title: Are Lifshitz and Berestetskii right in this case?

text: In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat {C}^{M} = \alpha_{2}$. Here (they started from the standart (Dirac) representation of the gamma-matrices) $$ \alpha_{2} = \begin{pmatrix} 0 & \sigma_{y}\\\ \sigma_{y} & 0\end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0\\\ 0 & -1\end{pmatrix}, $$ $$ \hat {C}^{M} = \hat {U}^{+} \hat {C} \hat {U}, \quad \hat {U} = \frac{1}{\sqrt{2}}(\alpha_{2} + \beta ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix} = \hat {U}^{+}. $$ I tried to get their result, but I only got $$ \hat {C}^{M} = \frac{1}{2}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix}\begin{pmatrix} 0 & \sigma_{y}\\\ \sigma_{y} & 0\end{pmatrix}\begin{pmatrix} 1 & \sigma_{y}\\\ \sigma_{y} & -1\end{pmatrix} = \begin{pmatrix} 1 & 0\\\ 0 & -1\end{pmatrix}, $$ because $\sigma_{y}^{2} = 1$. Where is the mistake? **An edit**. I found the mistake. I used the wrong definition of transformation of charge conjugation operator under unitary spinor transformation. The correct one is $\hat {C}^{'} = \hat {U}\hat {C}\hat {U}^{T}$.


title: Implementing a transformation as $UaU$ and not $UaU^{-1}$?

text: I know one associates to each symmetry transformation a unitary/antiunitary operater...etc. But equation 3.123 in Peskin and Schroeder (PS) says that parity is implemented as $(\mathbf{p}$ is the three momentum) $$P a_{\mathbf{p}}^sP = \eta_a a_{\mathbf{-p}}^s \tag{1}, $$ where $P$ is the operator implementing the parity transformation. My question is why the left hand side of $(1)$ not of the form $PaP^{-1}$?


title: Non-locality and topology

text: This is a purely speculative question: Has there been any work that describes non-locality/entanglement in QM by using exotic topologies in configuration space? The 'conceptual' picture that I have in mind is of two particles that are distant in the usual topology, but when entangled, being somehow part of the same 'irreducible' open set - so they can't be 'disentangled'. For this to have any chance of working, topologies would have to change with time.


title: Are there stable string theory vacua with non-minimal cosmological constant?

text: Naive reasoning suggests that a string theory vacuum with cosmological constant Lambda1 is always unstable as long as there is a string theory vacuum with cosmological constant Lambda2 < Lambda1 since the later is a state with lower energy density. If it's true, there should be very few stable vacua since these would be the vacua with the lowest possible cosmological constant. These vacua would probably have a Planckian-size cosmological constant and therefore be highly non-classical. However, I suspect there is something wrong with this line of reasoning. Probably I am thoroughly confused about something. Also, how to compare stability of vacua with different number of large dimensions? Comparing cosmological constants doesn't feel to make sense.


title: Explicit construction for unitary extensions of CPTP maps?

text: Given a completely positive and trace preserving map $\Phi : \textrm{L}(\mathcal{H})\to\textrm{L}(\mathcal{G})$, it is clear by the Kraus representation theorem that there exist $A_k \in \text{L}(\mathcal{H}, \mathcal{G})$ such that $\Phi(\rho) = \sum_k A_k \rho A_k^\dagger$ for all density operators $\rho$ on $\mathcal{H}$. (I'll consider the special case $\mathcal{H} = \mathcal{G}$ for simplicity.) If we use then the system+environment model to express this action as $\Phi(\rho)=\text{Tr}_{\mathcal{H}_E} (Y\rho Y^\dagger)$ for an isometry $Y$ from $\mathcal{H}$ to $\mathcal{H}\otimes\mathcal{H}_E$, where $\mathcal{H}_E$ is an ancilla modelling the environment, what is an explicit construction for a unitary $U$ that has the same action on inputs of the form $\rho\otimes\left|0\right>\left<0\right|_E$? That is, how can I construct an explicit dilation of the map to a unitary acting on a larger space? I understand that this is possible by Steinspring's dilation theorem, but actually constructing an explicit form for the dialated unitary I have had much less success with.


title: Comparison of different ab-initio codes

text: One may find on the web a lot of different computational packages to perform "ab-initio" calculations of electron structure of the solids. Usually, the documentation is not quite transparent about the disadvantages of the methods used. It takes some time to find out whether given package is suitable for you needs. And even if it is according to the documentation, it may turn out that it is not in reality. Is there a good comparison of different packages? Under "comparison" I mean not usability, etc. but comparison from the researcher's point of view. One aspect is the purpose of the package. I.e., some packages (like GAUSSIAN) are more chemistry-oriented, some focus on structural properties, and neither may be used if you want to calculate e.g. exchange interaction between the excitons in semiconductors. However, you may spend days realizing this fact from the documentation because authors did not know that such a problem even exists. Another aspect is methods which are implemented. The documentation is usually quite sparse on this subject also due to the fact that authors are not well aware of other methods which exist around. A good example is spin-related phenomena. Many packages introduce some limited tricks to handle these problems because they never thought that there are problems where these tricks do not work. Few examples: abinit, gaussian, quantum espresso, siesta, VASP, WIEN2k, Octopus


title: Supersymmetric Noether theorem and supercurrents -- invariance requirements

text: Consider $\mathcal{N}=1,d=4$ SUSY with $n$ chiral superfields $\Phi^i,$ Kaehler potential $K,$ superpotential $W$ and action ($\overline{\Phi}_i$ is complex conjugate of $\Phi^i$) $$ S= \int d^4x \left[ \int d^2\theta d^2\overline{\theta} K(\Phi,\overline{\Phi})+ \int d^2\theta W(\Phi) + \int d^2\overline{\theta} \overline{W}(\overline{\Phi})\right].$$ As in Weinberg (QFT, volume 3, page 89), requiring invariance of $K$ and $W$ under \begin{align}\delta\Phi^i &= i\epsilon Q^{i}_{~j}\Phi^j \\\ \delta\overline{\Phi}_i &=- i\epsilon \overline{\Phi}_jQ_{~i}^{j}\end{align} with $\epsilon$ small positive constant real parameter and $Q$ hermitian matrix, one obtains some conditions on $K$ and $W,$ namely (denoting $K_i=\frac{\partial K}{\partial\Phi^i}$ and $K^i=\frac{\partial K}{\partial\overline{\Phi}_i},$ and the same for $W$) $$ K_i Q^i_{~j} \Phi^j = \overline{\Phi}_j Q^j_{~i} K^i $$ $$ W_i Q^i_{~j}\Phi^j=0=\overline{\Phi}_j Q^j_{~i} \overline{W}^i $$ and can then compute the Noether current associated to such transformations, promote $\epsilon$ to a full chiral superfield $\epsilon\Lambda$, etc. The obtained invariance implies $\delta_{\epsilon}S=0$ for the above transformations. QUESTION: is the reverse also true? namely, by requiring $\delta_{\epsilon}S=0$ (instead of the apparently stronger invariance condition on $K$ and $W$), does one obtain the same constraints as above on $K$ and $W?$


title: Where do flying gas molecules get their energy from?

text: If you heat up water in a tank to boiling point, some of the water will turn to steam. Gas bubbles are created in the water. According to Archimedes principle, the gas bubbles will have a force equal to the weight of what they are occupying. Let's denote this force $F_g$. $$F_g = \text{Bouyant force of a gas bubble}.$$ Let's denote the height of the tank $h$. We also assume the steam bubbles are created at the bottom of the tank and also that the viscosity of water is nearly zero. So the work done when a bubble reaches the surface is $$\text{Energy} = (F_g-g m)h$$ $g$ is the gravitational acceleration and $m$ the mass of the bubble. Where did this energy come from? From the surrounding water? From the boiler? From the bubble itself? Can someone in detail explain to me this phenomena?


title: Does the doppler shift have any effect on the energy that the atom absorbs?

text: Does an atom moving towards the light that is falling on it, absorb more energy than when it is stationary? The atom should a see blue shift due to Doppler effect so the atom will see a light of greater frequency, causing it to absorb a greater amount of energy (or, equivalently, it will absorb less energy when it is moving away from the light due to red shift). Or will it absorb the same energy in all the cases? Does the atoms motion have any affect on the energy it absorbs?


title: Can I steal your electron?

text: The following paragraph has been extracted from the Wikipedia (Atomic orbitals): > Simple pictures showing orbital shapes are intended to describe the angular > forms of regions in space where the electrons occupying the orbital are > likely to be found. The diagrams cannot, however, show the entire region > where an electron can be found, since _according to quantum mechanics there > is a non-zero probability of finding the electron (almost) anywhere in > space_. Is the statement by Wikipedia correct? Since, there is a probability of finding electron at any distance from the nucleus, when the electron comes far from the nucleus, I will block it, so that it won't return to its parent atom. Am I not stealing the electron? I can steal even the electron of your body being in India, be careful! That's what we layman think from those statements. What's the actual meaning of the wikipedia statement?


title: Why are high voltage lines “high voltage?”

text: If I have two spheres of the same size and one sphere has a small amount of charge compared to the other that has a lot more charge, then clearly the sphere with the larger charge has a larger voltage (relative to the ground). My question is do high voltage power lines have a lot more charge that is placed on them? Is that what gives them the high voltage? I think I have a grasp of the step up stations that use transformers to kick up the voltage from power plants. This question seems almost silly to me but I have been struggling with this for a long time. I’ve done several searches online and I am not able to find answers. If there is a link that someone can provide, I much appreciate it.


title: Can you justify force calculations using more than one of Newton's laws?

text: If an object is at rest on a flat surface I remember my school teacher would say, "By Newton's 1st law the normal reaction, R, is mg (mass x g)". Ok, sure this is fine because the object is at constant velocity (velocity=zero) and so there are no net forces and so the two forces must balance. But what's wrong with saying "By Newton's 2nd law R-mg=0. Therefore R=mg." (The net force on the object is zero, therefore the acceleration on the object is zero) But can't you also say, "By Newton's 3rd law R=mg." (The object is pushing on the ground with a force of mg, therefore the ground is pushing on the object with a normal reaction of mg). Is this correct or have I erred somewhere?


title: Does alternating current (AC) require a complete circuit?

text: This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work _(ignoring resistance/inductance of the wire)_ " was recently asked on the Electronics SE. ![Picture edited from the one in the above link](http://i.stack.imgur.com/fcz1i.png)   _(Picture edited from the one in the above link)_ Though I respect the AC/DC experts there, I think _(with the exception of the top answer)_ they are all wrong. * * * My issue is that they all assume that AC requires a complete circuit in order to function. However, my understanding is that a complete circuit is necessary for DC, but not AC. My intuitive understanding is that AC is similar to two gas-filled rooms with a pump between them - the pump couldn't indefinitely pump gas from one room to another without a complete circuit _(DC)_ , but it could pump the gas back and forth indefinitely _(AC)_. In the latter case, not having a complete circuit just offers more resistance to the pump _(with smaller rooms causing a larger resistance)_. Is my understanding correct - **can AC circuits really function without a complete loop?**   More importantly, **what are the equations that govern this**?   If larger isolated conductors really offer less AC-resistance than smaller AC conductors, how is this resistance computed/quantified? Would its "cause" be considered inductance, or something else?


