title: How to show that the gravitational potential energy of a two particle system is -2 times the total kinetic energy, without using the virial theorem?

text: Consider a two-particle system with identical masses, orbiting in circles about their center of mass. I'm supposed to prove that: $$U_p = -2U_k$$ With $U_p$ potential energy of the system, and $U_k$ the total kinetic energy of the system. **I'm supposed to solve this pretending I have no knowledge of the virial theorem.** * * * So far this is what I got: * * * Starting off, in the radial direction, there is no net force on either of the particles. The two forces working on either of the particles are the centripetal force $F_c$ and the gravitational force $F_g$, in such a way that they must balance each other out: $$F_c = F_g$$ The relation for gravitational potential is given by: $$dU_p = - F_g dr$$ Rewriting this relation using our previous equation, I wrote: $$dU_p = - F_c dr$$ (This time with the centripetal rather than the gravitational force.) Integrating on both sides: $$U_p = - \int F_c dr$$ So far so good, though I don't have any expression for $F_c$ that have only $r$ in it but no other variables that depend on $r$. My first choice was to use $F_c = mr\omega^2$. $\omega$ depends on $r$, and I believe so does $v$ in the subsequent expression. No dice there. Sob! * * * So I thought, why not work from the second part of the initial relation I'm supposed to find. So I tried calculating the total kinetic energy: $$ U_k = \sum_N \frac{1}{2} m_n v_n^2$$ For an arbitrary particle $n$, with $N$ amount of molecules. Fine, so I have two molecules and $m_1=m_2$. Also, $v_1=v_2$. (Right?) That gets me to: $$ U_k = 2\cdot\frac{1}{2}mv^2=mv^2$$ And now what? I can't really squeeze out $U_p=-\frac{1}{2} mv^2$. * * * So, I was hoping to work backwards. Problem is, my expression doesn't even contain any $r$, so what do I have to go on to solve for $F_c$ in this expression: $$ U_p = -\frac{1}{2} mv^2 = - \int F_c dr$$ Any help much appreciated.


title: Does Schrödinger equation have dual-property with Heat equation?

text: I have experimental data that Schödinger equation maintains high frequencies, while heat equation low. **Does Schrödinger equation have some duality property with heat equation?**


Why is diamond structure the most stable structure? Is this mathematical issue or physics issue? Doesn't this relate to quantum physics?


title: Where to take $\theta$ (theta) in a triangle?

text: ![enter image description here](http://i.stack.imgur.com/fl0fF.jpg) This is an inclined surface having and unit vector A. The perpendicular and horizontal surfaces are the components of the inclined surface. We want to find out that how much flux will pass through the inclined surface. My question is that why don't we take θ at the point I have marked red? Why have we taken θ on the other point?


title: Can we measure an electromagnetic field?

text: As far as I can check, the Aharonov-Bohm effect is not -- contrary to what is claimed in the historical paper -- a demonstration that the vector potential $A$ has an intrinsic existence in quantum mechanics, neither that the magnetic field $B$ has an intrinsic existence in classical mechanics, but rather that the _magnetic flux_ is the relevant, measurable quantity. Indeed, the phase difference is either $\int A \cdot d\ell$ or $\int B \cdot dS$ (in obvious notations with surface $dS$ and line $d\ell$ infinitesimal elements), and these two quantities are always equal, thanks to the theorem by Stokes. Then, one could argue that classically one can measure the flux as defined above, whereas in QM one has only the _Wilson loop_ $\exp \left[\mathbf{i} \int A \cdot d\ell / \Phi_{0}\right]$, with $\Phi_{0}$ the flux quantum, as a relevant, measurable quantity. _I just elaborated above about the Aharonov-Bohm effect since it was the first time I realised the possible experimental difference between measuring a flux and a field. Let me now ask my question._ My question is pretty simple I believe: **Does anyone know a classical experiment measuring the magnetic field $B$ and not the flux $\int B \cdot dS$ ?** As long as we use circuits the answer is obvious, since one can only have an access to the flux and the voltage drop $\int E \cdot d\ell$ ($E$ the electric field), but do you know other experiment(s)? As an extra, I believe the same problem arises with force: * _Does anyone know a classical experiment probing the force field $F$, and not the work $\int F \cdot d\ell$ ? *_ **NB:** The question does not rely on how smart you can be to find a protocol that gives you an _approximate local value of the field_ , say $\int F \left(x \right) \cdot d\ell \rightarrow F\left(x_{0} \right)$ with $x_{0}$ a (space- time) position. Instead, one wants to know whether or not a classical measurement can _a priori_ gives a local value of the field, without _a posteriori_ protocol. Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ? _Post-Scriptum:_ As an aside, I would like to know if there is some people already pointing this question out (wether it's the flux or the local magnetic field or vector potential that one can measure), and if it exists some reference about that in the literature.


title: Light bending surfaces and energy funneling properties

text: In the last year there has been fuzz regarding metamaterials studied for the purpose of cloaking, and it seem to me they are somewhat glorified 2D waveguides, but in any case it seems a reasonable application of this to funnel radiation captured over a wide area of incidence into smaller area, and i thought that this could be a cheap way to produce ignition radiation for a fusion reactor. Reasons why this wouldn't work, or not produce enough energy density? All stuff i've read about fusion ignition with lasers doesn't seem to depend too heavily on the coherence of the radiation


title: Formula for electric permittivity

text: What is the correct formula for electric permittivity? ![Image here](http://i.stack.imgur.com/s34CV.png)


title: Bell's theorem and why nonlocality is problematic

text: I generally hear it assumed that Bell's inequality implies violation of counterfactual definiteness, because locality is considered sacrosanct. I understand of course that measurable violations of locality are logically inconsistent. But what is so bad about "hidden" violations of locality? What are the reasons nonlocal hidden variable theories are frowned upon? Is it just because the ontologies currently on the table (such as de Broglie–Bohm theory) are considered kind of ugly?


title: Possible colors of fire?

text: I have learnt that depending on the various gases those are involved in the reaction that produces fire, different colors (yellow, red or blue) of flames become visible. I have a question .. what are the possible colors that fire can possess? and do they (how, if yes) relate to the intensity? like I know blue is more intense than red, etc..


title: How does Delta baryon decay conserve angular momentum?

text: I'm a chemist so bear with me: I understand the Delta baryons $\Delta^{+}$ and $\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0. I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that pions decay into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$. This seems to suggest momenta of: $\Delta^{+} (J=\frac{3}{2}) \rightarrow{} p^{+} (J=\frac{1}{2}) + \pi^{0} (J=0)$ Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else.


title: What are some ways of inducing spin polarization?

text: I saw a talk today and they mentioned how nitrogen-vacancy diamond centers can be used to optically induce spin polarization and now I wonder what other ways there are to induce a spin polarization. Thank you!


title: Electric field singularity

text: What's a general proof for the log-singularity of electric field near the edge of a uniformly charged surface?


title: How much better is salt water at letting through microwaves than pure water?

text: > **Radio Wave Attenuation** > > There are two general types of matter (substances) in the universe that > affect electromagnetic waves, conductors and insulators which are called > dielectrics by scientists. Most, but not all, conductors are metals, such as > copper, aluminum, silver and gold. However, salt water is also a rather poor > conductor! Most, but not all, dielectrics are non metals. Examples of > dielectrics are paper, plastic, Teflon, glass, ceramic and dry wood. Pure > water is a good dielectric substance! Source: http://www.madsci.org/posts/archives/2002-03/1015162213.Eg.r.html I read this paragraph, and was truly intrigued. Is it really true that salt water (NaCl) passes radio waves (specifically 3cm) through better than pure water? Why is this, and how large is the difference? Would it be noticeable in an experiment?


title: Questions regarding standing waves

text: I have two questions regarding mechanical waves. 1) We know that standing waves are created when any wave traveling along the medium will reflect back when they reach the end. But in an open organ pipe there is nothing to oppose the wave and reflect it back. So then how are standing waves created in such a case. I did not find the answer in my textbook. 2)In standing waves where is the pressure greater - nodes or anti-nodes?


title: Physical Significance of Operator Norm/Spectral Norm of a Quantum Operator

text: Is there any physical significance of operator norm/spectral norm of a **hermitian** operator?


title: Describing the movement of the object in a particular situation in Lagrangian way

text: > _Suppose there is a object M, (sliding motion) moving by the initial speed > $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to > be nonexistent. It then meets a circular mold with the radius $R$ that the > object travels (perimeter). After traveling one round of the mold, it moves > onto the right straight path then meets with the path with friction, given > by friction coefficient $\mu_k$. When colliding with the wall at $x=d$, the > collision is perfectly elastic collision. The object moves back to the > original initial position, $x_0$, then bounces with some object, with > perfect elastic collision, back to continue the cycle._ ![the movement of the object](http://i.stack.imgur.com/4date.jpg) The question is, I do know how to describe the movement of the object in Newtonian way, but I would like to know how I would be able to describe the movement using Lagrangian mechanics (or Hamiltonian mechanics).


title: B-field and Magnetic forces, speed of a particle

text: Let's say I have a particle moving perpendicular to a uniform magnetic field of magnitude $x \ T$, and it moves in circle with a fixed radius. How do I find the speed of this electron? Initially I though to fusing the Biot-Savart law, but then the mass of the particle wouldn't affect it, so I figured that's not the right approach. Can anyone point me out to the right direction?


title: How old is SUN ☉?

text: How do we know/calculate the exact age of sun ☉ ? ie. 4.57 billion years. What is the way to calculate it?


title: Proof of $T=\sqrt{2y/a}$ in uniformaly accelerating object

text: Suppose that there is a object that does a y-axis-only free fall to ground. The initial distance from the ground is defined as $H$. How does one prove that time the object takes to reach the ground is $T=\sqrt{2H/g}$?


title: Why doesn't fusion contradict the 1st law of thermodynamics?

text: I was reading up on the 1st law of thermodynamics for my Chemistry exam and I was wondering why doesn't fusion contradict the 1st law of thermodynamics? The 1st law states that > The energy of an isolated system is constant or that whatever is put into the system, you get out of it, but in fusion you get more out of the initial reactions than you put in Hydrogen + Hydrogen = Helium etc. I am still a bit confused... Thanks for any help!


title: A change in the gravitational law

text: What would happen if the force of gravitation suddenly starts varying as $1$$/$$r^3$ instead of $1/r^2$ ? Would the symmetry of universe now seen be disrupted?


title: What is the definition of correlation length for the Ising model?

text: The correlation length $\xi$ is related to critical temperature $T_c$ as $$ \xi\sim|T-T_{c}|{}^{-\nu}, $$ where $\nu$ is the critical exponent.   1. Is this the **formal definition** of correlation length? If not, what is the formal definition of correlation length (for phase transition in the Ising model)?   2. Can you give a physical understanding of correlation length?


