Lorentz transformation matrix and its meaning in Minkowski diagram

I have sucessfully derived the Lorentz matrix for the boost in $x$ direction and its inverse. So I know how to get these two matrices: $$ \Lambda = \begin{bmatrix} \gamma & 0 & 0 & -\beta \gamma\\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ -\beta \gamma & 0 & 0 & \gamma \end{bmatrix} $$ $$ \Lambda^{-1} = \begin{bmatrix} \gamma & 0 & 0 & \beta \gamma\\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ \beta \gamma & 0 & 0 & \gamma \end{bmatrix} $$ What I lack is a geometrical meaning of those two. I have read, that it is some sort of a rotation, so I have already drawn the Minkowski diagram for steady _observer 1_ and one for the moving _observer 2_ as _observer 1_ sees it. Examples are drawn for $\beta = 0.5$. I have noticed that:   * _base cell_ is a square (picture 1) for a steady observer and a parallelogram (picture 2) for a moving observer   * light travels only diagonals of _base cells_   * axis $x'$ has coefficient $\beta$   * axis $ct'$ has coefficient $1/\beta$ **question 1:** Is there a way I can help myself with what I have figured out so far to show, how the Lorentz matrix deforms a square into a parallelogram and vice versa? **question 2:** I have heard of the invariant quantity $\Delta s$ in relativity for which I have stumbled upon an equation $\Delta s^2 = \Delta x^2 - c \Delta t^2$. Is it possible $\Delta s$ is an area of a base cell? **question 3:** Why is it so necessary for a scalar product to be defined with a minus sign for Minkowski spacetime? **picture 1:** ![Minkowski diagram for steady *observer 1*](http://i.stack.imgur.com/nmqKd.png) **picture 2:** ![enter image description here](http://i.stack.imgur.com/9KD3S.png) Thank you.


Differences of behaviour of a particle in a box in quantum theory between that in classic physics

Can anyone help me enlist 3 major differences between the quantum and classical physics of the behaviour of a particle in a box? I would like some insight into the differences without solving PDEs because I am not a physics student. Thank you very much


Does cosmological horizon grow or decrease as it radiates?

Ron Maimon in many posts claimed that cosmological horizon is like a big black hole. Black holes decrease as they evaporate and their radius decreases as well. So what is with a cosmological horizon? If cosmological horizon is just a black hole centered at the opposite side of the universe, we should see the radius of the external space groving as the radius of the BH decreases. But if the radiation decreases the horizon entropy we should see the area of the horizon decreasing. What is the correct conclusion?


Why does light in a room not form constructive and destructive interference patterns?

This is something that I have wondered for a long time. How come when I walk around why do I not see random black spots where light has collided destructively and bright spots where it has collided constructively?


LQG Demystified Book?

Does a similar easy to read introduction like David Mc Mohan's Demystified books exist for LQG? I'd like something easier to start with than a paper but deeper than a popular book.


Has the existence of anyons been experimentally verified?

I've been wondering whether there has been any experimental evidence for the existence of anyons or are they just objects of purely mathematical interest?


Why are noble gases used for lights?

I know that neon is used in advert signs due to its inertness. However, I am not entirely sure how the inertness is exploited. I think it is because Ne being inert means that after electricity frees an electron we can count on it returning to the ion left behind to emit a photon, whereas for more active elements, the gas may just exist as a plasma?


Pion Decay and Fractional electric Charge disappeared, why?

![Pion Plus Decay](http://hyperphysics.phy- astr.gsu.edu/hbase/particles/imgpar/pidec1.gif) Since the charged pions decay into two particles, a muon and a muon neutrino Fractional electric Charge disappeared, why? The decay proceeds by the weak interaction $W^{+}$ and can be visualized in terms of Feynman diagrams. my suggestion is that the Fractional electric Charge disappeared because it does not exist at all in the Quarks mechanism.


How 'Faster Than Light' Neutrino Was Product of Loose Cable at CERN?

A "faster than light" neutrino discovery was actually the result of a loose cable. A fiber-optic cable in a GPS receiver at the European Center for Particle Physics ( CERN) near Geneva Can someone show me the light here, What's Going On ? how long was fiber-optic cable?


Is a particular force different in different frames

Can a particular real force have different magnitude in different frames?


Entropy and the principle of least action

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?


What is an "inclined magnetic field"?

What is meant by an "inclined magnetic field"? How is it different from the usual magnetic field?


How organic impurities in the air affects laser?

We have a Nd YAG laser pumped Optical Parameter Oscillator in our lab. A technician came to our lab (my professor called him for its maintenance) and while he was working on it, he told me that because of the organic impurities in the air, the crystals got contaminated. He showed me a layer formation on the crystal where the laser hits. He explained this is due to the organics in the air which interacts with laser and get deposited. I am not clear what kind of organics (aromatic or some other?) which creates this and how it is get deposited there? Can anyone please explain me in detail how the atmospheric impurities (especially organics) can cause damage on lasers?


Oscillations and Majorana Neutrino

In neutrino oscillations, neutrinos can convert from one flavor to another. This implies individual lepton number is not conserved. Doesn't it also imply that, if the neutrinos have mass, the mass should be of Majorana type? But I'm confused. Can I define lepton number (and the states like $\nu_e ,\nu_\mu$ and $\nu_\tau$) if neutrinos are Majorana particles? Or in other words, can Dirac neutrinos exhibit oscillations? Is the total lepton number conserved?


Interpretation of $e|\psi|^2$ as electron density

In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to some bizarre consequences", and that Born's statistical interpretation of $|\psi|^2$ as a probability density is more satisfactory. I am aware that a position measurement of an electron leads to (in the Copenhagen interpretation) a collapse of the wave function into a position eigenstate $x$ with the probability given by $|\psi(x)|^2$, and that it is known from some scattering experiments that the electron behaves as a point- like particle. However, the electron density is experimentally observable, e.g. by X-ray scattering. One could argue that X-ray scattering is done with a large ensemble of atoms, so that what we actually observe is the average over many position measurements. I am wondering if this the only "bizarre consequence", since this distinction seems to be a minor difference to me. My questions aims at clarifying the difference between the probabilistic interpretation of $|\psi(x)|^2$ and the interpretation as an electron density.


Conservation laws in mSUGRA model

Can somebody list all the quantum numbers (beside R-parity) that are conserved in vertex for SUSY particles in mSUGRA model?


How many mirrors placed perpendicular to each other shall make a light ray completely reverse its path?

The answer is 3. But according to me 2 shall also work in 2 Dimensions? Does anyone has a theoretical proof?


How to combine measurement error with statistic error

We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times. I know that I will have a $\Delta t = 0.1 \, \mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1. What is the total error then? Do I add them up, like so? $$\sqrt{1^2 + 0.1^2}$$ Or is it just the 1 and I discard the (systematic?) error of my reaction time? I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1?


How to estimate a final error when you have N experiments with N errors?

if we have Xo±e1, X1±e2, X2±e3, ..., Xn±en when we know these, what is going to be our estimation for X, and it's error? for example if we have 344±1, 350±2, 345±5, 338 ± 2 estimation? error of our estimation?