title: Are neutrinos affected by gravity?

text: Layman here, but EE and BS physics. I know that light is affected by gravity. But are neutrinos? During the collapse of a star into a neutron star, as the electrons join protons to form neutrons (e.g., or the collapse of a star to a black hole?), I read that the only things that can get "out" instantaneously are the neutrinos. (this even applies to a normal star I presume, as the photons take "forever" to exit). But I know gravity is extreme in these instances, to say the least, so does gravity NOT affect the neutrinos? This would seem to be contradictory.


title: The measurement of electricity

text: I'm a student trying to understand electricity. As I learned from school, electricity is the flow of electrons, but I'm confused about the measurement of electricity.   * As I learned, voltage is the "pressure" of the electricity, but it's a bit abstract while I try to understand the particle world and atomic world. Is voltage the speed of the electrons moving, or the number of electrons moving in the flow? Electrons are particles, so how many electrons are moving in the flow if the flow is $5\textrm{ V}$ or $12\textrm{ V}$?   * What caused the different voltage in the atomic world between two points?    * What about amps – how do they relate to voltage?


title: Understanding velocity gradients in fluids

text: So I'm having trouble understanding velocity gradients conceptually, I have little physics training passed physics 101 (I'm a biologist), but I'm currently working in an endothelium research lab with a lot of fluid physics. I came up with an example in my kitchen and I tried to work it out based on youtube videos of fluid mechanics I was watching. If you have a tall glass and you fill it with water, then spin it around it's long axis (so the bottom doesn't move but it spins like a disc) it seems like the water inside doesn't move as quickly as you spin the glass. I'm guessing based on "no-slip" the water touching the glass is moving at the speed of the glass it touches... the water in the center of the column probably moves the least. If you stop spinning the glass the water continues to move (inertia) but it slowly stops (not sure why)... So my question is this: where is the water moving in fastest in that moment you stop spinning the cup/column... originally the velocity in the center was the lowest and the velocity on the glass was highest, but then wouldn't the glass be the source of friction and stop it? so is the velocity initially fasted on the periphery but then the layer with the fastest velocity moves away toward the center of the cup/column?


title: How can molecule of a few angstroms absorb visible light of a few hundred nanometers?

text: I guess visible light is visible, because it has the right wavelength to be absorbed (or not) and emitted (or not) by many different molecules. Now visible light has a wavelength in the order of a few hundreds nanometers, while the typical size of a molecule is rather in the order of a few angstrom. I guess the difference between the energy levels of the molecules will have the right magnitude that corresponds to the energy carried by a single photon. But it's still slightly strange, because a classical antenna would need a length of a quarter wavelength to be most effective at absorbing or emitting energy. I also wonder what the effective size of the scattering cross-section of a molecule to absorb a photon of an appropriate wavelength will be. Is it more in the order of the size of the molecule, or more in the order of the geometric mean between the wavelength and the size of the molecule?


title: Maybe photon energy is constant as the Universe expands?

text: This is a question following on from my previous post Time-like Killing vector in FRW metric? For simplicity I take the spatially flat FRW metric in cartesian co-ordinates given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ I transform the time co-ordinate $t$ to conformal time $\tau$ using the relationship: $$d\tau = \frac{dt}{a(t)}$$ so that the metric now becomes: $$ds^2 = a^2(t)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ We can introduce a function $\alpha(\tau)$ giving $$ds^2 = \alpha^2(\tau)(-d\tau^2 + dx^2 + dy^2 + dz^2)$$ One can now see that the FRW metric has a time-like conformal Killing vector field.. Now electromagnetic waves are conformally (or scale) invariant. If one makes the transformation: $$x \rightarrow \lambda x$$ $$t \rightarrow \lambda t$$ the electromagnetic wave equations are invariant. Thus it seems to me that there is a symmetry here that Nature could obey. If it did so then the implication would be that the energy of electromagnetic waves is constant as the Universe expands.Therefore wave lengths that scale with the Universe are associated with constant energy. However atoms have a fixed wavelength so that their energy would increase as the Universe expands. Thus the cosmological redshift would be explained not by electromagnetic waves losing energy in an expanding Universe but rather by atomic energies increasing. In conventional cosmology one tacitly assumes that atomic energies are constant as the Universe expands. But what is the basis for such an assumption? At least with photons one can offer a symmetry argument, consistent with general relativity, as to why their energy should stay constant.


title: Information encoded on the surface of a black hole

text: If an object that enters a black hole has its information content frozen at the event horizon, in what sense is it frozen? The usual analogy is of a hologram encoded in 2d which can be decoded into a 3d representation. The same analogy is used in describing the 3d universe we see as a representation of the holographic information encoded on the 2d "surface" of the universe.   1. What is meant by "information content"? Or have I stated this   incorrectly?   2. The analogy leads one to wonder if there is a way to   recreate the objects that have entered a black hole in any sense. Is the information frozen on the event horizon of a black hole   detectable or visible in any real sense from the outside? How is one to imagine this effect intuitively? I understand that in thinking about the holographic analogy I may have been going completely astray so please don't snipe at the question, simply correct me.


title: Heat Equation Equalities

text: While studying the heat equation, I ran into a few equalities that I cannot understand. For example, **Fourier's law of heat conduction** claims that $$\varphi(x,t)=-K_0\frac{\partial u}{\partial x},$$ where $\varphi$ is the heat flux (the amount of thermal energy per unit time flowing to the right per unit surface area), $K_0$ is the thermal conductivity, and $u$ is the temperature. How did Fourier end up with this? Another equality that I cannot understand is the relationship between thermal energy and temperature: $$e(x,t)=c(x)\rho(x)u(x,t),$$ where $e$ is the thermal energy density, $c$ is the specific heat (the heat energy that must be supplied to a unit mass of a substance to raise its temperature one unit), and $\rho$ is the mass density (mass per unit volume). Why is this true? I cannot seem to make the physical connection. Thanks in advance!


title: Problem with ring-method of du Nouy for calculating surface tension

text: I need to calculate the surface tension of a liquid with the ring-method of du Nouy and the correct formula to use is: > $\gamma = \frac{F}{4\pi R}$ I understand that the surface tension is the force that acts on a certain length which, in this case, is the circumference of the ring. Since the circumference of the ring is $2\pi R$ where does this extra $\times2$ come from? Thanks!


title: Derivation of differential scattering cross section - off-center target

text: This is a followup question to this pretty good answer regarding deriving the Boltzmann equation. What if the center of the target particle is actually not the same with the scattering center (or may not have a scattering center)? For example, see the classical problem of hard-sphere scattering.


title: Square bracket notation for dimensions and units: usage and conventions

text: One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them?


title: Notation for two variables with same dimensions

text: What symbol represents "has the dimensions of", as in "x has the dimensions of d"? Does such a symbol exist?


title: In which direction should flow of electric current be taken while solving problems?

text: Consider a simple circuit with a battery of $\theta\ \text V$s, and two resistors of $R_1 \ \Omega$s and $R_2\ \Omega$s connected in series. Let us assume that $R_1$ is connected nearer to the positive terminal of the battery. We are required to find the power $P_1$ and $P_2$, dissipated in these resistors. Now, here if we take the direction of electric current from negative to positive, we get the following result: $P_2 = I_2^2R_2$. Since $I_2 = \frac{\theta}{R_2}$, $P_2 = \frac{\theta^2}{R_2}$ Now, similarly, $P_1 = \frac{\theta^2}{R_1}$. But, there is a voltage drop across $R_2$, which equals $IR_2 = R_2\frac{\theta}{R_1 + R_2}$. Therefore, the voltage just before $R_1$ equals $\theta\left(1 -\frac{R_2}{R_1 + R_2}\right)$ and $P_1 = \frac{\theta^2\left(1 -\frac{R_2}{R_1 + R_2}\right)^2}{R_1}$. But, consider the case in which the direction of electric current is taken from positive to negative. We encounter $R_1$ first. In this case $P_1 = I_1^2R_1 = \frac{\theta^2}{R_1}$. There will be voltage drop between $R_1$ and $R_2$ which will equal $\theta\left(1 - \frac{R_1}{R_1 + R_2}\right)$, so $P_2 = \frac{\theta^2\left(1 - \frac{R_1}{R_1 + R_2}\right)^2}{R_2}$. These are two different results. Which direction should the electric current be taken in this case?


title: Is there a physical system that emulates mathematical cellular automata?

text: Theoretical cellular automata have been proposed as models for many physical phenomena from music to quantum mechanics. My question concerns the reverse: **Is there a _simple_ physical system that emulates a theoretical cellular automaton?** A computer running CA software such as Hashlife can be considered an (extremely complex) physical system that behaves according to the rules of the programmed CA. I am however looking for something much simpler, such as a set of dominoes that topple each other in a fashion conforming to the rules of a certain CA, wave interference patterns that model a CA etc. The point is that there have been enormous advances in CA theory lately (such as the discovery of the first self-replicating pattern in Conway's Game of Life in November 2013) that it would be interesting if these patterns could simply be "set up" in the physical world somehow and then they would evolve according to the CA rules without requiring a computer to simulate the rules. In the long run, this might even lead to the creation self-replicating physical systems.


title: Doubts about the Aharonov-Bohm effect

text: In _F. Schwabl_ , **Quantum Mechanics** p.148 it is explained that if we have a particle in an electromagnetic field given by potentials $\varphi$ and $\mathbf{A}$ with wave function $\psi$, then a gauge transformation $$\mathbf{A}'=\mathbf{A} + \nabla\chi\ \ \ \ \ \ \varphi'=\varphi - \frac{1}{c}\frac{\partial\chi}{\partial t}$$ induces a change on the wave function given by $$\psi' = e^{-\frac{ie}{\hbar c}\chi}\psi$$ Then it explains how this doesn't affect the physical meaning of $\varphi$ since it is only a change of phase and $|\varphi'|^2$ remains unchanged. A couple of pages later, the Aharonov-Bohm effect is presented. We have the classical setting ![Aharonov-Bohm experiment](http://i.stack.imgur.com/EODEf.png) The first thing Schwabl does is to compute (with the help of a gauge transformation) the wave function when one slit is closed. We get $$\psi = e^{\frac{ie}{\hbar c}\Lambda}\psi_0$$ where $\Lambda(\mathbf{x}) = \int_{\mathrm{origin}}^\mathbf{x}\mathbf{A}\cdot ds$ and $\psi_0$ is the wave function of the free particle. Then he sums the two wave functions (one for each slit) to get the interference on the screen. My problem is: How comes that a change of phase, which we said doesn't change the physical meaning of $\psi$ when speaking of gauge transformations, now affects the physical meaning of $\psi$ (namely producing an interference pattern)? Is this due to the fact that $\mathbb{R}^3$ with a straight line removed is not simply connected, and thus the integral defining $\Lambda$ is not path independent when both slits are open? Or is the cause something else entirely?


