title: POVMs that do not require enlargement of the Hilbert space

text: The usual justification for regarding POVMs as fundamental measurements is via Neumark's theorem, i.e., by showing that they can always be realized by a projective measurement in a larger Hilbert space. That justification is sometimes problematic because for some applications is important _not_ to enlarge the Hilbert space, so as to guarantee that the result you proved via POVMs is really about a Hilbert space of that dimension, and not just a shadow of a larger Hilbert space. So, my question is, how to implement POVMs without enlarging the Hilbert space? The only strategy I know is doing PVMs stochastically and grouping outcomes; for instance, the POVM $$\bigg\\{\frac{1}{2}|0\rangle\langle0|,\frac{1}{2}|1\rangle\langle1|,\frac{1}{2}|+\rangle\langle+|,\frac{1}{2}|-\rangle\langle-|\bigg\\}$$ can be implemented by measuring either $\\{|0\rangle\langle0|,|1\rangle\langle1|\\}$ or $\\{|+\rangle\langle+|,|-\rangle\langle-|\\}$ with probability $1/2$; by grouping the outcomes one can then measure the POVM $$\bigg\\{\frac{1}{2}|0\rangle\langle0|+\frac{1}{2}|+\rangle\langle+|,\frac{1}{2}|1\rangle\langle1|+\frac{1}{2}|-\rangle\langle-|\bigg\\}$$ or $\\{I/2,I/2\\}$ But this class can't be the whole set of POVMs; the Mercedes-Benz POVM (which has three outcomes proportional to $|0\rangle$ and $\frac{1}{2}|0\rangle \pm \frac{\sqrt{3}}{2}|1\rangle$) clearly can't be implemented this way. Is there a neat characterization of this class? Is there published research on it? Even better, is there another (more powerful) way of implementing POVMs without enlarging the Hilbert space?


title: General Relativity research and QFT in curved spacetime

text: A naive question: Are these subjects, i.e. classical GR and QFT in curved spacetime, being worked upon much anymore? Who is researching this and what are the problems within these fields? Any sources are greatly appreciated.


title: Why do 'dead' batteries work again after exchanging the places of the batteries in an electronic device?

text: My camera, which is powered by two AA batteries in series, would not power on. I removed the batteries, exchanged their locations, and the device worked again - for another 15 minutes or so. The temperature of the batteries did not change, and only about few minutes passed before I discovered this. Is there a physical explanation for this phenomenon?


title: Is relativistic motion equivalent to fluctuating gravitational fields?

text: The theory of relativity makes very precise predictions about how an object's motion through space-time affects the passage of time for both the object and observers in other frames of reference. I am curious, however, whether this motion is equivalent to changing / fluctuating gravitational fields? What if instead of an object moving through space/time we instead had the same object exposed to shifting gravitational fields, similar to what it would experience if it were actually moving relative to other masses? Can it be shown that the same set of effects would still exist from the object's frame of reference (eg time dialation, length- contraction, etc)?


title: In GR, how do particles know how to fall in instead of out of a gravitational well?

text: The geodesic equation (let's suppose that we're talking about massive particles, so I'll parameterize the path by proper time $\tau$) $\frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\rho \sigma}\frac{d x^\rho}{d \tau} \frac{d x^\sigma}{d \tau}=0$ is invariant under $\tau \rightarrow -\tau$. However, falling particles clearly have a direction, they always fall in instead of out. Formally, how does a falling particle 'know' which way to move in $\tau$?


title: If dark matter is a new type of particle, what does that imply?

text: My understanding is that dark matter cannot be (or is at least highly unlikely to be) an exotic form of any known particle. On the other hand, articles about particle accelerators seem to say that the Higgs is the last piece missing in the Standard Model jigsaw puzzle. _If_ dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery "stand to the side" of the Standard Model or would it certainly change the foundations? (Forgive my extremely lay understanding and vocabulary -- please feel free to correct my mistakes.)


title: Which will stop first a heavier car or a lighter car?

text: If the friction from brakes, wind resistance and all such factors remain constat, which will stop first? A heavier car or a lighter car? How will the momentum of the car and graviational pull on a heavier object influence the stopping of the car?


title: Do Category Theory and/or Quantum Logic add value in physics?

text: I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks? More particularly, is it worth investing the time to study these formalisms?


title: Calculate gravitational force between a sphere and a uniform rod

text: I want to calculate the gravitational force between a uniform rod of mass 2 kg, length 1 meter and a uniform sphere of mass 0.2 kg. Distance between the sphere and the rod is 0.1 meter ( The sphere is in the same line as the rod I.e. sphere is one meter from the near end of the rod). I thought of integrating $(G*m*M)/x^{2}$ from 0.1 to 1.1 meter which does yield the correct result but the solution in the textbook has a different procedure.


title: A universe of angular momentum?

text: I read this on Wikipedia: > _[...] That most tangible way of expressing the essence of quantum mechanics > is that we live in a universe of quantized angular momentum and the Planck > constant is the quantum. [...]_ And I'm wondering if anyone can explain this a little more. Is it saying that ALL action is made up of angular momentum which is quantised in units of Planck's constant? Or is it just saying that all matter is composed of particles which have quantised angular momentum? If the answer is the prior, this would mean that a photon's movement through space is as a result of angular momentum, which seems strange. Anyway, if you could help me out with a little explanation, that would be great.


title: Newtonian Gravity on a Riemannian $3$-Manifold

text: To solve the Poisson equation for the Newton Potential, say $\phi$, one can use the divergence theorem, such that $$\int_U \nabla^2 \phi \sqrt{g}~ dV= \int_{\partial U} <\nabla \phi,n> \sqrt{g_\Sigma}~ d\Sigma=1,$$ where $\sqrt{g_\Sigma}$ represents the induced metric on the border. Considering that $\phi$ is classic it will only depend on the relative position between the source and the point where we want to compute the potential, so, in order to use the divergence theorem, we need to find the set of points that are at the same distance of a given point (where we consider the source of the field to be) and that will be the border, $\partial U$, of the considered volume that contains the source. Well, if space is the three dimensional Euclidean space then the volume that contains the source is straightforwardly the 3-ball whose center is at the source position and the border $\partial U$ the corresponding 2-sphere. But if space is a different 3 Riemannian Manifold things are a "little" bit harder and finding the set of points that are at the same (geodesic) distance from the source is in general quite difficult or even impossible... But when searching for the geodesic curves on a manifold one can, in some circunstances, instead of searching for the curve that minimizes the distance $$d~=~\int_a^b [g(\dot{c}(s),\dot{c}(s))]^{1/2} ds,$$ search for the curve that minimizes the energy functional $$E~=~\int_a^b [g(\dot{c}(s),\dot{c}(s))] ds,$$ since in those circunstances the curves that minimize the energy are the same as the curves that minimize the distance. Well, finding the set of points that are at the same energy is much easier...I would then ask: what is the significance of the energy defined above and would it make sense, physically, to say that equipotetial surfaces are not the set of points that at the same distance from the source but the set of points that have the same energy. **PS:** Note that in the Euclidean Space, the set of points that are at the same distance is the same as the set of points that are at the same energy.