Does the heat death of the universe really imply a maximum entropy state *all* of the time? Or most of the time?

Statistically speaking, you're going to still encounter deviations from equilibrium, even though the expected value is equilibrium. But these rare deviations from equilibrium - which are inevitable - might have the power to do work. So does the universe inevitably descend towards a maximum-entropy state? Or is it only probabilistically destined towards a maximum-entropy state - that is - it will be in that state more than any other state. After all, someone has even hypothesized a Poincare recurrence time, as described below: http://en.wikipedia.org/wiki/Timeline_of_the_far_future > Scale of an estimated Poincaré recurrence time for the quantum state of a > hypothetical box containing an isolated black hole of stellar mass.[47] This > time assumes a statistical model subject to Poincaré recurrence. A much > simplified way of thinking about this time is that in a model in which > history repeats itself arbitrarily many times due to properties of > statistical mechanics, this is the time scale when it will first be somewhat > similar (for a reasonable choice of "similar") to its current state again.


Kolmogorov-Smirnov test vs Chi-squared test

What is the difference between the Kolmogorov-Smirnov test and the Chi-squared test? When should we use one instead of the other? I was reading this article, and I got confused a lot. It is hard to get a physics-related answer in the Cross Validated SE section, so this is why I am asking here.


How do you calculate the energy output if you know the volts, amps and time?


How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And as a related question, what is the motivation for using a Routhian?


How to compute observables from the boson field operator?

I think I understand that if given the two boson wavefunction of two different states \begin{align} \Psi(\boldsymbol{r}_1,\boldsymbol{r}_2) = \dfrac{ \psi_1(\boldsymbol{r}_1)\psi_2(\boldsymbol{r}_2) \+ \psi_1(\boldsymbol{r}_2)\psi_2(\boldsymbol{r}_1) } {\sqrt{2}} \end{align} the expectation value of an operator $\hat{Q}$ for $\Psi(\boldsymbol{r}_1,\boldsymbol{r}_2)$ is \begin{align} \langle \Psi | \hat{Q} | \Psi \rangle = \langle \psi_1(\boldsymbol{r}_1) | \hat{Q}_1 | \psi_1(\boldsymbol{r}_1) \rangle \+ \langle \psi_2(\boldsymbol{r}_2) | \hat{Q}_2 | \psi_2(\boldsymbol{r}_2) \rangle \end{align} If I add one more boson to state $\psi_1$, then the total wavefunction changes to \begin{align} \Psi(\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3) = \dfrac{ \psi_1(\boldsymbol{r}_1)\psi_1(\boldsymbol{r}_2)\psi_2(\boldsymbol{r}_3) \+ \psi_1(\boldsymbol{r}_2)\psi_1(\boldsymbol{r}_3)\psi_2(\boldsymbol{r}_1) \+ \psi_1(\boldsymbol{r}_3)\psi_1(\boldsymbol{r}_1)\psi_2(\boldsymbol{r}_2) } {\sqrt{6}} \end{align} The expectation value of $\hat{Q}$ for $\Psi(\boldsymbol{r}_1,\boldsymbol{r}_2,\boldsymbol{r}_3)$ becomes \begin{align} \langle \Psi | \hat{Q} | \Psi \rangle = 2\langle \psi_1(\boldsymbol{r}_1) | \hat{Q}_1 | \psi_1(\boldsymbol{r}_1) \rangle \+ \langle \psi_2(\boldsymbol{r}_2) | \hat{Q}_2 | \psi_2(\boldsymbol{r}_2) \rangle \end{align} I am stuck with understanding how to compute the expectation values of $\hat{Q}$ in terms of the boson field operator \begin{align} \Psi(\boldsymbol{r})=\sum_{\nu} \psi_{\nu} \left( \boldsymbol{r} \right) b_{\nu} \end{align} where \begin{align} \left[\Psi(\boldsymbol{r}_1),\Psi^\dagger(\boldsymbol{r}_2)\right]=\delta (\boldsymbol{r}_1-\boldsymbol{r}_2) \\\ [b_\alpha^\dagger,b_\beta^\dagger]=[b_\alpha,b_\beta]=0,\quad [b_\alpha,b_\beta^\dagger]=\delta_{\alpha\beta} \end{align} I am assuming that \begin{align} \langle \psi_{\mu}^\dagger \left( \boldsymbol{r} \right) | \psi_{\nu} \left( \boldsymbol{r} \right) \rangle = \int \psi_{\mu}^\dagger \left( \boldsymbol{r} \right) \psi_{\nu} \left( \boldsymbol{r} \right) d^3\boldsymbol{r} = \delta_{\mu\nu} \end{align}


How long does it take plasma to radiate its heat?

Lets say we have 1 gram of plasma (Argon) at 1 million kelvin confined in a vacuum with electromagnets. If we keep the magnets on but shut down the device that heated the plasma, how long will it take for it to cool off?


Neutrino mass with Dirac and Majorana

Why both Dirac mass and Majorana mass terms are needed to explain the mass of a neutrino?


Does mechanical resonance of an object changes in different surrounding medium?

When comparing mechanical resonance of an object (for example string) in air and in water, does the resonance frequency changes? My guess is that it does change because the surrounding medium will slow it down. Thank you.


How do we know photons have spin 1?

Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in spectroscopic measurements. On the other hand, a photon is neutral - how can one measure its spin if there's no magnetic moment? How do we know it has spin 1?


How can Magnets be used to pick up pieces of metal when the force from a magnetic field does no work?

I learned that the force from magnetic fields does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal list a refrigerator?


What is deep Fresnel region?

If I understand correctly, it has something to do with autocorrelation function, but can someone give me a definition or exact explanation? In case of scattering, if you wish to analyze pattern with Fourier transform, you can only do so within deep Fresnel region. Apparently (according to the text I've been reading) in practical experiment this would mean that detector has to be close to the sample. Say we want to observe scattering of light on some soft material and use laser to do so. Could this simply mean that if we don't go too far from source, light waves remain coherent? This article contains an example of the use of this terminology.


Definition regarding percolation

in a homework sheet studying bond-percolation on the Bethe lattice, a function $g(r)$ is introduced as "the probability of finding two nodes separated by a distance $r$ on the same cluster". Now just to see if I get this definition right, for the simple one-dimensional case (Bethe lattice with coordination number 2), would this mean that $$g(r) = p^r$$ if $p$ is the probability that a given bond exists, or would it be $$g(r) = 2p^r - p^{2r}$$ The former would mean: Given two nodes of distance $r$, what's the probability that they are connected, whereas the other would mean: Given one node, what's the probability that it's connected to _some_ node a distance $r$ away. And since in the one-dimensional case there are 2 nodes of distance $r$, we get add the probability for having a bond to each of them and then subtract the joint probability... I'm mainly asking because it will change the answer for the percolation threshold at coordination numbers larger than $2$. If the former definition is correct, I'd guess the correlation function would still be $p^r$ since there's exactly one path between two nodes. Whereas for the latter definition, I'd get something like $$g(r) = 1 - (1-p^r)^{N(r)}$$ where $N(r)$ counts how many nodes exist a distance $r$ away from a chosen "center" node, which is $N(r) = z(z-1)^{n-1}$ if $z$ is the coordination number.