title: Lie algebra of axial charges

text: Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} \partial_\mu\sigma\partial^\mu\sigma+\frac{1}{2}\partial_\mu\pi_a\partial^\mu\pi_a+g\bar N(\sigma+i\gamma_5\pi_a \tau_a)N$ we can construct conserved currents using Noether's Theorem applied to $SU(2)_L\otimes SU(2)_R$ symmetry: we get three currents for every $SU(2)$. By adding and subtracting them, we obtain vector and axial currents.   We could have obtained vector charges quickly by observing that they are just isospin charges, so nucleons behave as an $SU(2)$ doublet (fundamental representation), pions as a triplet (adjoint representation) and sigma as a singlet (so basically it does not transform): $V_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}N+\dot\pi_b(-i\epsilon_{abc})\pi_c]$ But if I wanted to do the same with axial charges, what Lie algebra/representation must I use for pions and sigma?   I mean, axial charges are $A_a=-i\int d^3x \,\,[iN^\dagger\frac{\tau_a}{2}\gamma_5N+i(\sigma\dot\pi_a-\dot\sigma\pi_a)]$ and I would like to reproduce the second term using a representation of Lie algebra generators of axial symmetry which act on $\sigma$ and $\pi$, but I don't know the algebra (I think it is $SU(2)$), neither the representation to use.   I tried to reproduce that form using the three matrices $T^1=\begin{bmatrix} 0&-i&0&0\\\i&0&0&0\\\0&0&0&0\\\0&0&0&0 \end{bmatrix}\quad T^2=\begin{bmatrix} 0&0&-i&0\\\0&0&0&0\\\i&0&0&0\\\0&0&0&0 \end{bmatrix}\quad T^3=\begin{bmatrix} 0&0&0&-i\\\0&0&0&0\\\0&0&0&0\\\i&0&0&0 \end{bmatrix}$ which should act on the vector $(\sigma,\pi_1,\pi_2,\pi_3)$, but I calculated their commutator and they don't form an algebra, so I think I'm getting wrong somewhere in my reasoning.


title: How can momentum changes from individual collisions be considered as a whole?

text: I have a question about deriving the equation of kinetic theory of ideal gas $$PV=\frac{1}{3} Nmc_r^2$$ where $N$ is number of atoms, $c_r$ is root mean square of atom speed and $m$ is mass of one atom. In deriving the equation, many text books consider the total rate of change of momentum within a certain time and calculate the pressure. However, in actual case the atoms collide on the wall of container separately at different position at different time, how can these changes of momentum be considered as a whole?


title: Magnetic field above center of current-carrying loop

text: A current $ I = 2A $ flows in a circular loop coil of radius $R = 10 cm$. Evaluate the magnitude of the magnetic field (in Tesla) at a point $P$ on the axis of the coil, and a distance $z=5cm$ away from the plane of the coil itself. ![enter image description here](http://i.stack.imgur.com/FDqJR.png) Using Biot-Savart's Law, $ \vec{dB} = \frac{\mu_{0}I}{4\pi r^{2}} (\vec{ds} \times \hat{r}) $ Due to symmetry, I expected the horizontal component to cancel out and leave us with the vertical component   $ B = \int \vec{dB}sin(\alpha) = \int \frac{\mu_{0}I}{4\pi r^{2}} (\vec{ds} \times \hat{r}) (\frac{z}{r}) = \frac{\mu_{0}I}{4\pi r^{2}} 2\pi R \frac{z}{r} $ However, the answer given uses $ \frac{R}{r} $ instead implying they're using $cos(\alpha)$ ? Isn't that the horizontal component? This seems so basic but I'm stuck.


title: Is Gravitational Red shift equal to $mgh$

text: Is gravitational shift - $\frac{gh}{c^2}$ (according to pound-rebka experiment) always equal to $PE=mgh$? because assume the gravitational pull, $g$, is equal to $1$ then we can say $g = 1$ similarly let us assume the height the photon wants to travel is also $1$ therefore $h=1$ and now let us assume the mass is $1$ so $m = 1kg$ now that we know if we plug in this into the following equations we get an imbalance as: $$\frac{gh}{c^2} = \frac{1}{c^2} = 1.11265006 * 10^{-17} J\approx Negligible$$ Now lets assume we convert this energy into mass again using $E=mc^2$ where $m = E/c^2$ which is $1kg$ with some while if we now plug in the values into the potential energy we get: $$PE = mgh = 1*1*1 = 1J$$ and we know $1 > (1.11265006 * 10^{-17})$ and since we know that potential energy is always equal to kinetic energy we can say that Gravitational Redshift creates an energy imbalance which goes against laws of Thermodynamics so what makes sure that gravitational redshift becomes equal to $mgh$?


title: Why can movies get away with 24 frames per second, but games can't?

text: Movies look reasonably smooth at 24 frames per second; unless the screen is huge and you use your peripheral vision, you usually can't see each frame individually. This has lead to a large amount of textbooks claiming that the visual response time of the retina is slower than 1/24 seconds. However, games at 24 frames per second absolutely suck. The motion is so choppy it is unplayable. For a game, you seem to need at least 40 frames per second, and even at that rate you still see individual frames if the objects are moving quickly. Why is this so? Is this related to the fact that movies have motion blur due to a non-zero shutter speed? Is "jerkiness" mostly perceived in terms of "gap between object in frame 1 and frame 2" rather than visual response time (the gap is smaller in films due to blurred objects "connecting" across frames)? If so, doesn't this prove that our visual response time is much faster than 1/24 seconds? To make the question more precise: is this response determined by physical mechanisms, such as the relaxation times of pigments in the eye or the firing times of neurons? If so, which ones? Or is it mostly due some sort of postprocessing done by the brain, i.e. more to do with the cognitive than the physical side of perception?


title: Will a prism widen a monochromatic beam?

text: The representation of the experiment which demonstrates that white light consists of many colors, invariably shows that the beam of white light is broadened inside the prism and is some what more broadened when it exits in the form of the separated colors. I am not able to figure out if this broadening of the beam will occur in the case of a strictly monochromatic light. For a more complete background, the following is what I am trying to achieve. I have a laser pointer out of which I am trying to get a sheet of laser. For this I am exploring the use of prisms (one or more prisms). Using this beam I plan to setup a crude preliminary demo of flow visualization. Any inputs are appreciated.


title: Could this open electric circuit work?

text: ![enter image description here](http://i.stack.imgur.com/FjmcZ.png) Two batteries are interfaced by a resistor. Parameters such as resistance can be set arbitrarily. Does it work?


title: Need an analytic model for describing a region with fast transition of magnetic field from a constant value to zero

text: I would like to describe analytically the electromagnetic fields in a region with the magnetic field changing rapidly from a constant value to zero. It's more or less like the fringe region of the an opposite pair of north and south poles of a magnet. Let's say the magnetic field has dominant $z$ direction and small $x$ direction components. It's exactly invariant with y and varying slowly with $z$, but drops almost abruptly from H0 to zero across $x = 0$ (in a narrow transition region). I think it's not difficult to imagine such a setting, but it seems not very easy to come up with elegant mathematical expressions to describe the fields here (at the same time being compatible with Maxwell's equations). I guess there should be something out there I could use for this, just couldn't find it. Do you guys have any idea?


title: Why do we not have spin greater than 2?

text: It is commonly asserted that no consistent, interacting quantum field theory can be constructed with fields that have spin greater than 2 (possibly with some allusion to renormalization). I've also seen (see Bailin and Love, Supersymmetry) that we cannot have helicity greater than 1, absenting gravity. I am yet to see an explanation as to why this is the case; so can anyone help? Thanks.


title: Minimum criteria for quantum state dissolution

text: What are the minimum conditions required to cause the colapse of the quantum state ? Or, what forces/equations determine when an object (for instance an electron) is forced out of its quantum state in to a determined position ?


title: Does Gödel preclude a workable ToE?

text: Gödel's incompleteness theorem prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a theory of everything for physics? * * * _Edit:_ I haven't before seen a formulation of Gödel that included time. The formulation I've seen is that any axiomatic systems capable of doing arithmetic can express statements that will be either 1) impossible to prove true or false or 2) possible to prove both true and false. This leads to the question: Are theories of (nearly) everything, axiomatic systems capable of doing arithmetic? (Given they are able to describe a digital computer, I think it's safe to say they are.) If so, it follows that such a theory will be able to describe something that the theory will be either unable to analyse or will result in an ambiguous result. (Might this be what forces things like the Heisenberg uncertainty principle?)


title: Generalization of a spin-glass order parameter definition

text: I am going through the paper, Solving the graph-isomorphism problem with a quantum annealer, by Hen et. al. Equation 4 on the second page gives the definition of the spin-glass order parameter as follows. $$ Q_2 = \left( \frac{1}{N \left(N-1\right)} \sum_{i\ne j} \langle \sigma^z_i \sigma^z_j \rangle^2 \right)^{\frac{1}{2}} $$ Now in Equation 5, it is generalized as follows. $$ Q_{2n} = \frac{1}{N^{2n}} \sum_{i_1, i_2, \ldots , i_{2n}} \langle \sigma^z_{i_1}, \sigma^z_{i_2} \ldots \sigma^z_{i_{2n}} \rangle^2 $$ I don't see how this generalization works. Any help will be appreciated.


title: Redshifting of light from a moving light source

text: We know that the expansion of space causes redshifting of light. I even read somewhere that the Doppler effect also happens in light. So imagine I switch on the headlights of my car in front of a wall. Now I reverse the car at a very high speed. Just assume that the road behind me is endless and straight and the headlight is very powerful. Now when I reverse my car, theoretically it will result in the redshifting of light. I understand the Doppler effect in sound but I just do not understand how is it possible with light. I mean if a wave of photons has already left the headlight then how will the movement of the car affect the wave. And just another question, if you somehow manage to get the headlights flicker (i.e. stop and start again) very fast, then will there be any redshift? **[Edit]Addition to the question** When the car moves backward, then only the distance between the car and the wall is increasing. How does that affect the energy of the light wave. I don't get that why should the motion of the car affect its wavelength. In the case of sound it is understandable because sound requires a medium to travel. Light can travel in vacuum as well. Please correct me and tell me where I am wrong.


title: Why do you hear an acute sound when you rub your shoe on the floor?

text: Sorry for the silly question, but I've been wondering about this for quite some time. After raining when I get home I rub my shoe's wet outsole on the floor and I hear an extremely acute (high pitch) sound (might be important to say my shoe's outsole is made of rubber and the floor is wooden). Why does a wet rubber on a wooden floor create such a high pitch sound? Does friction have anything to do with this? Thanks.


title: Delta normalization and density of states in the Golden rule of Fermi

text: In the text-book derivation of first order inelastic scattering amplitude, box normalization is usually used to calculate the result. This leads to a correct result through the Golden Rule of Fermi, but when I read Quantum Mechanics - A Modern Development by Ballentine, who avoids the box normalization completely and doesn't even mention Fermi's Golden Rule throughout the scattering theory chapter, I'm having difficulty deriving the usual ratio of momenta in $$\frac{\mathrm{d}\sigma_{if}}{\mathrm{d}\Omega} \propto \frac{k^\prime}{k} |V_{fi}|^2$$ using this approach. The only scattering amplitude that seems near to this is eq. 16.59/16.60 in his book, but those are both quite general expressions (so I would think filling in the correct wave functions here should give me the above). Any pointers in how to arrive at the above formula without applying box normalization is welcome.


title: Electric field of a charged spherical surface

text: Figure The dielectric shaped as on the figure has dielectric constant $\varepsilon=\varepsilon\left(r\right)$ and free charge density $\rho=\rho\left(r\right)$. What is the electric field and electrostatic potential as a function of $r$? The above is 3D object. We have $R_{1}$, $R_{2}$ and $R$. I really have no idea how to approach this kind problems. Does electric field in $r<R_{1}$ equal zero? Is the object a Gaussian surface? Does electric field in $R_{1} < r < R_{2}$ depend on dielectric constant?


title: Nature of Photons

text: Why is it that photons are emitted in bundles? My physics teacher's answer was "it's complicated"...


title: Does the Casimir effect allow to change the lifetime of a radiating atom?

text: Is it true that a spontaneously light-emitting atom changes its lifetime if it is put between two parallel plates that are so near that they attract each other through the Casimir effect? Thus: does the Casimir effect, and the changes in the vacuum state it induces, influence spontanenous emission? If so, who measured this effect for the first time? Where can I read about it?