title: What physical forces give rise to the peculiar bond angle of hydrogen peroxide?

text: In one of the Periodic Table videos, Prof. Poliakoff shows a model of a hydrogen peroxide molecule and claims that the H-O bonds will always be at right angles to each other. I have a rudimentary intuition for how molecules arrange themselves based on mutual attraction and repulsion - I've solved some Foldit puzzles - but the H2O2 defies that kind of common sense explanation. I get that there are forces pushing the hydrogens apart and forces pulling them together, but how does that fragile equilibrium of forces result in an _exact_ right angle?


title: How much money does an unused but plugged-in cellphone-charger waste in a year, if its not getting warm?

text: Is it right as xkcd states: > You can use heat flow to come up with simple rule of thumb: If an **unused > charger isn’t warm to the touch** , it’s using **less than a penny of > electricity a day**. Or, more importantly, because one has usually a lot of devices like that at home: > For a **small smartphone charger** , if it’s not warm to the touch, it’s > using **less than a penny a year**. Can one estimate by some _joule per time-unit_ calculation how much electricity is turned into heat to warm up a wall-plug cellphone-charger to, say, a maximum of 25°C? From there I would will calculate the wasted money by using .25€/kWh -- a calculation which does not involve physics, which I can therefore do myself ;-)


title: The number of mean free paths which a particle traverses is independent of the material traversed

text: This is a manual from a Monte Carlo physics manual: Computing the occurrence of a process Can anyone tell me why the number of mean free paths of a particle is independent of the material which it traverses? It does not make sense to me.


title: Optical Drive Physics

text: I have been recently wondering how is data stored on and retrieved from optical devices like CDs, DVDs, and Bluray. What makes these different storage types different from each other?


title: Is there any way to prove/disprove we are in a computer simulation?

text: Is there any way to prove/disprove we are in a computer simulation in some transcendental reality? Even if we are "really" not in a simulation, can we ever prove so? Even if some messages/evidence flashing "You are in a simulation by an Overlord" pops up, can we always dismiss it as a naturalistic hallucination? Can quantum mechanics apply to closed systems? MWI and consistent histories are candidates, but run into the preferred basis/choice of consistent histories problem. Copenhagen requires a transcendental observer and a Heisenberg cut, doesn't it? If there is a transcendental observer, doesn't that imply there has to be a transcendental realm? So, if our universe is a subsystem in some transcendental realm which can perform, and only perform, Turing computable computations, isn't that equivalent to saying our universe is a computer simulation in some transcendental realm? This doesn't necessarily imply there is some Overlord who is an Intelligent Designer Programmer. It could be an Overlord who is a "Blind Watchmaker" transcendental observer. This Overlord might be deist or an occasional interventionist. In Copenhagen, in order for something to exist, it has to be observed, right? That means we need an "all-seeing eye in the sky" transcendental Overlord, right? If this Overlord forgets about us, we die, and pop out of existence, right?


title: Question about laws of conservation in electrodynamics

text: One way of deriving laws of energy, impulse and angular momentum of electromagnetic field conservation is following:   1. Introduce two values below: $$ \mathbf P = \frac{c}{4 \pi}[\mathbf E \times \mathbf B], \quad W = \frac{1}{8 \pi}(\mathbf E^{2} + \mathbf B^{2}). $$   2. Using Maxwell's equations and time derivative of these values, get the laws. But how to argue an introduce of values from it. 1? Can Noether's theorem help to argue it?


title: If the universe can expand at a speed faster than that of light, why don't we feel the effect?

text: the universe, as I have read, is capable of inflating at a speed faster than that of light! How is that even possible if we don't feel the effect? or does it affect us?


title: What is a qualitative explanation for order of values of standard entropies of formation for isomers of hexane?

text: I have literature values for the standard enthalpy and Gibbs energy changes of formation at 298 K, $\Delta H_\text{f}^o$ and $\Delta G_\text{f}^o$, _for product in the ideal gas state_. Since $G=H-TS$ and by definition, the reactions are carried out at constant temperature I can solve for $\Delta S_\text{f}^o$ via: $$\Delta S_\text{f}^o=\frac{\Delta H_\text{f}^o-\Delta G_\text{f}^o}{T}$$ I've done this for three hexanes, n-hexane, 2-methylpentane, and 2,3-dimethylbutane. Below are the results: ![hexanes thermochemistry values](http://i.stack.imgur.com/Fzrd4.jpg) I guess the sign makes sense to me. It says that all three molecules represent more order than their reactants (C and H2). What is strange to me is the order of the standard entropies of formation. Since these are values in the ideal gas state, entropy effects due to intermolecular forces should not be included, correct? Hence, I would have thought that n-hexane would be the most "ordered" of the three molecules. Surely a linear molecule is less likely (requires higher order) than a branched molecule. What have I missed?


title: failing to see the conundrum in the Einstein hole argument

text: I've been reading about the Einstein hole argument, and i fail to understand what makes active diffeomorphisms "special" compared to passive diffeomorphismsm also known as good old coordinate transformations. Supposedly this article is a good overview of the problem. I've read this, but i feel that i must be missing something, since i can't feel surprised by the fact that two different coordinate atlas that are identical up to some global time slice and differ afterwards (in the article they call it coordinate transformations that are the same before a global time slice and differ later) will imply (through the field equations) evolutions of the metric that are identical but they will also differ later. Yes, i keep saying to myself while reading this, of course they must differ when the coordinates differ and be equal when the coordinates are equal, that is what happens when you change coordinates! Now, some other interpretation of the hole argument states that a solution to a GR coordinated system with matter will have a natural mapping to another solution to a completely different system with matter that so happens to have the same coordinate system. Not sure what that does even mean, a manifold is sufficiently defined with a collection of charts and coordinate transformations in the overlap of the charts, also known as a manifold atlas. ( _yes there are subleties for these assumptions when one is involved with exotic spheres or four dimensional spaces, where there are non-equivalent differentiable structures, but let's put a pin on that for now_ ) Some other interpretation is that you need matter (and actual observers) to actual pinpoint invariants that are preserved by these active diffeomorphisms, since all observables are done in relation with these. That is a fair statement, but seems to me pretty much devoid of any actual content, since it is natural that one will always make statements relative to, say, the earth, the sun, the milky way, even if the statements are written in manifestly covariant expressions, one will always take a collection of preferred charts to write fields in the given coordinates, which the measurement apparatus are calibrated and gauged against. Is there something am i missing? i feel like i'm going nuts over the silliness i perceive in this argument


title: Experimentally determining photon lifetime of a laser from transient response

text: At our lab, we have a DBR laser set up to some measuring equipment (oscilloscope, spectrum analyzer and random number generator). I have been tasked to experimentally obtain the photon lifetime of the laser. I suppose that analyzing the transient response on the oscilloscope would be a good way to begin. Perhaps seeing the relaxation oscillation frequency and using that in calculations. That is as far as I've been able to figure out and would like some pointers from the community. Thanks!


title: What is 'tilt locking' of a laser cavity?

