title: Zwiebach scalar product notation

text: I am currently working through Zwiebach's a First Course in String Theory. He seems to use dot-product notation interchangeably with the "down-up" notation. For example, on pg 176/section 9.1, he talks about the "gauge condiction $n\cdot X = \lambda \tau$ fixing the string...", "points $X^\mu$ that satisfy $n_\mu X^\mu=\lambda \tau$ are points that lie both on the world-sheet and on the hyperplane...", "the string with world-sheet time tau is the intersection of the world-sheet with the hyperplane $n \cdot x = \lambda \tau$ as illustrated...", and the illustration has an arrow pointing to said hyperplane with the $n_\mu x^\mu = \lambda \tau$ label. Does anybody know why he does this?


title: Pendulum in a Turning Airplane

text: My friend and I are having an argument over what happens with a pendulum hanging inside an airplane during a turn. Assuming that the airplane is making a standard 1 G coordinated turn, how does the pendulum react? I believe because the pendulum is attached to the string, it is pulled by the plane and will be in exact alignment with the bank angle of the plane. My friend says that the pendulum has a weight and will be between the bank angle of the plane and parallel with the earths gravity. My line of thinking is that since the plane produces lift which must be greater than the weight, that the net force acting on the pendulum is in line with the bank angle of the plane. Since the pendulum is attached to the plane I don't see how its weight can be factored differently even if it is being pulled by the string which is lifted by the plane. Can someone explain how the pendulum would react and the forces at work.


title: Molecules and electrons energy types

text: What are the types of energy that an atom or a molecule could have? For example they have kinetic energy, could they also have other types?


title: How can the reduction postulate be removed with the other postulates of QM still leading to correct predictions?

text: In the axiomatic presentation of QM, I've seen it stated many times that the reduction postulate is not needed and/or incorrect, and could be gotten rid of. However, without the reduction postulate, wouldn't tests of QM simply yield wrong results? If I measure a system that is in some linear superposition $\psi=\sum c_i |e_i\rangle$ of eigenstates of my observable $O$, the measurement postulate says I will get one of the eigenvalues as a result, with probability $|c_i|^2$. But now, since I no longer have the reduction postulate, I will calculate that as a result of the interaction with my measurement apparatus $\psi$ evolved linearly and unitarily into $\psi'$, and therefore still consists of (possibly a different) linear superposition $\sum c_i' |e_i\rangle$ of eigenstates of $O$. Therefore, again applying the measurement postulate, I calculate that I will now get $|e_i\rangle$'s eigenvalue with probability $|c_i'|^2$. After a sequence of $n$ such measurements, close together in time, the probability that I will get some particular $|e_j\rangle$'s eigenvalue in all the $n$ measurements is $(|c_j|^2 * |c_j'|^2 * ... * |c_j^{(n)}|^2) << 1$. However, we know from experience that making $n$ close in time measurements of the same observable will yield the same value with a probability of $1$ (or very close to $1$). So I don't see how the reduction postulate can just be dropped without either QM giving incorrect predictions, or without modifying some other postulates somehow?


title: Higgs Boson and its relation to the vacuum

text: The Higgs boson and the electroweak theory used symmetry breaking from condensed matter physics as its inspiration. The BCS theory of superconductivity is one such condensed matter symmetry breaking theory. I believe this theory uses some kind of model for the underlying atomic structure of the metal. In a similar manner the vacuum works in the same way with particles popping in and out with charges all the time. Is there a relation between the vacuum and the atomic structure of condensed matter. If so, could one define the vacuum with particular properties to match the Higgs potential? I know the Higgs potential is a vaccuum, but can properties be assigned to space itself and the fluctuating charged particles that pop in and out that are analogous to the charged atoms in a condensed matter metal and produce the Higgs fields this way? Maybe the vacuum is discrete like a metal lattice instead of continuous.


title: The relationship between the energy and amplitude of a wave? Derivation?

text: From multiple online sources I read that $$E \propto A^2$$ but when I mentioned this in class, my teacher told me I was wrong and that it was directly proportional to amplitude instead. As far as I know, every website I stumbled upon concerning this said that is the case. My teacher has a Ph.D and seems pretty experienced, so I don't see why he would make a mistake, are there cases where $E \propto A$? I also saw this derivation: $$\int_0^A {F(x)dx} = \int_0^A {kx dx} = \frac{1}{2} kA^2$$ located here, does anyone mind explaining it in a bit more detail? I have a basic understanding of what an integral is but I'm not sure what the poster in the link was saying. I know there is a pretty good explanation here, but it seems way too advanced for me (gave up once I saw partial derivatives, but I see that they're basically the same later on). The first one I linked seems like something I could understand.


title: How to get conductivity from Green function $\mathcal{G}(x_1,x_2,\tau)$ of inhomogeneous system?

text: I'd like to study an inhomogeneous system, i.e., momentum is not a good quantum number therein. Therefore, I tried to calculate temperature Green functions like $\mathcal{G}(x_1,x_2;\tau)$, or its twofold Fourier transformation $\mathcal{G}(p_1,p_2;\tau)$. But how can I get any transport property, e.g., conductivity, from these Green functions? I checked Mahan's overwhelming book, however, it only deals with the formalism of $\mathcal{G}(p;\tau)$ for homogeneous systems. Thanks in advance for any useful information.


title: What is the generalization, if any, of the weak and dominant energy conditions to SUGRA?

text: In standard general relativity, we have the null energy condition, the weak energy condition related to stability, and the dominant energy condition related to forbidding superluminal causal influences. With a cosmological constant added, the standard lore is to subtract the cosmological constant contribution first. However, strange things happen in anti de Sitter space with a negative cosmological constant. It can easily be shown that for an asymptotically AdS space, it's possible to include tachyonic scalers, as long as their mass squared isn't lower than $-ck^2$ where k is the radius of curvature and c is a precise coefficient which depends upon the precise units chosen. Such a model will still be stable, and the ADM energy still turns out to be nonnegative! Interestingly enough, this lower bound is satisfied naturally in SUGRA theories. Clearly, the weak energy condition is violated, but this doesn't rule out stability. Another example. Suppose a SUGRA theory has one or more chiral superfields and admits two or more stable supersymmetric phases. If the superpotential is W, they are characterized by $\partial W =0$. In SUGRA, the cosmological constant goes as $-|W|^2$. Suppose phase I has a smaller $|W|$. Then, it might appear at first glance that it is metastable, and will eventually decay to phase II after an exponentially long time. Certainly, with respect to phase I, WEC is violated. That's not so. Magically, the domain wall tension and hyperbolic geometry come to the rescue. The domain wall tension goes at least as $c\left[ |W_2|^2 - |W_1|^2 \right]$ as obtained from BPS analysis. Once again, the coefficient c in SUGRA is such that the total ADM energy still goes up. In fact, c saturates the bound. It differs from the flat spacetime case where the energy goes down by $R^3$ where R is the bubble radius and the tension contribution only as $R^2$ so that for large enough R, the total energy will go down. Because of hyperbolic geometry, asymptotically, both contributions scale exponentially in R. Naturally, the question is what is the proper generalization of WEC to SUGRA for purposes of proving stability, and the generalization of DEC for causality?


In positivism, what counts as empirical? Is it the readings of instruments, or the notes on a notebook or computer, or is it the sense perceptions of an experimenter? There certainly is the case we can have no direct experience of any of those things, not even the mind of a fellow experimenter. What about the objection that empirical observations -- whatever they "really" are -- are theory-laden and need to be _interpreted_ to make any sense?


How did these guys get the masses from Heim Theory? I won't even bother to ask about validity as I read the other post on this. So I know how we all feel about that. I hadn't even heard of it till today when I stumbled across it searching for something else. But I'd like to know: How did they get the subatomic particle masses to come out of their equations? Did they just "coach" them in the right directions- meaning are the equations just so vague that you could get any number out of them? I tried to read the papers but just don't get it. In discussions concerning it's invalidity I've found a lot of arguing over the true accuracy of the mass values. While I've seen tables to compare theoretical and actual values- the sources for such are all a group working on the theory. Has anyone seen the calculations and can answer? I think this could be informative as it can show how physics/math can be manipulated to show desired results


title: Quantum Hall effect for dummies

text: In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned. First, here are some random points that I've been able to gather   1. I(nteger)QHE occurs due to the presence of Landau levels   2. IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone   3. IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force   4. F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge   5. FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff   6. FQHE has something to do with hierarchy states So, here are the questions >   * Most importantly, do these points make sense? Please correct any > mistakes I made and/or fill in other important observations >   * How do explanations 1. and 2. of IQHE come together? Landau quantization > only talks about electron states while topological picture doesn't mention > them at all (they should be replaced by global topological states that are > stable w.r.t. perturbations) >   * How do explanations 4., 5. and 6. relate together >   * Is there any accessible introductory literature into these matters? >   * Do IQHE and FQHE have anything (besides last three letters) in common so > that e.g. IQHE can be treated as a special case? My understanding (based on > 3.) is that this is not the case but several points hint into opposite > direction. That's also why I ask about both QHE in a single question >


title: What is the duration of a snap?

text: What is the duration of a snap (of the fingers)? When someone says, "it's ridiculously fast, it's like [snap]" - where [snap] is them snapping their fingers - they would normally mean it is instantaneous, or near instantaneous. But what is the actual duration of the snap sound? That is from the start of the first snap-sound component to the end of the last? Disclaimer: obviously there will be variabilities here, but a general range of time would be gladly accepted. High speed video samples would be phenomenal.


title: What does the size of an object have to do with it's color?

text: Stephen Hawking mentions in his book 'A Brief History of Time' that quarks are much smaller than the wavelength of visible light so they do not have any color in the normal sense. What exactly does the size of the object with respect to the wavelength of visible light have to do with it's color? What is color int the normal sense? What other colors exist?


title: Total Momentum From a Standing Electromagnetic Wave

text: How does one show the momentum imparted to a perfect conducting resonance cavity (boundary) of any shape by a classical standing electromagnetic wave inside is zero? It should be by conservation of momentum. But I would like to see a detailed treatment argued specifically with the property of the electromagnetic wave, say with Poynting vector or electromagnetic stress-energy tensor. Specifically, given the boundary condition of the perfect conductor cavity, how does one derive $\frac{\partial \int\mathbf S dV}{\partial t} = 0$ or $\frac{\partial \int \mathbf<S>dV}{\partial t} = 0$ where $\mathbf S$ is the Poynting vector, the integral is over the space of the cavity, $<\cdot>$ denotes time average. One way of doing this could be to show the temporal spatial separated form of the Poynting vector $S(t,x)=S(x)e^{i\omega t}$ inside of the cavity. That is a Poynting vector of a standing wave. That form of $S(t,x)$ leads to its time average $<S>$ being zero. I would also suppose the average pressure on the boundary within the scale of the wave length is constant. How would one argue or describe that?