title: Reducing massive representation of the Poincare group to the massless one

text: I want to ask about the connection for massive and massless representation of the Poincare group. Sorry for the awkwardness. First I must to represent the formalism for both of cases. **Massive representation**. There is the formalism for an arbitrary spin-$s$ and massive representation of the Poincare group: $$ (\partial^{2} + m^{2})\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}}(x) = 0, \qquad (1) $$ $$ \partial^{a \dot b}\psi_{a...a_{n - 1}\dot {b}...\dot {b}_{k - 1}}(x) = 0, \qquad (2) $$ where $n + k = 2s$; field $\psi$ is symmetric by the all of indices and has the following transformation law under the Lorentz group $\left(\frac{n}{2}, \frac{k}{2}\right)$: $$ \quad \psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}}' = N_{a_{1}}^{\quad c_{1}}...N_{a_{n}}^{\quad c_{n}}{N^{*}}_{\dot {b}_{1}}^{\quad \dot {d}_{1}}...{N^{*}}_{\dot {b}_{k}}^{\quad \dot {d}_{k}}\psi_{c_{1}...c_{n}\dot {d}_{1}...\dot {d}_{k}}. $$ $(1)$ gives the irreducibility by mass (or the statement $P^{2}\psi = m^2\psi$ for first Casimir operator of the Poincare group), while $(2)$ gives the irreducibility by spin (or the statement $W^{2}\psi = -m^{2}s(s + 1)\psi$ for the second Casimir operator). This representation has $2s + 1$ spin degrees of freedom. For the case of an integer spin-$s$ representation refers to the traceless transverse symmetric tensor of rank s $A_{\mu_{1}...\mu_{s}}$: $$ A^{\mu}_{\quad \mu ...\mu_{s - 2}} = 0, \quad \partial_{\mu}A^{\mu ...\mu_{s - 1}} = 0, $$   $$ (\partial^{2} + m^{2})A_{\mu_{1}...\mu_{s}} = 0,\quad A_{\mu_{1}...\mu_{i}...\mu_{j}...\mu_{s}} = A_{\mu_{1}...\mu_{j}...\mu_{i}...\mu_{s}}. $$ **Massless representation**. There is the formalism for an arbitrary helicity-$\lambda$ and massless representations (for given helicity there is infinite number of the representations!) of the Poincare group: $$ \partial^{a \dot {d}}\psi_{a...a_{n - 1}\dot {b}_{1}...\dot {b}_{k}} = 0. \qquad (3) $$ $$ \partial^{\dot {b} c}\psi_{a_{1}...a_{n}\dot {b}...\dot {b}_{k - 1}} = 0. \qquad (4) $$ Both of them realize the irreducibility by mass $\partial^{2} \psi = 0$ and irreducibility by helicity, $$ W_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}} = \frac{n - k}{2}p_{c \dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}} = \lambda p_{c\dot {c}}\psi_{a_{1}...a_{n}\dot {b}_{1}...\dot {b}_{k}}. $$ The representation has only one independent component, as it must be for massless one. If the representation is real we must take the direct sum of representations $\left( \frac{n}{2}, \frac{k}{2} \right) + \left( \frac{k}{2} , \frac{n}{2} \right)$, so the number of degrees of freedom will increase from 1 to 2. **My question**. I'm interesting only in an integer spin representations. For spin-1 massive case and field (I choosed one of three possible variants) $$ \psi_{a \dot {b}} \to A_{\mu} = \frac{1}{2}\tilde {\sigma}_{\mu}^{\dot {b} a}\psi_{a \dot {b}} $$ there is three degrees of freedom, which refers to the transverse condition $\partial_{\mu}A^{\mu} = 0$. But in massless case we also can introduce gauge transformations $$ A_{\mu} \to A_{\mu}' = A_{\mu} + \partial_{\mu} \varphi $$ which don't change the transverse condition and Klein-Gordon equation: $$ \partial_{\mu}A^{\mu}{'} = \partial_{\mu}A^{\mu} + \partial^{2}\varphi = 0 \Rightarrow \partial^{2}\varphi = 0, $$ $$ \partial^{2}A_{\mu}' = \partial^{2}A_{\mu} = 0. $$ So we can also satisfy the condition $u_{\mu}A^{\mu} = 0$, where $u_{\mu}$ is some timelike vector. So the number of independent components reduce to two (we have the case when representation is real, so this representation gives one particle with two possible states of helicity). By the full analogy with spin-1 massive case I got the result with spin-2 massive case. In the beginning it has $$ 2s + 1 = 10 (symmetry) - 4 (transverse) - 1(traceless) = 5 $$ independent components. When $m = 0$ we have the gauge transformations $$ A_{\mu \nu} \to A_{\mu \nu}{'} = A_{\mu \nu} + \partial_{\mu}\varepsilon_{\nu} + \partial_{\nu}\varepsilon_{\mu}, $$ by which we can set the condition $u_{\mu}A^{\mu \nu} = 0$ which decreases the number of degrees of freedom by three (the another one is equal to one $k_{\mu}A^{\mu \nu}u_{\nu}$ from $\partial_{\mu} A^{\mu \nu} = 0$). So we also have 2 independent components. For the arbitrary integer spin representation I also can get two components, because there is gauge transformations $$ A_{\mu_{1}...\mu_{s}}{'} = A_{\mu_{1}...\mu_{s}} + \partial_{\mu_{1}}\varepsilon_{\mu_{2}} + ..., $$ by which I also can reduce the number of degrees of freedom of my field. Finally, my question (edited): can this reduction from massive to massless case be derived on the language of $(1)-(4)$?


title: Why was the conversion factor of the metric unit bar chosen the way it was?

text: The unit **bar** for pressure is clearly a metric unit, but its order of magnitude is a bit strange. In the centimeter–gram–second system of units we have:               1 bar = 1 000 000 baryes = 1 000 000 dyn/cm²      so the bar is not "coherent" with this system (the factor is not one). Also in the meter–kilogram–second (and SI) system we get:               1 bar = 100 000 pascals = 100 000 N/m²      while in the meter–tonne–second system:               1 bar = 100 pièzes = 100 sn/m²      So my question is simply, where does the conversion factor for bar come from, since it seems to not fit into usual systems? According to the Wikipedia article _bar_ this unit was created already in 1909 by British meteorologist Shaw, but not much detail is provided. Maybe the factor was simply chosen as the power of ten making the unit closest to the atmospherical pressure at sea level (which is 1.01325 bar by convention, and close to 1.01 bar on average)?


title: Classically efficient universal quantum computation (P=BQP) with magic and bound states

text: $\text P$ vs $\text {BQP}$ is an open question. That is, "can systems which require a polynomial number of qubits in the size of an input be described with only a polynomial number of bits?" If the answer is "yes", then the device you're reading this with can factor quickly and do other fun things, whoopee! Most quantum information scientists _assume_ that universal quantum computation is hard for Turing machines / classical computers, but there is no proof to show classical computers are quantum-weak. We know the following:   * Clifford circuits acting on stabilizer states are classically efficient (Gottesman-Knill theorem).   * All quantum states with a positive Wigner representation are classically efficient. This includes both stabilizer states and so-called bound states (see here). Bound states are interesting because they cannot be described trivially as combinations of stabilizer states (they lie outside the stabilizer polytope / octahedron), yet are still easy to classically simulate.   * Magic states promote Clifford circuits to full quantum universality by allowing for the application of a non-Clifford unitary through a process called 'gate teleportation' or 'gate injection' (see here). Magic states however, being pure non-stabilizer states are classically _inefficient_ to simulate.    * It has been shown that if one can simulate a _reusable_ magic state, then quantum power is classically realizable with software which takes advantage of the finding, ie. $\text P = \text{BQP}$ (see here). This is a big finding, but magic states are generally destroyed in the teleport/injection process so it seems hard to achieve magic-state-reusability. The reason why $\text P$ becomes $\text{BQP}$ when the magic state is reusable is because the number of terms introduced by the magic state is kept constant; whereas in the typical magic state scenario (where the magic state is destroyed during gate injection), the number of terms grows exponentially with each magic state introduced to the computation (generally $N$ magic states consumed per $N$ non-Clifford transformations required, so $d^N$ terms). Thus the challenge then becomes: _how can we keep the number of terms constant but not be burdened with devising an invincible magic state?_ Ie. how can we employ a linear number of magic states in our simulation but keep the terms introduced by such states constant as in the $\text{P}=\text{BQP}$ finding from this article. The proposed answer to the above problem is to compress the output of the gate injection qudit to the nearest 0 / neutral Wigner state (Wigner representations in this context are defined in this paper in the middle of p.5). Specifically, the gate injection circuit has 2 inputs: an info input $|\psi \rangle$, and a magic state $|T\rangle$. The output is the result state of a $T$ gate applied to $\psi$, $T|\psi\rangle$. In the event the output $T|\psi\rangle$ has a negative (inefficient) Wigner representation, we would then compress that qudit so that its state is shifted to the _nearest_ 0 Wigner representation. Your thought may be, well, the quantumness will be lost when you shift the output of the gate injection circuit to the nearest 0 Wigner state. But wait! 0 is neither a positive nor negative number, so we are not technically shifting it to a positive (weak and fast) state, but a neutral state -- by contrast, negative states are strong but seemingly slow. This is potentially an important distinction because while such 0 Wigner states are efficiently simulable, they _may also_ retain quantum power. The bottom line is we would have a circuit with Clifford gates, positive and neutral Wigner states (the latter sometimes called zero-sum-negative states), and unlimited use of magic states -- with the caveat that the output of the gate injection circuit can never be negative, and if so it must be shifted to the nearest 0/neutral Wigner state. **Is such a system universal for QC?**


title: Equations of motion of displacement field

text: We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\\} $$ Where $u_{ij} = (\partial_{i}u_{j} + \partial_{j} u_{i} )/2$. Where index $i = 1,2,3$ denotes the $x$ $y$ and $z$ axis and there is summation over repeated indices. This models a three dimensional lattice of connected springs in the continuum limit. I want to derive the equations of motion of this system and the dispersion relation. Using the EL equation I get: $$\mu\partial_{tt}u_{1} - \nu \partial_{11}u_{1} - \rho \partial_{12}u_{1} - \rho \partial_{13}u_{1} - \rho \partial_{21}u_{2} - \rho \partial_{31}u_{3} = 0. $$ And similarly for $u_{2}$ and $u_{3}$. Is this correct and if so, how can I solve these equations to get the equations of motion? Also, how do I find the dispersion relation for this system?


title: Is it possible to levitate a toy maglev train using only permanent magnets as long as the train stays in motion?

text: I am fascinated with magnets, and specifically the idea of magnetic levitation. I recently purchased a Levitron toy maglev train and have enjoyed finding the perfect sweet spot at which it levitates, and I understand that the Levitron does not violate Earnshaw's theorem because it is in rotating motion. I would like to build a toy maglev train that makes no contact with the track, or any guide rails, using only permanent magnets and their repulsive force. I understand that if I attempt to build a flat track with no guide rails, the magnets will not be able to stably levitate, and will shunt the train car to one side or the other. However, what if I built a trough, or a V-shaped track, with the train car sitting inside the track, and a corresponding v-shaped bottom, with repulsive magnets running the length of the track, inside the V? I realize that it wouldn't be stable while sitting still, but if the train were in motion, would that enable it to levitate stably? In my mind there should be a sweet spot where the train's motion overcomes any sideways shunting effects as it travels above the magnetic fields of the magnets in the track, while gravity's pull on the train car into the V would be negated by the repulsive force of the magnets. I haven't been able to find anyone who has tried this approach. Every toy maglev train that I've seen has either used guide rails or a superconductor. My gut tells me it's not possible, but I want to believe. :) Is a configuration like this possible? Is it possible to overcome the instability through one-directional motion?