text: In another question, jasonh mentions a scheme for locking a laser to a resonant cavity called 'tilt locking': > The signal is used for locking the frequency of a laser to a high finesse > cavity. The lock has to be stable enough for intensity and phase > fluctuations resulting from e.g. acoustic vibrations of the cavity to be > strongly suppressed. The error signal is created via the tilt locking > scheme, i.e. it is the difference of intensities on two halves of a split > photo diode. I'm familiar with Pound-Drever-Hall locking of a laser cavity, but I've never heard of 'tilt locking' before. How does it work?


title: Conformal transformation/ Weyl scaling are they two different things? Confused!

text: I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate transformations that effect the above metric transformation i.e $x \to x' \implies g_{\mu \nu}(x) \to g'_{\mu \nu}(x') = \Omega(x)g_{\mu \nu }(x)$. but any covariant action is clearly invariant under coordinate transformation? I see that what we mean by weyl transformation is just changing the metric by at a point by a scale factor $\Omega(x)$. So my question is why one needs to define these transformations via a coordinate transforms. Is it the case that these two transformations are different things. In flat space time I understand that conformal transformations contain lorentz transformations and lorentz invariant theory is not necessarily invariant under conformal transformations. But in a GR or in a covariant theory effecting weyl transformation via coordinate transformations is going to leave it invariant. Unless we restrict it to just rescaling the metric? I am really confused pls help.


title: Transition from coordinate space to momentum space for SHO

text: I am given that the ground state of the SHO in position space is given as $$\langle q|\psi_0\rangle=\frac{1}{a^{\frac12}\pi^{\frac14}}e^{-q/4a^2}$$ Where a is a constant with units of length. I am then asked to find the corresponding momentum space distribution. I know this will be related to a fourier transform, but I am unsure how to put it in this notation. I also know: $$\langle q|p \rangle=ce^{ipq/\hbar}.$$ Now to get from $\langle q|\psi_0 \rangle$ to $\langle p|\psi_0\rangle$ I am unsure of which properties to use.


title: Derive time difference of phase rotated wavelets

text: I have one question concerning phase rotation of signals and I am not sure wether it works or not: Let's assume we have a zero-phase wavelet x(t) i.e. Ricker wavelet. Unfortunately, in our experiment, the wavelet y(t) we use is not zero phase anymore but rotated by an phase angle $\Psi$. Yet, we still can clearly identify the peak of the wavelet at x(T). **Question** Is it possible to derive the position of the peak of the corresponding zero-phase wavelet x(t) from y(t)? First steps are: 1) equation Ricker wavelet: $$ x(t)=(1-2 \pi^2 f^2 t^2) e^{-\pi^2 f^2 t^2} $$ 2) equation rotated Ricker wavelet $$ y(\tau)=(1-2 \pi^2 f^2 \tau^2) e^{(-\pi^2 f^2 \tau^2 - \Psi \tau)} $$ My assumption was setting $y(\tau) = x(t)$ and solve for $t$ since $\tau,f,\Psi$ are known. Unfortunately the amplitude of the wavelet changes with the phase rotation. How can I account for this? please help!


title: Force needed to change momentum, from fixed position

text: I have a situation where I want to change the velocity of a mass, by applying a force from a fixed position. For example in the diagram below, the mass starts with the initial velocity in the upper left, and it must travel through the goal point with the final velocity. The catch is, this must be effected by a repelling force applied from the 'force origin' at the bottom of the diagram. ![enter image description here](http://i.stack.imgur.com/WjsVf.png) A real-life analogy would be a boat approaching a dock, and somebody on the boat reaches out with pole and pushes the boat away from the dock, with a single steady push and only touching the dock in one place. I need to calculate the force required at any given point during this transition. For example in this particular case if the mass was to follow the red dashed line, no force is necessary at all initially because it needs to move in a straight line for a while. During the red dashed line phase, the force will be needed, and will be strongest at the middle of that phase. The radius of the arc is not important, although it would ideally be as large as the situation will allow. The magnitude of the final velocity is also not important, and it does not seem like something that can be controlled anyway... It occurs to me that during the red dashed phase the mass undergoes a centripetal acceleration toward the focal point of the arc segment (which would be a heck of a lot easier to calculate!), but how to determine the force needed to accomplish that by pushing from a fixed point on the outside of the arc, I just cannot figure out.


title: Why does a magnetic field raise the ground state energy of an electrical particle?

text: I heard a statement that the ground state energy of a electrical particle in a magnetic field is larger than its ground state energy without the magnetic field. I just heard this statement. This statement shows an energy shift by a magnetic field. This energy shift is different from Zeeman splitting. Maybe I have some mistakes when I quote this statement. And I don't know why the ground state energy is different. I guess it has a quantum origin. If you have heard a similar conclusion, could you tell me the more details? Mathematics and proofs are much better.


title: Is vapour pressure based on partial pressures or just total pressure on the liquid?

text: The explanation for the boiling point of water is that at 100C, the vapour pressure becomes greater than atmospheric pressure. But say you had a a jar of water sealed in argon at 1atm, which is larger than the vapour pressure of water, wouldn't some water evaporate? I'm guessing yes, because there are no water molecules in the argon initially, so wouldn't some randomly leave the liquid, making the partial pressure on average non-zero? Another reason I can think of it being based on partial pressure is that an open jar of water in a windless dark, dry room will still eventually evaporate. So why is it that at 100C there is a sudden dramatic change? Isn't the vapour pressure at e.g. 95C more than the partial pressure of water in the atmosphere as well?


title: Sobolev norm for Schwarzschild metric

text: Considering a static spacetime of the metric form \begin{equation} \mathrm{d}s^{2}=-V^{2}\mathrm{d}t^{2}+h_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j} \end{equation} with a timelike killing field $\xi^{\mu}=(\partial_{t})^{\mu}$ we can choose a function space on each constant time hypersurface $\Sigma$ as$\mathcal{H}=\left\\{ f\,\mid\,\parallel f\parallel<\infty\right\\} $with the Sobolev norm $\parallel f\parallel$being given by \begin{eqnarray*} \parallel f\parallel^{2} & = & \frac{q^{2}}{2}\underset{\Sigma}{\int}\mathrm{d}\Sigma V^{-1}f^{*}f+\frac{1}{2}\underset{\Sigma}{\int}\mathrm{d}\Sigma Vh^{ij}D_{i}f^{*}D_{j}f. \end{eqnarray*} In (negative mass) Schwarzschild coordinates this norm reads reads \begin{equation} \parallel f\parallel^{2}=\frac{q^{2}}{2}\underset{\Sigma}{\int}\mathrm{d}\mu\mathrm{d}rR^{n}V^{-2}\mathcal{\mid}f\mid^{2}+\frac{1}{2}\underset{\Sigma}{\int}\mathrm{d}\mu\mathrm{d}rR^{n}V^{2}\mid f'\mid^{2} \end{equation} with $\mathrm{d}\mu$ being the volume element of unit $n$-sphere ($\mathrm{d}\Sigma=\mathrm{d}\mu\mathrm{d}rV^{-1}R^{n}$). My question is: How does this norm read for positive mass Schwarzschild spacetime? $V^2$ is assumed positive above (which is ok for $V=(1+2M/r)$ (M positive). But for $V=(1-2M/r)$ there will be the point of a change in sign so that the above norm cannot hold anymore.How does the norm then look like??? Thanks!


