title: Relationship between Schrodinger equation and string/membrane

text: In Sakurai's _Modern Quantum Mechanics_ (2nd ed) p.99, he says > We know from the theory of partial differential equations that ( _time- > independent Schrodinger equation_ ) subject to boundary condition > ($\psi(\mathbf{x}')\rightarrow 0$ as $|\mathbf{x}'|\rightarrow\infty$) > allows nontrivial solutions only for a discrete set of values of $E$. It is > in this sense that the time-independent Schrodinger equation yields the > quantization of energy levels. Once the partial differential equation ( > _time-independent Schrodinger equation_ ) is written, the problem of finding > the energy levels of microscopic physical systems is as straightforward as > that of finding the characteristic frequencies of vibrating **strings or > membranes**. Although I've found a paper (http://www.scientificexploration.org/journal/jse_21_1_hocking.pdf) about this, the relationship is still not very explicit. So would you please explain briefly about the relationship between the Schrodinger equation and string/membranes? Many thanks.


title: Festive physics: gold flake vodka

text: I have a bottle of vodka that has a load of gold flakes suspended in it. It has been sat still for over 24 hours and the flakes are all still suspended within the liquid: they have not risen to the surface or sunk to the bottom. Any ideas as to the physics behind this?


title: Electric field in conducting media

text: I know that the electric field is zero inside a perfect conductor. However, I am here discussing non-perfect conductors. So let's suppose we have a conducting medium with conductivity $\sigma_1$ and electric permittivity $\epsilon_1$. Another conducting medium is enclosed by the first medium and this one has conductivity $\sigma_2$ and electric permittivity $\epsilon_2$. Now both media are subjected to an external uniform electric field. The question is what is the electric field inside the first medium and what is the electric field inside the second medium. My understanding is that a steady- current is induced that flows in both media but I'm little confused and would appreciate some help.


title: s-wave, p-wave or d-wave collisions in scattering theory

text: In scattering theory, what is a good intuitive picture to think of s-wave, p-wave or d-wave collisions ? What is their importance and what are the examples where a particular one is assumed to be the only important one and rest are taken as negligible ?


title: How long until we fall into the Sun?

text: As a planet moves through the solar system, a bow shock is formed as the solar wind is decelerated by the magnetic field of the planet. Presumably the creation of this shock wave would cause drag on the planet, certainly in the direction of orbit but possibly rotation as well. Is there an estimate for the amount of drag on the Earth as it orbits the Sun? Based on the drag, how long would it take before the orbital velocity slows to the point that we spiral slowly into the Sun? Would any planets fall into the Sun prior to the Sun expanding into a Red Giant, gobbling them up?


title: Aerofoil Theory Project

text: I'm doing a project (dissertation) on the mathematics of Aerofoil Theory. I wonder if I could get some advice on a possible structure. I'm new to fluid dynamics, so it's quite hard to know where to start. So far I've looked at different types of flows; streamlines and velocity potentials. I've also covered a fair amount of complex variable theory - and can (loosely) see how conformal mappings are used to map for instance a cylinder to an aerofoil shape using Joukowski's transform. I would guess from here, I would want to look at Bernoulli's equation for lift and start looking at modelling an example. It would be useful to know if this is the right kind of structure to follow or if I'm approaching this the wrong way! So any help would be appreciated. Thanks!


title: About the most general (diff$\times$Weyl)-invariant and Poincare-invariant form of action

text: I have a question about the most general (diff$\times$Weyl)-invariant and Poincare-invariant form of action. In Polchinski's string theory p15, there is an action for manifold without boundary $$S_{\mathrm{p}}'= - \int_M d \tau d \sigma (-\gamma)^{1/2} \left( \frac{1}{4\pi \alpha'} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu} + \frac{ \lambda}{4\pi} R \right) \tag{1.2.33}$$ It is said > This is the most general (diff$\times$Weyl)-invariant and Poincare-invariant > action with these fields and symmetries. My question is, how to prove Eq. (1.2.33) is the most general one (without boundary)? Not saying what else could it be...


title: How do I show the existence of a conserved ghost number with BRST in bosonic string theory?

text: I have three questions about the BRST symmetry in Polchinski's string theory vol I p. 126-127, which happen together Given a path integral $$ \int [ d\phi_i dB_A db_A d c^{\alpha}] \exp(-S_1-S_2-S_3) \tag{4.2.3}$$ with $$ S_2 =-iB_A F^A (\phi) \tag{4.2.4}$$ $$ S_3 = b_A c^{\alpha} \delta_{\alpha} F^A(\phi) \tag{4.2.5} $$ the BRST transformation $$ \delta_B \phi_i = -i \epsilon c^{\alpha} \delta_{\alpha} \phi_i \tag{4.2.6a} $$ $$ \delta_B B_A=0 \tag{4.2.6b} $$ $$ \delta_B b_A = \epsilon B_A \tag{4.2.6c} $$ $$ \delta_B c^{\alpha} = \frac{i}{2} \epsilon f^{\alpha}_{\beta \gamma} c^{\beta} c^{\gamma} \tag{4.2.6d} $$ It is said > There is a conserved ghost number which is $+1$ for $c^{\alpha}$, $-1$ for > $b_A$ and $\epsilon$, and 0 for all other fields. How to see that? > The variation of $S_2$ cancels the variation of $b_A$ in $S_3$ I could get $i B_A \delta F^A$ in $\delta S_2$ and $(\delta_B b_A) c^{\alpha} \delta_{\alpha} F^A= \epsilon B_A c^{\alpha} \delta_{\alpha} F^A $ in $\delta S_3$. But there is a $c^{\alpha}$ in $\delta S_3$ > the variations of $\delta_{\alpha} F^A$ and $c^{\alpha}$ in $S_3$ cancel. How to see that?


title: Energy of electric field

text: In a semi conductor, for an electron in valence band to transfer to conduction band, it needs energy which can be provided by applying an electric field such that a force is applied on it until it breaks free from its parent atom. Now till the time it is at rest no change in kinetic energy is there. Hence, the energy that it gained and transfered to conduction band is electric potential energy. But this electric potential energy is stored in the field configuration i.e. The number at every point in space. And when u change that number you have to give off energy. N when the number is reverted back. It gives back to you the same energy. Now my question is that this case of the electron breaking free requires that the electric potential energy be localised at the electron like the electron's kinetic energy and not be stored in the field configuration. How is this happening? Is electric potential energy stored in electron or the electric field configuration at each point in space? And if in the field then how does the electron jump the energy gap and enter the conduction band, where does it get the energy from? Is kinetic energy locally stored in the particle or it is also a field configuration kind of thing? Is there a field associated with kinetic energy ? Or my understanding is wrong in which case please explain what is meant by electric potential energy is stored in electric field? And what is electric field energy density? edit: leaving the semiconductor example aside. Suppose i have an atom and i apply external electric field and that causes the atom to ionise. Where did the electron that flew away got the energy from?


title: question about the focul length and the position of a convex lens

text: How can I determine the focal length and position of a lens mounted in an inaccessible position inside a tube by a laboratory experiment? Please help.


title: Finite potential well, parity of solutions

text: I'm working through some problems for a QM exam and I've realised I don't really understand the concept of parity of solutions. I'm looking at a simple finite potential well problem: $$V(x)=0, \quad |x|>d$$ $$V(x) = V_0 \quad |x|<d$$ where $V_0$ is positive. I'm looking at bound states, so that $0<E<V_0$. I divide the x-axis into three regions, from left to right, in the standard way, so that my wavefunction is as follows: $$ \psi(x) = \begin{cases} Ce^{ax}, & x\in I\\\ A \sin kx + B\cos kx, & x\in II \\\ De^{-ax}, &x \in III \end{cases}$$ Where $a^2\equiv \frac{2m(V_0-E)}{\hbar^2}$ and $k^2\equiv \frac{2mE}{\hbar^2}$. Now this is the general wavefunction, but to make any progress I know I need to consider even and odd solutions (because the potential is symmetric). These I can find by imposing symmetry/antisymmetry on the wavefunction, so I get $$ \psi_S(x) = \begin{cases} Ce^{ax}, & x\in I\\\ B\cos kx, & x\in II \\\ Ce^{-ax}, &x \in III \end{cases}$$ $$ \psi_{AS}(x) = \begin{cases} Ce^{ax}, & x\in I\\\ A\sin kx, & x\in II \\\ -Ce^{-ax}, &x \in III \end{cases}$$ But what ARE those functions? Am I right in thinking that "half" the eigenvalues of energy correspond to eigenfunctions which are symmetric (and the other half - antisymmetric)? So that the entire energy spectrum looks like this: $$\\{ E^S_n , E^{AS}_n \\}_{n \in \mathbb{N}}$$ Is my understanding correct?


title: WKB approximation in two dimensions

text: Does anybody know how to implement the WKB approximation for the two- dimensional Schrodinger equation with a harmonic oscillator potential: $\frac{1}{2}\Biggl[-\biggl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\biggr)+x^2 + y^2 + 2xy\Biggr]\Psi(x,y) = E\Psi(x,y)$


title: Is contextuality required in quantum mechanics?

text: I still don't really understand what contextuality means in reference to quantum mechanics. If someone could give a clear definition that would be great. It sounds like it means you can't always relate the state of a system to its wavefunction or something? That confuses me since it seems to violate quantum mechanics. So a clear explanation would be great. The question is then, does any interpretation of quantum mechanics require us to accept contextuality? This paper seems to claim yes, but I don't know enough to evaluate it, so would appreciate a professional's eye: http://arxiv.com/abs/1006.0500 (This is a follow up on a previous question: What combinations of realism, non-locality, and contextuality are ruled out in quantum theory? )


title: Electric field for two coaxial cylindrical conductors of finite length with Gauss' Law

text: I'm confused with the Gauss' Law to calculate the electric field for two coaxial cylindrical conductors of finite length. I know that we can use the Gauss' Law to calculate the electric field for two coaxial cylindrical conductors of INFINITE length, but I don't see why we can use the Gauss' Law for the one of finite length. Ok here is the problem. > ![enter image description here](http://i.stack.imgur.com/GsLwh.gif) Two > coaxial cylindrical conductors are shown in perspective and cross-section > above. The inner cylinder has radius a = 2 cm, length L = 10 m and carries a > total charge of Qinner = + 8 nC (1 nC = 10-9 C). The outer cylinder has an > inner radius b = 6 cm, outer radius c = 7 cm, length L = 10 m and carries a > total charge of Qouter = - 16 nC (1 nC = 10-9 C). What is Ex, the > x-component of the electric field at point P which is located at the > midpoint of the length of the cylinders at a distance r = 4 cm from the > origin and makes an angle of 30o with the x-axis? The problem I have is that, I thought, in order to use the Gauss' law, the electric field has to be constant through out a gaussian surface, but I don't see why we can use a cylindrical shell of radius $4cm$ and length $10m$ to calculate the electric field at $P$ since I think the elctric field is not the same if I move from $0m$ to $10m$ on z-axis. Help me.