What is a Kustaanheimo-Stiefel transformation? Which applications has it in physics? Can you point me to a reference, where this transformation is explained?


title: regarding the infinite cross section in Rutherford scattering

text: The differential cross section of Rutherford scattering blows as $\theta\rightarrow 0$. People express this fact as "The Coulombic potential is of long range". But I am seeing the opposite: $\theta \rightarrow 0$ is when the incident particle is not affected by the potential, and therefore a blow when $\theta \rightarrow 0$ means short range potential. That was first. Secondly, I cannot see how the differential cross section is related to (some kind) of probabilities when infinities are involved.


title: Question Regarding Lorentz Transformation

text: I have question regarding the following velocity transform formula: $$v_b = \frac{v_a - u}{1-uv_a/c^2}$$ $u$ here refers to the speed that one frame is moving relative to another. How do I decide whether to use a positive or negative value of $v$. On top of that, when do we use the normal time dilation formula $t = \gamma t'$ compared to $t_B = \gamma(t_A −ux_A/c^2)$ and $l = l'/\gamma$ compared to $x_B =γ(x_A−ut_A)$.


title: Have CMB photons "cooled" or been "stretched"?

text: Introductory texts and popular accounts of why we see the "once hot" CMB as microwaves nearly always say something about the photons "cooling" since the Big Bang. But isn't that misleading? Don't those photons have long ("cool") wavelengths because space expanded since they were emitted. There's no separate "cooling" process, is there?


title: Do reversible processes exist in Nature?

text: Classical thermodynamics always discusses entropy in the light of reversible processes, and it lies at the heart of the definition of entropy. But do these reversible processes exist in Nature, or are they just a gedankenexperiment? Since if they do not exist, then classical thermodynamics gives a lower bound on the increase of entropy, but says nothing about the difference in entropy between the reversible and irreversible process. Doesn't this make the classical definition useless for a quantitative approach? Or is there proof that the entropy difference between the reversible and irreversible process is small for most everyday cases?


title: Covariance of rayleigh channel

text: **f** =[ _f1_ _f2_ _f3_ ]; **g** =[ _g1_ _g2_ _g3_ ]; _f1_ ,_f2_, _f3_ ,_g1_, _g2_ ,_g3_ are all independent identically distributed zero mean complex random variable. **h** = elementwise wise Schur-Hadamard product of **f** and **g**. What is the covariance matrix of **h**? Is that diagonal matrix?


title: If superposition is possible in QM, why do we often assume systems are already in their eigenstates?

text: My understanding is that an arbitrary quantum-mechanical wavefunction can be written as a linear combination of eigenfunctions of some Hermitian operator, most commonly the Hamiltonian; when a measurement corresponding to that operator is made on this superposition state, the wavefunction collapses and only specific values are observed -- namely, the eigenvalues of the particular eigenstates that comprised the wavefunction. (Moreover, the probability of measuring the eigenvalue $E_i$ is proportional to ${\|c_i\|}^2$, the square of the coefficient of that eigenstate in the linear combination, etc.) And yet, in many situations, it seems to be assumed that the system is already in an eigenstate and that superposition is not possible. For example:   * the electron in the hydrogen atom is said to be in, e.g., the 1s $^{2}S_{1/2}$ state or the 2s $^{2}S_{1/2}$ state, but never a superposition of the two.    * the possible angular momentum vectors for a QM rigid rotor with fixed $l$ are sometimes drawn as discrete "cones"... but couldn't the _average_ $\bf{L}$ point in any direction, since a rigid rotor might be in a superposition of states?    * when deriving Boltzmann statistics, we consider how to place $N_i$ particles into the level with energy $\epsilon_i$, but there is no consideration that a particle might occupy two (or more) energy levels simultaneously. Why do we ignore superposition in these and similar cases?


title: The synthesis of $^{254}\text{No}$

text: How is $^{254}\text{No}$ synthesised? Could you explain the reaction where it is preceded by $^{208}\text{Pb}(^{48}\text{Ca}, 2\text{n})$? References to articles are well enough—I was somehow unable to find anything sufficiently detailed and informative.


title: Pions as a superposition of quark states

text: in Isospin space there are two fundamental states called up and down quarks, which satisfy the following eigenvalue equations: $I u = (1/2) u$, $I d = (1/2) d$ and $I_3 u = (1/2) u, I_3 d = (-1/2) d$. The antiparticles have inverted signs in their 3-component of the isospin. This also raises my first question: How are anti-d and u not the same state? They satisfy the same eigenvalue equations for $I_3, I$ and if I recall correctly the space of isospin-$1/2$ particles is 2 dimensional, i.e. there's no room left for the antiparticles in a sense. How do I solve his paradoxon? Mesons are pairs of quarks and antiquarks. This means we can write the states of mesons as tensorproducts of an antiquark and a quark state. For example let's look at combinations of u and d quarks: $u^{(*)} = |1/2, \pm1/2>$, $d^{(*)} = |1/2, \mp 1/2>$ (the star denotes the antiparticle. The first component is total isospin, the second is 3rd component of Isospin) Using Clebsch-Gordan we can write the tensor of those two states as a direct sum: $|1/2, \pm1/2> \otimes$ $ |1/2, \mp 1/2> = |1,1> \oplus $ $|1,0> \oplus$ $ |1,-1> \otimes $ $|0,0>$ where the triplet of $I = 1$ states corresponds to the three pions. My second question is, how do I recover the ELECTRICAL charge of a pion state from these equations? In my script there is this operator given by $Q = e(I_3 + \frac{1}{2} id)$ which should return the electrical charge of an isospin state. This doesn't seem to give the right values as for example $Q |1,1> = e(1 + 1/2) |1,1>$ which doesn't give the charge of a pion. Cheers


title: SI units with more than one prefix in fractions

text: Is it (in the view of SI) correct to note units with more then one prefix? I discuss this since several months with friends, but we could not find a proper source for our statements yet. Examples for different notations of an electric field               E1 =  1 MV/m     E2 =  1 kV/mm     E3 = 10 kV/cm      In my research field is no standard established yet so we are continously multiplying.


title: Kinematics - projectile motion on inclined plane

text: Well I've tried everything to solve this problem, spent well over an hour, and have gotten no results. A particle is thrown over a triangle from one end of a horizontal base it grazes the top vertex of the triangle. It falls on the other end of the base. If $\alpha$ and $\beta$ are base angles that the sides of the triangle makes with the ground and $\theta$ is the angle of projection, prove that $\tan\theta$ = $\tan\alpha$ + $tan\beta$ After some more work and thinking, I got: $$\tan\alpha + \tan\beta = \frac{\cos(\theta + \alpha)\sin(\theta + \alpha)}{v\sin\theta(\cos(\theta + \alpha) - \sin\theta)}$$ Any way to simplify to get what I want?


title: electric-circuits

text: Why can't you connect the + of a device to one battery and the - to another battery (and have it work)?![enter image description here](http://i.stack.imgur.com/W9x5Q.png)


title: How can I find the motion equations of the 2-dim harmonic oscillator?

text: First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, I am no physicist. I've heard that ONE possibility is Hamilton mechanics, but to be honest, I hardly know what that is! Maybe you can give me help? I would be very thankful. **Update** : For the general spherical pendulum I get $$ \ddot{\theta}=\sin\theta\cos\theta\dot{\varphi}^2+\frac{g}{l}\sin\theta,~~\ddot{\varphi}=\frac{-2\cos\theta\dot{\theta}\dot{\varphi}}{\sin\theta}. $$ Isn't the 2-dim. harmonic oscillator a special spherical pendulum so that I maybe can use this results?


title: Direction of motion

text: What does the term direction of motion actually mean? Is it a direction where a particle is moving or the direction of its velocity? For example, what is the direction of motion of a projectile in each case when it is moving upward and then downward? In each case, the direction of its acceleration is downward.


title: Nuclear Physics: Eigenvalues of nucleus angular moment

text: In the shell model of nuclei, when we talk about collective motions, we describe any nucleus deformation, expanding its radius on spherical harmonics base like this $R(\theta,\phi) = R_0\left(1+\Sigma_{\lambda\mu}\alpha_{\lambda\mu}(t)Y_{\lambda}^{\mu}(\theta,\phi)\right)$ where $Y_{\lambda}^{\mu}(\theta,\phi)$ are the spherical harmonics and $\alpha_{\lambda\mu}$ time-dependents coefficients. We use this expression of the radius to calculate nucleus volume undergoing deformation and for imposing it to be constant. When we consider the surface tension and the coulomb potential, we obtain a multi-dimensional harmonic potential $V(\alpha_{\lambda\mu}) = \Sigma_{\lambda\mu}\frac{1}{2}c_{\lambda}|\alpha_{\lambda\mu}|^2$ The kinetic term is $T = \Sigma_{\lambda\mu}\frac{1}{2}B_\lambda|\dot{\alpha_{\lambda\mu}}|^2 $ and the Lagrangian of the system is $L = \Sigma_{\lambda\mu}\left[\frac{1}{2}B_\lambda|\dot{\alpha_{\lambda\mu}}|^2 -\frac{1}{2}c_{\lambda}|\alpha_{\lambda\mu}|^2 \right]$ To solve the problem in a quantum way, we introduce two operators $\hat{\alpha}_{\lambda\mu}$ and $\hat{\Pi}_{\lambda\mu}=-i\hbar\frac{\partial}{\partial\alpha_{\lambda\mu}}$ and defining the creation operator $\beta^{+}_{\lambda\mu}$ in the usual way, we can calculate the eigenvalues of the problem. Now it is coming my questions. We can define also the angular moment like $\vec{L}_{\lambda\mu} = \hat{\alpha}_{\lambda\mu}\times\hat{\Pi}_{\lambda\mu}$. I want to solve the eigenvalue-equation of angular moment. Experimentally, we find that angular moment associated to the ground state is $0^+$ and to the first excited it is $2^+$. I'd like to find these values solving the eigenvalue-equation. But, i don't know how to calculate them. How can i do? How can i write the matrix form of angular moment? (it is necessary or i can do it on other way?)