title: Gauge fixing of an arbitrary field

text: How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?


title: Time evolution in a static magnetic field of a 2-level system that is actually a 3 level system

text: What I want to do is describe the time evolution of only two states of a spin 1 particle on the bloch sphere. However, I'm running into trouble because I don't know how to construct the hamiltonian for those two states, such that the time evolution operator is correct. I'll describe what I mean in more detail below: I have a spin-1 particle, of which I'll describe the spin state as $m_S$. So there's three possibilities, $m_S = -1, 0, 1$. Now, if I want to describe how this spin behaves in a uniform static magnetic field, I simply write \begin{equation} {\cal H}_{0} = -\hbar \gamma_e B_0 \frac{\sigma_z}{2} = -\hbar \omega_0 \frac{\sigma_z}{2} = \begin{pmatrix} -\frac{\hbar \omega_0}{2} & 0 & 0\\\ 0 & 0 & 0\\\ 0 & 0 & \frac{\hbar \omega_0}{2} \end{pmatrix} \end{equation} Where $\sigma_z = \begin{pmatrix} 1 & 0 & 0\\\ 0 & 0 & 0\\\ 0 & 0 & -1 \end{pmatrix}$ Using the usual exponential for the time evolution, one has $U = e^{-i{\cal H}_{0}t}$ Up to this point I'm fairly certain I'm doing it right, but now I want to do something where it gets a little fuzzy for me. Instead of looking at the system as a three level system, I only want to look at two of the levels ($m_S = 0$ and $m_S = -1$) so that I can look at the system on the Bloch sphere. However, how do I do this? I suppose I can just choose my basis and define $ \left|0\right> = \begin{pmatrix} 1 \\\ 0 \end{pmatrix}$ $, \left|1\right> = \begin{pmatrix} 0 \\\ 1 \end{pmatrix}$ However, how does one now describe the time evolution of these two states in a magnetic field, like I did with the three states before? I can't simply take the spin 1/2 pauli matrix, because the spin 0 state does not evolve in a magnetic field. I suppose I could only take the part of the hamiltonian that has to do with the 0 and -1 states, but then how would this time evolution work in a tensor product, if my states are described with two coordinates and the hamiltonian is a 2x3 matrix?


title: Where does the $50^*$ in $SU(5): 10\otimes10= 5^*\oplus45^*\oplus 50^*$ in A. Zee QFT?

text: See A. Zee, _QFT in a nutshell,_ Appendix B, eq. (24) (p. 469 in first edition with a typo $55^*\to50^*$, cf. Zee errata; p. 530 in second edition.) Where does the $50^*$ in $SU(5)$: $$10\otimes10= 5^*\oplus 45^*\oplus 50^*\tag{24}$$ come from?


title: Why is the orbital angular momentum of a pi electron along the axis of two atoms' molecule one?

text: I'm reading quantum chemistry. The book says that the orbital angular momentum of a $\pi$ electron along the symmetry axis of a molecule made up of two atoms is $\pm 1$. I think this is a primary question, but I do not konw why. * * * I currently have a preliminary understanding of this:   ![$/pi$ orbit](http://i.stack.imgur.com/qxWju.png)   In a molecule inculding two atoms, potential energy axisymmetric about z' axis(the line connecting the two atoms). So, the angular momentum along z axis is quantized. That is to say $m_z$ is good quantum number. Let's consider $\pi$ orbit made up of two $p_z$. The orbital angular momentum along z axis(the symmetry axis of $p_z$) of an electron in $p_z$ is 0. So considering orbital angular momentum along z' axis of this electron, the electron is in $\frac 1 {\sqrt 2}(|+\rangle-|-\rangle)$. So the orbital angular momentum along z' axis is either 1 or -1. So the orbital angular momentum along z' axis of a $\pi$ electron is either 1 or -1.   Is my understanding right?


title: Enthalpy in thermodynamics!

text: What does enthalpy tell us about? According to definition; it is the total heat content of a system, if it is the total heat content of a system then what is Internal Energy? Since we know,   $$ H = U + PV $$ If we have one work done $d(PV)$ in Internal Energy itself,then what is the need of the second one?


title: Questions about four-momemtum

text: I am reading a note about Kerr metric, following is some screen shot where I have problems. * * * ![enter image description here](http://i.stack.imgur.com/RLNAq.png) ![enter image description here](http://i.stack.imgur.com/2p8Hn.png) * * * First of all, I think the author made a mistake. It should be $$ E=-u_\mu k^\mu=u^t\;\;\;L=u_\mu m^\mu=u_\phi $$ My questions are: 1) we have $E=u_t$, not $u^t$, why the author says > they are the energy at infinity per mass unit... 2) For massless particle, why can we choose properly the affine parameter so that the four-momentum coincide with the four velocity? I have done some google search about the four-momentum, such as this post. Are there two kinds of definitions? $$ P^\mu=mU^\mu=(-E,p_1,p_2,p_3) $$ and $$ P_\mu=mU_\mu=(E,p_1,p_2,p_3) $$ ? If so, how can these two be consistent? I don't think we have $$ -E=P^0=P_ag^{a0}=Eg^{00}+p_ag^{a0} $$


title: Mode groups in an optical fiber

text: I know what modes in an optical fiber mean but what are exactly mode groups in an optical fiber? From what I read until now, I have the impression that modes that have close propagation constants belong to the same group. But how 'exactly' do we group modes?


title: Help understanding proof in simultaneous diagonalization

text: The proof is from _Principles of Quantum Mechanics_ by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors that diagonalizes them both. The proof is: Consider first the case where at least one of the operators is nondegenerate, i.e. to a given eigenvalue, there is just one eigenvector, up to a scale. Let us assume $\Omega$ is nondegenerate. Consider any one of its eigenvectors: $$\Omega\vert\omega_i>=\omega_i\vert\omega_i>$$ $$\Lambda\Omega=\omega_i\Lambda\vert\omega_i>$$ Since $[\Omega,\Lambda]=0$ $$\Omega\Lambda\vert\omega_i>=\omega_i\Lambda\vert\omega_i>$$ i.e., $\Lambda\vert\omega_i>$ is an eigenvector with eigenvalue $\omega_i$. Since this vector is unique up to a scale, $$\Lambda\vert\omega_i>=\lambda_i\vert\omega_i>$$ Thus $\vert\omega_i>$ is also an eigenvector of $\Lambda$ with eigenvalue $\lambda_i$... What I do not understand is the statement/argument "Since this vector is unique up to a scale." I do not see how the argument allows to state the equation following it. What axiom or what other theorem is he using when he states "since this vector is unique up to a scale"?


title: Two slit experiment: Where does the energy go?

text: In Physics class we were doing the two slit experiment with a helium-neon red laser. We used this to work out the wavelength of the laser light to a high degree of accuracy. On the piece of paper the light shined on there were patterns of interference, both constructive and destructive. My question is, when the part of the paper appeared dark, where did the energy in the light go?