title: Are there measurable effects to scaling the action by a constant?

text: Classically, we obtain the equations of motion by finding a path which has an action that is stationary with respect to small changes in the path. That is the path for which: $\delta S =0$ Scaling the action by a constant should therefore do nothing. However, some books seem to consider the overall sign of the action important (since if we changed the sign we could find a path with arbitrarily negative action). Does a scaling factor or change in sign actually matter? For quantum mechanics we have: $K(x,y;T) = \langle x;T|y;0 \rangle = \int_{x(0)=x}^{x(T)=y} e^{i S /\hbar } Dx $ Now it looks like scaling the action $S$ will cause changes in how paths will interfere, but the overall sign of the action looks like it still won't matter. So now we can somehow measure the absolute scale of the action?


title: Thermodynamics of evaporation

text: If water is introduced in a container maintained at 20 °C in vacuum conditions, a gaseous phase will appear and the pressure will stabilize at the vapour pressure for the given temperature inside the container. So far so good. Now imagine the experiment is repeated but instead of vacuum conditions, the water is presurized with nitrogen at 1 atm. According to the phase diagram of water, liquid is the stable form of water in these conditions. Yet it is commonly observed that the water molecules with the highest kinetic energy will escape and form a gaseous phase. The partial pressure of gaseous water will be equal to the saturation pressure at this temperature. Why do we say that this is a liquid/vapour equilibrium in this case, since the liquid phase is not at pressure required for this equilibrium to appear? Edit: I just realized that my question seems to be answered by Raoult's law.


title: Why is the chiral symmetry $SU(2)_A$ not anomalous?

text: Using Fujikawa's path integral treatment of the triangle diagram, one can show that $$\mathrm{Tr} \gamma^5 = \int d^4 x\ \partial_{\mu}j^{\mu} $$ Where $j^{\mu}$ is the Noether current of $U(1)_A$. Thus, the $U(1)_A$ anomaly can be traced to to fact that the trace of $\mathrm{Tr} \gamma^5 \neq 0$ at loop order. My question is, why isn't $SU(2)_A$ anomalous too? I fail to understand why this applies only to $U(1)_A$.


title: Partition function for quantum harmonic oscillator

text: Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise: Exercise: > _A solid is composed of N atoms which are each localized at points in a > lattice. Excursions about the equilibrium position of each results in each > atom behaving as a 1-dimensional harmonic oscillator. Write down the > partition function for an individual atomic harmonic oscillator, and for the > collection, assuming that they have arrived in thermal equilibrium with each > other at temperature T._ $ Z_{SP} = \sum\nolimits_{n=1}^\infty e^{(-E_n \beta)}$ where $\beta$ is $1/(k_B T)$ and the Energy levels of the quantum harmonic oscillators are $E_n = \hbar \omega (n + 1/2)$. I now tried to use geometric series to evaluate the sum: $ Z_{SP} = e^{(-\beta \hbar \omega / 2)}\sum\nolimits_{n=1}^\infty e^{(-\beta \hbar \omega)n} = e^{(-\beta \hbar \omega / 2)} \frac{1}{(1 - e^{(-\beta \hbar \omega)})}$ With the substitution $\theta = \hbar \omega \beta $ it says the solution for $Z_{SP}$ should be $e^{(-\theta /2)} \frac{1}{e^{\theta} - 1)}$ which isn't quite the expression I got. Can anybody tell me where I went wrong or is the solution just false?


title: Orthonormality of Radial Wave Function in the Hydrogen Atom

text: `Question`: Show that the radial wave function for the hydrogen atom are orthogonal with no exceptions. `Attempt`: We know that the wave function for the hydrogen atom is orthonormal; that is, $$\langle\psi_{nml}\mid \psi_{n'm'l'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$ Splitting up $\psi$ into a radial component $R_{nl}$ and a angular component $Y_{lm}$, we have $$\langle R_{nl}Y_{lm}\mid R_{n'l'}Y_{l'm'}\rangle=\delta_{nn'}\delta_{mm'}\delta_{ll'}$$ We know that the angular component $Y_{lm}$ is orthonormal, and thus $$\langle Y_{lm}\mid Y_{m'l'}\rangle=\delta_{ll'}\delta_{mm'} $$ I believe that, if we do not have orthogonality in the radial component, we will not have orthogonality in the wave function. For example, if we do not have radial orthogonality, then at $n=n'$, we might have the radial component be zero. Thus, the radial wavefunction must be orthonormal. Is this correct reasoning? i' worried I' missing something.


title: Why Goldstone Bosons? (A Question about VEVs)

text: I understand how the mechanism of spontaneous symmetry breaking works, and why it produces Goldstone bosons (for global symmetries) and massive gauge bosons (for local ones). However, I'm confused as to why we identify Goldstone bosons as we do. Take the linear sigma model for example. Why do we say that the fields $\phi^n$ are unphysical, whereas the fields $\phi^n - v^n$ are? I get the feeling that it's a sort of definition that a physical field must have zero vacuum expectation value. At least in order to be interpreted as a particle, that is. But why is this the case? It's not a Wightman axiom, as far as I know! There's a related question here but the answers aren't really rigorous enough for my liking. In particular they both rely on "hand-waving" claims that "particles are small oscillations around a vacuum". I'd rather know that zero VEV is a physical principle, or see a proof that somehow zero VEV follows from a physical requirement. The other question seems to deal with the converse. Many thanks in advance for your help!


title: When is energy discrete/quantized for a potential well?

text: Specifically, my question is: Should one expect energy quantization for a particle in the following potential well? ![potential](http://i.stack.imgur.com/qhXAW.png) More generally, how can one tell whether or not energy should be discrete/quantized just by looking at the potential function?


title: Large object is pushed through a small wormhole, what happens?

text: A simple question about wormholes: What happens if a large object is pushed through a much smaller wormhole? For example, what happens if I push an elephant through a palm-sized wormhole? My intuition is that depending on the magnitude of the force applied, either a part of the elephant is ripped out, or the elephant sort of gets stuck in the wormhole, as if hitting a wall. Is this correct?


title: Finding max height of a mass launched by a spring?

text: I have the following problem on my homework. The actual problem has numbers, but I figure using symbolic notation will help me understand the concept in the long run. A block of mass $M$ is launched by a spring of negligible mass and constant $K$ along a frictionless surface. Prior to launch the spring is compressed a distance of $D_1$. Some time after launch mass then travels up an incline (the angle doesn't matter) that is also frictionless. However, at some point between leaving contact with the spring and entering the incline, the block passes over a rough patch of of distance $D_2$ where the coefficient of Kinetic friction is $C$. Find the max height the mass will travel up the incline. I've managed to work through the following: Looking at this I believe it is a conservation of energy problem. We have the Initial energy is equal to energy final - energy lost through friction. So, Initial Energy is $$E_i = \frac 1 2 kx^2 $$ Then our final Energy is simple $MGH$ since at it's max height it stops and all energy is potential again. So $MGH = \frac 1 2 kx^2 - \text{Friction}$ and this is where I get confused. I'm not to sure how I calculate the energy loss of friction. I believe it is the work friction does is equal to energy loss. So that would $W=FD$ distance is $D_2$ and the force would be $CMG$ since the normal force is equal to gravity on the flat surface. Am I correct in this? This would lead me to $$H = \frac{(1/2kx^2)-(CMGD_2)}{MG}$$ Which simplifies down to: $$H=\frac{1/2kx^2}{MG} - (CD_2)$$ This also confuses me, if my above calculations are correct, then the amound of energy loss due to friction does not depend on the mass of the object?


title: Person Pushing a Block vs. People Pushing off Each Other - Newton's Third Law

text: I have read so many forums on this and I still do not understand and it's affecting my ability to move forward with learning physics right now. Imagine the following scenario: a person on a frictionless surface pushes on a stationary object. Both the person and the object have the same mass. I understand that if the person pushes with 100N both the person and the object will feel that force (though in opposite directions) due to Newton's third law (though I'm not sure WHY the reaction occurs, but that is a question for another day). Now imagine the same scenario but instead of an object with the same mass it is another person with the same mass. Both people push off each other with 100N. I have been told that this will look the same; both people will feel 100N. However in my mind I am imagining that each person will feel the 100N push from the other person as well as the 100N reaction force from their own push. Should this not total to 200N? And if not, WHY NOT? I do not understand this and I can't stop ignoring it anymore. Thank you for your help, I'm sorry that this is probably a really obvious concept that I'm just not grasping but I cannot wrap my mind around it at all. Thank you.


title: Can Kirchhoff laws be applied to any circuit?

text: Kirchhoff's loop/current rule is just law of conservation of energy and Kirchhoff's junction rule is just law of conservation of charge.So, I think that these can be applied to any circuits unlike Ohm's law. Is there any exceptions?


title: Infinite period in Simple Harmonic Motion

text: I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period?   * When mass is 0.   * When mass is infinite. With $\tau=2\pi/\sqrt{k/m}$.


title: How did Eratosthenes know the suns rays are parallel?

text: Eratosthenes famously observed that the suns rays were perpendicular to the ground in one location, yet non-perpendicular to the ground at a location some miles to the north. On the assumption that the sun's rays are all parallel, this means the earth is round (or at least not flat). But how do we know the sun's rays are parallel? The rays of light from my desk lamp aren't parallel by the time they reach my desk. The spot directly below the lamp is hit perpendicularly, but the edge of the desk is hit obliquely. The issue is clearly determining whether or not the sun is far enough away that its rays can be assumed to be parallel. But how can we know that?


title: Is work done by how much velocity is changed or how much displacement is done?

text: > A sailboat is moving at a constant velocity. Is work being done by a net > external force acting on the boat? The answer key is "No" according to the work energy theorem about work is done when there is a change in final and initial velocity. But if I use Work = Force x displacement, even though the velocity is constant there is still some displacement done by the (constant) force acting on it. So I figured that the answer is "Yes". Which one is true?