Right Angle Circuitry

This question concerns the article Right Angle Circuitry. Having passed a grad E&M course, I feel like this should make perfect sense to me and I should be able to answer all the questions it raises. Sadly that is not yet the case. What are the answers to all the unanswered questions here? In particular:   * Why does the circuit in Figure 8 not act as a transformer?   * Is it possible to extract energy from a toroid with capacitor plates as shown in Figure 9? If not, why not?   * Have structures like that in Figure 10 ever been used in practice as transmission lines?   * Last question in the article: Is it possible to create "insulating electric circuitry which conducts only displacement current"?


How does one write eigenstates of field operators in terms of particle states in scalar field theory?

I am reading the first paper in Schwinger's QED anthology, where he discusses his action principle. In this, he writes down states that are simultaneous eigenkets of the field operators at all points of a space-like surface $\sigma $, $\lvert\Phi', \sigma\rangle$, which obey $\Phi(x)|\Phi', \sigma\rangle = \Phi'(x)|\Phi', \sigma\rangle$ for any $x \in \sigma$. I am aware that $\Phi(x)$ will commute with $\Phi(y)$ if the separation between x and y is spacelike, and hence what he does is doable, at least in principle. But how does one write down such a state in terms of particle states? I have also seen this construction somewhere else (I think Peskin), where functional integrals are first introduced to deal with amplitudes of the form $\langle\Phi'_1, t_1\vert\Phi''_2,t_2\rangle$, i.e. the propagation amplitude between a simultaneous eigenket of all $\Phi(x,t_1)$ and a simultaneous eigenket of all $\Phi(x,t_2)$, and the formalism is then extended to deal with the vacuum in far past to vacuum in far future amplitude.


Why is the Pegasus launched from a subsonic airplane?

Considering that the reason typically given for launching spacecraft from sea- level as opposed to mountains is that the limiting factor is _velocity_ , not _altitude_ , then why isn't the Pegasus rocket launched from a supersonic airplane? At ~50,000 lbs, a B-1 could take two of them to Mach 2, even if the plane had to be modified as is the case with the shuttle carriers. Other planes could do, especially the easily-available F-111 or some old Russian hardware such as the Tu-22 or Tu-160. Considering that NASA now launches humans on Russian hardware this is not entirely infeasible.


What does the $\Delta$ notation mean? E.g. for potential energy: $\Delta U$ vs just $U$? What is the difference?

I've seen in this article that potential energy is defined like this: $U=-\int _{\text{ref}}^r\overset{\rightharpoonup }{F}\cdot d\overset{\rightharpoonup }{r}$. However I've seen in other text books that potential energy is: $\text{$\Delta $U}$. What is the difference? Why is "Delta" used sometimes and sometimes not? How should I interpret the "Delta" form?


Definition of transmission and reflection probability

This is a basic question, but it does not seem to be well defined anywhere. Generally, two terms are mixed somewhat randomly: transmission PROBABILITY and transmission coefficient. So to be clear, generally for a scattering problem, one defines the transmission coefficient as: $T = \frac{C}{A}$ where $C$ is the amplitude of the "outgoing solution" and A of the "incoming one" and similarly for R, the reflection coefficient. But the square of these quantities are not probabilities they do not add up to 1. I found somewhere without explanation that the transmission probability, is: $ T^2 k_2/k_1 $ where $k_2$ is the outgoing wavenumber. So to sum up, in scattering problems: what is the definition of the transmission/reflection PROBABILITY?


Why can't a superconductor make a DC motor self sustaining?

Superconducting wire can host a low current magnetic field. I do not know if it supports a corresponding electrical field. Can a superconducting wire that sustains a current accelerate a DC motor? Where is the resistance in a superconducting homopolar motor? Please explain if I am way off of target. What am I getting wrong?


Island of Stability

When I was much younger, I remember being fascinated by the thought of an Island of Stability at very high atomic numbers. However, I have not heard much on this and I was wondering   * Did this idea ever have any meaningful support in the scientific community, or was it a fringe theory?   * Is there any validity to this considering how much more we actually know about atomic construction?


atmospheric phenomenon? What causes condensation trails to converge?

This air plane just caught my eye. Two contrails apparently are flowing backward, slightly off-centered and then ultimately converge, giving the overall shape of a very narrow rhomboid parallelogram, with one diagonal being on the order of about four plane-width's. Please explain what I am seeing here as detailed as possible. Some of you may even know what class of plane this may be and thus infer the scales involved, to give some hints of the scale of the contrail under investigation? I looked up other pictures, and if I am not mistaken, most contrails are slightly angled outward. Is this by design or due to expansion and convection of the gases involved? Can someone confirm this? ![enter image description here](http://i.stack.imgur.com/yzKW3.jpg) **Edit** :   * I did observe the described contrail-shape, with quite some consistency, over varying angles of ascension and declination.   * I have never heard of conspiracies regarding contrails (e.g. chemtrails) until after writing this post. It shows that good, hard explanations for _atmospheric contrail phenomena_ are in demand.   * Given the right circumstances (large distances) some apparent contrail convergence can be attributed to perspective alone. **Links** :   * Contrail Science (impressive resource, which I just peeked at so far)   * Contrail gallery   * Contrails - clouds (ppt)   * http://www.es.lancs.ac.uk/hazelrigg/amy/Introduction/The%20Science.htm   * http://www.epa.gov/oms/aviation.htm   * http://www-pm.larc.nasa.gov/newcontrail.html   * http://www.epa.gov/oar/oap.html   * http://www.history.com/shows.do?episodeId=464914&action=detail   * http://mynasadata.larc.nasa.gov/P4.html


Galilean invariance proof

I'm studying for a physics test, but I think I don't really understand Galilean invariance. In my textbook there is an example in which they prove that if you consider 2 frames S and S' in standard configuration that the second law of Newton is Galilean invariant by proving that if $x' = x - Vt$ than $F_x = F'_x$, so this law holds in both frames. So far I understand this. However, in the book there is one assignment in which they ask me to verify that the relationship between kinetic energy and momentum, $E = p^2/2m$, is Galilean invariant. I couldn't really figure it out by myself so I looked at the answers. The answer is as followed, according to my text book: > In S: > > $E = \frac{1}{2}m\dot{x}^2;$      $p=m\dot{x}.$ > > Substitute $\dot{x} = p/m$ in the equation for the energy: > > $E = \frac{1}{2}m(\frac{p}{m})^2=p^2/2m$ > > In S': > > > $E'=\frac{1}{2}m\dot{x}'^2-\frac{1}{2}m(\dot{x}-V)^2=\frac{1}{2}m\dot{x}^2-m\dot{x}V^2$ > > $p'=m\dot{x}'$ > > Assume the relationship holds: i.e., > > > $E'=\frac{p'^2}{2m}=\frac{1}{2m}(m\dot{x}-mV)^2=\frac{1}{2}m(\dot{x}^2-2\dot{x}V+V^2)=\frac{1}{2}m\dot{x}^2-m\dot{x}V+\frac{1}{2}mV^2,$ > > in agreement with the Galilean transformation of the kinetic energy Source: McComb, W. D., 1999. _Dynamics and Relativity._ New York: Oxford University Press Inc. I understand all of the equations, however I just don't understand why this verifies that this relationship is Galilean invariant. Can someone explain this to me?


Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?

The $SU(2)$ flux defined in the context of PSG is as follows: Consider the mean-field Hamiltonian $H_{MF}=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ description of a 2D lattice spin-model, the definition of $SU(2)$ flux $P_C$ for a loop $C=i\rightarrow j_1\rightarrow j_2\rightarrow ...\rightarrow j_n\rightarrow i$ with the base point $i$ is $P_C=\chi_{ij_1}\chi_{j_1j_2}...\chi_{j_ni}$. On the other hand, the two $SU(2)$ gauge-equivalent mean-field ansatz $\chi_{ij}$ and $\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$ describe the same projected spin- wavefunction. And the $SU(2)$ flux $P_C'$ for the same loop $C$ is given by $P_C'=\chi_{ij_1}'\chi_{j_1j_2}'...\chi_{j_ni}'=G_iP_CG_i^\dagger$, in general $P_C' \neq P_C$, but an observable quantity should be invariant under the $SU(2)$ gauge transformation $\chi_{ij}\rightarrow\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$,. Thus, does this mean that the $SU(2)$ flux $P_C$ is not an observable quantity? Thanks in advance.


Can an electron interact with itself to create interference?

I have been recently brushing up my elementary physics concepts, specifically quantum physics. So, if I set up a single photon emitter in the double slit experiment, it is possible for me to see interference (I am assuming), when I thought about this I realized that a photon is mass-less and travels at the speed of light, so time does not affect a photon, therefore a photon can interact with another instance of itself (which in our frame of reference exists in another time). My question is, is the same effect observable when a single electron is fired at speeds much lower than the speed of light? If so, how? because for this to happen, the electron (which has mass and experiences time the way we do) has to physically be present at two locations at the same time (both the slits)?


Why is the Hubble Space telescope able to see farthest in the universe?

What determines how far a telescope can see in the universe? How does recording data for a very long time (~10 years) help? If we could build a telescope which work at microwave region, will it be a better telescope (so that we could see even farther)?


Why does glass break at the line where you score it?

Why does it take such a small incision for the glass to break at that spot? Why is the structural strength of the material influenced by such a small imperfection?


How can I estimate relative humidity ($e/e_s$) from temperature and dewpoint?

## Background my goal is to estimate vapor pressure deficit ($vpd$) from Relative Humidity ($rh = \frac{e}{e_s}$): $$vpd = \frac{100 - \frac{e}{e_s}}{100} e_s$$ (from Hartmann "Global Physical Climatology) But it is surprisingly difficult to find values of climatological mean $rh$ (e.g. see motivating question at gis.SE. It turns out that WeatherUnderground provides values of dewpoint. ## Problem At first I thought it would be trivial to convert from dewpoint to $e$ or $rh$, but according to Lawrence 2005, there is no direct conversion, only a "highly accurate conversion" to get from water vapor pressure ($e$) to the dewpoint ($t_d$) (equation 7): $$t_d = \frac{B (ln\frac{e}{C})}{A - ln (\frac{e}{C})}$$ Where $[A,B,C]$ are empirical constants$[17.625, 243.04^oC,610.94Pa]$ ## Questions   1. why is there no exact relationship between $e$ and $t_d$?   2. how can I solve the last equation for $e = f(t_d)$ (or $rh$)?


Calculate temperature at isobaric or isochoric process

I've had some trouble in regards to calculate the temperatures in the different states. Considering the following pV-diagram: ![enter image description here](http://i.stack.imgur.com/j14ca.jpg) Here is the data I have: The adiabatic constant $\gamma$ is 1.4 $C_{mv}=20.8\frac{J}{mol\cdot K}$ $V_{2}=100\cdot 10^{-6}\ m^{3}$ $T_{2}=300\ K$ $p_{2}=100000\ Pa$ $V_{4}=4\cdot V_{3}$ Now my question is: How do I calculate the temperatures at the states 1, 3, 4 and 5. And how do I calculate the temperature or pressure at any given state in general? And how do I determine if for instance the initial state 1 is isobaric, isochoric or even both??


Hydrostatic pressure at lateral directions

I just read that, with respect to a stationary tiny cube, suspended in a fluid, that has a negligible weight and dimensions: > Pressure is the same in every direction in a fluid at a given depth, if it > weren't the fluid would start to move. This made sense to me the first time, but then I realised that the condition for the fluid not to move is that the net force is zero. However, the net force being zero doesn't necessarily require that forces in all directions are equal, just that two forces pointing in opposing directions are equal? So say, the vertical forces on a cube, pointing up and down, are 1N each, but then all the other forces pointing towards the sides of the cube, may as well be 10N each, and yet all the forces would cancel each other out, resulting in a net force of zero. Am I missing something here?


Are the intersections of past and future light cones spacelike?

Given a timelike reference worldline (not necessarily geodesic), we can define light-cone coordinates $\tau^+$ and $\tau^-$ so that the 3-D hypersurfaces of constant $\tau^+$ are past light cones of events on the reference worldline, the value of $\tau^+$ being reference proper time. Similarly $\tau^-$ for future light cones.   * Is there a common name for the 2-D surfaces of constant $\tau^+$ and $\tau^-$ induced by a timelike worldline?   * Is it correct that these are "locally spacelike" in the sense that any curve entirely within a surface is spacelike?   * Under what assumptions are they also "globaly spacelike" in the sense that there are no causal (timelike or null) curves between any two points on a surface?  From this we can also do a simple coordinate change to $r^\star = \frac{\tau^+ - \tau^-}{2c}$ and time $\tau^\star = \frac{\tau^+ + \tau^-}{2}$. In flat spacetime with a geodesic reference worldline, the 3-D hypersurfaces of constant $\tau^\star$ are just the usual orthogonal simultaneous spaces. [EDIT: should have been $r^\star = \frac{\tau^+ - \tau^-}{2}c$ ]   * Is there a common name for the hypersurfaces of constant $\tau^\star$ ?   * Are they also locally spacelike?   * Under what (very strong I expect) assumptions are they globally spacelike? ### Addendum In response to the request to clarify the co-ordinates: To illustrate, suppose spacetime is flat, and suppose the chosen reference worldline is geodesic. It can then be taken to be stationary at the spatial origin of Minkowski co-ordinates WLOG. In that case, $\tau^+$ is $t + r/c$ and $\tau^-$ is $t - r/c$, since an event at time $t$ and distance $r$ would meet the past and future lightcones originating from the reference worldline at those times, respectively. As a result $\tau^\star = t$ and $r^\star = r$. Of course this is not the case I am asking about. It is a general definition, not restricted to a geodesic reference worldline, nor to flat spacetime. Summary: if I am the reference worldline, then $\tau^+$ is what my watch is reading when I see the event. $\tau^-$ is what my watch is reading when the event sees me.