title: Relation between Force, Time, and Energy

text: 1. Is energy involved independent of force applied over time?   2. In other words, if I wish to apply a force for much more time would I require more energy?


title: Why are there more lunar maria on the near side than on the far side?

text: It is widely known that the side of the moon that faces us has a particularly large number of dark patches, which resemble seas of liquid, hence, "maria". It is strange, though, that the far side of the moon has a great lack of these features. Why does the far side of the moon have so few maria? If they maria formed due to impacts, is there some reason that the far side would have less collisions?


title: What objects look best in an O-III filter?

text: I've heard that an O-III (Oxygen III) filter is great for planetary nebulae. Is this true for all planetary nebulae, or just some or most? What other target types are often improved with an O-III filter?


title: What makes a good set of binoculars?

text: After trying a bunch of binoculars at Star Parties and the like, I have a pretty good feel for about what aperture and magnification I'd like in a new set binoculars. I'm an eyeglass wearer so a long eye relief would be nice, but what else should I be looking for in a good set of binos?


title: What's the deepest reason why QCD bound states have integer charge?

text: What's the deepest reason why QCD bound states have integer electric charge, i.e. equal to an integer times the electron charge? Given that the quarks have the fractional electric charges they do, this is a consequence of color confinement. The charges of the quarks are constrained in the context of the standard model by anomaly cancellation, and can be explained by grand unification. The GUT explanation for the charges doesn't care about the bound state spectrum of the QCD sector, so it just seems to be a coincidence that hadrons (which are composite) have integer charge, and that leptons (which are elementary) also have integer charge. Now maybe there's some anthropic argument for why such a coincidence is useful (in the case of proton and electron, it gives us atoms as we know them). Or maybe you can argue that GUTs naturally produce fractionally charged particles and strongly coupled sectors, and it's just not _much_ of a coincidence. But I remain curious as to whether Seiberg duality, anyons, some UV/IR relationship... could really produce something like the lepton-hadron charge coincidence, for deeper reasons. I suppose one is looking for a theory in which properties of bound states in one sector have a direct and nontrivial relationship to properties of elementary states in another sector. Is there anything like this out there? (This question was prompted by muster-mark's many recent questions about fractional charge, and by a remark of Ron Maimon's that the hadron-lepton charge coincidence is a "semi-coincidence", which assured me that I wasn't overlooking some obvious explanation.)


title: Figuring out which moons of Jupiter I saw through my telescope

text: Three days ago (Sunday 11st March) in London using my new telescope I was able to see Jupiter and three of its moons. It was very clear and easy to spot. How can I determine the names of the moons I saw (and which one was which)? I would assume they were the Galilean moons - as they are the largest, making the easily visible, however, I am not sure which moons I would have seen. This is how they lined up through the ‘scope : .                 .           O               .      where .=moon & O= Jupiter Any help would be appreciated. Thanks


title: What is behind the cosmological horizon barrier?

text: I'm wondering what is behind the cosmological horizon barrier?


title: Are we going to be able to travel trough space deforming the space-time?

text: I'm not talking about the speed of the spaceship. If we can deform space-time we needn't any type of propulsion. And how can the travel affect to it's pilots? Can they survive?


title: What is an approximation of the average number of supernovae every century in the Milky Way?

text: Besides giving out a number and quoted source I would appreciate a short derivation of this number/formula based on our current data and knowledge of supernovae and the Milky Way. Show some traceable reasoning that proofs the source. Aditionally, how many of these actually (on average) happening supernovae (also in neighbor galaxies) are spotted by amateur and professional astronomers?


title: If you focus in on one image, is combining two telescopes really equivalent to a larger mirror?

text: Have a look at this example: > In addition, the Keck I and Keck II telescopes can work together as the Keck > Interferometer. The 85-metre (279 ft) separation between the two telescopes > gives them the effective angular resolution in one direction of an 85-metre > (279 ft) mirror So, If you focus in on one image, is combining two telescopes (through optical interferometry) really equivalent to having a larger mirror? It's a follow-up question to an answer I got at here...


title: Looking for a resource that explains all colors and shapes of stars and galaxies

text: I've been looking at some of the astronomy pictures of the day from NASA. They all have content that is indicative of certain situations or events. Supernovas have the scattered lines look, new stars all seem to be blue, dust (I assume) takes the color of stars behind it, etc. Is there a resource that I can use that will explain all of the known colors and shapes given by these structures in space? Say I wanted to program a simulator to create images similar to these, I would need to know what causes the colors, the shapes, etc. I know there are infinite possibilities, but there are definitely repeated themes like the ones I mentioned. I get that some of these are false color, but that's irrelevant to me, the false colors usually mean something anyway, so just ignore the fact of there being false color.


title: How are galaxy filaments formed? And do they have any analogues in stellar formation?

text: > In physical cosmology, galaxy filaments, also called supercluster complexes > or great walls, are, so far, the largest known cosmic structures in the > universe. They are massive, thread-like structures with a typical length of > 50 to 80 megaparsecs h-1 that form the boundaries between large voids in the > universe.[3] Filaments consist of gravitationally-bound galaxies; parts > where a large number of galaxies are very close to each other are called > superclusters. So I've always wondered: how are these structures formed? One would (naively) expect that gravitationally bound structures would form as either spheres or discs.


title: Can spin and macroscopic angular momentum convert to each other?

text: Suppose an isolated system with a number of particles with parallel spins. Can the macroscopic angular momentum of the system increase at expense of the number of particles having parallel spins (that is by inverting the direction of the spins of some of them) or by converting all particles into spinless ones? Conversely in a rotating system of spinless particles can the macroscopic algular momentum be decreased by converting some of the particles into those having parallel spins? Can one choose a rotating frame of reference in such a way so in it a particle that has spin in inertial frame, does not have one?


title: World-sheet energy-momentum tensor and OPE

text: On p43 of Polchinski's book, it says that under the world-sheet translation $\sigma^a\rightarrow\sigma^a+\epsilon v^a$, $X^\mu\rightarrow X^\mu-\epsilon v^a\partial_a X^\mu$. And $$j^a=iv^b T_{ab},$$ $$T_{ab}=-\frac1{\alpha'}:(\partial_aX^\mu\partial_bX_\mu-\frac12\delta_{ab}\partial_c X^\mu\partial^cX_\mu):$$ Could someone please tell me where such expressions of $j^a$ and $T^{ab}$ come from? I assume that we start with the action $$S=\frac1{4\pi\alpha'}\int d^2\sigma\,(\partial_a X^\mu\partial_1 X_\mu+\partial_2 X^\mu\partial_2 X_\mu)=\frac1{2\pi\alpha'}\int d^2z\,\partial X^\mu\bar\partial X_\mu,$$ but I don't see how we can apply the definition of $T_{ab}$ from GR... Further, on the next page, when expressed in complex coordinates, it says $$T(z)=-\frac1{\alpha'}:\partial X^\mu\partial X_\mu:,~~~~~\tilde T(z)=-\frac1{\alpha'}:\bar\partial X^\mu\bar\partial X_\mu:$$ How do you get here from the $T_{ab}$ above?