title: Defining axes for conservation of energy

text: Problem: A bungee-cord jumper is on a bridge that is $y_1$ above water level $y_2$. The bungee cord obeys Hooke's law with spring constant $k$ and has a relaxed length $L$ The jumper stops before reaching the water, what height is her feet above the water at this point $d$? > I define the positive y direction as up and $y_2=0$. From this definition and given that the initial energy is $$E_i=mgy_1\tag{1}$$ I reason $\Delta h$ should be $y_1$ by a separate isolated case where a mass has $E_g =mgh$, when it is distance $h$ above a given 0 reference point. The final energy is a combination of $U_s=\frac{1}{2}kx^2$ and remaining $U_g=mgd$. Remembering the defined positive y direction, $d$ is positive by the same reasoning as for $y_1$. $x$ would be negative but can be ignored due to the even power. $g$ would be negative. The total initial energy in the system will be equivalent to the total final energy in the system given no non conservative forces apply. However, setting $E_i = E_f$ and solving for x to calculate the required $d$, $$\frac{1}{2}kx^2 - mgx - mgL=0\tag{2}$$ Yields a quadratic equation with no real solutions (as $g$ is negative). It is clear that my **interpretation** of my defined axes is incorrect. The problem is provided for situational awareness only, my question is given the convention I have used, why should $E_i=mg\Delta h$ be $mg(y_2-y_1)$ (a negative value as opposed to $y_1$). If this is the case wouldn't that also mean that $d$ should be negative and so the quadratic would still have no real solutions?


title: Is there some other name used for "ping rigidity"?

text: In MTW, p. 398, "Box 16.4 (continued)", there's an interesting sketch (which can also be seen on   p. 15 of this excerpt (www.pma.caltech.edu/~ph236/yr2008/readings/MTW_Chapter16.pdf).   (It's not the only sketch of this kind in MTW; but it happens to be the only sketch on p. 398.)   Shown are (cmp. text of p. 397)   * two open-ended lines representing two " _particles_ " (They are supposed to satisfy additional relations which are not of immediate interest here, however.),    * several line segments criss-crossing between the two open-ended lines, representing " _light rays bouncing back and forth between [the two particles]_ ", and   * two of the criss-crossing segments being extended such that they meet, with the coincidence point of these two segments denoted as $\mathscr Q$. In order to describe what's represented by that third feature (which, unfortunately, MTW don't seem to spell out explicitly) it is convenient to refer to the two explicitly shown particles by explicit distinct names; say "Paul" for one and "Eric" for the other (i.e. by names which have been suggested through this sketch by M. Goossens). Accordingly, it may be said that the sketch of MTW p. 398 shows that `(a)`: Paul observed $1$ _light ray bounce to, and back from_ event $\mathscr Q$ (or: $1$ "ping wrt." event $\mathscr Q$), and `(b)`: for his same initial/signal indication, and his same final/reception indication, Paul counted $3$ consecutive _back and fourth bounces_ (or: "pings") with respect to Eric. And likewise: `(c)`: Eric observed $1$ _light ray bounce to, and back from_ event $\mathscr Q$ (or: $1$ "ping wrt." event $\mathscr Q$), and `(d)`: for his same initial/signal indication, and his same final/reception indication, Eric counted $2$ consecutive _back and fourth bounces_ (or: "pings") with respect to Paul. Now, rather than considering only one such event $\mathscr Q$, we may ask about, and Paul and Eric may under certain conditions even actually find, a set $\cal L$ with several **additional events** (and including event $\mathscr Q$) such that the conditions `(a)` \- `(d)` had been satisfied by **all events of** $\cal L$. Taking again the suggestion from the already mentioned sketch by M. Goossens it may be found that there can be (at least) one particle identified who participated in all events of such a set $\cal L$; let's refer to it by the name "Lutz". If "Paul", "Eric" and "Lutz" have been identified accordingly, as having satisfied conditions `(a)` \- `(d)` throughout some sufficiently extended trial, then obviously `(1)`: Paul found any $3$ consecutive pings wrt. Eric same as the corresponding $1$ ping wrt. Lutz; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^EP{~}^L} = 3$ ", `(2)`: Eric found any $2$ consecutive pings wrt. Paul same as the corresponding $1$ ping wrt. Lutz; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^PE{~}^L} = 2$ ", and apparently (without further conditions or reservations) it follows that `(3)`: Lutz found any $3$ consecutive pings wrt. Eric same as the corresponding $2$ consecutive pings wrt. Paul; symbolically: " $\overset{\leftarrow ~ \rightarrow}{{}^EL{~}^P} = \frac{3}{2}$". This relation between such three participants together may be recognized as a general kind of " _rigidity_ "; which I'd call in default of any other terminology: "(triplewise) ping rigidity"; in the above example symbolically: " $\underset{\\{P, E, L\\}}{\shortmid \leftrightarrows \shortmid} = \\{2, 3, 1\\}$ ". **My question:** Have such relations, of three (or more) participants " _bouncing light rays back and forth between_ " each other and counting **constant integer or rational ratios** for the numbers of (consecutive, or individual) "corresponding pings" been recognized and already named elsewhere?


title: Showing Dirac equation's Lorentz invariance and use of unitary matrix $U$

text: Dirac equation is $i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 $ To show its Lorentz invariance, we convert spacetime into $x'$ and $t'$ from $x$ and $t$ and then $( iU^\dagger \gamma^\mu U\partial_\mu^\prime - m)\psi(x^\prime,t^\prime) = 0$ The question is, how does one show from the above equation the following equation follows?: $U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi(x^\prime,t^\prime) = 0$ where $U$ is some unitary matrix for lorentz transformation for $\psi$.


title: Non translation invariant correlator in CFT

text: I'm doing an exercise on vertex operators in the CFT book by Di Francesco & al.; exercise 9.2 p.329 : Using mode expansion show that: $$\langle\tilde{\phi(z)}\tilde{\phi(w)}\rangle= - \text{ln} \;(1-\frac{w}{z})$$ I managed to complete the calculation but I'm bothered by the fact that the right hand side is not apparently translation invariant. It seems that we have no charges (at infinity or wherever) and this should be a function of $(z-w)$. Am I missing out on something? The same way one line below we have: $$\langle V'_{\alpha_1}(z_1)\ldots V'_{\alpha_n}(z_n)\rangle = \Pi_{i<j} (z_i- z_j)^{2\alpha_i \alpha_j}z_i^{-2 \alpha_i \alpha_j}$$ which again doesn't seem translation invariant because of the last term.


title: How can I ensure my scale is not affected my surface beneath it

text: I have person weighing spring scale. I live in Japan where the house has wood floors which deflect significantly under weight. I weighed my self in the bathroom and in the tiled bath area. The difference was 2.5 kilograms greater in the bath area but it could still be affected by possibly an angled floor or deflection. So how you get as true a result as possible with that spring scale?


title: Commutability of two physical quantity matrices

text: Suppose that two matrices $A$ and $B$, representing real($\mathbb{R}$) physical quantity, can be multiplied commutatively with each other; i.e. $AB =BA$. However, each matrix cannot be multiplied commutatively with other matrices all the time (obviously). If two matrices represent physical quantity, can they be stated to have definite values togeether still? Or as two matrices cannot be multiplied commutatively with other matrices, is it not possible for two matrices to have definite values together? Also, suppose $C=A+B$. If A and B can be multiplied commutatively with each other, does $C$ have definite values? If $A$ and $B$ cannot be multiplied commutatively with each other, is there any way $C$ can have definite values?


title: If $L$ is a matrix that represents real physical quantity, why is $L^2$ non-negative real physical quantity?

text: In my textbook, it says that when $L$ is a matrix that represents real($\mathbb{R}$) physical quantity, $L^2$ represents non-negative real physical quantity. What would be the proof of this?


title: Do light and sound waves have mass

text: I have been reading Hawking's 'A Brief History of Time' and it has gotten me thinking about Einstein's theory of relativity, in that it assumes that an object must have infinite mass if it is to be traveling at the speed of light (please correct if I'm wrong in my beginner's knowledge of physics). But, do light waves have any sort of measurable mass? Or in that same vein, do sound waves? Is it somehow possible?


title: How does electron excitation relate to atomic orbitals?

text: Taking off from the Rutherford-Bohr model, ![Rutherford-Bohr model](http://i.stack.imgur.com/R8c0f.png)   **Figure 1.** _Rutherford-Bohr model._ when an electron absorbs energy (from light, heat, or otherwise) it jumps to a higher energy level, or shell. Then when the electron loses that energy, it reverts back to a lower-numbered shell. Now that the atomic orbital model is now more widely accepted, how does the above concept relate to it? ![atomic orbitals](http://i.stack.imgur.com/88Kwj.jpg)   **Figure 2.** _Atomic orbitals 1s, 2s, 2px, 2py, and 2pz._ I understand that the orbital model is based on the premise that we can never pinpoint the exact location of an electron at any time, hence the _electron cloud_ , or _probability density_ representation. Let's take a hydrogen atom for example. Its lone electron will be likely to be found in the volume defined by 1s. If that is excited to a higher energy level, will it be more likely to be found in the volume defined by 2s? What about for other elements with more electrons? When excited, will they follow the electron placement rules as shown below? (I wonder why I couldn't find 2px, 2py, 2pz in the illustration below, but that's probably for another question) ![Electron placement](http://i.stack.imgur.com/qPYcN.png)   **Figure 3.** _Electron placement._ I may have some terms and concepts mixed up, so corrections will be much appreciated. Also, please keep the jargon to a minimum, if possible. _All pictures are from Wikipedia._


title: Energy transfer to a driven oscillator

text: Why is the amount of energy transferred to a driven mechanical oscillator largest when the drive frequency is the same as the natural frequency of the oscillator? Why are they exactly out of phase by π/2? Clarifying example experimental setup: http://av.ph.tum.de/Experiment/1000/Grafik/b1605.gif


title: My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why?

text: I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it moves away from the "sun" and slows down as it approaches it. The best way I can describe it is that it's like the sun is at the wrong focus of the ellipse. I'm hoping someone can point me to what could cause this to happen? I've gone through all my code and I don't see any mistakes. My steps are:   1. Compute the Mean Motion of a satellite.   2. Use this to compute the Mean Anomaly   3. Calculate the Eccentric Anomaly from this (that was a doozy)   4. Calculate the True Anomaly from the Eccentric Anomaly   5. Calculate the Heliocentric Distance   6. Finally I take the Polar Coordinates and convert to Cartesian Coordinates which I then position over the top of my "sun" Somewhere in here I'm screwing up. I don't believe it's the final step as the calculated coordinates are derived from the angle and radius, which should be based on the correct focus. EDIT for the equations I'm using: **Mean Motion** $$ n = \sqrt{\frac{G(M+m)}{4\pi^2a^3}} $$ This is from Wikipedia. **Mean Anomaly** This is just $n \times t$ elapsed. **Eccentric Anomaly** I'm not really sure how to write this up for a non programmer. Basically the way I did this was based very much on some code that I found on the internet that I can no longer find. There's some recursion involved to gradually refine the answer. I'm convinced the issue isn't in here because I can enter my values into something like http://www.jgiesen.de/kepler/kepler.html and I have similar results. **True Anomaly** $$ \nu = 2\ \arg\left(\sqrt{1-e} \ \cos \left(\frac{E}{2}\right), \sqrt{1+e} \ \sin \left(\frac{E}{2} \right)\right) $$ $e$ is Eccentricity $E$ is Eccentric Anomaly from above Taken from here Wikipedia. **Heliocentric Distance** $$ r = a \frac{1-e^2}{1+e \ \cos(\nu)} $$ $a$ is my semi-major axis $\nu$ is the True Anomaly from above Taken from Wikipedia.