title: Is it possible to prove that projectiles with drag will not travel in a parabola under the theory of classical mechanics?

text: Can you prove that projectiles will travel in a parabola, provided the acceleration due to gravity $g$ does not vary significantly along the trajectory? This should take air resistance into account. I know how to prove a parabolic trajectory without air resistance (see my user page on Wikipedia, for instance, where I have successfully proven this), but with air resistance, the equations get more complicated and difficult to solve analytically.


title: A question on the Bousso-Polchinski paper

text: In this famous paper by Bousso and Polchinski, Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant an example in M-theory compactification is given in section 2.2, page 7. Since I am new to string theory, I wonder how exactly (2.16) $$ S= V_72\pi M_{11}^9\int d^4x \sqrt{-g_4}\left(R - {1 \over 2 \cdot 4!} F_4^2\right) + S_{branes}$$ is obtained from (2.12) $$ S= 2\pi M_{11}^9\int d^{11}x\sqrt{-g_{11}}\left(R-{1 \over 2 \cdot 4!} F_4^2 \right) +S_{{branes}}$$ Is there any assumption made which is not stated in the paper? Especially, why "eleven dimensional $\bf F_4$ is directly reduced to a four dimensional $\bf F_4$" Since I am new to string theory, I really want to have a thorough understanding of this work. Detailed explanation is very much appreciated!! Thank you very much!


title: Calculate the maximum variation in brightness of a star due to the presence of a planet in orbit

text: **Background** : As a planet orbits around a star, the star's brightness periodically dims as shown in the following picture. By measuring the variation of brightness, we are able to deduce information about the system. ![enter image description here](http://i.stack.imgur.com/1wAfD.gif) Source: http://blog.timesunion.com/weather/kepler62/2770/ **Problem description** : To calculate the maximum variation of brightness of a star with an orbiting planet. **Attempted solution** : Let $\ell_\text{max}$ be the brightness of the star (from the earth's POV) when the planet doesn't cross the star and $\ell_\text{min}$ when the planet crosses in its entirety the star. This is the max variation scenario and the variation is given by: \begin{align*} \left(\Delta \ell \right)_\text{max} = \frac{\ell_\text{min} - \ell_\text{max}}{\ell_\text{max}} \end{align*} When the planet doesn't cross the star: \begin{align*} \ell_\text{max} = \frac{L_\text{star}}{4\pi r_\text{es}^2} \end{align*} where $r_\text{es}$ is the distance between earth and star (from center to center). When the planet crosses the star, then the power per $m^2$ that reaches the planet's surface is: \begin{align*} \frac{L_\text{star}}{4\pi r_\text{ps}^2} \end{align*} where $r_\text{ps}$ is the distance between planet and star. Therefore, the power that is blocked by the entire surface of the planet is: \begin{align*} \frac{L_\text{star}}{4\pi r_\text{ps}^2} \pi R_\text{planet}^2 = \frac{L_\text{star}}{4} \left( \frac{R_\text{planet}}{r_\text{ps}} \right)^2 \end{align*} The apparent brightness of the star, as seen from earth, will be (EDIT: the way we spread the power loss to the surface of a sphere with radius $r_\text{es}$ is most likely wrong): \begin{align*} \ell_\text{min} &= \ell_\text{max} - \frac{L_\text{star}}{4} \left( \frac{R_\text{planet}}{r_\text{ps}} \right)^2 \frac{1}{4\pi r_\text{es}^2} \\\ &= \ell_\text{max} - \frac{\ell_\text{max}}{4} \left( \frac{R_\text{planet}}{r_\text{ps}} \right)^2 \end{align*} \begin{align*} \left( \Delta \ell \right)_\text{max} = \frac{\ell_\text{min} - \ell_\text{max}}{\ell_\text{max}} = -\frac{1}{4} \left( \frac{R_\text{planet}}{r_\text{ps}} \right)^2 \end{align*} **Question 1** : I've come accross some sites on the Internet and they give a result that is $\left(R_\text{planet}/R_\text{star}\right)^2$. Basically they take the ratio of planet's area to star's area without any more detailed calculations. Which makes me think that my calculations are wrong. Any insights ? **Qestion 2** (added afterwards): Why doesn't the transit depth depend on the distance between the planet and its star (or between us and the planet) ? If the planet was very very close to earth, wouldn't it cause an eclipse of the star ?


title: How does Ohm's law apply to superconductors?

text: As radio amateurs we've all learned the various relationships of power, voltage, current and resistance as expressed in Ohm's Law. My question relates to the following simple circuit showing an ideal power source capable of providing infinite current with no drop in the voltage it supplies. The resistor is made of a superconductor material. Let's ignore the effects of current density on superconductors for now. When the temperature of the superconductor material is lowered to the point at which its resistance drops to zero, how much power is being dissipated by the resistor? Is it given by... (Door #1) $P = I^{2}R\ = I^{2} \cdot 0 = 0 Watts$ or is it given by... (Door #2) $P = E^{2}/R = E^{2}/0 = \infty Watts$ or is it... (Door #3) some other equation or relationship? I've got to know the answer before I invest in converting all my shack's wiring to superconductors: if there's a short circuit will there suddenly be 0 Watts or $\infty Watts$ that will turn my neighborhood into an unsightly cinder?! ![schematic](http://i.stack.imgur.com/1uURY.png) simulate this circuit - Schematic created using CircuitLab * * * Don't get too hung up on the mathematics. This question is related to physics. I'll accept any answer that explains which "door" is the right one, and why. For those who doubt that the problem can be approximated mathematically, consider that it can be investigated experimentally. Imagine assembling a huge bank of fully-charged super-high capacitance capacitors, and connecting them via high-current wires to either end of a big thick hunk of high critical current density superconductor. We will, of course, select capacitors and connecting wires (and switch if desired) such that their total internal resistance will limit the current density through the superconductor to just below its critical current density so that it remains in its superconductive state throughout the experiment - and also keeps the capacitors and wires from melting. Just for the sake of argument let's say we generate a few million amps across the superconductor if only for a few milliseconds. That is a bit shy of infinite current, but quite enough to get a good idea as to which solution applies to the superconductor's behavior. If you like the answer provided behind door #1 then you should be happy to stand next to experimental apparatus while the experiment is conducted. If you like the answer provided behind door #2 then you probably want to remove yourself to the next county before the switch is thrown. If you like the answer provided behind door #3 then who knows? Whichever door you prefer, please explain the physics behind your selection. Since the circuit model can be approximated experimentally, we know that it must be possible to derive a mathematical approximation by simply modeling the experiment.


title: Nuclear forces and Chiral Effective field Theory

text: I am new to this field "nuclear forces and Chiral Effective field Theory". while reading this paper http://arxiv.org/abs/nucl-th/0006014 . I have got few question:   1. How many different version of CD-Bonn potential exists? CD-Bonn 2000 potential is most updated and correct version?   2. CD-Bonn, Nijmegen and Argonne V18 are constructed using chiral effective field theory?


title: What's the point of NASA's impossible space engine? Wouldn't a mirror work better?

text: My understanding (which is very little) is that the point of NASA's recent virtual-particle engine is to convert solar energy into momentum. That's fine, but what's the point? If the spacecraft is already receiving solar energy, can't it just carry around a mirror to reflect light and accelerate via radiation pressure? Or is the quantum version somehow injecting "more" momentum than is actually being delivered via radiation pressure?


title: Leonard Susskind said that the small value of the cosmological constant could mean that the universe is much bigger than we thought. Why?

text: He said that near the beginning of this Youtube video that it could mean that the universe is much bigger than he had thought that it was, and that the cosmological constant wasn't even a constant. I don't have a clue what his reasoning was though.


title: Intuitive reasoning for why short wavelength photons pack more energy?

text: The energy equation is straight forward, yes, but is there an intuitive reason for this?


title: How does exciting an electron's surrounding electromagnetic field cause 'electron excitation'?

text: In more meaningful words than the ones above, how does adding energy to the EM field cause the electron to to change orbitals or oscillate in a different pattern.


title: Spacetime, space observables and time observables

text: It appears to me that the concepts of space and time play a privileged role in Physical Theories. If we look at classical non-relativistic theories such as point particle mechanics, rigid body mechanics and fluid mechanics, we immediately see that these theories rely on the assumption of the existence of a Spacetime Continuum on which Physical System exist. For example point particle mechanics relies on the idea that point particles move in $\mathbb{R}^{3}$ (the "space component" of Spacetime), rigid body mechanics uses $\mathbb{SE}(3)$ as the configuration space of a rigid body and this, in turn, is derived by the assumption that rigid bodies move in $\mathbb{R}^{3}$ mainteining the relative distances of the point particles of which are "composed" fixed, fluid mechanics uses the Lie group of all the volume-preserving diffemorphisms as the configuration space and this choice, again, arises because of the assumption that fluids move in $\mathbb{R}^{3}$. Even non-relativistic quantum mechanics implicitely assume the existence of a Spacetime continuum otherwise the definition of the position observables would be meaningless. Nevertheless the assumption of the existence of a Spacetime Continuum on which Physical Systems are placed seems to me rather underestimated in that very little (if none) attention is devoted to the structure of the underlying Spacetime and the consequences related to it. I am particularly interested in the consequences related to the concepts of Space and Time arising after a Spacetime splitting is performed by means of a reference frame. I am led to think that, in a certain not yet precisely delined sense, there are observables directly connected with these concepts, and that such observables play a privileged role. I think that such an attitude is not new, for example in non-relativistic quantum mechanics the Heisenberg group arises from the idea that space observables and their canonically conjugated momentum observables are the observables in terms of which all other observables are expressed. Obviously nothing similar exists for a Time observable and this is precisely what I am interested in. In any case I am trying to work out something of concrete from these rough thoughts and I would like to hear if someone knows of a similar, or related, attitude toward observables in the literature. I apologyze anticipatedly if this question will not fit the rules of the forum. **Edit** I would like to know of any attempt to describe observables, both in classical and quantum theories, in terms of Spacetime. I know this is highly ambiguous but I am not able to put it differently. As a suggestion I found interesting the algebraic approach toward quantum theories of Araki, Haag and Kastler where the $C^{*}$-algebra of observables is thought of as being a net of subalgebras each of which is associated to a spacetime region.


title: Terminal Velocity

text: Let Suppose a bullet of 2 gram is falling with the speed of 200km/h. Now how can we know the time after which it gains it terminal velocity. Also when it gain its terminal velocity the force of friction and gravity will becomes equal and it stop accelarating do at this point the force of friction acting on it becomes constant or it go on increasing and also decrease the velocity of object.


