title: How two photons interfere in a double slit experiment

text: I have a very basic question about photons in double-slit experiment. I am not good at math, and have some Quantum mechanics knowledge. Math free explanation would be very good to me. When we do double slit experiment with single photon at a time, it will be detected at some location of screen. Over the time period the accumulation of detected locations show interference pattern. The reason for the interference pattern is that photon's location in not well defined, the presence of photon in particular place can be determined by prbability and this probability is presented as a wave. that probability wave is splitted in to two while passing the two slits, collide with each other and causes interference pattern. Question 1: Is this reason for single photon interference correct? Question 2: How is the probability wave behaving while two-photons passing double-slit at a time? Are 4 probability waves (2 for each) interfering?


title: A (mundane) CS analogy for quantum teleportation

text: From my limited understanding of quantum entanglement, it seems like qubits act the same way as pseudo-random-number-generators (except as far as we can tell, these ones really are random). When you entangle two qubits, you give them both the same seed. Then a measurement is like running an (otherwise deterministic) program. As long as Alice and Bob both run the same program on their two (identically seeded) PRNG's, they'll both get the same results. There's no teleportation here, they just both have access to the same information. Why is this referred to with the fantastic name "Quantum Teleportation"? Since Alice and Bob can't pass information to each other through their PRNG's (they're just copies after all, they're not really linked).


title: Are coherent states of light 'classical' or 'quantum'?

text: Coherent states of light, defined as $$|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{n!}|n\rangle$$ for a given complex number $\alpha$ and where $|n\rangle$ is a Fock state with $n$ photons, are usually referred to as the most classical states of light. On the other hand, many quantum protocols with no classical analog such as quantum key distribution and quantum computing can be implemented with coherent states. In what sense or in what regime should we think of coherent states as being 'classical' or 'quantum'?


title: What will happen after escaping earth's gravitational field?

text: Suppose that I escaped the gravitational field of earth. Then: am I going to be pulled by Sun's gravity?


title: Particle as a representation of the Lorentz group

text: In QFT one may refer to a particle as a representation of the Lorentz group (LG). More accurately - every particle is a quantum of some field $\phi(x)$ that belongs to some representation of the LG. _I wonder if that is really neccesary?_ Arguments are usually like following: Let $L$ be some Lorentz tranformation and $\Lambda(L)$ is the corresponding transformation of the field $\phi(x)$ , i.e. under the transformation $x\to x'=Lx$ field $\phi(x)$ goes to $\phi'(x')=\Lambda(L)\phi(x)$. Then under two succsessive transformations $L_2,L_1$ field $\phi(x)$ goes to $\Lambda(L_2)\Lambda(L_1)\phi(x)$. But one can also find himself in the same reference frame via transformation $L_2L_1$ and therefore observe the field $\Lambda(L_2L_1)\phi(x)$. Now demanding $\phi'(x')$ to be the same thing in both cases (since we arrived in the same reference frame) one concludes that $$\Lambda(L_2)\Lambda(L_1)=\Lambda(L_2L_1)\,\,\,\,\, (1)$$. In mathematical language one says that tranformations of the field $\phi(x)$ constitute the representation of the LG.   Well i do not see this to be neccesary. At least not within the "proof" that was been given. I've intentionally used the word "observe" above. Usually in physics not all the information that is encoded in the field value is virtually observable. For instance let $\phi(x)$ be a complex scalar field. Then an overall phase of the $\phi(x)$ is not observable. So i can assume that two ways of going to new reference frame may differ by the "gauge" tranformartion since it does not affect observables. In the case of the complex scalar field one can try to weaken (1) to something like $$\Lambda(L_2)\Lambda(L_1)=\Lambda(L_2L_1)e^{i\alpha(L_1,L_2)}$$ where $\alpha(L_1,L_2)$ is the real number depending on Lorentz tranformations $L_1,L_2$. My questions are:   1. is it really possible for the physical fields to behave as was described?   2. if so, does their behavior differ (at classical or quantum level) from the "ususal" fields behavior?


title: quantum theory of light

text: What's the scattering matrix for a PBS (polarization beam splitter) ? Is it just unitary ? If one polarization never couples into another polarization (then there's a lot of zeroes in that 4x4 matrix) - is that impossible somehow ?


title: Active gravitational mass of the electron

text: In PSE here electrons are added to a sphere and gravitational modifications are expected. My question is:   Is there any **experiment** that show that a negatively charged object is source of a stronger gravitational field than the same uncharged object? In other words:   The active gravitational mass of an electron is equal to his passive gravitational mass **by experiment**? added, for clarification:   from here activ/passive grav mass:   \- active gravitational mass: establishes the field   \- passive gravitational mass: responds to the field   \- by experiment: a carefully designed setup that evidences how electrons interact in the presence of a gravitational field. Does it exists already? If no, is it doable? etc.


title: Factors in predictions for Tolman's Surface Brightness Test?

text: http://en.wikipedia.org/wiki/Tolman_surface_brightness_test It says: > In a simple (static and flat) universe, the light received from an object > drops inversely with the square of its distance, but the apparent area of > the object also drops inversely with the square of the distance, so the > surface brightness would be independent of the distance. In an expanding > universe, however, there are two effects that reduce the power detected > coming from distant objects. First, the rate at which photons are received > is reduced because each photon has to travel a little farther than the one > before. Second, the energy of each photon observed is reduced by the > redshift. At the same time, distant objects appear larger than they really > are because the photons observed were emitted at a time when the object was > closer. Adding these effects together, the surface brightness in a simple > expanding universe (flat geometry and uniform expansion over the range of > redshifts observed) should decrease with the fourth power of (1+z). To date, > the best investigation of the relationship between surface brightness and > redshift was carried out using the 10m Keck telescope to measure nearly a > thousand galaxies' redshifts and the 2.4m Hubble Space Telescope to measure > those galaxies' surface brightness.[1] The exponent found is not 4 as > expected in the simplest expanding model, but 2.6 or 3.4, depending on the > frequency band. I'm trying to understand where $(1 + z)^4$ comes in. Specifically, it says: > there are two effects that reduce the power detected coming from distant > objects. **Question** Do the two effects each contribute by $(1 + z)^2$, which are then multiplied together? It also says > First, the rate at which photons are received is reduced because each photon > has to travel a little farther than the one before. Let's say just this factor was removed. Would the predicted brightness then be reduced by $(1 + z)^2$?


title: Entropy is constant. How to express this equation in terms of pressure and density?

text: In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:   1. The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$   2. The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$   3. A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass) My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately. PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics


How is the image in a mirror created without a lens or pinhole? From every point in the world there are infinitely many rays going out. How come the image on a mirror is ok, when there is no pinhole to capture exactly one ray from each world point, or a lens to focus all rays coming from one world point onto one image point? I mean, in computer vision we use these models to create images, so the mirror example here confuses me a lot.


How is the image in a mirror created? Are there infinitely many light rays? My motivation for the question is from image processing. We work with images as discrete 2D functions, as matrices. Spatial sampling is done, and also quantization. Then I thought, well that is far from perfect, I wonder how does God create images? And then I thought of a real example, well, the mirror! The 3D world is projected on a 2D plane, the mirror. So how many rays arrive on the mirror? Infinitely many? Does God do spatial sampling or is the image in the mirror continuous?


title: Is there a handwavy way to explain what quantum correlation means?

text: Is there a simple way to explain the difference between a classical and truly quantum correlation to a non-quantum person who has basic understanding classical correlation? I mean without invoking quantum mechanics, a simple CHSH type Bell inequity can be explained without it--perhaps with a little more math than I'm looking for, but more importantly non-classical correlation does not necessarily imply Bell violation. Has anyone found a successful approach to explaining quantum correlation to a non-expert audience? Even partially successful approaches would be helpful.


title: Explicit supersymmetry breaking fermion mass terms

text: I hope you can clear up my following confusions. In Girardello's and Grisaru's paper (Nuclear Physics B, 194, 65 (1982)) where they analysed the most general soft explicit supersymmetry breaking terms, they explicitly mentioned that $\mu \psi \psi$ is not soft, where $\psi$ is the fermionic component of a scalar superfield. This means that such a term will give rise to quadratic divergences. However, in Stephen Martin's SUSY primer, on page 49, he explictly says the following: > _One might wonder why we have not included possible soft mass terms for the > chiral supermultiplet fermions, like $L = −\frac{1}{2} m_{ij}\psi^i\psi^j + > c.c.$. Including such terms would be redundant; they can always be absorbed > into a redefinition of the superpotential and the terms ..._ This would seem to suggest that adding explicit chiral fermion mass terms is not problematic, since it is equivalent to a redefinition of the superpotential and the other soft breaking terms (e.g. scalar masses), both of which don't give rise to quadratic divergences. I seem to see a contradiction here, but I am sure it is due to some subtlety I am too blind to notice.


