title: What is the mathematical justification for the quadratic approximation to the energy of a spring in a one-dimensional lattice?

text: It follows easily from this draw, the length $l$ of this spring as a function of the vertical distance $x$, as $l(x)=\sqrt{1+x^{2}}$ ![enter image description here](http://i.stack.imgur.com/Cw2XT.gif) Now, $l$ can be expressed as a MacLaurin expansion: $$l(x) = 1+\frac{1}{2}x^{2}+\frac{-1}{8}x^{4}+\frac{1}{16}x^6+...$$ Now, _under the assumption that the natural length of the spring is equal to 1_ (which happens for $x=0$) we can write the variation in the length of the spring as a function of $x$, provided that $x$ does small oscillations around $x=0$, as: $$\Delta l = l - 1 \approx \frac{1}{2}x^{2}$$ There is no possible lower order approximation to $\Delta l$ (except saying that $\Delta l \approx 0$ which is of no use) because there is no first order term in the expansion. And so, I don't understand the justification that in the minute 46:30 of this lecture Prof. Susskind states that the longitude of a similarly arranged spring (belonging to a linear disposition of mass points connected by springs) is proportional to $x$ (he equivalently uses $q_{i+1} - q_{i}$ instead) ![enter image description here](http://i.stack.imgur.com/w9MdK.png) By the same reasons, I think that the potential energy of the spring $U = \frac{1}{2}k(\Delta l)^{2}$ for small oscillations of $x$ should be: $$U = \frac{1}{2}k(\frac{1}{2}x^2)^{2}$$ $$=\frac{1}{8}kx^4$$ You may or may not like a four order term in the potential energy but as mentioned, because of the absence of a lower order term in the expansion above, it is either that or nothing at all. But then, consequently with the statement of minute 46, the potential energy he derives (only vertical displacements are allowed) is quadratic and not of order 4 ![enter image description here](http://i.stack.imgur.com/7vlDm.png) This is of course not a mistake of Prof. Susskind but a well-known approximation and quite elementary, bread and butter for condensed matter physicists. But, **can anybody justify the validity of that quadratic approximation?** (needless to say, trying an expansion of the polynomial $x^4$ gives as a result... $x^4$ itself) I understand that such approximations always are (or at least can be justified as) introduced by neglecting terms in an expansion. But this one seems to me arbitrary.


title: Why is Gravitational force proportional to the masses?.

text: We know that two mass particles attract each other with a force $$F~=~\frac{G M_1 M_2}{r^2}.$$ But what is the reason behind that? Why does this happen?


title: Induced charge on sphere

text: I have a conducting sphere ($radius = a$) at potential $V_0$. It is enclosed by another thin shell ($radius = b, b > a$) which has a charge density $\sigma (\theta) = \sigma_0 \cos(\theta)$ for the polar angle. I'm supposed to find the potential everywhere, noting the field discontinuities at $r = a$ and $r = b$ (this probably involves Poisson's equation), as well as the surface charge density on the inner conducting sphere. So inside the conductor (r less than a), the potential should be constant. Additionally, integrating $\sigma (\theta)$ over all $\theta$ gives me $0$ total charge (very positive at the top of the sphere and negative at the bottom of the sphere), so the potential for $r > b$ should be constant as well because there is zero total charge inside (using Gauss's law with $Q_{enclosed} = 0$). I'm really stuck how to do the potential in the in-between shell, as well as finding the surface charge densities (though once I know the former, the latter mat be doable via the discontinuity of the field $E_{in} - E_{out} = \sigma / 2$). Any ideas?


title: How do forces work

text: Is there a mechanistic-type explanation for how forces work? For example, two electrons repel each other. How does that happen? Other than saying that there are force fields that exert forces, how does the electromagnetic force accomplish its effects. What is the interface/link/connection between the force (field) and the objects on which it acts. Or is all we can say is that it just happens: it's a physics primitive? A similar question was asked here, but I'd like something more intuitive if possible.


title: Gas Cylinder Life

text: Say I have a gas cylinder filled with nitrogen to some volume at some pressure. On that cylinder I have a regulator so that I get a steady pressure out. I wish to discharge the cylinder. How many hours until that cylinder is exhausted? I would assume the 'new volume' would be Vl=(Ph*Vh)/Pl. However, I don't understand how the flow rate is determined.


title: Conjugate Transpose of Hamiltonian Matrix

text: I read some notes saying, $$i\hbar \frac{dC_{i}(t)}{dt} = \sum_{j}^{} H_{ij}(t)C_{j}(t)\tag{1}$$ where $C_{i}(t) = \langle i|\psi(t)\rangle$ and $H_{ij}$ is hamiltonian matrix. However, what is obscure to me is the way to deduce $$H^*_{ij} = H_{ji}\tag{2}$$ in the notes. They explain (2) as > (2) follows from the condition that the total probability that the system is > in some state does not change. If you start with a particle then you have > still got it as time goes on. The total probability of finding it somewhere > is > > $$\sum_{i}^{}\left|C_{i}^2 \right|$$ > > which must not vary with time. If this is to be true for any starting > condition $\phi$, then (2) must also be true. I cannot understand the relationship between the total amplitude and (2). Well.. though I omit other contents/derivative of (1) in the notes to simplify my question, it will be very helpful for me if you figure out the explanation of the notes above.


title: Relativity of simultaneity - An example

text: I am trying to understand the relativity of simultaneity in different frames, and I am trying to work out an example. Suppose along the x-axis there are two points 2000m apart. Event A happens at t=0 and event B happens at $t=2\cdot10^{-6}s$ after event A happens in the rest frame. If there is an observer moving along the positive x-axis, how fast must he move in order to see the two events happen simultaneously? So a diagram might look like this: `A------------------------------B`   `Observer ->` My understanding is that the observer should "see" the two events simultaneously if the information reaches him at the same time. Thus, if he stands somewhere between A and B such that after A happens, the information of A and B happening reach him at the same time, then he would think that A and B are simultaneous events. If my logic is correct, then he should stand somewhere to the right of the middle point between A and B. But the example is asking for the velocity of the observer. I am not sure how to do this since I think the initial position of the observer should matter as well (if it does matter, can we assume he starts at the left point?).


title: Does a photon have a north and south pole?

text: A photon has an oscillating magnetic and electric field. Is the magnetic field a dipole?


title: Average Neighbouring Impurity Separation in a Random 1D chain

text: ![](http://i.imgur.com/MwtfT.jpg) I have a finite and discrete 1D chain (edit: linear chain, i.e. a straight line) of atoms, with unit separation, with a set number of impurities randomly distributed in the place of these atoms in the system. What I would like to do is describe the separation between neighbouring impurities (call it "D" which will always be an integer) statistically, and also to work out the average separation . For example, the plot below was calculated from several thousand simulations of a chain of length 200 atoms and 10 impurities where the y-axis is the probability $P(D)$ of finding an impurity at distance D, and the x-axis is nearest impurity distance $D$. It kind of looks like a Poisson distributon, which one would expect since the system is discrete and random and a kind of counting exercise, but it doesn't work to well as a fit to the data points. It has been a long time since I did any statistics so I'm not sure how to start expressing what I found mathematically. Since I know the system length ($L = 200$) and the number of impurities ($N_i$) is a fair starting point the impurity density $\rho = N_i/L$ ? _EDIT: The chain isn't allowed to self-intersect, it's a straight line in each case. The system I'm using above is a straight line of 200 evenly spaced atoms, and I'm distributing 10 impurities in the place of random atoms (e.g. at sites 4, 11, 54,...so there are still discrete steps between sites). The graph above is the result of finding the spacings between these impurity sites._ _EDIT 2: Attached a picture at the top_ ![](http://i.imgur.com/Rqgn6.jpg) EDIT 3: Okay so it seems it could be my PRNG code causing problems. I'm using Fortran 95, here is the code:               `CALL RANDOM_SEED(size = n)     ALLOCATE(seed(n))          CALL SYSTEM_CLOCK(COUNT=clock) !!! intentionally slows it down to prevent succesive calls from returning the same number          do i = 1 , 1000000          end do     seed = clock + 37 * (/ (i - 1, i = 1, n) /)     CALL RANDOM_SEED(PUT = seed)          call random_number(x)`      EDIT 4!: Repeated this for a system size of 50 with 3 impurities (10,000 iterations), fitted a geometric distribution to it, as one can see immediately there is a huge variation. Is this explained by the fact for small n and L when we go from our first impurity (at site j) the probabilities to find the next impurity change drastically since effectively we are now looking for a system with n=2 and L= 50-j, which would be really sensitive to the location of the first impurity hence the huge variation. When we go to a huge system we can essentially treat the remainder of the chain as still being really long, and n and L are pretty much unchanged so this effect is masked and we see a geometric distribution arise? EDIT 5: Just did a bit of reading around Bernoulli Trials, is what I'm seeing and describing a binomial distribution hence the peak I see with smaller systems? Still that wouldn't explain why my 200 system is different from Ilmari's below... ![](http://i.imgur.com/ybsJa.jpg)


title: How to properly read a measurement result if it is a number?

text: If the result of a measurement is i.e. $3.2 \pm 0.7$, what is 0.7? At which confidence level we know that the real result is inside of this interval?


title: What is the work done below proportional to?

text: A raindrop falls from a certain height to the ground, with a terminal velocity u. The viscous force is F = 6*3.14*n*r*v, where n is the coefficient of viscosity, r is the radius of the raindrop, and v is the speed. Then, the work done by all the forces acting on the raindrop till it reaches the ground will be proportional to   1. r^7   2. r^2   3. r^5   4. r^3 ?


title: What meaning does the slope of the efficiency path on a Mollier diagram have in terms of temperature?

text: Let's say I have a steam turbine that I have modified to increase its isentropic efficiency. As a specific case, consider the modification outlined in the Mollier diagram below. The arrows represent the before and after modification paths of the turbine on the Mollier diagram. The slope of the arrows represent how efficient the turbine is. A horizontal slope is a throttling process whereas a vertical slope is 100% isentropic efficiency. Slopes in between obviously are something less than 100%. The slope is $\Delta{H} \over \Delta{S}$ which has units of temperature (degR). Does the fact that the units of the slope are temperature hold any special meaning? Why would the value of a temperature inherently represent a performance level of isentropic efficiency? Does the fact that the values are always negative mean anything? Is the reason 100% efficiency is impossible related to the fact that the value of the temperature is $-\infty$? ![turbine mod](http://i.stack.imgur.com/hO5bN.jpg)


title: Ampere's law and external currents

text: In Ampere's law the current outside the curve taken is not included in the expression. Does this mean that only the currents crossing the area bounded by the curve taken contribute to the magnetic field one calculates? P.S. I am new to stack exchange. I posted similar question earlier but it was put on hold as it was unclear. Hope that this one is clearer.