title: Is ultradense deuterium real?

text: I've found several articles discussing experimental evidence of a deuterium state of densities over $140 \textrm{ kg}/\textrm{cm}^3$:   * F. Winterberg. Ultradense Deuterium. arXiv.   * Shahriar Badiei, Patrik U. Andersson, Leif Holmlid. High-energy Coulomb explosions in ultra-dense deuterium: Time-of-flight-mass spectrometry with variable energy and flight length. _International Journal of Mass Spectrometry, Volume 282, Issues 1–2, 15 April 2009, Pages 70-76._ DOI.   * Patrik U. Andersson, Leif Holmlid. Superfluid ultra-dense deuterium D(−1) at room temperature. _Physics Letters A, Volume 375, Issue 10, 7 March 2011, Pages 1344-1347._ DOI. Have these super-dense states been replicated by third parties? Have these super-dense states yet to be replicated by third-parties?


title: Closest model for relationship between velocity and distance

text: Which is the model that best shows the relationship between velocity and distance:   * $v \propto d$   * $v \propto d^2$   * $v^2 \propto d$   * $v^2 \propto d^2$ * * * What I think is from: $$v=at \Rightarrow t=\frac{t}{a}$$ Then substitute into: $$\begin{aligned} d &= \frac{1}{2} at^2 \\\ &= \frac{1}{2} \frac{v^2}{a^2} \\\ &= \frac{1}{2} \frac{v^2}{a} \\\ \end{aligned}$$ So can I say closest model should be $d \propto v^2$? But my experimental data shows that $d^2 \propto v$. That is by plotting a least linear squares fit with excel and using the $r^2$ value. Could use `LINEST()` too. ![enter image description here](http://i.stack.imgur.com/rbr2o.png) Is my data wrong then?


title: "All energy gets taken from the particles and put into the fields". What does that mean?

text: In the The Emperor's New Mind book. While author explains why classical physics fails to explain our universe, says: > The root cause of these is that two kinds of physical object must coexist: > particles, each described by a small finite number (six) of parameters > (three positions and three momenta); and fields, requiring an infinite > number of parameters. This dichotomy is not really physically consistent. > For a system with both particles and fields to be in equilibrium (i.e. > 'fully settled down'), all energy gets taken from the particles and put into > the fields. This is a result of the phenomenon called 'equipartition of > energy': at equilibrium, the energy is spread evenly among all the degrees > of freedom of the system. Since the fields have infinitely many degrees of > freedom, the poor particles get left with none at all. > http://www.amazon.com/The-Emperors-New-Mind-Concerning/dp/0192861980 My questions: What does "energy gets taken from the particles and put into the fields" mean? How does this relate with classical physics failure?


title: IIA and IIB Compact on 8D

text: How can compactifying IIA (non-Chiral) and IIB (Chiral) Superstring on $T^2$ (2-torus) gives rise to ($2$ dual descriptions of) the same $\mathcal N = 2$ supergravity in $8$ dimensions? I don't see it. Could you please explain it to me or recommend me some literature to read about it? Thank you!


title: Will perfect black hole apply force on matter?

text: While standing on a planet its gravitation force is noticeable and manifests only through the normal force, e.g. if jupiter had rocky surface, standing on the surface will apply about 2.5 times more force than when standing on earth. However sky diving on earth and on jupiter (without atmosphere) would apply the same amount of "destructive" force, that is 0. Thus gravitation force cannot really destruct matter without something for the matter to interact with.   If a black hole with no matter falling into it exists, will a person with space suite die if he would "jump" into it? he will have nothing applying "destructive" force on it, just kinetic energy that cannot manifest without something to hit.


Why are white holes the same thing as black holes in quantum gravity? Their Penrose diagrams in semiclassical gravity are utterly different.


title: Can a vortex be self-sustaining?

text: Can vortices be self-sustaining? I suppose vortices in water are not self sustaining, since they need constant supply of water at speed i.e. kinetic energy. But is the same case applicable to space time vortices? like wormholes?? or do they need some energy source for sustenance? Please try to give a intuitive explanation.


title: Can I run a pulse tube cooler in reverse to generate pressure oscillations?

text: Since a pulse tube cooler is basically a Stirling engine without moving parts running "in reverse", does that imply that when I keep the cold end at ambient temperature and apply heat to the hot end, I get pressure oscillations? Or do I need the pistons and crankshaft of a standard Stirling engine for that? I think the question can be boiled down to how the working fluid gets back to the hot end after cooling, and the reason I don't seem to understand this might be that I have not yet fully understood how the buffer volume and orifice work in the pulse tube cooler.


title: Finding particles in the classically forbidden regions

text: Have particles ever been found in the classically forbidden regions of potentials? For example, in a square well: has an experiment been able to find an electron _outside_ the rectangular well (i.e. in the exponential fall-off regions) ? And more importantly, has anyone ever observed a particle **while** tunnelling?


title: Quantum mechanics potential barrier problem

text: While reviewing some quantum mechanics, I cam across a very interesting situation. For a potential barrier, if a particle has an energy $E$ less than the potential barrier $V_0$, it is possible to measure it inside the potential barrier or the _classically forbidden region_ quantum mechanically. But, if we calculate reflection and transmission coefficients for the wave, we find out $R=1$ and $T=0$ which means the wave will be fully reflected and no transmission will take place. But still there is probability of finding it inside the potential barrier. How?


title: What is the minimum density required for nuclear fission?

text: This is a followup to a **prior question** I asked about gaseous nuclear fission. What is the lowest nuclear fuel density required to initiate fission? This is critical for a gas core reactor with a solenoid magnetic bottle containment system in terms of how strong the magnetic field should be keep the fuel at a high enough density to trigger fission. Of course, the reactor vessel would be a cylinder with an inner jacket made of beryllium to reflect neutrons back into the core.


title: One terminal of an ideal voltage source connected to earth->Massive current will flow?

text: Assume we have a "somewhat" ideal voltage source like a DC power supply powered by mains. Take just one terminal and form a conductive path between it and earth ground. Assuming that no conductive path exists going back to the power plant ground spike, and a fuse isn't blown, wouldn't a huge amount of current flow into the earth due to the massive capacitance of the earth? Once again, I am assuming that this voltage source can maintain its voltage indefinitely.


title: X-Ray transparent material

text: I am looking for a material which is fully transparent to X-rays. In addition, it should be solid enough to serve as a substrate on which I will deposit some biomolecules.


title: Calculate mass of air in a tyre from pressure

text: How can one calculate the mass of air inside a tyre, given a particular tyre size; a pressure, in $kPa = \frac{1000kg}{m\cdot s^2}$; and assuming room temperature, and normal air composition? I can't quite work out what part of the equations I'm missing to remove the $s^2$ component. I realise that surface/volume ratio is important, but for the purposes and approximation to a torus would be fine. For example, assuming a 700cx25 bicycle tyre, we might assume a torus where the diameter between centre of the two cross-section circles is about 630mm, and the diameter of the circles themselves is about 30mm. Let's assume a tyre pressure of 500kPa (~73psi). Rough volume would be $630$mm$\times\pi \times \pi (15$mm$)^2 = 1.4\times10^6$mm$^3 = 0.0014$m$^3$ Rough surface area: $630$mm$\times\pi \times 30$mm$\times\pi = 1.87\times10^5$mm$^2 = 0.187$m$^2$


title: what does holographic principle from string theory say about the possibilities of wormhole travel?

text: Is travel through stable macroscopic wormholes between remote points of spacetime going to be possible in a definitive theory of gravity, be it string theory or something beyond it? Physicists level of understanding about the holographic principle and AdS/CFT correspondence has improved significantly and continues to evolve, as it is suggested by the current ongoing debate regarding black hole firewalls. I understand that the semiclassical picture of wormholes cannot survive because string theory demands that the required positive energy conditions are satisfied, so nontrivial classical topologies are censored from geodesically- complete world lines, just like naked singularities. But i cannot help but wonder (and hence my reason to do this question) if all the thorny subject about information being conserved by unitarity, while at the same time, holography implying there is a connection between all the boundaries of the bulk spacetime (understanding as such the union of the cosmological horizon and all the black holes in its interior), _could allow room for solutions where the infalling information in black holes survives by coming out, scrambled or otherwise, out of some other distant event horizon_ (behaving at times as a white hole, at others like a black hole)


title: What is the physical interpretation of force times area?

text: I know that $\text{Force} \times \text{Distance = Work}$.   But, what would be the physical meaning of $\text Force \times \text Area?$ Is such a quantity used in physics?


title: Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?

text: I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude: $$F(r, t) = \frac{k}{r^2}e^{-at}$$   * $r$ is the distance from the force center,   * $t$ is time,   * and $k$, $a$ are constants. I am concerned about the $t$ in the exponent. I did more than half of the question already (which involves finding lagrangian/hamiltonian equations) using a potential I'm not even sure is correct. I calculated it by taking the integral of $F(r, t)$ with respect to $r$. It is just that there is a $t$ in the magnitude so I know I shouldn't be able to do that.


title: Tensor product of Hilbert spaces and non-interacting particles

text: Consider a system of N quantum mechanical particles described on Hilbert spaces $\mathcal{H}_1,...,\mathcal{H}_N$ and with Hamiltonians $H_1,...,H_N$. The Hamiltonian operator $H_1$ acts on the Hilbert space $\mathcal{H}_1$ etc. These particles are non-interacting if the Hamiltonian of the system can be written as the sum of the Hamiltonians of each particle: $$H=H_1+...+H_N$$ Why the Hilbert space of the combined system of these non-interacting particles is considered to be a tensor product of the Hilbert spaces of each particle? $$\mathcal{H}=\mathcal{H}_1\otimes...\otimes \mathcal{H}_N$$ For interacting particles is it a tensor product too?


title: How do I get the focal length from a camera matrix?

text: Given the camera matrix, can I find the focal length of this camera ?


title: The role of resistor in e.g. an AND gate

text: What is the role of the resistor in e.g. an AND gate like this one? : ![AND Gate](http://i.stack.imgur.com/LirVA.png) One often sees lots of resistors in electric circuits, but I haven't really understood their role.


title: How Is Entanglement Created Among Qubits?

text: **How are qubits entangled?** I understand the basics of entanglement but what I do not get is how it occurs in nature or in the lab. What causes entanglement to occur or what is done to the particle to make it occur.


title: Is the total energy of a quantum system the sum of the particle energies?