On earth, using earth-sheltering techniques can significantly reduce the temperature fluctuations on a structure. Would the same statement be true as well on the Moon? Does the Moon's core still contain significant heat?


title: What frame(s) of reference are used to measure the rotation of the Sun around the galaxy ?

text: I can find various speeds and estimated durations listed at numerous places but none specifically describe the frame of reference. Possible options as example of kind of answer I expect.   * Local Galactic cluster    * Distance quasars    * The cosmic background radiation? **\--------- UPDATE ---------** Thanks AIB and voithos. Lot of reading for me. Though technically, I still don't have an answer that meets the following criteria.   1. rotational velocity(average preferably) of sol around best estimate of center of Milky-way galaxy.    2. publicly available reference(I don't have immediate access to some of the books given)    3. frame of reference external to Milky-way galaxy. As I note below the only reference (wmap5basic_reprint.pdf) I can read that uses an external frame of reference doesn't specifically state the vector is rotational (despite wikipedia article assuming such). The topic is barely touched on in that paper. I realised the speeds are variable. What I had not realised is that the whole idea of a (relatively) clearly defined x orbiting y system doesn't really scale up well from the local solar system to the galactic scale. The galaxy is more like a whirlpool or tornado compared to the "clockwork" appearance of the solar system. Although both the solar system and galaxy are constantly(very slowly) changing "fluid" rotational systems, the galaxy is obviously far more fluid than the solar system. Also we have not yet been able to observe anything about it's center. Or in other words, we are not "orbiting" the galaxy, we are part of the galaxy. I suspect the topic is more in the realm of "fluid dynamics" than "orbital mechanics" I've accepted AIB's answer as the most enlightening to me personally. Also, it would appear I have wiki-sidebar blindness. Apologies for that. FYI   The paper referencing the speed relative to the CMB, as mentioned in wikipedia article can be found here http://cmbdata.gsfc.nasa.gov/product/map/dr3/pub_papers/fiveyear/basic_results/wmap5basic_reprint.pdf The relevant section appears to be 7.3.1. "... implies a Solar System peculiar velocity of 369.0 ± 0.9kms-1 with respect to the CMB rest frame." Although it's not obvious to me what vector that velocity is along. Though Dipole Anisotropy in the COBE DMR First-Year Sky Maps gives a specific velocity(including vector) for the local galactic group in relation to the CMB rest frame "implied velocity of the Local Group with respect to the CMB rest frame is 627 +/- 22 km/s toward (l,b) = (276 +/- 3 deg, 30 +/- 3 deg)." FYI   Other reference frames that are external to the local galaxy are "The Supergalactic coordinate system"


title: Homogeneity of space

text: How do I verify homogeneity and isotropy of space, for example for hyperbolic space? My idea is to verify that Lorentz transformations in 4-vectors can move any desired point on the hyperboloid into the origin, and can rotate the hyperboloid about the origin by an arbitrary angle. The problem is I do not know how to map these statements into a precise mathematical language. Can you please help me?


title: Seeking help on simplified mathematical (biophysical) model of serpentine locomotion in viscoelastic environment

text: I'm trying to build a mathematical model of C. _elegans_ locomotion under overdamped viscous environment. In order to simplify this problem, I use six discrete pieces of rigid cylinders to represent worm body, and this chain of rigid pieces are connected by torsional springs this way:                                         -----*-----*-----*-----*-----*-----      Notation: ------ is one piece of rigid body; * is torsional spring. Suppose that 1) external drag force is applied on the center point of each piece, 2) we know the length of each piece, spring stiffness, and damping coefficient(which should be very small and negligible), and external viscous drag coefficients, 3) I input a periodically changing torque to the first torsional spring (counting from head), and supposedly posterior spring torque is coupled both to the angle displacement of the anterior spring and its current torque. Can someone help me write a set of dynamics equations that describe the locomotion of the worm? Or I also appreciate if someone can suggest me some book or article to read about. Thanks, Frank


title: Proton gas density

text: As far as I know the lightest gas is hydrogen due to low mass of its nucleus, but what if we were to somehow strip hydrogen atoms of electrons and enclose protons in a container made of teflon (high electronegativity). Would such a proton-only gas be stable? And what would be its density under atmospheric pressure (assume teflon envelope around the gas)? Would electrostatic effects between positively charged protons contribute to lowering density of such gas?


title: Would gravity on the surface of a planet which is being consumed by a black hole change?

text: Assuming that the black hole starts grow in the exact center of the planet and that the general structure of the planet does not degrade as it is eaten from the inside, would the gravity on the surface of the planet be affected by the black hole growing. My uneducated guess is that as the matter becomes more compacted the gravity on the surface would actually decrease due to the fact that the distance to the mass, with respect to the surface, is increasing. Also would the evaporation of the black hole actually decrease the total amount of mass in the center of the planet?


title: Transmissivity of Liquid Crystals in the near-infrared range

text: The gist of my query is in the title. I have been searching for papers and studies that could provide transmission curves (in nematic liquid crystals) across the visible to near-IR range (or from 300 nm to 2200 nm). I would thus like to know if anyone has encountered, or happens to know, of how a layer of nematic liquid crystals (for example in an LCD) deals with incident infrared light - whether it allows the majority through, or if it absorbs/reflects most of the waves.


title: Why do we need 2. Quantization of the Dirac Equation

text: As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The **Dirac equation** , is given by $ i \hbar \gamma^\mu \partial_\mu \psi = m c\cdot \psi $ where the Dirac matrices $\gamma^\mu$ are defined by $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = \eta^{\mu\nu}$ and where $\psi$ is a "solution". Ok, the first deal of confusion already starts with the $\psi$'s. It seems that people freely see them as spinor valued functios or as "operator fields". But if I understand this correctly, seeing them as operators, is not part of the original picture, but was later added as the so called **second quantization**. Right? Now my question is the following: Why do we need this second quantization of the Dirac equation? What experiments can not be described by the original Dirac equation? Maybe there is a list somewhere or such.


title: Feeling the Breeze

text: This question is based on an odd phenomenon I noticed last week. Since then, I've been collecting more data to try to get an accurate picture of the facts. Each morning, I go to my bus stop. It's on a main road, and when I get there, there is a lot of traffic. Every couple minutes a large truck goes by, and I suddenly feel a breeze go by me. The catch here is that there is about a one- second delay between the truck passing by and the start of the breeze. What is causing this phenomenon? Here's some more information: I'm about three meters from the road; The breeze begins roughly one second after the truck passes by, and last for one and a half seconds; The breeze seems to be blowing diagonally, from my rear left to my front right side (I'm in America, so the trucks go from left to right); The breeze does not seem to depend on truck size (although small vehicles do not cause it). I do have two possible explanations for the breeze: 1) The truck goes by at about 30 mph. This could create a tiny, short-lived vacuum at the back as it moves forward, thereby drawing air towards the rear of the truck. 2) The breeze is the result of some sort of "bow shock" from the truck. Neither of these explanations seem to satisfy me, though, because they seem a bit far-fetched, and I am certainly no expert in fluid mechanics. What could be the cause of the breeze? As a last note, I have no idea what tags to use for this question, because I don't know what the cause of the breeze is. Suggestions for these and other edits are welcome.


title: Rotational symmetry in integration

text: Can someone please tell me why $$4\int d^4x \, x^\mu x^\nu ~=~\int d^4x \, g^{\mu\nu}x^2 $$ by some rotational symmetry argument?


title: What algorithms can be used to compute the binodal in a ternary Flory-Huggins theory?

text: What are the most popular algorithms used to obtain a binodal curve for the ternary mixture (starting from Flory-Huggins theory)? I would like to obtain a plot similar to the one calculated here http://physics.stackexchange.com/a/112972/46158. I am aware that the binodal consists of pairs of points, say $\phi=(\phi_A,\phi_B,\phi_C)$ and $\bar\phi=(\bar \phi_A,\bar \phi_B,\bar \phi_C)$, for which the chemical potentials are equal \begin{align} \mu_A(\phi_A,\phi_B,\phi_C)=\mu_A(\bar \phi_A,\bar \phi_B,\bar \phi_C)\\\ \mu_B(\phi_A,\phi_B,\phi_C)=\mu_B(\bar \phi_A,\bar \phi_B,\bar \phi_C)\\\ \mu_C(\phi_A,\phi_B,\phi_C)=\mu_C(\bar \phi_A,\bar \phi_B,\bar \phi_C) \end{align} (here $\phi_A$, $\phi_B$ and $\phi_C$ are relative volumetric concentrations of components $A$, $B$ and $C$). Because the volumetric concentrations must sum to unity ($\phi_A+\phi_B+\phi_C=1$ and $\bar\phi_A+\bar\phi_B+\bar\phi_C=1$) so we have a total of five conditions for six unknowns. To recover the binodal it suffices to solve these five equations for pairs of points $\phi$ and $\bar\phi$. This brute force approach was developed in 70s [1] but there is little to no information in more recent experimental papers concerning the algorithms used by the authors. I am unsure whether the algorithm from [1] is still used or some alternative has been found. [1] Hsu C C and Prausnitz J M 1974 Thermodynamics of Polymer Compatibility in Ternary Systems Macromolecules 7 320–4


title: Purcell and the Magnetic Field

text: When showing that the magnetic field derived from the curl being proportional to current and the divergence being zero is unique, Purcell says that at some sufficiently remote enclosing boundary the difference of two magnetic fields (two possible fields from the curl), D, would take a constant value. Why is this true?


title: Hyperopia, Far Sightedness

text: With hyperopia, the focal point is behind the retina, shouldn't this mean that the image is flipped on the retina itself from what is usual? I must be drawing my ray diagrams wrong. A little something different from: Optics, lenses and our eyes


title: How to find speed when accelerating down a slanted wire

text: I saw this picture on one of my social media sites with the caption, "I'd do this in a heart beat! Who's with me!" ![enter image description here](http://i.stack.imgur.com/GMzJi.jpg) I was about to go balls to the walls and say, "I'm in! When and where??" But then I got to thinking, how fast would I be going when I hit the water? If I were going too fast, would it hurt me? SO I was trying to figure this out, and I'm not very good at physics so I was wondering if you guys could help me out. I estimate the guy is 90 kg in mass, the wire is angled pi/6 from the horizontal and he's about 50 meters above the water when he starts (all estimates...). What is the formulas I need to figure out the speed the guy will be going once he hits the water? I know there's some calculus in there, and I'm pretty good at calculus.


title: Underlying physical basis of an exponential distribution

text: My data set of upper atmospheric cloud occurrences $N$ versus their thickness (or optical brightness, say $B$) show an exponential variation over more than two orders of magnitude - that is $N$ varies as $\mathrm{e}^{-cB}$. The quantity $c$ can vary from location to location. More than one first-principle model shows the identical exponential law. This behavior is clearly robust, but I have no idea as to what would cause such a beautiful distribution of such complicated animals, like clouds. Where would I start to try to understand the underlying physical principle? There must be something basic here that I am missing.


title: Why is the optimum wheel size of a bicycle about the same as that of a car?

text: The optimum wheel diameter of cars and bikes appear to be roughly the same, certainly well within an order of magnitude. This is despite very different average speeds and propulsion mechanisms. Can anyone come up with a dimensional analysis-type argument explaining why this is?


title: How is capacitance defined for three concentric spheres?

text: If we have a configuration of metal concentric spheres (each of negligible thickness) of radii $r_1,r_2,r_3$ respectively and $r_1<r_2<r_3$, and we are given the potentials of the spheres to be $0, constant,0$ respectively. How might we find the capacitance of the configuration? _**EDIT: My question: How is capacitance defined for such a system?_** Added: I think I have managed to find the charge on each shell :) What is missing now is just a definition of capacitance for such a system.


title: Why does snow seem to reflect red light best?

text: I've noticed that snow seems to reflect red light more than any other wavelength at night. It's seen in the sky, as well as the snow itself, both of which take on a reddish tint. Why exactly does it do this? Additionally, why is this only noticeable under moonlight? If it's the moonlight that's tinted red, then alternatively, why is that? My apologies for the rather basic question, but I can't seem to find information on the matter elsewhere.


title: Electric motors and change in flux

text: "When a current passes through an electric motor, the magnetic force on the motor causes a torque on the loop of wire causing it to turn". However, when the loop rotates, there should also be a change in magnetic flux causing an induced emf (Faraday's law). However, my book makes no such reference. my questions:   1. Is there going to be an induced emf opposing the applied emf of the DC source in the electric motor?   2. If yes, wouldn't the induced emf cancel the applied emf causing no magnetic force to be exerted and thus no longer converting electrical energy to mechanical energy?


title: The velocity of light changes from one medium to another, but does its energy change?

text: Does energy of light change when a light travels from a medium to another of different optical density?