title: Electron in the proximity of a magnetic monopole

text: I am puzzling about an exercise in the book "Electromagnetic Theory" by Ferraro (p.543). An electron (mass $m$, charge $-e$) in a monopole magnetic field $\vec{B}\left(\vec{r}\right)=g\frac{\vec{r}}{r^3}$ has due to the Lorentz force $\vec{F}_L=-e\vec{v}\times\vec{B}$ the following equation of motion: $$m\ddot{\vec{r}}=-\frac{eg}{r^3}\dot{\vec{r}}\times\vec{r}.$$ Since $\vec{J}=\vec{r}\times\vec{p}+eg\frac{\vec{r}}{r}$ is constant, the electron moves on the surface of a cone, which tip is in the origin at $\vec{r}=0$. Mathematically, $\vec{r}\cdot\vec{J}=egr=rJ\cos\vartheta$, thus the angle $\vartheta$ between $\vec{r}$ and $\vec{J}$ is constant. Now I use spherical coordinates: $$\vec{r}=r\mathbf{e}_{r} $$ $$\vec{v}=\dot{\vec{r}}=\dot{r}\mathbf{e}_{r}+r\dot{\vartheta}\mathbf{e}_{\vartheta}+r\dot{\varphi}\sin\vartheta\mathbf{e}_{\varphi}$$ $$\vec{a}=\ddot{\vec{r}}=\left(\ddot{r}-r\dot{\vartheta}^{2}-r\dot{\varphi}^{2}\sin^{2}\vartheta\right)\mathbf{e}_{r}+\left(2\dot{r}\dot{\vartheta}+r\ddot{\vartheta}-r\dot{\varphi}^{2}\sin\vartheta\cos\vartheta\right)\mathbf{e}_{\vartheta}+\left(2\dot{r}\dot{\varphi}\sin\vartheta+2r\dot{\vartheta}\dot{\varphi}\cos\vartheta+r\ddot{\varphi}\sin\vartheta\right)\mathbf{e}_{\varphi} $$ and obtain the following equations of motion (using $\dot{\vartheta}=0$): $$\ddot{r}=r\dot{\varphi}^2\sin^2\vartheta \qquad(1)$$ $$r\dot{\varphi}^2\cos\vartheta=\frac{eg}{mr}\dot{\varphi} \qquad(2)$$ $$2\dot{r}\dot{\varphi}+r\ddot{\varphi}=0 \qquad(3)$$ with the boundary conditions: $$r\left(0\right)=r_0\qquad\varphi\left(0\right)=0.$$ In order to solve (1) for $r\left(t\right)$, one has to use the fact that the kinetic energy $\frac{1}{2}m\vec{v}^2$ is conserved. This holds due to the vector potential $\vec{A}=\frac{1}{2}\vec{B}\times\vec{r}=0$. Therefore, $$\vec{v}^2=\dot{r}^2+r^2\dot{\varphi}^2\sin^2\vartheta\equiv u^2=\text{const}$$ Using this in (1) one gets the simplified differential equation: $$r\ddot{r}=u^2-\dot{r}^2\qquad(4)$$ Does anyone know how to derive the following solution of (4)? $$r\left(t\right)=r_0\sqrt{1+\left(\frac{ut}{r_0}\right)^2}\qquad(5)$$ Solving (2) for $\dot{\varphi}$: $$\dot{\varphi}=\frac{eg}{mr^2\cos\vartheta}$$ and inserting (5) I obtain: $$\varphi\left(t\right)=\frac{eg}{mr_0u\cos\vartheta}\arctan{\frac{ut}{r_0}}.$$ Now the solutions says: $$r\left(\varphi\right)=\frac{r_0}{\cos\left(\varphi\sin\vartheta\right)}.$$ But I get something similar, but not equal: $$r\left(\varphi\right)=\frac{r_0}{\cos\left(\frac{mr_0u\cos\vartheta}{eg}\varphi\right)}.$$ Has anyone an idea where I could have done something wrong?


title: Relative error of equivalent resistance of resistors in parallel

text: I just saw a formula in my book for relative error in equivalent resistance of two resistors connected in parallel. > $\frac{\Delta R}{R^2} = \frac{\Delta R_1}{R_1^2}+\frac{\Delta R_2}{R_2^2}$ How do I get derive this formula?


title: Photon as the carrier of the electromagnetic force

text: My physics background goes as "far" as reading popsci books on QM, Particle Physics, and Cosmology so pardon my ignorance in the below questions. I've read that the photon is the particle (quanta in QFT) exchanged between electrons, and that is how the electromagnetic force is explained in QFT.   1. How do we know about these virtual photon exchanges given the fact that we can't observe them directly? How did we arrive at " ** _electrons exchange virtual photons and that's the cause of the electromagnetic force between them_** " from merely observing electrons absorbing or emitting photons?    2. If electrons throw photons at each other doesn't that mean that they should only scatter (repel)? If that is so why do magnets and opposite charged particles attract ?   3. I've read that because photons don't posses electric charge therefore they can't exchange between themselves other photons so they can't collide (scatter) in the usual way, unlike the W+, W-, Z0, and gluons which posses the charge of the force they mediate. The only collision processes between photons that I've heard of are Delbruck scattering and the phenomenon of conjuring matter (1 electron , 1 positron) out of 2 photons that collide at a close angle between their trajectories. My third question is: Do these two rare phenomena of photon "scattering" happen here on earth naturally (no LHC or other particle accelerators), in the upper atmosphere, or only in the deep dark space? Thanks!


title: A sphere, a simple object?

text: In this video, the woman says that a sphere is a _pretty simple object_. What intrigues me is the use of a sphere for such a calculation. First of all, the sphere wouldn't be perfect as a perfect sphere doesn't exist in nature (first source of imprecision). Second, one will have to use π for calculating the volume. As π is irrational, one will have to use an approximation for it (second source imprecision). So why not using a simple cube? The shape would be closer to a cube due it's harder to make a sphere and it's volume would be calculated more precisely. Why they choose to do this calculation using a sphere?


title: Kinetic energy of a gas molecule

text: I am working through the derivation of the equation $$pV=\frac13mNC_{rms}^2,$$ but relating to this formula my textbook states that the kinetic energy of an average molecule of gas is given by $$KE=\frac12mC_{rms}^2$$ but how is this formula right since the mean squared speed ($C_{rms}^2$) is not the same as the average speed of the molecule squared, $v^2$, so why do we use the mean squared speed for kinetic energy and not the means speed squared? Is it just that it is easier to calculate?


title: Most efficient type of heat pump

text: Can I make a heat pump beat the Carnot efficiency? Why is the Carnot process the most efficient one? If I have a heatpump that is sphere shaped, and cascaded in layers like a onion can I beat Carnot efficiency? Heat would be transferred and concentrated from the outermost layer that is in contact with ambient air, and brought towards the center to heat a fluid.


title: Why don't gravitational waves free stream

text: Simply hot dark matter is not allowed due to free streaming. So do gravitational waves: a) free stream if not why not?-Surely they can since they are relativistic an weakly interacting. b) if they free stream is it allowed because they don't dominate the dark matter content?


title: WIMPs and Gravitinos

text: I was just wandering if a gravitino could be in the WIMP class, because they are both massive (admittedly WIMP's more so) and are both weakly interacting and predicted by R-parity. i.e Gravitinos are a sub class of WIMPs.


title: Forces acting on a point mass in a spinning rigid body

text: I have learned that all spinning objects will continue spinning even if no force is acting on it, and the tendency to do so is called moment of inertia. But I wonder about the fact that a single point mass in a spinning rigid body changes direction, even though no force is acting on it. How this is possible? This violates Newton's laws. How a point mass changes direction without force?


title: Running a stirling engine in reverse?

text: I just bought a small stirling engine which converts heat flow down a temperature gradient into circular motion of a little wheel. If I turn the wheel with my finger, will the engine generate a temperature gradient?


title: Accelerating onto and over inclined plane

text: Consider an RC car going off a jump. What angle is optimal to achieve the greatest distance? The height of each jump is the same. The angle of the jump and the related length of the jump changes. This is an experiment my son is doing for his 6th grade science fair. The RC car used is a hobbyist RC car with a top speed of approximately 30 mph. It has shocks, and the tires are rubber with foam inside. We've used a Netduino to control acceleration so that the each run is identical.   1. The optimal angle for ballistic trajectory is 45 degrees. This is based on his research as he hasn't quite reached calculus yet. http://en.wikipedia.org/wiki/Range_of_a_projectile   2. My understanding is that the Work to climb the inclined is identical for each jump. They are all the same height, and climbing them produces the same potential energy. Am I completely misunderstanding this?   3. The tires do not slip.   4. The car starts from 20 feet from the high end of the jump. Given this information, would the 45 degrees be the optimal angle of the jump? We haven't completed the test runs yet, but initial tests show the 45 degree jump performs poorly. We reduced the test speed such that the car does not scrape the bottom when it hits the ramp. Other than the trajectory and the Work required to climb the jump, what aspects are we missing? Where is the energy going when it hits the 45 degree jump?


title: Gravitational collapse and free fall time (spherical, pressure-free)

text: A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (This was originally from a multiple-choice exam - I solved the problem via dimensional analysis on the options then. I'm wondering how it might be solved directly now). The answer is $$t = \sqrt{\frac{3\pi}{32G\rho_0}} $$


title: Magnetic Induction - How would different coil configurations affect the outcome?

text: This is a branch question (from this question) related to magnetic induction: These deal with how different configurations of coiled layers might affect the outcome:   * If one were to bundle multiple layers of coiling to up the coils, would it be more effective to have each layer connected in parallel and running in the same direction? (example - 2 ins and 2 outs)   * If one were to coil 'loosely' such that the wire is wrapping around the coil at a 45 degree angle - would it generate less induction than the same number of coils tighter coiled at nearly a 90 degree angle (pretty sure this is a yes - but wondering if there is any silver lining in terms of resistance - perhaps with a deluge of parallel-connected layers))?   * In the event that two or more layers are connected in parallel - would the energy created across each layer likely be significantly less? In the event of (say) five layers of coiling connecting, each with differing currents (or voltages), how might the resultant current be combined once reunited (presuming a similar wire is receiving the energy)?  I am keenly interested in the variables at play being defined - as well as any information on how such might play out. Thank you for any guidance.