title: What is the intuition behind the Lorentz factor from Special Relativity

text: Time Dilation = $1/\sqrt{ 1-v^2/c^2}$ but why? How do you get to that conclusion? I know you use Pythagoras'theorem and my current understand goes like this: $$vt^2+ct^2=cT^2$$ then you take the square root of $Ct^2$ and from there I think you divide by $c$ to get the time dilation but I'm not sure. And if this is correct what steps do you take to get to the formula used now


title: Finding the correct units for the energy-momentum tensor?

text: I'm trying to understand the energy-momentum tensor $T^{\mu\nu}$ but I'm confused about the units. My textbook says the components of $T^{\mu\nu}$ are $\mathrm{Jm^{-3}}$. Four-momentum is is given by$$P^{\mu}=\left(E/c,\mathbf{p}\right)=\left(E/c,p_{x},p_{y},p_{z}\right)$$ The $E/c$ component of $P^{\mu}$ has units $\mathrm{Jsm^{-1}}$. The definition of $T^{\mu\nu}$ is “the rate of flow of the $\mu$ component of four-momentum across a surface of constant $\nu$.” Using this definition, how do you get the rate of flow of the $E/c$ component of four-momentum across a surface of constant time (ie the $T^{00}$ component) to have the correct units of $\mathrm{Jm^{-3}}$? Surely you need to multiply $E/c$ by something with $\mathrm{s^{-1}m^{-2}}$ units, but what exactly? Thank you


title: Why is there a space between the flame and the nozzle on the space shuttle?

text: ![Space shuttle nozzle flame](http://i.stack.imgur.com/j4LQ6.jpg) Why is there some space between the flame and the nozzle on the space shuttle? (see above picture)


title: Determine the electric field

text: A loop of wire is put in a changing magnetic field. The magnetic flux through the loop is given by $4t(t+2)$. The loop is connected to a parallel plate capacitor that has a plate separation of 15mm. Determine the electic field between the plates at time $t=3$ s. Can anyone go through the relevent equations for this question? Thanks


title: What determines the rate at which an item will freeze in the given context?

text: Assume you have two pieces of meat in two rooms and all factors are constant e.g. meat volume, meat temp, room size and so on. The only difference is that the temp in one room is -5c and the other is -20c. Obviously the meat in the -20c room will freeze faster because the air will chill the meat faster. Therefore the rate of freezing in this case is dependant on the temperature of the room. Now, in the context of a freezer(which is similar to a chilled room) I understand that 'the rate of cooling is determined by how fast the air inside the freezer can extract the heat from the new (warmer) food'. Correct me if I'm wrong, but a freezer does this by pumping out cool air constantly until the set temperature is reached in the freezer. It is the increasingly cool air/or temperature of the air inside the freezer which determines how fast the item is cooling. Therefore besides the fact that the air temp is fluctuating, it is the same as the above example I gave. Am I correct? 1) In other words although using a bigger load, hotter item may take longer, in reality it is the cold air in the freezer(temperature) reached that will determine how fast the item freezes? Now someone told me that it is not the temperature rather the 'cooler capacity and load' that determines the rate of pull down\freezing. 2) I take it a bigger load means it will bring the temp down/or less cool air will circulate however doesn't this just mean that other factors are being changed which will change the temperature in the freezer to decrease. Even then doesn't this mean it is the new lower temperature that freezes the food and hence the freeze rate or is there another factor I am missing? 3) Now if the freezer cooler capacity says it can freezer 2kg of food in 12 hours what does that mean exactly? Does it mean it will bring the temp of less the 2kg of food to the set temp e.g. -18c in 12 hours? Most freezers have a minimum setting of -18c. Assume we lowered the same freezer to -5c, does this mean it will feeze the food completely up to -5c in 12 hours or sooner? I know at -5c the all water will not freeze as only higher temps can do that. Really what I am wondering is, imagine you set a freezer to -18c and it took 12 hours to bring that food to -18c and during this time the food(not freezer) only spent 1 hours going from -2c to -5c. If we now lower the temp to -5c would the item get to -5c within an hour(since that what it spends on the higher setting), or would only the temp in the freezer get to -5c in 1 hours and not the food? Would the -5c case cause the food to freeze much slower and take much more time, perhaps hours to get to -5c or would it be as usual. Hope you understand what I am saying. 4) Do you expect that if a 1kg of meat is frozen at -3c it will take a long time to freeze in a freezer which has a capacity of 2kg and 12 hours? Would it take many days to get to that temp? Ideally I want an answer in layman terms. Many thanks.


title: If an Electrical Field can jump over a point on its stright path of propagation?

text: Consider point B between points A and C on a stright line in vaccum(or any other environment). If the electrical fild $\vec E$ (or an EM wave) should necessarily pass through B to affect C and appear there on C? I mean if on a path of a propagating field we should always have a continuous set of points which the mentioned field affect and pass through them? Why/Why not?


title: Mercury's orbit

text: Mercury has a weird orbit, it's elliptical orbit rotates around an axis, but for no reason. My question is why? What is causing this?


title: Air resistance of a general shape object

text: I'm trying to code a simple physics simulator in C++ but I'm stuck on the air resistance issue. Surfing the Internet I only find a drag coefficient for very basical area shapes when the air is perpendicular to the area. However, even when simulating a falling parallelepiped, I found this not sufficient to describe the motion of the falling body, as the air is not always perpendicular to each face. More precisely, I'm able to calculate the air drag on each face of the object, obtaining a 3D vector which describes the force acting on the body. However it's obvious that when such an object falls through the air, even an angular momentum is induced to the body. I'm not able to determine it. I was thinking about describing the air flow as a matrix of parallels vectors, thus obtaining a resultant force as a weighted sum of the effect of each vector hitting the body. However in this case I'm not able to determine the force of each single wind vector hitting the surfaces of the body. What is the simplest way to obtain object rotation due to air resistance? And what is the finest way?


title: How do I find the tension in the cable from this problem?

text: ![enter image description here](http://i.stack.imgur.com/wMTwy.png) I am trying to find the tension of the cable but I don't know what to do.


title: Please explain these equations relating to Newtonian gravity

text: Here is an extract from some badly-written slides:   * A planet in an orbit with radius $R$ having a period $T$ falls through a distance of $R$ in a time $T/4$.   * A planet has a speed of $R/T$ and an acceleration of $R/T^2$.   * Comparing gravity on Earth $g$ with the moon $g_m$ at a distance $R_m=4\times10^8m$: $g/g_m=9.81T^2/(16R_m)\approx10^4$.   * Compare this with $(R_e/R_m)^2\approx5\times10^3$.   * Newtown abandoned the inverse square law for a time because this test failed. The problem was the incorrect value for $R_m$. Can someone please explain where $T/4$ comes from in the first line, how the formula in the third line is constructed, and what the comparison is supposed to show? What is the person who wrote these slides actually trying to say?


title: When mutual inductance is occurring between two coils, is self inductance always occurring in each individual coil?

text: When a coil connected to an AC generator creates an EMF in another nearby coil (mutual inductance), is self inductance simultaneously occurring in both coils?


title: How do we measure a quantum system properties? (length, mass and time)

text: How do we measure a quantum particle properties? (length, mass and time)


title: Regarding space launch hazard and near earth debris

text: I have saw this picture in wikipedia ![enter image description here](http://i.stack.imgur.com/ofR4p.jpg) and I begin to wonder why doesn't current space launches take consideration of avoiding space debris, as say you want to sent a robotic mission to mars, during the launching phase do you have to calculate your trajectory to avoid colliding your spaceship with near earth debris, or is this collision not a concerning issue, in addition why don't scientist seems to be worry about deep space collision with small rocks, it seems to me that if a spaceship travels to mars there seems to be a significant probability of colliding with small rocks?


title: Should normalisation factor in a QM always be positive?

text: I have a fairly simple question about a normalisation factor. After normalising a wavefunction for a particle in an infinite square well on an interval $-L/2<x<L/2$ I got a quadratic equation for a normalisation factor $A_0$ which has a solution like this: $$A_0 = - \frac{8\pi}{6} \pm \sqrt{\frac{8^2\pi^2}{36}+\frac{3}{4}}$$ The sign $\pm$ gives me two options while a normalisation factor can only be one value. So I want to know if there are any criteria on which i could decide which value to choose. Is it possible that normalisation factor should always be a positive value? * * * I added the procedure I used to derive this eq. : ![enter image description here](http://i.stack.imgur.com/eqq5T.jpg)


title: How does laser induced fluorescence in excited atoms work?

text: Atomic clocks use laser induced fluorescence in order to detect excited atoms, how does this work? Apparently the clocks need to detect the states of nearly 100% of the atoms being examined. In an atomic fountain clock, a gas of Cesium atoms is excited and then a probe laser causes the altered atoms to fluoresce. I don't know how this would work or how they could be so accurate in knowing how many atoms were excited.


title: Modelling multiple heat sources?

text: I am working on a project to model heat transfer in a data center. I have basic models for a one dimensional heat transfer using conduction and convection. Such as, taking a server and modelling the temperature decrease over time and models taking distance into account based Fourier's equations. However I am now looking to step the model into 2 dimensions and this involves considering a stack of servers with multiple heat sources stacked on each other (such as a server rack). I am struggling to find any useful journals that detail the interactions of multiple heat sources, would people mind pointing me in the direction of some useful sources or explain the basics to me so I can research and expand on them?