Tricky operator identity: $[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$?

This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \\{ L^2, \vec{r}\\}$$ The curly brackets denote the anticommutator, $AB+BA$. The $\vec{r}$ operator is the position operator. The $L^2$ operator is given by: $$L^2 = -\hbar ^2 \left( \frac{1}{\sin \theta} {\partial\over\partial\theta} (\sin \theta {\partial\over\partial\theta}) + \frac{1}{\sin^2 \theta} {\partial^2\over\partial\phi^2}\right)$$ Is there a way of proving this identity without tediously expanding all the commutators? I've been trying to find one but was unable to.


Kinematics - concept question

A child tosses a ball directly upward. Its total time in the air is T. Its maximum height is H. What is its height after it has been in the air a time T/4? Neglect air resistance. Ok so I know that there is no x-component to the velocity: $$D=v_0 t+\frac{1}{2}at^2$$ $$\frac{1}{2}D=H$$ so, $$\frac{1}{2}H=v_0(\frac{1}{4}t)+[(4.9)(\frac{1}{4}t)^2]$$ I am confused as to where to go from here? I am overcomplicating this, missing the underlying concept. I don't want the answer, but just some guidance.


How are electromagnetic waves differentiated?

I would like to know how the signals for remote controlled cars, radios, etc.. That use radio waves are told apart from each other. I know that the radio waves are modulated to encode data and the frequency or amplitude are changed, so then the waves are propagated through the air and received at another location via a receiver that is tuned to a certain frequency that the waves were emitted, but I'm sure in most places in the world by now there are numerous amounts of waves traversing at any point, why doesn't the receiver of this device happen to catch another wave of the same frequency instead of the one that was intended? or is there anything that stops me from having a device that emits a wide range of frequencies or amplitudes that would manipulate nearby electronics?


Which planet in the solar system is it?

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


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

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


Electrons, spins, and degeneracy

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


Conservation of mass in fluid flow

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


Mechanics Landau Galilean Principle

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


Eigenstates of an observable

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


how to calculate acceleration?

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


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

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


Sommerfeld results & van Hove singularities

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


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

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


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

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


Finding the equation of motion of anharmonic potential

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


CPT invariance of Dirac equation

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


Uniformly charged sphere's electric field

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


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

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


Bleaching groundstate

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


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

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


What does it take to understand Maxwell's equations?

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


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

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


energy of deflection of a moving object

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


intensifying a light source

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


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

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


about field gradient

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


Apparent dimensional mismatch after taking derivative

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


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

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


Simple argument for unexpected behavior in SUSY model

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


Is there a momentum for charge?

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


Variable area manometer duct

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


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

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


can one introduce magnetic monopoles without Dirac strings?

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


A fly in an accelerating car

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


Mosquito in a Bus

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


Butterfly inside the accelerated train

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


Does the Milky Way have dark matter satellite galaxies?

This recent paper by Weinberg et al. discusses that one potential problem with our current model of Cold Dark Matter (CDM) is that is predicts a greater number of satellite galaxies for the Milky Way than are actually observed, and satellite galaxies with larger masses than those the Milky Way has (although the paper does say that many of the Milky Way's satellite galaxies may be too dim for us to currently observe). I know that at least dark matter galaxies are believed to have been observed (see this article). Is there any observational or other evidence for the existence of dark matter satellite galaxies to the Milky Way? And how have they been found? Edit: As has been pointed out, this is a question we likely don't have the answer to yet (otherwise there wouldn't be recent papers posing the question still), so perhaps a better question would be how could/would you find dark matter satellite galaxies of the Milky Way?


Speed of Magnetic Signal over Large Distance

If I had a very strong magnet on Earth and a very sensitive compass on Mars (just using planets to illustrate large distance), how long would the compass take to notice if I turned the magnet 180deg? I assume it can't beat the speed of light. Do the outer reaches of the magnetic field move as a rigid body with the turning magnet or do they lag in motion?


What are the limits of validity for the magnetic field of a solenoid?

1) The field outside a solenoid is approximated to be zero, because of opposites points "cancelling out". Does this approximation of the field being almost zero become worse as the diameter of the solenoid increases? (since the opposite sides are further apart) I'm looking for an answer for two cases: 1) Only the diameter increases, but all other dimensions remain the same. 2) All dimensions increase proportionally, keeping the aspect ratio the same. 2) Are solenoids approximated as a series of perfect circles (since opposite sides cancel only in a circle)? And consequently shouldn't a stretched out solenoid's field be completely different since its nothing like a series of circles? **EDIT:** ![enter image description here](http://i.stack.imgur.com/IHvMQ.png) The field at P is the superposition of the fields from the opposite "points". If the diameter of the solenoid increases, the bottom "points" are further away from $P$, so their contribution to the field decreases. Therefore the assumption that the fields from the top points cancel with the bottom points becomes falser/less valid.


How do inductors produce current?

This is how I understand batteries and capacitors. The terminals produce an electric field which causes the electrons to move in random directions. The surface charge builds up causing new electric fields to be created, which forces the electrons to move in the direction of the wire, creating current. But with inductors, there is no negative or positive terminal (as in an end with a surplus or deficit of charge), so, on a microscopic level, how does it produce current?


If the wave function can be collapsed, can we collapse the particle function?

No doubt it's been questioned before so what is the theory or experiment? If there is an experiment.


Symmetry breaking

What is a good place to learn the details of symmetry breaking? What I am looking for is a more serious exposition than the wiki-article, which explains the details, especially the mathematical part, but at the same time less detail than the typical quantum field theory book. Preferably I would like to see a clean made up example with all the details, but not necessarily a real example, what one can see in most books, which naturally has a lot of details that are there because that's what the world is, and not to illustrate the idea. I hope it is clear what I am looking for. EDIT: Just to clarify. I am looking for a reference, but if somebody is willing to write an example here, I would be more than happy.


Orbifold CFT of SU(2)/G and SO(3)/G

In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed. In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. (i.e. "... it would be very interesting to devise operations that act on the space of RCFT's and generate new theories from a given one. As an example one can think of the construction of nondiagonal modular invariant combinations of characters, coset models, etc. Another operation of that kind is the concept of **an orbifold**.") In p.521, Table 6 of this paper, they provide an example of SU(2)/H$_8$ $\simeq$ SU(2)/($Z_2 \times Z_2$) orbifold. (here H$_8$ is a quaternion group.) **Question** : $\bullet$ Here I am interested in **orbifold CFT** of **SU(2)/G and SO(3)/G** kinds (with G as its normal subgroup) . Can someone provide some more examples and data (i.e. S and T matrix, fusions of quasi-particles) of **SU(2)/G and SO(3)/G** **orbifold CFT**? $\bullet$ What are their **central charges** $c$ of this 1+1D CFT? (e.g. what is the central charge of this SU(2)/D$_2$ model?) What is their **ground state degeneracy** of 2+1D bulk TQFT on the spatial $T^2$ torus? $\bullet$ If there is also information on its corresponding **twisted quantum doubles** , in the form of $D^\omega(G')$ of some certain group $G'$. It will be nicer. Reference and books are welcome, but please **offer summaries** instead of just giving Reference. (ps. Francesco yellow-book CFT Sec 17.8.4 has some discussions, but I could not find orbifold CFT of $SO(3)/D_n$ of $D_n$ dihedral group or $SU(2)/H$ of $H$ as a quaternion.)


