


title: Archimedes' principle for two liquid layers

text: Problem: I have a cylindrical vessel of height $H$ and radius $R$. There are two liquid layers in the vessel. The first has density $D_1$ and height $h_1$, the second has density $D_2$ and height $h_2$. The second liquid is floating on the first liquid (thus $D_2 < D_1$) and they are both entirely within the vessel ($h_1+h_2 \le H$). Now I have a cube of density $D$ and edge length $n$. The cube fits into the vessel ($n \le \sqrt2R$) but it might not be completely inside it ($n>H$ is possible). I place it into the vessel such that one side is horizontal (the cube is not lopsided). Some liquid might overflow as a result. How do I use Archimedes' principle to calculate the topmost liquid level after placing the cube? I can ignore damping and other physical effects. EDIT: Note that the only values I know are $H, R, D_1, h_1, D_2, h_2, n$ and their abovementioned constraints. My workings: I tried using the formula directly, where $\text{weight of cube}=\text{weight of displaced liquid}$. I first get $Dn^3=D_2n^2s_2$ where $s_2$ is the height submerged in the second liquid. If $s_2 \le h_2$ then the cube is only submerged in the second liquid. So the answer is $\min(H, h_1+h_2+{n^2s_2\over\pi R^2})$, accounting for overflows. This case is simple. But if $s_2>h_2$ then the cube can be submerged in the first liquid too. I thought of subtracting the weight of the second liquid displaced to obtain the height submerged in first liquid, but realize that I don't know that. The weight displaced depends on the final liquid height, which can be affected by overflows, whether the cube fits totally within the two liquids, etc. In fact, it can be the case that after the liquid level rises, the cube now displaces more of the second liquid, causing it to not be submerged in the first (is that even possible?) It seems very messy and I have no idea how to start. :( Does anyone have a nice solution to this problem?


title: Faradays law for a falling magnet

text: If experimentally it can be proven that the velocity of a falling magnet through a coil is proportional to the emf induced, has it also been proven that dB/dt is proportional to the emf induced and how so ?


title: Does electricity flow on the surface of a wire or in the interior?

text: I was having a conversation with my father and father-in-law, both of whom are in electric related work, and we came to a point where none of us knew how to proceed. I was under the impression that electricity travels on the surface while they thought it traveled through the interior. I said that traveling over the surface would make the fact that they regularly use stranded wire instead of a single large wire to transport electricity make sense. If anyone could please explain this for some non-physics but electricly incline people, I would be very appreciated.


title: Why does inertia happen?

text: In this video R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true?


title: Renormalization of worldsheet energy-momentum tensor

text: At the end of section 2.3, Polchinski (in his volume 1) derives the energy- momentum tensor for free massless scalars on worldsheet. He adds a footnote that "the only possible ambiguity introduced by the renormalization is a constant times $\delta_{ab}$, from the subtraction". What does he mean? How does it relate to the familiar renormalization in field theory?


title: About Divergence in polar coordinates

text: I've got a conductor in a cylinder shape who is rotating with angular velocity $\omega$ around its axis, that correspond to the $z$ azis I want to calculate the electric field and the density of electric charge. I have no problem finding $$E = \frac{m_e \omega^2r}{|q_e|} u_r$$ where $m_e$ and $q_e$ are the mass and electirc charge of the electron, and $u_r$ is the versor directed towards the center of the circle (and is thereofore perpendicular to the $z$ axis) (I know that I had to use several approximation here, which are pretty standard to be honest, but that's not the point as the result appear to be right according to the textbook) To determin now the density of eletric charge, I would use maxwell's equation $$\nabla \cdot E = \frac{\rho}{\epsilon_0}$$ $E$ is expressed in cylindrical coordinates, though (am I right?) So the divergence would be $$\nabla \cdot E = \frac{1}{r} \frac{\partial rE}{\partial r} = \frac{2m_e \omega^2}{|q_e|} = \frac{\rho}{\epsilon_0}$$ On the book there is written though that > for a vector field that depends solely upon $r$ directed toward the centre > of the circle, the divergence is just $$\nabla \cdot E = \frac{dE}{dr} = > \frac{m_e \omega^2}{|q_e|}$$ I don't undestand what he means and why I'm wrong. Thank you!