title: Ephemeris calculations: Light time correction of the moon

text: I am currently trying to calculate apparent positions from raw JPL data.   I've got it pretty much figured out, but there is one thing that's bugging me: Has the light time correction of the moon to be done heliocentric or geocentric? Light time correction is needed because the speed of light is finite and therefore the bodies of the solar system are not seen where they are, but where they have been before (See: Has everything we see happened in the past?) A simple algorithm (discarding light retardation due to the sun) for the light time correction works like this:   1. Calculate barycentric cartesian positions of earth and planet at time $T$   2. Calculate distance $s$ and light time $\Delta T = \frac s c$   3. Calculate position of planet at time $T' = T - \Delta T$   4. Repeat steps 2 & 3 until $T'$ doesn't change significantly. Note: The movement of the solar system is not taken into account because the calculation is done with the solar system as the inertial frame of reference. If I calculate the same thing for earths moon, do I have to take the movement of the system earth-moon into account; Do I have to use a heliocentric frame of reference?   If not, the movement of the moon is a (well, actually not so) simple orbit around the earth that keeps the same radius - more or less.   But if so, the movement of the moon is a combination of moon orbit and earth orbit and calculating the position at time $T'$ therefore changes the distance of earth and moon. The "Explanatory Supplement to the Astronomical Almanac" does not mention any special handling of the moon so I assume one has indeed to take the movement of the system earth-moon into account when calculating moons position at time $T'$ and do the calculation in a heliocentric frame of reference. But why can't I do these calculations with "earth-moon" as the inertial frame of reference (please excuse if that's wrong nomenclature) instead of the solar system? **Edit**   After Seans post I have noticed I was imprecise.   I know different frames of reference will give different results and as JPLs data is fitted to observations, so obviously one has to use exactly the calculations that JPL used to derive the planetary ephemerides. To explain my question further, I will assume Moon is trying to send a light ray to Earth (center of Earth, to be precise):   If Earth is moving, Moon has to adjust the direction of the light ray so it will hit a moving target.   If Earth is not moving, Moon can send the light ray directly downwards from its orbit. So, obviously, Earth is moving though the solar system and Moon has to hit a moving target.   **But** , the solar system itself is moving, too. Every light ray in the solar system has to hit a moving target. This is not taken into account. I think it all boils down to this:   Without gravitational influences, does every light ray move in a straight line with the speed of light $c$ in every valid frame of reference?   Wikipedia seems to say so. If so, why is the moving solar system a valid frame of reference (so the movement of light can be calculated as written in the inital question) for earth and moon while earth-moon is not? **Edit 2** According to Sean, a frame of reference with the barycenter of earth+moon would be valid. But the results of both ways of calculation differ. This is because the position of earth at time $T'$ is never calculated, so in the solar system, the position of moon at time $T'$ relative to the position of earth at time $T$ changes with two components: Movement of the moon on its orbit around the earth + movement of the barycenter of earth and moon on its orbit around the sun. Calculated in the frame of reference of earth+moon however, the movement of moon only consists of its orbit around the earth. Therefore, there is a difference in calculation. This leads me to my confusion: Only one frame of reference can be valid (different results for the apparent position of moon) while it seems as if both frames of reference should be valid. **Edit 3**   Why is it wrong to define the movement of the moon as an orbit around earth (and neglecting the orbit of earth+moon around the sun), when calculating apparent positions for an observer on earth? **Edit 4**   According to Sean, it is not. Strange. I assume that the Astronomical Almanac is confirmed through observations and all other results therefore must be wrong.   The difference is 0" to 20".   Sadly I could not find any exact observational data of moon - its apparent diameter is ~30', so they would have to be exact to the level of 1% of its disc.   It's still a riddle to me.   If Sean is correct, the Explanatory Supplement is wrong, as are practically all other ephemerides. Which leads me to the suspicion there might be more to it than Sean (and I) think.


title: "A Crash Course in Lagrangian Dynamics". Is it still available online?

text: In an Amazon review of "Schaum's Outline of Lagrangian Dynamics" I found this: > _I recommend that you type"Lagrangian Dynamics" into Google and look at some > of the excellent sets of lecture notes available on-line. The online > tutorial "A Crash Course in Lagrangian Dynamics" is particularly helpful. At > only 18 pages it gets to the heart of the matter and contains some solved > numerical examples._ This intrigues me, but following his advice doesn't give the tutorial, save for a couple of incredibly dodgy looking download sites. Anyone got any ideas on how I can get hold of it?


title: Understanding action reaction in an example

text: When I move an object that object should move me as well. I tried standing on a skateboard to reduce friction, and holding a heavy barbell, move myself away from it (while still holding it) but I haven't noticed any movements of a skateboard. Am I missing something here?


title: Penetration and skin depth in 'metallic' magnetic induction tomography (classical electromagnetism)

text: Does anyone know how AC magnetic fields behave in 'metallic' magnetic induction tomography (MIT). Do they form a magnetic circuit around a metallic specimen or penetrate through it, or both? This would be in respect of the skin effect limiting the penetration of the AC magnetic driving field (primary). MIT works by an AC magnetic driving field inducing eddy currents in a metallic specimen (to be imaged) and sensors (coils) on the other side of the specimen picking up the secondary magnetic field due to the eddy currents induced in it. It actually senses or picks up the total field (secondary and primary magnetic fields). Sensor coils measure the spatially distributed total fields to be tomographically represented as an image.


title: Is 'amp' a technically invalid term?

text: I've been told to use the term ampere in exams and class (I'm in high school), instead of amp as it's not a valid unit, although I've been using the amp for years along with all of my friends who do electronics. Is this technically invalid or acceptable as far as the scientific consensus goes, and what would you think if you saw it used in a scientific paper?


title: Dissipating light pollution

text: If all the lights in a city, and the area surrounding it were shut off. How long would it take for the light pollution to clear, so the light from stars could be seen?


title: Synchronisation of clocks

text: How can two clocks be synchronised with each other at some instant without being at the same place and same time $?$ considering that simultaneity is a relative concept .


title: Fixed lever arm spinning under gravity, why am I getting these results?

text: Suppose there is a lever arm of length $L$, a mass $m$, and it is fixed at one end. The lever is parallel to the ground. So the force acting on the center of mass of the lever would be $mg$. Now let's look at the moment when the lever is released. Equating the equations for torque, we have $Fr=I \alpha$, so in this case $\dfrac {mgL}{2}=\dfrac {ML^2 \alpha}{3}$ , or $\alpha=\dfrac {3g}{2L}$ . To find the tangential acceleration at the center of mass, we can use the formula $a=r \alpha$, so in this case $a= \dfrac {3g}{4}$, so $ma=\dfrac {3mg}{4}$. Why doesn't $ma$ equal $mg$?


What's an atomic superstate/superposition, and how is it possible? I understand the basics - being something can be moving and staying still at the same time; the observer changes the behaviour - but I'd like to know a little more, as it interests me, and how is it possible without actually interfering with said thing?


title: Is conformal time observable?

text: The standard FRW metric with cosmic time is $$ ds^2 = -dt^2 + a^2(t)(\gamma_{ij}dx^i dx^j),$$ and we can measure $t$ as the proper time for comoving observers. Thus it makes sense to talk about the age of the universe $t_0$ in terms of the coordinate $t$. When using conformal time, the metric becomes $$ds^2 = a^2(\eta)(-d\eta^2+\gamma_{ij}dx^idx^j).$$ Suppose I know the evolution of the scale factor $a(t)$ for all time, in fact lets just assume it's matter dominated, $a(t)\propto t^{2/3}$. Is there any way to make sense of the quantity $\eta_0$, the conformal time today? Is this quantity measurable? If not, and I compute an "observable" where $\eta_0$ appears explicitly, are there any ideas about what could have gone wrong?


title: Problem understanding a step in derivation of Lorentz Transformation

text: I need to understand a step in the derivation of Lorentz Transformation. ![taken from thecuriousastronomer.wordpress.com](http://i.stack.imgur.com/9dtdK.jpg) ![taken from thecuriousastronomer.wordpress.com](http://i.stack.imgur.com/uBQNm.png) I cannot understand how can we equate the equations 1 and 2. I am talking about the relation between ct, x, y, z. Where did it come from?


title: Information paradox and spacelike slices

text: I'm reading S. Mathur's paper on the information paradox and I can't seem to understand the reason why we choose spacelike slices. Is it because we want to have a global timelike vector so that we will be able to define the vacuum state "canonically"? If that's the case what whould happen in Kruskal coordinates? If the horizon is non-singular, then there whould be no need for connector. Where would the "streching" take place, and where would the particles be created? Does the paradox even exist in this case?


title: How far we can see in the thickest possible fog?

text: There is quite foggy day today here, I cannot see end of the street. This made me think about how much worse can it be. So here is my question: how thick a fog can be? What's the smallest visibility distance possible? By visibility distance I mean how far I can see a black object against a white background in the fog. UPDATE: There were some nitpicks, so let me clarify a bit:   * I think of water/ice fog.   * It's said that angular size matters, so let's say an object whose angular size as big as the full moon.    * Whiteout doesn't count, assume the soil remains dark.


Is supercritical CO$_2$ a newtonian fluid? I can't seem to find any articles describing the stress-strain relationship for carbon-dioxide in a supercritical state (high temperature/pressure combination). I know it's newtonian in its gaseous state and I presume that being in a supercritical state would not alter its stress-strain relationship, but I can't find any references stating one way or the other. Any help would be greatly appreciated! :)


title: Show that the circumference of the Bohr orbit of hydrogen is an integral multiple of the wavelength of the electron in it

text: We have an electron revolving in a the $n$th Bohr orbit of an atom of hydrogen. The problem is to prove that the circumference of the orbit is an integral multiple of wavelength of this electron $\lambda_e$. $$2\pi r_n = k\lambda_e, \ k \in \mathbb{N}$$ where $r_n$ is the radius of the $n$th Bohr orbit. I started out by figuring out by expanding out the different terms in the equation: The $n$th Bohr radius for hydrogen is given by: $$r_n = 5.29n^2 \rm{pm}$$ $$\implies 2\pi r_n = 10.58n^2\pi\tag1$$ The wavelength of an electron is given by: $$\lambda_e = \frac{h}{p_e}$$ The angular momentum of an electron is given by: $$p_e = k_1\hbar, \ k_1 \in \mathbb{N}$$ $$\implies \lambda_e = \frac{h}{k_1\hbar} = \frac{2\pi}{k_1}\tag2$$ How do I show that $(1)$ is an integral multiple of $(2)$?


title: FQH Edge Theory as decoupled chiral bosons

text: The action describing the edge theory of the Fractional Quantum Hall effect is given by \begin{equation} S = \frac{1}{4\pi} \int \mathrm{d}x \ \mathrm{d}t \left[ K_{IJ} \ \partial_{t}\phi_{RI} \partial_{x}\phi_{RJ} - V_{IJ} \partial_{x}\phi_{RI}\partial_{x}\phi_{RJ} \right] \end{equation} for scalar fields $\phi_{RI}$ with $I=1....dim(K)$ and some symmetric,invertible matrix K and some positive definite, symmetric "velocity matrix" V. The equations of motion for the fields read \begin{equation} \partial_{t}\partial_{x} K_{IJ}\phi_{RJ} - \partial_{x}^{2}V_{IJ}\phi_{RJ} = 0. \end{equation} This smells already a lot like the chiral boson theory in 1+1D Minkowski space which this action is supposed to be equivalent to. However I do not see the transformation that "decouples" the fields. Can anyone help? Thanks in advance for your responses!