If the quarks in a neutron are (up,down,down), why isn't it negatively charged? Excuse the silly question, just wondering.


title: spread of fock state distribution and infinite revival time of rabi oscillation in spontaneous emission

text: In cavity QED for a 2-level atom, the revival time for oscillation b/w the states $\left|\ e\ 0\right\rangle$ and $\left|\ g\ 1\right\rangle$ (absorbing the same photon that is emitted) is said to be infinity because in general, Rabi oscillation revival time is proportional to the square root of the mean of the occupancies of Fock states. Why should the spread of the Fock states distribution go to large values or close to infinity in the spontaneous emission case, to make the revival time infinity ? Refer :- http://prl.aps.org/abstract/PRL/v94/i1/e010401 for a description of Rabi Oscillations Revival


title: Wick's theorem for calculating OPE

text: I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, $$T(z)\partial_{w}\phi(w)=-2\pi:\partial_{z}\phi(z)\partial_{z}\phi(z):\partial_{w}\phi(w).$$ Using Wick's theorem, we know that $$:\partial_{z}\phi(z)\partial_{z}\phi(z):\partial_{w}\phi(w)=:\partial_{z}\phi(z)\partial_{z}\phi(z)\partial_{w}\phi(w):+2\langle \partial_{z}\phi(z)\partial_{w}\phi(w)\rangle :\partial_{z}\phi(z):$$. Then why is this just equal to $$2\langle \partial_{z}\phi(z)\partial_{w}\phi(w)\rangle \partial_{z}\phi(z)?$$ as stated in many CFT books?


title: Lecture Notes confusion: Constructing the Einstein Equation

text: This question is on the construction of the Einstein Field Equation. In my notes, it is said that > The most general form of the Ricci tensor $R_{ab}$ is > $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ where $R$ is the Ricci scalar. Why is this the most general form (involving up to the second derivative of the metric --- by definition of $R_{ab}$)? I suppose there are symmetry and degrees of freedom arguments. But I can only see why the LHS is a _possible_ form. I can't se why it is the most general form... > Taking the covariant derivative $\nabla_a$ of the expression above gives > $$C=\frac{1}{2}$$. This I understand. > Compare the resulting expression with the Poisson equation gives > $$A=\frac{8\pi G}{c^4}$$. This I _don't_ understand --- perhaps I am being silly again... but still. I assume the "Poisson equation" referred to here is $$E^i{}_i=4\pi\rho G$$ where $E^i{}_i$ is the tidal tensor and may be expressed as $$E^i{}_i=R^i{}_{aib}T^aT^b$$ where $T^a$ is the tangent vector of the geodesic. So contracting $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ with $T^aT^b$ gives $$4\pi\rho G=AT_{ab}T^aT^b$$. But then what? Or perhaps I have already made a mistake? I only know the energy-momentum tensor $T_{ab}$ to be of the form $$\begin{pmatrix}H&\pi_i\\\\\frac{s_i}{c}&T_{ij}\end{pmatrix}$$ where $H$ is the energy density, $\pi_i$ is the momentum density, $s_i$ is the energy flux. But I don't understand how it leads to $$A=\frac{8\pi G}{c^4}$$.


title: Hollow gold bar

text: A scammer got a hollow gold bar and fills it with a combination of lead and air, with the same average density as gold. What's the simplest way of discovering the fraud? I know that x-rays will see it, but are there simple means for analyzing it? (without destroying it)


title: Conservation of momentum leading to damage

text: What would be an intuitive way to damage objects in a physics game using impulses? Since momentum is conserved, so is impulse (the change in momentum for any two time periods) in a closed system. So if I have the impulse of a collision (that is, the change in momentum during a time interval for one of the two bodies) what would be the intuitive way to damage both the objects. Should each object receive the same damage based on the impulse of the overall system, or should the damage incurred be split unevenly based on the relative mass of each object? My intuition is that objects with larger mass would have higher hit-points, and so would be less affected if the same damage was incurred to both the objects in the collision.


title: Radiation from farther galaxies

text: I've read many facts from NASA's webpage.. Sometimes they tell, (for example) "NASA's Chandra X-Ray Observatory discovered this ultra-luminous X-Ray source (about 15 million LY) which shows an extraordinary outburst..." My question is.. Is it possible to obtain those electromagnetic radiation (Infra-red, Visible, UV or X-Rays) from such a long range? If so.. How?


title: Maxwells Equation from Electromagnetic Lagrangian

text: In Heaviside-Lorentz units the Maxwell's equations are: $$\nabla \cdot \vec{E} = \rho $$ $$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$ \nabla \times \vec{E} + \frac{\partial \vec{B}}{\partial t} = 0 $$ $$ \nabla \cdot \vec{B} = 0$$ From EM Lagrangian: $$\mathcal{L} = \frac{-1}{4} F^{\mu \nu}F_{\mu\nu} + J^\mu A_\mu$$ I can derive the first two equations from the variation of the action integral: $S[A] = \int \mathcal{L} \, d^4x$. Is it possible to derive the last two equations from it?


title: What is a chemical potential good for?

text: I read that the definition of the chemical potential is, that it is the partial derivative of the Free energy with respect to the number of particles, $$\mu=\frac{\partial F}{\partial N}.$$ Unfortunately, I don't get this definition, wherefore I wanted to choose an example: Let me think of a system for example a water in oil emulsion. How would one evaluate the chemical potential of this mixture? There the chemical potential would definitely depend on the question: Where do you want to insert which sort of particles?


title: Observable: possible outcome of measurement vs (linear) transformation

text: One of the postulates of quantum mechanics is that every physical observable corresponds to a Hermitian operator $H$, that the possible outcomes of the measurements are eigenvalues of the operator, and that after a measurement the state collapses into the eigenfunction corresponding to the observed eigenvalue. Some authors state that this is the reason quantum mechanics is "built on" linear algebra. However from a linear algebra standpoint this postulate seems strange. In linear algebra operators are linear transformations that take vectors to vectors; hence I would expect an operator also be a linear transformation, physically realised by a black box that takes in a particle in a state $\psi$ and emits the particle in the transformed state $H \psi$. In fact, if I look at the creation operator (from say an electron in a harmonic oscillator potential), this clearly can't correspond to an observable because it's not hermitian. I can imagine the black box to be implemented by a machine that fires a photon into the electron. I can also imagine operators that for example rotate the angular momentum of an electron without measuring it. Can I create a physical apparatus that "implements" the position operator (in position space, it takes a state $\psi(x)$ and returns $x \psi(x)$)? Why does QM use operators in these two different senses and how do I distinguish them?


title: Infinite-range 1D Ising model

text: The Hamiltonian for this system is given by \begin{equation} \mathcal{H} \\{S\\} = -H\sum_i S_i - \frac{J_0}{2} \sum_{ij} S_i S_j, \end{equation} where $H$ is the external magnetic field and there is no restriction to nearest neighbour interaction. The spin at each site, $S_i$, may take value $+1$ or $-1$. Now my question is: **Why does this model only make sense if $J_0 = J/N $, where $N$ is the number of spins in the system?**


i've problem with the last terms of uncertainty principle: 1\. are these equalities true: $$\langle \\{\Delta \hat A,\Delta \hat B\\} \rangle=\langle \\{\hat A, \hat B \\}\rangle$$ $$\langle [\Delta \hat A,\Delta \hat B] \rangle=\langle [\hat A, \hat B] \rangle.$$   1. what is result of product commutator with anti commutator?


title: Elimination of velocities from momenta equations for singular Lagrangian

text: this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac. Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$, (where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of $q_i$)($L$ is the lagrangian, $q$ represent degrees of freedom in configuration space) Now Dirac says : "If the $n$ quantities $∂L/∂v_i$ on the right-hand side of the given equations are NOT independent functions of the velocities, we can eliminate the $v$'s from (the above-given) set of equations and obtain one or more equations: $\phi_j(q, p) = 0$, ($j = 1, 2, ... , m$ if there are $m$ such constraints)" Could anyone please explain how this comes about? I can't understand how the $v$'s can be necessarily eliminated, and if the $p$'s are not all independent, then we can simply obtain relations like $\sum_i a_ip_i = 0$ (where $a$'s are non-zero coefficients), not involving $q$'s at all. Thanks (and apologies in case I'm missing something obvious).


title: How can radioactivity change depending on the number of nuclei?

text: I came across this problem. Please help me solve it. > A radioactive sample S1 having an activity of $5\ \mathrm{\mu Ci}$ has twice > the number of nuclei as another sample S2 which has an activity of $10\ > \mathrm{\mu Ci}$. The half lives of S1 and S2 can be > > (A) 20 years and 5 years, respectively > > (B) 20 years and 10 years, respectively > > (C) 10 years each > > (D) 5 years each All through out my high school i was never introduced to the idea that radioactivity could change depending on the number of nuclei. Can someone throw some light on this?


title: Equation that tells me the rpm and mass of a spinning disk needed to keep a second large mass stable using gyroscopic effects

text: I am trying to figure out how large of a mass and how quickly I need to spin said mass to keep a two-wheeled robot stable. Ideally, I am looking for a formula that relates _m_ 1=mass of robot, _m_ 2=mass of spinning disk, _v_ =rotational speed of disk (rpm), and some sort of stability factor _s_ (or amount of mass _m_ 1 that it can counteract). Is there any such formula, and what would it be? Thanks!


title: Is velocity of light constant?