title: What is the physical meaning of equivalence of 1st and 2nd quantization formalism?

text: Ref (Superstring theory (Green, Schwarz, Witten)) Take an $n$ dimensional euclidean space-time $x_0,x_1...x_{n -1}$, a relativist real scalar field, with a propagator $G_E(x,y)$. The propagator could be written : $G_E(x,y) = <x|\Box^{-1}|y>$ (with $<x|y> = \delta (x - y))$ We could write $<x|\Box^{-1}|y> = \int_0^{+\infty} ~d\tau~ <x|e^{- \tau \Box}|y>$ Now, we can consider a $n+1$ dimensional space-time, considering non- relativist particles. The hamiltonian is then $H_{n+1} = \frac{p^2}{2m} = \frac{\Box}{2m}$ By taking $m = \frac{1}{2}$, we get : $<x|\Box^{-1}|y> = \int_0^{+\infty} ~d\tau~ <x|e^{- \tau H_{n+1}}|y>$ Now, we can use standard Quantum mechanics path integrals, that is : $<x|e^{- \tau H_{n+1}}|y> = \int_{x(0) = x}^{x(\tau) = y} Dx(u) e^{-\frac{1}{4} \int du ~\dot x^2(u)}$, where $x(u)$ represents a path from $x$ to $y$, done in a "proper time" $\tau$. So, finally : $G_E(x,y) = \int_0^{+\infty} ~d\tau~ \int_{x(0) = x}^{x(\tau) = y} Dx(u) e^{-\frac{1}{4} \int du ~\dot x^2(u)}$ Now, the question is : > What is the philosophy of these equivalence between 1st and 2nd quantization > formalism ?


title: Cosmological constant vs. quintessence scalar field

text: I understand the basic idea of these two concepts, but I have question about what they really "are" (a little hard for me to put into words). When one talks about the cosmological constant, it seems as though it is some intrinsic property of space like an innate energy density. But quintessence is described by a scalar field, unlike the cosmological constant, and it appears to be different in some way. I think what I am really trying to wrap my head around is this idea of a scalar field. What is a scalar field _really_? Is it nothing more than a cute piece of mathematics to describe something that we don't understand at a more fundamental level? Or is there more to it? Forgive me if this doesn't make sense, but the idea of a scalar field is tossed around a lot and I can't really _feel_ what the scalar field is.


title: Are electric field lines due to Faradays law closed?

text: Are the field lines of electric field produced by Faraday's law of induction (and assuming $\rho = 0$) necessarily closed? If not what would be a counterexample. If it's true, how to prove it in mathematical rigorous way? How does the situation changes, if you consider the net field produced by a charge distribution $\rho$ _and_ by Faraday's law of induction (where the latter one must not be zero)? Again, what would be a counterexample or how to prove it? Do you have any references for this? **Edit** How does the situation changes if you consider only cases where the induced electirc field is time independent? Is the claim true for this case or is there also a counterexample as given by @arccosh?


title: How to understand exciton?

text: I know that exciton is an electron and hole orbiting around each other(at least that is what I have heard). So if the hole is a hypothetical particle with a plus sign, mining that it is a lack of an electron jumping around in a certain material, how should I imagine hole jumping out to the gap in a semiconductor and there orbiting around with an electron. Is that the one- electron-less atom that does it or is that just made out idea like a hole?


title: Physical position eigenfunction normalisation

text: We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ Our professor told us that for any value $a>0$, the physical position eigenfunction is $$\psi_{x_0}(x)\cong N_1\delta_a(x-x_0).$$ How can I show that $\psi_{x_0}$ is a physical position eigenfunction?