title: Index Notation Double Curl

text: My question is about Einstein notation. It does not matter the specifics of this example (the del operator could be another random vector), I just want to know if my assumption about notation is correct. I know that cross product or curl $$\vec b=\vec\nabla \times \vec V = \nabla_iV_j\epsilon_{ijk}\hat e_k$$ but when I take another cross product, what do I get? I believe that $$\left(\vec\nabla \times \vec V\right)\times \vec b = \left(\nabla_iV_j\epsilon_{ijk}\hat e_k\right)V_l\hat e_k\epsilon_{lkm}\hat e_m$$ which would then equal $$\nabla_iV_jV_l\epsilon_{ijk}\epsilon_{lkm}\hat e_m$$ Am I correct? If not, where did I go wrong?


title: Lippmann-Schwinger solution

text: What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\psi_k \rangle=|\phi_k \rangle+G_k V|\psi_k \rangle $$ Taking the inner product with $\langle\phi_{k'}|$ \begin{align} \langle \phi_{k'}|\psi_k \rangle&=\langle \phi_{k'} |\phi_k \rangle+\langle \phi_{k'} |G_k V|\psi_k \rangle \\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |V|\psi_k \rangle\\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |H-H_0|\psi_k \rangle\\\ &=\delta(k'-k)+g(E_k-E_{k'})\langle \phi_{k'} |E_k-E_{k'}|\psi_k \rangle \end{align} Basic algebra then gives us $$\langle \phi_{k'}|\psi_k \rangle[1-g(E_k-E_{k'})(E_k-E_{k'})]=\delta(k'-k)$$ and isolating the inner product on the left hand side gives us $$\langle \phi_{k'}|\psi_k \rangle=\delta(k'-k) [1-g(E_k-E_{k'})(E_k-E_{k'})]^{-1}$$ where $g(E_k- E_{k'})$ is the eigenvalue of $G_k$ resulting from hitting $\langle \phi_{k'} |$ on the left? This seems to suggest that the eigenstate $|\psi_k \rangle $ of the full Hamiltonian $H$ is the same as the eigenstate $|\phi_k \rangle$ of $H_0$, up to a proportionality factor, since $|\psi_k \rangle $ has no amplitude to be in $|\phi_{k'\neq k} \rangle$.


title: Electromagnet, ideal turns depending on ohm

text: More turns -> stronger field   more turns -> longer copper wire   longer copper wire -> more resistance(ohm)      at what turn does the resistance make the electromagnet weaker? - I want to make an ideal electromagnet.      (Sorry for being vague but I'm looking for some kind of formula or an example)


title: Equation of state not linear in energy or system size

text: Which of these two equations of state are valid? $$S_1 = L_0 \gamma (\theta E/L_0)^{1/2} - L_0\gamma\left[\frac{1}{2} \left(\frac{L}{L_0}\right)^2 + \frac{L_0}{L} - \frac{3}{2}\right]$$ $$S_2 = L_0 \gamma e^{\theta n E/L_0} - L_0\gamma\left[\frac{1}{2} \left(\frac{L}{L_0}\right)^2 + \frac{L_0}{L} - \frac{3}{2}\right]$$ where $\gamma$, $\theta$ are constants, $L_0$ is a function of $n$ (which is the mole number). $S$ has to be monotonically increasing and linear as a function of $E$ since it's extensive. Clearly $S_2$ can't fit that description, but $S_1 \propto \sqrt{E}$ which is not linear either, and it also scales that way for the system length too. This is exercise 1.2 from David Chandler's introduction to statistical mechanics book.


title: About the seesaw mechanism

text: I was reading about the seesaw mechanism in my Lecture notes and got a technical question. See for example http://www.lhep.unibe.ch/img/lectureslides/9_2007-11-30_SeeSawMechanism.pdf page 13. There it says than in the limit $M_R >> M_D$ and $M_L=0$ then the second mass is $m_2=M_D^2/M_R$. But when I apply the previous limit to the solution $$m_{1/2}=(M_L+M_R)/2 \pm \sqrt{(M_L-M_R)^2/4+M_D^2}$$ I get $m_2=-M_D^2/M_R$. What am I doing wrong here?


title: Units for physical constants

text: Someone told me that units for $G$ and $\epsilon_0$ (gravitational constant and Coulomb's constant) are placed there simply to make equations work dimensionally and that there is no real physical interpretation for them. Specifically he said $G$ was developed as the number such that when multiplied by the remainder of the expression for gravitational force creates the proper force and that its units are given to balance with the units of force. The same goes for $\epsilon_0$. Is this true?


title: light absorbing diodes characteristics

text: Recently, there was a publication regarding some new materials that would let thru light in one direction and either absorb it or reflect it in the other direction; the publication does not clarify on which one. It does not get as well into other important details as the bandwidth where the diode is effective. This is interesting because if this really does what they say it does, it is potentially a way to drive a spaceship without any propellant or propulsion energy expenditure. Assuming the effective bandwidth is not in some dark region of the cosmic spectra, you could conceive a radiation sail with this material, and reach significant speeds over a few years of acceleration. But for doing any concrete estimations, we need hard numbers on the transmittivity and dielectric eigenvalues of these things or something that would allow to calculate the expected impulse. Do anyone knows how to find more info on this material? that MIT article does provide a reference to the paper, but i don't have access to that journal gateway, and the abstract is really scant on the interesting details (probably because they are orienting its application for optical computing!)


title: Are there any well-known theories successfully unifying the inertial and gravitational mass?

text: From what little I know of general relativity, the equality of inertial and gravitational mass is an axiom of the theory. I suspect that this precludes GR from unifying them in the same sense as electric, magnetic and weak interactions are unified as the electroweak force. But since the two appear to be equal under staggeringly precise measurements, there must be some interest in constructing a theory of mass and motion which fundamentally unifies the two, that is, makes the separation nonsensical / impossible. Are there any particularly outstanding attempts in this regard, even if the resulting theories are known to fail to correspond to our universe in some other important way?


title: Inverse square law in 2+1 dimensional universe from a Yukawa coupling?

text: There is a nice result that in 3+1 space time, a Yukawa coupling leads to an inverse square law force as the mass of the scalar field goes to zero. I was wondering what the corresponding force in a 2+1 dimensional Universe would be? Geometrically, I would guess that it should be $1/r$ rather than $1/r^2$, but my explicit calculation for a Yukawa coupling gives that it is $1/r^2$ in 2+1 space as well. I was wondering if that was correct?


title: Concave Sphere Habitat characteristics

text: I'm writing a short story set in an artificial planet-sized sphere with an ecosystem in its inner surface, whose "gravity" is created through spinning. Energy sources aside, what other interesting physical characteristics should I take into consideration, apart from varying force levels as one moves relative to the direction of the spin (may be minute given the size of the structure), as well as increasing "weightlessness" as one gets closer to the axis? Also, with sufficient mass in its shell, what is the effect of (real) gravity on objects on the inner surface?


title: Can single DC 1.5V battery generates 1500 Watts power?

text: Suppose there is a DC $1.5\mathrm{V}$ battery connected with $1.5\times10^{-3}\mathrm{\Omega}$ resistor. Then the amount of circuit current is $I=V/R=10^3\mathrm{A}$. (according to Ohm's law) I know the formula $P=IV$ which is electrical power. Finally I got the electrical power of $1500 \mathrm{W}$ in this case. I can't believe this! How can single DC 1.5V battery generates 1500 Watts power? What am I missing!


title: Why does a ball bounce forever?

text: In short: Do bouncing balls keep bouncing forever? If not, does it have to do more with energy than velocity? > Learner.org: Energy > >> If energy is conserved, why do bouncing balls, pendulums, and other cyclic systems eventually stop moving? The energy doesn’t disappear but rather it is transferred to the surroundings, where it is unavailable to keep the cycle going. The result is that most objects moving in cycles eventually move less and less and then finally stop. But that wouldn't make sense mathematically: although the limit converges to zero, it doesn't fully ever approach it. Do the mathematical laws break down at the micro physical level much like they do in general relativity? Does the ball keep bouncing up and down but reaching some height in space $\epsilon > 0$?


title: What would happen to Earth if Saturn were to exit the Solar System?

text: Suppose Saturn simply vanished. How would that affect the rest of the Solar System, specifically Earth?


title: Next crescent moon?

text: How do I find the date of next crescent moon on wolframalpha or any other site which tells the appearance date time of next crescent on a specific location? _I am searching for it to look for 1st of Ramadan_


title: How to find the Andromeda galaxy without using a go-to telescope?

text: In other words, what is the proper technique (star-hopping or other?) in order to find and properly point a telescope to this target? Would a star atlas or other tool/reference help? Can I use the R.A. and Dec. coordinates to find such deep-space objects? I can recognize the constellations around which M31 is located, but I could never figure out a way to pin-point it. Any help or explanation is much appreciated. Thank you!


title: The Sun as a gravitational lens

text: Since the Sun is a gravitational lens with as focal length of 550 AU for visible light, with an immense amplification factor, shouldn't it light up objects hanging out there? We should get solar sails up there to finally boost them somewhere interesting, opposite to the star that is emitting the light. For neutrinos, I've heard that the focal length is 110 AU. Wouldn't a massive neutrino bombardement trigger interesting nuclear reactions (maybe that's not an astronomy question)?


title: Is the Milky Way significantly easier to see by astronauts?

text: On Earth the Milky Way appears very ghostly when looked at with the naked eye, somewhat easier to see when NOT looking directly at it. Is it substantially easier to see by astronauts in space or is it largely the same?


title: Are there any known neutron stars just below the Tolman–Oppenheimer–Volkoff limit?

text: Do we know any neutron stars just below the Tolman–Oppenheimer–Volkoff limit? And how are they different from most other neutron stars?


title: Have Sterling Engines Been Used In Outer Space?

text: I was reading about Stirling Generators having potential applications for power generation in outer space. Have Stirling engines ever been sent into space?


title: Density of stars near the center of the Milky Way

text: At night, I can look up and see the Milky Way across the sky. My question is, supposing our solar system was, instead of way out on an 'arm' of the galaxy, if we were near the galactic center. Would the night sky be much brighter?? If so, how close would we have to be to never have darkness of night?? Thanks!


title: Does the "Andromeda Paradox" (Rietdijk–Putnam-Penrose) imply a completely deterministic universe?

text: **Wikipedia article:** http://en.wikipedia.org/wiki/Rietdijk–Putnam_argument **Abstract of 1966 Rietdijk paper:** > A proof is given that there does not exist an event, that is not already in > the past for some possible distant observer at the (our) moment that the > latter is "now" for us. Such event is as "legally" past for that distant > observer as is the moment five minutes ago on the sun for us (irrespective > of the circumstance that the light of the sun cannot reach us in a period of > five minutes). Only an extreme positivism: "that which cannot yet be > observed does not yet exist", can possibly withstand the conclusion > concerned. Therefore, there is determinism, also in micro-physics. **Question:** I realize that this is a very philosophically-heavy topic, but at the same time, it is physical in the strictest sense (in that it's not a "What if" question, but more of a thought experiment, like the Twin Paradox, that follows the logical implications of relativity to some extreme scenario). So, setting aside matters of "free will" and whatnot, which I'm not interested in... If I understand this article correctly: the content of these papers implies a 100% deterministic universe — one in which the future of any object is "set in stone"; **but,** at the same time, this future is completely inaccessible to any observer that is in its past (i.e: impossible to predict even given initial conditions, because of small-scale uncertainty, chaos, etc.). Are the proofs in these articles are well-accepted? (Meaning that this "everything has already happened depending on the observer" universe is the only conclusion one can arrive at?). I was not aware of the Andromeda Paradox before. I can't imagine it being very useful, but it's interesting enough as a mind-bender that I'd've thought one would hear about it more often.