text: There is a difference between the concept of the speed of light and the velocity of light. are both of them constant ($dc=0$ and $dv_c=0$)? if yes, why?


title: What quantum states can be measured using entanglement?

text: I have been reading about quantum entanglement, and I was wondering what quantum states can be "sent" using entanglement. I know you can measure the spin of one of the particles, and know the spin of the other, but can you also do this with other quantum states, such as polarization, or energy? Also, could you theoretically entangle molecules, or even larger structures? What is happening in this field of physics? Finally, if you were able to measure the energy levels of the entangled particles, when you changed the energy state of one of the particles, would the other? Could this violate the first law of thermodynamics, or is the change so small that it does not matter, since this is the quantum world?


title: Making an equal amount of positive and negative energy?

text: I was wondering if it would be possible to create an amount of positive energy out of a vacuum, in addition to an equal amount of negative energy, thus not violating the first law of thermodynamics, since the total amount of energy in the universe would remain the same. It could be like an "energy loan," with the positive energy being like the "loan money," and the negative energy being like the "debt." Is this theoretically (or even practically) possible? For example, I am asking if you could create 200 units of positive energy out of a vacuum, as long as you also created 200 units of negative energy too, so as not to violate the second law of thermodynamics. By negative energy, I mean energy that is below 0 degrees Kelvin. I came across the idea as I was reading about the zero-energy universe theory. It suggested that this mechanism was what made the universe, and that it could have been started by an unusually large quantum fluctuation. Does this idea conflict with the second law of thermodynamics in any way? Would entropy have to be considered, since negative energy reduces entropy, violating the second law? If so, is there a way of changing the idea a bit so there as to avoid the problem? Could "quantum interest" account for the missing entropy? Do any restrictions apply on the negative and positive energy, such as quantum inequalities? Can the positive and negative energy be separated? Is there a way to get around the restrictions, if there are any? Finally, if all of this is possible, could it be done? Would a possible method be to recreate an unusually large quantum fluctuation, as described in the second paragraph? Are there other ways of making the "energy loan"?


title: How was physics taught in the past?

text: Were there physics teaching aids used in the past? Or did professors just basically follow the textbook the students read? In other words: How was physics was taught in the past?


title: Origin of induced emf

text: What is the origin of induced emf in a conductor which is produced by changing magnetic flux.I mean why changing magnetic flux causes the production of induced emf. Is there some kind of energy involved?Is it.


title: Electric Field at surface/side of cylinder

text: I know I can use Gauss's law to find the Electric Field inside and outside the cylinder very easily. We can select Gaussian surfaces for different cases (i.e. $r \lt R$ and $r \gt R$, where $R$ is the radius of the cylinder and $r$ is the radius of Gaussian surface). But I want to know Electric fields at the bottom and the upper circular surface of the cylinder. Edit: Charge of the cylinder can be anything ($\lambda, \sigma, \rho$). For clarification there is one cylinder and I want to find the E field on the top and bottom circular surfaces rather than inside or outside.


title: Is Ehrenfest theorem equivalent to Bohr's Correspondence Principle?

text: Ehrenfest theorem is usually dubbed as the quantum mechanical equivalent of Newton's law and Griffiths states, in the first chapter of his textbook, that Ehrenfests theorem enables us to work with expectation values in quantum mechanics as _Ehrenfest theorem tells us that expectation values obey classical laws._ On the other hand, the correspondence principle states that _the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers._ Are the two logically equivalent? References: http://farside.ph.utexas.edu/teaching/qm/lectures/node31.html


title: Is gravity reversed during inflation?

text: I am not clear on that. Is gravity repulsive during inflation?


title: Is integrability necessary for the Amplituhedron?

text: It is well known that there exist mappings between operators in N = 4 Super Yang–Mills and spin chain states making the theory Bethe Ansatz integrable. Is integrability a necessity for the Amplituhedron? Most QFTs are not integrable. Does Nima Arkani-Hamed, et al. claim to be able to extend the Amplituhedron to generic QFTs? Is then maybe integrability somehow hidden in the structure of all QFTs and the ones which are not integrable emerge as certain limits of the underlying Amplituhedron?


title: Functional integral aproach for Feynman rules

text: I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or electrodynamics). Could you provide a list of good books, articles, lecture notes and/or any other sources (except Peskin), where Wick's theorem and the Feynman rules are derived from the functional integral **in detail** , preferably with examples?


title: Gravity force and dark energy

text: If gravity is a fundamental force which bends spacetime and dark energy is energy which stretches spacetime, what is the difference between the terms force and energy?


title: dual variables for lattice fermions

text: I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I can't seem to find literature on dual variables for lattice fermions (systems with inner products of grassmann fields). For the spin and gauge cases we do harmonic analysis and fourier expanded the exponential of the action (or a character expansion for the gauge case). Is there an analogous process for lattice actions with grassmann fields? Thanks.


title: Why does Joule heating not occur when no current flows through a conductor?

text: Joule heating happens every time when the conduction electrons transfer kinetic energy to the conductor's atoms through collisions, causing these conductor's atoms to increase their kinetic and vibrational energy which manifests as heat. Then, why wouldn't it happen when no current is flowing through the conductor? When there is no current, the electrons are still moving randomly at a speed of $\mathrm{~10^5\ m/s}$, but at a zero average velocity. Then, why don't these electrons collide with the atomic ions making up the system and transfer energy to them causing them to heat up even when no net current is flowing?


title: String vibration

text: In many textbooks an illustration of a vibrating string at a fundamental mode shown (and wikipedia) shown like this http://en.wikipedia.org/wiki/File:Standing_waves_on_a_string.gif). However if you watch youtube videos of string vibrating in slow motion you see something like this (http://projects.kmi.open.ac.uk/role/moodle/pluginfile.php/1239/mod_page/content/1/ta212_2_008i.jpg) So which one is this? Is the first one just a schematic version of the second? Also the notion of a wave propagation in a string sort of relevant only to the second pictire only? Or am I wrong? Please clarify, I think I am missing something fundamental here. Thanks


title: Focusing laser & off-axis illumination & diffraction limit

text: I've thought I had a good understanding how resolution enhancement tricks works for projection lithography, until I tried to understand if it's possible to get sub-diffraction performance for focused laser beam: Let's assume we have a laser with nice and shiny perfect aspherical focusing lens (with performance greatly exceeding diffraction limit for it's NA). This system moves around and draw stuff. The question is - can we get enhanced resolution if we apply usual lithography trick of annular/quadruple illumination? Obviously, we won't reduce aberrations thanks to off-axis illumination, as there are none.


title: Casimir Forces and its associated Feynman Propagator

text: This is a continuation to my previous question, in which I began an attempt solve the Casimir Force problem using path integrals. As one of the answers there suggest I solve the Feynman propagator subject to the boundary conditions $x=0$ and $x=L$ at the plate boundaries. The equation for Feynman propagator is $$ (\Box^2+m^2)\Delta_F(x-x') = -\delta(x-x') $$ The solution to the free field is $$ \Delta_F(x-x') = \lim_{\epsilon\rightarrow0}\int \frac{d^4p}{(2\pi)^4}\frac{e^{ip_{\mu}(x^{\mu}-x'^{\mu})}}{p^2-m^2+i\epsilon} $$ What would be the boundary conditions that I have to exactly impose ? Imposing a boundary condition would mean, I think we might have to introduce the a new function (I don't if am right, but this is in general true for Green's function I guess) $$ \Delta_F(x-x') \rightarrow \Delta_F(x-x') + F(x-x') $$ where $F(x-x')$ is such that it satisfies the Boundary condition. Now my question is in case I have boundary condition (like below) how do I solve the differential equation for the boundary conditions like, (take plates to be at $z=0$ and $z=L$) $$ \Delta_F(x-x')\bigg|_{z=0} = \Delta_F(x-x')\bigg|_{z=L} = 0 $$ EDIT 1: It just occurred to me that there might be short route to this problem with some conceptual reasoning, I gave this a try.. Considering the region between the plates, I know the momentum is quantised in the z-direction, so I have (which is some sense imposed by the boundary condtions) $$ p_z = \frac{n\pi}{L} $$ Now using the Feynman propagator in momentum representation, which is $$\widetilde\Delta_F(p) = \frac{1}{(p^0)^2-(\textbf p^2+m^2)+i\epsilon}$$ In this I can substitute for, $p_z$, which will give me $$ \widetilde\Delta_F(p) = \frac{1}{(p^0)^2-(p_x^2+p_y^2+\Big(\frac{n\pi}{L}\Big)^2)+m^2)+i\epsilon} $$ Now can I get back to position representation, but with integral on $p_z$ replaced by a sum over $n$. Am I right in doing this procedure ? EDIT 2 : Following the procedure that I have mentioned, for a simple (1+1) case of the Feynman propagator in position representation, I have $$ \Delta_F(x-x') = \sum_{n=1}^\infty\int\frac{dp_0}{(2\pi)^2}\frac{e^{ip_0(x^0-x'^0)}e^{i\frac{n\pi}{L}(z-z')}}{(p^0)^2-\big(\big(\frac{n\pi}{L}\big)^2+m^2\big)} $$ EDIT 3 : $$ \text{Tr}\log{\Delta} = - \sum_n \int dp_0 \log{\bigg(p_0^2 - \bigg(\frac{n\pi}{L}\bigg)^2 + m^2\bigg)} $$ But this term seems to diverge, how does one obtain a cutoff in the context of this problem. (A cutoff for $p_0$ integral is also needed I guess).


title: Simple description of exchange interaction?

text: What is a simple bare-bones description of exchange interaction between two electrons? For instance, it seems to me that the only necessary ingredients are the Coulomb interaction and the requirement that the total wavefunction be antisymmetric.


title: Who is right? Force exerted on a body. Me or my friend?

text: > Three masses $m_1 = 3\text{ kg}$, $m_2 = 9\text{ kg}$ and $m_3 = 6\text{ > kg}$ hang from three identical springs in a motionless elevator. The > elevator is moving downward with a velocity of $v = -2\text{ m/s}$ but > accelerating upward with an acceleration of $a = 5\ \mathrm{m/s^2}$. (Note: > an upward acceleration when the elevator is moving down means the elevator > is slowing down.) What is the force the bottom spring exerts on the bottom > mass? Take $g = 10\ \mathrm{m/s^2}$. **My argument** The elevator is steadily slowing down and it get an upward acceleration. So as it goes down the net force is $$\begin{gather}F_s - mg = ma \\\ F_s = 6(10 + 5) = 6(15) = 90\text{ N}\end{gather}$$ But this is the force the spring has. It is the same as the force EXERTED on the body attached? Is there a Newton's third law involved? **Friend's argument** He says that BECAUSE it has an upward acceleration and it is going down that we need to be concern with $$Fs = mg - ma = 90 - 60 = 30\text{ N}$$


title: Does opening the oven door heat the house more than if the door is closed?