Renormalization Group for non-equilibrium

For equilibrium/ground state systems, a (Wilson) renormalization group transformation produces a series of systems (flow of Hamiltonians/couplings $H_{\Lambda}$ where $\Lambda$ is the cut-off) such that long-wave/asymptotic behaviour of $H_{\Lambda}$ is the same as of $H_{\Lambda'}$ after rescaling by $\Lambda/\Lambda'$. The idea of this definition implies an exact starting point for RG formalisms, with technical details varying between the fields and approximation methods. (For examples, see arXiv:1012.5604 and Wikipedia article). Now, for **non-equilibrium condensed-matter systems** there is research direction aiming at generalization of the RG approach to a steady state, e.g., a voltage-biased strongly interacting quatum dot (or Kondo impuryity). For examples, see arXiv:0902.1446 and related references. I would like to understand the conceptional foundations for the non-equlibrium RG. > What is the definition of an RG transofrmation in a non-equilibirum, steady > state ensemble? I see a problem in the fact that the non-equilibirum desnity matrix which is used to define the problem is not a function of the Hamiltonian alone, thus it is not clear to me how is the effect of the change in the cut-off is split between the Hamiltonian (running couplings) and the density matrix (renormaltizaton of the boundary/external conditions?)


Which microscopy technique is for wet condition?

I am searching for a microscopy technique in wet conditions. It must be possible to see through the specimen. ESEM is good but can't see through. TEM needs high vacuum which will boil away water. The specimen is embedded inside a gel. The size of the specimen is between 0.5 µm and 20 µm.


Does spin of the earth vary at different locations?

Does the spin of the earth around its axis vary at different distance from the sun?,if so how does it correspond to the length of day and night,?


Conjugate of an operator applied to a function

In section 6.3.1 of the following MIT Open Course Ware (PDF) (22-02, Introduction to Applied Nuclear Engineering), the author, Prof Paola Cappellaro, derives Heisenberg's equation using the definition of expectation values, $$ \frac{d}{dt}\langle\hat{A}\rangle=\frac{d}{dt}\int d^3x\psi^*(x,t)\hat{A}\psi(x,t)\\\ =\int d^3x\left(\frac{\partial\psi^*}{\partial t}\hat{A}\psi+\psi^*\frac{\partial\hat{A}}{\partial t}\psi+\psi^*\hat{A}\frac{\partial\psi}{\partial t}\right) $$ and Schrodinger's equation, $$ \frac{\partial\psi}{\partial t}=-\frac{i}{\hbar}\hat{H}\psi,\qquad \frac{\partial\psi^*}{\partial t}=\frac{i}{\hbar}\left(\hat{H}\psi\right)^* $$ Then the author uses $\left(\hat{H}\psi\right)^*=\psi^*\hat{H}^*=\psi^*\hat{H}$ to get $$ \frac{d}{dt}\langle\hat{A}\rangle=\int d^3x\left(\frac{i}{\hbar}\psi^*\left[\hat{H}\hat{A}-\hat{A}\hat{H}\right]\psi+\psi^*\frac{\partial\hat{A}}{\partial t}\psi\right)\\\ =\frac{i}{\hbar}\langle\left[\hat{H},\,\hat{A}\right]\rangle+\langle\frac{\partial\hat{A}}{\partial t}\rangle $$ There are two things I don't understand in this derivation. The first is that the total time derivative $\frac{d}{dt}$ turns into a partial time derivative $\frac{\partial}{\partial t}$ when inside the integral. The second is that the author states $$(\hat{H}\psi)^*=\psi^*\hat{H}$$ This confuses me as $\hat H$ is the Hamiltonian operator, and contains an $x$ derivative. Thus the left hand side of the above equation is another state $\phi^*$ such that $\phi=\hat H\psi$, but the right hand side is an operator: the derivative is yet to be applied to anything. This equation by itself just makes no sense to me. Any help with either of these two issues would be greatly appreciated!


What is the simplest way to prove the Earth is round?

Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin. How would you prove that the Earth is, in fact, round?


Proofs that Earth is not flat?

Well, I do not believe that Earth is flat, but I met some conspirationistas believing that all physics we learn in school is twisted, all the Earth's pictures from space are photoshopped by NASA, etc, etc. So those guys wanted to put a satellite into orbit, take some pics, and see for themselves. Of course, a satellite won't stay above a flat Earth, but what other very simple means to prove/check that earth is round (well, geoid) do we have? I am looking for simple means, available to anyone, that **do not need any external tool or knowledge** that a naysayer would reject. Some things I've considered:   * Meteorological balloon with camera and a phone with GPS, with a parachute to recover it. It has been done by amateurs, and in good conditions, you can see the Earth's curvature.   * Measure the Sun's height above horizon at noon at two different latitudes. The difference, along with some simple geometry would give you Earth's diameter.


What if Correspondence principle in QM happening in classical sense?

I assume a simple set up (hope to generalize it later)... suppose there is a position pdf (need not be position but any) (probability density function) which is the magnitude of a $\Psi(x)$ predicted by QM (lets call it P_quantum) and let there be a classical prediction pdf (P_classic), according to CP, for large quantum numbers, P_quantum agree with P_classic in an average sense, but my question is for suppose, what if it agress to converge to P_classic(x) in pointwise convergence, while still explaining all the microscopic things like hydrogen atom..blah blah, what would that mean philosophically? for example see Particle in a 1-D box and the correspondence principle For supose What if classical mechanics originate from QM in a classical sense ( it currently does in only statistical sense) but still explaining all the microscopic things like Hydrogen atoms, atomic spectra etc..., what would that mean? Right now they converge only in statistical sense, hence they say you need not converge pointwise as anyway you lack measuring device at that microscopic level for macroscopic things. What if they did converge pointwise, would that mean the lack of measuring device argument disappear from the QM scene?


How to calculate the charge on a capacitor

![enter image description here](http://i.stack.imgur.com/YiGA9.png)I have a network in which $C_1$ ($8.73 \;\rm \mu F$) and $C_2$ ($2.46 \;\rm \mu F$) are connected in parallel and $C_3$ ($7.45 \;\rm \mu F$) and $C_4$ ($2.22 \;\rm \mu F$) are connected in parallel. $C_{12}$ and $C_{34}$ are connected in series. The network is connected to a $21.9\;\rm V$ battery. I calculated the total capacitance to be $5.187 \rm \;\mu F$. How do I find the charge on $C_1$? I use the equation $Q=CV$ right? So I did $5.187\;\mu F$ multiplied by $21.9\;\rm V$ and got $113.6\;\rm\mu C$ but that is not the right answer? What am I doing wrong?