title: Image charges, laplace equation and uniqueness theorem

text: Consider a well-known problem of the electric field generated by a system composed of a point charge in proximity of a large earthed conductor. It is said that the potential due to an image charge satifies all the boundary conditions - it this case, constant potential on the surface of a conductor - and therefore, by the uniqueness theorem, it is the only possible potential distribution. However, we can of course imagine other non-trivial fields satisfying the boundary condition of constant potential on the surface, for example generated by three imaginary charges such that the whole system is again symmetric. So my question is: what is the full set of boundary conditions needed to find the unique potential distribution? This specific example is just an illustration of a more general question: how to use Laplace/Poisson equations to describe the field of point charges? The divergence of electric field has a singularity at the point charge. Another incarnation of this problem occurs when we want to use the formula for the energy stored in the electric field - we find the energy of the electric field generated by a point charge to be infinite. Does this mean that the concept of a point charge is too idealised?


title: What specifically are the measurements that correspond to fixing the infinite number of counterterms in quantum gravity?

text: I understand that quantum gravity is nonrenormalizable because there are an infinite number of counterterms. In QED the counterterms correspond to the loop corrections to the vertex function as well as the electron and photon self-energy. This corresponds to corrections between the bare vs measured mass and charge. What is the analog of this in quantum gravity? Why isn't there just a gravition self-energy correction, etc? What do each of the infinite counterterms correspond to? Are there diagrams corresponding to each one?


title: Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`?

text: `geometric algebra` gives geometric meaning to linear algebra and much more. it can provide a coordinate free geometric interpretation of spaces. those who learn of it, tend to be dismayed they weren't taught physics in this framework. what are the pro's and con's of replacing linear algebra and vector calcunningulus with geometric algebra? Good books on geometric algebra:   * Geometric Algebra for Computer Science has exercises to self tests, and features prettyfied pictures for extra clarity.   * Books by Hestenes have an incredible signal to noise ratio (extreme compression of information to Shannon Limit) and are definitely a must...   * many introductions floating around, remember search for "`geometric algebra`" and not " ~~algebraic geometry~~ " Also try to see the Pascal triangle structure of number of k vector basis blades in n dimensions!


title: Feynman Diagram for Bragg Equation

text: Did Feynman ever derive the Bragg equation as a particle representation using Feynman diagrams? And where is it available? I spoke to Freeman Dyson and he couldn't recall.


title: Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$

text: I just finished deriving the commutators: \begin{align} [\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\\ [\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\\ \end{align} On the Wikipedia it is said that **these commutators can be used to find energy eigenstates** of Quant. harm. oscillator, but explanation is a bit too fast there. Anyway i strive to be able to derive the equation $W_n = \hbar \omega \left(n + \tfrac{1}{2}\right)$ in full, but first i need to clarify why theese two relations hold: \begin{align} \hat{H}\hat{a} \psi_n &= (W_n - \hbar \omega) \hat{a} \psi_n\\\ \hat{H}\hat{a}^\dagger \psi_n &= (W_n + \hbar \omega) \hat{a}^\dagger \psi_n \end{align} **I can't see any commutators in above relations** , so how do the commutators i just calculated help us to get and solve these two relations? I am sorry for asking such a basic questions. I am a self-taught and a real freshman to commutators algebra.


title: M1 vs M2 on a frictionless surface

text: I was wondering, if I had a frictionless surface and I had an M1 that was 100kg and an M2 that was 1kg and I pushed each separately with an equal force F would they move at the same speed or am I assuming incorrectly?


title: Occam's razor on spin statistics theorem?

text: Highly related to A reading list to build up to the spin statistics theorem I see 2 parts to the spin statistics theorem:   * (spin $n$ or $n+\frac{1}{2}$)   * step 1 given that a spin is integral or non-integral conclude symmetry or antisymmetry of wave function   * ([anti]symmetry)   * step 2 using symmetry or antisymmetry of wavefunction conclude that it must obey BE or FD statistics.   * (BE/FD statistics) I have never seen a clear explanation of 1 nor 2... The referenced question author seems to be satisfied with just 1... and seems to believe 2 by naming association: symmetric or assymetric wavefunction corresponding to boson or fermion, and BE statistics corresponds to boson, FD statistics to fermion. i.e. enough to be able to decide what to use without necessarily understanding how statistics is associated with wave function