title: Entropy and dark matter structure

text: Inspired by this question. I believe that the usual explanation that preserves the second law of thermodynamics as an astrophysical gas cloud collapses under gravity is that the gas must heat and radiate, and while the entropy of the collapsed gas may be lower than the entropy of the uncollapsed gas, the entropy in the emitted radiation is more than enough to compensate. However, dark matter is thought to undergo a similar collapse process, and it does _not_ radiate by definition. I recall hearing that there is still no contradiction of the second law here, but I can't recall the explanation. What saves the second law here? Is it simply that some mass must be ejected by the collapsing system, i.e. the collapsing halo is "radiating mass" rather than radiating photons?


title: Why does the majority of smoke leave when flames appear?

text: When I burn wood, paper, or other plant matter, where there is a flame, there generally isn't too much smoke, but it I blow the fire out, smoke starts billowing up. It billows thicker and thicker as the material gets hotter, and suddenly a flame springs up. At the same moment, the smoke dissipates quickly. I've also noticed this in wood stoves. Overnight, the fire dies down and there are no flames, only some red coals. After I pile more wood on, it begins to smoke. After a time, there is a huge amount of smoke coming up, with no flames. Wait a little longer, or blow the coals, and flames will appear, and the smoke will mostly disappear. Why does this happen? Are the visible flames really just glowing soot, as described here? Because I don't see why glowing soot and non-glowing soot would make a difference in smoke production. Is that claim false?


title: What is a potential in De-Broglie-Bohm theory?

text: As I have so far understood, the De-Broglie-Bohm theory is based on two equations:   1. The Schrödinger equation.   2. The Guiding equation of the De-Broglie-Bohm theory. In the Schrödinger equation (as always) there is a potential term. But what is a potential in the terms of the theory? Is it an electrostatic potential or a gravitational potential? Does the De-Broglie-Bohm theory need another theory to stand on or does it stand on its own? I hope my question is clear enough.


title: Fluctuation spectrum of lipid bilayer membranes

text: I am interestend in calculating the fluctuation spectrum of a thermally fluctuating 2d membane which is only subject to a surface tension $\sigma$. ($\mathcal{H}=\sigma\int\mathrm{d}A$) Depending in which form I write my Hamiltonian to begin with my results differ by a factor 2, which should not be the case. Therefore I assume I am messing something up whithout noticing. Below I explain my calculation so far. I use the Monge presentation and work in the limit of almost flat height profiles $h(x,y)$. By using a Fourier transform I can write the Hamiltonian of the system in the following way: \begin{equation} \mathcal{H} = (2\pi)^2 \frac{\sigma}{2} \int k^2\ \left|\tilde{h}(\mathbf{k})\right|^2 \ \mathrm{d}\mathbf{k} \end{equation} where $\sigma$ is the surface tension of the membrane. Since the integrand is invariant under the mapping $\mathbf{k}\mapsto -\mathbf{k}$ one can also rewrite this Hamiltonian as \begin{equation} \mathcal{H} = (2\pi)^2 \sigma \int_{k_y>0} k^2\ \left|\tilde{h}(\mathbf{k})\right|^2 \ \mathrm{d}\mathbf{k} \end{equation} Where one gets a factor 2 by restricting the integration to the upper half plane of $\mathbf{k}$ - space. The $(2\pi)^2$ stems from the convention of my Fourier transform which is \begin{align} h(\mathbf{x}) &= \int \tilde{h}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{x}}\mathrm{d}\mathbf{k} \\\ \tilde{h}(\mathbf{k}) &= \frac{1}{(2\pi)^2}\int h(\mathbf{x}) e^{-i\mathbf{k}\cdot \mathbf{x}} \mathrm{d}\mathbf{x} \end{align} I want to calculate $\langle |\tilde{h}(\mathbf{k})|^2\rangle$. Therefore I use $\tilde{h} = a+ib$ And since $h$ is real I know that $\tilde{h}(\mathbf{k})=\tilde{h}(-\mathbf{k})^*$ which leads to $a(\mathbf{k}) = a(-\mathbf{k})$ and $b(\mathbf{k}) = -b(-\mathbf{k})$. Hence I start by calculating $\langle a^2(\mathbf{k})\rangle$. Therefore I establish the partition sum using the functional integral over all height profiles $h$. \begin{align} Z &= \int \mathcal{D}\tilde{h} \ e^{-\beta \mathcal{H}(\tilde{h}(\mathbf{k})} \\\ &= \prod_{k_x}\prod_{k_y}\int_{-\infty}^{\infty}\mathrm{d}a(\mathbf{k})\int_{-\infty}^{\infty}\mathrm{d}b(\mathbf{k})\ \exp\left[-\beta (2\pi)^2 \frac{\sigma}{2} k^2(a^2(\mathbf{k})+b^2(\mathbf{k}))\right] \end{align} Since I evaluate $\langle a^2(\mathbf{k})\rangle$ for a psecific fixed $\mathbf{k}$ the average can be computed by the Gaussian integrals \begin{align} \langle a^2(\mathbf{k})\rangle = \frac{\int_{-\infty}^{\infty} a^2(\mathbf{k}) \exp(-\beta (2\pi)^2\sigma/2 k^2 a^2(\mathbf{k}))\mathrm{d}a(\mathbf{k})}{\int_{-\infty}^{\infty} \exp(-\beta (2\pi)^2\sigma/2 k^2 a^2(\mathbf{k}))\mathrm{d}a(\mathbf{k})} \end{align} And by using the result for the standard exponential integrals this evaluates to \begin{equation} \langle a^2(\mathbf{k})\rangle = \frac{\frac{\sqrt{\pi}}{2(\beta(2\pi)^2 \sigma/2 k^2)^{3/2}}}{\frac{\sqrt{\pi}}{\sqrt{\beta (2\pi)^2\sigma/2 k^2}}} = \frac{1}{(2\pi)^2\beta \sigma k^2} \end{equation} So far so good. But if I had used my second form of the Hamiltonian where I only integrate of half of $\mathbf{k}$ space I would have obtained a result that differs by a factor of 2 which should not be the case. The partition sum would then be \begin{align} Z &= \int \mathcal{D}\tilde{h} \ e^{-\beta \mathcal{H}(\tilde{h}(\mathbf{k})} \\\ &= \prod_{k_x}\prod_{k_y>0}\int_{-\infty}^{\infty}\mathrm{d}a(\mathbf{k})\int_{-\infty}^{\infty}\mathrm{d}b(\mathbf{k})\ \exp\left[-\beta (2\pi)^2 \frac{\sigma}{2} k^2(a^2(\mathbf{k})+b^2(\mathbf{k}))\right] \end{align} But as soon as I calculate $\langle a^2(\mathbf{k})^2\rangle$ I can again cancel many terms and obtain the two integrals \begin{align} \langle a^2(\mathbf{k})\rangle = \frac{\int_{-\infty}^{\infty} a^2(\mathbf{k}) \exp(-\beta (2\pi)^2\sigma k^2 a^2(\mathbf{k}))\mathrm{d}a(\mathbf{k})}{\int_{-\infty}^{\infty} \exp(-\beta (2\pi)^2\sigma k^2 a^2(\mathbf{k}))\mathrm{d}a(\mathbf{k})} \end{align} which are the same as above, except that $\sigma /2$ is now replace by $\sigma$ only. This would yield the result: \begin{equation} \langle a^2(\mathbf{k})\rangle = \frac{\frac{\sqrt{\pi}}{2(\beta(2\pi)^2 \sigma k^2)^{3/2}}}{\frac{\sqrt{\pi}}{\sqrt{\beta (2\pi)^2\sigma k^2}}} = \frac{1}{2 (2\pi)^2\beta \sigma k^2} \end{equation} Obviously there must be a mistake, because the result should not depend on in which form I write the same Hamiltonian. Can someone point out what is going wrong in my current approach. Thanks and best regards khx0


title: Is Newtonian physics invariant under fast-forward?

text: I am aware of how the laws of physics are invariant with a time reversal transformation, so that (ignoring entropy) a film of billiard balls running backwards cannot be differentiated from one running forwards. What about a film of billiard balls running in fast-forward motion? Ignoring relativistic effects, I think that there would be no way to tell the playback speed. OK, then imagine that one of the billiards were made out of delicate, thin glass, and that in the regular play-back film the motions are slow enough that it does not break. Playing the film in fast-forward motion, the delicate ball still would not break despite being hit by very fast-moving ones. Does this mean that Newtonian physics is not invariant under fast-forwarding? Is there a way of expressing this mathematically (analogous to saying t goes to -t for time reversal)? And finally, since the glass ball does not break, is this violation of intuition an illusion having to do with entropy, like with running a film in reverse, or is there some kind of modification that needs be made to the forces of the molecular bonds under fast-forward motion to prevent the glass from shattering?


title: Which is stronger, wire rope or chain?

text: Assume they are the same material with the same weight, which is stronger?


title: From where comes the raindrop

text: First off all, I've no knowledge of how all this works. I've only got some common sense of "how things work"... But I've been wondering this for quite a long time now, so maybe you guys can help me. The question is rather simple: Where does the raindrop come from? And with that I mean: The raindrop that just hit the ground, where did that come from? From the clouds that are right above me, in front of me, or back of me? So let's say that we've got these 3 possibilities, which one is the most likely one? The first one, where the rain is traveling with the same speed horizontal as the cloud it came from: (so it came from the cloud directly above me) ![Raindrop 1](http://i.stack.imgur.com/FLkOU.jpg) The second one, where the rain is traveling faster than the clouds (due to wind?). So that the rain came from the cloud that yet has to pass me. ![Raindrop 2](http://i.stack.imgur.com/oHvYk.jpg) The third one, (And i do think that this is the right one) is that the rain is falling slower (horizontally seen) than the cloud (because the cloud is lighter). So that the rain that hits the ground came from the cloud that already past me. ![Raindrop 3](http://i.stack.imgur.com/0U8uE.jpg) An another possibility is that the rain is going left and right, (keeping it 2D for simplicity) because of the different wind regions (in the high clouds, the wind goes from left to right, but lower it goes from right to left). Is there even an "right" answer for this? Or are there too many variables that could change everything? And could you please explain it as simple as possible? I've got zero knowledge in this area... Thank you in advantage! -Also, i don't know if this is the right stackexchange site to ask...


title: Does this Zeilinger group result provide experimental proof of backward-in-time causation?

text: Does this recent Zeilinger group delayed choice entanglement experiment imply backward-in-time influences? http://arxiv.org/abs/1203.4834   From the abstract: "This can also be viewed as “quantum steering into the past ”."


title: Does Super Mario physics work in reality?

text: This illustration summarizes my question: ![Illustration](http://i.stack.imgur.com/Q8sEK.png) Please assume no air resistance (unless that makes a large difference). Here's what we concluded, but I could use both confirmation and also help with the math: Yes, jumping off of a plank will move you "up", but the timing of the jump makes a big difference. If you jump earlier in the fall, you'll move up, but you'll also add to the distance of the fall.. If you jump at the very end, you're essentially not adding to the fall at all, so you're getting essentially the maximum effect. In order to minimize the velocity at which you hit the ground, the best strategy would be to jump at the moment before impact. To keep the math simple, let's say you can jump one meter off of solid ground, and also that the bridge is ten meters above the ground. If you jump the instant the bridge crumbles, then you're actually doing yourself harm, since you will then fall from 11 meters. If you jump a less than a meter into the fall, you may get back up to the height of the bridge, get to "rest", but then still fall from 10 meters, as if you had done nothing. The closer to the ground that you jump, the larger the share of your jumping velocity that will be subtracted from your downward velocity without later contributing to your downward acceleration.


