title: Which planet in the solar system is it?

text: > It is known that the period of revolution of the earth about the sun is > $365.3$ days and that the semimajor axis of the earth's orbit is $1.495 > \times 10^{11}$ m. An astronomer notes that the period of a distant planet > is $29.7$ earth years. What is the semimajor exis of the distant planet's > orbit? Check in a reference source (encyclopedia,etc.) what planet of the > solar system this might be. This problem is from Anderson's _Introduction to Flight_ , 4th edition, Exercise 8.4. I was presented a formula $$\left( \frac{\tau_2}{\tau_1} \right)^2=\left(\frac{a_2}{a_1} \right)^2$$ where $\tau_1, \tau_2$ denote the periods of earth and the other planet, respectively, and $a_1,a_2$ denote the periods of earth and the other planet, also respectively. When I plug in $a_1=1.495 \times 10^{11}$ m, $\tau_1=365.3$ days, and $\tau_2=29.1 \cdot 365.3$ days, I see the equation like this: $$\left( \frac{29.1 \cdot 365.3}{365.3} \right)^2=\left(\frac{a_2}{1.495 \times 10^{11}} \right)^2$$ Solving for $a_2$, I get $a_2 = 4.44 \times 10^{12}$. I looked up for the planet with this semi-ajor axis on Wikipedia, and it turns out to be Neptune. Am I right on this?


title: Wave Function Integral I need help conceptually and Mathematically

text: $$\int_{-\infty}^{\infty}\frac{\partial^2\bar{\psi}}{\partial{x^2}}\frac{\partial\psi}{\partial{x}}~dx.$$ I have read that this is equal to Zero. Only problem is that what I am reading about doesn't really explain why. I think it is that at positive and negative infinity the $\frac{\partial\psi}{\partial{x}}$ approaches 0. I am thinking about the Gaussian when I think about this, but then how do I show it mathematically, as well. A note I tried to search this online, but it s hard to find anything on my specific problem and I am not sure what it is I need to search to get the result I need.


title: A question on quantum computing and de Broglie's pilot wave theory

text: I don't know much about quantum computing except what I have read about on wiki and popsci. I have been reading about the de Broglie-Bohm pilot wave theory and how they describe quantum mechanics in terms of discrete particle traveling a pilot wave and it seems produce the same quantum mechanical statistics as the Copenhagen version. some experiments by fluid dynamicists from MIT also seem to support this alternate view. My question is if this alternate view is the correct view of reality would that effect quantum computing in any way? ie. dealing with discrete particles instead of probabilistic "particles". both theories explain the superposition of waves and both produce the same stats so I would guess not.


A strange particle, $X$, decays in the following way: $X → π^– + p$. State what interaction is involved in this decay. I know the answer to be weak interaction, but why is it weak interaction? What is the reasoning? Why could it not be a strong interaction? Is it merely because the particle is a strange particle?


title: Temperature in a Voltaic Cell

text: The potential difference across a voltaic cell varies with temperature. But my question is whether the voltage increases or decreases as temperature rises. According to the Nernst equation, the two are inversely proportional, however if one were to use Gibb's Free Energy equation, as temperature increases, so does the potential difference across the cell. It would make sense that as temperature increases, there would be an increase in kinetic energy, which would probably add to the resistance and therefore make the voltage increase as well (assuming current stays the same) Do you think the potential difference would increase or decrease with an increase in temperature?


title: Is the edge of our Hubble Sphere within our Cosmic Event Horizon and why?

text: I was recently shown a pretty cool video about common cosmological misconceptions. It got me reviewing the different between event horizon (current distance within which we **_will see/interact_** ), particle horizon (current distance within which we **_have seen/interacted_** , and the Hubble Sphere (distance at which things are currently receding at the speed of light). I've found the following figure extremely helpful: ![enter image description here](http://i.stack.imgur.com/6yzwk.jpg) But I'm very confused about the event horizon currently lying outside of the Hubble Sphere. If the universe is accelerating in its expansion, then it seems like everything outside of the Hubble Sphere at this moment - will never be in causal contact, never be within our light-cone, and thus be outside of our event horizon. Is this figure correct?


title: How dangerous is a parallel plate capacitor which is used in physics demo experiments?

text: How dangerous may it be when experimenting with a parallel plate capacitor typically used in physics demonstration experiments? Take for example this one (that's what I use): http://www.ld-didactic.de/ga/5/544/54422/54422de.pdf Depending on the plate distance the capacity varies. I tried to estimate if this could be dangerous but I am not sure about it. Here is what I did: From the manual, the capacity $C$ for a plate distance $1\,\mathrm{mm}$ is about $C = 550\,\mathrm{pF}$. Suppose you charge it using $U_0 = 25\,\mathrm{kV}$. And suppose your fingers are a bit sweaty, thus I assume a body resistance of $R = 500\,\mathrm{\Omega}$. The discharging equation is: $$ I(t) = \frac{U_0}{R} \mathrm{e}^{-\frac{t}{RC}} $$ This gives me an initial current $I_0 = I(0\,\mathrm{s}) = \frac{U_0}{R} = 50\,\mathrm{A}$ which would be extremely dangerous if applied over a certain time. However in a very short time, in $10^{-6}\,\mathrm{s}$ the current drops below $0,02\,\mathrm{A}$, which should be save. I am not quite sure if my considerations are correct and appropriate and I am not sure which conclusion I should draw, how dangerous it is (or not). I guess that it doesn't make much difference in this case if $2\,\mathrm{kV}$ or $25\,\mathrm{kV}$ were used. The key factor seems to be the time in this case (because in each case the initial currents are extremely dangerous if applied long enough, so the question seems to be: is the time short enough to make no harm?) ![plot](http://i.stack.imgur.com/IPEzT.png) **Edit** If you charge it say with $25\,\mathrm{kV}$ at $1\,\mathrm{mm}$ and then enlarge the space to the maximum i.e. to $70\,\mathrm{mm}$, the voltage would potentially raise to $1750\,\mathrm{kV}$, however the capacitance would drop to $7,8\,\mathrm{pF}$. Would doing this operation change anything significantly concerning the safety of the experiment? Would be great to get a detailed and comprehensive analysis of the safety of such a kind of capacitor.


title: Why is Clapeyron equation so important?

text: **Context** : I'm studying basic thermodynamics. My textbook has a chapter on the _Clapeyron_ equation which, as a reminder, is given by the following formula: \begin{equation} \frac{dP}{dT} = \frac{\Delta H_{a,b}}{T\Delta V} \end{equation} (a,b being the different phases) **Question** : The equation describes the tangents of the coexistence curves in $P,T$ diagrams, but why is it so important ?   1. Does indeed encompass some fundamental thermodynamics knowledge in it (that I am obviously missing) and/or   2. Is it useful at calculating thermodynamic properties of substances[*] and/or   3. Has it been artificially emphasized in our course, so that it serves as a question in the exams? **Possible answer** : I've found in my lecture notes that "Clapeyron equation is the most important experimental confirmation of the 2nd law of thermodynamics". How would you comment on that ? * * * [*] I watched an online MIT lecture on Clapeyron (that SE won't let me link) and it had a nice example with an hypothetical RDX explosive detector and the least number of molecules that should detect, based on the vapor pressure of RDX in room temperature.


title: How do I calculate the work done by snow when stopping a person from falling from a plane?

text: I have the following data:   Assume a man falling from a plane from a height of 370 m ($=h$) high (Straight down). The man falls into a pile of snow and creates a 1.1 m ($=d$) deep crater. The man has a mass of 88kg ($=m$) and his velocity right before hitting the snow was 45 m/s ($=v$). What is the amount of work done by the snow when stopping the man? I've tried the following: Since $W_{net}=\Delta K$   I can say that $W_G + W_S = K_1-K_0$   Where $W_G$ is the amount of work done by gravity, $W_S$ is the amount of work done by snow and $K_1$ and $K_0$ is respectively the amount of Kinetic energy after and before the person falls into the snow.   This gives me the equation:   $mg(h+d)+W_S=0- \frac{1}{2}mv^2$ This gives me the result of approx. $-409463$ $J$ for $W_S$.   Is this correct? My book tells me it should be $-9,0.10^4$ $J$ so either I'm wrong or my book is... Thanks in advance!


What is the defining property for two quantum states to become entangled? Is it just that the combined system cannot be in a product state? Why is this?


title: Electrons, spins, and degeneracy

text: In an atom, two electrons can have the same set of $n,\ell,m$ quantum numbers as long as they have opposite spins. My introductory physics and chemistry courses have all introduced this as two electrons occupying the same energy state; however, the spins of the electrons cause splitting of the energy levels they occupy. This leads me to think two electrons with opposite spins are only able to share the same set of $n,\ell,m$ quantum numbers because they actually occupy slightly different energy levels. My question is: if there were no splitting of energy levels due to spin, would two electrons still be able to occupy the now-exactly the same energy level or do they need to have different energies, no matter how small the difference?


title: Import of Celestial Effects on Satellite Radio Interference

text: Some internet (among other) infrastructure comprises satellites, which beam communications in radio frequencies. These satellites, to ground observers, appear as very small solid angles in the sky. This makes me wonder if there is significant interference between other skyward radio sources (e.g. stars (incl. sun), CMBR, ionospheric reflections, etc.). These satellites are solar powered, so they can't exactly drown out everything. I assume one calculates the satellites' orbits sufficiently to aim transmit/receive dishes precisely. But surely this can't ensure that other radio sources don't influence the outcome. My question is essentially: how much of this makes a practical difference? Do we just aim very carefully? Is there a lot of signal processing error correction that happens? Are the effects just negligible? Do they just transmit on very narrow frequencies, thus producing high intensities with low power? What relative orders of magnitude are interfering signals versus communications signals?


title: Contradicting forces on a circular loop under current in magnetic field?

text: I have the following general conceptual concern. Think of a thin conducting loop of radius $R$ placed in the $x$-$y$-plane at $z=0$. There is a homogeneous current density $\vec{j}$ running through this loop: $$\vec{j}(\vec{r})=|j|\delta(z)\delta(x^2+y^2-R^2)\frac{-y\,\vec{e}_x+x\,\vec{e}_y}{\sqrt{x^2+y^2}}$$ Per definition, this loop has a magnetic moment of: $$\vec{m}=\frac{1}{2}\int d^3r\,\left(\vec{r}\times\vec{j}(\vec{r})\right)=-|j|\pi R\,\vec{e}_z$$ Now, imagine there appears a magnetic field that can be locally described as: $$\vec{B}=b_0 z\,\vec{e}_z$$ If we ask ourselves what the electromagnetic force on the loop will be, we have two equations that can give us the answer. (The two answers should be the same, but curiously they are not). First equation is the straightforward definition of Lorentz force: $$\vec{F}_1=\int d^3r\,\left(\vec{j}(\vec{r})\times\vec{B}\right)$$ And the second equation makes use of the magnetic moment (and is also exact for this simple magnetic field): $$\vec{F}_2=\nabla(\vec{m}\cdot\vec{B})$$ It is now straightforward to see that the first force has the structure $\vec{F}_1=A\vec{e}_x+B\vec{e}_y$, while the second force must clearly look like $\vec{F}_2=C\vec{e}_z\neq 0$ (evidently non-zero due to inhomogenous $\vec{B}$ field). My question is - what went wrong and how to fix this?