title: Density Operator, Expectation Value, Coherent States

text: How would I go about evaluating expectation values like $\langle X \rangle$ and $\langle P \rangle$? Work I've done: I've done the integration over $\phi$ and rewrote $\rho$ as: $\rho = e^{-|\alpha|^2} \sum_n \frac{|\alpha|^{2n}}{n !} \lvert n \rangle \langle n \rvert$, where $n$ is the number of states. My intuition says to calculate expectation values using $\langle X \rangle = Tr(\rho X)$, but I'm having some difficulty with the calculation. Could someone help flesh out the details? Since this is for a coherent state, is $\langle X \rangle$ going to be what you normally get for coherent states or will it be different since the state is a mixture?


title: Why does string theory (in case it is true ) have NO divergencies?

text: Why is string theory in 10 or 26 dimension not divergent? Due to the high number of spacetime dimension (10 or 26) it should have a lot of UV divergencies of the form $ \int k^{n}dk $ and gravity within the approach of the string theory should be non-renormalizable too, or shouldn't it?


title: How would you detect Hawking radiation?

text: Hawking theorized that a black hole must radiate and therefore lose mass (Hawking radiation). According to classical relativity though, nothing can escape a black hole, the hawking radiation would just fall back inside the black hole. Therefore how can one measure Hawking radiation or will this hypothesis remain a hypothesis forever?


title: What is the theoretical/intutive meaning of $x(t) = x_0 \cos ( \omega t + f )$?

text: What is the meaning of $x(t)=x_0 \cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ the phase constant? I know how to solve the mathematical problems which use this expression, but what I don't understand is how sine and cosine functions are used to express AC waves. I know the math but I do not understand the theory behind it. I do not have an intuitive understanding of these expressions. If you cannot explain at least guide me to a resource that can. i want to understand the math in a theoretical manner like are we going to take the sine/cosine of the wt or the solved value of it, since sine/cos is a ratio so basically are we multiplying the since/cos fraction value to the variables, oh i am so confused please explain .


title: Is there a second order differential wave equation that only allows a finite set of discrete eigenvalues?

text: I tried constructing a second order differential wave equation that only allows a finite set of discrete eigenvalues by using the power series expansion such as \begin{align} A_{j+2} = \dfrac{j-m}{(j+1)(j+2)-n(n+1)} A_{j} \end{align} to obtain the follwing wave equation \begin{align} \dfrac{\partial^2}{\partial x^2} \- \dfrac{n(n+1)}{x^2} = x \dfrac{\partial}{\partial x} \- m , \end{align} where $n$ is any integer and $m$ is the eigenvalue in question. I realized that this equation does not force $m$ to be bounded for finite discrete values, because given a maximum value of $j_{ \text{max}}$, one can work the recursion backwards if $m$ is greater than $j_{ \text{max}}$. Does this suggest that it is not possible to construct a second order differential wave equation that only allows a finite set of discrete eigenvalues?


title: Are there massless bosons at scales above electroweak scale?

text: Spontaneous electroweak symmetry breaking (i.e. $SU(2)\times U(1)\to U(1)_{em}$ ) is at scale about 100 Gev. So, for Higgs mechanism, gauge bosons $Z$ & $W$ have masses about 100 GeV. But before this spontaneous symmetry breaking ( i.e. Energy > 100 GeV) the symmetry $SU(2)\times U(1)$ is not broken, and therefore gauge bosons are massless. The same thing happens when we go around energy about $10^{16}$ GeV, where we have the Grand Unification between electroweak and strong interactions, in some bigger group ($SU(5)$, $SO(10)$ or others). So theoretically we should find gauge bosons $X$ and $Y$ with masses about $10^{16}$ GeV after GUT symmetry breaks into the Standard Model gauge group $SU(3)\times SU(2)\times U(1)$, and we should find massless X and Y bosons at bigger energies (where GUT isn't broken). So this is what happened in the early universe: when temperature decreased, spontaneous symmetry breaking happened and firstly $X$ & $Y$ gauge bosons obtained mass and finally $Z$ & $W$ bosons obtained mass. Now, I ask: have I understood this correctly? In other words, if we make experiments at energy above the electroweak scale (100 GeV) we are where $SU(2)\times U(1)$ isn't broken and then we should (experimentally) find $SU(2)$ and $U(1)$ massless gauge bosons, i.e. $W^1$, $W^2$, $W^3$ and $B$ with zero mass? But this is strange, because if I remember well in LHC we have just make experiments at energy about 1 TeV, but we haven't discovered any massless gauge bosons.