title: CFTs and formalizing quantum field theory

text: Moshe's recent questions on formalizing quantum field theory and lattices as a definition of field theory remind me of something I occasionally idly wonder about, and maybe this site can tell me the answer. Are there any mathematicians working on defining quantum field theory by beginning with a rigorous definition of CFTs, and working from there? The reason I ask is that I think this is how most of us in physics think about quantum field theory (that is, in a Wilsonian way): to define a QFT, you start with a UV fixed point, and deform it with some relevant operator. So if you had a general theory of CFTs, you would know how to understand how CFTs respond to external sources for operators, and getting a more general QFT would "just" mean turning on a spatially homogeneous source for some operator and seeing how it responds. The other object we study is "effective field theory," which you might imagine in this language is a CFT that serves as the IR fixed point, together with some notion of equivalence class of irrelevant operators deforming "up" away from that point (being agnostic about whether you ever reach a UV fixed point). Very (extremely) naively, I would suspect mathematicians might have better luck studying the space of CFTs rather than trying to begin with all QFTs. And you might imagine this approach would be well-suited for questions physicists might care about (like, say, whether there is an " _a_ -theorem" or something similar, analogous to the _c_ -theorem in 2 dimensions, characterizing RG flows as irreversible). Axiomatic/algebraic/constructive field theory seems to worry about _all_ kinds of field theory at once, and other mathematicians seem to be trying to dig up interesting structure in perturbation theory, which I'm not sure will ever lead to progress in nonperturbatively understanding field theory. I know there are some mathematicians who work on CFTs. (I found this MathOverflow question that has a lot of links to work by mathematicians on CFTs, for instance.) But I wonder if any have tried to work on CFTs as a _route_ to understanding QFT more generally.


title: Space as "flat" plane

text: I was watching the documentary Carl Sagan did about gravity (I believe it's quite old though) and wondered about space being "flat" and that mass creates dents in this plane as shown at about 3 minutes in this clip (http://www.youtube.com/watch?v=Y-db4iC0aHw) Is this simply a metaphor or is it something more than that? Does gravity follow this just instead of being flat the dent is "rotated" in all directions?


title: Can we compute the magnitude of the stress caused by sound waves on a wall?

text: As a follow up to this question, Could we _really_ compute the magnitude of the stress caused by sound waves on a wall? If so, How do we do that?


title: Why is the answer to this diffusion example unintuitive?

text: Imagine a linear decrease in concentration from left to right. Using Fick's first law, $J = -D \frac{d \psi}{d x}$ for all x, from left to right, we have the same flux amount because the decrease is linear. So $J(x) = m$ According to Fick's second law, $\frac{d \psi}{d t} = -D \frac{d^2 \psi}{d x^2} = \frac{d J}{d x}$ So $\frac{d J(x)}{d x} = 0$ so dJ/dx is just 0 since the 2nd derivative of a line is 0. Yet this seems unintuitive. I would expect as long as there is a concentration gradient, there should be a change in concentration at each point until the concentration is completely uniform. There must be an error in my math or reasoning, where is it? EDIT: To clarify the boundary conditions, imagine a closed box with no out flow or in flow at the edges.


title: How do you find the velocity function of a mechanical wave?

text: With the form $y(x,t)=A\sin(kx-\omega t+\phi_0)$, there are two variables, How do I find the velocity? I don't know I can apply derivative with two variables.


title: Is there really such a thing as an irreversible process?

text: If an isolated system goes from a state A to B, will it always eventually fluctuate back to state A? If not, give an simple example. Is it right to say that entropy only says that the probability for reversal is very low, but not actually 0, ie that it will take a very long time for some processes to reverse themselves, but they will infact eventually happen with probability 100%? So will the universe eventually fluctuate back into yesterdays state? Or do we add some other physical law to make it irreversible?


title: Single trace partition function

text: I would be glad if someone can help me understand the argument in appendix B.1 and B.2 (page 76 to 80) of this paper. The argument in B.1 supposedly helps understand how the authors in that paper got from equation 3.6 to 3.8 on page 18 of the same paper. These 3 equations form the crux of the calculation done in this paper and unfortunately I am unable to see this. In the appendix B.2 they calculate certain rational functions which are completely mysterious to me. Like I can't understand what it means to say that $\frac{1}{1-x}$ is the partition function of the operator $\partial$. Similarly one can see such functions in the equations B.7, B.8 and B.10 of that paper. Curiously these polynomials had also appeared on page 6 and 7 in this paper long before the above paper. I am completely unable to understand how the series of polynomials between equation 15 to 20 of this paper were gotten and what they mean. I haven't seen any book ever discuss the methods being used here. I would be glad to hear of some pedagogic and expository references on the background of all this.


title: Shaking a jar of balls

text: A jar is filled with two types of balls, red and green. Red balls have radius $r_1$ and mass $m_1$, green balls have radius $r_2$ and mass $m_2$. If initially the balls are randomly placed throughout the jar, and we are to shake this jar, generally one type of balls will tend to gather at the top. Why does this happen? What are necessary conditions on $r_1$, $r_2$, $m_1$, $m_2$ such that green balls are more likely to occupy the upper part of the jar after shaking?


title: Is a periodic force capable of transporting a particle to large distances?

text: I have a particle at rest. At $t = 0$ a periodic force like $F_0 \sin\omega t$ starts acting on my particle. Can such a force transport my particle to infinity when $t \to \infty$? Please answer this question without solving the mechanical problem, just by intuition.


title: Trebuchet launch into space

text: I was reading some questions on your site about using a Trebuchet to launch a payload into space. Would it be possible to use the same sort of principle, but to have, say a 100 mile steel cable attached to the earth, or tower, on the other end have a rocket launching straight up, and use the rotation of the earth to help launch a craft into space. the cable would suffer at the fast end, but the space craft would only have to carry a fraction of the fuel needed at the moment. Is any of this possible?


title: Does a 4-current J determine a unique maxwell-faraday F tensor up to isometry?

text: Maxwell's equations on a pseudo-Riemannian manifold $(M,g_{ab})$ say, $$d_a F_{bc} = \nabla_{[a}F_{bc]} = 0,$$ $$\nabla_a F^{ab} = J^b,$$ where $d_a$ is the exterior derivative, $\nabla_a$ is the covariant derivative compatible with $g_{ab}$ (i.e. $\nabla_a g_{bc}=0$), and $F_{ab}$ is antisymmetric ($F_{ab}=-F_{ba}$). Let $J^a, F_{ab}$ and $\tilde{J}_a, \tilde{F}_{ab}$ be two solutions to Maxwell's equations in this form. Suppose that $J^a$ and $\tilde{J}^a$ are equivalent up to isometry, in that $\varphi_*J^a = \tilde{J}^a$ for some isometry $\varphi:M\rightarrow M$. **Question** : Does it follow that $F_{ab}$ and $\tilde{F}_{ab}$ are equivalent up to isometry, in that $\varphi_*F_{ab} = \tilde{F}_{ab}$? Wald's _General Relativity_ (1984, Chapter 10 Problem 2) shows a sense in which the answer is yes in the source-free case ($J^a=\mathbf{0}$). I wonder what is known about the non-source-free case.


title: The inverse square law of sound through solids?

text: We all know about the inverse square law of sound. In short the power of the wave will get evenly spread on an ever increasing spherical expansion and this will dissipate the power of the wave at a rate of inverse squared to the distance travelled. ![enter image description here](http://i.stack.imgur.com/ETY90.gif) But what about sound waves travelling through a solid like a rectangular table? Is there such a law that relates the power of a signal to the distance travelled? Even a approximation of the relationship would help; is this an inverse square relations as well? ![enter image description here](http://i.stack.imgur.com/uapig.jpg)


title: Gravity on supermassive black hole's event horizon

text: M = black hole mass Gravitation is about 1/r^2 Schwarzschild radius r is ~ to M Greater BH -> weaker gravitation on its horizon. Lets take black hole so enormous that the gravitation on its horizon is negligible. Person A is 1 meter 'outside' the horizon, person B is inside (1 meter from the horizon aswell) person B throws a ball to person A. Both just started accelerating towards BH very slowly, so why person A wont catch the ball? Why A wont even ever see person B granted A will somehow escape later on? EDIT: http://en.wikipedia.org/wiki/Schwarzschild_radius EDIT2: http://mathpages.com/rr/s7-03/7-03.htm


title: Random Hankel matrix and eigenvalues distribution

text: I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.


title: Why the involution condition is imposed in the definition of integrability?

text: For an $N$-degree-of-freedom system to be integrable, the usual definition imposes the existence of $N$ independent conserved quantities, which must be in involution to each other, i.e., $$\\{ F_i, F_j \\} ~=~ 0\tag{1}$$ Why this extra condition? Why is $$\\{ H, F_i \\}~=~0 \tag{2} $$ insufficient?


title: How to derive this formula?

text: as far as i know the range of an obliquely projected body which is projected with a velocity of **u** with some angle $\theta$ with the horizontal under the action of gravity is R=$u^2sin2\theta$/g by which we can calculate the range of projected body when it comes to the same level of the point of projection but how to calculate the remaining distance which it covered below the point of projection i got this formula on the web but i cant prove this ![enter image description here](http://i.stack.imgur.com/WRlGA.png) Can any one help me by suggesting me what concept I am missing to calculate the range?


title: Why do plastic bags make less noise when immersed in water and crushed?

text: When a plastic bag is crushed in air it makes noise. But when immersed in water and then crushed we hear very little noise. Why is it so? Even if I go inside water, I do not hear the noise the same as I hear in the air.


title: when photons can be trapped in a cavity and manipulated. How they can be observed without being destroyed?

text: An observer is anything that can cause a wave function to collapse. That is an interpretation of wave function collapse (usually referred to as **the measurement problem** ). Now,   1. when can photons be trapped in a cavity and manipulated?    2. How they can be observed without being destroyed!?   3. Through interactions with atoms in cleverly designed experiments!?