title: Why $\Delta x \Delta p_x$ for stationary states increase linearly with n?

text: **Harmonic Oscillator**   * $\displaystyle \Delta x\Delta p_x = \hbar (n+\frac{1}{2})$ **Particle in a box**   * $\displaystyle \Delta x\Delta p_x = \frac{\hbar}{2} \sqrt{(\frac{n^2\pi^2}{3}-2)}$ We notice that for small n the product is of the same order of magnitude as $\hbar$ and for large n it grows linearly with it:   * Is that behavior expected? if yes then why?    * Is this a general behavior for any stationary states of any system?


title: Factorization of fermionic scattering integral in 2d momentum rep

text: the scattering integrals for fermions involves both momentum ($k$) and energy ($k^2$) conservation and a nonlinear phase space factor of a distribution function $f(k)$. $$\begin{multline}I(k) = \sum_{k_1, k_2, k_3} \delta(k^2+k_1^2+k_2^2+k_3^2) \delta(k+k_1+k_2+k_3)\times \\\ \Bigl[f(k)f(k_1)\bigl(1-f(k_2)\bigr)\bigl(1-f(k_3)\bigr) - f(k_2)f(k_3)\bigl(1-f(k)\bigr)\bigl(1-f(k_1)\bigr)\Bigr]\end{multline}$$ In energy space, energy conservation is linear and powerful factorizations are possible to compute the integrals fast. In momentum space, the nonlinear energy conservation constraint complicates factorizations. Trivially, in 2 dim one can work in polar coordinates separating off the radial part. But has anybody seen a more efficient reduction of numerical complexity (e.g. mapping to FFT,...) Thanks for any hint!


title: Does the weight of an magnet raise when balancing an object?

text: I was wondering if say I use an magnet which is facing positive and then I expel an another magnet above it which is facing the positive with its positive side and I ensure the magnet does not rotate or change its face via say and small string so its always facing positive will face positive. As a result (I hypothesize) my both magnets attempt to repel and with this they create an force pushing down upon the bottom magnet due to repelling. The question here is while I am doing this experiment I use an weight measure to see if the weight increases and I would think it would increase but not sure, so can someone tell me if my hypothesis is correct?


title: Can the path of a charged particle under the influence of a magnetic field be considered piecewise linear?

text: Ordinarily we consider the path of a charged particle under the influence of a magnetic field to be curved. However, in order for the trajectory of the particle to change, it must emit a photon. Therefore, in principle, if we were able to view the path of the particle at sufficiently high resolution, would its path actually be polygonal, with a photon emitted at each vertex?


title: Is any percentage of the regular solar wind that penetrates into Earth's atmosphere able to reach Earth's surface?

text: Is any percentage of the regular solar wind charged ions able to penetrate the earth atmosphere? if so, can it be detected or measured on the surface of earth at sea level?


title: How capacitive is the human body?

text: Testing a capacitive screen, it's clear that it can tell the difference between a celery touching it (human holding it with an isolating material) and a human touching it. Touching it holding a celery, but no isolating material works too. So, what is the range of 'normal' capacitive potential of a human body? (I am not interested in capacitive screens, but rather on electrical properties of human bodies, however, the capacitive screen got me interesting on it). How could we measure it?


title: What libraries are available for solving problems at the college physics level?

text: Are there any libraries available for solving problems at the level of college physics? In particular, I'm interested in libraries for general purpose programming languages that can calculate results both numerically _and_ symbolically. Libraries which could be used for implementing something like this are SymPy (Python), GiNaC (C++), or SymbolicC++ (C++). I'm asking this question because I've started cooking up a C# library for symbolic computation and have used it to solve a few projectile motion problems, but would like to review other work out there. Here's an example of a program which solves such a problem. Here's what the program outputs when run: ![photo](http://i.stack.imgur.com/ZX14W.png) Any pointers or suggestions welcome. Thanks!


title: How long does it take a magnetic field to join both ends?

text: My question is: when a electromagnet is activated, the field must take time to form at both ends and stabilize. What is this time? Or, to put it differently, what is the speed at which the magnetic field forms?


title: What's the difference between an entangled state, a superposed state and a cat state?

text: 1) Can a state be entangled without also being a superposition? (Please give an example.) 2) Can a state be a superposition without being entangled? (Again, an example please.) 3) And what about a cat state? I am royally confused, and a little bit of googling didn't help.   This question is partly motivated by the recent news announcement (Nature, NYTimes) that ten billion entangled pairs have been created. 4) Are these spin pairs also in a superposition? 5) Does this count as a cat state?


title: Does any particle ever reach any singularity inside the black hole?

text: I am not a professional physicist, so I may say something rubbish in here, but this question has always popped in my mind every time I read or hear anyone speak of particles hitting singularities and "weird things happen". Now to the question at hand, please follow my slow reasoning... As far as I've learned, to reach a black hole singularity, you must first cross an event horizon. The event horizon has this particular property of putting the external universe on an infinite speed to the falling observer. Now due to the Hawking radiation, and knowing that the cosmic background radiation is slowly dimming, sooner or later every black hole in this particular instance of inflation we are living in will evaporate, according to an external observer of said black holes. This means that every black hole has a finite timespan, as long as this universe survives that long. Now if we go back to the falling observer, we had already established that such an observer would see the outside universe "speed up" infinitely. This means that when the falling observer "hits" the event horizon, he will (or it if we speak about particles, which is clearer in this case), be immediately transported in time towards the final evaporation moment of the black hole. Either this or the particle gets some weird treatment. My point is, such a particle never gets to the singularity, because it has no time to get to it. The moment it crosses the event horizon, the black hole itself evaporates. Where am I wrong here?


title: What do the modes of fermion fields look like?

text: A boson field can be understood as a collection of stationary modes (e.g. plane waves of various polarizations), and for each mode there is a quantum harmonic oscillator. If the QHO for some mode is in an energy eigenstate $|n\rangle$, we interpret this as meaning that the field constains $n$ particles in that mode. Since a QHO has infinitely many energy eigenstates, a boson field can contain arbitrarily many particles in any mode. Can a fermion field also be understood as a collection of modes? What do the individual modes look like? Because of the Pauli exclusion principle, I presume that each mode is a two-state system with an empty and a filled state. Are these any different from other two-state quantum systems, e.g. a single electron spin?


Just a simple question. Does a muon detector on Earth's surface correctly measure the mean lifetime of a muon? I would think the answer is no because most muons detected are created about 15 km above Earth's surface, so on average they will have survived for some amount of time calculated using special relativity before reaching the detector. The detector I'm talking about is a scintillator, so it detects when the muon enters the detector with a flash, then if the muon decays it detects another flash. The time between flashes is said to be the muon's lifetime. Is this muon's lifetime the machine giving me the actual lifetime (2.2$\mu s$), or just the lifetime of it on Earth's surface? here is a plot of my data showing a histogram of decay times ![enter image description here](http://i.stack.imgur.com/Oc1yB.png)


title: Concentration of fluid pumped through a tube?

text: I have a bottle filled with a fluid A, and a tube of volume $V_0$ filled with a fluid B leading away from it. When I now start pumping fluid A out of the bottle, I am interested in the concentration $c(V)$ of fluid A in fluid B at the end of the tube after the volume $V$ exited the tube. In my second situation (I am afraid there are two), after having pumped $V_0$, a valve at position $V_1$ switches and the pump is reversed. The mixed fluids are now pumped into a second tube of volume $V_2$. Again, I'd like to know what drips out of there. **Situation 1**               |    A    |     |         |      V0     |      ====================    <- B     |         |     |         |     |_________|      **Situation 2**               |    A    |     |         |          V0     |      =========++==========    <- B     |         | V1  ||     |         |     || V2     |_________|     ||                           ^- B      I've tried using the Hagen-Poiseuille equation (or its proof as given in the Wikipedia), but that grows ugly fast. Is there a simple(r) way to approach this?


title: How does particle physics compare to other fields of physics?

text: Okay, so I just read through huge sections of http://gordonwatts.wordpress.com, which I found quite interesting. There are some questions that really came to my mind. Do a lot of particle physicists use windows computers? (relative to other physicists?) And why? In most other academic departments, the professors seem to gravitate towards Linux/MACs. But Watts writes a lot of his programs for Windows only, which shows that there's definitely a lot of people who use it there. Does particle physics research always stall until we get the sexiest new supercollider? (like, is it pretty much stalled until CERN starts putting out substantial output?) We haven't heard of anything from particle physics for quite some time (of course, we will soon, from CERN). But from the likes of particle physics blogs, I'm just getting the impression that everyone is talking about what's possible, rather than what results came out. It seems that condensed matter physics, biophysics, and astrophysics are the fields that are getting the most press. If you're an experimental particle physicist, does that mean that you just use computational techniques to reduce/analyze the data now? In other words, could you be an experimental physicist without even knowing how to use laboratory apparatus? (just like how you can be an astrophysicist without knowing how to use a telescope these days). Thanks!


title: Algorithm to simulate impulse response of a solid?

text: There are well-documented methods of calculating the impulse response of a room (ex. image method, ray-tracing method); however, I have not been able to find anything similar for the impulse response of an arbitrary solid given its size, density, etc. Can anyone point me in the right direction?


title: Tachyon vertex operator (Polchinski's book)

text: * I would like to know how does Polchinski in his book "derive" what is the "tachyon vertex operator" (..as say stated in equation 3.6.25, 6.2.11..) I can't locate a "derivation" of the fact that $:e^{ikX}:$ is the tachyon vertex operator.  (..I understand that it follows from some application of the state-operator map but I can't put it together..)   * And then what is the meaning of the ``higher vertex operators" - which are of the form of arbitrary number of either operators of the above kind or the derivatives of $X$ w.r.t either $z$ or $\bar{z}$. (..like in equation 6.2.18..)


title: What happens if you x-ray an x-ray?

text: The title largely sums up my question, what does happen if you either x-ray an x-ray, or point two x-ray generators at each other?


title: How do collisions of fundamental particles produce different fundamental particles?

text: When considering fundamental particles as waves in fields, it seems like any collision of two particles of some fundamental type could only create energy within that type's field. Why do we expect that this energy is transferred to a different field, creating a different fundamental particle? What is the mechanism of this energy transfer and what, if any, is the physical interpretation of the energy's "path"?