title: water pressure in the oceans or great lakes

text: Is there a way to create energy from water depth pressure? Convert the pressure to any type of energy? Does there need to be movement? Can someone comment on this process? Thanks


title: Lightcone coordinates

text: The Light cone coordinates are defined as $$x^+ = x^0 + x^3$$ $$x^- = x^0 - x^3$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach in his A First Course in String Theory [Second Edition,page-25] says that: There is no Lorentz transformation that takes the coordinates $(x^0, x^1 , x^2 , x^3)$ into coordinates $(x^+, x^-, x^1, x^2)$. Why? What is the reason behind this?


title: How long does it take to scan a typical scanning electron microscope image?

text: I suppose the answer depends on resolution and imaging area, but can you provide some ball park measures of imaging times with an SEM?


title: Projectile motion, what is the relation between initial angle and speed at a given distance

text: I am trying to understand the relation between the initial angle of a projectile and its initial speed, given that the distance this projectile will travel is fixed and at the same height as the origin. So far, I've used the equations of motion in two dimensions, combining $D=v_{i}^{x}\times t$ with $t_{max}=\frac{|\vec{v_{i}}|}{g}\times \sin\theta$ to get: $$D=|\vec{v_{i}}|\times \cos\theta \times 2 \times \frac{|\vec{v_{i}}|}{g}\times \sin\theta$$ which I then simplify to: $$g\times D = \sin2\theta\times|\vec{v_{i}}|^2$$ I'm aware this can also be seen as just a mathematical question, but I'm trying to understand the relation between the angle and the speed, and I just can't get passed this point.


title: Isn't the front edge of a wave, kind of "information" which travels faster than light?

text: Considering the definition of phase and group velocities, We know group velocity can't exceed C but phase velocity can be infinitely high. Assume a monochromatic electromagnetic wave traveling with a phase velocity greater than C in the space,   1. In terms of existence or inexistence of a wave, why don't we consider "changing the wave from being absent to being present" as a kind of "information" in an area of space at which wave arrives?   2. How much is the value of velocity at which the leading edge of the wave progresses in the space? Isn't it equal to the phase velocity of the wave? Why/why not?    3. Assume a monochromatic source of sinusoidal electromagnetic wave (like $A\left( {t,x} \right){\rm{ }} = {\rm{ }}{A_0}cos(kx - wt)$ ) in vaccum, located at the position of ${x_0} = 0$ and a receiver at ${x_1}$ , $\,3 \times {10^8}\,\,meters\,$ distant from the source. We turn the source on at the time ${t_0}$ , calculate the time interval between ${t_0}$ and the first instant (${t_1}$) at which the receiver feels an electromagnetic wave from the source.(assume $\,\omega = 6 \times {10^8}\,{{rad} \over {\sec }}\,$ , $\,k = 1\,{{rad} \over m}\,$ )


title: Heating metal with laser?

text: Can a laser be used to heat metal's to the point they glow red? How are laser cutters able to cut metal very easily? How much heat can a laser generate?


title: Is $w=mg$ the right way to calculate mass?

text: How can you find the mass of an object by using a weight and a gravitational acceleration from another planet then earth? I was given $5.8\text{ N}$ and an acceleration of $4.0\ \mathrm{m/s^2}$. The answer I keep getting is $1.45\text{ kg}$, but the real answer is supposed to be $.58\text{ kg}$. I'm using the formula $w=mg$. Am I doing something wrong?


title: Energy is quantized

text: How can energy be quantized if we can have energy be measured like in 1.56364, 5.7535, 6423.654 kilo joules, with decimals? Thanks Also isnt it quantization means energy is represented in bit quantities meaning you can not divide, lets say 1 bit of energy


title: Are black holes really singularities?

text: A popular assumption about black holes is that their gravity grows beyond any limit so it beats all repulsive forces and the matter collapses into a singularity. Is there any evidence for this assumption? Why can't some black holes be just bigger neutron stars with bigger gravity with no substantial difference except for preventing light to escape? And if neutrons collapse, can they transform into some denser matter (like quark-gluon plasma) with strong interaction powerful enough to stop the gravity? In this video stars are approaching supermassive black hole in the center of our galaxy in a fraction of parsec. The tidal force should tear them apart, but it doesn't. Can there be some kind of repulsive force creating limits for attractive forces?


title: Percentage increase in the length of pendulum

text: I'm struggling with a physics question : What should be the percentage increase in the length of the chord of a pendulum for the period increased by 1%? The answer is 2%. I tried with $w = 2 \pi/T = \sqrt{ g/L}$ but no success.


title: Factor of two differences for free field Green's functions in conformal field theory

text: I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given > $$ \frac{1}{\pi \alpha'} \partial_z \partial_{\bar{z}} X^{\mu}(z,\bar{z}) > X^{\nu} (z',\bar{z}') = -\eta^{\mu \nu} \delta^2 (z-\bar{z}', > \bar{z}-\bar{z}') \tag{2.1.20} $$ 2) In David Tong's string theory lecture note, p 186 (in the number at the bottom) or p 193 (counting by acrobat reader), > $$\langle Y^a(z,\bar{z}) Y^b(\omega,\bar{\omega}) \rangle = G^{ab} > (z,\bar{z}; \omega,\bar{\omega} )$$ $$ \partial \bar{\partial } G^{ab} > (z,\bar{z}) = - 2 \pi \delta^{ab} \delta (z,\bar{z}) \tag{7.33} $$ here $\alpha'$ has been scaled by $X^a = \bar{x}^a + \sqrt{\alpha'} Y^a$ in p 185/192. Why there is a factor of 2 difference in the RHS of (2.1.20) (Polchinski) and (7.33) (Tong)? In page 77, it is given > $$ \langle \partial^2 X(\sigma) X(\sigma') \rangle = - 2\pi \alpha' > \delta(\sigma-\sigma') \tag{4.20}$$ I am not sure what is the definition of $\partial^2$. Since $\partial \bar{\partial}=\frac{1}{4} (\partial_1^2 + \partial_2^2) $ and $\delta^2(z,\bar{z})=\frac{1}{2} \delta(\sigma^1) \delta(\sigma^2)$ (Polchinski p33),the factor at RHS of Eq. (4.20) in David Tong's lecture note seems to agree with Polchinski (2.1.20), but not Tong's (7.33) In addition, David Tong's lecture is related to > We'll be fairly explicit here, but if you want to see more details then the > best place to look is the original paper by Abouelsaood, Callan, Nappi and > Yost, \Open Strings in Background Gauge Fields", Nucl. Phys. B280 (1987) 599 3) In the original paper, it is given > $$\frac{1}{2\pi\alpha'} \square G(z,z')= - \delta(z-z') \tag{2.7}$$ where $\square=\partial_{\tau}^2 + \partial_{\sigma}^2$. Again, a factor of two difference between (2.7) and (2.1.20) of Polchinski. In Eq. (2.1) of that paper states > $$ S= \frac{1}{2\pi \alpha'} \left[ \frac{1}{2} \int_{M^2} d^2 z \partial^a > X_{\mu} \partial_a X^{\mu} + i \int_{\partial M} d \tau A_{\mu} > \partial_{\tau} X^{\nu} \right] $$ In Polchinski's string theory volume I, p32, it is given > $$ S = \frac{1}{4\pi \alpha'} \int d^2 \sigma \left( \partial_1 X^{\mu} > \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) > \tag{2.1.1}$$ Since $d^2 z = 2 d \sigma^1 d \sigma^2 \tag{2.1.7}$ , is that the origin of factor of 2 difference? Because combininig Eqs.(2.1.1) and (2.1.7) leads to $$ S = \frac{1}{2\pi \alpha'} \int d^2 z \left( \partial_1 X^{\mu} \partial_1 X_{\mu} + \partial_2 X^{\mu} \partial_2 X_{\mu} \right) $$


title: Why can't we destroy energy?

text: From a wikipedia article: In physics, the law of conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time. Energy can be neither created nor destroyed, but can change form; for instance, chemical energy can be converted to kinetic energy.   What is the reason behind this Law: is there a proof that we can't destroy energy? I mean if we couldn't destroy energy in our universe maybe we can do it in other universes the point is why is it a law and not a theory


title: Space translation of operators, states, and particle densities

text: In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation of $P=-i\frac{d}{dx}$ and $x$, I got $$\tag{2} e^{ia P}\rho(x) e^{-ia P} ~=~ e^{ia [P,~\cdot~]}\rho(x) ~=~e^{a\frac{d}{dx}}\rho(x)~=~\rho(x+a)$$ with a plus instead of $\rho(x-a)$ with a minus as Coleman (1). This was on Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture. Does anyone see where I may have gone wrong? References:   1. Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture.   2. S. Coleman, _Notes from Sidney Coleman's Physics 253a,_ arXiv:1110.5013, p. 23.


title: Why does the amount of dark matter increase the further away from the galaxy's center?

text: Why isn't the dark matter just randomly distributed? How does it know where to go?


title: Do electrons have a radius when they behave like a particle?

text: I know sometimes electrons behave like waves, but it sometimes can be seen as a particle. while it's a particle, does it have a radius? or, a volume? If it doesn't even have a volume, how can we still call it a particle?


title: Are Lagrangians and Hamiltonians used by Engineers?

text: Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges or buildings?