Loudspeaker frequency response dip

I'm trying to determine the reason for a dip in the frequency response curve of my loudspeaker. I have a loudspeaker mounted on the inside of an enclosure, and a 100mm ID tube mounted in-front of the speaker on the outside of the enclosure. The measured frequency response is: ![enter image description here](http://i.stack.imgur.com/AsZbD.png) At roughly 4100 Hz, a large dip in the response is noted. This dip corresponds to the cut-off frequency of the (0, 1) cross mode. This leads me to believe that it has something to do with the higher-order mode propagation, but I don't understand what. Could someone help me understand this problem please.


Interpretation of the wave function in quantum mechanics

I just started watching the coursera lectures on the basics of quantum mechanics and one of the first lectures were on deriving Schrodinger's equation and its interpretation it under Born's interpretation. What I want to ask is what the wave function \begin{equation} \psi({\bf r},t) \end{equation} return and represent. Now I know that \begin{equation} |\psi({\bf r},t)|^2dxdydz \end{equation} is the probability of finding the quantum particle described by \begin{equation} \psi({\bf r},t) \end{equation} in the volument element \begin{equation} dV = dxdydz \end{equation} at time T. But im not sure what the wave function $\psi$ returns. Could someone please explain in laymans term the return type of the $\psi$ function and what the lone $\psi$ function represents.


Why does water have several different solid phase but only one liquid and gas phase? Is there any meaning? or any reason behind it? Or is it just the way the nature behaves?


Reflection positivity in general

In the Euclidean QFT obtained by "Wick-rotating" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field. What's the correct formulation if you have chiral fermions and/or terms like the QCD theta angle? Could someone give the references?


Is there a simulator for observing the force between two charges comprehensively?

I basically want to observe the electrostatic force between two charges. I want to know if is there a simulator which could both help me to observe and calculate the force between two charges. I found this simulator which is pretty good I think. However, that would be great if there is a more comprehensive simulator with greater distances, more parameters etc. I know this is a duplicate but nobody answered the other question and I really want to know if there is a simulator like that. I am sorry if I broke the rules. Thank you.


Can all quantum superpositions be realized experimentally?

When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally. On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can "engineer" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\Psi(x)=\frac{4}{5}\phi_1(x)+\frac{3}{5}\phi_5(x)$, where {$\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\Psi=\sum a_n\phi_n(x)$. That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation? It is kind of frustrating to study for long hours/read/solve problems/HW on all kinds of potentials and on superpositions without knowing how/if they can be realized in experiment. If someone knows references in which this issue is discussed it would be greatly appreciated.


Radiation pressure at high temperature

I am reading a book on the history of the hydrogen bomb. The author describes the Teller-Ulam design and explains that the liquid deuterium charge was compressed by the radiation pressure generated by a vaporized plasma of polyethylene. This would allow the generation of pressures orders of magnitude higher than with high explosives. Is it possible to quickly estimate this pressure? If I assume a temperature of $T=1,000,000K$ and use $P=\frac{4\sigma}{3c}T^4$ I calculate a pressure of about $2.5$ million bars. Is this realistic? How does it compare with explosives?


How can the Lorentz transformation cause two objects' masses to both increase?

Suppose there are two people **A** and **B**. A is standing on the earth and **B** is on a rocket in space. Now **B** passes by the earth at a speed of 0.9c and **A** sees him. According to **A** , **B's** mass would have increased. But since the lorentz factor considers only relative velocity, we can say that even **A** is moving at a speed of 0.9c relative to **B**. So will **A's** mass also increase according to **B** after applying the the lorentz transformation formula? If yes then my doubt is how can the mass of both increase?


How strong is electron degeneracy pressure?

I'm trying to get some specific numbers for electron degeneracy that I can understand, using a concrete example. Take for example this portion of carbon crystal: ![carbon crystal lattice](http://i.stack.imgur.com/WODZS.gif)   1. Exactly how much energy would be required to overcome the electron degeneracy pressure of these 18 carbon atoms, in a theoretical laboratory capable of such things.   2. I believe that neutrinos would be emitted as a byproduct of the collapse, is that correct? What other reactions if any would be needed to fully account for conservation of energy/momentum etc?


Group rings in Physics

In my institute a few mathematicians work on Group Ring. Since it has close connection with representation theory, I thought that there must be some interesting connections of it with Physics. However I did not manage to find any information from our local experts; perhaps for one reason, they are more interested on finite groups and its structure. Hence I am asking the same question to the community. Does there exists any connection between group rings and physics? (I am having a feeling, that it can be used as a tool for solving representation theoretic problems. But does its usefulness (if there is any) end there only?) Advanced thanks for any reply. Also please provide some references (paper or books), which describes such connection(s), if exists,


How to calculate specific heat capacity of an alloy

I have done an experiment in which I measured the specific heat capacity of a coin. I would like to compare the results to the actual value. I have found out that the coin is made of Cupronickel alloy (for simplicity let's assume it's just 75% copper and 25% nickel). Knowing the SHC values of the two elements (copper and nickel) how would I go about calculating the theoretical value of the SHC of the Cupronickel alloy, so that I can compare it to my experiment's results?


Equation for Hubble Value as a function of time

I am trying to write the equation for the situation where the Hubble parameter $H$ would be changing over time. In other words, it would represent an accelerated expansion of the Universe. That is, $H$ can no longer be the simple $H=1/t$. In the new equation, I should be able to plug a future time and see what the Hubble Value will be in that future. I think I got most of the concepts right. First of all I understand that the key to the problem is in $H=a˙(t)/a(t)$. Where $a(t)$ is the scale factor from the Friedman equations. I also understand that if $H$ is changing then $a¨(t)> 0$ and also $H'(a)>0$. But I'm still very uncomfortable when my pencil meets the paper. The Friedman equations are not stated as a function of $t$, but as a function of $a$ where $a$ is the time scale, and frankly, I don't know how to use the time scale factor. In any event, here is my poor attempt to do it. According to Wikipedia, one of the solutions of Friedman equation is (assume flat space k=0): $a(t) = a_0 t ^{2/3(w+1)}$ Therefore: $a'(t) = d(a_0 t ^{2/3(w+1)}) / dt$ $a'(t) = (2a_0/3(w+1)) t^{ -(1+3w)/3(w+1)}$ And I suppose that we could now substitute: $H=a˙(t)/a(t)$ with the above: $H = (2a_0/3(w+1)) t^{ -(1+3w)/3(w+1)} / a_0 t ^{2/3(w+1)}$ Simplified: $H = (2/3(w+1)) t^{-(3w+2)/3(w+1)}$ And $w$ is typically known from observation. I will appreciate if someone can let me know if I am in the right path or totally derailed. I have a feeling that $a(t) = a_0 t ^{2/3(w+1)}$ was not the right place to start because if $w=-1$, then everything goes down the drain. But then again, in an accelerated expansion, $w$ would not equal -1. It would always be less than -1. Also, in the final equation, if $w<-1$ then H<0, which could not be right. So I'm not sure what to think. Many thanks in advance, Luis


Rotation, cats landing on their feet, and conservation of angular momentum

Let _θ_ be the orientation (angle) of a body (such as a cat), and let _ω_ be its angular velocity. It is well-known that _θ_ can change even when the body is not rotating, using the conservation of angular momentum; that is, even when _ω_ = d _θ_ /d _t_ = 0. That's how cats land on their feet so well. But how can _θ_ possibly ever change, when its derivative is zero?! What's wrong with the math?