title: What is the difference between orbital angular momentum of photons and their polarization

text: What is the difference of OAM of photons and their polarization?


title: Conservation of mass in fluid flow

text: When deriving the continuity equation in physics class for two immiscible fluids flowing in succession we used the principle of conservation of mass. My question is, shouldn't volume be conserved instead of the mass? To further elaborate, assume section 1 of a pipe has cross-section area $A_1$ and section 2 has area $A_2$. At the point where liquid 1 with density $r_1$ is completely in section 2 and liquid 2 with density $r_2$ is just about to enter section 2. Shouldn't the volume of liquid in section 2 be conserved, i.e., $V_\mathrm{in}$ = $V_\mathrm{out}$ instead of the mass? (Assume ideal fluids and ideal flow)


title: Mechanics Landau Galilean Principle

text: I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle. What does Landau mean by saying space to be homogenous and isotropic and time is homogenous for an inertial observer ? Also, how does he conclude using this that, the Lagrangian must be independent of $\vec{r}$, time and direction of $\vec{v}$ . I am not understand what he wants to convey. Also a side note, are there any lecture notes/books/videos etc. that explain in a bit more detail what Landau wishes to convey to the reader at times in his book.


title: $\mathrm{CO_2}$ rate of deposition

text: Parts of Antarctica average -127°F (-83°C). The freezing point of $\mathrm{CO_2}$ is -109.3 °F (-78.5 °C). If one could construct a large container at such a place shielded from the sun, would atmospheric $\mathrm{CO_2}$ be expected to freeze out inside the container? If so, what rate of deposition could be expected? If practical, this might be a solution to the excess atmospheric $\mathrm{CO_2}$ problem. However, I suspect that the rate of deposition is too small, the energy cost of increasing it with extra pressure or other means is too large, and sundry engineering considerations mean it is not viable.


title: Eigenstates of an observable

text: Can we use eigenstates of ANY observable as base of the Hilbert space? If we can, is this equal to the statement that those eigenstates are orthogonal to each other and normalizable?


title: how to calculate acceleration?

text: The problem is as follows. A searchlight casts a spot of light along the face of a horizontal wall located 100m above from the searchlight. It rotates at a constant $\omega$ of 4rads/s. Determine the magnitude of the velocity and acceleration at which the spot appears to travel across the wall at the instant $\theta$ =45 degrees. ($\theta$ = the angle of the spot from the vertical) I managed to find the velocity as 800ms/s but I can't figure out the way to find the acceleration. Any help?


title: Virial theorem and variational method: an exercise (re-edited)

text: I have a hydrogen atom, knowing that its Hamiltonian has been modified turning the standard potential $$ V_{0}(r) = -\frac{Z}{r} $$ into $$ V(r) = -\frac{g}{r^{\frac{3}{2}}} $$ with $g$ a positive constant. **An upper bound to the ground-state energy is to be determined.** I thought of using the variational method, with a trial function $$ \psi(\xi) = A\cdot e^{-\xi r} $$ $A = \frac{\xi^3}{\pi}$ (normalization condition). The idea for such a wave-function comes from having regarded $V(r)$ as a standard hydrogenic potential with a "space-dependent atomic number", $Z(r) = g\cdot r^{-5/2}$. ## _FIRST QUESTION: Is this a good choice for the trial function? If not, why?_ So, just like in a usual hydrogenic atom, $$ \langle T\rangle = \langle\psi(\xi)|-\frac{\nabla^2}{2}|\psi(\xi)\rangle = \frac{\xi^2}{2} $$ > I thought of using the virial theorem (being $V(r)$ spherically symmetric > and $\propto r^{-3/2}$) to calculate the part $\langle V\rangle$: $$ \langle > V\rangle = -\frac{4}{3}\langle T\rangle $$ But I wasn't able to apply it > without obtaining something illogical! Calculating $\langle V\rangle$ explicitly I found   $$ \langle V\rangle = \langle \psi(\xi)|-\frac{g}{r^{-3/2}}|\psi(\xi)\rangle = 4\pi A^2 \int_0^{\infty} \\! r^2 e^{-2\xi r}(-\frac{g}{r^{3/2}}) \, \mathrm{d}r = $$ $$ -4g\xi^3\int_0^{\infty} \\! r^{1/2} e^{-2\xi r}\, \mathrm{d}r \xrightarrow[{d}r \rightarrow 2t{d}t]{r \rightarrow t^2} -8g\xi^3\int_0^{\infty} \\! t^2 e^{-2\xi t^2}\, \mathrm{d}t = $$ $$ -8g\xi^3\frac{1}{4{(2\xi)}^{3/2}}\sqrt{\pi} = -\sqrt{\frac{\pi}{2}}g\xi^{3/2} $$ So, going on applying the variational method, I have to minimize $$ E(\xi) = \langle\psi(\xi)|H|\psi(\xi)\rangle = \langle\psi(\xi)|-\frac{\nabla^2}{2}-\frac{g}{r^{-3/2}}|\psi(\xi)\rangle = $$ $$ \langle T\rangle + \langle V\rangle = \frac{\xi^2}{2} -\sqrt{\frac{\pi}{2}}g\xi^{3/2} $$ calculating $$ \frac{{d}E(\xi)}{{d}\xi} = \xi - \frac{3}{2}\sqrt{\frac{\pi}{2}}g\xi^{1/2} = 0 \Rightarrow \begin{cases} \xi = 0 \\\ \xi = \frac{9\pi}{8}g^2 = \bar{\xi} \end{cases} $$ **The only acceptable solution is the positive one, $\bar{\xi}$, so that an upper bound to the ground-state energy is**   $$ E(\bar{\xi}) = \frac{\bar{\xi}^2}{2} -\sqrt{\frac{\pi}{2}}g\bar{\xi}^{3/2} $$ ## _SECOND QUESTION: How can I properly use the virial theorem?_ I asked something like this before and I thought I understood it but - obviously - I have not. Please make this clear to me; thank you in advance!


title: Why does smoke go inside the car if the back door is open?

text: Inspired by another question I wanted to ask about phenomenon that I've experienced. We had a van with separate trunk deprtment (similar looking one below) to transport things. But some of the things were too big, so we couldn't really close the rear door (the door was actually opening up, not like in the pic before) so we left it partially open (some 30-50cm gap). That's why I had to get inside and hold some boxes so they wouldn't fall out. My experience felt like a gas van. The smoke from exhaust pipe was going straight in through the open gap. Why did the smoke have to go through the trunk department? I've also heard similar stories of friends driving a car with open trunk door - smoke seems to flow inside and through all the car. ![enter image description here](http://i.stack.imgur.com/0iokv.jpg)


title: Why is the spring constant $W_p''(0)$?

text: According to my physics book, the spring constant can be calculated from knowing the potential energy, with the formula $k = W_p''(0)$. I don't really understand why, and the book doesn't explain it any further. How do I know the two are equal?


title: Can the technology behind Particle accelerators can be used for energy storage?

text: As I understand, the kinetic energy of the proton beam in a hadron collider is quite large. Can you build a energy storage system that is based on accelerating a proton beam to relativistic speeds to retrieve the energy later? Here is my related question: Can the technology behind Particle accelerators can be used for space propulsion?


title: Master Field Large N limit

text: I would like to ask a question about the so-called ''Master Field''. As far as I understand, this represents a classical configuration in the large n limit (saddle point solution) but there is no systematic way to derive master fields for gauge theories. Is anybody trying to solve this problem anymore? Has the Ads/CFT correspondence partially solved it or hinted towards the right direction by considering the gravity dual of the gauge theory as the ''Master Field''? If you know any relevant review papers please post!


title: Materials with directional friction property?

text: At the macroscopic level a simple metal file or rasp creates a large amount of friction in one direction and a much smaller amount in the opposit direction. Shark skin displays the same properties due to very fine scales. Are there any materials with a surface that displays similar behaviour at a microscopic level? also what would this effect/property be called? I have been searching using 'anisotropic friction' and 'directional friction' but sofar haven't found anything that achieves this due to microscopic effects.


title: Harmonic oscillator

text: Let $|0\rangle,...$ be the states of the harmonic oscillator. Then a squeezed state was defined as $|\xi\rangle =S(\xi)|0\rangle $, where $S(\xi):=e^{\frac{1}{2}( \xi (a^{ \dagger ^2}-a^2))}$, where $a$ and $a^{\dagger}$ are the canonical operators linked with the harmonic oscillator problem in QM. Now we were supposed to show that $S^{\dagger}(\xi) X S(\xi)=Xe^{-\xi}$. A hint says that we are supposed to differentiate $F(\xi):=S^{\dagger}(\xi) X S(\xi)$ and reexpress this expression in terms of $F$. Finally, there shall be a differential equation that does it. The problem is: In my opinion the derivative is just: $F'(\xi):=\frac{1}{2}S^{\dagger}(\xi)(a^{ \dagger ^2}-a^2) X S(\xi)+\frac{1}{2}S^{\dagger}(\xi) X S(\xi)(a^2-a^{ \dagger ^2}).$ Now I do not see how to proceed. If something is unclear, please let me know.


title: Acceleration vector for an object moving in a elliptic trajectory

text: I am interested in deriving what the radial and tangential components of the acceleration vector should be for an object following an elliptical trajectory centered on the origin, in which the relation between the speed $v$ (module) is related to the distance of the object to the origin ($r$) by $v=kr^\beta$ where $k$ and $\beta$ are constants. I tried to find information online, but much of the information about elliptical motion is devoted to objects subject to gravitation. I know that the components of acceleration in polar coordinates are $$a_r = \frac{d^2r}{dt^2}-r \left(\frac{d \theta}{dt}\right)^2 $$ and $$a_\theta = r\frac{d^2\theta}{dt^2}+2 \frac{dr}{dt} \frac{d\theta}{dt}$$ but I do not know how to implement the constraint $v=kr^\beta$ there or whether that is the way to go.


title: How to deal with mean field method in antiferromagnetism?

text: There are lots of ways to apply the mean field method to deal with the Ising model whose ground state is a ferromagnetic state. Hence, it is easy to find the order parameter named magnetization to describe the mean behavior of spin- spin interaction. But, in the antiferromagnetic Ising model, the ground state is an antiferromagnetic state, and I realize it is difficult to find a parameter which effectively describes the mean interaction. If I choose magnetization similar to what we did in Ising model, then, I find it is zero, the mean field is always zero. I get nothing from this mean field. I cannot find a direct physical parameter describing the system to replace the spin-spin interaction. Does this imply that the mean behavior of spin-spin interaction is always zero? Is there a mean field method to deal with the antiferromagnetic Ising model? I hope you can help me, thanks!