title: Derivative chain rule in a triangle, confusing but interesting problem

text: I asked the question in math.stackexchange. But I think it is better to ask here again. I am new to these sites. Please forgive me if it is not polite. http://math.stackexchange.com/q/921001 You can reply any poster. I paste it again here. ![variables in triangle](http://i.stack.imgur.com/RmuHH.png) Refer to the above figure. Assuming the length of the 3 edges of triangle are $r_0,z_0,\xi_0$. And we have $\xi=\sqrt{r^2+z^2}$ (Eqn.1)and $\xi_0=\sqrt{r_0^2+z_0^2}$. The normal vector on the hypotenuse is $\hat{n}$, and the components are $n_r=\frac{z_0}{\xi_0}$ and $n_z=-\frac{r_0}{\xi_0}$ Meanwile$\frac{\xi}{r}=\frac{\xi_0}{r_0}=\frac{1}{\cos{\theta}}$(Eqn.2) and $\frac{\xi}{z}=\frac{\xi_0}{z_0}=\frac{1}{\sin{\theta}}$(Eqn.3). Now I have a funtion $T=T(r,z)$ on the hypotenuse, and I need to calculate $f=\frac{\partial{T}}{\partial{z}}n_r-\frac{\partial{T}}{\partial{r}}n_z=\frac{\partial{T}}{\partial{\xi}}\frac{\partial{\xi}}{\partial{z}}n_r-\frac{\partial{T}}{\partial{\xi}}\frac{\partial{\xi}}{\partial{r}}n_z$ (Eqn.4) If using Eqn.1, then $\frac{\partial{\xi}}{\partial{z}}=\frac{z}{\sqrt{r^2+z^2}}$ (Eqn.1*a)and $\frac{\partial{\xi}}{\partial{r}}=\frac{r}{\sqrt{r^2+z^2}}$(Eqn.1*b) and $f=\frac{\partial{T}}{\partial{\xi}}(\frac{z_0z}{\xi_0\sqrt{r^2+z^2}}+\frac{r_0r}{\xi_0\sqrt{r^2+z^2}})$. (Eqn.5) If using Eqn.2 and Eqn.3, $\frac{\partial{\xi}}{\partial{z}}=\frac{\xi_0}{z_0}$ (Eqn.6)and$\frac{\partial{\xi}}{\partial{r}}=\frac{\xi_0}{r_0}$(Eqn.7) and $f=\frac{\partial{T}}{\partial{\xi}}(\frac{\xi_0z_0}{z_0\xi_0}+\frac{\xi_0r_0}{r_0\xi_0})=2\frac{\partial{T}}{\partial{\xi}}$(Eqn.8) It looks to me that Eqn.6 and Eqn.7 are right. but Eqn.8 is wrong. because when $\theta$ goes to 90degree,it should be $f=\frac{\partial{T}}{\partial{z}}=\frac{\partial{T}}{\partial{\xi}}$ So what is the right expression for $f$, and why? ($f$ is azimuthal component of the $\nabla\times({T\hat{n}})$) as somebody asked.


title: Difference between Hamiltonian in classical Mechanics and in quantum Mechanics

text: I have a question about difference between Hamiltonian function (the description of system in classical physics) and the Hamiltonian operator (quantum mechanics). I think that there two different points of view: a physical one and mathematical (more technical) one. In classical Hamiltonian mechanics state of the system (just for sake of simplicity let's consider one dimensional case) is determined by the variables $p, q$. It actually means that if one has defined the initial values of $p$ and $ q$ in arbitrary point of time $t$ then one can find their values in subsequent moment of time $t + \Delta t$ $$ q(t + \Delta t) = q(t) + \dot q(t) \Delta t $$ $$ p(t + \Delta t) = p(t) + \dot p(t) \Delta t $$ using canonical equations: $$\dot q = \partial H / \partial p$$ $$\dot p = - \partial H / \partial q$$ where $H$ classical Hamiltonian function. In case of quantum mechanics. The state of the system is defined by $\Psi(q, t)$. And if we know $\Psi$ at given moment of time $t$ we can calculate it at subsequent moment $t + \Delta t$: $$\Psi(q, t + \Delta t) = \Psi(q, t) + \dot \Psi(q, t) \Delta t$$ where $i \hbar \dot \Psi = \hat H \Psi$ and $\hat H$ is the Hamiltonian operator. As for me this leads to the following consequences   * In classical physics Hamiltonian defines canonical variables, but in QM Hamiltonian operator defines only one quantity (psi function)   * Classical motion is defined by canonical equation (principle of the least action), QM Hamiltonian constructed in such a way to satisfy Schrodinger equation (it is not derived from principle of least action)   * Mathematically Hamiltonian in CM is just a function of $q,p$ variables but in quantum mechanics it is a Hermitian operator Please could you tell me if I am right or if I have something missed here? I am actually interested in what is the difference between quantum and classical Hamiltonian? I will be very pleased because it is very interesting topic for me.


title: Can photons have negative energy?

text: Apparently there are 2 electron self-energy graphs possible. The first, the more "familiar", where the incoming electron at time $t_1$ splits up in a photon and an virtual electron. At $t_2>t_1$ the virtual photon joins the electron again. But the Feynman-propagator also allows $t_1>t_2$, where apparently the incoming electron hits a positron at $t_1$ which was created at time $t_2<t_1$. But for the creation of the electron-positron pair at $t_2$ the photon has to provide the positive energy at $t_2$ for the creation process, so this photon seems to be created at $t_1>t_2$, so the photon is apparently moving backward in time. The other possibility is of course apply Feynman's saying: Backward in time running particles with energy $E$ can be interpreted as forward in time running particles with $-E$ (assuming $E$ can have positive or negative sign in general). Therefore, it would be equivalent to say that for the electron self-energy diagram where $t_1>t_2$, at $t_2$ a photon is created with energy $-E$ ($E$ being the energy needed for the creation of the electron-positron pair) and then moving in time forward to $t_1$ to deliver this negative energy to destroy the (incoming) electron-(virtual) positron pair. Therefore I come to the conclusion that virtual photons can have negative energy in running in time forward or positive energy running in time backward. What about real photons? I would be astonished about real negative energy photons. Or would the dispersion relation $\omega^2=k^2$ allow for negative frequencies (keeping in mind that these negative frequency solutions would again correspond to photons as anti-photons and photons are the same)?


title: Power delivered by a plug

text: Ok so i know that (in the UK) the voltage of a mains supply is 230v. But when we plug a item e.g. a tv in to it what determines how much power is drawn from this supply. I presume it is the current but what prevents all the current in the mains going to your device? Also what happens to the power that is not used?


title: Invariance of Temperature in Classical Physics

text: How can we explain that Temperature is a classically frame-independent quantity to high school kids?


title: Stresses in asteroid during close flyby

text: The acceleration of an asteroid (such as 2012DA14) as it approaches earth is proportional to the reciprocal of distance $r$ from earth center, squared. the derivative of the acceleration, or jerk, is then proportional to the cube of the reciprocal of distance $r$, and the speed $v$. Thus the jerk is rapidly increasing as the asteroid approaches. Will the asteroid experience stresses from the jerk that could possibly cause it to break up?


title: Is it possible to construct a quantum "computer" using laser light similar to the double-slit experiment?

text: Is it possible to construct an arrangement of optical devices (lasers, mirrors, slits, splitters) such that the construction could carry out a single quantum "computation"? I understand that such a setup for arbitrary computations is quite difficult and is being worked on, but from the perspective of a fun demo or simple project could you construct a setup where the input and resulting operations (maybe a CNOT or n few bits of addition) are uniquely determined by the arrangement of slits and the output can be read by measuring the resulting interference pattern? If not are there any other simplifying assumptions one could make in order to carry out something which would still qualify as a single quantum "computation", even in a trivial manner for a DIY demo?


title: Why does a large train cause the ground to shake?

text: I work in a 4 story building that is approx. 150 feet away from a set of train tracks. When a large (40+ car) freight train goes by, the shaking in the building is perceptible. As I've watched the train go by, there does not appear to be any side to side movement and the speed is constant. What exactly is causing the vibrations in the ground? Is it simply the train's traversal over each segment of track? Surely the train itself is not "bouncing" along the track with enough energy transfer to shake the ground, right?


title: What is the relationship between Coriolis forces and Angular momentum?

text: Recently I did an assignment about helicopters, and during the oral exam we talked about Coriolis forces. I know where it comes from, and I've seen the derivation: Example In the case of helicopters, it increases the blade speed as a consequence of flapping, like explained in Coriolis helicopter ![Helicopter Coriolis](http://i.stack.imgur.com/WAs1j.jpg) I understand that through flapping, the center of gravity moves inward, and conservation of angular moment dictates that the speed should increase. I mentioned to my lecturer that we do not really need Coriolis forces to explain this effect, as it can be explained by conservation of angular momentum only. Only when you state that the Coriolis force are the forces that ensure conservation of momentum, it makes sense. However, she said that the Coriolis forces are actual forces as a consequence of a rotating reference system, and not a consequence of angular momentum. **To me, it seems that the Coriolis force is a force that ensures conservation of momentum. E.g. we know that the speed increases when the distance to the C.o.G. decreases, but the Coriolis forces are the actual forces that enforce the acceleration.** Is this the right way to look at it?


title: Natural electricity in the atmosphere - lightning

text: If air is such a poor electrical conductor, how can lightning be produced that appears to travel through it?


title: Time dilation derivation of special relativity

text: In almost all of the derivations using the postulates of special relativity (SR), we use experiments involving light signals. For example, we make a clock using a light signal or measure lengths using light signals, etc. The reason for doing this is never stated. Why do we do this? Aren't there any other thought experiments that could help us achieve the expressions for time dilation and length contractions without using light signals? Is it even possible to derive expressions for the Lorentz transformation without using experiments involving light signals in some form? Please don't start off with space-time metric. I want to know if such a thing can be done using the two postulates given by Einstein.


title: Is gravitational potential energy proportional or inversely proportional to distance?

text: We know that if an object has been lifted a distance $h$ from the ground then it has a potential energy change: $$\Delta U = mgh $$ so $h$ is proportional to $\Delta U$. However, we have also the gravitational potential energy law: $$ U= -\frac{G M m}{r} $$ where the distance is inversely proportional to the potential energy. What did I miss? Is the distance of the object proportional or inversely proportional to the potential energy?


title: Inverse Fourier Transform Of K-space Image...what is the object space scale?

text: Checked around a buch and could not find any help. But I needed help with: Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length? More detailed explanation in the below image. ![enter image description here](http://i.stack.imgur.com/re3eq.png)


title: 2nd order phase transition trouble deriving coefficient in fluctuations analysis

text: I can't get one of the coefficients in the equation for $T < T_c$ in the bottom, specifically the equation with the factor of two. any help appreciated. Consider an ising type expansion of the free energy density in the order parameter $\psi$. (the mean values of $\psi$ corresponding to a uniform configuration.) \begin{equation} f(T,\psi) = f_0 + G(\nabla\psi)^2+ A\psi^2 + B\psi^4 \end{equation} \begin{equation} \bar{\psi} = \begin{cases} \pm\sqrt{\frac{-A}{2B}} & \qquad T < T_c \\\ 0 & \qquad T > T_c \end{cases} \end{equation} these solutions come from minimizing the free energy density. Expanding the coefficients near the critical temperature, we have \begin{equation} A = at + ... \end{equation} \begin{equation} B = B_0 + ... \end{equation} \begin{equation} G= G_0 + ... \end{equation} Where $t=T-T_c$. Then evaluating the free energy density at $\psi = \bar\psi$ we get \begin{equation} \left. f(T,\psi)\right|_{\psi=\bar{\psi}} = \begin{cases} f_0 - \frac{a^2t^2}{4B_0} & \qquad T < T_c \\\ f_0 & \qquad T>T_C \end{cases} \end{equation} To study fluctuations in the order paramater define, $\delta\psi = \psi -\bar{\psi}$. \begin{equation} \delta\psi = \begin{cases} \psi & \text{in symmetrical phase} \\\ \psi - \sqrt{\frac{-at}{2B_0}} & \text{in disordered phase} \end{cases} \end{equation} We now calculate the change in free energy though I am not sure how exactly to calculate this, for some reason I am missing something simple I'm sure (!). \begin{equation} \Delta F(T,\delta\psi) = \begin{cases} \int d^{d}r\,\left( G(\nabla\delta \psi)^2 + at\delta \psi^2\right) & \qquad T > T_c \\\ \int d^{d}r\,\left( G(\nabla\delta \psi)^2 -2at\delta \psi^2\right) & \qquad T < T_c \end{cases} \end{equation} I have spent some time trying to calculate the coefficient above for $\delta\psi^2$ for $T < T_c$. I am clearly not understanding something here, since when I calculate $f(\psi) - f(\bar{\psi})$ I do not get the above expression. In fact I am not sure exactly how to go about calculating this. Landau and Lifshitz gives the above coefficient in section 146 "Fluctuations of the Order Parameter". I am also confused because it seems like there are two values of the mean $\bar{\psi} = \pm (\frac{-at}{2B_0})^\frac{1}{2} $ however in the book I only see a reference to the positive square root. This topic is a bit obscure so I'm not sure anybody really is gonna be able to help, but I'd appreciate any insight.