text: In the winter I am in the habit of opening my oven door once I am done baking so that I can add the heat to the house. However I recently thought about it and it would seem that even if the door is closed, and the oven insulated, the only way for the oven to cool would be to heat the house. So this would mean that there is no benefit to opening the door in the winter and keeping it closed in the summer. So my question is: Does opening the door of a hot oven heat the house more effectively than allowing it to cool with the door closed?


title: Shouldn't the electric field in a solid insulating sphere be linear with radius?

text: I am a senior in High School who is taking the course AP Physics Electricity and Magnetism. I was studying Gauss's laws and I found this problem: > A solid insulating sphere of radius R contains a positive charge that is > distrubuted with a volume charge density that does not depend on angle but > does increase with distance from the sphere center. Which of the graphs > below correctly gives the magnitude E of the electric field as a function of > the distance r from the center of the sphere? > > ![Choices A through E](http://i.stack.imgur.com/gR7kV.png) The correct answer is given to be choice D but I cannot see why the answer is D. Isn't the equation for electric field in this case just $E = \frac{q \cdot r}{4 \pi \epsilon _{0} \cdot R^{3} } $ if $r \le R$? This formula occurs for spherical insulators as given by the textbook Fundamentals of Physics by Halliday/Resnick. According to this equation for the electric field, the graph should clearly be linear until $r=R$. That is why I think the answer is C. I believe this is a problem with the textbook, am I correct? If I am wrong can someone please explain why I am wrong? Thank you very much for your time!


title: Using Quantum Teleportation in a way to have the effect of matter teleportation

text: So, I understand that quantum teleportation is the transfer of a quantum state from one particle or system of particles and its correlations to another receiving system consisting of particle(s). Can't this be used in a way to achieve teleportation as depicted in sci-fi? You start out with a translationally entangled particle pair (position and momentum are correlated) as described by this paper: http://pra.aps.org/abstract/PRA/v61/i5/e052104 Now according to these papers, you can theoretically teleport the position and momentum information of particle. Atomic teleportation of the external degrees of freedom, (Their position and momentum) http://pra.aps.org/abstract/PRA/v49/i2/p1473_1 http://prl.aps.org/abstract/PRL/v86/i14/p3180_1 http://iopscience.iop.org/0295-5075/75/6/847 Firstly, I need someone to help me understand **exactly what is being teleported in regards to the papers I've listed**. I understand completely the discrete case of quantum teleportation with spin or polarization, I'm not asking for that. This is what I know: 1) Start out with a translationally entangled pair. 2) Interact an "input" particle with one of the pairs. 3) Make a measurement of the input particle with one of the pairs' position and momentum. (This is the step I don't really understand) But I know that this is analogous to the Bell state measurement in discrete quantum teleportation that is widely described everywhere. 4) This result is communicated to the other laboratory where appropriate "shifts" of position and momenta are done to the entangled pair just like in the discrete case again. So, if position and momentum information is teleported, then does this mean that if the input particle was propagating in the X-direction like a wave- packet, then after teleportation, the receiving particle will now move in the X-direction relative to its original position? This is quite confusing to me. I want to understand this because I want to ask the trillion dollar question, "What if you are to replace the input particle with a input MOLECULE?" For a diatomic molecule, its simply two atoms of the same type. According to this wikipedia article: http://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals I know there are vibrational states in molecules which ARE essentially position and momenta information right? If we are to set up TWO translationally entangled pairs of atoms of the same type... And have them simultaneously interact with the molecule in step 3), _then can't we do the appropriate SHIFTS of the entangled pairs of atoms to make them turn into the input molecule?!_ So, the overall effect is that if you start out with H2 in Lab A, and had a translationally entangled pair for each atom, then you'll end up with H2 in Lab B after teleportation which is exactly what we want if we want to stick close to the sci-fi sense. If what I am saying makes sense, then can't this be in principle be extrapolated to larger molecules and eventually cells, and organs, to an entire human? I know that this is a large jump... However, what I'm trying to say is that according to this kinda scheme I'm asking about, its sticking to real physics, and its essentially accomplishing teleportation in the sense people know about. Do I make sense?


title: Is there any other mathematical tool to measure velocity, instead useing derivative?

text: To measure velocity we use derivative $$v=\frac {dr}{dt}.$$ Is the any other mathematical tool to do this?


title: A few questions related to frame dragging

text: I am trying to get my head around a few concepts related to frame dragging and related physics. In regards to black holes that have no charge and all their mass is tied up in rotational kinetic energy then the ergosphere is a maximum size. By definition, as I see it defined, the outer boundary of the ergosphere is the point at which a photon becomes stationary with respect to outside observation if it is **orbiting** retrograde. The radius at the **equator** of this maximal case is $2GM$ making it twice the inner event horizon radius if I am understanding correctly. The innermost stable orbit for massive particles is at GM in a pro-grade direction. If any of this is incorrect please let me know. So my questions are: >   1. Pretty much by definition it is impossible for a massive particle to go > slower than the speed of light relative to outside observers if it is within > the ergosphere - true? What is the limit of this speed relative to outside > observers? I'm looking for the limit before it passes into the inner > horizon. > >   2. Since the frame dragging extends outward (presumably to infinity with > extremely negligible effect) what would happen in the case where you had a > ring of orbiting black holes equally spaced with spin aligned such that the > axis of rotation was tangential to the circle? Without touching event > horizons it would appear to be able to pull massive particles through the > center of the circle in a straight line rather than a curved path as above - > correct? And if the holes were close enough would that allow the ergospheres > to merge (temporarily as I'd assume everything would merge eventually) would > that allow faster than light travel for massive particles in a straight > line? I'm neglecting that the orbits of the holes are unstable and would > radiate gravitational waves and that they are massive enough so tidal forces > are 'small' compared to the particle or mass of particles. > >   3. If the holes above are arranged as stated would this extend the > ergosphere inward toward the center if they are closely spaced at some > distance? I've seen cases for general non-maximal spin cases of two holes > and am unable to determine how this would play out. > > Thanks for taking the time to read and if I put too much in a single question I apologize but didn't want to spam several closely related questions.


title: Simulating the evolution of a wavepacket through a crystal lattice

text: I am interested simulating the evolution of an electronic wave packet through a crystal lattice which does not exhibit perfect translational symmetry. Specifically, in the Hamiltonian below, the frequency of each site $\omega_n$ is not constant. Suppose the lattice is specified by a certain tight-binding Hamiltonian $$ H = \sum_n \omega_n a^\dagger_n a_n + t \sum_{<n>} a^\dagger_n a_{n+1} +\text{all nearest neighbor interactions} + \text{h.c}. $$ We prepare a wavepacket, and for simplicity, we express the wavepacket in the fock basis of each lattice site $$ | \psi \rangle = \sum_i |b_1\rangle |b_2\rangle \ldots |b_n\rangle. $$ Thus, there are $b_1$ electrons in the $1$st lattice site. Of course, electrons are fermions and $b_1$ may be either $0$ or $1$. Suppose we treat this problem purely quantum mechanically. Then we will need to prepare a vector of length $2^n$, which is computationally intractable for any significant $n$. I am interested in physical techniques that may be employed to simplify this problem. Is it possible to attempt the problem in a semiclassical manner?


title: Freezing point depression - cooling my drink with the same method as salt on a highway?

text: I understand that adding/sprinkling, say NaCl, on a highway depresses the freezing point by making any moisture on the road harder to freeze as the NaCl molecules get in the way of phase transition. So now we need lower temperatures to freeze the moisture on the road. However, why can I cool my drink "better" by placing it in a salt + ice bath instead of just an ice bath? It would seem that a salt + ice bath only keeps the water from freezing or in other words, the ice from melting for longer duration of time. Where does the lowering of temperature come into this picture? I am not able to connect these two phenomenon although people tell me they are the same. Links to what I have already seen: http://antoine.frostburg.edu/chem/senese/101/solutions/faq/why-salt-melts- ice.shtml http://revision3.com/forum/showthread.php?t=34351 I need someone to rephrase something and I cant put my finger on it. ## Edit: I was thinking more about it and here is what I can further add to my question:   1. Sprinkling salt on the highway PREVENTS water to ice transition by intereference due to NaCl molecules. It is actually preventing the liquid-solid transition and it would seem to people that NaCl is _melting_ the ice.   2. When I place my drink in a ice + salt (NaCl) mix, the NaCl molecules dissociate into Na+ and Cl- ions into the water and this is an endothermic reaction where the NaCl molecules remove heat from the water molecules decreasing the temperature of water molecules and converting them to ice. I am assuming that the NaCl molecules strip the ice of heat as well further decreasing their temperature. Any comments?


title: Degrees of freedom in the infinite momentum frame

text: Lenny Susskind explains in this video at about 40min, as an extended object (for example a relativistic string) is boosted to the infinite momentum frame (sometimes called light cone frame), it has no non-relativistic degrees of freedom in the boost direction. Instead, these degrees of freedom are completely determined by the (non-relativistic) motions in the plane perpendicular to the boost direction. I dont see why this is, so can somebody explain to me how the degrees of freedom are described in this infinite momentum frame?


title: What exactly heats a pool of water on a sunny day

text: On a sunny day an outdoor swimming pool will heat up fairly quickly. My question is, what is the exact mechanism for this and can we put numerical figures on it? Given that water is clear and colorless, it seems that it will absorb very little direct radiation from the sun. The sides and the bottom of the pool can however heat it by conduction. Is it possible to estimate the influence of radiation versus conduction in this process or in general to give a full explanation?