text: I really should know this off by heart (this is my field...) but I never really grasped the difference between the total wavefunction of a system and the wavefunctions of particles within it, so it only just dawned on me that perhaps the total energy of a system was simply the sum of the energies of the individual particles. It's true, isn't it? By 'energy' here, I really mean $\langle E \rangle$. So would the expectation value of the total energy equal the sum of the expectation values of all the component particles? Or is there some conditionality to it? I.e. in a quantum computer, if the states of two qubits are opposites, then the expectation value of the total energy would be the sum of the component energies for each possible scenario $|0\rangle|1\rangle$ and $|1\rangle|0\rangle$, multiplied by the probability of each. Took me a long time to learn to ask the stupid questions. [Edit: brief mathsing for two qubits: $ \hat{H} \Psi = E \Psi $ $ \Psi = \psi_m \psi_n $ $ \psi_m = a_m |0\rangle + b_m |1\rangle \quad\quad\mathrm{(ditto}~~\psi_n \mathrm{)} $ $ \Psi = a_m a_n |0\rangle |0\rangle + a_m b_n |0\rangle |1\rangle \dots $ $ \langle E \rangle = a_m a_n E_{00} + a_m b_n E_{01} \dots $ or time-independently $ \langle E(t) \rangle = a_m(t) a_n(t) E_{00}(t) + a_m(t) b_n(t) E_{01}(t) $ ]


title: Random walks on resistive network

text: I have been referring to a paper http://arxiv.org/abs/physics/0405135 to determine the effective resistance using random walks for an infinite square resistive lattice Though the author seems to indicate this as a simple problem (maybe i am missing something) i have been unable to prove this $∆_{AB}$ = $\frac{1}{2p_{AB}}$ where, $∆_{AB}$ = $\sum_{n=0}^\infty$ $(P_{n}(A) − P{n}(B))$ $P_{n}(x):$Probability that a Random walker after n steps is found at x $p_{AB}:$Probability that a random walker, starting at A, gets to B before returning to A Could someone please help me with this ? (For more detailed description refer to the link)


title: Commutator of Momentum with a Position dependent function

text: I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a derivation, or provide one here?


title: Evaluating propagator without the epsilon trick

text: Consider the Klein–Gordon equation and its propagator: $$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$ I'd like to see a method of evaluating explicit form of $G$ which does not involve avoiding singularities by the $\varepsilon$ trick. Can you provide such a method?


title: Possibility for contact lenses that enhance the vibrancy of color

text: Would it be possible to develop oculars that would enhance the vibrancy of color? I know there are many digital filters to improve vibrancy, but are there physical devices able to produce the same effect? Would it ever be physically possible to produce a device small enough to be worn like a contact lens? This is quetion I developed while watching a Korean drama as I noticed the contrasting colors compared to the shows I normally watched. I feel environment and perception play a big part in emotional well-being. It is my thought that a device with this capability might serve as an aid to better emotional health.


title: Classical dynamics with Schrodinger equation

text: What are some interesting classical systems for which the dynamics can be reduced to a many-body Schrodinger equation, at least in some useful regions of phase space, and in particular, with many variables.


title: It seems to me that superpotentials can be defined in a theory with or without supersymmetry. Is this true?

text: I recently read "An Introduction to Supersymmetry in Quantum Mechanical Systems" by T. Wellman (amongst other sources) in an effort to find out what a superpotential actually is and how it relates to the potentials of particles/fields). Here's the link: http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDkQFjAA&url=http%3A%2F%2Fphysics.brown.edu%2Fphysics%2Fundergradpages%2Ftheses%2FSeniorThesis_Wellman.pdf&ei=ulm- UPSCLZOY1AWjwYHYDw&usg=AFQjCNGrg_2jv5NZ7b6k4Fs7er34jgtw3w&sig2=yjQYy1Lf_gZVS- RRefUCsQ It occurred to me that the Fermionic Hamiltonian and the Bosonic Hamiltonian can be formulated without supersymmetry. On page 13, Wellman expresses these Hamiltonians in terms of the superpotential W in a way that is purely algebraic and doesn't require supersymmetry. In other words we have a bunch of terms that give us the two Hamiltonians; these terms are then simply replaced by W's (see equations 3.2 to 3.9). We only actually get supersymmetry when the separate assertion is made that Q operators exist that transform between our fermionic and bosonic states. So would it be true to say that non-supersymmetric theories contain superpotentials, W, within their Hamiltonians in the same way that supersymmetric theories do? If this is the case the superpotential is just a useful function that is especially helpfully when we consider supersymmetric theories? I.e. superpotentials exist with or without supersymmetry.


title: Can an experiment be designed to distinguish between actual wave interference and probability or pilot wave results?

text: I am a physics groupie, so please excuse me if this question is stupid, but I am trying to better understand the particle/wave duality in quantum physics. It would seem that, in the double slit experiment or any similar experiment, there should be some method to tell the difference between two actual waves interfering (Schrodinger) and a probability wave (Einsteinian proposal) or a pilot wave (de Broglie proposal). If there are actually two waves interfering, there is both destructive and constructive interference. If not, there is just a distribution of particle hits. Is there not some way to design an experiment that would show which is actually happening?


title: Dirichlet's work on gravity in non-Euclidean space?

text: In the book _The Norton History of Astronomy and Cosmology_ by the late John North I have found the following statement (page 514): "The German mathematician Lejeune Dirichlet studied the law of gravitation in non-Euclidean space towards the end of 1850." Does anyone perhaps know more about that?


title: Vacuum energy and perpetual motion

text: The part of the Einstein equations of general relativity referred to vacuum energy, introduce a repulsive term in gravity. This means that as the space become bigger and bigger, vacuum part become more and more important, leading to an undefined accelerated growth of the universe. Why vacuum energy does not violate the principles of thermodynamic? This is a sort of perpetual motion, isn't it?


title: Basic questions about voltage drop in DC circuit

text: I understand all the concepts of what voltage is using all the analogies but some things related to the drop of voltage across a circuit confuses me.   1. If I had a short circuit and attached a voltmeter I would get a potential difference reading of 0 volts. How is current then going through the wire if it is 'X' volts at any point in the wire?   2. Let's say I have a DC 9V battery with a load in the middle. I read somewhere (probably mistaken here) that the voltage drop in this situation must be 9V or rather, that the sum of load resistances must be equal to voltage of the source. I mean, there is a variable amount of resistance to each load so at the base of the load I might have 9V and at the end of the load I might have 5V with a 4V potential difference.


title: Is it possible to build a huge spaceship like the one of of Star Trek or Prometheus?

text: Ok, so theoretically, is it possible to build a spaceship like the ones we see off of Star Trek or Prometheus or will physics not allow it? What are some challenge faces such an task. Also, is there other ways we can use to launch heavier payload from earth or are we just stuck with using tons of fuel to launch small object?


title: Why can't a spaceship accelerate for ever? Since there is no friction in space

text: I have seen many answers like: because we don't have infinite energy, because of gravity, because it is impossible, because of physics. But they don't really answer my question.. I mean if there is no friction and at some points, even if just in theory, you could be so far away from other objects or at the right distance between many so that the total force vector is null... why is it still considered impossible?


title: How do they point a camera on a space probe?

text: When a probe like Voyager 1 is sent into deep space they are able to point the camera at the planet and its moons and take pictures.   * How is the camera pointed at the object?   * Is the camera fixed to the probe or do motors orient the camera independently?   * Do they know the exact position of the probe with respect to the planet and pre-program orientations and exact times to take pictures?   * Is there something on board to point the camera at a close gravitational mass of some kind?   * Some other sensor to orient the probe? Light or infrared? How do they make sure it doesn't get confused by the sun?


title: Dynamics of a Rocket

text: I am interested in modelling the trajectory of a rocket from the Earth to the Moon by solving a differential equation numerically. Below are some key facts and assumptions I am using. I want to make sure that I have not made any serious mistakes, nor disregarded any necessary facts. We will consider the following equation, $$ \vec{T} + \vec{c}(\vec{r})\dot{\vec{x}} + \vec{G}(\vec{r})= m(t) \ddot{\vec{x}}, $$ where $T$ is the _**constant**_ rocket thrust, $c$ denotes air resistance and is a function of radial distance from the earth, and the rocket has mass that drops at a rate that is _**constant**_ with respect to time (we are assuming that a constant amount of fuel is always used for constant rocket thrust -- is this a valid assumption?). Now a question:   * The trajectory of the rocket is not straight; how do we incorporate parabolic motion into the numerics?


title: Lie derivative of Dirac Delta

text: In the setting of general relativity, I came across a source term of the wave equation of the following form: $$ \frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t)) $$ where $p\in M$ is a point in our 4d spacetime and $\gamma(t)$ is a trajectory that the source takes in the 4d spacetime. $\sqrt{q}$ is the 3d metric determinant of a preferred 3d slicing of $M$. $\delta^{(3)}(p-\gamma(t))$ is a 3d Delta distribution which means that we should have $$ \int_M\,\delta^{(3)}(p-\gamma(t))\,f(p)\,d^3x=f(\gamma(t))\,. $$ Of course, this is rather the physics short hand notation that $\delta^{(3)}(p-\gamma(t))$ is a map $C^\infty_c(M)\to C^\infty(\mathbb{R})$ that maps a function $f$ to $f\circ \gamma$. We would like to show that the Lie derivative of the source along a certain vector field $T$ vanishes if $\gamma(t)$ is a Killing trajectory, that is $\gamma$ is an integral curve of $T$. However, we are very confused about the rigorous treatment of this expression. Our intuition states the following:   1. The delta distribution should transform as scalar density of weight 1 under changes of the 3d frames and as a scalar under changes of the frame along the forth direction. However, we are not sure how to make this rigorous, nor how to find the Lie derivative of such a combined expression.   2. The object $1/\sqrt{q}$ should be an inverse object, that is it is a scalar density of weight $-1$ under changes of the 3d frame and a regular scalar along the forth direction.   3. Combining these two statements, it would make sense that the original object $$ \frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t)) $$ is a regular 4d scalar. This would also make a lot of sense because it serves as source term of a regular 4d scalar wave equation.   4. Finally, it makes somehow sense that the Delta distribution is invariant under the Killing vector field $T$ iff the trajectory $\gamma$ is an integral curve of $T$. But we are not sure how to prove this and how to deal rigorously with the delta distribution.


title: Is it possible to travel at precisely the speed of sound?

text: I've been talking to a friend, and he said that it's impossible to travel at exactly the same speed as the speed of sound is. He argued that it's only possible to break through the sound barrier using enough acceleration, but it's impossible to maintain speed exactly equal to that of sound. Is it true? And if it's true, why?