title: Is the exponential of the distribution $i\Delta^+(x)$, the 2-point function of a free quantized Klein-Gordon field theory, a distribution?

text: From answers to a previous question, a finite degree polynomial in the distribution $i\Delta^+(x)$, with Fourier transform $2\pi\delta(k^2-m^2)\theta(k_0)$, _is_ a distribution, even though a product of distributions is not in general well-defined. So far I've not been able to find, or be myself sure enough, whether $\exp{\\!\left(i\Delta^+(x)\right)}$ is a distribution. It seems that the exponential is the boundary value of the function $\exp{\\!\left(\frac{1}{z^2-m^2}\right)}$, but it's not clear to me yet whether that's enough. There seems to be nothing on this in Streater & Wightman (which I own, but don't yet understand well enough to apply the methods), and I don't have heavy lifting books like Reed & Simon, etc. More specifically, what I currently think I'm _really_ interested in is whether a bounded function of a distribution, such as $\tan{\\!{}^{-1}\left(\exp{\left(i\Delta(x)\right)}\right)}$, is a distribution.


title: What is the fundamental difference between ghost and auxiliary fields?

text: I am somehow confused by the notion of auxiliary fields, such as for example the fields F and D which appear in supersymmetry, and the notion of ghost fields which appear for example in the BRST formalism. Both kinds of fields are deemed to be not physical. Can somebody once and for all in a clear cut way explain to me what the fundamental differences (and similarities) between ghost and auxiliary fields are?


title: Why are exothermic reactions easier than endothermic?

text: If we talk about electric heating then we have simple elements in which we apply electrical energy which gets converted into the kinetic energy of the electrons which heats it up. But while cooling, we need to take away energy from those electrons doing their dances, since we do not even have to apply energy and rather take it away it should have been easier but we know it is not. Why is it easier to give free electrons kinetic energy than it is to take it away? Similarly if we see on the reaction front exothermic reactions are much more easier to sustain and perform than endothermic ones, this also seems to show that endothermic reactions must somehow be difficult than exothermic ones in general. Why does it seems so?


title: Force required to move a fluid out of piston at depth

text: I have a cylinder filled with water in it at the bottom of the ocean. Say 100 meters down. In this cylinder is a piston that moves up and down. Its job is to squish the water out the bottom of the cylinder to create an air space in the top half of the cylinder. There are lots of holes to let the water move freely out of the bottom of the piston. There is also plenty of air lines to the surface of the ocean that will let air freely into the top half of the cylinder. Assuming we are at sea level and the density of sea water to be 1025 kg/m³, pressure increases by 1 atm with each 10 m of depth. So the cylinder is under about 10.9204 atm of pressure from the surrounding water at 100 meters down. It is my understanding to get the piston to move down and squish the water out of the cylinder you need to overcome the 10.9204 atm of surrounding water pressure. My question is this. If you were to push on the ocean bottom piston with another piston of the same displacement located at the surface of the ocean would that not make the 10.9204 atm of pressure irrelevant? (the pistons would we connected with hydraulic lines) My thinking is the Elevation head due to the fluid's weight, the gravitational force acting on a column of fluid in the hydraulic line from the piston above is already exerting 10.9204 atm of pressure. Is this correct or am I missing something?


title: Sommerfeld results & van Hove singularities

text: According to Sommerfeld the derivative of the density of states $g'(\varepsilon)$ apears in several thermodynamic quantities. Will this also be the case if one use the correct dispersion relation of crystal? If yes don't we encounter the divergence in these quantities when we pass a van Hove singularity (where $g'(\varepsilon)\rightarrow \infty $)?


title: What happens to the neighboring star of a type Ia supernova?

text: Supernovae of type "Ia" are those without helium present, but with evidence of silicon present in the spectrum. The most accepted theory is that this type of supernova is the result of mass accretion on a carbon-oxygen white dwarf from a companion star, usually a red giant. This can happen in very close binary star systems. Both stars have the same age and models indicate that they almost always have a similar mass. But usually one of the stars is more massive than the other and the more massive star evolves faster (leave the main sequence) before the lower mass star does. A star with less than 8-9 solar masses evolves at the end of its life into a white dwarf, binary systems would consist of a white dwarf and a red giant which has greatly expanded its outer layers. During the explosion an amount of carbon undergoes fusion that a normal star would take centuries to use up. This enormous release of energy creates a powerful shockwave that destroys the star, ejecting all its mass at speeds of around 10,000 km / s. The energy released in the explosion also causes an extreme increase in brightness, so these supernovae become the brightest of all, emitting around 10^44 J (1 foe). Normally there are no traces of the star that caused the cataclysm, but only traces of superheated gas and dust that is rapidly expanding. **What happens to the neighboring star?**


title: What physical interactions actually make single stars leave their binary companions at formation?

text: From an interesting ScienceDaily article, I read this > Before the groups of stars disperse, binary stars move through their birth > sites and the group studied how they interact with other stars > gravitationally. "In many cases the pairs are torn apart into two single > stars, in the same way that a pair of dancers might be separated after > colliding with another couple on a crowded dance floor," explains Michael > Marks, a PhD student and member of the International Max-Planck Research > School for Astronomy and Astrophysics. The population of binaries is > therefore diminished before the stars spread out into the wider Galaxy. > > The stellar nurseries do not all look the same and are crowded to different > extents, something described by the density of the group. The more binaries > form within the same space (higher density groups), the more interaction > will take place between them and the more binary systems will be split up > into single stars. This means that every group has a different composition > of single and binary stars when the group disperses, depending on the > initial density of stars. Here's the question: what are the types of gravitational interactions that make binary systems split up into single stars?


title: Where does the light of the Big Bang come from?

text: I'm wondering whether the **residual light of theBig Bang** comes from one particular **direction** and what possibilities do we have to detect its position? ![Enter image description here](http://i.stack.imgur.com/VJboT.jpg)


title: Can $M_3$ move upwards and $M_2$ move to the right?

text: ![enter image description here](http://i.stack.imgur.com/vxYJd.png) Can M3 move upwards and M2 move to the right? in the solutions they assume this, but I find it impossible, it doesn't make sense physically. I get the answer wrong if I assume M3 moving down and M2 moving right in the next question, which is the one that makes sense.


title: Do Gravitational Waves Actually Repel Spacetime?

text: I read something explaining that g-waves actually repel space-time. It described the following. Mass bends space-time creating gravity. An object like the sun, if it suddenly disappeared the space-time it bent would spring back up and the space it compressed (it's gravity well) would form a gravity wave that would hit earth sending it flying into space. It concluded that g-waves squeeze and stretch space-time and actually repel it. Is this valid?


title: Finding the equation of motion of anharmonic potential

text: If I have a potential given by: $$U=U_0\left[2\left(\frac xa\right)^2-\left(\frac xa\right)^4\right]$$ It says that at $t=0$, the particle is at the origin ($x=0$) and the velocity is positive and equal to the escape velocity, which I found to be $\sqrt {2U_0/m}$ I have the differential equation: $$m\ddot x=-\nabla U=-U_0\left[\frac {4x}{a^2}-\frac {4x^3}{a^4}\right]$$ EDIT:: So I have the initial $(x,\dot x)$. Now to find $x(t)$, I use conservation of energy. $$E=K+U=\frac 12mv^2+U_0\left[2\left(\frac xa\right)^2-\left(\frac xa\right)^4\right]$$ The energy of the system is $U_0$, so I can change the above equation to: $$U_0=\frac 12 m \left (\frac {dx}{dt}\right)^2+U_0\left[2\left(\frac xa\right)^2-\left(\frac xa\right)^4\right]$$ $$\left(\frac{dx}{dt}\right)^2=\frac 2m U_0\left[1-2\left( \frac xa \right)^2+\left( \frac xa\right)^4 \right]$$ I found that the value in brackets reduces to $\frac{1}{a^2} (x^2-a^2)^2$ So: $$\frac {dx}{dt}=\sqrt{\frac{2U_0}{m}} \frac {x^2-a^2}{a^2}$$ So I want $$\int^{x}_{x_0} \frac{dx}{x^2-a^2}=\int^t_0 \sqrt{\frac {2U_0}{m}}\frac{dt}{a^2}$$ This ends up being a hyperbolic arctan function, which could potentially make sense, but am I going in the right direction?


title: CPT invariance of Dirac equation

text: We know that Dirac equation is \begin{equation} ( i \partial _\mu \gamma ^\mu - m ) \psi ~=~0. \end{equation} How can we show that Dirac equation is invariant under CPT transformation?


title: Why do larger planets cool more slowly than smaller ones?

text: I learned it during my environment class. My professor introduced the concept--- ratio of surface area and volume. The numerator has connections with the radiative loss, while the denominator is associated with the heat content. The ratio is inversely proportional to the radius of the planet. The conclusion is if I get smaller ratio, the cooling is more slowly. But I cannot find the relationship between the radius with the cooling speed because I think the longer radius, the larger surface area and bigger volume, which leads to greater radiative loss and more heat content. They are varying in the same direction... So I hope someone can give me an explanation to correct my idea.


title: Does a force do work on an object with constant velocity?

text: I know that a force does no work on an object if the object's displacement is zero, but if an object is moving at a constant velocity $\bar{v}$, and a force $\bar f$ (let's say that $\bar f$ and $\bar v$ have the same direction) acts upon it over a linear distance $d$, but the velocity remains the same (perhaps there is an equal and opposing force $\bar f_{opp}$), does $\bar f$ do work on the object?


title: Uniformly charged sphere's electric field

text: I am facing this topic for the umpteenth time in my college career and, of course, every teacher has explained it in a different way. In this course, to find the expression of the electric field of a uniformly charged sphere, the professor started from this equation. $$ \frac{1}{r^2}\ \frac{\partial}{\partial r}\ \lgroup r^2 \frac{\partial u}{\partial r}\ \rgroup = -\frac{q}{\epsilon}\ \delta(r) $$ some idea of what he is doing?