title: Pressure at a Given Depth: Impossible Result

text: Let's say there's a static fluid in an arbitrarily deep vessel with constant non-zero compressibility ($k=\frac{1}{B}=-\frac{dV}{Vdp}=\frac{d\rho}{\rho dp}$). Let $z$ be the vertical distance below the surface of the fluid. Then $dp=g\rho dz$ ($g$ here is positive and assumed to be constant). As $k$ is constant, $\rho$ can be expressed in terms of $p$: $\rho=C_{0}e^{kp}$. If $\rho=\rho_{0}$ and $p=p_{0}$ at $z=0$, then $C_{0}=\rho_{0}e^{-kp_{0}}$, which is a positive quantity. So $e^{-kp}dp=C_{0}gdz$, which can be solved to get $p(z)=-\frac{1}{k}\ln(C_{1}k-C_{0}kgz)$. This doesn't make sense, because the coefficient of $z$ is negative, so there exists a positive value of $z$ for which $p=\infty$. What did I do wrong?


title: Can a rock be considered frozen

text: Water usually comes to mind when thinking about freezing. Once it reaches a certain temperature, water freezes, becoming a solid. However could you make the same statement about a rock? Is a rock at room temperature considered frozen because it is in a solid state?


title: Diff(M) as a gauge group and local observables in theories with gravity

text: In a gauge theory like QED a gauge transformation transforms one mathematical representation of a physical system to another mathematical representation of the same system, where the two mathematical representations don't differ at all with respect to observables. Gauge transformations are therefore manifestations of a true redundandcy of the mathematical descriptions. In GR the role of diffeomorphisms is different, a diffeomorphism represents a change of the reference frame. Different observers in different reference frames will of course have different results when they measure the same observable/event. In this sense I agree with Raymond Streater (see Diff(M)) that it is misleading to say that Diff(M) is a gauge group (if you disagree with me please explain). In AQFT one associates observables (selfadjoint operators) with bounded open subsets of a spacetime, these observables represent what is observable in the given domain of space and time. A detector that is operated for two hours in a laboratory would be, for example, represented by such an observable (this is only approximately true, because the Reeh-Schlieder theorem says that it is not possible to use a truly localized observable, but one has to use an "approximately local observable" instead). I think that this line of reasoning will stay true even if one day there is a theory of quantum gravitation. But from time to time I read statements like "local observables are not gauge invariant in a theory with (quantum) gravity and therefore cannot exist/ are not valid observables". (If my phrasing of the statement is wrong, please explain and correct it.) I have never read about an explanation of this statement and would like to hear about one. Isn't a detector, for example, a (approximatly) local observable and won't detectors exist within a theoretical framework of (quantum) gravity? Edit: A little of explanation of "observables" in GR: I am aware that in GR only "events" make sense as an observable, but of course not, for example, the spacetime coordinates of a point of spacetime, see the discussion of Einstein's hole argument on the nLab:   * spacetime, see paragraph "Einstein's hole argument". When a detector detects a particle, I assume that this is an event that is observable because it is defined by the proximity of a localized field excitation and the detector, and the fact that the detecter makes "ping" is a fact that all observers in all reference frames agree upon.


title: Can the Universe create itself?--Is Gott's Use of CTC in Planck time or less valid?

text: My question is the title of a 1991 paper by Richard Gott and Li-Xin Li: http://arxiv.org/abs/astro-ph/9712344 and is also a subject of his popular book, "Time Travel in Einstein's Universe" Ultimately with cosmology, the chicken and egg question reduces to "How can something come from nothing?" Gott and Li use the concept of a "jinn" (or djinn-genie)---that is something that loops back in time in a closed timelike curve--In the movie, "Somewhere in Time", an elderly Jane Seymour places a pocket watch in young Christopher Reeve's jacket and tells him to come back to her Later, he travels back in time and gives her the watch. The watch is a djinn. Where did it come from? Gott argues that in say the first Planck time, the universe could have pulled a closed timeline curve and created itself. Other than the immense fun value of this thought, is using the concept of CLC in this way valid if it is done in less than the Planck time without violating causality?


title: Local measurement of entangled particle pairs and interpretation of state

text: This just started to bother me after reading yet another entangled particle question, so I hate to ask one myself, but... If we have two entangled particles and take a measurement of one, we know, with certainty, what the outcome of same measurement of the other particle will be when we measure it. But, as with all fundamental QM, we also say the second particle does not have a definite value for the measurement in advance of making it. Of course, the particle pairs are always described by an appropriate wave function, and measurement of half the pair changes the probabilities associated with each term. But, and this is the part that bothers me, with knowledge of the first measurement, we know the outcome of the second measurement. There's nothing strange or forbidden or anything about this, no non-local affects, no FTL communication, etc. But does it still make sense to talk about the second particle _not_ being in a definite state prior to the measurement? Or can we consider a local measurement of one half of an entangled pair to be a measurement of the other half? The latter option feels distasteful because it implies communication between the particles when communication is possible, which is obviously wrong, but I find it hard to pretend the second particle in the pair, whose state I know by correlation, is somehow not in that state before I confirm it by measurement. I should clarify that my uneasiness comes from a comparison with the usual "large separation" questions of entangled particles, where our ignorance of the first measurement leads us to write an "incorrect" wavefunction that includes a superposition of states. Although this doesn't affect the measurement outcome, it seems there is a definite state the second particle is in, regardless of our ignorance. But in this more commonly used scenario, it seems more fair to say the second particle really isn't in a definite state prior to measurement. As a side note, I realize there's no contradiction in two different observers having different descriptions of the same thing from different reference frames, so I could accept that a local measurement of half the pair does count as a measurement of the other half, while a non-local measurement doesn't, but the mathematics of the situation clearly point to the existence of a definite state for the second particle, whereas we always imply it doesn't have a well- defined state (at least in the non-local case) before measurement. At this point, my own thinking is that the second particle is in a definite state, even if that state could, in the non-local separation case, be unknown to the observer. But that answer feels too much like a hidden variables theory, which we know is wrong. Any thoughts?


title: How to measure the effects of the rotation around the Sun?

text: It is well know that we can measure the spinning of the Earth with a Foucault pendulum. But, is there a similar experiment for the rotation of the Earth around the Sun? I would like to know if we could prove that the Earth rotates the Sun without astronomical observation.


title: Where is the Higgs Field?

text: Where is the Higgs Field? I get the idea of how it works but I couldn't find anything explaining where it is.


title: Out-of-Plane Phonons

text: I am trying to derive the out-of-plane phonon dispersion relation for a membrane. As far as I can tell, one of the simplest ways to do so is with a Lagrangian of the form: $$L_{bending}=\frac{1}{2}\rho \dot{h}^2-\frac{1}{2}\kappa(\nabla^2h)^2$$ Where $h$ is the out-of-plane displacement, $\rho$ the mass density and $\kappa$ the bending rigidity. I tried to proceed by using the Euler-Lagrange equations with the substitution: $$h=h_o \exp\left[-i(\omega t + qa)\right]$$ as per standard-solid-state phonon treatments, but I don't know how to deal with the $(\nabla^2 h)$ term following the above substitution (perhaps that is the incorrect part). I know the answer should turn out to be: $$\omega^2=\frac{\kappa q^4}{\rho}$$ Note: Many of the papers I look at point to the following paper for the answer - but Google gives me nothing for it. But I feel the answer is simple and I am not doing the math correctly. There are various more complicated treatments in the literature, including QM formulations, none of which I can follow very well. I. M. Lifshitz, Zh. Eksp. Teor. Fiz, vol. 22, p. 475, 1952.


title: Confusion about units of angular momentum

text: According to multiple sources the SI units for angular momentum are kg * m$^2$ / sec I am confused about the derivation for this. Here is what I have done: $$L = I \cdot \omega \\\ = m \cdot r^2 \cdot r \cdot v(translational) \\\ = kg \cdot m^3 \cdot m/s \\\ = (kg \cdot m^4)/s$$ What are my doing wrong here


title: what kind of system respects $SU(N)$ symmetry?

text: I read this post, Is the symmetry group of two spin 1/2 particles $SU(2) \times SU(2)$ or $SU(4)$? If the picked answer is correct, can I believe that an $N$-degenerate system respects $SU(N)$ symmetry? But if so, I am confused in this way: can we understand that a spin $j$ particle contains $2j+1$ internal degenerate states? Then the symmetry of this particle is $SU(2j+1)$, right? If $j>2$, then it is larger than $SU(2)$! I must be wrong somehow since I know little about group theory. So, basically, what kind of system respects $SU(N)$? * * * **Update 1** Since I have not enough reputation, I could not add comment on the linked post, which I am still confused, so I may also post here(related to this issue): There are $M\geq 2$ particles, if each particle respects $SU(N)$ symmetry, and if   * A. there is no interaction between them, then what is the symmetry group, if   *     * A1.they are non-identical particles?     *     * A2. they are identical particles?     * B. there is pairwise interaction between them, and the interaction is $SU(N)$ invariant,   *     * B0. what do I mean by $SU(N)$ invariant?     *     * B1. symmetry group for non-identical case?     *     * B2. for identical case?


title: Can a scalar field model gravity? How accurate would be the results? Are there any difficulties with such a model?

text: Newtonian gravity can be described by the equation: $$ \nabla^2 \phi = 4 \pi \rho G $$ where $\rho$ is the mass density, $\phi$ is the gravitational potential, and G is the universal gravitational constant. Of course, one of the shortcomings is that it is not consistent with special relativity. General relativity handles this shortfall but is a tensor theory. However,the above equation can be modified as follows: $$ (\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{{\partial t}^2}) \phi = 4 \pi \rho G $$ This would be consistent with special relativity. Would this model give results close to experiment? Would it give correct results in some observed situations where Newtonian gravity fails? Why or why not?


title: Stern-Gerlach and Hund's second rule

text: According to Hund's second rule, the spin tends to be maximal. That would, in my understanding, imply that, regarding the Stern-Gerlach experiment, the important electron in a silver atom has spin 1/2 (or "up"). This should hold for all atoms. Every atom would have the same spin magnetic moment ( _smm_ ) - because the smm points in direction of the spin - and thus be affected by the **B** -field just like any other atom. But, if every electron has the same _smm_ w.r.t. the magnetic field, why do we get two "blobs" on the screen and not just one "blob" above or below the z-axis? Where is the error in my thought process?


title: $t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$

text: What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\\{t_1,t_2,t_3\\}=\left\\{\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{pmatrix},\,\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\\ i & 0 & -i \\\ 0 & i & 0 \end{pmatrix},\,\begin{pmatrix} 1 & 0 & 0 \\\ 0 & 0 & 0 \\\ 0 & 0 & -1 \end{pmatrix} \right\\} \end{equation} I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation?