title: gaussian fitting of xrd peaks

text: I am reading a paper regarding the nano crystalline material with hexagonal crystal structure. In that paper in order to find out lattice parameter of hexagonal crystal structure they have fitted a Gaussian curve from the xrd pattern. They have not mentioned in the paper whether they manually or using a program fitted the data. For my purposes I finds out the lattice parameters of a cubic crystal structure without any Gaussian curve fittings. My question is what is the advantage of using a Gaussian fitted data to find lattice parameters?


title: Why is the rainy season in the winter for some (mediterranean?) climates and in the summer for other (tropical?) climates?

text: Generally speaking, the rainy season is in the summer in tropical climates, and in the winter in mediterranean climates. (I admit that is a gross oversimplification) What is the cause for this difference? Purely speculatively, it seems intuitive that hot weather would lead to higher humidity, which in turn would lead to more spontaneous rain clouds. This would obviously be maximized by summer weather. But why is the rainy season in the winter in colder climates?


title: How many stars did people think there were in the 11th century (or thereabouts)?

text: I hope this isn't too off-topic. Someone showed me a reference to a French, 11th century biblical commentator who implied that there were over 600,000 stars. This got me thinking, how many stars _did_ people/astronomers think there were back then? As I understand it For the actual reference, it is Rashi on Deut. 1:10: > **The Lord, your God, has multiplied you, and behold, you are today as the > stars of the heavens in abundance.** (Deut 1:10) > > And, behold, you are today as the stars of the heavens: But were they [the > Israelites] on that day as [many as] the stars of the heavens? Were they not > only six hundred thousand? Assuming they only counted stars they could see, **how many stars can a sharp- eyed person see with the naked eye** , assuming ideal observing conditions of course? On top of that, **were there any theories in ancient astronomy that extrapolated the number of stars to include ones that couldn't be seen**? Other links: Wikipedia on Astronomy in Medieval Islam Wikipedia on Astronomy in Medieval Europe


title: Vector Displacement

text: A novice golfer on the green takes three strokes to sink the ball. The successive displacements are 4.10 m to the north, 1.90 m northeast, and 1.00 m at 25.0° west of south. Starting at the same initial point, an expert golfer could make the hole in what single displacement? _**____** m _**_____** north of east I'm stuck on this problem and don't know how I would go about solving this problem. Could someone explain how to solve it with steps for me, I'm mostly interested in understanding how to do this rather than just getting the answer.


title: Can stimulated emitted photons be absorbed?

text: Typically a stimulated photon will be one of a pair with its stimulating photon. If the leading photon is absorbed by a particle in the ground state, will it then be re-emited by the stimulated emission caused by the second photon? If it causes a stimulated emission from an excited particle, will there then be a train of three photons, or will the new pair leave the excited particle excited? I am interested in the case where there is not a population inversion, but presumably the same rules would apply to lasers.


title: Why do we need to prove the gauge invariance of QED (or all of the gauge theories) on the Feynman diagrams language?

text: Let's have the QED lagrangian. It has explicit gauge invariance, so, by the naive thinking, all of the EM processes must satisfy the property of gauge invariance. So why do we need to recheck of gauge invariance on the Feynman diagrams language? Is is connected with fact that after renormalizing the propagators its poles may shift (so the photon aquire the mass)? Also what's about non-abelian gauge theories?


title: Water Jet Cutter and Laser

text: I watched a video of "Glass cutting" which uses a **Water Jet Cutter**. It was said that Cutting glass simply by machines would eventually crack it... So, They're using grains of sand (by placing a sand paper beneath) to cut glass using a Water Jet. It also has enough force to cut through _steel_. ![Water Jet](http://i.stack.imgur.com/0NK0W.png)   * What exactly is the property of GLASS so that it requires a "Water Jet" to cut it..?   * Also, Why can't we use a powerful laser and some kinda coolant to cut through glass..?


title: Charge and the Dirac field

text: In Zee's quantum field theory in a nutshell, 2nd edition, pg 550 he has > $Q=\int {d^3p \over (2\pi)^3(E_p/m)} \sum_s > \\{b^\dagger(p,s)b(p,s)-d^\dagger(p,s)d(p,s)\\}$ > > showing clearly that $b$ annihilates a negative charge and $d$ a positive > charge. I would very much appreciate an explanation of why its not the other way round. As the $d^\dagger(p,s)d(p,s)$ term has the negative sign why doesn't that mean it is associated with the negative charge particle?


title: Is the fuel burn for a satellite launch affected by the position of the moon relative to the launch site?

text: The gross mass of a satellite rocket is tiny compared to that of Earth, and Luna. Between them, however, the two bodies set up tides in bodies of water which itself is again considerable mass. At the point of launch a satellite rocket might not be affected by the Moon's gravitation attraction. As it goes higher/ more distant from earth/closer to escape/ target orbit do it's fuel burn statistics change depending upon the position of the moon vis-a-vis launch location/payload release location?


title: Maxwell equations invariant under Lorentz transformation but not Galilean transformations

text: Why Maxwell equations are not invariant under Galilean transformations, but invariant under Lorentz transformations? What is the deep physical meaning behind it?


title: Spatial Resolution in Magnetocardiography

text: In the science paper called Development of an optical cardio-magnetometer by George Bison, it is mentioned that spatial resolution of 10-20mm is required for mapping MCG signal(page 52). What does the spatial resolution means here and does it depends upon the number of sensors only or could depend on the size of the sensor also?


title: Dynamics of counter-rotating flywheels

text: I've wondered about this for ages. If we create a pair of flywheels that rotate in the opposite direction with the same angular momentum, but are co- located and have the same mass and inertial moment (one can imagine various ways to accomplish this, at least approximately) -- it is clear to me that there cannot be any precession forces, but, if we try to rotate the entire assembly around an axis perpendicular to the axis of flywheel rotation, will the force needed to produce this secondary rotation be the same as if the flywheels were stationary, or will it require a proportionally greater force to rotate in this way, as it does with a single flywheel?


title: Reflectance of Titanium as Function of Thin Film Thickness

text: As far as I know, transmittance equals $e^{-\alpha x}$, where $\alpha$ is absorption coefficient and $x$ is thin film thickness($100-300\,nm$). My team and I have engineered a way to find absorption. Transmittance, T= (Output intensity)/(initial intensity). And, absorption A=(initial intensity-output intensity)/(initial intensity). After simplifying the equation, one should get $$A= 1-T$$ The problem I am facing is I do not know any equation, which will give me reflectance of Ti as a function of thin film thickness at $808\,nm$ wavelength. If someone gives me an equation of transmittance, please include the effects of absorption so that I can calculate the reflectance afterwards.


title: Entropy of a de Sitter horizon

text: The cosmological event horizon found in a de Sitter universe has some interesting similarities to that of a black hole. For example, since we can find a temperature at the horizon, we are able to use the relation $S = A/4G$ to calculate the entropy of the horizon related to it's surface. I am looking for a method of directly calculating the entropy, instead of using the famous relation given above. One example I found was mentioned in Les Houches Lectures on De Sitter Space (pages 17,18). Unfortunately, the method is given as an exercise, and the conclusion given is far from obvious (to me). They start by looking at the static Schwarzschild-de Sitter metric in three dimensions given by $$\mathrm{d}s^2 = -(1 - 8GE - r^2)\mathrm{d}t^2 + \dfrac{\mathrm{d}r^2}{1-8GE-r^2} + r^2 \mathrm{d}\theta^2 $$ The paper states that finding a Green function for $SdS_3$ by analytic continuation from the smooth Euclidian solution periodic in $\tau \rightarrow \tau + \dfrac{2 \pi i}{\sqrt{1 - 8GE}}$ will give all required answers to find the entropy (the author continues summing up temperature, invoking a thermodynamic identity and finishes off with $S$. I have yet to find another paper that uses this derivation, but I am quite interested in the part missing. Is there someone who can shed a light on how finding such a Green function leads to temperature/entropy?


title: Why is jumping into water from high altitude fatal?

text: If I jump from an airplane straight positioned upright into the ocean, why is it the same as jumping straight on the ground? Water is a liquid as opposed to the ground, so I would expect that by plunging straight in the water, I would enter it aerodynamically and then be slowed in the water.


title: How to calculate magnification

text: I have various magnifying glasses and I'm using them when I take macro photos with a phone or a camera. I want to group/label my magnifying glasses by their magnification power. And by magnification power I mean something like 10x. To calculate their magnification power, I've read several Wikipedia pages, Lens (Optics) for calculating focal length, and Magnification for calculating magnification power, but I end up with wrong results. I must be doing something wrong. For example, I have this biconvex magnifying glass ![enter image description here](http://i1115.photobucket.com/albums/k544/akinuri/SAM_0195_s-1.jpg) Some of its properties diameter = 6 cm   thickness at the edge = 3 mm   thickness at the center = 7 mm   radius of the curvature = 225 mm (1)   I figured the focal length of magnifying glass by experimenting with a light source, but I want to calculate it on a paper. How can I calculate the focal length of a magnifying glass and its magnification power? For example, in this page Amazon - SE Folding Pocket Magnifier, it says 10x for the magnifier. How do the companies that produce magnifying glasses calculate it? I'd appreciate if you could provide an example calculation in your answer, preferably with the values I've given. * * * (1) I didn't calculate it, I just drew the magnifying glass in Illustrator with its actual values   Index of refraction doesn't matter. The result doesn't have to be 100% correct. 1.52 can be used as the index.


title: How to calculate upward force of a container filled with a less dense gas than the surroundings where it is placed?

text: I'm trying to determine the calculation which would state the upward force of a container where the density inside the container is less then that in which it sits. Does this calculation change with shape? I.e. sphere vs cube?


title: Harmonic Oscillator Expectation Value

text: In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\left \langle x \right \rangle$ for the coherent state $\left| \alpha \right \rangle$ $$x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_- )$$ $$\sqrt{\frac{\hbar}{2m\omega}}\left \langle \alpha \right| (a_+ + a_- ) \left| \alpha \right \rangle =\sqrt{\frac{\hbar}{2m\omega}} \alpha ^* (a_+ \alpha + a_- \alpha )$$ This doesn't agree with the result my book has given, which is $\sqrt{\frac{\hbar}{2m\omega}} (\alpha + \alpha ^* )$. I'm not used to working in bra-ket notation, should it's extremely likely that I'm missing something small here. The general principle is that $\left \langle \beta \right| A \left| \beta \right \rangle = \beta ^* (A\beta)$, right?