title: Time-dependent Schrödinger equation with $V=V(x,t)$

text: I was wondering about the following: If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t),$$ where the potential is also time dependent. What is the general strategy to solve this one? Separation of Variables or are there better techniques available? Especially if $V(x,t) = V_1(t)V_2(x)$. For example if you know the solution to $$E_n = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V_2(x) \psi(x),$$ does this help to find the general solution?


title: Equation of state of a rubber band

text: I have the following question that I attached in png format. ![enter image description here](http://i.stack.imgur.com/v5MpZ.png) I have done part (a), but I am having difficulties in part (b) when I proceed according to the book. I have non zero tension at equilibrium length. I reached that conclusion by taking the derivate of entropy with respect to L and n at the same time. What is the physical principle underlying this?


title: What happened to potential energy?

text: ![](http://i.stack.imgur.com/2uHzp.gif) I was learning how charge can be virtue of a body's potential energy.Meanwhile,I was hung by this question. [gravitational and other forces except coulombic,are assumed to be not acting on the charged bodies in the following cases] What I thought was,if we have two charged bodies in the space interacting each other with only coulombic force(force due to property of charge).And if we suppose eliminate charge property(either by removing electrons and protons or any one of them) of one of the body,then the body from which the property of charge was eliminated gets thrown away for certain distance.Thus,converting energy stored in it by the virtue of it's charge into kinetic energy(I don't know energy get's converted into other energy or not). Now,consider a single charged body not interacting with any other body in the space.Even if you remove charge from it,the body will not get thrown away. So,   **is there any other energy into which potential energy stored in it by the virtue of charge got converted?**   Or   **is it that charged body has charge as the virtue of it's potential energy only when it has another charge interacting with it?** **NOTE:** I have heard so many of them speaking about mass-energy equivalence.But I have a naive idea about it.Anyway,you can notice in the above explanation,if you eliminate the charge property of a body,by removing protons and electrons or any one of them,mass is going to reduce by certain amount.I think there might be some link with mass-energy equivalence.


title: Is Higgs mechanism necessary in QCD?

text: As we known, we introduce Higgs mechanism into Standard model to explain mass of Gauge boson, lepton and neutrino(maybe?). Higgs mechanism is necessary for electroweak theory because gauge symmetry and chiral symmetry. But in QCD, Is Higgs mechanism still necessary? It seems there is no chiral symmetry in QCD so that mass of quark not necessarily to be zero. And on the other hand, gluon seems to be massless(maybe I am wrong?mass of proton). It seems like Higgs mechanism is not a necessary in QCD. Is my understanding right? Thanks a lot!


title: When light is only considered as a particle, is it still considersed to be oscillating electic and magetic waves?

text: I have my head around wave-particle duality, however people tend to refer to light as either a wave or a particle in different situations. If I were to consider light as a particle am I still regarding it as an oscillating electric wave and magnetic wave? And are these waves a reason we can consider light a wave or is it purely a quantum effect?


title: Why is it common to plot $xG(x,Q^2)$ and not simply $G(x,Q^2)$?