title: Conserved topological charge for d=3 Yang-Mills. G=U(2)

text: Consider a pure Yang-Mills lagrangian density $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu}$$ with gauge group $U(2)$. Take the generators for $U(2)$ to be $t_0$, $t_i \ i=1,...,3$ with commutation relations given by $$[t_0,t_i]=0$$ $$[t_i,t_j]=i\epsilon_{ijk}t_k$$ In particular $t_0$ is the generator of the $\mathfrak{u(1)}$ factor in the expansion $\mathfrak{u(2)}\simeq \mathfrak{u(1)}\times \mathfrak{su(2)}$ and $t_i$ are the generators of the Lie Algebra $\mathfrak{su(2)}$. Now, in $d=3$ the field strenght Hodge-dual is a current $j^{\mu}:=\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\nu\rho}$ and is conserved in virtue of the Bianchi Identity. The questions are: 1) What is it meant when they say the current is conserved? Is it covariantly conserved (ie $D_{\mu}j^{\mu}$=0) or simply conserved (i.e $\partial_{\mu}j^{\mu}=0$) 2)Do I have just one vector current, or one for each generator of the gauge group? (i.e 4 in this case) 3) Can you explicitly carry out the computation of the conserved current and charge? 4) I am asked to state if the conserved charge arises because of the factor $U(1)$ of the gauge group (which has an algebra generated by $t_0$), because of the factor $U(1)$ which is the cartan subalgebra of $SU(2)$ (generated by $t_3$), or because both of them. [I really don't understand this question, what would you answer? Thanks.] The part of the computation I did is the following. $$F^0_{\mu\nu}=\partial_{\mu}A^0_{\nu}-\partial_{\nu}A^0_{\mu}$$ $$F^i_{\mu\nu}=\partial_{\mu}A^i_{\nu}-\partial_{\nu}A^i_{\mu}+g\epsilon^{ijk}A_{\mu}^jA^k_{\nu}$$ Therefore using Bianchi I have $$0=D_\mu\epsilon^{\mu\nu\rho}F^0_{\nu\rho}=(\partial_{\mu}-igA_{\mu})\epsilon^{\mu\nu\rho}(\partial_{\nu}A^0_{\rho}-\partial_{\rho}A^0_{\nu})$$ while for the other side $$0=D_\mu\epsilon^{\mu\nu\rho}F^i_{\nu\rho}=(\partial_{\mu}-igA_{\mu})\epsilon^{\mu\nu\rho}(\partial_{\nu}A^i_{\rho}-\partial_{\rho}A^i_{\nu}+g\epsilon^{ijk}A_{\mu}^jA^k_{\nu})$$ What can I do from here? It seems to me that the currents $$j^{\mu}_0=\epsilon^{\mu\nu\rho}(\partial_{\nu}A^0_{\rho}-\partial_{\rho}A^0_{\nu})$$ and $$j^{\mu}_i=\epsilon^{\mu\nu\rho}(\partial_{\nu}A^i_{\rho}-\partial_{\rho}A^i_{\nu}+g\epsilon^{ijk}A_{\nu}^jA^k_{\rho})$$ are both covariantly conserved... Thanks a lot for answers and clarifications.


title: Why are other universes in the picture of Dark Flow?

text: It seems to me that the popular suggestive explanation of dark flow is another universe interacting with ours. Isn't there other alternative explanations?


title: Warning: Gruesome. Which one cuts faster? Guillotine with a slanted blade or with a flat blade?

text: The following image shows Guillotine that is a device designed for carrying out executions by decapitation. ![enter image description here](http://i.stack.imgur.com/vT3RX.jpg) Intuitively, slicing with pushing is less effective than slicing with pushing and sliding. When slicing, the contact area varies, which in turns make the pressure varies even though the force is kept unchanged. Shortly speaking, what is the effect of slanted blade? Does it cut faster? Bonus question: Is slicing with pushing is less effective than slicing with pushing and sliding? What makes the difference?


title: Assuming space is infinite can our observable universe be an island amongst an archipelego?

text: According to recent measurements our observable universe is roughly 93 billion light years in diameter; also it appears (according to WMAP measurements) that spacetime is flat. Supposing space is infinite. It seems to me that it isn't outside logical possibility that there is another observable universe completely outside of our observational range and so far away it has no appreciable effect on the curvature of our universe. Note, I'm not using the word universe here as everything in space.


title: What is the maximum theoretical limit on how much a HDD can store

text: I was wounding what was the maximum theoretical limit on how much a HDD(with 5 platters and a 3.5 inch size drive) can store because they have 6TB (not to be confuse with TiB) Because most people do. There is a limit on how close they can put the hard to the platter due to Quantum tunnelling.


title: Describing physical constants in alternate wording; c = there can only be 671million miles of space for every second of time

text: This spawns from part of an answer to a question I asked. > All sorts of things go to 0 and/or ∞ if you start boosting at c, and so you > cannot boost into and out of a photon's frame. It somewhat relates to a video from the discovery channel where they talk about space, time, and reality. They describe the speed of light and its implications specifically they describe the speed of light in a counter intuitive definition. (This is what I remember its not a quote exactly, something along the lines of) > The speed of light means there must be so much space for every moment of > time What they mean by this is that the speed of light is constant so there must be so much space for every moment in time. This is because something moving at light speed covers so much distance but never more in one unit of time. Though I don't think that covers the whole idea, I feel I am missing a part about a particle moving at a fraction of the speed of light and how much space it is allotted? I think this is valuable to phrase physical constants like this. Are there any other units, constants, ideas in physics that can be stated this way? Maybe..   * Planck scales   * Time   * Relativity, or parts of it   * golden ratio, e, pi


title: Why is the angle of impact complementary to the angle of launch in the simple equations for the range of a projectile?

text: I'm using the standard equation for the range of a projectile: \begin{align} d &= \frac{v\ \text{cos}\theta}{g} \left( v\ \text{sin}\theta + \sqrt{v^2\ \text{sin}^2\theta + 2gy_0}\right) \end{align} where $d$ is the horizontal distance traveled, along with the associated equation for the angle $\psi$ of the projectile's trajectory in the moment before impact: \begin{align} \text{tan}\psi &= \frac{\sqrt{v^2\ \text{sin}^2\theta + 2gy_0}} {v\ \text{cos}\theta} \end{align} In the section on angle of impact, wiki provides a derivation for the fact that $\theta + \psi = 90^\circ$, and I failed when attempting to derive this equation symbolically, possibly because I don't remember a useful trig identity. I wasn't entirely clear on some of the substitutions the wiki article made, so I made my own attempt: \begin{align} \text{tan}\psi &= \frac{\sqrt{v^2\ \text{sin}^2\theta + 2gy_0}} {v\ \text{cos}\theta} \\\ \text{tan}^2\psi &= \frac{v^2\ \text{sin}^2\theta + 2gy_0} {v^2\ \text{cos}^2\theta} \\\ &= \text{tan}^2\theta + \frac{2gy_0}{v^2\ \text{cos}^2\theta} \\\ \text{tan}^2\psi - \text{tan}^2\theta &= \frac{2gy_0}{v^2\ \text{cos}^2\theta} \\\ (\text{tan}\psi + \text{tan}\theta)(\text{tan}\psi - \text{tan}\theta) &= \frac{2gy_0}{v^2\ \text{cos}^2\theta} \end{align} At this point, I was tempted to use the trig identities: \begin{align} \text{tan}(\psi + \theta) &= \frac{\text{tan}\psi + \text{tan}\theta}{1-\text{tan}\psi\ \text{tan}\theta} \\\ \text{tan}(\psi - \theta) &= \frac{\text{tan}\psi - \text{tan}\theta}{1+\text{tan}\psi\ \text{tan}\theta} \end{align} but making that substitution didn't seem to help anything, so I checked that I could at least get this result numerically. No such luck, however. As an example, I used $g = 9.81\ \text{m/s}^2, \theta=30^\circ,v=10 \ \text{m/s},$ and $y_0=3 \ \text{m}.$ This gives me: \begin{align} \text{tan}\psi &= \frac{\sqrt{10^2\ \text{sin}^2 30 + 2(-9.81)(3)}} {10\ \text{cos}30} \\\ &= 1.05742 \\\ \therefore \ \ \ \psi &= 46.5985^\circ \\\ \end{align} I don't see any errors in the exceedingly simple Matlab code I used to calculate this:               g = 9.81;     theta = 30;     v = 10;     y0 = 3;     psiAng = atand(sqrt((v*sind(theta))^2 + 2*g*y0)/(v*cosd(theta)));      If I assume an initial height $y_0 = 0$ but keep the other values, I get $\psi \approx 30^\circ = \theta$, which doesn't satisfy this property either. Is $\theta + \psi = 90^\circ$, or have I committed one or more errors in the above calculations? I'm ignoring all other factors not present in these equations, e.g. air resistance.


title: What Speed Would an object need to leave the earth at to reach L1?

text: Let's say the Earth is an airless sphere. What speed would an object weighing 1 kg need to leave the surface at in order to get to and be motionless at L1, where the Moon's gravity becomes stronger than the Earth's gravity?


title: Myths in the history of Physics

text: As an example of such a historical myth that we all learned, there is the story that in his confrontation with Cardinal Bellarmine in 1632, Galileo had all the evidence on his side, won the intellectual argument by the weight of that evidence, and was defeated only by the ignorant prejudices the trial judges. However, historical studies have also advanced as more detailed information was obtained and reflected upon. The actual situation was much more nuanced. The version of Copernican theory defended by Galileo assumed 1. a very small solar system, the size of which should clearly be seen by measurements of planetary parallax and 2. circular orbits, not elliptical (although Galileo knew of the work of Kepler, but declined to incorporate it). Both Copernicus and Galileo had attempted to measure parallax in the solar system, and failed to get a result > 0. (It was only measured by Bessell in 1838.) Furthermore, because it used circles for orbits, the Copernican model was actually less accurate than the old Ptolemaic epicycle system of approximations. In addition, Galileo predicted only a single tide/day, not the actual two. On the other hand his unambiguous observation of Venus in all four phases in 1610 definitely ruled out the Aristotelian model of heavens, since in this model the sphere of the sun was always beyond the sphere of Venus, although the Tychonic, Capellan and Extended Capellan models were still possible, each either with or without a daily rotating Earth. These latter models had the virtue of explaining the phases of Venus without the vice of the 'refutation' of full Copernican heliocentrism’s prediction of stellar parallax. From Wikipedia's article on Galileo, confirmed by Weinberg. Conclusion: on the basis of the evidence available to them, the Church had a pretty strong case on the evidence against Copernican heliocentrism, arguably stronger than Galileo. Furthermore, on the basis of contemporary documents and doctrines going back to Augustine and the 4th century, if the church had seen conclusive proof in favor of the Copernican system, it was already formally committed to changing its interpretation. The Church, being engaged in a power struggle during the reformation, would likely have followed conclusive evidence had it been presented. Refs: see the historically accurate and nuanced video at http://online.itp.ucsb.edu/online/colloq/tutino1/rm/qt.html, for slides see http://online.itp.ucsb.edu/online/colloq/tutino1/, book at Google. This is not an obscure, idiosyncratic view of the Galileo affair, but rather the generally accepted view of knowledgeable scholars. What other famous oversimplified or false myths of this sort do you know and can document and or reference corrections to?