title: Units in time dilation calculation

text: I am currently working through Brian Greene's "World Science U" course on special relativity, and I have a question regarding one of the calculations performed for an exercise on time dilation (MODULE 12: Time Dilation: Examples - Problem 1). The question is as follows: > Imagine that today is your birthday and you decide that you want to > celebrate your next birthday on planet Zaxtar, 100 million light-years away. > How fast would you need to travel if you leave immediately? I have no difficulties with the setup of the equation: If $t_E$ is the time it take the ship to travel to planet Zaxtar from the perspective of someone on Earth, $t_S$ is the time it takes the ship to travel to planet Zaxtar from the perspective of someone on the ship, and if the ship moves at a constant velocity $v$, then we should simply solve the equation: $$ t_S =t_E /\gamma \implies 1 = \frac{10^8}{v}\sqrt{1 - \frac{v^2}{c^2}}. $$ Here is my difficulty: If one solves this equation for $v$, then we see that: $$\begin{align} v^2 = 10^{16} (1 - v^2/c^2) &\implies v^2 = 10^{16} - 10^{16} v^2/c^2 \\\ &\implies v^2 \left( 1 + \frac{10^{16}}{c^2}\right) = 10^{16} \\\ &\implies v^2 = \frac{c^2 \cdot 10^{16}}{c^2 + 10^{16}} \\\ &\implies v = c \frac{10^8}{\sqrt{10^{16} + c^2}}. \end{align}$$ However, at the beginning of his solution (video available here) equation, Brian sets $c = 1$ by using appropriate units: $c = 1$ light-yr/yr. He proceeds to solve for $v$, getting: $$ v = \frac{10^8}{\sqrt{10^{16} + 1}}, $$ and then he argues using dimensional analysis, that we should have $$ v = c \frac{10^8}{\sqrt{10^{16} + 1}}, $$ which follows since the units for $10^8$ and $\sqrt{10^{16} + 1}$ are light-yr, so that they cancel. My question is this: what goes wrong with the first method used to solve the problem? It is clear that if $c = 1$, then trivially $c^2 = 1$, but where is the mistake regarding the units? It seems that the units are not quite right due to the fact that $10^{16}$ and $c^2$ have different units.


title: What are orbifolds and why are they useful and interesting for physics?

text: Just what the title says.   1. What's the basic definition of an orbifold?    2. How do they arise in physics and why are they interesting?


title: What happens to this energy of the capacitor?

text: Let's say you have a charged, isolated parallel plate capacitor with a certain voltage. My book says that if you insert a dielectric, the voltage drops. However, that would require the energy between the plates to go down. What is this energy converted to?


title: Introduction to Anderson localization

text: I find Anderson's original paper too terse. I am looking for something that introduces me gently to the subject so that I can understand Anderson's paper and other literature. What references are out there that introduce Anderson localization? Anything will work for me: lecture notes, review papers, introductory papers ...


title: Determine the distance from time-velocity graph

text: A small car that we made in class was travelling and its time was recorded at 1,2,3, and 4 meters. The points in the form (seconds, meters) are as follows:               pt1 = (2.13,1)     pt2 = (3.48,2)     pt3 = (5.09,3)     pt4 = (6.75,4)      The question is to determine the distance traveled after 3.48 s, from the velocity-time graph. I divided the graph into three parts and multiplied the time by the velocity for each section. ![enter image description here](http://i.stack.imgur.com/eFExR.png)               A1 = (2.13)(0.46948) / 2 = 0.4999962 m     A2 = (3.48-2.13)(0.74074-0.46948) / 2 = (1.35)(0.27126) / 2 = 0.1831005 m     A3 = (0.46948)(1.35) = 0.633798 m          Total A = 0.4999962 m + 0.1831005 m + 0.633798 m = 1.3168947 m      I think this is wrong because in the point list, the distance traveled at 3.48 seconds is 2 m.


title: Integration by parts to derive relativistic kinetic energy

text: I have come across a weird integration during derivation of relativistic kinetic energy. Our professor states that i can get RHS out of LHS using integration by parts: $$ \int\limits_0^x \\! \frac{d}{d t}\Big[ mv \gamma(v)\Big]\,\, d x = v \\!\cdot\\! mv \gamma(v) - \int\limits_0^v \\! m v \gamma(v)\, dv $$ * * * I know a standard formula for integration by parts which i wrote below, but I don't know how to choose $\frac{dg}{dx}$ or $f$ in my case. $$ \boxed{\int\\! \frac{dg}{dx} f\,\, d x = f \\!\cdot\\! g -\\!\\! \int\\! \frac{df}{dx}~g\,\, d x} $$ How did my profesor do this? Any help will be appreciated.


title: Simple QM question about Sy matrix

text: Given a spin 1/2 particle in state $|\alpha\rangle=\begin{bmatrix}a \\\b\end{bmatrix}$, what is the probability of it being measured in the $S_{y+}$ state. Is this equivalent to, if $S_y$ is measured on this particle, what is the probability the result being $\hbar/2$? I think it's supposed to be like $|\langle S_{y+}|\alpha\rangle|^2$ right? And $|S_{y\pm}\rangle=\frac{1}{\sqrt{2}} \begin{bmatrix}1 \\\\\pm i\end{bmatrix}$ But then I get $a^2/2 +b^2/2$ for both. That doesn't make sense. What am I doing wrong?


title: Flow of supercurrent in a superconductor

text: I have two questions one practical and one theoretical. Even though I have a decent understanding of superconductivity both phenomenological as well as theoretical (i.e. BCS), some things just slipped through the cracks. First question: Say you have a decent quality superconducting ring completely isolated from external magnetic fields. Then a supercurrent is supposed to flow (with no voltage applied) which can can last for years without much dissipation. Will this current just automatically set in the moment I cool my sample below $T_c$? Or do I have to apply an external perturbation for a _short_ duration, remove the perturbation, and let the current go around the ring indefinitely? If the latter is true then what is this perturbation? It can't be voltage (can it?) since no voltage is required to make the current flow. If it is voltage (or any other perturbation) then how do I apply it? Now, the second question is a bit theoretical. It even (I think) _superficially_ contradicts what I described above. Since cooper pairs (in at least $s$-wave pairing) are formed by $\psi_{\mathbf{k},\uparrow}$ and $\psi_{-\mathbf{k},\downarrow}$ their net momentum is zero. How do they give rise to a supercurrent then? According to BCS they're not supposed to move, right? Can someone please help me reconcile this?


title: What does it mean that an inspiral is 'adiabatic'?

text: Binary systems emit gravitational radiation. This causes the system to lose energy, which results in a shrinking of the semi-major axis. I have read on countless occasions that this 'inspiral' is _adiabatic_ (here for example). What does this mean that the shrinking is adiabatic?


title: What is the period of a physical pendulum without using small-angle approximation?

text: What is the expression for the period of a physical pendulum without the $\sin\theta\approx\theta$ approximation? i.e. a pendulum described by this equation: $$ mgd\sin(\theta)=-I\ddot\theta $$ **Motivation for my question:** I'm asking this because I had an assignment in which I had to measure the period of oscillation of a physical pendulum and we always had to release it precisely from the same angle. So I'm wondering whether that precaution is irrelevant because it feels to me that the measurement error and error created by ignoring air drag is more significant than those that would emerge by not caring about being precise about the initial angle, as long as it is smaller than circa 30°. Am I right or wrong?


title: Slow light and gravitational lensing

text: It has been proposed that the Sun's gravitational lens be used to observe distant objects, but according to relativity a spacecraft would need to be 550 AU away in order to take advantage of the effect. Would it be possible to instead slow down the light while it is being deflected using, for example, a Bose-Einstein condensate such as a large quantity of very cold sodium placed into orbit around an asteroid? An answer to this question mentioned that it is useless to do the same thing using an ordinary material with a high refractive index because of dispersion, but as I understand it the mechanism for producing slow light changes the group velocity, which is the speed at which energy is carried, instead of the phase velocity.


title: Hydrogen ionization energy

text: Proton and electron have a distance equal to the Bohr radius apart in the hydrogen atom. Knowing this, what's the ionization energy of the atom? So we know $U = -kq^2/a_B$, but this is potential energy, not ionization. How should I proceed?


title: Null Geodesics in Einstein Universe

text: I am currently taking a course in General Relativity, and I've hit a bit of a roadblock with a homework assignment. We are given the metric for Einstein's universe to be (forgive me, this is meant to be formatted for LaTeX, but I'm new at this whole stackexchange business) $ds^2 = c^2 dt^2 - \frac{1}{1-kr^2} dr^2 - r^2 d{\theta}^2 - r^2 \sin^2{\theta} d{\phi}^2$ and are asked to obtain the null geodesic equations. I understand that $ds = 0$ along null geodesics, and I am familiar with the equation forms involving the Christoffel symbols. What confuses me is, how are we to specify what we take our $t, r, \theta$, and $\phi$ derivatives with respect to? If I treat it like when we worked through a non-null case (we used the Schwarzschild metric in class), then I feel like I'm either dividing by zero or doing some serious hand-waving, neither of which is appealing. A couple of caveats: I have read in other textbooks that another method of working through this problem is using Killing vectors... we weren't taught that method, and I'd prefer to use the technique developed in class. If anyone can point me in the right direction, that would be greatly appreciated.


title: In superluminal phase velocities, what is it that is traveling faster than light?

text: I understand that _information_ cannot be transmitted at a velocity greater than speed of light. I think of this in terms of the radio broadcast: the station sends out carrier frequencies $\omega_c$ but the actual content is carried in the modulated wave, the sidebands $\omega_c\pm\omega_m$. The modulation envelop has its group velocity and this is the speed at which information is being transmitted. We also know, for example, that x-rays in glass have a phase velocity which is greater than the speed of light in vacuum. My question is, what _exactly_ is it that is travelling faster than the speed of light? **EDIT** : I know it couldn't be anything physical as its superluminal. My problem is, what is it that has absolutely no information content yet we associate a velocity with it.


title: What's the shocks mach number in this specific problem?

text: The shock wave I'm interested in occurs for a special Riemann problem, known as **1D Noh's problem**. It means basically that two streams of air (equal density) are colliding at $x = 0$ for $t = 0$; I will show my calculation and hope that someone could verify or correct it; I'm very new to the concept of shocks and maybe I got something wrong. The initial values are:   $$\rho(x, t = 0) = \rho_0$$ $$v(x, t = 0) = \begin{cases}v_0 & {\rm for\,} x < 0 \\\ -v_0 &{\rm for}\, x > 0\end{cases}$$ At a later time $t_1$, we find: $$ \rho\left(x,\,t_1\right)=\begin{cases}3\rho_0&{\rm for}\,|x|<x_0 \\\ \rho_0 &{\rm otherwise}\end{cases} $$ $$ v\left(x,\,t_1\right)=\begin{cases}0&{\rm for}\,|x|<x_0 \\\ -v_0 &{\rm for}\,x>x_0 \\\ +v_0&{\rm for}\,x<x_0\end{cases} $$ This picture clarifies the problem: ![enter image description here](http://i.stack.imgur.com/DgMpr.jpg) (source) One shock wave is propagating to the left, one to the right. It is sufficie nt to look at one of them and I'll pick the right one. In order to apply the jump conditions, I'll write down what I consider the left and the right values (note that: $$\rho_l = 4 \rho_0 \\\ \rho_r = \rho_0$$ When it comes to the velocities, there is the question which frame of refer ence to choose. Of course, in the rest frame of the "tube", we have $$v_l = 0\\\ v_r = -v_0$$ But normally we will stay in the rest-frame of the shock wave, right? So if $v_{\rm shock}$ is the velocity of the shock wave, we find $$v_l = -v_{\mathrm{shock}}\\\ v_r = -v_0 - v_{\mathrm{shock}}$$ Now I have two different approaches how to go on. With $$\rho_l v_l = \rho_r v_r$$ $$v_l = - v_{\mathrm{shock}} = \frac{\rho_l v_0}{\rho_l - \rho_r}$$ Now I want to calculate the pre-shock mach number. This is: $$M_l = v_l/c_{s,l}$$ where $c_s$ is the speed of sound. In my given example (not the same as in the picture), $\rho_0 = 1$, $v_0 = 3$ and $c_{s,l} = c_{s,r} = 2$; So my result is: $$M_l = 0.5$$ The other approach is to use the formula $$\frac{\rho_l}{\rho_r} = \frac{v_l}{v_r} = \frac{(\gamma +1)M_l^2}{(\gamma -1)M_l^2 + 2}$$ I should obtain the same value for $M_{l}$, right? I would be grateful for any clarification!