title: What is the phenomenological logic behind Fermi liquid theory

text: I am a super beginner when it comes to Solid State Physics and when wanting to learn more on the subject, I end up reading on Landau's Fermi liquid theory that supposedly justifies the quasi-free electron model I was taught at the undergraduate level. I have spanned now few .pdf lectures that I could find online and find them very confusing as I don't even understand really the meaning of the first sentence which more or less always starts by: > in a Fermi gas, the excitations satisfy blablaba... Basically, the problematic is presented as being that of interactions between electrons (or phonons) in a metal and to account for them in the statistical thermodynamic treatment of charge carriers in metals to rationalize, among other things, the effective mass one needs to consider when looking for quantitative agreement between the quasi-free electron model and experiments. My questions are super simple here:   * What is an excitation here? I do know what is an excitation if I have a ground state: it is an excited state full stop. But the terminology used in Fermi liquid theory suggests excitations of the equilibrium state...what is that?   * Why do we even look at these excitation things? I come from a soft matter background and if you have to deal with interactions, you do plenty of things like cluster expansions, path integral formulations followed by Saddle point or perturbative expansions, Gibbs-Bogoliubov etc... So why is it apparently more clever to look at these objects?   * From what I understand, the logic seems to be inspired by particle physics or rather by what I learnt in basic QFT where a vacuum is the ground state of the field. Then having "particles" means that you have excitations of this ground state. The ingenious part of Landau's theory seems here to look at the statistics of excitations of the Fermi sea (instead of the statistics of the particles in the sea) and assume that they will still follow a Fermi-Dirac distribution at equilibrium. The reason why it may be nice, is because interactions will only change the ground state (energy, pressure etc...) but not necessarily the way excitations equilibrate; which will always be Fermi-Dirac like...is that it?  I understand that there are multiple questions here but it is to be more specific about my primary question which is on the "logical reasoning behind the Fermi liquid theory".


title: Jet rapidity vs. Pseudorapidity

text: thanks to this post Why is pseudorapidity defined as $-\log \tan \theta/2$ I got a better understanding of what makes (pseudo) rapidity an interesting an often-used concept. Tangentially working with jet physics (the spray of hadrons kind of jet :), I am wondering why people work with rapidity (no pseudo here) when it comes to jets. What is the reason for this (doesn't pseudo rapidity offer the same convenient properties of Lorentz invariance wrt z boosts)? More importantly, how does one determine the rapidity (or rapidity difference) in a measurement at a hadron collider, where one does not know the the full four vector of any physics object? I suppose that the above question might be related to the topic of jet mass, which I have read in papers before. Any hints on the definition/usefullness of jet mass would also be appreciated. Thanks


title: Is there a physically motivated "trick" to evaluate this convolution?

text: I'm working on adapting some of the formalism in this paper to a system I'm working with. The part I'm interested in amounts to convolving a density profile $\rho(r)$ with a smoothing kernel $S(r,\epsilon)$. The smoothing kernel used in this particular case is Plummer smoothing: $$S(r;\epsilon) = \frac{3}{4\pi}\frac{\epsilon^2}{(r^2+\epsilon^2)^{5/2}}$$ and the density profile is left as a general (but spherically symmetric) function. The convolution (incidentally, I'm pretty sure this is a correct use of the term convolution, but would appreciate a confirmation/correction if I'm using it incorrectly) I want to evaluate is: $$\rho(\vec{r};\epsilon) = \int d\vec{r}' \rho(\vec{r}') S(|\vec{r}-\vec{r}'|;\epsilon)$$ Following along with the paper, this is going to be a lot easier to evaluate in cylindrical coordinates with the vector $\vec{r}$ lying along the $z$-axis, and without too much difficulty I arrive at the first part of his equation (8), but get stuck trying to get the second part. Equation (8) reads: $$\rho(r,\epsilon) = \frac{3\epsilon^2}{2}\int_{-\infty}^{\infty}dz\int_0^{\infty}dRR\frac{\rho\left(\sqrt{R^2+z^2}\right)}{(R^2+(z-r)^2+\epsilon^2)^{5/2}} \\\= \frac{3\epsilon^2}{2}\int_{-\infty}^{\infty}dz\int_0^{\infty}dRR\frac{\rho\left(\sqrt{R^2+(z-r)^2}\right)}{(R^2+z^2+\epsilon^2)^{5/2}}.\tag{8}$$ He points out that "the equality holds because the outer integral is taken over the entire $z$ axis", but I don't see how this follows. A couple details on the geometry. The cylindrical coordinates are $R$ and $z$ (and $\phi$ I guess, but that just gives a $2\pi$). $r$ is the distance from the origin of the spherically symmetric $\rho(r)$. It follows that: $$|\vec{r}'| = \sqrt{R^2+z^2}$$ and $$|\vec{r}'-\vec{r}|^2 = R^2 + (z-r)^2.$$ I thought the solution might be as simple as translating everything along the $z$-axis a bit, something like $\tilde{z}=z+r$, but that doesn't seem to work out. My next hunch is that the "trick" that I need to see has something to do with the behavior of $\lim_{r\to\infty}\rho(r)$: this had better be zero to impose a finite total mass (of course this does not guarantee finite mass, and indeed there are several commonly used density profiles that have unbounded total mass, in practice they get truncated at some $r$, and they always still limit to 0 as $r\to\infty$), but I'm still stuck. So, anyone see what makes that equality true?


title: Modeling and Simulating Pendulum Motion

text: I've been having difficulty creating a mathematical model of a pendulum in my code. While there is plenty of info on general equations describing a the motion of a pendulum, I seem to be having difficulty translating those into code, or more specifically cartesian coordinates. These difficulties no doubt stem from my poor math skills, thus, I am looking for help. My initial thought before even researching the matter was to simply use two independent Sine Wave oscillators. I used a basic equation `y(t) = A*sin(ω*t+φ)`. I think that its probably worth mentioning here that while I _understand what that equation does_. I'm not 100% clear one why it behaves periodically. Of course have dealt with trigonometric ratios before, but I don't feel like I have a grasp on the actual concepts behind it. I don't fully appreciate all of its consequences I guess. Anyways, I initially figured I could determine the x coordinate and the y coordinate using 2 separate sine wave oscillators, while having the period of the y oscillator be half of the value of the x oscillator. This however (possibly obvious to you...) resulted in something closely resembling lissajous curves. Now I've been reading explanations on the movement of a pendulum, yet while reading I get lost at two points regardless of the source I am reading from. First of all, everyone refers to a differential equation that I don't understand, probably due to never taking calculus. Here is a link to them all... http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3 And secondly, how can I derive two separate coordinates from one of these equations?


title: Conservation of angular momentum tensor $L^{\mu\nu}$ in special relativity

text: I have edited this question because I don't think that the related post answers my question fully. It refers to Noether's theorem but I would like an explicit illustration in an easier fashion: The angular momentum tensor is defined: $$L^{\mu\nu}~=~x^\mu p^\nu-x^\nu p^\mu$$ I would like to show that if a particle is not acted on by an external force that the angular momentum is conserved. Despite it being a direct consequence of the theorem I want to compute it in an inferior way, would this be sufficient? Without loss of generality take the rest frame of the particle $$\frac{dx^\mu}{d\tau}=(c,\underline{0})$$ Since the particle is not acted on by an external force, we have $$\frac{dp^\mu}{d\tau}=\left(\frac{1}{c}\frac{dE}{ d\tau},\underline{0}\right)$$ $$L^{\mu\nu}L_{\mu\nu}=\left( 2 (x\cdot x) (p\cdot p) - 2 (x \cdot p)^2 \right)$$ Can I show the conservation by $$\frac{d(L^2)}{d\tau}=0$$


title: Hysteresis and dissipation

text: Hysteretic phenomena are often linked to dissipation. When there is an hysteresis loop, the dissipated energy can usually be computed as the area of the cycle. For example, in ferromagnetic materials, the relationship between the magnetization and the magnetic field can exhibit an hysteresis loop, corresponding to the microscopic dissipation by Joule effect ; in elastic materials, there is an hysteresis in the relation between the constraint and the extension, corresponding to the internal friction. There are lots of other examples where I do not know exactly the dissipation processes : in all first-order phase transition (e.g. liquid-gas), in the contact angle, and so on. I feel like hysteretic phenomena cannot appear without dissipation, because hysteresis needs memory as well as the possibility to loose this memory (which is an irreversible process). However, perhaps I miss some other possibility. So, is hysteresis _always_ linked to a dissipation? Is it _due to_ irreversibly? Is there a means to prove that formally?


title: Trajectories using Polar Coordinates

text: Once I asked my teacher how to find the trajectory of any particle that is acted upon any force.(Generally) He Told me that I couldn't do it as I did not know polar coordinate geometry as of then but now I've finally realized the effectiveness of polar coordinates and can solve simple polar geometry problems.Like finding the polar equations for electric equipotentials or the polar equation of electric field lines etc etc. Now I want to know how to use polar coordinates to find the trajectory of any given particle acted upon by a set amount of varying/steady forces. For example the trajectory of a particle kept in the vicinity of two charged particles.   * I know i begin by selecting the origin .   * then i select any assumed trajectory .   * then on that trajectory i need to take a small element $dl$ and find its components along the direction of the position vector $\vec{r}$ ,$\vec{dl}_r$ and one along the direction of increasing angle $\theta$, $\vec{dl}_{\theta}$ .   * then i need to find the forces along the given directional vectors.    * finally i relate $\vec{dl}_r , \vec{dl}_{\theta},\vec{F}_r \mbox{&} \vec{F}_{\theta}$   * finally i get a relation between theta and r which will be the equation of the trajectory. are my steps correct if not can you guide me to any reference on the net helping me to gain knowledge as to how to go about my problem.


title: Why and how, in QED, can excited atoms emit photons?

text: The quantum mechanics of the structure of atoms as determined by the electromagnetic forces inside them correctly describes the location and coupling of the different energy levels in essentially all atoms and molecules. However, "classical" quantum mechanics predicts that atoms in excited states will remain there for all time, and it does not provide a mechanism for atoms in excited states to radiate a photon and move to a lower level. How does QED justify this process?


title: Why is 'the period' marked as letter T?

text: I'm not a native English speaker and I was wondering, why 'the period' got the letter $T$. I've asked myself the question when I was thinking about stuff related to the frequency. I.e.:   * $f$ - frequency   * $v$ - velocity of a wave   * $a$ - acceleration all of those have 'generic' markings, but not 'the period'. Was the letter $T$ used to somehow relate to 'time' the _period_ describes? (I just need an acknowledgement really)


title: Reference needed for Iron-based superconductors

text: Iron-based superconductor is a class of high-$T_c$ superconductors discovered in 2008. Are there any review papers about these superconductors yet? If not, which are the key papers in the field?