title: Bleaching groundstate

text: I'm reading an article about two-dimensional infrared spectroscopy and I don't understand the following sentence. > Bleach or stimulated emission contributions yield negative signals. What are "bleach contributions"? I have never heard of it and cannot find a suitable explanation on the Internet. Can somebody explain that to me? The article I'm referring to is J. Chem. Phys. 121, 5935 (2004)


title: Why does $U = T S - P V + \sum_i \mu_i N_i$?

text: From the first law of thermodynamics: $$ dU = TdS - PdV + \sum_i\mu_i dN_i.$$ Quoting Wikipedia: > As conjugate variables to the composition $N_{i}$ the chemical potentials > are intensive properties, intrinsically characteristic of the system, and > not dependent on its extent. _Because of the extensive nature of $U$ and its > variables,_ the differential $dU$ may be integrated and yields an expression > for the internal energy: > > $$ U = T S - P V + \sum_i \mu_i N_i.$$ Emphasis mine. $U$, $S$, $V$, and $N$ are extensive while $T$, $P$, and $\mu$ are intensive. How does this fact allow $U$ to be integrated this way?


title: How is the speed of light constant in all directions for all observers?

text: **Please imagine the following thought-experiment:** ![light on a train moving at .5c](http://i.stack.imgur.com/KFfV7.jpg) * * * **Order of Events:**   1. **Pulse** \- A single pulse of light is emitted from the light towards the mirror   2. **Reflect** \- The pulse hits the mirror and is reflected back towards the light    3. **Return** \- The pulse returns to the light. **Observers:**   1. **BoxGuy** \- An observer on the boxcar   2. **PlatGirl** \- An observer on the platform * * * **Question:** With the above configuration, how can the speed of light be constant for both observers in both directions? **Analysis:** Assuming the speed of light is constant for BoxGuy relative to himself, the time between Pulse and Reflect is equal to the time between Reflect and Return. This is because the distance the light travels relative to him is **d** in both cases. With the same assumptions for PlatGirl, the time between Pulse and Reflect is less than the time between Reflect and Return. This is because the mirror will travel **2 * d** on the away trip (because when light has traveled **2 * d** , the mirror will be **d** farther to the left, so both the mirror and pulse will be in the same location), but only **2/3 * d** on the return trip (using similar logic). Assuming that the light pulse is in the same location for all observers at any given moment, Pulse has to occur simultaneously for both BoxGuy and PlatGirl, Reflect has to occur simultaneously for BoxGuy and PlatGirl, and Return has to occur simultaneously for BoxGuy and PlatGirl. Finally, if we try to figure out the relative passage of time for BoxGuy and PlatGirl with the above, we get that time travels faster for PlatGirl than for BoxGuy during Pulse-Reflect. This is because light travels farther for her ( **2*d** ) than him ( **d** ) during that time. With similar logic, we get that time travels slower for PlatGirl than for BoxGuy during Reflect-Return. The last conclusions do not make sense, since the coming or going of a beam of light should not affect the relative time-lapse for two observers. For example, if this were the case what would happen if another pulse was emitted the moment the first pulse is reflected? Time cannot move faster AND slower for both of them. Thus, either the speed of light is not constant, the same light beam can simultaneously be in different locations at once for different observers, or there is another flaw in the analysis. Which is it and why? **Notes:**   * As mentioned by other users, **d** will be shorter for PlatGirl than for BoxBoy according to SR. However, the duration of Pulse-Reflect is still shorter than Reflect-Return for PlatGirl, and the durations are equal for BoxBoy.   * In response to my question on Mark's answer, we can use the Lorentz Transform to calculate PlatGirl's space-time coordinate for BoxGuy's Reflect observed event, which happens at (d,d/c) in his frame of reference: $\lambda = (1/\sqrt{1-.5^2}) = (1/\sqrt{.75}) = \sqrt{4}/\sqrt{3} = \frac{2\sqrt{3}}{3}$ $t' = \lambda (t - vx/c^2) = \lambda (d/c - (-.5) \cdot d/c) = \frac{2\sqrt{3}}{3} \cdot (1.5d/c) = \sqrt{3}d/c$ $x' = \lambda (x - vt) = \lambda (d + .5c \cdot d/c) = \frac{2\sqrt{3}}{3}*1.5d = \sqrt{3}d$   * Similarly for (0, 2d/c): $t' = \lambda (t - vx/c^2) = \frac{2\sqrt{3}}{3} (2d/c) = \frac{4\sqrt{3}}{3} d/c$ $x' = \lambda (x - vt) = \frac{2\sqrt{3}}{3} (.5c \cdot 2d/c) = \frac{2\sqrt{3}}{3}d$


title: Active versus passive transformations

text: I am a bit confused by the concepts of active and passive transformations. In all the courses I am doing at the moment we do transformations of the form: $$ \phi(x) \rightarrow\phi'(x') = \phi(x) $$ and $$ \partial_{\mu}\phi(x)\rightarrow\partial' _{\nu}\phi(x') = \frac{\partial x^{\alpha}}{\partial x' ^{\nu}}\partial_{\alpha}\phi(x) $$ This is all perfectly clear to me. However, I am reading Peskin and Schoder at the moment, and they adapt an "active" point of view (their words), such that the above transformations are: $$ \phi(x) \rightarrow\phi'(x) = \phi(\Lambda^{-1}x) $$ and $$ \partial_{\mu}\phi(x)\rightarrow\partial _{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x). $$ I don't understand how to interpret this and especially how to derive the second equation.


title: why drag cause mass in higgs field ? how could drag cause mass?

text: why in higgs field drag cause mass? drag is force in general not mass Higgs field- Inquiring Minds - Questions About Physics how drag of higgs field cause mass?


title: Why can we say there is no tangential acceleration in this specific problem?

text: I'm looking at a problem involving a simplified rollercoaster with only 3 cars (connected with weightless rods) for which the friction and distance from the carts to the rails can be ignored. The part I was thinking about is the following: at a certain moment cart 2 is situated on top of an arc as I have drawn here. ![rollercoaster](http://i.stack.imgur.com/QVKc4.png)      At a certain point in the set of questions, we seem to rely on the fact that all carts share the same acceleration $a = \frac{{v_2}^2}{R}$. Now I understand that they all share the same normal acceleration since they are performing a circular motion, but what is stopping them from having a tangential acceleration? In other words: why do we know $a_t = 0$ for each cart?


title: Collapse in Quantum Field Theory?

text: I do not want answers telling me that wave-function collapse is not real and decoherence is the answer (I know the situation with that). I am asking a question purely on the basis **if** wave-function collapse is the correct method. My question is: in normal quantum mechanics superposition of the state (position, momentum) exists until the wave-function collapses (how or why it collapses is not important in this question), now in quantum field theory we can also have superposition as in the superposition of Fock space states with different particle number. Can the superposition also collapse here under collapse interpretations of quantum mechanics/quantum field theory? Laymans answers would be mainly appreciated...


title: Why do disconnected diagrams not contribute to the S matrix?

text: I've read somewhere that disconnected diagrams do not contribute to the S-matrix. I don't see why this is the case. I do know why vacuum bubbles do not contribute: given a generating functional for a scalar field the n-point correlation function follows from: $$ G(x_1, \dots,x_n) = \frac{1}{i^n}\frac{\delta}{\delta J(x_1)}\frac{\delta}{\delta J(x_n)}Z_0[J]|_{J=0} $$ If you treat the generating functional in perturbation theory, expaning out the denominator of $$ Z[J] = \frac{\int\mathcal{D}\phi\exp(iS + i\int J\phi dx)}{\int\mathcal{D}\phi\exp(iS)}$$ will cancel all vacuum bubbles from the numerator. Does something similar hold for the disconnected diagrams?


title: What if the Earth stops in its orbit and instead falls straight towards the Sun?

text: The Earth orbits around the Sun because it has angular momentum. If we stopped the Earth in orbit and then let it fall straight towards the Sun, how long would it take to reach the Sun in seconds? Details and assumptions               The mass of the sun is 2*10^30 Kg.     The mass of the earth is 6*10^24 Kg.     The earth is 149,600,000 Km from the Sun.     You may treat the Earth and Sun as point masses.


title: Coexistence of light and dark

text: There are two things light and dark. Is there any system where the thing is neutral, i.e we see light and dark together in same ratio. Is it possible practically/theoretically ?


title: What does it take to understand Maxwell's equations?

text: Assume I want to learn math and physics enough to reach a level where I understand Maxwell's equations (The terms and reasoning in the equations I.e. why they "work"). What would I have to learn in order to have the tools I need to make sense out of it? I'm kindof looking for a road map which I can use to get started and know what to focus on. The fields are pretty big so some pruning would be very helpful (if possible)


title: Explanation of a Foucault pendulum

text: The equations of motion of a Foucault pendulum is given by: $$\ddot{x} = 2\omega \sin\lambda \dot{y} - \frac{g}{L}x$$ $$\ddot{y} = -2\omega \sin\lambda \dot{x} - \frac{g}{L}y$$ where $\omega$ is the rotational frequency of the earth which has a value of $7.27 x 10^-5$, $\lambda$ is the latitude of where the pendulum is, $g$ is the acceleration due to gravity, $L$ is the length of the pendulum's string. What I don't know is what does $x$ and $y$ represent? I have read some derivations of these equations but I really cant figure out what they are trying to say.


title: Where do positron and electron get K.E from in Pair production

text: In pair production, we get a positron and electron from a photon. The photon should have 1.02Mev energy to carry out this process When the energy of positron and electron is added, we get the energy of photon. hf = E- + E+ = (m0c2 + K.E-) + (m0c2 + K.E+) = K.E- + K.E+ + 2m0c2 mc2 for electron or positron is = 0.51 MeV so, hf = 2 x 0.51 + K.E- + K.E+ hf = 1.02 + K.E- + K.E+ so, where do the positron and electron get K.E from because the the sum of their energy is equal to photon.


title: Given a radius and velocity calculate position of an aircraft banking to make a turn in three dimensional space

text: Suppose an aircraft is moving at a certain fixed altitude above the ground. It follows a path defined by latitude and longitude. Now if we want to define the position of an aircraft at any point in the air, three variables are required for example X for latitude, Y for longitude and Z for the altitude. Suppose an aircraft flies and reach a certain fixed altitude Z, it then follows a route defined by X and Y. Now suppose that at any stage during the flight the aircraft decides to take a turn.The aircraft does a maneuvre in such a way that it turns either to the left or right and increases or decreases it's altitude in the same time. Assuming knowing the speed and the current three Dimensional position of the aircraft, how can the future position of the aircraft after a known time t can be predicted? Also assume that other aircraft related constant parameters are also known like radius of the turn, Bank angle etc


title: Meaning of "Grounded"

text: In my opinion, "grounded" means having the same potential as the potential at infinity, which is usually set to zero. Now if we consider a conductor inside a uniform electric field, what is the meaning of a conductor being "grounded"?