title: Renormalizibility by power counting

text: When testing a theory for its renormalizability, in practice one always calculates the mass dimension of the coupling constants $g_i$. If $[g_i]>0$ for any $i$ the theory is not renormalizable. I am wondering where this criterion/trick comes from? Is there an easy way to see that a coupling constant with positive mass dimension will yield a non-renormalizable theory?


title: Ising Hamiltonian for relativistic particles

text: An Ising system is described by the simple Hamiltonian: $$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units of $\hbar$), the $c_{1i}$ are local magnetic fields and the $c_{2ij}$ are (nearest neighbor) interaction constants. So one usually only keeps constants $c_{2ij}$ if $x_i$ and $x_j$ are located close enough in space; after a certain spatial range the $c_{2ij}$ are supposed to become negligible. Note that (1) can be interpreted as two low-order terms of a general Taylor expansion of a general Hamiltonian H($x_1$,$x_2$,…,$x_n$): $$H = c_0 + \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j + \sum\limits_{i,j,k} c_{3ijk} x_i x_j x_k + … \,\,\,\,\,\,(2)$$   My questions are the following:   Q1) Does (1) also hold if the particles are moving ? Looking at Feynman’s book on stat mech it seems that (1) is then also valid: the form (1) arises according to Feynman due to magnetic dipole interaction or due to a “Coulomb potential combined with Pauli’s exclusion principle”; nothing in the derivation demands that the particles are static.   Q2) Suppose all particles are on a 2D lattice, except two particles (1 and 2) which are emitted from a source in the middle of the lattice and move in opposite direction through the lattice at relativistic speeds. Can we apply (1) here or do we need a corrected version ? (One way to correct (1) if some particles are moving might be to include higher order terms as in (2).)


title: Why vary the action with respect to the inverse metric?

text: Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to the inverse of the metric, i.e. indices are in superscript. Why is this? Why not vary with respect to simply the metric tensor? Or is it just a convention/preference?


title: Angular velocity of precession

text: So in my textbook they say this ${\rm d}\theta$ = |$d\vec{L}$|/|$\vec{L}$| $d\vec{L}$ is the change in angular momentum caused by a torque whose vector is perpendicular to $\vec{L}$, which is the angular momentum vector of a spinning disk. Think of a gyroscope. Imagine a right-angle triangle whose vertical component is $d\vec{L}$ and horizontal component is $\vec{L}$. Next to the hypotenuse and the horizontal component is angle $\theta$. I assume they say ${\rm d}\theta$ = |$d\vec{L}$|/|$\vec{L}$| because d$\theta$ is so small and close to zero. BUT HOW IS IT THAT IT IS EQUAL? I mean I can see they are roughly equal but why are they equivalent? I can't find a proof anywhere. If anything I'm talking about the angular velocity of precession. That is, how the original axis of angular momentum is changing with respect to time, due to some torque acting on the system. I kind of feel we're treating differentials too freely in physics. But it is true that I only know first-year calculus so I don't know anything about differentials. Thank you.


title: Why do sand dunes, hills and mountains assume the shape they normally do?

text: What about something taken in a tablespoon? This shape, as I understand, could be explained by gravity, friction and similar concepts, but why the shape isn't cubical or cylindrical couldn't be explained in easier terms. Please help me sort out this conundrum.


title: Is this oscillator driven?

text: > A mass $m$ is attached to a vertical massless spring or a spring constant > $k$. Originally, the spring was relaxed because the mass was held by a clip. > Suddenly the clip was released. THe mass dropped down and the maximal > elongation of the spring was recorded as $l$. What is the resonant frequency > of the system? The gravitational constant is $g = 9.8 \frac{m}{s}$. I assume (based on instructions from the professor) the intention of the question is that there is some damping going on here (damping constant $b$). What I'm confused about is if the gravitational force here makes this a driven damped oscillator or not. The "driving" force is a constant, so it isn't changing the oscillations at all (I think?). Meaning, is my equation (for $\omega = \sqrt{\frac{k}{m}}$ and $\beta = \frac{b}{2m}$): $$ \ddot{x} + 2\beta\dot{x} +\omega^2x = 0 $$ or $$ \ddot{x} + 2\beta\dot{x} +\omega^2x = g $$ If it is driven, is my resonant frequency $\omega_r = \sqrt{\omega_0^2 - 2\beta^2}$? What else would resonant frequency mean in this case?


title: Effects of space mining on Earth's orbit

text: I was reading a post about space mining, specially lunar mining. I was thinking about what would change in Earth's orbit if we start bringing tons of rocks to it? I mean, in a huge scale. So, would space mining change Earth's orbit in a way to produce any dangerous effects? If the system in question is not a planet/satellite like Earth/Moon, would that change anything? Would bringing tons of rocks from Mars be any different than Moon?


title: How to Perform Wick Rotation in the Lagrangian of a Gauge Theory (like QCD)?

text: I'm studying Lattice QCD and got stuck in understanding the process of going from a Minkwoski space-time to an Euclidean space-time. My procedure is the following: I considered the Wick rotation in quantum mechanincs $x_0 \to -i x_4$. From this, I thought it would be reasonable to assume that for the potential vector, the Wick rotation would be $A_0 \to -i A_4$, since $A_\mu$ is a four- vector like $x_\mu$. This implies $F_{0 i}F^{0 i} \to -F_{4 i}F_{4 i}$ and assuming a metric $g^{\mu \nu} = \;\mbox{diag}(1,-1,-1,-1)$, this results in $F_{\mu \nu}F^{\mu \nu} \to -F_{\mu \nu}F_{\mu \nu}$. Now, considering that $d^4x = dt\, d^3x \to -i d\tau\, d^3x$ the action should transform as \begin{equation} i S = -\frac{i}{2}\int d^4x \;\mbox{Tr}(F_{\mu \nu}F^{\mu \nu}) \to \frac{1}{2}\int d^4x \;\mbox{Tr}(F_{\mu \nu}F_{\mu \nu}) = S_E\,, \end{equation} where $S_E$ is the Euclidean action, which is a positive number. So, $iS \to S_E$ instead of the expected $iS \to -S_E$. I am obviously doing something wrong. I suspect it could be in the transformation of $d^4x$, but I cannot see why it would be wrong. One thing that I noticed is that if I use the metric $g^{\mu \nu} = \;\mbox{diag}(-1,1,1,1)$, then I get the proper signal. But this is changing the metric in the middle of the calculation, which would be wrong without compensating with an appropriate minus signal and then the issue would persist. I have issues with the Fermionic sector as well. I considered $\partial_0 \to -i\partial_4$ following the transformation of $x_0$. Also, I saw in the books (Gattringer, Rothe) that it was needed that $\gamma^0 \to \gamma_4$ and $\gamma^i \to i \gamma_i$ so the definition for the $\gamma$ matrices could change from $\\{\gamma^\mu,\gamma^\nu\\} = 2 g^{\mu \nu} \to \\{\gamma_\mu, \gamma_\nu\\} = 2 \delta_{\mu \nu}$. It seens reasonable. The problem is that the transformation in the action becomes \begin{equation} iS = i\int d^4x \; \bar{\psi}(i\gamma^\mu \partial_\mu + g_0 \gamma^\mu A_\mu - m)\psi \to \int d^4x \;\bar{\psi}(\gamma_\mu \partial_\mu - i g_0 \gamma_\mu A_\mu - m)\,, \end{equation} which is not the Eucliden action. I tried using $A_0 \to i A_4$ in the hope I could have made some mistake in the logic above, but with no luck. So what is the prescription to perform the Wick rotation? How to figure out which transformations I should perform in a wick rotation? Recomendations of papers and books are welcome. Thanks for the help.