title: Is there a formalism for talking about diagonality/commutativity of operators with respect to an overcomplete basis?

text: Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The coherent superposition of two wavepackets well-separated in phase space is decidedly non-classical. Is there a formalism I can use to call this density matrix "approximately diagonal in the overcomplete basis of wavepackets"? (For the sake of argument, we can consider a specific class of wavepackets, e.g. of a fixed width $\sigma$ and instantaneously not spreading or contracting.) I am aware of the Wigner phase space representation, but I want something that I can use for other bases, and that I can use for operators that aren't density matrices e.g. observables. For instance: $X$, $P$, and $XP$ are all approximately diagonal in the basis of wavepackets, but $RXR^\dagger$ is not, where $R$ is the unitary operator which maps $\vert x \rangle \to (\vert x \rangle + \mathrm{sign}(x) \vert - x \rangle) / \sqrt{2}$. (This operator creates a Schrodinger's cat state by reflecting about $x=0$.) For two different states $\vert a \rangle$ and $\vert b \rangle$ in the basis, we want to require an approximately diagonal operator $A$ to satisfy $\langle a \vert A \vert b \rangle \approx 0$, but we only want to do this if $\langle a \vert b \rangle \approx 0$. For $\langle a \vert b \rangle \approx 1$, we sensibly expect $\langle a \vert A \vert b \rangle$ to be proportional to a typical eigenvalue.


title: How to calculate preassure at solar focus?

text: As starlight passes the sun it gets bent by gravity and will be focused. I am interested in see how strong the pressure from the starlight would be at a focal point. I was thinking if the pressure was very strong it would be use for interstellar transport with a sail. I did some calculations a while back and got a very high value for one point (like 100,000 N/m^2) but they were sketchy calculations. To make this a very specific question: If a point is on the opposite side of the sun from a star, what is the signal gain of the starlight from gravitation lensing as a function of distance from the sun?


title: How to derive or justify the expressions of momentum operator and energy operator?

text: It has been noted here$\\! { \, }^{\text(1, 2)}$, for instance, that $$\mathbf{F} = \frac{d}{dt}\\!\\!\biggl[ \, \mathbf{p} \, \biggr]$$ is true in all contexts. Likewise, in notable contexts it is apparently true that $$\mathbf{F} = - \nabla \Phi := - \frac{d}{d\mathbf{r}}\\!\\!\biggl[ \, \Phi \, \biggr].$$ Is this, in a nutshell, a sufficient and valid justification for setting (in the corresponding suitable contexts)   the momentum operator as$\\! { \, }^{\text(3)}$ $$\mathbf{\hat p } \propto -i \nabla := -i\frac{d}{d\mathbf{r}}$$ and setting the (potential) energy operator as$\\! { \, }^{\text(4)}$ $$\hat \Phi \propto i\frac{d}{dt}$$ and both with the same constant of proportionality, $\hbar$, whereby $$\mathbf{\hat F} = \frac{d}{dt}\biggl[-i\hbar\frac{d}{d\mathbf{r}}\biggr] = -\frac{d}{d\mathbf{r}} \biggl[i\hbar\frac{d}{dt}\biggr] \sim \frac{d^2}{dt \, d\mathbf{r}} = \frac{d^2}{d\mathbf{r} \, dt}$$ ? **EDIT (related merely to formalities):** (${ \, }^{\text 1}$: Please note that the assertion to be noted _here_ had been expressed explicitly in the form > $F = \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}$ is true in all contexts. However, since it appears admissible to take note of an assertion without strictly quoting and clinging to its original literal expression (as had been tacitly presumed already in the initial statement of my question, and as it seems to be thus confirmed) I'd like, as far as unambiguously possible, to express the operation of " _differentiation_ " consistently by using (a form of) _Leibniz's notation_.) (${ \, }^{\text 2}$: Please note that the question whose notable answer to which reference has been made above had been tagged (primarily) as has " _http://physics.stackexchange.com/questions/tagged/newtonian-mechanics_ ".) ($ { \, }^{\text 3}$: Please note that the indicated expression of the _momentum operator_ is explicitly stated there as > ${\bf \hat p } = -i \hbar \nabla$ and > In one spatial dimension this becomes: $\hat{p}=\hat{p}_x=-i\hbar{\partial > \over \partial x}$, where the nabla symbol ($\nabla$) is related to http://en.wikipedia.org/wiki/Directional_derivative#Notation .) (${ \, }^{\text 4}$: Please note that the indicated expression of the _energy operator_ is explicitly stated there as > $\hat{E} = i\hbar\frac{\partial }{\partial t}$. )


title: Universal formulation of Kepler problem for the hyperbolic case

text: I'm trying to derive the universal formulation of the time-of-flight equation that appears in the Kepler problem (following Bate _Fundamentals of astrodynamics_ and Vallado _Fundamentals of astrodynamics and applications_ ), and I don't see a point about the hyperbolic case. Starting from the energy equation: $$ \dot{r}^2 = -\frac{\mu p}{r^2} + \frac{2\mu}{r} -\frac{\mu}{a} $$ after a Sundman transformation ($\dot{\chi} = \frac{\sqrt{\mu}}{r}$ ) and separating we have $$ d\chi = \frac{dr}{\sqrt{-p + 2 r - \frac{r^2}{a}}}$$ Then both Bate and Vallado proceed and integrate assuming the elliptic case ($a > 0$), yielding $$ \chi + c_0 = \sqrt{a} \arcsin{\frac{\frac{r}{a} - 1}{\sqrt{1 - \frac{p}{a}}}}$$ **Later it is justified that this formulation is valid for elliptical, parabolic and hyperbolic orbits** using the Stumpff functions, even though the integration was not performed for the case $a < 0$ ($a = 0$ is trivial). Furthermore, Vallado states "this case results in a hyperbolic sine solution, which we won't use" but after several trials I got a hyperbolic cosine solution instead: $$ \chi + c_0 = \sqrt{-a} \cosh^{-1}{\frac{\frac{r}{-a} + 1}{\sqrt{1 - \frac{p}{a}}}}$$ I expected a sinh though, to keep simmetry with the other case. My main question is: **how come can you start from a certain case and then prove the formulation is valid for all of them**? And as an aside: how would the hyperbolic derivation be? ~~Note: I know Battin _An Introduction to the Mathematics and Methods of Astrodynamics_ has probably more detailed math but there's no way I can get one until the end of the holidays.~~


title: Sun: When its closest and when its farther away?

text: If the sun is closest to the North American Hemisphere in the winter than it is in the summer, why doesn't our faces and hands tan and/or blister quicker? Please provide two hyperlink sources with your answer. Thank you.


title: Why Negative refractive index is negative

text: $n=\sqrt{\epsilon \mu }$ Negative refractive index happens when permittivity ($\epsilon $) and permeability ($\mu $)of a material is negative. My question is, if permittivity and permeabilitity of a material is negative then the end product, according to the above equation, should be positive. If two negative numbers multiplied should give a positive number. Then why in here two negative numbers ($\epsilon $ and $\mu $) gives a negative refractive index?


title: Is the Schwarzschild black hole unphysical?

text: To obtain the Schwarzschild metric from Einstein equations of general relativity, we suppose that the energy density is a distribution : $$ \rho (\vec{r}) = M \delta(\vec{r})$$ The Schwarzschild radius, corresponding to the horizon, is $R_s = 2 M$, in units $G=c=1$. A physical interpretation of this, is that you cannot put more energy than $M$ in a sphere of radius $R_s$. The way I understand this, is because the total energy of the black hole, sum of its mass energy (positive) and its auto-gravitational energy (negative), should remain non-negative. However, for the Schwarzschild black hole, for each radius $r$ between $0$ and $R_s$, the total mass in the sphere of radius r, is : $$ m(r) = \int_0^r \rho(\vec{u}) d^3u =\int_0^r \delta(\vec{u}) d^3u = M$$ So it seems to violate the principles cited above, and so the Schwarzschild black hole should be unphysical. If it makes sense, a physical black hole (static, with spherical symmetry) should have a mass/energy density $\rho (r)$, such that, whatever the value of $r$ is, we have the inequality (in units $G=c=1$): $$m(r) = \int_0^r \rho (u) 4\pi u^2du \le \frac{r}{2}$$


title: Is there a symmetry associated to the conservation of information?

text: Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory. We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information.


title: Second order energies of a quartic pertubation of a harmonic oscillator

text: A homework exercise was to calculate the second order perturbation of a a quantum anharmonic oscillator with the potential $$ V(x) = \frac{1}{2}x^2 + \lambda x^4 $$ We set $\hbar = 1$, $m=1$, etc. Using the harmonic oscillator $H = \frac{1}{2}p^2 + \frac{1}{2}x^2$ as my basis hamiltonian, I calculated the perturbed ground state energy multiplication factors as $$ E_0(\lambda) = 1 + \frac{3}{4} \lambda - \frac{21}{8} \lambda^2 + \mathcal{O}(\lambda^3) $$ while our lecture notes state $$ E_0(\lambda) = 1 + \frac{3}{4} \lambda - \frac{21}{16} \lambda^2 + \mathcal{O}(\lambda^3) $$ I did not find any sources in literature, neither did I find a mistake in my calculations yet. Which one is correct?


title: two electron spins line up in opposite direction and the Ising model

text: What cofuses me with the Ising Model is the claim that these nearby little magnets want to face the same direction. facing the same direction, they have lower energy. This seems to conflict with the fact that north poles of magnets repel each other. So little magnets act different from big everyday magnets? But then, it's said that if you bring two electrons near enough that they can feel each other's magnetic field, then if you wait enough, a photon will be emitted and they will get into an entangled state such that their spins are facing opposite directions. This seems to conflict with the claim that little magnets facing the same direction have lower energy. I am confused. It just looks like there are two kinds of little magnets and one kind's north poles repel each other, and another kind's north poles attract each other. What is really going on?


title: Wigner friend experiment

text: Let's supposed we take the Wigner's friend experiment from the metaphysical arena and try to implement it as an actual physical experiment Assuming Wigner's lab friend is kept as a coherent superposition of states until the external experimenter tries to open the decoherence-insulated lab door, then the following check to detect coherence should work, even after the enclosed physicists has interacted with the dead/alive cat, but before the sealed lab is allowed to interact with the external experimenter:   * the sealed lab has two lasers (A and B) inside, that mix at a half-mirror, and then the laser goes out of the lab from the half-mirror. This is the only way the state in the inner lab can affect the external universe. If both lasers are on, only one of the external paths has constructive interference, while the other has full destructive interference.    * if the cat is dead, the sealed lab experimenter turns on laser A and keeps B off. Both branches of the outward paths of the half-mirror give half intensities when the single laser A is on while laser B is off   * if the cat is alive, the sealed lab experimenter turns on laser B and keeps A off. Both branches of the outward paths of the half-mirror give half intensities when the single laser B is on while laser A is off   * After some synchronization time, the external experimenter measures photons from each outward path of the half-mirror, and he sees two possible outcomes: 1) one of the paths has 100% intensity, the other has 0% (coherence is kept) 2) both paths have 50% intensity (coherence was lost previously) Thoughts? What you expect to be the result of this experiment? why?