title: Recession of galaxies

text: when you take such a given galaxy and it is said that you are moving away from us with a speed $v$, what do you mean really? Let me explain: is the galaxy itself which has an intrinsic velocity of recession , or the galaxy is at rest (sorry for the forcing), but is that the whole universe is expanding (see the famous example of the balloon that is inflated ) and then we see the Milky Way Galaxy get away with speed ( boh ) of 20,000 km / s? Or these 20,000 km / s are the combined action of both ( ie that the Universe is expanding and an actual speed of removal of the galaxy ) .. Always referring to the example of a separation speed of 20,000 km / s , and pretending that someone has already replied saying that this speed is due solely to the expansion of the Universe , I can say that the 20,000 km / s are due to the sum of 10,000 km / s + 10,000 km / s speed of removal of Milky Way Galaxy and that I'm looking at a point equidistant between the two galaxies ?


title: What are examples of Solutions of Newton's Laws that have analogs to Solutions of the Wave Equation?

text: The idea is to give examples of processes that deal with properties of a particle that have clear wave analogues.


title: Bernoulli's principle

text: In steady flow, the air that flows faster causes a lower pressure, the lower pressure causes it to move faster, or exactly both at the same time? Also how is it that we're talking about the pressure of the fluid, if in reality that term for pressure comes from a pressure that's outside the immediate system (according to the derivation; see below). $$dW=p_1A_1ds_1 - p_2A_2ds_2$$   * $dW$ is work done on the system   * $p_1A_1ds_1$ is work caused by a pressure that's outside the fluid and in the same direction as the flow   * $p_2A_2ds_2$ is work caused by a pressure at the other ending of the system that opposes the direction of the flow.   * If this is not immediately clear, I'm considering a closed pipe and my system is an element of fluid within that pipe. One last thing: how do you solve any relativistic effect for the Bernoulli equation? Like, suppose I'm in a car with a certain velocity, is the pressure inside my car greater or less than the pressure outside? Well that depends on the frame of reference, etc. EDIT: The car is not sealed.


title: EPR Experiments and Monogamy

text: Normally in an EPR experiment two measurements are performed on entangled particle pair. Only the particle pair is treated quantum mechanically and it is usually prepared in a state like $$ (\,\left|\uparrow \downarrow\right>+\left|\downarrow\uparrow\right>\,)/\sqrt{2} $$ In modern approaches to QM it is common to also treat the detectors as a part of the experiment that the rest of the experiment becomes entangled with. The measurement of the left spin in the parallel direction can then be represented by the transition $$ \left|0\right>\,(\,\left|\uparrow\downarrow\right>+\left|\downarrow\uparrow\right>\,)/\sqrt{2} \leadsto (\,\left|\uparrow\uparrow\downarrow\right>+\left|\downarrow\downarrow\uparrow\right>\,)/\sqrt{2}$$ where $\left|0\right>$ represents the ready state of the detector. In a typical EPR experiment we would measure the right spin. If the measurement was again in the parallel direction this would determine the state of all of the components. In what sense is the entanglementafter the first measurement monogamous? Is there an error in the above description?


title: QM: Commutation relations between irreducible vectors and angular momentum $[J^2,T_q^k]$

text: reading about the irreducible tensors and its commutation relations with the angular momentum one can find relations for $J_{z}$, $J_{+}$, $J_{-}$, but I was wondering, what about $J^2$ ? from sakurai: (revised edition p.236) (sakurai napolitano edition p.232) $$ [\bar{J}.\hat{n}, T_{q}^{k} ] = \sum_{q'} T_{q'}^{k} \langle {kq'} | \bar{J}.\hat{n} | {kq} \rangle $$ and for evaluating with $\hat{n}=\hat{z}$, $\hat{x}$, $\hat{y}$ $$ [J_{z}, T_{q}^{k} ] = \hbar q T_{q}^{k} $$ $$ [J_{\pm}, T_{q}^{k} ] = \hbar \sqrt{(k \mp q)(k \pm q + 1)} T_{q \pm 1}^{k} $$ But is there an expression for $ [ \bar{J}^{2}, T_{q}^{k} ] $ ? $$ [ \bar{J}^2, T_{q}^{k} ] = [ J_{x}^2+J_{y}^2+J_{z}^2, T_{q}^{k} ] = [ \frac{J_{+}J_{-} + J_{-}J_{+}}{2} + J_{z}^2 , T_{q}^{k} ] $$ $$ = \frac{1}{2} J_{+}[J_{-},T_{q}^{k}] + \frac{1}{2} [J_{+},T_{q}^{k}]J_{-} + \frac{1}{2} J_{-}[J_{+},T_{q}^{k}] + \frac{1}{2} [J_{-},T_{q}^{k}]J_{+} \+ J_{z}[ J_{z} , T_{q}^{k} ] + [ J_{z} , T_{q}^{k} ]J_{z} $$ $$ = \frac{1}{2} J_{+} \hbar \sqrt{(k + q)(k - q + 1)} T_{q - 1}^{k} \+ \frac{1}{2} \hbar \sqrt{(k - q)(k + q + 1)} T_{q + 1}^{k} J_{-} $$ $$ \+ \frac{1}{2} J_{-}\hbar \sqrt{(k - q)(k + q + 1)} T_{q + 1}^{k} + \frac{1}{2} \hbar \sqrt{(k + q)(k - q + 1)} T_{q - 1}^{k} J_{+} $$ $$ \+ \hbar q J_{z} T_{q}^{k} + \hbar q T_{q}^{k} J_{z} $$ I cant figure out what should i "sandwich" this with in order to obtain a useful expresion, if something. Any other idea? thanks in advance


title: Do electrons collapse into nucleus, if electrons in the atom are constantly excited?

text: From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels. When the electron is present as close to the nucleus as possible, the atom has the minimum possible energy and is said to be in the _ground state_. When energy from some outside source is supplied to it, it can absorb a definite amount of energy and jumps to higher energy state. Such a state of an atom in which the atom possesses more energy than possessed in the ground state is called _excited state_. These excited states are unstable and the electron tends to come back to lower energy level. This transition (change) from upper to lower energy level occurs with a jump and energy is emitted in the form of a quantum equal to difference in energies between the two levels. I have a doubt here, if electron absorbs energy from the outside source and jumps to the higher energy state, it will be now storing absorbed energy, as potential energy at the excited state. If electron emits all the absorbed energy (quanta-difference in energies between the two levels). By virtue of what energy does electron come down? I mean, in common situations, we say that an object at any height comes down converting it's potential energy into kinetic energy. Here in case of electron, it has already emitted absorbed energy as quanta. So, is it that electron losses some energy other than the energy absorbed from the source, to come down to ground state. I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus. But, this not we really observe. I think there might be some misunderstanding by me or there might be any of the existing model like quantum mechanical model, which could account for this. If any were the case, please explain.


title: Irreversible heat engines strictly less efficient than reversible ones

text: I undertand how Canot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it isless clear why they need to be less efficient, as I have seen stated in some places. If they could be equally efficient then an irreversible engine engine could be used to drive a reversible engine operating between the same heat reservoirs, without any net energy transfer between the reservoirs. It would then be unclear what is irreversible about the irreversible engine. Does that constitute an actual contradiction though? If so can the argument be stated more tightly? It feels a litle sloppy as is. It could also be a quation of how an irreversible engine is defined in Carnot's theorem. I understood it to mean one that cannot be run in reverse as a heat pump, which could presumably include a Carnot engine with a one way ratchet attached. If it really means an engine whose thermodynamic effects can not be undone, the implication would be trivial.


title: What does an applied force on an already accelerating object do?

text: I'm a big beginner having only taken Grade 11 high-school physics. Imagine this situation: an object is accelerating north, and while it's accelerating, it's 'hit' by a northward force. To me it would make sense that the acceleration of the object would increase - something like this:               aNew = aOld + (f/m)      I got the (f/m) from _f=ma_. But then think of gravity - while an object is accelerating downward at 9.8m/s2, it has the force of gravity pulling it too. So then, based on my previous logic, it's acceleration would constantly be increasing!               aNew = aOld + (fGravity/mass)      And I know that you don't increase your acceleration while you fall, so now I'm confused. **In a nutshell:** What exactly are the rules for how an already accelerating object reacts to forces? Am I wrong that your acceleration would increase? **EDIT:** I realized I've made a big mistake; I never knew that the acceleration drops to 0 the second the net force is 0. This makes sense now.


title: What is the smallest distance measured that the muon is considered pointlike?

text: What is the smallest distance measured that the muon is considered pointlike. I know for the electron it is clost to 0.001 of a femtometer. It seems to be assumed that this is the same for the muon. However, I can't find this stat for the muon in a Google search. My thinking on this is that if their is a muon radius, obviously suggesting substructure that is in the ball park of 0.04 femtometeres, could this affect the lambshift and possible explain the discrepancy in the recent measurements of the proton radius and possibly the anamolous muon magnetic moment?


title: Kinetic energy of a mass rotating about two axes

text: I'm working on a question where I have to find an equation for the kinetic energy of a mass being rotated by a fold-able arm. The equation for $m_1$ is obvious, however, the equation I derived for $m_2$ is not entirely consistent with the solutions. ![image for question](http://i.stack.imgur.com/tdnEn.jpg) To derive the equation I figured that I could use the resultant instantaneous velocity acting on $m_2$ to find its kinetic energy. If I say $v_1$ to be the instantaneous velocity due to $\dot \theta_1$ and $v_2$ to be the instantaneous velocity due to $\dot \theta_2$ then the resultant instantaneous velocity can be found using the cosine rule: $V_{r}^2 = V_1^2+V_2^2-2V_1V_2 \cos(\theta_d)$ $V_r$ is the resultant velocity, $\theta_d$ is the angle difference between the two velocity vectors and can be substituted with $\theta_1 - \theta_2$. And hence the kinetic energy for $m_2$ can be said to be: $K_2 = \frac{m_2}{2}\left(l_1^2\dot\theta_1^2+l_2^2\dot\theta_2^2-2l_1\dot\theta_1l_2\dot\theta_2 \cos(\theta_1-\theta_2)\right)$ However, the solution suggests a slightly different answer: $K_2 = \frac{m_2}{2}\left(l_1^2\dot\theta_1^2+l_2^2\dot\theta_2^2+2l_1\dot\theta_1l_2\dot\theta_2 \cos(\theta_1-\theta_2)\right)$ I can't think of any reason why the sign on the last term should be positive, would appreciate any help.


title: Spring Constant - Waves

text: I tried to prove that for the same length of spring , a one dimensional wave that moves faster in spring 2 than spring 1, its spring constant will be lower than the spring constant of spring 1. That's the begining of the proof I tried to make: $$w = \sqrt{\frac km} = 2π \cdot \nu$$ $$\nu = \frac{1}{2π} \cdot \sqrt{\frac km}$$ $$k = m \cdot \nu^ 2 \cdot 4π ^ 2$$ What am I missing?