title: How to send a spacecraft to the moon without any acceleration

text: I have two (maybe equal) question   1- Is this possible to send a spacecraft to the moon without any acceleration(Constant speed)?   2-Is this possible to throw a ball on the top of tree without any acceleration(Constant speed)?      Thanks


title: What cosmic event would cause Sun rising from the west?

text: I'm wondering what kind of astronomical or geological events would cause the reversal of Earth's rotation. For instance, Is a meteorite passing very close to Earth able to reverse its rotation? Can the presence of Earth inside extreme magnetic field cause such event?


title: quantum field theoretic models of decoherence

text: I am interested in whether there is a field theoretic description (there is, so what is it?) of the tensor product (aka density matrix) model of open quantum systems. In particular, I am interested in how QFT might express the model of decoherence where the environment has some probability per time of "measuring" the state. For example, a qubit has classical states 0 and 1, which form a favored basis of the quantum Hilbert space. Let $A$ be an observable with this as eigenbasis and distinct eigenvalues. A photon interacting with the qubit causing it to decohere with probability $p$ can be modeled by saying the photon measures $A$ on the qubit with probability $p$. In other words, we have the unitary evolution $|0\rangle_S \otimes |un\rangle_E\rightarrow\sqrt{1-p}|0\rangle_S\otimes |un\rangle_E+\sqrt{p}|0\rangle_S\otimes |0\rangle_E$ $|1\rangle_S \otimes |un\rangle_E\rightarrow\sqrt{1-p}|1\rangle_S\otimes |un\rangle_E+\sqrt{p}|1\rangle_S\otimes |1\rangle_E$, where $|un\rangle_E$ is the state of the environment $E$ which nows nothing about the qubit $S$, and $|0\rangle_E$ is the state that knows the qubit is in state 0, and so on ($|un\rangle_E, |0\rangle_E,$ and $|1\rangle_E$ are all orthogonal). This whole model seems to conceptually rest on the environment's photon being some discrete quantity. The thing is, if the photon is really just an excitation in the all-permeating electromagnetic field, it is more like there is a little bit of interaction between the environment and system all the time, not just at Poisson-random times. It is easy to change the model above to have the probability instead be a rate write down a time evolution. It seems from the field theory perspective that this is more conceptually accurate. _Can one intuit (or calculate!) what this rate is from the electromagnetic field?_


title: Can an arbitrary RLC-circuit network have non-minimum phase zeros?

text: I am working with certain input-output maps that can be thought of as large RLC-networks. I thought maybe this might be a place to get some thoughts/ideas/answers. My basic question is, given some large connected RLC network (all linear and ideal elements) and two ports in the network, say 'a' and 'b', can the transfer-function between the ports have non-minimum phase zeros? More generally, is it true that every transfer function representing an RLC- circuit network is minimum phase? I suspect the answer is true, but I am having a hard time proving it. Thanks!


title: Two ways of calculating the expectation value of momentum

text: The expectation value of momentum is given by: $$ \langle p\rangle = \int_{-\infty}^{\infty}\psi^{*}(x)\left(-i\hbar\frac{\partial}{\partial x}\right)\psi(x)dx $$ How can I show that the above expression is equivalent to this? $$ \langle p\rangle = \int_{-\infty}^{\infty}p|\tilde\psi(p)|^{2}dp $$ I have tried to use that $$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde\psi(p)e^{ipx / \hbar}dp$$ and $$\psi^{*}(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde\psi^{*}(p)e^{-ipx / \hbar}dp$$ Then $$ \langle p \rangle = \int_{-\infty}^{\infty} \left[\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde\psi^{*}(p)e^{-ipx / \hbar}dp \right ]\left(-i\hbar\frac{\partial}{\partial x}\right )\left[\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\tilde\psi(p)e^{ipx / \hbar}dp \right]dx$$ $$=\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}\left [ \left (\int_{-\infty}^{\infty} \tilde\psi^{*}(p)e^{-ipx / \hbar}dp \right) (-i\hbar) \left (\int_{-\infty}^{\infty} \frac{\partial}{\partial x}\tilde\psi(p)e^{ipx / \hbar}dp \right)\right]dx$$ $$=\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}\left [ \left (\int_{-\infty}^{\infty} \tilde\psi^{*}(p)e^{-ipx / \hbar}dp \right) (-i\hbar) \left (\int_{-\infty}^{\infty} \frac{ip}{\hbar}\tilde\psi(p)e^{ipx / \hbar}dp \right)\right]dx $$ But I don't know if this is the right approach or if I'm doing the right thing.


title: No-answers-found Li-Ion battery questions

text: **Background** Ive spent almost 7 hours researching about li-ion batteries because my Galaxy S2 only lasts 2 hours with all turned off and only 40 apps installed (removed system apps and bloatware) and screen brightness below lowest normally possible in settings (I use ScreenFilter and set its opacity to >80% which means the slider is at less than 20% and I can barely see anything) when playing not CPU stressing game or just installing and removing apps and setting settings! **Questions not answered yet after almost 7 hours of searching** **After putting battery in freezer overnight** (in ziplock bag to avoid moisture (according to sites battery loses >20% Capacity in >3 months to a year) ** does the rapid heating up cause by turning phone on and games harm battery?** **Exactly how much is it safe to discharge?** (Ive found to avoid full discharge on several sites but no site says exactly how much of a discharge isnt harmful except about 20% but with no sources to back it up and I dont know how accurate is that "about") Why does my phone discharge so fast? (2hours even when not gaming; see background)


title: Phonon scattering process in raman spectroscopy

text: I am trying to understand diagrams that involve showing phonon scattering processes that contribute to raman spectroscopy peaks. For example, I drew the one at the bottom of this post. This is supposed to the scattering process responsible for the D peak in graphene. I'm trying to understand exactly what the arrows represent. At initial glance it looks almost as if the phonon scatters and increases energy of magnitude $E_L$ and transitions to another branch, then scatters again, losing energy of magnitude $D$, which is somehow indicative of the D peak? ![enter image description here](http://i.stack.imgur.com/dXi9j.png) Thanks!


title: Getting nonphysical results when solving for the index of refraction of a slab?

text: I'm trying to computationally find the refractive indices (real and imaginary) for a thin slab suspended in air (so the only indices to deal with are air and my material's). I've experimentally taken transmission and reflection measurements of intensity in a certain wavelength range. From a textbook chapter I read, $$r_{ij}[n_i, n_j]= (n_i - n_j)/(n_i + n_j)$$ $$t_{ij}[n_i, n_j]= (2 n_i)/(n_i + n_j)$$ $$r = (r_{ij}[n_1, n_2] + r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})$$ $$t = \sqrt{n3/n1} (t_{ij}[n_1, n_2] t_{ij}[n2, n3] e^{ 2i \pi n_2 d/\lambda})/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \pi n_2 d/\lambda})$$ and $$T = |t|^2$$, $$R=|r|^2$$ for transmission and reflection of a plane wave through a 'slab' (though I may be missing a factor of 2 in a couple places. I've used Mathematica style brackets for the functions). From here, $T$ and $R$ are both functions of $n_1$ (the air, ~1) and $n_2$ my material ($n_3$ is air again, here). So mathematically I should be able to figure out $n_2$ for each $T$ and $R$ pair, at each wavelength. So I've done this. I have Mathematica (MM) find the $n_2$ that minimizes $$\sqrt{(T-T_{exp})^2+(R-R_{exp})^2}$$ at each wavelength. Then, to see how close it is, I plug $n_2$ back into the original equation and plot it against my experimental results to see how close they match up. The problem is, they're matching up incredibly well -- but the results I'm getting for $n_2$ aren't realistic (more specifically, the real part of $n_2$ is negative, and this is not a negative refractive index material...). Here is a graphical example of what I mean (the horizontal axis is wavelength, in units of microns. The vertical axis is unitless for all): ![expT](http://i.stack.imgur.com/ziXv4.png) ![ns](http://i.stack.imgur.com/VhNxv.png) And here I've plotted the values of $T$ and $R$ from my computed $n_2$'s (green and black) over the first graph. As you can see, they're so similar you can't tell, except at the leftmost tails. ![exptheoryT](http://i.stack.imgur.com/xNzzQ.png) What could be going on? One possibility is that though my solution for $n_2$ gives very close values for $T$ and $R$, there are very different values of $n_2$ that give even closer values. Someone I talked to told me that the plane wave model doesn't always apply at some scales -- which blew my mind because I have always seen it used. He said it was the far field solution to dipole radiation but that it might not apply at my length scale. Could anyone verify or refute this?


title: How to estimate the Kolmogorov length scale

text: My understanding of Kolmogorov scales doesn't really go beyond this poem: > _Big whirls have little whirls that feed on their velocity, and little > whirls have lesser whirls and so on to viscosity._ \- Lewis Fry Richardson Th smallest whirl according to Wikipedia would be that big: $\eta = (\frac{\nu^3}{\varepsilon})^\frac{1}{4}$ ... with $\nu$ beeing kinematic viscosity and $\epsilon$ the rate of energy disspiation. Since I find no straightforward way to calculate $\epsilon$, I'm completely at loss at what orders of magnitude to expect. Since I imagine this to be an important factor in some technical or biological processes, I assume that someone measured or calculated these microscales for real life flow regimes. Can anyone point me to these numbers? I'm mostly interested in non-compressible fluids, but will take anything I get. Processes where I believe the microscales to be relevant are communities of synthropic bacteria (different species needing each others metabolism and thus close neighborhood) or dispersing something in a mixture.


title: Calculating acceleration increase

text: ^     |     |     |     |                   +     |                  +|     |                +  |     |              +    |     |            +      |     |          +        |     |        +          |     |      +            |     |    +              |     |  +                |     |+                  |     +----------------------------->     o                  x1      (X axis is distance and Y is acceleration) Excuse my crappy ASCII graph but I can't post images so please imagine that the line made of + is actually a concave curve My question is: knowing x1 how would I calculate the acceleration increase needed to get from O to x1 without overpassing x1?


title: Why is water not used as a dielectric in the condenser?

text: The dielectric constant of water is very high. Then why is it not used as a dielectric in the condenser?