text: I'm trying to understand the modern description of high-energy scattering processes involving hadrons in the initial states. The phenomenological parton distributions functions play a central role, and as I understand it at the moment, if we are e.g. talking about gluons, the function $G(x, Q^2)$ is the probability of finding a gluon with momentum fraction $x$ inside the hadron if the transmitted four-momentum is $Q^2$. When these functions are plotted, I often encounter plots showing $x G(x, Q^2)$ instead of simply $G(x, Q^2)$. Why is this so? Is this just because the plots look a lot nicer if plotted this way? Or is there some deeper reason behind this that I haven't figured out yet? As an example, take a look at the plot used on Wikipedia. ![http://en.wikipedia.org/wiki/File:CTEQ6_parton_distribution_functions.png](http://i.stack.imgur.com/1es60.png) (Picture from: http://en.wikipedia.org/wiki/File:CTEQ6_parton_distribution_functions.png)


title: Nucleon-nucleon collision density

text: Can someone help me derive expression for nucleon-nucleon collision density as a function of transverse plane coordinates (ultra-relativistic heavy ion collisions) using Wood-Saxon nuclear densities( $1/(1 + e^{(r-c)/d})$) in Glauber model (c, d are constants)?


title: What is the relationship between $V(t)$ and $V(x,y,z)$

text: I was recently asked this by a friend. He told me that coming from a physics background, he does not understand $V(t)$ and he believes it is purely theoretical construct made up by circuit theorists. Because in his mind a voltage $V$ is always a field with $<x,y,z>$ components. You can take the gradient of it, the laplacian of it to get meaningful results. But in circuit theory, we instead defines a quantity $V(t)$, which conflicts with a lot of results that you would get from elementary physics (i.e. the integral of $E$ field does not result in a time component!) So, what is the relationship between $V(t)$ and $V(x,y,z)$ and how can this be intuitively understood?


title: AC circuit analysis: Why does current have the same frequency as the voltage?

text: ![enter image description here](http://i.stack.imgur.com/j2uDv.png) The above example we have a 50hz AC voltage source. Why is that when we solve for I, I must have frequency of 50 Hz? Is there any phyiscal (non-mathematical) reason to explain this? Thank you


title: Can someone explain to me the concept of atmosphere opacity?

text: ![enter image description here](http://img1.wikia.nocookie.net/__cb20071104233556/psychology/images/8/83/Atmospheric_electromagnetic_transmittance_or_opacity.jpg) On this diagram, why is the atmospheric opacity shaped as it is? If we are able to see lights due to low atmospheric opacity, why can't we see radio waves, when the atmospheric opacity is so low for these kinds of waves? What does the author mean by "observable"?


title: how much heat i need to reach constant temperature

text: Is there an equation I can use to calculate the heat required (as a function of time) to sustain water at some specif temperature? I already know how much heat I need to reach the desired temparature using the formula Q=m*cp *(T2-T1). My question is how much heat do I need to keep 30,000lts of water at 70degress celcius for 30min once i have reached such temperature. Any help would be greatly appreciated


title: If the entire universe is the same age, can one assume that any potential life elsewhere is at the same stage of development as us?

text: There was some discussion in another question regarding whether you can consider all parts of the universe to have existed for the same length of time, and the conclusion seemed to be that, because of relativity, it depends. I'd like to know, in a sort of practical sense, if we imagine life to exist elsewhere in the universe, can we assume it to be at a similar stage of development as ourselves, if that development took a similar path as ours on Earth?


title: Infinitesimal input, macroscopic output

text: I must admit that I never got well how physicists handle _infinitesimal_ quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and Shubin's _The Schrödinger Equation_ ) destabilize me somewhat: > It is in the nonlinear world that _infinitesimal inputs may result in > macroscopic outputs_. To appreciate what I'm hinting at consider [...] > electronics, [...] transistors, [...] TV[...]. Can you provide me with some simple example of such a phenomenon? In my opinion this could not be possible, because as soon as you start introducing _infinitesimal_ quantities, all subsequent equations need be truncated to first-order. So an equation like $$\text{macroscopic output}=F(\text{macroscopic input} + \text{infinitesimal variation}) $$ cannot be correct. I must be wrong, but why? Thank you.


title: Difference between 1PI effective action and Wilsonian effective action

text: What is the simplest ay to describe the difference between these two concepts, that often go by the same name?


title: Is Bose-Einstein condensate a good example of a classical massive boson field?