title: Flux through side of a cube

text: I am looking at Griffiths introduction to Electrodynamics 3rd ED. Problem 2.10 asks for the flux of $E$ through the right face of the cube, when a charge $q$ is in the back left corner of the cube. The solution manual says it should be $\frac q {24\epsilon _0}$ but I think it should be $\frac q {6\epsilon _0}$ because each side should experience the same flux, and the face of the cube represents 1/6th of the cube.


title: Rotating point in 3D space by Unknown rotation vector

text: I have a random point defined by:               int x = rand.nextInt(9)-4;//-4 to +4     int y = rand.nextInt(4)+2;//+2 to +5     int z = rand.nextInt(9)-4;//-4 to +4      Y happens to be the "up" vector, but I suspect that's irrelevant. I want to rotate this point so that it ends up relative to a vector that passes through (i,j,k) rather than relative to the vector that passes through (0,1,0). Thus if (x,y,z) is (0,6,0) and (i,j,k) is (1,1,0) the result should be about (5,5,0). Essentially I'm trying to draw a random deflection vector based on the initial input force vector. In this case, fractures through 3D voxel rock, but bullet deflection lines off armor make for a good visualization.


title: Help with calculating resistance at a given temperature

text: > The resistance of a bulb filament is $100\Omega$ at a temperature of > $100^\circ \text{C}$. If its temperature coefficient of resistance be $0.005 > \space \text{per} ^\circ \text{C}$, its resistance will become $200\Omega$ > at a temperature of > >   1. $300^\circ\text{C}$ >   2. $400^\circ\text{C}$ >   3. $500^\circ\text{C}$ >   4. $200^\circ\text{C}$ > Now the linear approximation is $R_t=R_0(1+\alpha T)$ Therefore \begin{align*} 100\Omega &= R_0(1+0.005\times 100) \\\ \therefore R_0 &= \frac{100}{1.5} \\\ \text{Now } 200 &= \frac{100}{1.5}(1+0.005\times t_2) \\\ 0.005\times t_2 &= 2 \\\ t_2 &=400^\circ\text{C} \end{align*} But there is also this formula $\alpha=\frac{R_2-R_1}{R_1(t_2-t_1)}$ \begin{align*} 0.005 &=\frac{200-100}{100(t_2-100)} \\\ t_2 &= 1/0.005 + 100 \\\ t_2 &= 300^\circ\text{C} \end{align*} Why are these answers differing?


title: Why do smaller objects become harder to break?

text: When grabbing a typical tree branch of at least two feet, it's so easy to snap with a less than one inch circumference that even a toddler can do it. However, after breaking it, the smaller halves with multiply from the center, going like this: 1 - BB (before break) 2 - AB (after break) 4 - AB (half of second break equals four) 8 - AB (continuing multiplication factors of two) And so on.... But what I want to know is why this is ... in quantum physics. I want to know why the more you break it, the harder it is to keep breaking it from the center of each half. This applies to tearing as well ... if you tear a cardboard box in half down the middle it's easy, but if you turn it over and continue on it keeps getting harder as the pieces get smaller.


title: Transparency of solids using bandgaps and relation to conduction and valence bands

text: I think I understand how a solid can appear transparent as long as the energy of the photons travelling through it are not absorbed in the material's bandgap. But how does this band gap relate to conduction and valence bands which explain insulators, semconductors, and conductors as described in the 'band theory for solids' (say) here: http://hyperphysics.phy- astr.gsu.edu/hbase/solids/band.html ? Specifically I wish to use answers to this question to construct a useful visualisation of this aspect - trying to explain to lay people why some solids are transparent and some are semiconductors (or both) and using the bandgap to do so. See a basic, perhaps rushed, explanation of transparency started here: http://www.youtube.com/watch?v=Omr0JNyDBI0 There are two good reference questions here: Transparency of materials and followup here: Why isn't light scattered through transparency? So of course its not just about bandgap but also how classical waves are built up from a quantum substrate and materials with structures smaller than the wavelength of light. That's my visualisation challenge... I intend to show:   * high energy X-rays make many solids appear to be transparent because they are not absorbed vs the energies in IR, visible, and UV light for some materials.   * photons vs electrons being fired into a material   * glass structure vs crystal lattics or random structure solids   * opaque thin carbon vs transparent thick diamond and glass - why is paper opaque at visible light frequencies   * QED to classical wave showing how one leads to the other and how each model explains transparency   * Conduction, semiconduction, insulation and bandgaps and their relation to visible light energy transparency   * and probably some other aspects. To reiterate my question: How does the bandgap theory of solids link transparency to conduction and valence bands (or other bands) ? Feel free to suggest mechanisms for visualisation or any other contributing factors to explain this property of QED.


title: Why is optical orbital angular momentum (OAM) called "topological charge"?

text: The terminology "topological charge" is frequent in lots of research papers related to optical vortex or optical OAM, it is used to represent the optical OAM. Why? How to comprehend it?


title: Conservation of linear momentum, when is it conserved?

text: 1. Will Linear momentum be conserved in a non-inertial frame of reference?    2. In other words what is the fundamental condition for linear momentum to be conserved?   3. Also which is more fundamental- Newton's Third Law or Conservation of Linear Momentum? (as in Newton's third law does not hold true when we go to the quantum/atomic level or at relativistic speeds whereas linear momentum to my knowledge holds true)


title: an example where changing the frame of reference of an observer changes the outcome of events!

text: consider two identical charges moving with uniform velocity. There will be a magnetic force of attraction between them as two currents in the same direction attract each other. If I sit on one of the charges, according to me the other charge is not moving. So there wont be any magnetic attraction. How does changing the frame of reference change the outcome of the interaction? What is it that will actually happen?


title: Why doesn't a bike/bicycle fall if going with a high speed?

text: Why does a bike/bicycle fall when its speed is very low or close to zero and is balanced when going with a high speed?


title: Why does a bicycle (without any support of stand) falls down being at rest, but not under motion?

text: I have always seen a bicycle not standing without any support, it either falls down to the right or to the left, may even to some other direction. But,the same two wheeler when under motion,moves balanced, without falling to either side,even it not falls, when the two wheeler is bent to left or right by the driver, like the one which we could see in the bike races. **So, what makes the bicycle at rest to fall? And what makes it not to fall when under motion?** _MY VIEW ON THE CONCEPT_   If the bicycle is at rest, it will be acted upon by downward force ($mg$,where $m$ is mass and $g$ is acceleration due to gravity). If it is affected just by this force, it should have stood without falling, because there is no force which pulls it to either side to make it fall. I think, it might fall, either due to the direction of wind, or due to the nonuniform distribution of mass, or due to other external forces. If we consider the same bicycle to be under motion, I don't know what makes it not to fall,even affected by nonuniform distribution of force,or direction of wind. I think there might be something related with vectors, angular momentum,and forces like centripetal and centrifugal. (What ever I said, is just my opinion about the concept, I don't claim it to be true. Correct me,if I am wrong anywhere.)


title: Why do we fall when we ride bicycle slowly but we don't when riding it fastly

text: > **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? Why is it that, when we ride bicycle at good speed, it is perfectly balanced and we don't fall. But we lose balance when we ride bicycle at lower speeds.


title: How do bikes stay upright?

text: > **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? How does a bike stay upright? I know there are various theories involving castor effects, and gyroscopic motion but as far as I'm aware no one knows for certain how a bike stays up right. What is the best theory right now?


title: Is Conformal Symmetry Local or Global?

text: I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense. Obviously when the metric is viewed as dynamical then the symmetry is local, because essentially then we're dealing with a change of variables under which the metric transforms with a local scale factor $\Omega = \Omega(x)$. Usually, however, we think of the metric as fixed. David Tong's excellent notes suggest that in this case the symmetry should be thought of as global. But I'm not sure I agree. Say we work in 2D and have general conformal transformation given by holomorphic function $f(z)$. Under a conformal transformation $z\to w$ say, viewed actively, a general field $\Phi$ will transform to $$(\frac{dw}{dz})^h \Phi$$ where the prefactor is clearly dependent on the spacetime point. This would suggest that the transformation is local in the physics sense. Perhaps the distinction he is trying to make is between physical and gauge transformations. But then again I might be wrong because I thought that only global transformations had nonzero conserved quantities, and there's definitely a conserved current for conformal symmetry. Could anyone help to clarify this for me?


title: What is $\langle \sigma_\mu \rangle$ $\langle \sigma_\mu \rangle$ for the Pauli Matrices?

text: What is \begin{align} \sum_{\mu=0}^{3} \langle \sigma_{\mu} \rangle^2 = ? \end{align} $\sigma_{\mu}$ are the Pauli matrices. The Bra-Ket notation is used in this question: \begin{align} \langle \sigma_{\mu} \rangle = \langle \Psi \lvert \sigma_{\mu} \lvert \Psi \rangle , \end{align} where $\Psi$ is the Pauli spinor of two complex components.


title: Holographic Image

text: In a holography set-up, as shown in the figure below, Illumination beam and reference beam both are in phase. The interference pattern generated at the detector contains the whole information about the object. When illumination beam will incident on to the object its phase and amplitude will change and further object beam will recombines with reference beam at the detector. Question is how the Interference pattern provides complete 3D information of the object ? ![Typical Holography setup](http://i.stack.imgur.com/9SQa1.png)


title: All possible electromagnetic Lorentz invariants that can be built into the electromagnetic Lagrangian?

text: Given the electromagnetic Lagrangian density $$ \mathcal{L}~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}~=~\frac{1}{2}(E^2-B^2) $$ is a Lorentz invariant, how many other electromagnetic invariants exists that can be incorporated into the electromagnetic Lagrangian?


title: Is it feasible to transfer energy from power stations to communities via photons instead of electrons?

text: Electrical wires are relatively inefficient in transferring energy--especially when the place of production is quite far from communities. Would it be possible to transfer that energy via photons? I realize that to some degree photons react with atoms, but might there be a specific condition in which this would be feasible and even more efficient--such as a specific wavelength of the electromagentic spectrum which encounters little resistance from the atoms in air (and thus low energy loss). This is often a hurdle for utilities when the source of power is far away from the communities, such as hydro electric dams. I'm not speaking about removing metal wires altogether, just when it may be practical.