title: Why isn't the sunset/sunrise rainbow-colored

text: When the sun is rising/setting, it goes through a phase where the light is bending from the atmosphere. I believe this image will explain much better than I ever could. ![enter image description here](http://i.stack.imgur.com/eEbN8.png) Now, if light goes through a prism, it is bent and the different wavelengths bend different amounts, resulting in the rainbow output. If the light from the sunrise/sunset is bending, then shouldn't we see a rainbow instead of the red tint? At the very least, shouldn't the top of the sun appear more red and the bottom more purple?


title: Finite rod aligned with a time varying magnetic field

text: > A finite length rod is aligned parallel with a magnetic field that has a > changing magnitude (let's say a sinusoidal change), given the conductivity > and permeability of the rod and assume a large skin depth, derive an > expression for the average power dissipation in the rod. My train of thought begins with trying to calculate the Poynting factor over the surface of the rod. The magnetic field inside the material will not be a complex one given the skin depth is assumed large because no attenuation is considered. However, I am stuck on deriving an expression for the electric field. Any thoughts?


title: Why don't you always use a high sampling rate?

text: I was just curious, as I was reading about aliasing. As far as I understand, aliasing comes from the fact, that you use a bad sampling rate, resulting in getting a wrong waveform compared to the one you had from the beginning. So why is it you just don't use a fast sampling rate all the time ? Or maybe you do that, and I just don't know ?


title: Obtain the eigenfunction of Jz for the wave function of an electron in a hydrogen atom?

text: > The wave function of an electron in a hydrogen atom is given by ![Wavefunction](http://i.stack.imgur.com/uHfLn.png) >   1. Is this wave function an eigenfunction of _Jz_ , the _z_ -component of > the electron’s total angular momentum? If yes, find the eigenvalue. (Hint: > For this, you need to calculate Jz Psi21*mlms*.) >   2. If you measure the _z_ -component of the electron’s spin angular > momentum, what values will you obtain? What are the corresponding > probabilities? >   3. If you measure J^2, what values will you obtain? What are the > corresponding probabilities? > How can I solve this problem or with which rules can be obtains.


title: Does a black hole have enough time to actually form a singularity?

text: I am trying to wrap my head around black holes, singularties and hawking radiation. Physics.se contains many intresting questions and answers, but from none I could so far read about the interaction between formation of a singularity and hawking radiation (most seem to talk about the point at which matter plunges into a black hole that already has a singularity). From what I understand, from the perspective of a star collapsing into a singularity, the perceived time is just like in classical mechanics: the mass is accelerated towards the center of gravity, and when "everything" is there, it is a singularity. Now this takes a finite time, and the matter itself perceives just that. Part of the question is: how long is this time? I am assuming in the magnitude of milliseconds. Now as I understand hawking radiation, as soon as there is an event horizon, it will start emitting negative mass/energy into the direction of the singularity. Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy/mass that are already on their way to the singularity. Since approaching the singularity, time dilation becomes kind of infinite, even the $10^{100}$ years or so it is supposed to take for hawking radiation to evaporate a black hole seem to be enough to send the necessary amount of negative energy/mass particles down to the singularity so that they can "chase" the positive energy/mass paticles very closely. But will they reach them before a singularity forms, so that they cancel out each other to the point there is not enough gravitational force? If this is just a matter of calculating "$n$s until matter reaches singularity" and "$m$s until negative mass/energy reaches singularity" and calculation shows that always $n<m$ then I would be intrested in the calculation (if possible presented in a way that doesn't require too exotic concepts). If this is a qualitative matter, in that some of my ideas are fundamentally wrong, please point those out (again if possible in a way that doesn't require too exotic concepts).


title: Do negative numbers have any physical meaning?

text: So, mindlessly wandering off into space, thinking about quantum and how cool physics is, I came to a realization that... well.. negative numbers to me make 0 sense. You have either something, or not something. It's always a yes or no answer, with everything And everything always has an opposite; it's nonexistence. I either have a red scarf, or not. If there are 3 doors, labeled 1 2 3. I have to choose one of them. The choice isn't me choosing which door, the choice is what door i'll pick. I'll either pick door 1, or not pick door 1. The opposite of door 1 is no door 1. Lets go really small. I have 2 grains of salt. I can't take 3 grains away. I believe negative numbers were invented by man to create debt. Because it doesn't make any sense in physics. Can someone please explain this to me!!


title: Balloon tethering cable tension model (catenary tether)

text: If a moving balloon (let's say it's a blimp system with some rotors for movement) is connected to the ground by a cable that describes a catenary curve, how can we model the tension being applied on the blimp by the cable? ![MSpaint draft](http://i.stack.imgur.com/mmAmG.png) This force has a vertical component that will be related to the mass of the cable and a horizontal component with the opposite direction of the movement of the blimp. Is there any model for this kind of behavior?


title: Is it possible for an object to stop sinking after a while?

text: Suppose you drop an object into a body of fluid (on a planet with constant gravity). Suppose the fluid is pure (as in, there are no solid granulates or bubbles present in it), and finally suppose that there are no waves in the fluid, and that its density is constant everywhere. Is it possible for said object to start sinking, but eventually slow down and completely stop sinking further (as in, it will stop and remain suspended in the liquid without hitting the bottom)?


title: Does the Earth have a semi-solid body at its core because of lower pressure at the center of the Earth?

text: I have read that the Earth has a semi-solid iron body at its core, even though that object is surrounded by liquid magma.... I'm wondering if this is because of a pressure (and resulting temperature) differential -- or if there is another reason. 1) Since the forces of gravity of a sphere are at equilibrium at the center of the sphere -- doesn't that mean that the pressure at the mass center of a planet would be zero -- since there would be no "force" to be distributed at that area? If the answer to #1 is true, then it leads to these further questions.... 2) At what point between the surface and the center of the planet would the pressure be the greatest? 3) Would the depth in the Earth, at the max. pressure point specified in #2, be where we would expect to find the central layer of the Earth's magma? -- (Since pressure generates heat, wouldn't that be the hottest point; from a "pressure variable only" view?). 4) Shouldn't it actually be cooler at the Earth's mass center, than at the max. pressure point specified in #2 -- since the pressure is lower at the mass center? 4.a) And could that center temperature be enough cooler that it allows a core body to semi-solidify?


title: Workdone in an equipotential surface is zero?

text: An equipotential surface is one in which all the points are at the same electric potential. If a charge is to be moved between any two points (say from point A to point B) on an equipotential surface, according to the formula $dW = q\cdot dV$, the work done becomes zero. My question is, how to move a particle without doing any work?


title: Concerning the curvature of an airfoil (shape)

text: I am wondering about the reasons for the shape of a turbine blade airfoil, see here Do you know the reason for this shape? Usually, very large curvatures like this are to extract high lift from LOW speed environments. But the conditions inside a turbine is the most strenuous for any object, and the speeds are certainly not low (right?)


title: Observing Jupiter's non-Galilean moons

text: What strength of telescope is required to observe some of the non-Galilean moons of Jupiter? My current telescope at 50 magnification resolves the Galilean moons well, but I'm guessing it's far below what's needed to observe any of the smaller moons. Given that the largest is 5% of the size of the smallest Galilean moon (according to what I found on Wikipedia, in _Moons of Jupiter_ ), is it even possible to observe them with an amateur telescope?


title: How large is the universe?

text: We know that the age of the universe (or, at least the time since the Big Bang) is roughly 13.75 billion years. I have heard that the size of the universe is much larger than what we can see, in other words, much larger than the observable universe. Is this true? Why would this not conflict with what is given by Special Relativity (namely, that faster-than-light travel is prohibited)? If it is true, then what is the estimated size of the universe, and how can we know?


title: Quantum mechanics and everyday nature

text: Is there a phenomenon visible to the naked eye that requires quantum mechanics to be satisfactorily explained? I am looking for a sort of quantic Newtonian apple.


title: Equivalent RC circuit to a RRC circuit?

text: I'm in doubt about a situation that I've seen sometimes: imagine we have a resistor in parallel with a resistor and a capacitor in series. Since I don't know how to generate figures of circuits to post here, the situation can be described as: a single resistor on the right, and on the left a resistor and a capacitor in series. ![enter image description here](http://i.stack.imgur.com/cwNW7.jpg) If there was no capacitor, I know I could replace the resistors by an equivalent one. My doubt is, do this continues to be true in this case? I mean, can I replace this configuration by one capacitor with one resistor in series such that this resistor is equivalent to the other two? If we can, what's the argument beyond this?


title: Cauchy's Second Law of Motion understanding

text: I'm told that Cauchy's Second law of motion is $$\tau_{ik} ~=~ \tau_{ki}.$$ Now my notes don't go into great details for simplicity sake, but they mention that the order of magnitude of body force $\rho F \delta V$ and inertia $\frac{D}{Dt}(\rho \delta V v)$ on a volume is smaller than that of stress $t(n) \delta S$ on a surface. What I'm not understanding is the order of magnitude stuff? I'm not sure why stress has a larger order of magnitude.


title: Can't find the mass scale; calculation using the modified minimal subtraction scheme and dimensional regularisation