text: Physically, we know that a BEC has formed if a macroscopic number of bosons occupy a single quantum state. The wave-function $\Psi(x)$ of the latter, normalized to the total number of condensed atoms $N \gg 1$, gives the macroscopic description of the condensate. $\Psi(x)$ will satisfy the ususal Schrodinger euqation in the trapping potential (which is the special case of the Gross-Pitaevskii equation with zero interactions). On the other had, a macroscopic limit of a boson field should be described by the (classical) Klein-Gordon equation. In the relevant non-realtivistic limit, dispersion of the Klein-Gordon field is quadratic, as in the Schrodinger equation, but I struggle to derive the former from the latter. Despite this technical difficulty of mine, is it a valid claim that BEC is a physical realization of the non-relativistic, classical-wave-limit of a boson field? I see it in the same vein as electric-field description of the laser radiation being a classic limit of the quantum coherent state. (The difference between number states and coherents states in the limit $N \gg 1$ is, hopefully, a mere technicality here).


title: Where does the excess energy emitted by a microwave go?

text: If there is nothing in the microwave, where does the excess radiation go? Why doesn't the radiation accumulate and blow it up? Should I cook two pieces of Canadian Bacon twice as long as I cook one?


title: how could I exist where I did not exist in the past?

text: If I was able to go back in time via Machine or whatever, would traveling back in time not create a paradox? I was not there, so how could I exist where I did not exist in the passed? i would have to assume that we could never go backwards in time in any way shape or form.


title: Diagram-like perturbation theory in quantum mechanics

text: There seems to be a formalism of quantum mechanics perturbation that involve something like Feynman diagrams. The advantage is that contrary to the complicated formulas in standard texts, this formalism is intuitive and takes almost zero effort to remember (to arbitrary orders). For example, consider a two level atom $\\{|g\rangle, |e\rangle\\}$ coupled to an external ac electric field of frequency $\omega$. Denote the perturbation by $\hat V$, with nonzero matrix element $\langle e|\hat V |g\rangle$. Then the second order energy correction reads $$E^{(2)} = \langle e|\hat V |g\rangle\frac{1}{\omega_g - \omega_e +\omega} \langle g|\hat V |e\rangle + \langle e|\hat V |g\rangle\frac{1}{\omega_g - \omega_e -\omega} \langle g|\hat V |e\rangle $$ where the first term corresponds to the process absorb a photon then emit a photon while the second process is emit a photon then absorb a photon. Does anybody know the name of this formalism? And why it is equivalent to the formalism found in standard texts?


title: Modeling the probability of a photodiode measuring photons targets at a neighbor

text: In current digital cameras, sensors are arrays of photodiodes which "transform" photons energy to electrons. I am aware that the probability of a photon to generate an electron is modeled by a Poisson distribution. However, I can not find any precise information on how the hit point of a photon on the sensor surface will be measured depending on its position :   * Does the energy measured on a given photodiode (which is a finite surface) depends on the distance from the center of the surface to the photon impact point?    * I can imagine that a photon hitting the surface generate energy which propagates in space (isn't it?). Can this "energy area" be modeled? For example (this is why I ask this question), how much energy is received by a neighbor photodiode when a photo hit another photodiode. An answer with detailed equations or reference to academic publications will be greatly appreciated.


title: Partial Differentiation of a Tensor

text: I have doubts in the statement that the partial or ordinary differentiation of tensor is not a tensor. The argument for this is that the partial differentiation of the tensor involves evaluating the transformation matrix at two neighbourhood points (say $P$ and $Q$) in the manifold and by the definition of tensor (the set of quantities that transform according to that rule where transformation matrix is evaluated at a point $P$) this is not the case. Thus, the partial differentiation of tensor is not tensor. Now my doubt is: what if the transformation matrices at those points are equal?


title: Where do the high order rainbows appear?

text: This site shows locations of the high order rainbows up to sixth order. My thought is that, There should be even higher order bows like 7th, 8th, etc. (Though they would be practically impossible to observe.) So, Where would they appear on sky?