title: Derivation of Boundary Conditions

text: ![Picture](http://i.stack.imgur.com/7vhGa.jpg) ![Equations 1 and 2](http://i.stack.imgur.com/ofSbh.jpg) Source: http://my.ece.ucsb.edu/York/Bobsclass/201C/Handouts/Chap1.pdf (page 6). I am trying to make sense of the derivation on the right side of these two integrals. The first one which says that the surface integral of $D$ on the boundary (that is when $\Delta$h and $\Delta$A become infinitesimal) is the difference in the $D$ between surface 1 and 2 multiplied by the area $\Delta A$. But how? At the boundary, I do not understand what is going on. The second integral value is just the charge enclosed on the surface, but $\rho_{se}$ is the electric charge density on which surface (that is surface in region 1 or region 2)? I followed a couple of text books, my teachers' lecture slides, plus other resources but I am unable to understand this. Also, if someone could please teach me how to fish rather than give me the fish (find answers to my questions on my own rather than give the direct answer..like where did my thinking go wrong) that would be great.


title: How fast does it take to create a magnetic field in a solenoid?

text: If a solenoid/electromagnet has current flowing, it creates a magnetic field. Electricity is very very fast, I believe close to the speed of light? So, when power is given to a solenoid with $n$ turns, how fast is the magnetic field created? Similarly the same question, but for the case of a wire.


title: Beryllium Vacuum Sphere Boat/Aircraft

text: Is it possible to make a solid rigid evacuated "balloon" out of Beryllium or other elements or alloys? The critical buckling pressure at which an evacuated sphere is given as $$ P_1=\frac{2E\cdot\left(h/R\right)^2}{\sqrt{3\left(1-\nu^2\right)}} $$ where $E$ is Young's modulus, $h$ the shell's thickness, $R$ the radius to the midsurface shell, and $\nu$ is Poisson's ratio. Of course this is theoretical and experimental and literature values for submersibles hover around 1/4 of the theoretical pressure. Beryllium has the following properties:   * Density=1.85 g/cm$^3$   * Young's Modulus=287*10$^9$ Pa   * Poisson Ratio=0.032 Calculating the upward force, buoyancy at 101.3 kPa atmospheric pressure and Radius of 10m gives 4.24*10$^8$ N. The weight of this sphere with shell thickness 1mm has 2.32*10$^7$ N downward force of Beryllium material, so it will theoretically float at ground level. Calculating the critical buckling stress of this beryllium sphere is 331.57 kPa, which is 3.27 times the atmospheric pressure of 101.3 kPa, so theoretically it won't buckle. But in experimentation done in earlier experiments as stated before, due to construction/material non-ideal situations, the critical buckling formula predicts about 4 times stronger than actual, so it would need about 400 kPa, not 331 kPa as Beryllium sphere with this specification would hold. Increasing the strength capability also increases weight and therefore lowers it's ability to float, but Beryllium is the closest element with this situation I have calculated that could work if ideal (I have only been using periodic table elements and household materials) Is it possible with an alloy of elements? Using the equations for buckling and buoyancy, the following is needed. What alloys/materials come to your mind? Material considerations:   1. Material with larger Poisson ratio   2. Material with larger Young's Modulus   3. Less dense material   4. Construction (is it possible) Material considerations affect on lift   1. Thickness can be larger, increases strength by the squared. But also increases weight. If a material with lower density with similar structural properties as Beryllium is used, it would help.   2. Radius can be lower for same reason, but also lowers buoyancy (volume) My other designs (don't worry about these just yet, just if you are curious is a vacuum sphere with a pressurized "outer tube" with spokes in tension (carbon fiber nanotubes maybe) to keep the 2 spheres in position. Another is a spherical truss out of carbon fiber nanotubes with a light membrane around. The truss prevents the outer membrane from collapsing due to the atmospheric pressure The "point" of this if you ask, as to why not use a traditional balloon is it can be used as a boat by filling the sphere with air, and then used as a aircraft at will by evacuating the sphere. An alternative mode of transportation for sea vessels etc.


title: Can you create mass with $E=mc^2$?

text: If you use the equation $E=mc^2$ could you make matter by dividing the $c^2$? I'm sorry if this is a really stupid sounding question or if it shouldn't be asked here.


title: Light traveling through a medium

text: Does the frequency of light change when it travels across an interface between two media? What happens to the light wavelength and the light velocity at the interface? I've gotten different answers some say all three change and some say frequency is constant or some say that the wavelength is constant... What's the right answer?


title: On this infinite grid of resistors, what's the equivalent resistance?

text: I searched and couldn't find it on the site, so here it is (quoted to the letter): > **On this infinite grid of ideal one-ohm resistors, what's the equivalent > resistance between the two marked nodes?** > > ![Nerd Sniping](http://i.stack.imgur.com/gUKoJ.png) With a link to the source. I'm not really sure if there is an answer for this question. However, given my lack of expertise with basic electronics, it could even be an easy one.


title: Driving on snowy roads

text: 'tis the season as they say! It seems to me obvious that it's better to drive in high gear on snowy roads to reduce the torque. However, there are completely opposite advices being given on different sites:   * weather.com says "Use low gears to keep traction"   * bbc.co.uk says "Stay in a higher gear for better control" Who is right? You can use a simplified model of the car:   * the torque of the engine is directly controlled through a smooth limited function ($0\leq T\leq T_{max}$ ) whose time derivative is also limited.   * torque is transmitted through a choice of gears to the wheels (ratios are givens)   * the wheels need not slip on the road ($\mu$ is given, circumference is given) Questions:   1. What is the relationship between gear ratio, torque function and car acceleration?   2. Is it true that there is a preferred gear ratio (given a set) that always maximises acceleration, independent of the choice of function? If so, which?   3. What is the best choice of function and gear to maximise acceleration?


title: Klein–Gordon equation

text: It is said that Klein-Gordon equation is a relativistic version of the Schrodinger equation. In Schrodinger equation, it is straightforward to include potential energy. But for K-G eqn things seem to be more complicated. To put it specifically, how can one find the energy eigenfunctions of a particle in a finite square well via K-G eqn?