title: What does the notation $c = [1:\beta]$ mean?

text: I have been reading a online-book/blog/material on Quantum Mechanics, when I encountered a notation on a page and I have no idea what it means. See if you can help. Here's the link and follows the paragraph where I am stuck. > Observe that exchanging either the incoming or the outgoing particles is > tantamount to exchanging the two alternatives and, correspondingly, > the two amplitudes, so that A2 takes the place of A1 and vice versa. Since > the two amplitudes have the same magnitude, there is a complex number c > of unit magnitude such that A2 = A1 c. In other words, multiplication > by c = **[1:β]** represents an exchange of the incoming or outgoing > particles. > > If the incoming or outgoing particles are exchanged twice, then (i) A1 > gets multiplied by $c^2$ and (ii) the original situation is restored. > Thus A1 = A1 $c^2$, whence it follows that $c^2$ = **[1:2β]** = 1. This > means that 2β must be equal to an integral multiple of 360°, and this > leaves us with two possibilities: β = 0°, in which case A2 = A1, or β = > 180°, in which case A2 = −A1. I have put the notation in bold. What is it that the writer means exactly by $[1:\beta] \ \ and \ [1: 2 \beta]$ ? :)


title: Finding the equation of motion regarding a block pulling a uniform disk

text: ![enter image description here](http://i.stack.imgur.com/QkMLC.png) My problem is finding the equation of motion for the block. Using Newton's Second Law, I did: $$4g-T-R=4\frac{dw}{dt}$$ But the mark scheme says: $$4g-T-R=0.2*4\frac{dw}{dt}$$ And I don't understand why the radius has anything to do with this. Can somebody please explain?


title: energy of deflection of a moving object

text: Imagine an object moving in a straight line in space, perhaps shaped like a yo-yo (to make rotational effects negligible). Suddenly a constant force is exerted at right angles to the direction of motion, such that the object is now deflected into a circular path. What is the energy required (or work done) to maintain the constant deflection of this circular path? Start over, and imagine a second identical object moving in the opposite direction, but offset so that when a similar force is applied to it, it travels in the same circular path, but offset by 180 degrees. Clearly, the energy required to keep both objects moving in the path is twice that for just one of them. Now, start again, but this time have the two objects instead link up with a cable and snap fastener, as they go by each other. The cable tension causes both objects to travel in the same circular path as before. Does the cable do the same work that was done before to keep the two objects in the circular path? If not (and besides, how can a cable do work?), then was the energy wasted before, to do what a simple cable can do without energy?


title: electrical forces of two charges repelling each other

text: If I place two positive charges of different magnitudes on the x-axis (one at the origin, one at the some position x), as the two charges repel each other are the forces that they impart on each other the same? Why? Edit: I'm working on a problem that has that situation, and the problem asks for me to find the position of one of the charges if the other one is at a given position. Why can't I just set the two forces equal to each other and use that to find the position of the other charge?


title: What Causes the Noise Floor of Modern CCDs?

text: CCDs are getting pretty good these days, but all systems are subject to noise. A typical value for a CCD seems to be 20 electrons RMS per pixel. This article from qsimaging.com says > CCD Read Noise is a fundamental trait of CCDs, from the one in an > inexpensive webcam to the CCDs in the Hubble Space Telescope. CCD read noise > ultimately limits a camera’s signal to noise ratio, but as long as this > noise exhibits a Gaussian, or normal distribution, its influence on your > images can be reduced by combining frames and standard image processing > techniques. It doesn't really describe the source of the noise however. I know that astronomers like to cool their CCDs, so I'm thinking it is from thermal phonons being converted to electrons? I'm specifically asking about the CCD so lets assume that the shot noise of the incoming light is well below the typical CCD noise quoted above.


title: Dirac bracket and second class constraints in first-order gravity formalism

text: In the first order formulation of general relativity, the frame field $e_{\mu}^a$ and $\mathrm{SO}(3,1)$ spin connection $\omega_{\mu c}^b$ are independent variables. In the Hamiltonian formulation of this theory, one finds that there are second-class constraints. According to Dirac, the way to deal with these second-class constraints when quantising is to first define the Dirac bracket, which is essentially a new Poisson bracket that 'respects the constraints', in the sense that the Dirac bracket of any two constraints is another constraint, and then proceed with the quantisation procedure. After looking a little bit in the literature, I have been unable to find any paper that actually attempts to construct the Dirac bracket for the first- order formulation of general relativity. And indeed it seems people go to lengths to reformulate gravity so that it doesn't have any second class constraints from the get-go (e.g. using the Ashtekar variables). My question is, has the Dirac bracket for first-order gravity been constructed? If so, a reference would be great.


title: There are two definitions of S operator (or S matrix) in quantum field theory. Are they equivalent?

text: I read several textbooks of QFT and found that there are two kinds of definition of $S$ operator (or S matrix).   * First kind: Define $\hat{S}$ is map from out space to in space $$\hat{S}\left|\beta,\text{out}\right\rangle：=\left|\beta,\text{in}\right\rangle,$$ so that $$S_{\beta\alpha}:= \left \langle \beta,\text{out} | \alpha,\text{in}\right\rangle= \left \langle \beta,\text{out} \middle |\hat{S}\middle | \alpha,\text{out}\right\rangle= \left \langle \beta,\text{in}\middle |\hat{S}\middle | \alpha,\text{in}\right\rangle.$$ I understand that all these vectors are defined in the Heisenberg picture.   * Second definition: $$S_{\beta\alpha}:={}_\text{in}\\!\left \langle \beta,\text{out} \middle |\hat{S}\middle | \alpha,\text{in} \right\rangle_\text{in}$$ where subscript $,\text{in}$ means vector are in interacting picture. In this definition, then, $$\hat{S}=U(+\infty,-\infty),$$ where $U(+\infty,-\infty)$ is the evolution operator in interacting picture. Are these two definitions equivalent? I am confused about it. Remark: I konw that the matrix element $S_{\beta\alpha}$ is the same in these two pictures, what I want to ask is whether the operator $\hat{S}$ is same in these two definitions. Thanks!


title: Admixtures of longitudinal and timelike photons!

text: In the quantization of electromagnetic field the physical states $|\psi\rangle$ are found to obey the following relation: $[a^{(0)}(k)-a^{(3)}(k)]|\psi\rangle=0$ It is explained as the physical states are admixtures of longitudinal and timelike photons. What does longitudinal and timelike photons physically mean? Why the polarizations, $\epsilon^{(0)}$ and $\epsilon^{(3)}$, timelike and longitudinal photons, are called unphysical?


title: lifetime of a PN junction in a diode

text: diodes are made of silicon, which have been enriched in phosphorus and boron to create the PN junction. Since the electrons are always going in the same direction, from N to P in the junction, I was wondering whether a diode had a given lifetime after which there is no electrons anymore to move in the PN junction (no deficiency nor excess in electrons in any side of the PN junction) ? Thanks!


title: intensifying a light source

text: When trying to intensify a beam of light by refracting it through a lens, (as in a lighthouse fresnel system or similar railroad style switch lamp from years past), is the beam intensity increased by moving the lens closer to the beam? That is, refracting it as close as possible to the light source? Or of a similar condition, would an old kerosene barn lantern be brighter if the globe was an inch away from the flame instead of 3 to 4 inches? TE


title: Gases at different pressures reaching uniform pressure

text: First of all, sorry if my question is dumb, I have very little understanding of physics. Also, English is not my native language. If two gases, initially with different pressures, are mixed in a closed vessel, do they (the mixture being treated as a single gas) generally reach uniform pressure? Now, atmospheric pressure reduces with altitude. However, how can the air that is above exert pressure on the air below without them reaching the same density? How can it be that the less dense air pushes the denser air and the system is stable?


title: Spinor reps in $\mathbb{R}^{1,3}\times{}B$ space-times

text: I am considering spinors in a space-time which is $\mathbb{R}^{1,3}\times{}B$ being $B$ a compact manifold of $D$ dimensions. I know that in ordinary 4 dimensional space-time spinors are representations of $O(1,3)$. Now, in my case, are spinors expected to be reps of $O(1,3+D)$? Does the compactness of $B$ impose some restrictions to this? I have the feeling that we have to expect spinors to be reps of $O(1,3)\times{}O(D)$ since I don't feel that making a boost in the compact space is allowed but I am not sure. Any clarification on the reps spinors are in in the mentioned space-time will be greatly appreciated.


title: why evaluate at lambda = 0

text: I am trying to understand Herbert Goldstein's introduction to 4-vectors. He describes a 1-D curve in spacetime $ P_(\lambda) $ then he says a 4 vector is defined as the tangent vector to this curve $$ v = \biggr ( \frac {dP} {d\lambda}\biggr)_{\lambda =0} $$ why is $ \lambda $=0? what does that have to do with anything? I have been staring at this for like 20 minutes I still don't understand what he is talking about... it's giving me problems because i need to understand this part later because it is relevant to how tensors transform also he says $ \lambda $ is a measure of a length along the curve... i don't really follow that point either... i though $ \lambda $ could be any parameter like proper time etc. any help on this??


title: Numeric method to calculate the charge distribution on a conducting surface?

text: If I have an arbitrary (closed?) conducting surface and a nearby charge density, is there a simple numeric way of computing the induced charge distribution on the surface?


title: Compound lenses and virtual images

text: Ok so I have this problem where I have a system of two lenses. All I know is that the distance between the object and the first lens is 30cm, the distance between the object and the final image is 70cm, the focal distance of the first lens is 20 and the total magnification is -0.666. I need to find the focal distance of the second lens and the distance between lenses 1 and 2. Now I've tried calculating the parameters of the first lens' image (it should be 60cm past the first lens and the magnification should be -2). My problem is that if I now use this image as an object for the second lens and solve the equations, I obtain absurd results (values should put the lens between the first lens and the final image but they don't). How should I go about solving such a problem?


title: about field gradient

text: I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? To my understanding, I think a gradient field means the magnitude of field is position dependent, is that right? I read an article online introducing the strength of the magnetic field of some objects. The earth's magnetic field strength is about 0.5Gauss and a small magnetic bar (toy) produce about 5Gauss field. Let take the magnetic toy as example, I wonder how to estimate the gradient of the field? From the math, it seems that we should estimate how does the field changed in a unit of distance, so for the bar producing field of 5Gauss, roughly how big is the gradient of that field?