title: Proof of quantization of magnetic charge of monopoles using homotopy groups

text: Suppose we place a monopole at the origin $\\{{\bf 0}\\}$, and the gauge field is well-definded in region $\mathbb R^3-\\{0\\}$ which is homomorphic to a sphere $S^2$. Then the total manifold is $U(1)$ fibers attached to base $S^2$. We may ask how many types of "phase texture" are there on a sphere? Then I use $\pi_2 (U(1)) = \pi_2(S^1)$ since we established the map $S^2\mapsto U(1) \sim S^1$. But $\pi_2 (S^1) = 0$ not $\mathbb Z$!! Where I got wrong? According to the famous book Topology and Geometry for Physicists, the correct formula should be $$ \pi_1(S^1) = \mathbb Z$$ Why $\pi_1$?


title: How to calculate the highest theoretical artificial hill?

text: The biggest peak in the world is Mount Everest. Imagine someone starting to make an artificial hill (like pyramide) from soil (earth). So, when starting with an 200x200 Km base area, with 45degree slope, its mathematical height is 100km (Low orbital space height). Is it possible to make such an artificial peak? (Without taking into account financial issues and so on.) If not, why not? What would happen? Is there a height limit?


title: Why does a "speed of sound" exist?

text: I've recently read that wind cannot be faster than the speed of sound (german source). But why is there a speed of sound? I understand (well, mostly accept to be honest) that the speed of light in vacuum is a maximal speed for all matter. And I understand that you need more energy the more you accelerate particles. But why can't you make wind faster?


title: Proof that Statistical Mechanics is a model of Themodynamics

text: The laws of thermodynamics are essentially four axioms of a mathematical theory. The expectation values of a statistical ensemble are supposed to satisfy the axioms of thermodynamics (under the assumption that the ergodic hypothesis holds). **How is it proved that statistical mechanics satisfies the axioms of thermodynamics?** The question is important to me because the abstract theory of thermodynamics is used to develop things like the Maxwell relations, which are then applied to statistical mechanics ensembles. For this to give valid results it statistical mechanics must be a model of thermodynamics. I would appreciate pointers to proofs of this and references if you have any. Thanks a lot.


title: Photoelectric effect - calculating a current in the photocell in which only $5\%$ of photons manage to cause the effect

text: I have been trying to solve a problem about photocell and have obtained the wrong result. Can anyone tell me where did I go wrong? So it is easier I will write down all my calculations. > We have a photocell and we shine a light with of $\lambda=550nm$ on a > cathode with a surface $4mm^2$. On the surface of the cathode > $j=1\times10^3W/m^2$. > > **(a)** What is the work function $A_0$ of the metal from which cathode is > made if the stopping voltage equals $U_0=1.2V$? > > **(b)** What current flows through the photocell if we connect it to a > voltage (not the stopping voltage) and only $5\%$ of the incoming photons > manage to trigger the photoelectric effect. * * * First I calculated the energy of an incoming photon: \begin{align} W = \frac{hc}{\lambda} = \frac{1.602\times 10^{-19}Js\cdot 2.99\times10^{8}\tfrac{m}{s}}{550\times10^{-9}m} = 3.602\times10^{-19}J \end{align} Then I calculated the work function for question (a): \begin{align} W &= W_k + A_0\\\ W&= U_0 e_0 + A_0\\\ A_0 &= W - U_0 e_0\\\ A_0 &= 3.602\times10^{-19}J - 1.2V \cdot 1.602\times10^{-19}As\\\ A_0 &= 1.680\times 10^{-19}J\\\ A_0 &= 1.05 eV \end{align} Until now, it went well and my results match with the book, but when I try to calculate the answer for (b) i don't get the right result. This is what I did: \begin{align} j=\smash{\frac{dP}{dS}} \longrightarrow P &= j \cdot S \\\ P&= 1\times10^3 W/m^2 \cdot 4\times10^{-6}m^2\\\ P&= 4 \times 10 ^{-3} W \end{align} This is the power of all of the incoming photons while only $5\%$ manage to cause the photoelectric effect. But the power of the photons who manage to cause the photoelectric effect is smaller so $P'= 0.05 \cdot P = 2\times10^{-4}W$. Using $P'$ i now try to calculate the current flowing in the circuit and i get the wrong result _(the right one is $I=89\mu A$)_ : \begin{align} P' &= UI\\\ I &= \frac{P'}{U}\longleftarrow \substack{\text{I am not sure about this step where}\\\\\text{I inserted $U=1.2V$. Does $U$ effect}\\\\\text{the current $I$?}}\\\ I &= \frac{2\times10^{-4}W}{1.2V}\\\ I &= 166.6\overline{6} \mu A \end{align} I get about twice as much as is the solution but it is unfortunately wrong. A hint or some explanation would be helpful.


title: metamaterials for gravitational waves

text: Is there such a thing as refractive index of a material for gravitational waves ? How would the index of a material depend on density - Say density of interstellar cloud vs planet vs star vs neutron star vs black hole ? Could you get an object with negative refractive index - What would this do to a gravitational wave ?


title: interaction at a distance: how it works

text: In classical physics, I know the forces due to gravity and electrostatics. I know a lot of work has been done, but don't know the status. Is there a generally accepted theory of how the force is exchanged? If I understand correctly, electromagnetism works by exchange of photons. But electrical conductors interfere with the propagation of electrical fields, is there something similar for gravity? Are the weak and strong forces analogous to electromagnetism, exchanging gluons? If so, in a nucleus, I can understand how two adjacent protons could latch onto each other, but is the attraction between more distant particles cut off by intervening particles? Any references to a short paper summarizing what is known in lay language would be very helpful.


how safe is using the remote car toy from radiation, especially for kids, The manufacturers claiming that it will be safe, but is that 100 % safe over kids


title: Ratio of radiation sources in the uranium cycle

text: Sorry, I know y'all are probably getting a lot of questions re: Fukushima, but I had a very specific one and no-one has been able to answer it. I am specialised in medical radiation, and have been explaining the risks to workers in this accident, but I need to know what are the rough proportions of alpha, beta (+/-) and gamma radiation flying around. If I have this right, most of the decays in the uranium cycle are alpha or beta+ and beta-. Some isomeric transitions (and obviously the beta+ decays) will release gammas. Obviously there will be a small amount of xrays too, from Bremsstrahlungs and such. But what are the percentages of this in general uranium fuel material (I guess 'spent' fuel is the real problem here, not fuel undergoing chain reaction)? Is 50% of the radiation alphas? Is 80% gammas? It makes a huge difference from a biophysics perspective as alphas are easy to protect against (for the workers) but gammas are unshieldable. Beta- decays are in between, pretty easy to deal with. Anyone know the uranium fuel cycle well enough to give a rough estimate here? EDIT: Added some thoughts in my response to Bars, who helped a lot. Any other thoughts that follow on from those insights?


title: How to solve this Schrödinger equation?

text: I am taking an intro level quantum mechanics class. Our textbook gives a problem like this: > The deuteron is a nucleus of "heavy hydrogen" consisting of one proton and > one neutron. As a simple model for this nucleus, consider a single particle > of mass $m$ moving in a fixed spherically -symmetric potential $V(r)$, > defined by $V(r)=-V_{0}$ for $r<r_{0}$ and $V(r)=0$ for $r>r_{0}$. This is > called a spherical square-well potential. Assume that the particle is in a > bound state with $l=0$. > > (a) Find the general solution $R(r)$ of the Schrodinger equation for > $r<r_{0}$ and $r>r_{0}$. Use the fact that the wave equation must be finite > at 0 and $\infty$ to simplify the solution as much as possible. (You do not > need to normalize the solutions). > > (b) The deuteron is only just bound: ie., $E$ is nearly equal to $0$. Take > $m$ to be the proton mass, $m=1.67*10^{-27}kg$. and take $r_{0}$ to be a > typical nuclear radius, $r_{0}=1*10^{-15}m$. Find the value of $V_{0}$ (the > depth of the potential well) in MeV (1 MeV=$1.6*10^{-13}$J (Hint: The > continuity conditions at $r_{0}$ must be used. The radial wave function > $R(r)$ and its dervivative $R'(r)$ must both be continuous at $r_{0}$; this > is equivalent to requiring $u(r)$ and $u'(r)$ must both be continuous at > $r_{0}$, where $u(r)=rR(r)$. The resulting equations cannot be solved > explicitly but can be used to derive the value of $V_{0}$. I found I need to solve an equation of this form when $r<r_{0}$: $$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial {r}^{2}}(rR(r))-V_{0}R(r)=ER(r)$$ After expansion this becomes a second order ODE I am not able to solve. But I am not able to use the condition $r=0$ to obtain any good results. At $r=0$ we should have $-V_{0}R(r)=ER(r)$, which implies $E=-V_{0}$. (Is this allowed?). Then I can get $$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial {r}^{2}}(rR(r))=0$$ yet solving this implies $rR(r)=c_{0}r+c_{1}$, which I "feel" is impossible since it would imply $R(r)=c_{0}+\frac{c_{1}}{r}$, which has a singularity at $r=0$ unless $c_{1}=0$. On the other hand at $r=\infty$ we should have the equation become: $$-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial {r}^{2}}(rR(r))=ER(r)$$ Hence naming $rR(r)=S(r)$, we should have $$\frac{\partial}{\partial {r}^{2}}S(r)=kS(r), k=-\frac{2mE}{{\hbar}^{2}}$$ and solve it we have $S(r)=e^{\sqrt{k}r}\Rightarrow R(r)=\frac{e^{\sqrt{k}r}}{r}$. This function is obviously non-constant at the origin and since $k$ is negative, it actually "blow up". I do not know how to reconcile this desperate situation. So I need some help. Hence I cannot continue to part $(b)$ as well. I feel there is something very wrong in my approach. I also want to ask what does it mean "the deuteron is only just bound", why this implies $E$ is nearly equal to $0$? Also, if I cannot solve the equation explicitly, how can I get $V_{0}$'s value? I am very confused.


title: Depends river stream on river level?

text: If we have referral particle in the middle of the river, will it's velocity increase with higher river level?


title: High speed and low speed photons

text: Looking at the discovery of the neutron, and I came across this page: http://www-outreach.phy.cam.ac.uk/camphy/neutron/neutron3_1.htm The animation on the left, talks about low energy photons and high energy photons. And it implies the more energy a photon has the faster it moves. I don't understand the whole light is photons and waves at the same time thing, but I thought the speed of light was constant, that Gamma rays travel at the same speed as visible light, infrared, microwaves, etc. So how is that some photons can move faster than others (with more energy)?