title: Gupta-Bleuler Formalism

text: In the Gupta-Bleuler formalism we have a problem with two states (scalar photons and longitudinal photons), because here $\langle \vec{k}_a|\vec{k}_b\rangle $ is negative or zero. However, I thought that only $|\langle \vec{k}_a|\vec{k}_b\rangle |^2$ corresponds to probabilities, a quantity which would be non-negative anyway. What am I doing wrong?


title: Derivation of normal shear stress

text: I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is $$ \tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx} \tag{23} $$ and $$ \tau'_{yy} = \frac{\tau_{xx}+\tau_{yy}}{2}-\tau_{yx}\tag{24} $$ Can someone elaborate on the note to make it clearer?


title: Why Liouville theory is interesting?

text: What makes Liouville theory subject to relatively intense research field?


title: Are "confinement" and "asymptotic freedom" two sides of the same coin?

text: On Wikipedia it says that the two peculiar properties of quantum chromodynamics (QCD) are: confinement and asymptotic freedom. Asymptotic freedom is the idea that at low energies we _cannot_ use perturbation theory because the coupling becomes extremely strong. One the other hand, at sufficiently high energies, the coupling becomes very small and so we can use perturbation theory. This can all be determined by investigating the $\beta$-function. My question is: > Why is the asymptotic freedom of quarks not considered to be a sufficient > explanation for confinement? My logic: According to asymptotic freedom: at low energies the coupling becomes so strong that quarks cannot be isolated which explains confinement. If I am right (which I'm probably not), then why is confinement still considered to be such a mystery?


title: How to do linear stability analysis on this system of ODEs?

text: I was trying to do linear stability analysis of spring pendulum. I arrived at the differential equations which describe the system. But I am unable to proceed to linear stability analysis. Is it possible to do linear stability analysis on 2nd order differential equations by finding eigen values of Jacobian matrix? The equations are as below : $mr'' - m(L+r)(\theta')^2-mgcos(\theta)+kr' = 0$ and $(L+r)\theta''+2r'\theta'+gsin(\theta)=0$ * * * UPDATE : The fixed points are $(mg/k,0)$ and $(-mg/k,\pi)$


title: Recoil from a shotgun

text: First of all I'd like to point out that I'm a complete layman when it comes to physics and I apologise if this isn't the correct forum for this. If I were to sit on an office chair on a laminated floor and fired a shotgun out in front of me roughly how far backwards would I recoil, imagine I weight 170lbs, I'll let you choose both the office chair and shotgun being used (I haven't got a shotgun to use as an example, this is purely hypothetical). Also if I was to fire the shotgun in the same situation as above but facing a wall would I be projected further?


title: Why in the relativistic quantum mechanics $ \gamma_4$ name is not used instead of $ \gamma_5$?

text: I have seen in the in the Dirac equation $$\gamma_0,\gamma_1,\gamma_2,\gamma_3.$$ Then I have seen the definition of a new matrix $$\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3.$$ Now my question is why the name of the new matrix has not been given as $$\gamma_4.$$ Is there any historical reason behind this or it is simply taken as $$\gamma_5$$ without any special reason skipping $$\gamma_4~?$$


title: The effects of reversing the leads of an ammeter

text: I was playing around with a battery, an ammeter, and a light bulb. The ammeter originally read 1.99 A, but after reversing the leads going into and out of the ammeter the ammeter read -1.98A. I know this was because the leads were reversed but I am not sure why reversing the leads led to the ammeter readind a negative value. Why?


title: Photons straight into black hole

text: What happens to a photon shot straight into a black hole? Does it gain infinite momentum before it crosses the horizon? If it has a finite momentum going in, then it would seem that a photon of the same momentum could escape the BH by time reversal. Does the photon enter the BH at the speed of light as we perceive it from the outside? Or does time dilation take place as the photon gains energy? Does the photon add to the mass of the black hole an amount of mass m = e/c^2, where e is photon energy? If so, is it the energy the photon had at the beginning, or the energy the blue shifted photon has as it crosses the horizon?


title: Is a spinning object heavier than a non-spinning object?

text: Does spinning an object make it heavier? A real-world example: I was mowing the lawn in front of my house, a lawn that tends to have some steep inclines. I realized that the lawn-mower was easier to move, and much easier to pull up inclines when it was off than when it was running. The only difference between a lawn-mower that's running and one that is isn't is that the blade is rotating and there's some electricity running through it as it uses up gas. I guess the most significant of those would be the spinning blade. Does the spinning blade make the lawn-mower heavier as a whole object, and would that mean that spinning objects are heavier than non-spinning objects?


title: How does (kg/m^3) * (m/s^2) * (m) come out to be units of N/m^2?

text: To me it seems to come out to be kg/(m*s^2). Is this somehow equivalent to N/m^2?


title: Electron propagator in a strong background B field

text: Near a magnetar with $B>B_\rm{QED}$, the strong B field will suppress the Compton scattering cross section of photons with a specific polarization (E-mode). Some references I know deal this problem using time-dependent perturbation in quantum mechanics. My question is there an intuitive way in QED to see how this takes place and why polarization is important in this case. The electron propagator will be modified. How can we include the background field in the filed theory calculation and derive it?


title: How many oxygen molecules touch you in your lifetime?

text: Following up on a comment by BlueRaja to this beautiful answer of Ilmari Karonen, I would like to phrase this follow up question: How many air molecules hit your average human's skin during their lifetime? Actually, make that **how many oxygen molecules do you breathe in your lifetime?** (In case you want some motivation: Ilmari calculates that the chance for a random oxygen molecule in the atmosphere (though I note that local effects could actually play a role for this one) to have existed in that form since the time of _Homo erectus_ is about one in 1014. However, there's a lot of them molecules, so maybe you _do_ brush up with quite a lot of them often.) Any takers?


title: If a paper disc is cut into a spiral, does its moment of inertia change?

text: It is obvious that there is no change in the mass of it and its radius. But the shape of the object does change. Does it mean its moment of inertia will also change?


title: Calculating uncertinity in measurements

text: I have a question relating to uncertinity.The equation used is period of a pendulum.      T=2π √(l/g)   For example consider time to complete 10 cycles by the pendulum as 19.6 (+/- 0.2)s.   I want to know how to calculate the **uncertinity** of   1.period (T)   2.T^2   3.log T       Any help would be appriciated :)


title: outward pressure in an confined electron gas

text: Suppose i have some electrons stored in a empty shell container with a negative ion layer in the inner surface so the electrons keep bouncing inside without being able to leave the inner cavity. I want to compute the electrostatic pressure that the electrons exert on the shell container. So i assume the electrons have a uniform volumetric charge density $\rho$, and the container has a radius $R_0$. My derivation is as follows: electric field at radius $r$ is $$\frac{1}{3 \epsilon_0} \frac{\rho r^3}{r^2} = \frac{1}{3 \epsilon_0} \rho r$$ force on an element of charge $dQ = \rho dr dA$ is $$F = p dA = \frac{1}{3 \epsilon_0} \rho^2 r dr dA $$ which means that the pressure satisfies $$ \frac{dp}{dr} = \frac{1}{3 \epsilon_0} \rho^2 r $$ which by simple integration leads to $$ p = \frac{1}{6 \epsilon_0} \rho^2 r^2 $$ evaluated at the shell this gives $$ p(R_0) = \frac{1}{6 \epsilon_0} \rho^2 R_0^2 $$ Which looks reasonable. But the part that i'm not sure is if this result is sensitive to the assumption that the charge density is uniform inside the volume. The relaxed density of the electrons will tend to be maximum near $R_0$. > How can i estimate the relaxed density radial function and/or how will this > change my result?


title: Particles vs Waves

text: As I remember long ago, in my physics classes, I always had a great trouble understanding the concept of waves. Our professor used to explain, as if everything in this world is made up of waves. However, with any logic, a normal person would always conclude that probably world is created of very small point like particles. So what are waves exactly? I think it's just a type of disturbance that is able to make particles jump up and down? So how come they become an entity? I would appreciate explanation. (A simple explanation, in lay-man language. :) )


title: When moving fast Time slows down Vs speeds up

text: I was watching an old cartoon movie where a scientist makes a gadget, which when bound on the wrist, freezes the movement of the whole world. So, that one may do 100s of things in a single second. (The beagle boys later use the gadget to rob a whole bank in 1 second). The scientist explains the working that the gadget helps in moving "very very fast". So fast that the person who wears it, sees the world, he finds everything frozen. As whatever he is doing is going on 1000s of times faster than the people in the world are doing. The above explanation is quite logical and easily understandable actually. So, with this logic, if movement occurs faster, time freezes. More the movement, more the time freezes. With this thing in mind, how come then theory of relativity claims that "moving faster may transport you ahead in time". What I am assuming here is "transporting ahead in time" is probably something exactly opposite of "freezing the time". Am I right?


title: Amateur thoughts on QM theory at celestial scale

text: Some amateur scientist asked me that why can't one just simply apply the entire theory of QM at atomic scale to "quantize" celestial system with a different choice of $\hslash$, which he believed can be used to understand celestial orbital configurations, for example, the "discrete" distances from various Solar System objects to the sun, and to somehow "unify" the fundamental forces. Well I'm pretty sure this has been done before and must turn out to be mathematically invalid. However, can anyone give me some intuitive arguments (not too technical) about why this would not work at all.


title: Can an underdensity in space act as a negative matter density powering a warp drive?

text: Consider a homogeneous isotropic universe filled with a perfect fluid with density $\rho$ and pressure $P = \rho/w$. E.g. for $w=-1$ we get a universe equivalent to one with "vacuum energy" or with the cosmological constant. However, consider now that it is really a perfect fluid with which we can manipulate and $w$ is more or less arbitrary. Considering General relativity, could a _relative underdensity_ of the fluid in the shape of a ring act as exotic matter powering a warp drive of an Alcubierre type? Note that I want to ask only about the formal solution of Einstein equations and whether a similar metric to the one proposed by Alcubierre can be achieved without negative matter density. * * * This question is one of the questions breaking up the discussion of this question into smaller pieces.


title: Temperature of a neutron star

text: In our everyday experience termperature is due to the motion of atoms, molecules, etc. A neutron star, where protons and electrons are fused together to form neutrons, is nothing but a huge nucleus made up of neutrons. So, how does the concept of temperature arise?