text: I am taking a course on quantum field theory where there is some confusion regarding the renormalisation scheme we are using (and a corresponding one in my mind). Apparently the lecturer meant MS-bar when he said MS. Anyway, on with the question. Consider the* 1-loop contribution to the 1PI 4-point correlation function of phi-4 theory in $d=4-\epsilon$ dimensions, i.e the Lagrangian is $$L=-\frac{1}{2}(\partial \phi)^2 -\frac{1}{2}m^2 \phi^2-\frac{1}{4!} \mu^\epsilon \lambda \phi^4$$ where $\mu$ is a mass scale required to make $\lambda$ dimensionless as we vary $\epsilon$. The Feynman rules yield an expression of the form $$\frac{(\mu^{ \epsilon}\lambda)^2}{2} \int \frac{d^d k}{(2 \pi)^d}\frac{1}{(k^2 +m^2)((k+p)^2+m^2)}$$where $p$ is a certain sum of external momenta, which eventually reduces to $$\frac{\lambda^2}{2 (4 \pi)^2} \int_0^1 d\alpha \ (m^2 + p^2 \alpha (1-\alpha))^{-\frac{\epsilon}{2}} \Gamma\left(\frac{\epsilon}{2}\right) \mu^{2 \epsilon} [4\pi]^{\frac{\epsilon}{2}}$$I can clearly pull one factor of $\mu^\epsilon$ into $ (m^2 + p^2 \alpha (1-\alpha))^{-\frac{\epsilon}{2}}$ to get $\left(\frac{m^2 + p^2 \alpha (1-\alpha)}{\mu^2}\right)^{-\frac{\epsilon}{2}}$ so that the final, finite result at 1 loop looks like $$ -\frac{\lambda^2}{2 (4 \pi)^2} \int_0^1 d\alpha \log\left(\frac{m^2 + p^2 \alpha (1-\alpha)}{\mu^2}\right) +\text{2 other terms*}$$ _But where did the other factor of $\mu^\epsilon$ disappear to?_ Was it implicitly 'countertermed' away along with the factor $ [4\pi e^{-\gamma}]^{\frac{\epsilon}{2}}$? It looks as if this can be done at this order, but does this generalise to higher loop orders, i.e, is there a prescription telling us how many factors $\mu^\epsilon$ to counterterm away? Note that in phi-3 theory we get an identical loop integral by considering the 1-loop contribution to the 1PI 2-point correlation function, the difference being that $(\mu^{ \epsilon}\lambda)^2 \to (\mu^{ \frac{\epsilon}{2}} g)^2$ (where $g$ is the phi-3 coupling constant) in front of the integral. So that diagram does not have this problem somehow. *There are two other contributions obtained by permuting the external momenta. These have identical integral expressions. EDIT: I'll write this down fully. The term inside the integral can be written as $$(m^2 + p^2 \alpha (1-\alpha))^{-\frac{\epsilon}{2}} \Gamma\left(\frac{\epsilon}{2}\right) \mu^{2 \epsilon} [4\pi]^{\frac{\epsilon}{2}} =\left(\frac{m^2 + p^2 \alpha (1-\alpha)}{\mu^4}\right)^{-\frac{\epsilon}{2}}\Gamma\left(\frac{\epsilon}{2}\right) [4\pi]^{\frac{\epsilon}{2}}=\left(1-\frac{\epsilon}{2}\log\left(\frac{m^2 + p^2 \alpha (1-\alpha)}{\mu^4}\right)\right) \frac{2}{\epsilon} [4\pi e^{-\gamma}]^{\frac{\epsilon}{2}}$$ OR as $$=\left(1-\frac{\epsilon}{2}\log\left(\frac{m^2 + p^2 \alpha (1-\alpha)}{\mu^2}\right)\right) \frac{2}{\epsilon} [4\pi e^{-\gamma}\mu^2]^{\frac{\epsilon}{2}}$$ where the last two equalities are both asymptotic expansions in $\epsilon$ as $\epsilon \to 0$. I can absorb that last square bracket into the counterterm if I want to, at this order, and effectively ignore it.


title: When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?

text: Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation   * $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \mathcal{D} x e^{\frac{i}{\hbar}S}$ ($S$ is the classical action). It is often said that one gets classical mechanics in the limit $\hbar \rightarrow 0$. Then only the classical action is contributing, since the terms with non-classical $S$ cancel each other out because of the heavily oscillating phase. This sounds reasonable. But when we look at the Heisenberg equation of motion for an operator $A$   * $\frac{dA}{dt} = \frac{1}{i \hbar} [A,H]$ the limit $\hbar \rightarrow 0$ does not make any sense (in my opinion) and does not reproduce classical mechanics. Basically, the whole procedure of canonical quantization does not make sense:   * $\\{\cdots,\cdots\\} \rightarrow \frac{1}{i \hbar} [\cdots,\cdots]$ I don't understand, when $\hbar \rightarrow 0$ gives a reasonable result and when not. The question was hinted at here: Classical limit of quantum mechanics. But the discussion was only dealing with one particular example of this transition. Does anyone has more general knowledge about the limit $\hbar \rightarrow 0$?


title: What is the physical basis of the $\rho \bf{u} \times \bf{\omega}$ force on a fluid vortex line?

text: In fluid dynamics, the force density on a vortex line is $\bf{f} = \rho \bf{u} \times \bf{\omega}$. In Faber, Fluid Dynamics for Physicists, ch. 4, this is "derived" by analogy with magnetostatics, but the analogy does not appear to scan (top of p. 129 if you're following along at home; Faber appears to be using Bernoulli's principle even though the flow in fig. 4.4 is not irrotational across the boundary. There are some other implied steps in the analogy I don't think are well-justified either, but that's an aside.) Can you derive this force from basic fluids principles, say from Euler's equations for incompressible flow? I've tried writing $\rho \bf{u}\cdot\nabla\bf{u} = \nabla(\frac{1}{2}\rho u^2) - \rho \bf{u}\times\bf{\omega}$, but I can't find an argument to move the $\rho\bf{u}\times\bf{\omega}$ term to the force side of the equation.


title: Why don't X-rays/Gamma rays ionize all the atoms at the surface of a material?

text: Recently I've been wondering why certain materials are transparent or opaque to different wavelengths of light. The most common explanation for why a material, like glass, is transparent (in the visible spectrum) is because the photons of those wavelengths don't have enough energy to excite electrons to a higher energy state, so the photons move through the material unaffected. Once you reach a critical frequency, however, the photons will begin to be absorbed because they have enough energy to excite the electrons (which is why glass is opaque in ultra-violet). By that logic, one would expect that ALL frequencies higher than the critical point would be opaque. That is not what happens, however, as x-rays/gamma-rays penetrate deeply into certain surfaces (like your skin). Furthermore, x-rays/gamma-rays can ionize atoms by providing an electron with enough energy to escape the atom (which I believe is the cause of the photoelectric effect). Finally, an escaped electron can release some of its remaining energy through Compton Scattering, producing a lower frequency wave than that of the incident one. Compton Scattering seems to offer a candidate solution; i.e. that it's the scattered, lower frequency waves that make it through. However, according to this article, and more specifically this graphic (shown below), Compton Scattering starts occurring at a certain energy threshold, so it doesn't explain the relatively high penetration depth of lower frequencies. Furthermore, the graph also demonstrates that the penetration depth is proportional to the photon's energy, which again contradicts the trend discussed in the first paragraph, where one would expect the penetration depth to decrease as the photon's energy increases. ![Attenuation length vs. Photon energy](http://i.stack.imgur.com/yu9t0.png) I have looked for an answer on several sites, but none have provided me with a fully satisfying answer. These sites tend to either simplify the material, or simply gloss over this particular aspect of the more general question: "Why are materials transparent?". So, to summarize, my question is: Why does the penetration depth of high frequency waves increase for higher energies, and why is this not (apparently) applicable to lower frequencies? Given a high enough frequency, but still lower than the Compton Scattering frequency, why aren't all surface atoms ionized, effectively making the material opaque? (i.e. photoelectric absorption)


title: Fermi's weak interaction theory

text: In Fermi's theory, we have energy squared in the numerator of the cross- section which makes it diverge as energy increases. But isn't that the Fermi constant suppresses it with increasing order?


title: Liouville's theorem and gravitationally deflected lightpaths

text: It is customary in gravitational lensing problems, to project both the background source and the deflecting mass (e.g. a background quasar, and a foreground galaxy acting as a lens) in a plane. Then, the lensing problem can be regarded as a mapping between the unlensed source plane, and the lensed image plane. In such transformations, the Jacobian evaluated at a point of the source plane, expresses how an infinitesimal area located around that point increases. Lens mass and mass distribution, relative positions and distances involved give rise to different scenarios. The special case in which the distortions are too small to be resolved by telescopes, is called _microlensing regime_. Typically, a dark, unseen object like a floating planet, happens to cross transversally in front of a background star. The image of the background star suffers amplification and distortions that are unresolved, but a change in brightness is detected, with a very typical light curve shape. The measured light curve of a microlensing event can be related to physical parameters of the problem, because the change in brightness of a lensed image can be modelled simply by dividing the area of the lensed image by that of the unlensed source image. If that can be done, it is because the mean surface flux of the image equals that of the source. That is, gravitational lensing can make a tiny source appear bigger in the sky but in plain terms, every square inch of the image has the same brightness of every square inch of the source. Here comes my question, because that seems to me rather counter-intuitive and, when I try to find a rigorous justification to it, I find the same arcane sentence _in each book, in each review, in each paper_ I have seen: > **'Because of Liouville's theorem, gravitational lensing conserves surface > brightness'** (... and therefore the magnification is found by dividing the subtended area of the image by that of the source). Every single author I have read, drops that sentence as if it were something very obvious, and quickly goes into other questions. I have tried to trace-back the origin of the idea, by consulting the bibliography of every book or document in which that thing is stated. Interestingly, I have recognized sort of a fingerprint of obscure points like this one, a patter that is repeated in many of the documents, as if some authors didn't understand and merely copied from each other, developing and personalizing only the parts they understand in between. I have rigorously developed and resolved every one of the dark points in that pattern, but this one remains unresolved. Is it perhaps something obvious? **How is Liouville's theorem applied to photons along null geodesics?** I will accept an appropiate link or paper reference as a good answer.


title: Cylinder rolling down slope problem

text: A uniform cylinder of mass $m$ and radius $r$ is rolling down a slope of inclination $\theta$. The cylinder rolls without slipping. You may take the acceleration due to gravity to be $g$. At what rate does the cylinder accelerate down the slope? The answer is $\frac23 g \sin\theta$ How do you get to this answer?


title: Force on impact in case of a free fall

text: In a free fall, on impact with the ground, the body experiences a massive force because of a huge acceleration since the velocity goes to 0 rather quickly. Since this force has to come as a result of reaction, can it be assumed that the body applied a massive force on impact far greater than its weight?


title: Are oscillations of electron chirality experimentally observable?

text: Is there any plausible experiment by which chirality oscillations in electrons could be observed experimentally, such as through some analogy to neutrino oscillation experiments?


title: Where are we : On level ground or on a ramp - moving in a train?

text: Let's say we are traveling in a train. The path has two parts: one at ground- level and the other moving up on the ramp. The ramp has an inclination of $\arctan\frac{a}{g}$ with the horizontal, where $a$ is the acceleration of the train on level ground and $g$ is the acceleration due to gravity. The train does not accelerate on the ramp, but moves with a constant velocity. Can we comment where we are (sitting inside the train of course!) when we have only a pendulum hanging on the roof to observe. ( _windows are blackened_ ) ![Image](http://i.stack.imgur.com/uCujR.png)


title: Why is $dL = L d\epsilon$?

text: Let's say there's a random elastic material. It's length is $L$ and it's tensile strain $\epsilon= (L-L_0)/L_0$ Now, when one pulls on it the following is true: $dW_{tot}=FdL =\sigma AdL=\sigma A L d\epsilon$ Why can we make that last step or in other words: why is $dL = L d\epsilon$?


title: Is it possible to transform chemical energy in electrical energy by modifying the equilibrium constant?

text: Considering a Daniell cell, is it possible to create endlessly electrical energy by using atmospheric temperature (for exemple) variations to modify the reaction's eqilibrium constant ? I apologize if this question is trivial.


title: Current without Voltage and Voltage without Current?

text: At school I've always learned that you can view Current and Voltage like this: **The current** is the flow of charge per second and **the Voltage** is how badly the current 'wants' to flow. But I'm having some trouble with this view. _How can we have a Voltage without a current?_ There is nothing to 'flow', so how can it be there? Or is it 'latent' voltage, I mean is the voltage just always there and if a current is introduced it flows? Also, I believe you can't have current without voltage. This to me seems logical from the very definition of current. _But if you have a 'charge' without a voltage, doesn't it just stay in 1 place? Can you view it like that? If you introduce a charge in a circuit without a voltage it just doesn't move?_