title: Good book on the history of Quantum Mechanics?

text: Can anyone recommend a good book on the history of Quantum Mechanics, preferably one that is technical and not afraid to explain the maths (I did a degree in Physics many years ago) and also that explains the developments using the technical language of the time rather than using treatments as we know understand them. If it also explained the input from experiments that would be great too.


title: Effect of introducing magnetic charge on use of vector potential

text: It is well known that Maxwell equations can be made symmetric w.r.t. $E$ and $B$ by introducing non-zero magnetic charge density/flux. In this case we have $div B = \rho_m$, where $\rho_m$ is a magnetic charge density. But this means that $B$ cannot be expressed as the curl of vector potential anymore. Does it mean that it is impossible to develop electrodynamics of non-zero electric and magnetic charges in terms of vector and scalar potentials? What happens to $U(1)$ gauge invariance then? P.S.: I know that for the point-like magnetic charge vector potential can still be introduced, but not in the entire space. My question is related to non point-like magnetic charge densities.


title: how does a lizard walk on vertical walls?

text: One day I was observing a lizard walking on walls as easily as we walk on horizontal surfaces. I got curious to know the physics behind this but I couldn't get to any useful conclusion. I tried to Google it but all the hypothesis like lizards create vacuum while they walk or they secrete some sort of chemical (and few others) were rejected due to some or the other reason, so I thought maybe here I could get some help on the subject.


title: How many grams of explosive are equivalent to the energy in a battery?

text: In the movie _Terminator 3: Rise of the Machines_ , Arnie's fuel cells were twice shown to be extremely devastating when their stored energy was released. In real life, how many grams of, say, gunpowder or TNT are equivalent to the energy stored in   1. An AA alkaline battery   2. A laptop battery   3. An electric car battery Would the batteries be as devastating as the explosives if the energy could be released all at once (within milliseconds, say)? Presumably the power of chemical explosives come from the volume of gas they release...


title: confusion about finite line charge

text: My physics knowledge is (sadly) very infinitesimal and I have a question about a calculation which I think is wrong, but want to be sure. I tried to calculate the potential of a finite line charge. According to my understanding, this is simply done by integrating point charges along the line. When I do that, I get something that looks like this (lines are the equipotential lines) ![line charge](http://i.stack.imgur.com/tDolf.png) This plot makes intuitive sense to me since the potential computation is basically convolving a $1/\|x\|$ function with a "wide" delta pulse which I expect to look like this. The integration result is $$\frac{\lambda}{4\pi\varepsilon_{0}}\log\left(\frac{\sqrt{x_{2}^{2}+\left(a-x_{1}\right)^{2}}+x_{1}-a}{\sqrt{x_{2}^{2}+\left(b-x_{1}\right)^{2}}+x_{1}-b}\right)$$ where $(x_1,x_2)$ are the coordinates where the potenial is computed and (a,b) are the limits of the line charge that lives on the x-axis. However, many calculations that I found on the web (e.g. this one) obtain a result that is independent of the abscissa. First I thought that their result might only hold for x-values between the ends of the line, but even then the potential should not be independent of the abscissa. Am I getting something important wrong here? Thanks in advance!


title: What happens if an object has more kinetic energy than the Gravitational Binding Energy?

text: So the binding energy of an object is the amount of energy needed to move it an infinite distance away from another mass to essentially “escape” its gravitational field. But what happens if you give the object more energy than the binding energy. Will the object go “past infinity”?


title: How air humidity affects dry process of a washing?

text: I've noticed that during wet weather my washing does not want to dry. Is it because of lower temperature, or because of air humidity?


title: What's the most efficient strategy to shake a bottle to create a global flow?

text: Consider a bottle of orange juice with pieces of the fruit gathered at the bottom of the bottle. You need to shake the bottle to evenly distribute the pieces in the bottle. The first part of the process seems straight forward, just giving the bottle a few sharp shakes to create turbulence and destroy the conglomerate of fruity pieces. However, what's the most energy-efficient strategy to create a global flow to distribute the stuff? Since at the start of the process, the interesting parts are gathered at the bottom of the bottle, a strong axial component would be neccesary, for energy reasons a laminar flow would be interesting (assuming the dispersal has allready happened). Edit to add: Efficient means the least energy input from me. If your mixing method involves throwing the bottle off a cliff, the energy for cyrrying it up there counts.   Where I live, juice comes in round bottles or in boxy containers.


title: bound charges in materials

text: I have a question regarding the bound charges in electrostatics, I think I am a bit confused, on one side I have read that bound charges in a capacitor with a dielectric inside the plates are on the surface of the dielectric material. On the other side, in other books bound charges refer to the charges of molecules which cannot move as it happens in metals (free electrons). So, if I put a solid dielectric inside a capacitor the bound charges will be only on the surface? what about the charges on the molecules which compose the solid? If I take for instance a glass of water, are there bound charges?


title: Rate of evaporation of water

text: In our everyday life, does the temperature or humidity have a greater effect on how fast things dry? Are there any theories/models that explain this quantitatively?


title: How does heating cause motion?

text: W.R.T. Newton's laws a force is required to change the state of rest of a body. Then how does heating cause a particular object's atoms and molecules to wiggle ? What is the force being applied when you heat something ? What is thermal interaction?


title: Is "analog" quantum-computation not useful?

text: I understand what a qubit-based quantum computer by the current definition is and how they are constructed. I read another thread where someone suggested encoding a computation into a dual-slit experiment and reading out the answer as interference vs non-interference, and this was mostly shot down since it's not "useful", because the system ceased to be a quantum computer when decoherence occurred. But this reminded me of an old idea I had on other ways to encode computations in physics than the "classical" qubit-based schemes. As a striking example, consider one of the canonical uses for a quantum-computer - to efficiently simulate quantum-level systems. Well - to do this without a quantum-computer, you could actually assemble the system itself and let it "run", and then read out what you were looking for to simulate in the first place in many cases, in a vastly more simple way than digitally calculating it with qubits. Is this then not also a "quantum computation", really? The closest analogue I can come up with is the way analog computers were designed before digital computers took over, and that they were indeed much more efficient at a certain class of problems. Even a system of 3 particles performs a calculation (of sorts) which is difficult to simulate traditionally. More generally, if you could encode your problem into a hamiltonian which you subsequently map onto a physical structure wouldn't this be very useful? You might not be able to run _algorithms_ on it, but you might be able to format the problems the algorithms were meant to solve in the first place. I'd be interested in hearing if this approach is pursued and I have missed it or if it is deemed to be useless for some reason, or if it's useful but "digital" quantum computation is simply more attractive in the long-run so most of the focus is on that. Maybe there is a measure on the computational complexity that can be mapped or performed by a general quantum structure (non-qubit- based)? I seem to remember a Canadian company called D-Wave which I think do this more or less, and there was (is) much heated discussions on if it was "real quantum computation" or not. EDIT: Nature published a special Nature Insight review on this in April 2012, especially about Quantum Simulators, where the issues I brought up in this question are discussed, including the "digital vs analog quantum computation" issue.


title: Exchange operator in terms of rotation operator

text: I have studied about exchange operators and rotation operators and I know that an exchange between 2 particles in a combined state is the same as rotating each particle 180 degrees (according to http://en.wikipedia.org/wiki/Exchange_symmetry). Could someone explain the mathematical equation behind this? I have tried to search but to no avail.


title: Dark-energy sources and respective implications

text: Firstly, I want this to be a very layman/simple level. According to Sean Carroll, we have certain possibilities for where dark-energy comes from: http://www.youtube.com/watch?v=v6o2bUPdxV0 Including   * A modification of general relativity   * Vacuum energy   * A modification of vacuum energy And he goes on to say that **if vacuum energy is the right answer the universe will continue to expand**. What would be the implications if the right answer is a modification of general relativity? Are there any other theories that explain dark energy that make predictions about the future like this?


Does physics address the topic of consciousness? For instance, does physics say anything about how it _might_ arise or what might be its qualitative properties? I'm wondering because it's interesting how certain combinations of particles are conscious (at least apparently) while others are not.


title: Image formation (lenses)

text: I have a question regarding image formation by lenses. It is the following: A lense with a focal length of $-48.0cm$ forms an image $17.0cm$ to the right of the lense. Where is the object positioned? **My attempt at a solution:** Well, since the focal length is negative, we know that the lense is divergent (concave). The problem statement says that the image is to the right of the lense, so $s_i=17cm (>0)$ We can compute the position of the object by using the formula below: $$\dfrac{1}{d_0}+\dfrac{1}{d_i}=\dfrac{1}{f}$$ Changing what we know for the values, we get: $$\dfrac{1}{d_0}+\dfrac{1}{17}=- \dfrac{1}{48}$$ And doing the calculations, it comes down to: $$s_0=-12.55$$ However, the solutions say the solution is $s_0=26.3$. One can get that if we take $s_i$ to be negative, which is wrong, in my opinion. (?) So, why am I wrong? Or am I correct?


title: Speed of Water in a River

text: Does a river flow faster the higher the water level - or does it go at exactly the same speed? I was recently tubing in a river and the water seemed to be going faster with the water level being higher; this may have just been an error in judgement. Intuitively, perhaps a higher water level is associated with a higher water pressure, which would lead the river to go faster as in a hose. But the river is not limited to a space constraint.


title: Atomic physics - lattice energy

text: Question: Why is ionic lattice energy inversely proportional to the radius of the atom? Most heterogeneous covalent molecules are polar to some extent. The degree of polarity, or the dipole moment, depends on the difference in electronegativity difference between the two atoms. The larger the dipole moment, the higher the ionic character. What I know: Electronegativity decreases as you go downward in a group, however, the size increases, usually, as you go downwards in a group. Thus, ionic character will increase upon going downward, but the ionic lattice energy will decrease? This seems contradictory. Is this true? And if it is, why is it so?


title: Rapids in a River

text: Can I expect to have bigger or smaller rapids in a river that has a water level that is higher than average? I am not sure which way this would go. With the lower water, one is closer to the rocks, which, I think, generally are the things that create the rapids. But perhaps a higher water level is associated with a faster current, which would seem to make rapids greater.


title: Do we have a quantum field theory of monopoles?

text: Recently, I read a review on magnetic monopole published in late 1970s, wherein some conjectures of properties possibly possessed by a longingly desired quantum field theory of monopoles are stated. My question is what our contemporary understanding of the quantum field theory of monopoles is. Do we have a fully developed one? Any useful ref. is also helpful.


title: A problem about special relativity

text: ![how should i understand the meaning of the apparent earth angular size, is it just the same as the size at question a ](http://i.stack.imgur.com/6JdnO.png) i can't understand it clearly, so please help me!


title: Is Coulomb's law accurate for moving charges?

text: Can we use Coulomb's law to calculate the force between two charges which are not at rest? If not, what formula should be used to calculate the force? I searched it, but I couldn't find a clear answer.


title: Why doesn't matter pass through other matter if atoms are 99.999% empty space?

text: The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions/intersections from all possible directions?