title: The philosophy behind the mathematics of quantum mechanics

text: My field of study is computer science, and I recently had some readings on quantum physics and computation. This is surely a basic question for the physics researcher, but the answer helps me a lot to get a better understanding of the formulas, rather than regarding them "as is." * * * Whenever I read an introductory text on quantum mechanics, it says that the states are demonstrated by vectors, and the operators are Hermitian matrices. It then describes the algebra of vector and matrix spaces, and proceeds. I don't have any problem with the mathematics of quantum mechanics, but I don't understand the philosophy behind this math. To be more clear, I have the following questions (and the like) in my mind (all related to quantum mechanics):   * Why vector/Hilbert spaces?   * Why Hermitian matrices?   * Why tensor products?   * Why complex numbers? (and a different question):   * When we talk of an n-dimensional space, what is "n" in the nature? For instance, when measuring the spin of an electron, n is 2. Why 2 and not 3? What does it mean? Is the answer just "because the nature behaves this way," or there's a more profound explanation?


title: Causal and Global structure of Penrose Diagrams

text: What kind of global and causal structures does a Penrose diagram reveal? How do I see (using a Penrose diagram) that two different spacetimes have a similar global and causal structure? Also, I have the following metric $$ds^2 ~=~ Tdv^2 + 2dTdv,$$ defined for $$(v,T)~\in~ S^1\times \mathbb{R},$$ e.g. $v$ is periodic. This is the according Penrose diagram: ![](http://www.matheplanet.de/matheplanet/nuke/html/uploads/8/8403_misner_penrose_2.jpg) Is the Penrose diagram that I have drawn correct?


title: 2 Component Spinor Formalism

text: In Chapters 34-36 of the Srednicki QFT book, 2 component spinors and their combinations in Dirac and Majorana spinors are carefully constructed. Specifically, in equations 36.14 and 36.15 the following left-handed spinors are defined: $$ \begin{equation} \begin{split} &\chi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2)\\\ &\xi = \frac{1}{\sqrt{2}}(\psi_1 - i\psi_2) \end{split} \end{equation} $$ A couple of pages later, charge conjugation is then defined as "Charge conjugation simply exchanges $\chi$ and $\xi$". Now, I have read in other books (e.g. Georgi) that if a particle is defined in one SU(2) representation of the Lorentz group (e.g. as left handed spinor), then the corresponding anti-particle is in the other representation (e.g. as right handed spinor).However, in the above definition in Srednicki, both fermion and anti-fermion are left-handed spinors. Have I got the this right? How can both of these statements be correct? Any help to clear up my confusion would be greatly appreciated. As an aside, I have a more general issue understanding the intuition behind Dirac spinors. The state of a general fermion or anti-fermion is often given with a Dirac spinor $\Psi$. If this fermion has mass, then the fermion is a combination of a left-handed and right handed spinor, as in Srednicki: $$ \Psi = \begin{pmatrix} \chi_c \\\ \xi^{\dagger\dot{c}} \end{pmatrix}. $$ As helicity and handedness are frame depenendent, I can kind of understand how the Dirac fermion can be a combination of the two above. However, Dirac fermions can also by viewed as a combination of a fermion and anti-fermion in much the same way. This I do not fully understand. How can a fermion with mass be described by a field that is part particle and part anti-particle? Again, any insights that the community could provide would be much appreciated.


title: Information about an expanding event horizon

text: Assume an observer sent a beam of photons close to an event horizon, say at some distance x (a distance far enough to avoid the photons falling in.) This light would still be observable, albeit red shifted and with it's path curved appropriately. Now assume the black hole absorbs enough mass to expand it's event horizon beyond the distance x. This stream of photons would stop. Does the observer not have information about a process that occurred exactly at the event horizon, that is it's expansion? Isn't this a region that one should not be able to get any information about due to the fact that nothing could contact the event horizon and return to the observer?


title: Are antiparticles just particle-shaped holes?

text: If particles are simply regions of space where certain quantum fields have non-zero divergence, are anti-particles simply the corresponding regions of opposite divergence? This seems like the intuitive answer, especially when considering the process of annihilation. I have heard before that anti-particles are analogous to particles moving through time in reverse, which seems to indicate that an annihilation event is just a point of symmetry in that (arguably, single) particle's history. This analogy breaks down, however, when it comes to gravitation, since there seems to be no evidence of a negative-mass particle. So what then is the meaning of an anti-particle, when their symmetry is preserved across certain fields, but not other?


title: Is Brian Cox right to claim that Gravity is a strong force for large masses, is it wrong, or is it only a matter of interpretation?

text: I watched a program of his in which it was claimed that since mass bends space in accordance to General Relativity, then in the case of very large stars it becomes a strong force to the point of being able to crush a star to a single nucleus (Neutron Stars) or less (Black Holes). His argument is that Gravity is a force that **scales** and that it is not simply a matter of adding individual components and hence to claim it's weak, but that since space is bent in those areas, then gravity as a _fundamental force of nature_ becomes stronger. Now, I wonder not only about the claim's accuracy, but also if it's only a matter of interpretation and nobody is really wrong or right, as long as the discussion is framed properly.


title: Why are some extra dimension theories known as strongly coupled and others as weakly coupled?

text: I was looking at pdf file of the presentation of a conference talk. The speaker discusses two types of "mechanisms" for stabilizing the weak scale and calls them "weakly coupled" and "strongly coupled". The examples are: Weakly coupled: SM with a light Higgs, SUSY, Little Higgs, Twin Higgs, Large extra dimensions, Universal extra dimensions. Strongly Coupled: Technicolor, Topcolor, Top See Saw, Composite Higgs, Randall-Sundurm warped extra dimension models. I need to understand what makes one such beyond standard model theory to be weakly coupled and another to be strongly coupled, in general and especially why large/universal extra dimension and Randall-Sundrum models belong to two separate groups. Perhaps I should emphasize where my doubt is: I understand why theories like technicolour are thought to be "strongly coupled" but I do not at all understand why theories involving say Randall-Sundrum extra dimensional models are also so. When we say Randall-Sundrum warped extra dimension model, we mean a particular kind of background on which different types of interactions take place. But why should all interactions taking place in such a background will have to be strongly coupled? How does background spacetime decide whether an (or all) interaction(s) will be strongly coupled? Or does this have anything to do with the fact that in (classical) GR, spacetime is dynamically determined and is coupled with matter and energy content of the universe and may be due to quantum gravity effects this strong/weak nature of interactions are/is manifested. But what is the way to explain that?


title: Regarding formulation of a multipoint model of fluid dynamics

text: Suppose I am trying to formulate a multipoint model of fluid dynamics. I have a procedure for doing so the details of which is not important to this question, but only that it is based on a series expansion, the higher order terms in which series being dependent on larger number of spacial points. The equations for the first two terms in a Burger formulation is something like shown below: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 u^0}{\partial x^2}(x,t) \+ u^0(x,t)\frac{\partial u^0}{\partial x}(x,t)\\\ &\qquad\qquad\qquad\quad\;\;+\int_0^1 u^1(x,t;x_1)\frac{\partial u^1}{\partial x}(x,t;x_1) dx=0\\\ &\frac{\partial u^1}{\partial t}(x,t;x_1)+\frac{\partial^2 u^1}{\partial x^2}(x,t;x_1)+u^0(x,t)\frac{\partial u^1}{\partial x}(x,t;x_1)+u^1(x,t;x_1)\frac{\partial u^0}{\partial x}(x,t)=0 \end{align} it is however an odd formulation, the point $x_1$ appears as a mere parameter in the second formulation while it appears only in the integrand of the first equation, so that apparently there is no enough restriction on the behavior of the function $u^1$ with respect to $x_1$, so it seems there s no unique solution for $u_1$. If the flow is a homogeneous flow but $u^1$ can be written as $u^1=u^1(x-x_1,t)$, so that derivation with respect to $x$ will also take into play the point $x_1$, so that $x_1$ will now behave like it is a more important variable. This will get more importance if we further consider the higher order terms like $u^2(x,t;x_1,x_2)$ which in a homogeneous flow will be writable as $u^2(x-x_1,x-x_2,t)$ and whose $x$-derivative becomes: $$\frac{\partial u^2}{\partial x}=\frac{\partial u^2}{\partial(x-x_1)}+\frac{\partial u^2}{\partial(x-x_2)}$$ Such a treatment imports the variables $x_1$, $x_2$ and etc. in the second and higher order equations in a more proper manner, in such a way that hope to find unique solutions increase.   1. The problem is that how the first formulation which was more general is lame in giving such unique solutions as the second formulation is apparently capable of?   2. Or maybe there is a must in every multi-point modeling of fluid dynamics (actually turbulence) to consider one of the following two procedures? a- To expand the series in terms of functions of the form: $u^0(x,t)$, $u^1(x,x-x_1,t)$, $u^2(x,x-x_1,x-x_2,t)$ and etc. ? b- To write the equations once at the point $x$, then once at the point $x_1$ available also in the arguments of $u^1$ and higher order functions, then once at the point $x^2$ available also in the arguments of $u^2$ and higher order functions, and etc., then consider all those equations in one place, for example by adding them together. As the series is truncated somewhere at a function $u^n$ this will not contain infinitely of equations but I am not very hopeful that this is the right path to take.     * any idea about these or other methods for multipoint formulations?


This is a confused part ever since I started learning electricity. What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)? All of them have the same SI unit of Volt, right? I would appreciate an answer.


title: Binding Energy of He

text: The graph of nuclear binding energy is relatively smooth going from H to U, except for He4 (alpha particle). Why is He4's binding energy so anomalously high compared to its neighboring isotopes?


title: A water drop in a falling lift

text: Consider a lift, which is at rest in an homogeneous gravity field. There is a thin layer( **with thickness** $h$) of water on the floor of the lift. At some moment a single cable, supporting the lift, breaks and the lift begins free falling(forever). It is easy to describe qualitatively what happens with water, i think: the formation of a drop begins. During this process the drop jumps up from the floor and once the total kinetic energy of the drop is dissipated into the heat, the center of the drop stands at some height $H$ from the floor. (All the kinetic energy of the drop is coming from the difference in surface tension energy of water between initial and end moments.) Question: How to determine(approximately) $H$ ? For simplicity let's assume that the cross-section of the lift is a disc- shaped with radius $R$. $(h<<R)$ Remark: **A key point is to determine how the total kinetic energy distributes between internal kinetic energy and translational kinetic energy of the drop, i think.** With this the problem will be solved. So it is convenient at the early stage of formation of the drop ignore viscosity effects of water and air resistance. **Edit:** I would like to clarify that by internal kinetic energy i mean the kinetic energy of macroscopic motion of water inside the drop. At early stage there is no energy loss assumed, for simplicity.


title: What is the source of high-frequency squeal in faulty CRT monitors and TV, and what is physics of generating this sound?

text: Sometimes old faulty CRT monitors generate nasty high-frequency squeal sound. What element might be responsible for generating such sound? I have heard that it might be dry electrolytic capacitor; is it? What is the physics of generating this sound (how it is generated)?