title: Eigenvectors of the angular momentum operator $S_x$

text: For a spin of $\frac{1}{2}$ the angular momentum operator can be written as $\vec{S} = \frac{\hbar}{2} \vec{\sigma}$ in matrix form. Find the eigenvalues and eigenvectors of $S_x$ where $\sigma_x = \begin{pmatrix} 0 & 1 \\\ 1 & 0 \\\\\end{pmatrix}$ **Attempt:**   * _**Finding the eigenvalues:_** $|S_x - \lambda I|=det\begin{bmatrix} -\lambda & \hbar/2 \\\ \hbar/2 & -\lambda\\\\\end{bmatrix} = 0 \Rightarrow \lambda = \pm\hbar/2$ As far as I know that is correct so far.   * _**Finding the eigenvectors:_** Using $\lambda = -\hbar/2$ first: (I don't get a correct answer for this one, but I do for the other eigenvalue) $\begin{bmatrix} \hbar/2 &\hbar/2 \\\ \hbar/2 & \hbar/2\\\\\end{bmatrix}\begin{bmatrix} \alpha \\\ \beta \end{bmatrix}=\begin{bmatrix} 0 \\\ 0 \end{bmatrix} $ $\Rightarrow (\hbar/2)\alpha + (\hbar/2)\beta = 0$ $\Rightarrow (\hbar/2)\alpha + (\hbar/2)\beta = 0$ Therefore the corresponding eigenvector is $c \begin{bmatrix} -1 \\\ -1 \end{bmatrix} $ where $c$ is some constant. I know that the answer should be $c\begin{bmatrix} 1 \\\ -1 \end{bmatrix} $ where after normalisation $c$ is found to be $1/\sqrt{2}$. **Where have I gone wrong? The eigenvalues are correct so it can't be that.**


title: Matrix order in Dirac equations

text: The trace of matrix is always sum of its eigen values , which can be seen if $\hat{U}$ transforms the matrix $\alpha_i$ into it's diagonal form . $$ \begin{pmatrix} A_1 & 0 & \cdots & 0 \\\ 0 & A_2 & \cdots & 0 \\\ \vdots & \vdots & \ddots & \vdots \\\ 0 & 0 & \cdots & A_N \end{pmatrix}= \hat{U}\alpha_i\hat{U}^{-1} =tr\alpha_i\hat{U}\hat{U}^{-1} =tr\alpha_i $$ Therefore we can say that order of matrices have to be even. We also have following requirements from dirac equations $$(\alpha_i \alpha_j+ \alpha_j \alpha_i)= 2\delta_{i\,j} I$$ $$\left\\{ \alpha_i , \alpha_i \right\\}= 2\delta_{i\,j} I$$ $$(\alpha_i \beta+ \beta \alpha_i) =0 $$ Now how can we prove that $\alpha$ and $\beta$ are 4*4 matrices? Please, don't skip too much.


title: Potential Gradient & Electric field...

text: Potential gradient is the negative of the electric field. Does the _negative_ (here) means that its direction is opposite to electric field.? If it does mean this, How is the direction of the potential gradient _opposite_ to that of electric field..? Explanations would be helpful...


title: Internal forces in a truss and its geometry

text: I'm to work out the internal forces in a truss, but I can't get my head around the geometry of the truss itself. I'm starting to think there may have been information on the diagram which I missed. Could I get some directions on how to find x? ![](http://i.imgur.com/aesgDl.png)


title: Does infinite energy imply Infinite mass (and vice versa)?

text: If some kind of source was able to supply an infinite amount of energy, does that imply that it also must have an infinite mass? Is the contrary also true?


title: Edge channels in Quantum Hall effect

text: Why is the value of Hall conductance directly proportional to the number of edge channels in the sample?


title: Do metals have their distinctive look because of the electron sea which surrounds the metal atoms?

text: are metals shiny because of the electron sea which surrounds the atomic lattice of the metal sample. are metals more shiny because the electron are more evenly distributed on the surface?


title: Torque per unit length on infinite rotating charged cylinder

text: For homework I have the following question, but I am stuck on finding the torque on the cylinder. An infinite cylinder of radius $R$ carries a uniform surface charge $\sigma$. We start rotating the cylinder to a final rotating speed of $\omega_{f}$. How much work will this take per unit length? Do it in the following two ways: a) Find the magnetic field and the induced electric field (quasistatic approximation). Then calculate the torque you must exert and from that obtain the work done per unit length. ($W=\int N d\theta$) b) Use the energy of the final magnetic field to find the energy per unit length stored in the field. So far I have been able to calculate the magnetic and induced electric field: $$\oint\vec{B}\cdot d\vec{l}=\mu_{0}I_{encl}\Longrightarrow \vec{B}(t)=\mu_{0}\sigma R\omega(t)\hat{z}$$ Which gives rise to flux $\Phi=\mu_{0}\pi r^{2}\sigma R \omega(t)$ for $r<R$ and $\Phi=\mu_{0}\pi R^{3}\sigma \omega(t)$ for $r\geq R$. Then I can calculate the induced electric field with: $$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi}{dt}\Longrightarrow \vec{E}= \left\\{ \begin{array}{ll} -\frac{1}{2}\mu_{0}r\sigma R\frac{d\omega}{dt}\hat{\theta} & \mbox{if } r<R \\\ -\frac{\mu_{0}\sigma R^{3}}{2r}\frac{d\omega}{dt}\hat{\theta} & \mbox{if } r > R \end{array} \right.$$ But I am not sure how to find the torque on the cylinder from this. My guess is that $dF=dqE$ with $dq=\sigma Rd\theta dz$ and just forget about $dz$ because we are interested in the answer per unit length. But everything I try I always have the factor $\frac{d\omega}{dt}$ which should not be in the final answer because one can calculate the energy stored in the field per unit length easily by (which is also the answer to b): $$W=\frac{1}{2\mu_{0}}\int^{2\pi}_{0}\int^{R}_{0}(\mu_{0}\sigma R \omega_{f})^{2}rdrd\theta=\frac{1}{2}\pi \mu_{0} \sigma^{2} R^{4} \omega_{f}^{2}$$


title: Current without voltage?

text: Suppose you have a 120 volt, 20 amp circuit which only has a light bulb connected. When measuring the voltage going to the light bulb my meter reads 120 volts. The basics taught me that a circuit is a circle, so the back and forth movement of electrons continues after the light bulb along the neutral wire back to the socket etc. So, why isn't there voltage on the neutral wire? I'm assuming there has to be current in order for the circuit to work. If so, is there current without voltage? Thanks for any help with this!


title: Why time is considered a dimension?

text: Why time is considered to be a dimension? And the other 7 (except the 3 dimensions of space, and the dimension of time) dimensions that string theory suggests, why can't they be realized?


title: Electrostatics Equilibrium problem

text: ![enter image description here](http://i.stack.imgur.com/vsPmJ.png) What should be the value of Q? I am doing the basics of coulomb's law,can anyone explain this to me?


title: Will I be able to push a small object in front of me in the outer space?

text: Imagine I am standing on Earth, and pushing a tennis ball away from me. The ball moves. If it is very heavy, I will move back instead of the ball. Now consider the same scenario in outer space, where both the ball and I are hanging without any third body to support/assert gravity on us. What happens in this case? Will the ball move or will it make me move backwards, despite being very small? Or will both of us move equally away from each other? How exactly does inertia come into picture here?


title: T duality for lower codimension branes or ALF spaces

text: These are purely mathematical questions in some sense but I believe this is relevant in string theory as the title says. I wonder if it is possible to perform T-duality of Taub-NUT space not along the $S^1$-fibration (maybe usually seen cases) but along one of directions in $\mathbb{R}^3$. It looks terrible to me because of the follwoing naive reason: In the long distance limit (compared to the radius of the compactified circle), the Taub-NUT monopoles become vortices in two dimension and this allows some monodromy. (Of course, I know the system of 7-branes allowing monodromies usually used in F-theory.) I heard talks about the objects called exotic branes which appears to be related to the question but I do not know how related not being familiar to those. In addition, I do not know whether the T-duality works well or not preserving the backgroud supersymmetry. The another point I would like to care about is $S^1$ compactification breaks the $SO(3)$ isometry of $\mathbb{R}^3$ where the monopoles live and also does the trisymplectic structures associated to the space rotation. I think this doesn't break the SUSYs but does R-symmetry seemingly (for the dimensionally reduced theory), is it true ? (This is the same as the T-duality on flat space-time case but I'm confused now.) Furthemore, can we perform another T-duality in other direction in $\mathbb{R}^2$ ? (I think third times T-dualities operation is impossible because of the compactness (3-torus $T^3$) for the monopoles i.e. maybe anomalous.) I'm sorry it is rambling. Summarize the main points of questions, 1). Is it possible to carry out such T-dualities for Taub-NUT spaces ? 2). If so, is the dual space hyper-Kähler (or 16 SUSYs preserved in the compactification in type IIA/B) and what are expressions of triholomorphic forms? Is the quaternionic structure nontrivial? I'd be pleased to also be given some references in addition to direct answers.


title: What is the spin of an electron along the x-axis?

text: I know that an electron or any other particle for that matter, has a measured spin which is either up or down. This spin is along the z-axis. But what if we do not measure it along the z-axis and do it along the x-axis instead. What will the result be? Will it get deflected along the x-axis in a Stern Gerlach experiment? Lastly, will it always get deflected along the axis we measure its spin?


title: Statistical mechanics: Meaning of "accessible" in "accessible microstates"

text: What does "accessibility" mean in statistical mechanics? Is it an equivalent concept to accessibility in mathematical control theory? I'll provide an example: When two systems A and B interact on a subspace of their respective spaces, i.e. where they overlap in space, does accessibility of states of B from A mean the part where they overlap?


title: Why isn't the 3 pts function vanishing in the Ising model by Z_2 symmetry?

text: The Ising model with vanishing external field possesses the $Z_2$ symmetry: $$\sigma_i \rightarrow - \sigma_i$$ implying that the 1 pt function vanishes: $$<\sigma_i> \;= 0$$ In the same way, shouldn't all the n point functions vanish for n odd? But at its critical point, the 3 pt function obeys the following by conformal symmetry: $$<\sigma_i\,\sigma_j\,\sigma_k> \;\propto \; \frac{1}{|i-j|^{\frac14}|k-j|^{\frac14}|k-i|^{\frac14}}$$ and so, except if the coefficient factor is zero doesn't vanish. Where is the mistake in this reasoning?


title: Introductory exercise set with solutions for string theory

text: I am trying to self-study string theory, beginning with Ibanez and Uranga (String Theory and Particle Physics). Unfortunately, the book contains no problems, so I'm not sure whether I am fully understanding the material. Are there any pedagogical string theory problem sets, for a beginner, preferably with solutions, available on the web that would accompany my book? I.e. questions initially covering the quantization and symmetries of the Nambu-Goto and Polyakov actions, the wordsheet etc.


title: Ground State Wavefunction of Two Particles in a Harmonic Oscillator Potential

text: `Question:` > Two identical, non-interacting spin-$1/2$ particles are in a 1D Harmonic > Oscillator Potential. Their Hamiltonian is given by > > $$H=\frac{p_{1x}^2}{2m}+\frac{1}{2}m\omega^2 > x_1^2+\frac{p^2_{2x}}{2m}+\frac{1}{2}m\omega^2x_2^2 $$ > > What is the ground state wave function for the two particles in the singlet > and triplet states; i.e., when $S=0$ and $S=1$, respectively. `Attempt:` I believe that, for non-interacting indistinguishable particles, we have $$\psi=\frac{1}{\sqrt{2}}\left \\{\psi_1 (x_1)\psi_2(x_2)+\psi_1(x_2)\psi_2(x_1)\right \\}$$ As well, the ground state of a single particle in a 1D Harmonic Oscillator Potential is $$\psi_0(x)=\left ( \frac{m\omega}{\pi \hbar}\right )^{1/4}\exp \left \\{-\frac{m\omega}{2\hbar}x^2 \right\\}$$ Therefore, would our $\psi$ for the two particle system just be $$\psi=\frac{2}{\sqrt{2}}\left ( \frac{m\omega}{\pi \hbar}\right )^{1/2} \left (\exp \left \\{-\frac{m\omega}{2\hbar}(x_1^2+x_2^2) \right\\} \right )$$ I feel like I'm missing something. Also, how do i account for the $S=0$ and $S=1$ cases? I'm very confused how I incorporate them into my general case above, which does not consider spin. Any help would be appreciated.