Determining $y$ coordinate of charged particle motion

I've tried several different ways, with a tutor, and with other colleagues. One of my colleagues has even gone to the professor with whom he worked out an equation for the component, and it was later confirmed to be incorrect. Here is the question: > A proton ($q = 1.6 \times 10^{-19}$ C, $m = 1.67 \times 10^{-27}$ kg) moving > with constant velocity enters a region containing a constant magnetic field > that is directed along the $z$-axis at $(x,y) = (0,0)$ as shown. The > magnetic field extends for a distance $D = 0.55 m$ in the $x$-direction. The > proton leaves the field having a velocity vector $(v_x, v_y) = (4.8 \times > 10^5, 1.8 \times 10^5)m/s$. This is the diagram: ![enter image description here](http://i.stack.imgur.com/EDPe3.png) The question we have is how to determine $h$, the $y$ coordinate of the proton as it leaves the magnetic field. I've already determined that the velocity $v$ is equal to 512640.2247 m/s, and that the radius $R$ is 1.533 m. The trajectory of the proton is curved on the inside, so we can't use anything involving any derivation of similar triangles. I've tried solving for $v$ as a function of time, but then realized that $v_y$ is always changing anyway, which lead me to calculate the acceleration so the time term that I used incorporated the change, but that didn't work either. Edit: Really sorry folks, I wrote this in a hurry to get to work on time. Yes, the radius I mentioned is indeed the radius of the circular trajectory of the proton after it enters the magnetic field. Also, I do understand to keep things variable, and I mentioned the radius and velocity only as to sort of show what I have available to work with. I've drawn the imagined circle suggested below, and I've made the realization that it would be helpful to relate the radius, the angle, and the velocity, but I'm not sure how to do so accurately.


What areas of physics depend on the sum $1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12$?

This youtube video from Numberphile, http://youtu.be/w-I6XTVZXww shows how the value is derived. In the video, one interviewee claims that "this result is used in many areas of physics". In the video, only string theory is mentioned. Which areas of physics use or depend on the sum $$1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12?$$


Pair production - mathematically?

Allover the web i am only seeing a statement similar to this: > Pair production is not possible in vaccum, 3rd particle is needed so that > conservation of momentum holds. Well noone out of many writers shows, how to prove this matematically. So this is what interests me here. * * * **First i wanted to know if pair production really cannot happen in vacuum** , so i drew a picture and used equations for conservation of energy and conservation of momentum to calculate energy of a photon $(h \nu)$ needed for pair production. It turns out $h\nu$ is different if i calculate it out of conservation of energy or conservation of momentum. And even more! It can never be the same because equallity would mean parts $v_1 \cos \alpha$ and $v_2 \cos \beta$ should equall speed of light $c$. Well that cannot happen. Below is my derivation. * * * ![pair production](http://i.stack.imgur.com/wGF1v.png) CONSERVATION OF ENERGY: $$ \scriptsize \begin{split} W_{1} &= W_{2}\\\ W_f &= W_{e^-} + W_{e^+}\\\ h\nu &= W_{ke^-} + W_{0e^-} + W_{ke^+} + W_{0e^+}\\\ h\nu &=\left[m_ec^2 \gamma(v_1) - m_ec^2\right] + m_ec^2 + \left[m_ec^2 \gamma(v_2) - m_ec^2\right] + m_ec^2\\\ h\nu &=m_ec^2 \gamma(v_1) + m_ec^2 \gamma(v_2)\\\ h\nu &=m_ec^2 \left[\gamma(v_1) + \gamma(v_2) \right]\\\ \end{split} $$ * * * CONSERVATION OF MOMENTUM: $y$ direction: $$ \scriptsize \begin{split} p_{1} &= p_{2}\\\ 0 &= p_{e^-} \sin \alpha - p_{e^+} \sin \beta \\\ 0 &= m_e v_{1} \gamma(v_{1}) \sin \alpha - m_e v_{2} \gamma(v_{2}) \sin \beta\\\ &\text{If $\boxed{\alpha = \beta} \Longrightarrow \boxed{v_1 = v_2}$ and:} \end{split} $$ $$ \begin{split} \scriptsize 0 = 0 \end{split} $$ $x$ direction: $$ \scriptsize \begin{split} p_{1} &= p_{2}\\\ \frac{h}{\lambda} &= p_{e^-} \cos \alpha + p_{e^+} \cos \beta \\\ \frac{h \nu}{c} &= m_e v_{1} \gamma(v_{1}) \cos \alpha + m_e v_{2} \gamma(v_{2}) \cos \beta\\\ h \nu &= m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big] \end{split} $$ Alltogether: Because momentum in $y$ direction equals 0 (holds for some combinations of $\alpha, \beta, v_1, v_2$) whole momentum equals just the momentum in $x$ direction. So if i add them i get: $$ \scriptsize h \nu = m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big] $$ * * * From this i can conclude only that i cannot sucessfully apply conservation of energy and conservation of momentum at the same time and therefore pair production in vacuum cannot happen. **QUESTION1:** Why do writers state that 3rd particle is needed so that conservation of momentum holds? What if conservation of momentum holds and conservation of energy doesn't? How can we say which one holds and which one doesnt? **QUESTION2:** Do writters actually mean that if a 3rd particle is included we can achieve $h \nu$ to match in both cases? **QUESTION3:** Can someone show me mathematically how this is done? I mean it should right?


Induced charge on sphere

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?


How do forces work

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.


Relativity of simultaneity - An example

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?).


Does a photon have a north and south pole?

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


Average Neighbouring Impurity Separation in a Random 1D chain

![](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)


Finding the correct units for the energy-momentum tensor?

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


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

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


Modelling multiple heat sources?

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?


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

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.


Dissipating light pollution

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?


How far we can see in the thickest possible fog?

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.


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

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 $(2)$?


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

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?


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

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?


Why do arrows point backwards in time for Feynman Diagrams?

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!


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

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?


splitting spin representation when reducible

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.


transfer of electricity through electromagnetic waves

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


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

> **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.


Quantum mechanics with non-cartesian coordinates

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?


Numerical tests of Titius–Bode formula for artificial planetary orbits

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?


"Rocket in a box" thought experiment

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?