title: What was so surprising about accelerating cosmological expansion?

text: The 2011 Nobel Prize in Physics, as far as I understand, concerns the expanding universe -- galaxies moving away from each other at ever increasing speed (that's what I think I read in newspapers). Now, the idea of expanding universe is not new. What is the novelty in this study? What was unexpected in it?


title: what are the direct real life applications of general relativity and quantum physics

text: What are the direct real life applications of general relativity other than nuclear technology? What I meant was, was there any technology developed based on general relativity that can benefit mankind today? Secondly, are there any adverse effects quantum technology today?


title: Solar Cycle UV Variation

text: The average energy we receive from the Sun is 1,366 w/m^2, and this only varies by 0.1% from the activity peak to trough of its 11 year cycle. About 9% of the energy comes from wavelengths less than 400 nm. The 200-300 nm region varies on the order of a few percent, 150-200 nm by 10-20% and shorter regions by over 50% with maximums at the cycles peak activity. Energy above 400 nm is almost constant over the cycle. Why does solar energy variation over the Sun's activity cycle vary inversely with wavelength? http://astro.ic.ac.uk/research/solar-irradiance-variation


title: Distinguishability in Quantum Ensembles

text: Inspired by this question: Are these two quantum systems distinguishable? and discussion therein. Given an ensemble of states, the randomness of a measurement outcome can be due to classical reasons (classical probability distribution of states in ensemble) and quantum reasons (an _individual_ state can have a superposition of states). Because a classical system cannot be in a superposition of states, and in principle the state can be directly measured, the probability distribution is directly measurable. So any differing probability distributions are distinguishable. However in quantum mechanics, an infinite number of different ensembles can have the same density matrix. What assumptions are necessary to show that if two ensembles initially have the same density matrix, that there is no way to apply the same procedure to both ensembles and achieve different density matrices? (ie. that the 'redundant' information regarding what part of Hilbert space is represented in the ensemble is never retrievable even in principle) To relate to the referenced question, for example if we could generate an interaction that evolved: 1) an ensemble of states $|0\rangle + e^{i\theta}|1\rangle$ with a uniform distribution in $\theta$ to 2) an ensemble of states $|0\rangle + e^{i\phi}|1\rangle$ with a _non-uniform_ distribution in $\phi$ such an mapping of vectors in Hilbert space can be 1-to-1. But it doesn't appear it can be done with a linear operator. So it hints that we can probably prove an answer to the question using only the assumption that states are vectors in a Hilbert space, and the evolution is a linear operator. Can someone list a simple proof showing that two ensembles with initially the same density matrix, can never evolve to two different density matrices? Please be explicit with what assumptions you make. Update: I guess to prove they are indistinguishable, we'd also need to show that non-unitary evolution like the projection from a measurement, can't eventually allow one to distinguish the underlying ensemble either. Such as perhaps using correlation between multiple measurements or possibly instead of asking something with only two answers, asking something with more that two so that finally the distribution of answers needs more than just the expectation value to characterize the results.


title: How to communicate institutional knowledge in a big physics collaboration?

text: I am a graduate student in a large experimental physics collaboration. Newcomers to the collaboration invariably complain about excessive use of jargon and insufficient documentation as barriers to their understanding and quick integration into the collaboration. Of course, jargon is unavoidable in any technical pursuit, and one should not expect to find comprehensive, polished documentation of what is essentially a work-in-progress, but there must be some effective techniques to communicate this stuff to new students with a minimum of frustration? **What are some "best practices" in communicating institutional knowledge to new (graduate student and post-doc) members of a large scientific collaboration?** Alternatively, what is the best way to prepare students for interaction with a big collaboration? Perhaps a few bits of advice at the beginning could help them become integrated and acquire institutional knowledge much more quickly(?).


title: Homework question about electric field between two spheres

text: I was helping a friend of mine with the following question from Knight's book and I was not able to answer part (c). ![enter image description here](http://i.stack.imgur.com/G2AdB.jpg) Here is what I think I know: (a) I expect $V_1 = V_2$; the two spheres are in equilibrium so no current flows between the two spheres. (b) If $V_1 = V_2$, then $Q_1/R_1 = Q_2/R_2$. This implies that $Q_1 = Q_2 (R_1/R_2) > Q_2$. That is, $Q_1$ is larger than $Q_2$. (c) Here is were I am unsure: I think that $E_2 > E_1$. Question: Mathematically it makes since because if $V_1 = V_2$, then $E_1 = V_1/R_1$ and $E_2 = V_2/R_2$, then $E_2 > E_1$. But I don't see this physically. Can someone explain this?


title: Dyson Air Blade as a propulsion system?

text: I've read from many sources that Dyson Air Multipliers are more efficient and quieter than normal fans. Now, with the proof of concept, is it possible to use its principles as a propulsion system for, say, a quieter helicopter?


title: Energy conservation and interference

text: I have a problem with energy conservation in case of interfering waves. Imagine two harmonic waves with amplitudes $A$. They both carry energy that is proportional to $A^2$, so the total energy is proportional to $2A^2$. When they interfere, the amplitude raises to $2A$, so energy is now proportional to $4A^2$ and bigger than before. The equivalent question is what happens to the energy with the superposition of two waves that interfere destructively. Also, if someone could comment on the statement about this problem in my physics book (Bykow, Butikow, Kondratiew): the sources of the waves work with increased power during the interference because they feel the wave from the other source.


title: Does quantum mechanics prohibit extraterrestrial life?

text: There is a known thought experiment, connected to quantum immortality: a duel between physicist and a philosopher. Each turn physicist and philosopher fire at each other with a pistol. The quantum immortality predicts that each of the participants will find themselves alive and the opponent dead after a number of shots. We can modify the experiment in the following ways:   1. They do not fire at each other, but it is lightning that shots them. The outcome should not change.   2. There are not two but more participants.   3. They are located at separate islands in the ocean. It seems that even if there are hundreds of participants, each of them will eventually find all others killed by lightning. If he could not observe other isles directly, after he discovers and explores other isles, he will find them uninhabitable. Now pretend that the participants actually live on separate planets, and in each million years there happens disaster that kills all inhabitants on 1% of all planets. It seems that after years the physicist that lives on one of the planets should discover that all other planets are uninhabitable. Even more: there is even no chance that organic matter on each other planet could actually evolve into anything resembling actual life. That said the physicist will find out that there is no other inhabitable planet in the Uinverse. On the other hand, this does not mean that extraterrestrial life does not exist. It actually cannot not to exist. But it exists in parallel universes (in terms of MWI) and thus unobservable. Both physicist on planet A and philosopher on planet B exist, but will never meet in the same universe. Thus, QI it seems predicts one concrete observable consequence that extraterrestrial life can be never detected. What is your opinion?


title: Definitions and usage of Covariant, Form-invariant, Invariant?

text: Just wondering about the definitions and usage of these three terms. To my understanding so far, "covariant" and "form-invariant" are used when referring to physical _laws_ , and these words are synonyms? "Invariant" on the other hand refers to physical _quantities_? Would you ever use "invariant" when talking about a law? I ask as I'm slightly confused over a sentence in my undergrad modern physics textbook: > "In general, Newton's laws must be replaced by Einstein's relativistic > laws...which hold for all speeds and are **invariant** , as are all physical > laws, under the Lorentz transformations." [emphasis added] > > ~ Serway, Moses & Moyer. Modern Physics, 3rd ed. Did they just use the wrong word?


title: Could moving land mass alter Earths gravity?

text: This is potentially a very stupid question but I'm going to ask it anyway. With all these huge buildings such as the Abu Dhabi Mosque, where an unbelievable amount of materials such as marble was moved from one side of the earth to the other, is it possible that if we 'shift enough stuff' that we could change the earths centre of gravity, and potentially alter its orbit?


title: Cooling induction current generation

text: We know that conducting materials can be heated by electromagnetic induction. Is it possible to generate current using a cooling process?


title: Fresnel Zones-How are they Formed?

text: How are Fresnel Zones formed? What phenomena of light allow ellipsoid areas to be in phase? I've tried reading articles, but they more or less introduce me to characteristics of light, and then tell me that Fresnel Zones exist. How does the circular wave coming out of one antenna "bend back" to the other one in the shape of an ellipse?


title: How to validate a new physics theory and keep the credit for it

text: For the last few years, My friend worked on figuring out the "theory of everything". She is afraid of sharing her theory with well known physicists, because she doesn't want other people to take credit for her idea. So, my here's what I'm wondering: **1) What is the best way for her to validate her theory with experienced physicists?** and **2) If the theory ends up being worth publishing, how can she go about publishing it considering the fact that she has no reputation in the field at the moment?**


title: Could the phenomenon of vortex bursting be exploited to reduce wake turbulence?

text: One of the classic stories in the annals of aerospace engineering is the development (and subsequent redesign) of the F-18 and its Leading Edge Extensions (LEX) due to fatigue problems, problems that were ultimately traced to the poorly-understood process of vortex bursting. This aptly named phenomenon is characterized by rapid increases in vortex diameter, mixing, and turbulent kinetic energy. ![enter image description here](http://i.stack.imgur.com/PMjQz.jpg) > As a vortex travels downstream, it enlarges and becomes weaker. If the > rotational velocity of the vortex drops low enough, the increasing pressure > within the vortex causes it to lose its tornado-like structure and break > apart. This bursting behavior was found to occur just ahead of the F-18 > vertical tails. The resulting air flow impinged directly on the tails > causing severe buffeting and structural damage. (Aerospaceweb) For a highly maneuverable fighter like the F-18, it was imperative that the LEX vortex remain intact and persist long enough in the flow-field to clear the vertical stabilizers, so vortex bursting was definitely an issue. For a commercial airliner like a 777, on the other hand, we have the exact opposite problem. The vortex pair trailing an aircraft in flight presents a very real and perilous hazard to any aircraft which happens to follow it. As a result, the more expeditiously these vortices can be dissipated the better. Vortex bursting breaks up the vortex and (presumably) reduces its presence downstream. ![enter image description here](http://i.stack.imgur.com/fpAIS.jpg) My question is, what are the physical variables which promote or inhibit vortex bursting and, if they can be controlled, can this phenomenon perhaps be used to reduce wake turbulence? Do all vortices experience bursting? If not, why not?


title: How hot does the upstairs need to be before it will heat the basement through convection?

text: During the winter we keep the basement door closed so that the cold air from upstairs won't flow into the basement. Then we heat the basement and the upstairs gets heated partially by conduction through the floor. But during the summer, the upstairs is heated while the basement (unfinished) is around 62 degrees F (16 degrees C). How hot does it need to be upstairs before opening the basement door will actually heat the basement? I know hot air rises, but it seems that if there were a massive heat difference between upstairs and downstairs that air turbulence or diffusion would carry some of the heat. There's an 8' 7" height difference between upstairs and downstairs.