title: The Energy-Momentum Tensor and the Ward Identity

text: I have a question regarding a homework problem for my quantum field theory assignment. For the purposes of the question, we can just assume the Lagrangian is that of a real scalar field: $$\mathcal{L}=\frac{1}{2}\left( \partial _\mu \phi\right) ^2-\frac{1}{2}m^2\phi ^2$$. The first part of the problem was to calculate the energy-momentum tensor corresponding to the translation symmetry $x\rightarrow x+a$. I obtained: $$ T_\mu ^\nu =(\partial _\mu \phi )(\partial ^\nu \phi )-\delta _\mu ^\nu \mathcal{L}. $$ So far, so good (at least I hope so). The second part of the problem states: "Write the corresponding Ward Identity." What does he mean by this? I looked up "Ward Identity" in the index of our text (Peskin and Scroeder), but was unable to find something that seemed relevant to the problem. Where do I even begin?


title: Will magnetic field lines change when a magnet gets closer to other magnets?

text: As many books tells that the magnetic field can be illustrated by a bunch of field lines, which start from one pole and end up at the other. If I have a strong uniform field with field lines along horizontal direction. What happen if I place a small magnet somewhere away from the field line, will it change the direction of the source field (lines)? I am thinking that some parallel lines should detour so to end up at the south pole of the magnet. But how much will it change the uniform field?


title: Blimp challenge. Doubts about volume at different heights

text: I want to design and build a small semi-rigid airship/dirigible. It will be controlled with RC so I will install motors, arduino, battery etc. Filled with helium. Weight of the equipment is about 2kg and weight of the dirigible itself (envelope, light internal scaffold...) is unknown since I don't know how big it has to be. Now, I was all happy calculating the lift force minus weight equals 0, so it stays still:               volume * airDensity - volume * heliumDensity + weigh = 0      Since I want it to stay still at 3000m, I used 0.91kg/m3 for airDensity (source: Wikipedia) and 0.1256 kg/m3 for heliumDensity (source: WolframAlpha). **First questions:** are those density values good? Should I count in air temperature? If so, where do I find that data? I want to make the airship fly over some small mountains but the climate there is very cold so I guess air temperature is going to change a lot throughout its flight. **More questions:** what volume should the airship be design to hold? I don't want it to explode or be dangerously overinflated, so I'm guessing I should design to hold the volume helium will have at 3000m, but what volume is that? Is it affected by airDensity at 3000m since it's surrounding the airship? I asked a physicist friend of mine but honestly it only confused me more with weird formulas and doubts (as you can see, I'm not a physicist, I wish to be one day though) If you can't answer **all** questions, any information, examples or whatever that may help me is very welcome. Thanks!


title: Thermodynamics in a closed room - Why does the stale smell depend on Temperature?

text: Why does the stale smell disappears in a crowded room if you cool it via air- condition ? What's the physics ?


title: When and how did the idea of the tensor product originate in the history quantum mechanics?

text: At some point in the history of quantum mechanics, it was accepted that a single particle is described by a wavefunction which is a function of the position of the particle $\mathbf{r}$, denoted: $$\psi(\mathbf{r})\,.$$ At some (possibly later) point it was also accepted that _two_ particles are described by a wavefunction which is a function of the positions of each one of the particles, $\mathbf{r}_1$ and $\mathbf{r}_2$, denoted: $$\psi(\mathbf{r}_1,\mathbf{r}_2)\,.$$ In other words, the Hilbert space describing the two-particle system is the _tensor product_ of the Hilbert spaces describing the system of each particle.   * **When was this idea originated and how?** I understand that there are important consequences of the tensor product structure, mainly _entanglement_ , but as far as I know these consequences were explored _after_ this structure was already accepted.   For example in the EPR paper (1935) the tensor product structure was already taken for granted. To make my question clearer, here is an alternative (false) way that could have been used to describe a system of two particles. If the particle is thought of as a "wave-like" object (like was the case in these days I think), why not describe two particles with a single wavefunction $\psi(\mathbf{r})$, which is normalized: $$ \int d\mathbf{r} |\psi(\mathbf{r})|^2 =2~, $$ i.e. "double" the quantity of the particle. Of course a lot of information is missing in this description when compared to the correct description, but how did the originators of quantum mechanics know that this information is supposed to be there in the first place?   [ **Edit:** As Luboš says in his answer, a better alternative description for the purpose of this question is to use two wavefunctions $\psi(\mathbf{r}_1)$ and $\psi(\mathbf{r}_2)$. Here too there is information missing, and the rest of the question is the same.]   * **Was there an experimental result at that time that could not have been explained by this alternative description?**   * **Was it mere physical intuition that led to the tensor product structure?**


title: Why do safety helmets have a softer inner layer nearer the head?

text: I know that when an object collides onto the helmet, it causes an inelastic collision so that energy is absorbed by the structure of the helmet, so what exactly does the softer inner layer do? Does it prevent the energy from being transferred into the head?


title: Logical understanding of the canonical probability distribution (canonical ensemble)

text: I am having problems in understanding the logic of this distribution: $P(\Psi_{j})=\displaystyle\frac{e^{-E_{j}/kT}}{\displaystyle\sum_{j'}e^{-E_{j'}/kT}}$ The book I am studying use the case of a sample in contact with a reservoir at thermal equilibrium to derive this distribution. I understand the derivation, but I don't understand the logic of the distribution itself. The aspect I'm having problem with is the fact that the lowest the energy of the sample, the highest the probability. What I don't understand is why this happens even thought there is an average energy given by the temperature which I thought should be more probable then any energy lower than this for a given particle. This doubt implies that I am looking at $P(\Psi)$ as the probability for a given particle, wich does not seem to be the case, but if I think of it as the probability for the hole sample, it makes even less sense for me since the energy should be totaly given by the temperature, so it wouldn't make sense to make a distribution of it if the temperature is considered constant. Thanks in advance


title: RMS Free Path vs Mean Free Path

text: I am trying to determine the mathematical difference between mean free path and root-mean-square free path. For an ideal gas, the relaxation time is $$\tau=\frac{1}{\sqrt2 \pi nd^2 \bar v}$$ and the mean free path is $$\Lambda=\tau \bar v $$ so the velocities cancel. When I am calculating the RMS free path, I am assuming I would use $$\Lambda_{rms}=v_{rms}\tau_{rms}$$ and I am assuming $$\tau_{rms}=\frac{1}{\sqrt2 n \pi d^2 v_{rms}}$$ This would again cause $v_{rms}$ to cancel leaving the RMS free path as the same value as the mean free path which seems odd to me. Should I just use regular $\tau$ instead of $\tau_{rms}$?


title: Space time a function of itself, objects in it, or both?

text: Is spacetime a function of itself, objects within it, or both? I am struggling to understand just what is spacetime without objects in it (or theoretical reference points) and thus no frame of reference except itself. It seems that spacetime without objects (on its own) is more of an abstraction and would be hard if not impossible to quantify without someTHING in it by which it could be measured. I feel like I must be missing something.


title: Resources on Master Equations

text: Presently I am reading about "Introduction to dynamical process theory and simulation" which uses the notion of Master Equations to solve Markov process. I am very new to this. Can someone provide me the references so that I can learn about master equations from scratch.


title: What exactly does Aaron D. O'Connell's experiment show?

text: I watched a TED talk by the scientist Aaron D. O'Connell about actually seeing quantum superposition. The link to the talk is :- http://www.ted.com/talks/aaron_o_connell_making_sense_of_a_visible_quantum_object I researched on it a bit but I am still not able to understand how did his experiment work. Ultimately while measuring, the state of superposition would have been destroyed. What was it that made the experiment successful. I didn't even understand how could he conclude that everything is in two places at the same time. Especially when you see the talk, he mentions that his device was both vibrating and not vibrating at the **"same time"** which seems a bit ambiguous to me. How can you take two different measurements at the same time? Here is a picture of the "diving board" his team used that was in a vibrating superposition: ![enter image description here](http://i.stack.imgur.com/8g7GY.jpg)


title: How does gravity affect sound waves?

text: Someone asked me this question and I don't think I gave him an adequate answer (I was trying to think of the extreme case - that of neutron stars)


title: Apparent dimensional mismatch after taking derivative

text: Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them. Now, we wish to take the derivative of $u = log (\frac{x^2}{a^2})-log (\frac{a^2}{x^2})$. Here, we have taken the natural logarithm. It is clear that $u$ is a dimensionless function. $$\frac{du}{dx} = \frac{a^2}{x^2}.\frac{2x}{a^2} - \frac{x^2}{a^2}.(-2a^2).\frac{2x}{x^3} \\\ = \frac{1}{x} - 4. $$ Here, the dimensions of the two terms on the right do not match. The dimension of the first term is what I expected. Where am I going wrong?


title: Beam waist at extremely long distances

text: A beam of light of wavelength $\lambda$ and width $W$ needs to be focused at a distance $D$ to a spot not bigger than $w_S$, which stands for 'width of sail'. Now, the diffraction limit says it clearly that a coherent beam will diverge at a rate no less than $\frac{\lambda}{W}$, and that the minimal target spot width will be $\sqrt{D \lambda}$. But my understanding from the comments in this question is that this applies in the far-field limit _far away_ from the focal point, so my doubt is that if this hard bound on the spot size can be avoided by replacing the collimated beam by a focusing beam with an extremely long focal length. So, my question boils down to the following: can the fundamental diffraction limit on spot size, be replaced with the extremely difficult, but possible in principle engineering problem of creating a focusing element with extremely high focal lengths, in such a way that for a specific length, the target spot size can be made smaller A concrete scenario to make things more clear: the wavelength is $\lambda = 10^{-6}$ meters. The distance to the target is $D = 10^{12}$ meters ($10^{18}$ wavelengths). If the beam converges at focal length $D$, my hope is to keep the beam spot (which in this case coincides with the beam waist) at $10^2$ meters (which is 10 times smaller than the target beam spot if the laser would have been collimated, instead of focusing at $D$)


title: Do I need to have an airtight container to siphon boiling water?

text: I'm attempting to siphon boiling water into another container for a school project. I have researched the physics of a siphon but I can't seem to make my boiling water siphon out of its initial pot. Does the pot need to be airtight to increase the pressure? Is 3/8" tubing too thick for this to work?


title: negative energy cancelling energy

text: In what circumstances can negative energy cancel out energy? If a positron and electron are created they cancel out as one of them has negative energy. Are there any other ways that negative energy and energy cancel out? Is negative energy everywhere around us or somewhere else?


title: What will be the unit of mass in $E=mc^2$ when energy units change?

text: I know that conventionally the unit of $E$ in $E=mc^2$ is taken as joules when mass is in $\mathrm{kg}$. But I am doing some calculations of my own with different units of energy ($\mathrm{keV}$,$\mathrm{MeV}$ etc.). So my question is whether we have to convert $\mathrm{keV}$ or $\mathrm{MeV}$ to joules in order to calculate mass or we can keep the values of $\mathrm{KeV}$, $\mathrm{MeV}$ units and then find mass (I think this method is easier)? If we are keeping $\mathrm{keV}$ and $\mathrm{MeV}$ then what will be the unit of mass, will it still be $\mathrm{kg}$? I am a chemist so please forgive if I asked an irrational question. Advanced thanks for your help


title: $WW\to t\bar{t}$ growth

text: I was told recently that "it is well known that processes like $WW\to t\bar{t}$ ($t$ being a top, or any massive fermion) grows linearly with the energy in the absence of an Higgs boson." Does anyone know of a reference? The only ones I can find regarding the growth of $WW \to f\bar{f}$ are about the old argument for having a neutral current, i.e. $WW \to f\bar{f}$ cross- section grows if $Z$-mediated diagrams are not included. Many thanks in advance.