title: Why do arrows point backwards in time for Feynman Diagrams?

text: I've been looking into Feynman Diagram for quite some time, and the fact that anti-particles point backwards in time after an interaction has been puzzling me. I understand this to be some convention regarding charges and conservation, yet I wish to know what does this all mean. Does the fact that the pair of particles have lines at, say t=0, and an interaction occurs at t=5 mean that both particles existed at t=0 and we are just measuring the effects of a future interaction? Thanks!


title: Direction of the induced electric field

text: Assuming there is a time varying magnetic field (B), how to determine the direction of the induced electric field due to B. and to which parameters does it depend? P.S. determining the electric field in an arbitrary point in space such as P


title: Calculating force absorbed by compression of a material (i.e. rubber)

text: Say I was trying to design padding to absorb impacts using polymers or similar materials, such as rubber. I have a good understanding of any relevant physics, most likely, but I wouldn't know where to start in calculating the degree of impact that can be absorbed by a material as a function of the dimensions, density, properties, velocity, forces, etc. of or acting on the material. Note the following points:   * The material should not be permanently deformed by the impact more than negligibly.   * The material is expected to be either one chunk or a series of thin layers, whichever is more effective, a total of less than 10cm thick.   * The object being protected will have a smooth but curved surface, to some degree. Any calculus this fact introduces is fine by me.   * Impacts are vertical.


title: "Light" states in critical $O(N)$ model in $2+1$ (and holography)

text: Let me split the question in a few parts,   * Can someone give me a reference which explains the CFT properties of the critical $O(N)$ model in $2+1$? Like how are the CFT correlators (in a $1/N$ expansion?) and the central charge in that calculated?    * I guess that by "light" operators one means operators with conformal dimensions smaller than the central charge. Can someone give me a reference which explains as to what are these light states that exist in the critical $O(N)$ model and in what sense are they like a free field? (..as I have often heard being said..)  {..Are there other CFTs out there which have similar properties like the above and do for all of them the central charge grows with some $N$ (the same $N$ in whose large limit the system is having a lot of free-field like light states) ?...}   * For any CFT is it true that if it doesn't have "too many" light states then it is probably holographic? (...by "too many" I think it is meant that the number of states (primaries?) with scaling dimensions below the central charge don't grow exponentially in the central charge - may be they can grow at most polynomially?..)


title: How to handle D-brane particles in the S-matrix using string worldsheets?

text: In string theory, it's possible to collide a number of strings only in the asymptotic past, and get a final asymptotic state which contains a D0-brane, an anti D0-brane, and a number of strings. If space is compactified, the D0-brane can be replaced by some wrapped D-brane particle, or a stable bound state of a number of them. How is this handled using string worldsheets only? It's true if the string coupling is weak, many strings have to collide to produce a pair of D-branes, but the string coupling is always nonzero, even if tiny. Does this mean a worldsheet formalism only is incomplete? It's true this is a nonperturbative effect, but does this mean perturbation theory is incomplete?


title: How to write Schrodinger equation when a particle with some spin quantity and orbital angular momentum

text: Quantum mechanics: Suppose that there is a particle with orbital angular momentum $|L|$. But the particle also has spin quantity $|S|$. The question is, how do I reflect this into Schrodinger equation? I do know how Schrodinger equation becomes for each case - when a particle has particular orbital angular momentum and when a particle has some spin, but not when both occur.


title: How quantum field transforms in case of some particular spin

text: Except when a particle is spin-0, field of all particles transforms when frame of reference is changed, and this defines what spin is. The question is, specifically how does the quantum field transform in the corresponding equation, and how does this relate to the quantity of spin? I want to see how this works mathematically (specifically, I want to know spin-1/2 and spin-1, obviously.)


title: splitting spin representation when reducible

text: I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example http://hitoshi.berkeley.edu/230A/clifford.pdf). It is also well known that this is reducible into two $2^{m/2}$ dimensional irredible reps. I'd like to get an explicit construction of these two irreps as matrices... similar to the construction in the link above. Does anyone know where that's give.


title: about use CST to calculate drude model of gold in different wavelength

text: from hand book of optical constant of soild, i can know the different reflective of gold in different wavelength of light. but when u wanna set a drude model in CST, it also needs the eps infinity to modify the model.. the equation is shown below ε=ε(∞)−ω2pω2+iγω. How do I get the value of ϵ(∞) for gold? Is there any reference by using which we can directly convert the free electron theory Drude model to the modified Drude model? can someone tell me the detaill process of get ϵ(∞) for gold? the relate link is Optical constants of noble metals: the Drude model for microwave modelling


title: Why are dark matter explanations more apt to consider different mechanics than emissivity?

text: I've seen some treatments in the scientific popular press (podcasts) that consider the possibility that dark matter can be explained by an error (or new term) in our understanding of gravitation & mechanics. I don't see anyone (in the podcasts) considering whether our understanding of "emission" or blackbody radiation were complete. Why are scientists more willing to consider new laws of mechanics than consider we don't know everything about how "regular matter" reacts with or emits photons. I'll admit that some of my "gut feel" is in a guttural reaction in college to the implausibility of a T to the fourth power in Stefan–Boltzmann law. If the full answer where to be found in this P.SE question, I'll study it in greater detail.


title: Where do the bipolar jets of black holes come from?

text: How are they formed? And why are they so bright?


title: Why Silver atoms were used in Stern-Gerlach experiment?

text: For the Stern-Gerlach experiment done in 1922:   1. Why were silver atoms used?   2. Silver atoms contain many electrons in different shells (with different angular momemtum quantum numbers. Why are those not affected by the magnetic field like the $5s$ electron?


title: How do we visualise antenna reception of individua radiowave photons building up to a resonant AC current on the antenna?

text: I am a chemical/biological scientist by trade and wish to understand how quantum EM phenomena translates to our more recognizable classical world. In particular I want to get a mechanistic picture of what is going on when a tuned antenna is interacting with a photon of the desired frequency? I believe an individual electron on the antenna (many electrons) accepts a photon but how does the eventual process of a measurable AC current build up on the dipole (or 1/4 wavelength for example) to be fed with no reactance onto the transmission line? Is there a good text that discusses quantum EM with classical antenna theory? "When photon meets antenna" is a great meeting ground for a quantum/classical bridge. Unfortunately I do not have a serious maths background but will try anything suggested. I have read and listened to many of the Feynman's popular quantum discussions which only increases my thirst for a better understanding of how quantum EM translates to our more visible world.


title: short circuit an alkaline battery

text: i'm not doing anything related to physics, but i'm just curious : What really happen when i short circuit an alkaline battery ? some article in the net shown that fire/explosion can be happened when short circuit the battery. How can this happen? coz i got some article said that there is an 'internal' resistance of the battery that prevent very high current is drawn (exceed the maximum current), so i think this become the safety tool , so current or temperature won't be a problem. But why can still produce fire or explosion ? And why the manufacture of the battery (like for example some of AA alkaline) doesn't provide info about maximum c rating or maximum current to be drawn (so where can i find it?). Ussually only Voltage and capacity(mAH) is printed. Is that kind of info is not important so it's ignored. Sometime the mAH is not even printed on AA batery. I can only see it on lipo or nimh batt. Thanx


title: How does negative energy from Hawking Radiation cause a Black Hole to shrink?

text: Hello this is expanding upon a question that was previously asked on stack exchange that I linked below. From what I understand of Black Holes is that overtime Hawking Radiation results in the Black Hole shrinking. This being do to virtual particle pairs. The point I find confusing, and I may be misunderstanding it, but the black shrinks due to negative energy of the particle it absorbs when the other particle escapes but it does not matter whether or not it was the matter of the antimatter that was absorbed. Is the Black Hole losing mass if it absorbs only the normal matter half of a virtual particle and the antimatter half escapes? Hawking Radiation - Why it causes black holes to die?


title: Why is it more likely that the particle with negative mass falls into a black hole?

text: If particle-antiparticle pairs pop into existence at the event horizon, and presumably their orientation to the black hole is random, why is it the particle with negative mass is more often (always?) the one that falls in, leading to hawking radiation with positive mass? Why is it not equally likely for positive mass and negative mass particles to fall in, leading to no net mass change of the black hole?


title: hawking radiation

text: My question is regarding Hawking Radiation and its effect on black hole. Now as we know that particle and particle pairs create at every place in space and it is true near black hole horizon. Then why it is like that each time gravity of black hole sucks antimatter particle and its mass decreases . Is it true to think that on average black hole sucks an equal amount of particle and antimatter particle in some given time and its mass remains constant?


title: Why does Hawking Radiation not add up to zero?

text: First off: I am not familiar with the details of quantum mechanics or relativity. My understanding of Hawking radiation is as follows: Pairs of particles and anti-particles can and will spontaneously form in space and when that happens near the event horizon of a black hole the anti-particle might fall into the black hole and the particle not, effectively causing the black hole to lose mass. Now, this is totally fine with me. The thing I do not understand is: Why do more anti-particles fall into the black hole than particles? If these pairs pop up randomly, shouldn't the effects of Hawking-radiation and anti-Hawking- radiation cancel out?


title: warp drive with gravitational waves in the nonlinear regime

text: gravitational waves are strictly transversal (in the linear regime at least), also their amplitudes are tiny even for cosmic scale events like supernovas or binary black holes (at least far away, maybe we should ask some physicists located a bit closer to the center of the galaxy), but lets put all those facts aside for a second and consider a gravitational source big enough to generate gravitational waves with amplitudes of the order of the galaxy. For instance consider a planar wave like in my mediocre drawing: $$ h_{\alpha \beta} e^{i (k_{y} y - \omega t)} $$ where $$ h_{\alpha \beta} \approx 1 $$ so the perturbation is in the nonlinear regime ![gravitational wave reducing distances between stars](http://i.stack.imgur.com/V95L2.png) i draw two far away objects in three different time slices (this is why they are repeated 3 times), the topmost is the objects without the gravitational wave, the one in the middle represents the objects in the crest of the gravitational wave, and the one in the bottom represents the objects in the valley of the wave. So, my point is that people would only have to travel an arbitrarily small distance when the wave is on the valley (assuming circular polarization) even if the "normal" distance (i.e: $h_{\mu \nu} = 0$) is several light-years away Besides being slightly impractical to set up such a mammoth gravitational source, this kind of warp drive is valid from a physical standpoint? Are there any physical limits to gravitational wave amplitudes in such nonlinear regime?