What was the density of the universe when it was only the size of our solar system? Did it approach neutron star density? Is it physically correct to even ask such a question?


title: Repulsion and attraction of electric currents

text: Now, I understand that when a an electron travels, it creates a magnetic field. If you put two wires with current traveling in the same direction they repel, and current traveling in opposite directions attract. I (somewhat) understand how this is caused by electrostatics and special relativity, but when I tried to get a better idea of the forces at work, I ended up with a result that said the force of attraction between to opposite wires was stronger than the force of repulsion between to wires with the same direction of current. Is this correct, or did I just mess up somewhere?


title: On Thermodynamics of Spontaneity of the Reduction-Oxidation Chemical Reactions

text: According to the first and second law for a closed system containing different chemicals we have \begin{align} &\delta Q - \delta W = dU = T dS - p dV +\sum_i \mu_i d N_i\\\ &\Rightarrow\;\delta W - p dV + \sum_i \mu_i dN_i = \delta Q - T d S \le 0\qquad\because\text{2nd law}\\\ &\Rightarrow\;\delta W - p dV + dG\bigr|_{p,T} \le 0\\\ &\Rightarrow\;\delta W - p dV \le -dG\bigr|_{p,T}\\\ \end{align} If $\delta W = p dV$ then $dG\bigr|_{p,T}\le 0$, that is, the condition $dG\bigr|_{p,T}\le 0$ coincides with the second law only if the only work done by the system is the pressure work and no other kind. In addition, if $\delta W = p dV + \text{``other works"}$, then $\text{``other works"}\le -dG\bigr|_{p,T}$. This means the change in the "minus Gibbs function" is the maximum work attainable from the system beside the pressure work. This extra work can be positive or negative, in the form of electric work, friction work etc. Therefore, it is clear that:   1. A system cannot at the same be derived by $dG\bigr|_{p,T}\le 0$ and does e.g. an electric work $\delta W_{Electric}$;   2. If the system does have a $\delta W_{Electric}$ then its maximum value would be equal to $dG\bigr|_{p,T}$ but if so, then the process is already assumed reversible and all the inequalities should be substituted by equalities. However, according to this Mc Graw-Hill link during a spontaneous reduction- oxidation chemical reaction the Gibbs Free Enthalpy must decrease and at the same time the change in the Gibbs Free Enthalpy is the maximum _electric work_ that the reaction can do: $$\Delta G = W_{max}\le 0$$ I cannot understand this and have a number of problems with this derivation:   1. First of all, the maximum work that a system can do on its surrounding equals minus the change of the Gibbs Free Enthalpy, so the equality $\Delta G = W_{max}$ doesn't hold?   2. If the work has attained its maximum value, then the process must be assumed as reversible, but the inequality in the formula above holds for irreversible processes!   3. If there is an Electric work done by the system then decreasing Gibbs free energy is no longer necessary due to the second law?   4. When both the donor and acceptor of electron in the chemical reaction lay inside the system the electric work will nowhere enter the formulation as the electric work is an internal work that does not crosses the boundary of the system!? But the whole formulation is here to study $W_{electric}$ in the system as spontaneity of the reaction should be related to the electromotive force of the reaction which has a chance to appear in $W_{electric}$ only! So what should be taken as system here? **Hint.** In the books on thermodynamics that I have seen that discuss the electric works they are usually dealing with the problem of Electrochemical cells. But electrochemical cells work in an outer electric circuit and so if one assume the cell as the closed system yet the electric work will enter the discussion. Only one book was talking about open-circuit Emf but again I have the problems listed above with that as well. [ _I have asked the same question at the Chemistry.SEhere but have not been convinced with the single answer given there._]


title: Do photons occupy space?

text: Total noob here. I realize that photons do not have a mass. However, they must somehow occupy space, as I've read that light waves can collide with one another. **Do photons occupy space?** and if so, does that mean there is a theoretically maximum brightness in which no additional amount of photons could be present in the same volume?


title: Does time travel violate conservation of mass/energy?

text: Imagine I exist at time $t_1$ and my mass is $m$. At time $t_2$ I time travel back to $t_1$. At time $t_1$ there is now a net increase of mass/energy in the universe by $m$. At time $t_3 = t_2 - x$ where $x < t_2 - t_1$, I travel back to t1 again. The net mass in the universe has now increased by $2 \times m$. Properly qualified, I can do this an arbitrary $n$ number of times, increasing the mass in the universe by $n \times m$. This extra mass, of course, can be converted to energy for a net increase in energy. Does this argument show that traveling back in time violates the conservation of mass/energy?


title: Does the weight of fluid in a conical container act entirely on the base?

text: # Scenario description Lets assume both containers have a capacity of 300 litres. One is a vertical tube as shown in pic 1 Other one is more or less a V shaped containers as shown in pic 2 Both are at ground level # Question No doubt that whole weight of 300 litres will act on the base of the tube (pic 1). But will the weight of the whole 300 litres act on the base of the container as shown in pic 2? I.e if you keep both the containers on a weighing scale, for sure, it will show a weight of approx. 300 kg each. But what I'm asking is whether the weight of whole 300 litres acts on the **base** of container 2 (since the container's walls are slanting, as opposed to the walls of the tube in pic 1). If the whole weight will not act on the base of the container, can you state the reasoning and also the principle, so that I can further get to know about it? Pic 1                          |         |                        |         |                |         |                |         |                |         |                |         |                |_________|      Pic 2                        \              /               \            /                \          /                 \________/


title: Which interaction violates T symmetry?

text: While reading Peskin and Schroeder (page 64) I come across this > _Although any relativistic field theory must be invariant under the proper > orthocronous Lorentz group, it need not be invariant under P, T, or C. What > is the status of these symmetry operations in the real world? From > experiment, we know that three of the forces of Nature the gravitational, > electromagnetic, and strong interactions are symmetric with respect to P, C, > and T. The weak interactions violate C and P separately, but preserve CP and > T. But certain rare processes (all so far observed involve neutral K mesons) > also show CP and T violation. All observations indicate that the combination > CPT is a perfect symmetry of Nature._ The question is this. As is stated above all known interactions preserve T, yet it is said that there are processes that violate it. How is this possible? Via which interactions do these processes happen?


title: Solving particle on a ring problem using momentum eigenvalue equation instead of energy eigenvalue equation

text: I have read somewhere that for particle on a ring problem you don't have to solve eigenvalue equation $H\psi=E\psi$ you can instead solve eigenvalue equation $P\psi=p\psi$ where P is momentum operator. Why?


title: Action for $p-p'$ strings (equation 13.5.21 in Polchinski's textbook)

text: This action reads $$S=-\frac{1}{4g_{D9}^2}\int d^{10}x F_{MN} F^{MN}-\frac{1}{4g_{D5}^2}\int d^{6}x F'_{MN} F'^{MN}- \int d^6 x \left[ D_{\mu} \chi^{\dagger} D^{\mu} \chi + \frac{g_{D5}^2}{2}\sum\limits_{A=1}^{3}(\chi_i^{\dagger}\sigma_{ij}^A \chi_j)^2\right]$$ My question is about the $\chi$. The field content of the theory is : 2 vector multiplets (one from the $5-5$ strings and one from the $9-9$ strings) and 3 hypermultiplets (one from the $5-5$ strings, one from the $9-9$ strings, one half from the $5-9$ and another half from the $9-5$). In the action, the first two integrals are the kinetic terms for the vector multiplets. If I understand correctly, $\chi$ is a doublet describing the $5-9$ and $9-5$ hypermultiplet scalars. The first term in the square brakets is the kinetic term for those, but then the second term looks like the potential **for the $5-5$ hypermultiplet** (see eq. B.7.3) ! So here is my question : what is $\chi$ really, and how is the action above obtained ? Edit : my guess would be that $\chi$ is a generic notation for the scalars of all the hypermultiplets. But then there would be something like another index, because I assume that the range of $i$ is $1,2$ if $\sigma_{ij}^A$ are the usual Pauli matrices.


title: Demostrating possible equivalence of two tensors

text: Is there anyway to see by inspection that a form like $$a(x^2 )^{-3} (g _{μσ} x_{\rho} x_{ ν} + g_{μρ} x_{σ} x_{ ν} +g_{νσ} x_{ρ} x_{ μ} + g_{ νρ} x_{ σ} x_{ μ} ) $$ may be equivalent to (i.e reduced down to or reexpressed) $$ b(g _{μν} x_{ ρ} x_{ σ} + g_{ ρσ} x_{ μ} x_{ ν} )(x^2 )^{ −3} ?$$ where $g_{\mu \nu}$ is the metric tensor (diagonal). I have tried to put in various permutations of $\mu \nu \rho \sigma$ and from $1111$ and $2222$ for example, I obtained the constraint that $a/b = 2$ but I am not really sure what this means. If I try the combination $1221$ e.g then it implies $b=0$, which seems to contradict my first result. Does this mean that the two forms are not equivalent?


title: How does friction and mass of the string influence the tension force?

text: According to my book, "tension is the reaction of a rope when it is stressed". Then it also said when the string is massless, tension is same everywhere. However in a,let, pulley-rope system,when there is friction between the pulley & the rope, the tensions are not same. Why the tension is not same in this case? Does tension arise due to Newton's third law? When a block is attached to the rope ,the rope gets streched and can I say tension then arises due to Newton's third law or only to undo the deformation? If Newton's 3rd law is cause, then how can there be different tensions in a same rope? Plz explain.


title: What is the mathematical definition of work?

text: I'm looking for the pure mathematical definition of work, but I haven't yet learned line integrals. My book says that the work due to a force ${\bf F}$ from point $A$ to point $B$ is $$ W= |AB|\cdot |{\bf F}|\cdot\cos(\angle AB,{\bf F}) $$ but it also says this only applies for constant forces. I am assigned a problem which asks me to determine the work of from a point $A$ to $B$ with gravitational force $$F=GmM/R^2.$$ I don't think that I can apply the normal rule above, since it only works for forces that don't change their pointing. Am I mistaken?


title: How would the universe change?

text: How would the universe be modified if protons (as we know them) have negative charge and electrons (as de know them) have positive charge.


title: Aren't all electrons the same? So what about electron that absorbs photon?

text: I learned that electron absorbs a photon and goes into higher energy state. But also all electrons are identical. What is a difference between the electron in low orbital energy state? and the high one?


title: Was the firewall paradox resolved?

text: For quite sometime there has been a claim that the firewall paradox has been resolved (via lasers). For instance, http://global.ofweek.com/news/Lasers-to- solve-the-black-hole-information-paradox-9867 . I was wondering what the physics community thought about it? I haven't heard much of a reaction of the physics community. Why is this? His paper: http://arxiv.org/abs/1310.7914