title: With a box that has perfect mirrors on the inside would it be possible to trap light?

text: With a box that has many perfect relective mirrors, would it be possible to trap a beam of light in the box indefinetly?


title: What math do I need for mathematical physics? In what manner should I learn math?

text: I'm a freshman undergraduate. I've got my sight on mathematical physics. I love math but I don't have the talent nor the inclination for purely abstract mathematics. I also love physics. The only math for physics I know is cook-book calculus 1. (I have deep interest in mathematics, I have finished an intro to proof book, some introductory set theory, cardinality of infinite sets. Right now I'm working on mathematical logic.) I have this compulsion to prove every result and understand why some math thing is defined so. As such, I am very slow in learning mathematical methods. Cook-book calculus sucks. But where I come from we're now allowed to take math classes. I'm worried that if I don't study the pure maths version of calculus I might miss some important result or the understanding which might be really useful. Self-studying mathematics really takes time. I want to be as efficient as possible. My question is, how should I learn the math behind the math methods? What mathematics should I learn for mathematical physics? Is learning the mathematics really that important? If possible, please recommend some good books.


title: Simple argument for unexpected behavior in SUSY model

text: Consider a supersymmetric theory with 3 chiral superfields, $X, \Phi_1$ and $\Phi_2,$ with canonical Kahler potential and superpotential $$ W= \frac12 h_1 X\Phi_1^2 +\frac12 h_2 \Phi_2\Phi_1^2 + fX.$$ One can show, by doing calculations, that (i) supersymmetry is spontaneously broken, but (ii) one- loop corrections do not lift the classical pseudo-moduli space. QUESTION: is it possible to say (ii) without looking at the explicit form of Coleman-Weinberg potential, e.g. making some field redefinition which shows that this is not an interacting theory and it is very close to Polonyi model?


title: Power generator powered exclusively by gravity is possible?

text: According to this site, it seems so. Power generator powered exclusively by gravity: ![enter image description here](http://i.stack.imgur.com/scWcR.jpg) ![enter image description here](http://i.stack.imgur.com/L7ObN.jpg) ![enter image description here](http://i.stack.imgur.com/GIMxY.jpg) ![enter image description here](http://i.stack.imgur.com/nw1M2.jpg)


title: Lookback Time & Age of the Universe Calculations

text: I try to calculate the age of the universe with the FLRW model: $$ H(a) = H_0 \sqrt{\Omega_{\mathrm{R},0} \left(\frac{a_0}{a}\right)^4 + \Omega_{\mathrm{M},0} \left(\frac{a_0}{a}\right)^3 + (1-\Omega_{\mathrm{T},0}) \left(\frac{a_0}{a}\right)^2 + \Omega_{\Lambda,0}}. $$ I set $\Omega_{\mathrm{M},0} = 0.317$ (matter density) and $\Omega_{\Lambda,0} = 0.683$ (dark energy), as delivered by Planck 2013; $\Omega_{\mathrm{T},0} = 1.02$ (space curvature), according to this site; and $\Omega_{\mathrm{R},0} = 4.8\times10^{-5}$ (radiation density), according to this document. For the time $t(a)$ I take the scale factor $a$ and divide it through the integrated recessional velocity $$ t(a) = \frac{a}{\int_0^a{H(a')a'\ \mathrm{d}a'}/(a-0)} $$ and finally simplify to $$ t(a) = \frac{a^2}{\int_0^a{H(a')a'\ \mathrm{d}a'}}. $$ But the problem is, I then get about $8\times10^9$ years for the age of the universe, but it should be around $12\times10^9$ years (which I get when I set $\Omega_{\mathrm{R},0}$ to zero): $\Omega_{\mathrm{R},0} = 4.8\times10^{-5}$: ![plot for first value of radiation content](http://i.stack.imgur.com/32C94.png) $\Omega_{\mathrm{R},0} = 0 \to 0.00001$: ![plots for changing value of radiation content](http://i.stack.imgur.com/UqXId.gif) Do I have to use some other models than FLRW/ΛCDM, or is one of my parameters outdated?


title: Is there a momentum for charge?

text: Since mass and charge behave similarly, so, just like center of mass, I define a point center of charge, that is defined by $$\vec r_{qm} = \frac {\sum{q_i \vec r_i}} {\sum{q_i}}$$ where $\vec r_i$ is a position vector w.r.t. the origin. Now suppose just like momentum,there is a quantity we call _charge-momentum_ $q \vec v$ for a system which is changed iff there is an external quantity denoted by $q \vec a$ where $\vec a= \frac{d\vec v}{dt}$ . Let us name this new quantity $ q\vec a$ as _charge-force_. Now I conjecture, for every _charge force_ , there is an equal and opposite _charge force_. And that charge-momentum of an isolated system will be conserved. Just like momentum, can this be a conserved quantity of the universe? Does this make any sense ?


title: maybe the universe has already 'ended'?

text: I aks you if this reasoning has a base in what we know of universe. There are many articles about so called "vacuum metastability event". As I understand this happens (can happen) with an enormous low probability in every place of the universe. A bubble is produced that then expands with speed of light 'eating' all matter without this matter knowing what is about to happen. (Because we can't have any data in advance from phenomena that are coming towards us with c). And I understand it like this: Probability of such event can be attribute of the space itself, like every Planck frame has a very tiny probability of going in different vacuum state like in radioactive decay but here is a difference that it once changes has the ability to 'ignite' or 'infect' its neighbors and propagate indefinitely. Or it may be some rare event than could happen only a few times in the history of universe like very energetic collision of particles. But here is a key thing : universe is VAST. Even very tiny probability on universe volumes becomes certainty. So maybe we live in a universe where some number of this bubbles has been created and are coming towards us. So universe (if we could see faster than light) looks like a growing swiss cheese with bubbles constantly created and growing). And we are simply luck to be in place not yet engulfed by it. An if we were able to get some faster than light information (tachyons ?) we could pinpoint our (that is in our frame, the Earth... of course destroying of whole Solar System would take hours of course) demise to a certain definite time in the future like 1 242 137 331 years + days + seconds.


title: Variable area manometer duct

text: How does one measure the pressure at the end of the manometer tube when the manometer tube itself is having a variable area? ![manometer with variable area tube](http://i.stack.imgur.com/ZMAcK.gif)


title: Does Coulomb's Law, with Gauss's Law, imply the existence of only three spatial dimensions?

text: Coulomb's Law states that the fall-off of the strength of the electrostatic force is inversely proportional to the distance squared of the charges. Gauss's law implies that a the total flux through a surface completely enclosing a charge is proportional to the total amount of charge. If we imagine a two-dimensional world of people who knew Gauss's law, they would imagine a surface completely enclosing a charge as a flat circle around the charge. Integrating the flux, they would find that the electrostatic force _should_ be inversely proportional to the distance of the charges, if Gauss's law were true in a two-dimensional world. However, if they observed a 1/r^2 fall-off, this implies a two-dimensional world is not all there is. Is this argument correct? Does the 1/r^2 fall-off imply that there are only three spatial dimensions we live in? I want to make sure this is right before I tell this to my friends and they laugh at me.


title: Why are so many forces explainable using inverse squares when space is three dimensional?

text: It seems paradoxical that the strength of so many phenomena (Newtonian gravity, Coulomb force) are calculable by the **inverse square** of distance. However, since volume is determined by **three** dimensions and presumably these phenomena have to travel through all three, how is it possible that their strengths are governed by the inverse of the distance **squared**? The gravitational force and intensity of light is merely **4 times** weaker at **2 times** the distance, but the volume of a sphere between the two is **8 times larger**. Since presumably these phenomena would affect all objects in a spherical shell surrounding the source with equal intensity, they travel in all three dimensions. How come these laws do not obey an **inverse-cube** relationship while traveling through space?


title: can one introduce magnetic monopoles without Dirac strings?

text: To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization of charge. Okay, it looks great. But is this the only way to introduce magnetic monopoles?


title: Energy band diagram of a system of Silicon Quantum dots

text: Suppose that we have a system of Silicon nanoparticles embedded in ZnO dielectric matrix. i'm thinking about how to construct the energy band structure of this system , suppose that we already have all the values of electron affinity, work function and energy bandgap of c-Si and ZnO. So i will align the Fermi level of c-Si and ZnO first, and repeat the process to get a periodic pattern for the energy band of the whole system. c-Si and ZnO used in this case are all intrinsic so there are little carriers in the system at equilibrium.   My question is will there be a field at the interface of c-Si and ZnO in the energy band structure like in the case of p-n junction ? In p-n junction there is diffusion of minor carriers (e into p side and hole into n side) which results in the recombination of e and holes at the interface and thus there is a depletion region with a lot of charged ions, so there is a field at the interface of p-n junction. In c-Si and ZnO the situation is different, there's no charged ions at the interface and thus no field. But i'm quite confused, should the energy band be abrupt at the interface, like in metal-metal contact ? I am trying drawing the detail of the energy band diagram, there is a lot of things i don't know about this, so please give me some ideas on what i should take note of ? (band offset, quantum effects etc. )


title: A fly in an accelerating car

text: A fly is flying around in a car, the fly never touches any surface in the car only fly’s around in the air inside the car. The car accelerates. does the fly slam in to the rear window. or does the fly continue to fly uninterrupted?


title: Mosquito in a Bus

text: My Question goes like this: If a mosquito inside the bus wants to move from back to front, will there be any difference if the bus is stationary or moving (ie extra effort in the latter case?)? Consider it as a closed bus & mosquito at a fixed point in bus, ie the air column will be constant throughout the process, & the mosquito will always be in that air column & never touch any part of the bus. Here's another question which has the similar answer as above question : Suppose the bus is at rest, & mosquito is also at rest (wrt to both ground & bus) and is in the air column, & not touching any part of bus. If bus starts moving, will the mosquito stay at rest wrt ground, ie go to back, or will it stay at rest wrt bus? For both questions, consider that when bus starts,it instaneously starts moving in constant velocity.(no acceleration) Thanks in advance to everyone who tries to answer it.


title: Butterfly inside the accelerated train

text: Suppose there is a train moving right with constant acceleration and there is a butterfly inside the train which moves opposite to direction of motion of train i.e. towards left. The butterfly always moves with constant velocity outside the train. Does the butterfly retard or accelerate inside the train due to acceleration of train or does it move with constant velocity easily? Please explain.