title: transfer of electricity through electromagnetic waves

text: Is it possible to transfer electricity from one place to another without the help of physical wires.


title: What is wrong with the De Broglie–Bohm theory a.k.a "Causal Interpretation" of quantum theory?

text: > **Possible Duplicate:**   >  Why do people still talk about bohmian mechanics/hidden variables I've heard of De Broglie–Bohm theory a.k.a causal interpretation of quantum theory. The predictions match accurately with with the nondeterministic quantum theory. As a philosophy buff this one seems just like the classical universe. No funny new-age religion style gimmicks. So why don't people use this interpretation instead of the Copenhagen "magic" interpretation? P.S. I'm not an expert. I possess only superficial knowledge of quantum physics.


title: Effective mass in Aluminium lattice?

text: How do we calculate the effective mass of an electron in an Aluminium lattice? Is there any simple analytical way to work it out?


title: Why did the Hydrogen in Fukushima explode *outside* the reactor vessel?

text: As pointed out before, it orginated inside the reactor vessel but exploded outside: how did it get there? Did it pass the shell by diffusion or was it released by the engineers?


title: Quantum mechanics with non-cartesian coordinates

text: Let say we have the classical hamiltonian of a harmonic oscillator: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ and we want to find the hamiltonian operator in quantum mechanics, we use the transformations $x\rightarrow x$, $y\rightarrow y$, $z\rightarrow z$, $p_x\rightarrow -i\hbar\frac{\partial}{\partial x}$, $p_y\rightarrow -i\hbar\frac{\partial}{\partial y}$ and $p_z\rightarrow -i\hbar\frac{\partial}{\partial z}$ to obtain: $$\hat{H}=\frac{-\hbar^2}{2m}\Delta+\frac{k_1x^2+k_2y^2+k_3z^2}{2}=\frac{-\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)+\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ Now suppose I want to treat this system in sphercal coordinates, should I transform the classical hamiltonian in spherical coordinates ($p_r\rightarrow \frac{\partial}{\partial r}$, $p_{\theta}\rightarrow \frac{\partial}{\partial \theta}$, $p_{\varphi}\rightarrow \frac{\partial}{\partial \varphi}$) or should I transform the hamiltonian operator to spherical coordinates using $$\Delta=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2}$$ and why?


title: Is Gravity Energy?

text: This might be stupid, but is gravity a form of energy? And, if so, couldn't we use it for power?


title: Theory with interaction and the birth of bound states during propagation

text: Suppose we want to calculate vacuum expectation $$ \tag 1 D_{lm}(x - y) = \langle \Omega | \hat {T}\left( \hat {\Psi}_{l}(x)\hat {\Psi}_{m}^{\dagger}(y)\right)| \Omega\rangle = \langle \Omega| \hat {T}\left( \hat {\Psi}_{l}(x)\hat {\Psi}_{m}^{\dagger}(y)\right)| \Omega\rangle , $$ where $\hat {\Psi}_{a}(x)$ refers to the irreducible representation of the Poincare group, and we have lagrangian containing interaction $\hat {\Psi}(x)$ with some other fields (including interaction "with itself"). We may insert full basis $1 = \sum_{\mathbf p , \sigma , i}| (\mathbf p , \sigma )_{i}\rangle \langle (\mathbf p , \sigma )_{i}|$ ($i$ numerates the number of particles) between fields operator in $(1)$: $$ \tag 2 D_{lm}(x - y) = \sum_{i, \mathbf p , \sigma}\hat {T}\langle \Omega| \hat {\Psi}_{l}(x) | (\mathbf p , \sigma )_{i}\rangle \langle (\mathbf p , \sigma )_{i}| \hat {\Psi}_{m}^{\dagger}(y)| \Omega\rangle $$ Let's then separate one-particle states from the full basis in $(2)$: $$ \tag 3 D_{lm}(x - y) = \sum_{\mathbf p , \sigma}\hat {T}\langle \Omega| \hat {\Psi}_{l}(x) | (\mathbf p , \sigma )\rangle \langle (\mathbf p , \sigma )| \hat {\Psi}_{m}^{\dagger}(y)| \Omega\rangle + ..., $$ where $...$ marks the contribution of the other (multiparticle) states. Here is one remark: in the theory with interaction we have also the bound states, where mass $m$ refers to the bound states of two and more particles. Even for states of one field $\Psi $ we have masses between $(m, 2m)$, where $m$ refers to the one-particle state. So, the question: are $...$ summands in $(3)$ not equal to zero in theory with interaction? I.e., the propagator $(1)$ contains not just the $[D_{lm}(x - y)]_{free}$ line, but also the lines of the bound states, doesn't it?


title: How does my kitchen scale know I'm holding it?

text: If I switch on my electronic kitchen scale while it's on the table, it reads 0 grams. If I pick it up, it reads around -500 grams. Conversely if I switch it on in my hand it tares to 0g and then reads about 500 grams when I put it down. What's the difference, physically, between resting on my (not moving) hand, and on the table?


title: What happens if we put together a proton and an antineutron?

text: A hydrogen nucleus consists of a single proton. A 2-hydrogen (deuterium) nucleus consists of a proton and a neutron. A tritium nucleus consists of a proton and two neutrons. This makes me wonder how an atomic nucleus made of a proton and a "minus one neutron" would look like, and the closest thing to a "minus one neutron" I can imagine is an antineutron. > What happens if we combine a proton and an antineutron? Are things like this > even possible? > > If such a thing is an atomic nucleus, can we add an electron and get an > atom? **Edit:** in the comments below, I also asked this more specific question (I suppose it's useful to also mention it here in order not to create a complete chaos): > If the proton and antineutron annihilate, is it still possible that the > thing they annihilate to remains somehow stable enough to behave like an > atomic nucleus?


title: How can a paramagnetic element be a permanent magnet?

text: I'm just starting with magnetism and stuff and I was reading about different types of magnetism. Such as ferromagnetism, anti-ferromagnetism and paramagnetism. If I understand it correctly, examples of ferromagnetic elements are Iron, Cobalt and Nickel. They are permanent magnets (if the crystal structure is correct). But paramagnetic elements (and anti- ferromagnetic elements) will act like a magnet if there is an external magnetic field applied to it. But how can a Neodymium magnet, very well known permanent magnet, be one when its a paramagnetic element? (Source properties Neodymium)


title: how many times have the first photons lapped the universe?

text: I'm not really interested in the actual number of times, but rather an explanation of a closed Universe. How do we still measure the background radiation of the Big Bang? Those electromagnetic waves should have all passed this point in space a long time ago. Unless the Universe is closed and those Electromagnetic waves pass over the same region of space over and over again (like a bug running around the hemisphere of an expanding ball). If I recall correctly, at this point in time, the (circumference) of the Universe is increasing faster than the speed of light so it would be impossible for any current photons to ever complete another lap, but was this always so? Is the background radiation also continuously lowering in frequency as it red-shifts through ever expanding space?


title: Numerical tests of Titius–Bode formula for artificial planetary orbits

text: Titius–Bode law is an empirical formula stating that positions of planets in Solar system is described by: $$a=0.4+0.3 \times 2^m [AU],$$ where $m$ are natural numbers. All planets, with exception of Neptune, fall within a few percent of it. Moreover, it predicted the existence of Uranus (so, at least to me, it does not sound like a coincidence). There are simulators of solar system formation (even some online). Are there numerical simulations generating such orbits (as expected from Titius–Bode law). Or are they showing that there is no preference for such orbits? See also:   * Why do the planets' orbital distances fall on an exponential curve?


title: Questions on Penrose's paper - Conformal Treatment of Infinity

text: _I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. On suggestion, I will modify this post._ I am reading Penrose's paper on the conformal treatment of infinity (find here). The basic concept that underlies this is the fact that in order to study asymptotic behaviour of a space-time with metric ${\tilde g}_{\mu\nu}$, we may instead study a space-time that is conformally related to it by defining an unphysical metric on a compact manifold $g_{\mu\nu} = \Omega^2 {\tilde g}_{\mu\nu}$. He then goes on to say that the asymptotic properties of fields can then be investigated by studying local behaviour of fields at infinity on this unphysical manifold **provided that the relevant concepts can be put into a conformally invariant form**   1. What are the relevant conformally invariance concepts that one can study? What I can think of is the causal structure of space-time, gravitational waves (since they are described by a conformally invariant Weyl tensor). What else is there?   2. I have also often heard that massless fields satisfy conformally invariant equations in curved spacetimes. (see this question). Indeed Penrose claims that this can be done if "" **interpreted suitably** ". What does he mean by this? Further, since massless particles can only reach $\mathscr{I}^\pm$, does his formalism only apply to null infinity?   3. What about massive particles? Surely, the equations for such particles are not going to be conformally invariant. Further, these particles would reach $i^\pm$. One can't apply the above formalism to massive particles right? Is there an alternative way of constructing the asymptotic structures of $i^\pm$?


title: Since everything with mass exerts a gravity force on everything else, why do objects float in outer space?

text: For example, if you were to go out into deep space, and just slow down and stop your rocket. Everything inside the rocket that's not strapped in, starts floating. Why is that if every object has mass and thus attracts everything else. If a book was accelerated to the center of mass of the whole rocket + everything in it, why would it not stay there instead of floating back and forth like they show in movies? Does this have to do with how weak the gravity is, so when it falls it's actually not enough force to keep it from bouncing back like on planets? Thanks


title: "Rocket in a box" thought experiment

text: Here's a thought experiment that I came up with a while ago. It might be hard to visualize, so I'll describe it as best I can. Take a rocket in a vacuum, in outer space. Attach a large "box" to the end, so that the exhaust gases from the nozzle will shoot straight into it/ Let's make the box large. Really large - for our purposes, about a mile long, wide and tall, with the nozzle stuck on in the center of one side. When the rocket is turned on, the exhaust gases will shoot out the back end, and the rocket - and thus the box - will move forward. But what about when the gases hit the other side of the box? For the first scenario, imagine that the exhaust gases are in a single, quick pulse (all effects involving sideways motion can be ignored). In an (admittedly poor) attempt to model it, we can imagine it is a small object - like an oddly shaped cube. It will go in the opposite direction as the rocket. Now, when it hits the other side of the box, it will collide. A few things could happen here: 1) The collision is a gaseous "elastic" collision - in other words, the box will rebound in the other direction, and the box/rocket will go _backward_ while the gases go _forward_. When the gases hit the front of the box, the rocket will go forward and the box will go backward. And so on. 2) In an inelastic collision, some energy will be lost during each impact, and the oscillations will grow smaller and smaller. 3) The gases and the box will undergo an "inelastic" collision, and the rocket and gases will stop moving. In both these scenarios, the gases are in a small pulse. I have an idea of what will happen (thought using a small cube is a very poor analogy). But what if the gases continue to emanate from the nozzle (and thus the mission had been better planned!)? So my question) are this: 1) In the first scenario, am I correct in saying that the rocket and gases will oscillate back and forth, or will something completely different happen? 2) What will happen in the second scenario?


title: Do the physics in the FlyBoard video make sense?

text: If you haven't seen the video of the FlyBoard, please have a look: http://www.youtube.com/watch?v=Cd6C1vIyQ3w&feature=youtube_gdata_player Yes, it's amazing, but do the physics make sense or is this another "doctored" video? Specifically, I'm wondering how Newton's 3rd law comes into play here. I'm not an expert in jetskis or physics, but I understand that a jetski sucks in water (not sure from which direction) and expels it out of a rear exhaust. Surely that should create an opposite force on the jetski itself and pushes it forward (regardless of the host that is now attached)? However, in the videos, the jetskis are relatively stationary. Also, as we know from our garden hoses, when it's full of pressure, it tends to want to "straighten", but these hoses seem to be totally content at being full of pressure but constantly "curved". Lastly, I think you'll need unparalleled balance to actually bring this off, almost like trying to balance a marble on top of a balloon. But I could be wrong on this. Your inputs please?


title: Quantum Mechanics- Antenna emitting electromagnetic radiation

text: Radio signals are being transmitted in a frequency of $ 8.4 \times 10^9 s^{-1} $ and being received by an antenne that is capable of receiveing power of $ 4 \times 10^{-21} watt $ ($ 1 watt = 1 J s^{-1} $ ) . Estimate that number of photons per second of this electromagnetic radiation that this antenne is capable of receiving. I can easily calculate the number of photons per second that this radio produces, dividing the emitted power by the energy of one photon, $\hbar\omega=h\nu = h\times8.4 \times 10^9 s^{-1}$ but have no idea how to calculate the number of photons that this antenne can receive. Can you help me? Thanks in advance


title: Why is the partition function called ''partition function''?

text: The partition function plays a central role in statistical mechanics. But why is it called ''partition function''?


title: What is the CFT dual to pure gravity on AdS$_3$?

text: Pure $2+1$-dimensional gravity in $AdS_3$ (parametrized as $S= \int d^3 x \frac{1}{16 \pi G} \sqrt{-g} (R+\frac{2}{l^2})$) is a topological field theory closely related to Chern-Simons theory, and at least naively seems like it may be renormalizable on-shell for certain values of $l/G$. This is a theory which has been studied by many authors, but I can't seem to find a consensus as to what the CFT dual is. Here's what I've gathered from a cursory literature search: Witten (2007) suggests that the dual is the monster theory of Frenkel, Lepowsky, and Meurman for a certain value of $l/G$; his argument applies when the central charges $c_L$ and $c_R$ are both multiplies of $24$. In his argument, he assumes holomorphic factorization of the boundary CFT, which seems to be fairly controversial. His argument does produce approximately correct entropy for BTZ black holes, but a case can be made that black hole states shouldn't exist at all if the CFT is holomorphically factorized. He also gave a PiTP talk on the subject. Witten himself is unsure if this work is correct. In a recent 2013 paper, McGough and H. Verlinde claim that "The edge states of 2+1-D gravity are described by Liouville theory", citing 5 papers to justify this claim. All of those are before Witten's 2007 work. Witten's work does mention Liouville theory, and has some discussion, but he doesn't seem to believe that this is the correct boundary theory, and Liouville theory is at any rate not compatible with holomorphic factorization. This paper also claims that "pure quantum gravity...is unlikely to give rise to a complete theory." Similar assertions are made in a few other papers. Another proposal was made in Castro _et.al_ (2011), relating this to minimal models such as the Ising model. Specifically, they claim that the partition function for the Ising model is equal to that of pure gravity $l=3G$, and make certain claims about higher spin cases. It doesn't seem to me that all of these can simultaneously be true. There could be some way to mitigate the differences between the proposals, but my scan of the literature didn't point to anything. It seems to me that no one agrees on the correct theory. I'm not even sure if these are the only proposals, but they're the ones that I'm aware of. First, are my above statements regarding the three proposals accurate? Also, is there any consensus in the majority of the HET community as to whether pure quantum gravity theories in $AdS_3$ exist, and if so what their CFT duals are? Finally, if there is no consensus, what are the necessary conditions for each of the proposals to be correct?


title: Determinant expansion

text: I've seen in a few textbooks now a useful looking expansion procedure for determinants, but I don't understand the details of it. Here is precisely the example I'm thinking of (Ex. 7.6). I don't understand the notation \begin{equation}\det(M+\Lambda)=\det(M)\left[\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{i_{1},i_{2}...i_{n}} \lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{n}}\det(\Delta_{i_{l}i_{k}})\right]\end{equation} $\Delta=M^{-1}$ Does this mean $\det(\Delta_{i_{l}i_{k}})$ is some sort of co- factor determinant? Also, this was done in the context of Grassmann variables, so would that indicate that $M$ must be antisymmetric? Or does this hold for any $M$ (invertible)? If someone could walk me through the notation of this I would be very grateful. EDIT: I think my main source of confusion is what the author means by $\det(\Delta_{i_{l}i_{k}})$.


title: Are Uncertainties in Measurements Important?

text: In the first lecture of MIT's Classical Mechanics Prof. Lewin highlights the importance of uncertainties in measurements by quoting "Any measurements, without the knowledge of uncertainty is meaningless." He measures the length of a student and an aluminium bar both in their vertical and horizontal positions respectively and notes their lengths in centimeters and calculates the difference between the two positions. Question:Is it so important to make such serious considerations while making measurements? Do professional physicists make such considerations when making real world measurements? Aren't we well equipped to just ignore these uncertainties?


title: physics , magnetic lines of a magnetic

text: Why magnetic lines comes from north to south out side of the magnet is any magnetic lines comes from south to north if so in which direction What is the reason of magnetic lineS


title: Vertical and horizontal components of forces and vectors

text: I'm getting a bit confused when finding components of vectors and forces. In problems for vectors, I've always known that if you want to get the components of a vector, you would use the following: ![enter image description here](http://i.stack.imgur.com/UWNQM.png) * * * Recently we've started working with Forces in homework problems. This diagram is from one of my homework problems, which they want me to draw the free body diagram: ![enter image description here](http://i.stack.imgur.com/4zGQN.gif) Here is the free body diagram given by the homework solution: ![enter image description here](http://i.stack.imgur.com/KGMsO.gif) * * * What I don't understand is why would I be using $m_{2}gsin(\theta)$ instead of $m_{2}gcos(\theta)$ when finding the X-component of the gravitational force of $m_{2}$. Force _is_ a vector, so why wouldn't you use $m_{2}gcos(\theta)$? Are the rules different when dealing with forces?


title: Creating small polarised scientific lenses?

text: How would I go about creating small polarised scientific lenses? Thinking 5mm by 5mm, with a light-source behind it for the purposes of reducing (polarised) surface reflection (assisted with a cross-polarised viewer). Is the idea that I just need to apply magnesium fluoride; and if so, at what consistency and against what type of lens?


title: Don't understand the integral over the square of the Dirac delta function

text: In Griffiths' _Intro to QM_ [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on p. 101). He then says that these eigenfunctions are not square integrable because $$\int_{-\infty}^{\infty}g_{\lambda}\left(x\right)^{*}g_{\lambda}\left(x\right)dx ~=~\left|B_{\lambda}\right|^{2}\int_{-\infty}^{\infty}\delta\left(x-\lambda\right)\delta\left(x-\lambda\right)dx ~=~\left|B_{\lambda}\right|^{2}\delta\left(\lambda-\lambda\right) ~\rightarrow~\infty$$ (cf. second formula on p. 102). My question is, how does he arrive at the final term, more specifically, where does the $\delta\left(\lambda-\lambda\right)$ bit come from? My total knowledge of the Dirac delta function was gleaned earlier on in Griffiths and extends to just about understanding $$\tag{2.95}\int_{-\infty}^{\infty}f\left(x\right)\delta\left(x-a\right)dx~=~f\left(a\right)$$ (cf. second formula on p. 53). References:   1. D.J. Griffiths, _Introduction to Quantum Mechanics,_ (1995) p. 101-102.


title: Stephen Hawking's theory of future backreaction

text: Several years ago, New Scientist featured an article on a new theory by Stephen Hawking that involved the future having some effect or "backreaction" on the present. As it would be very tedious and time consuming to root through several years of back issues to find it again, I was hoping someone might be able to provide a reference, or just put me on the right track. I tried a search myself, but could find nothing on the ArXiv. I thought for a moment it might be http://arxiv.org/abs/astro-ph/0305562 ("Cosmology from the Top Down"); but although that is interesting, it doesn't match what I recall of the article. Please note, I'm not interested in starting a discussion of the merits or otherwise of this theory, merely seeking a reference.


title: Electric Field inside a regular polygon with corner charges

text: If we have equal charges located at the corners of a regular polygon, then the electric field at its center is zero. Are there other points inside a polygon where the field vanishes? The simplest case would be an equilateral triangle of equal charges. I thought it would be easy to find the location of such other points, if they exist. I am having a hard time proving or disproving their existence.


title: What is the basic idea behind how quantum computation is supposed to work?

text: As far as I know that QM is probability-based and Feynman proposed a quantum computer (QC) as an emulator for fast computation. Reading the introductory articles, I just do not get the essence of the speedup. I understand the q-bits and such but basically, what is the idea? Does Feynman simply mean that instead of computing the function of how something behaves (the frequency of oscillations for instance) from a complex model we just build the model physically and observe the behaviour? I see that even simple probabilistic problems may produce involved computations. Marginally more complex problem (its model) can demand prohibitively more complex computation. Does QC just builds a physical model and replaces the computation with measurement?


title: What kind of curve would describe the rate of air coming out of a balloon?

text: If one graphed the volume per time of air being expelled from a common elastic balloon out of a hole of constant size relative to the balloon's surface area, would the curve of the graph be exponential or polynomial? I have neither the means to perform the experiment nor the brains to theorize, but I'm very interested.