title: Question about an example of non-integrable constraints

text: The example is a thin disk rotating of an inclined plane. The disk can roll not only down the plane, but also "sideways". Let $(x,y)$ be the position of the CM, where the $y$ axis is down the slope and the $x$ axis is horizontal on the plane, the angle the disk has rolled w.r.t its symmetry axis be $\phi$, and the angle of the disk plane made with the $y$ axis be $\theta$. Then the constraints are $$\dot{x}=R\dot{\phi}\sin\theta,\quad \dot{y}=R\dot{\phi}\cos\theta$$ or $$dx=R\sin\theta d\phi,\quad dy=R\cos\theta d\phi$$ Then the author argued that the constraints are non-integrable as follows: Suppose there were a constraint of the form $$f(x,y,\phi,\theta,t)=0$$ then $$0=df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial \phi}d\phi+\frac{\partial f}{\partial \theta}d\theta+\frac{\partial f}{\partial t}dt$$ Then $$dx=R\sin\theta d\phi, \quad dy=R\cos\theta d\phi$$ implies $$\frac{\partial f}{\partial x}=1,\quad \frac{\partial f}{\partial \phi}=R\sin\theta, \quad\frac{\partial f}{\partial y}=0,\quad \frac{\partial f}{\partial \theta}=0.$$ ... I can't see how to get the last conclusion above.


title: under what conditions happen the anti-Zeno effect?

text: As you might know, the Zeno effect is intuitively expressed as what happens when a system is measured in intervals smaller than the half life of the state it is currently on. As a consequence, the state has a negligible probability of doing a transition and is kept 'stuck' in its current state, making the effective evolution operator the identity. I don't know an equivalent picture for the Anti-Zeno effect. Under what conditions does it happen and why? **does the above picture is merely interpretational or is fully accurate at a fundamental level?**


title: When is many-body perturbation theory valid?

text: I'm calculating expectation values (thermal, time-independent) using many-body perturbation theory, but I'm unsure how to work out what values the parameter I'm expanding the perturbation series in can take. I read that it's when the matrix elements, $\langle i | H_{pert} | j \rangle$ where $H_{pert}$ is the perturbative term of the Hamiltonian and $| i \rangle$ and $| j \rangle$ are the eigenvectors of the unperturbed Hamiltonian, are much smaller than the energy difference between $i$ and $j$. But I don't really understand what that means, or how it helps me to calculate what values my perturbation parameter can take - Is there a method I can use to figure it out? **Edit:** As requested, to make this concrete, I have a one dimensional fermionic Hubbard model with Hamiltonian \begin{equation} H= -t \sum_{\langle l,m \rangle} (c^\dagger_l c_m + h.c.) + U \sum_l (n_{l\uparrow} - 1/2)(n_{l\downarrow}-1/2) \end{equation} I have a special case where I know that $U$ is very small and I want to use many-body perturbation theory to see its effects on correlation functions. I calculate the correlation functions using the functional integral method (i.e. calculating a functional partition function). For this case, how would I go about finding out how small $U$ has to be in order for perturbation theory to be valid? Secondly, (if this should be a separate question, please let me know!) if instead I have a random $U$, dependent on its position in the lattice, \begin{equation} H= -t \sum_{\langle l,m \rangle} (c^\dagger_l c_m + h.c.) + \sum_l U_l (n_{l\uparrow} - 1/2)(n_{l\downarrow}-1/2) \end{equation} I can then use a similar functional integral technique, but take an average over the functional partition function (e.g. over a Gaussian distribution). This average removes the $U_l$ and leaves $\Delta$, the variance of the distribution we've averaged over. In this case, it is $\Delta$ which the perturbation series is expanded in. How would I go about finding how small $\Delta$ has to be for the perturbation series to be valid? I don't want an answer that's true for any system, I just want to understand how to go about finding it for any system. So if anyone knows of another system where it is shown how small the expansion term has to be, please let me know. Thanks.


title: Find the real height of an object by eyes

text: I wonder if I can get the height of an object (tree or a building) with the knowledge of:   * The distance between me and the object?    * The apparent height of that object?


title: How can a portable 5V battery charge another 5V battery?

text: For a device that is portable and can charge an iPhone 4 or Nintendo 3DS: http://www.amazon.com/New-Trent-iCruiser-IMP1000-Blackberry/dp/B003ZBZ64Q I wonder, if this device outputs 5V, and iPhone or the 3DS also has a 5V battery, then the electromotive force (EMF) at both sides should be the same. If the force is the same at both sides, just like we we push a block on the table with 1 Newton force to the left, and there is also a 1 Newton force pushing the block to the right, the block will not accelerate to the left or right. So when the iPhone is slightly charged, when it reaches 5V, then how can it be charged further? Also, say, if the external device is 11,000 mAh, then, what if at some state, such as when it reaches 3,000 mAh, the voltage falls to 4.3V, then can it still charge another device that is 5V? Or can the full 11,000 mAh energy all go to the other device? A third related questions is, say if the device reaches 3,000 mAh, while the iPhone or 3DS or any other device reaches 6,000 mAh, won't the iPhone "reverse charge" the charger, or the energy keep on going both ways, and dissipate gradually as heat in the copper USB cable that connects them?


title: in SUSY, does WW scattering unitarisation needs the higgs boson?

text: One of the arguments of LHC "win-win situation" is that the scattering of W particles needs to include new terms to preserve unitatity begond 500 GeV or so. In the SM, this is realized by the higgs boson, and the same argument extends to the MSSM. Now, one could consider just a "phenomenology Susy standard model" where simply the superpartners to the known SM are added. This implies that, even without a Higgs mechanism, for each of the W and Z one needs to add an extra scalar boson, to get the right multiplcity; a massive gauge supermultiplet must contain four bosonic helicities. Of course, in the MSSM, two of these come from the Higgses. But from the point of view of SUSY, they are independent. My question is, are these scalars, alone or jointly with the sleptons and squarks, enough to preserve the unitarity of WW scattering? A similar question could be, if you remove two higgses from the five that come in the MSSM, is it still possible to unitarize WW scattering?


title: Decay Amplitudes Notation

text: This question is mostly about how to interpret notation used in Particle Physics. I am given that at lowest order the rate of $b\rightarrow s\gamma$ is proportional to $\langle B_p|b^\dagger b|B_p\rangle$ where $p$ is either u or d labels and $b$, $b^\dagger$ are creation annhilation operators. However, I am unable to understand this notation. For some initial state transforming to a final state isn't the amplitude given by $\langle \mbox{final}|\hat{O}|\mbox{initial}\rangle$. Shouldn't the amplitude be in this case $\langle s\gamma|b^\dagger b|b\rangle$ or something? So what does the provided matrix element mean, exactly?


title: Quantum entanglement on cosmological scales

text: This may be a foolish question given my limited understanding of QM but here it is. As I understand quantum entanglement basically means that two particles evolve as a single "unit", i.e., are described by a single wave function. Now, it seems to me that production of entangled pairs (or n-tuples?) of particles must be a common occurrence and assuming that these particles may then travel a long way away from each other, does this not mean that the whole universe must be a mess of entanglements, effectively breaking the principle of local action? I believe, I understand the reasons why information cannot propagate faster than light from one region of space to another in spite of entanglement but it still seems to me that a universe of "disentangled" particles would behave quite differently from the universe where many particles are entangled. Is there a flaw in this reasoning?


title: A basic confusion about what is an atom

text: Wikipedia defines atom as > _The atom is a basic unit of matter that consists of a dense central nucleus > surrounded by a cloud of negatively charged electrons._ and defines electron as: > _The electron is a subatomic particle with a negative elementary electric > charge. An electron has no known components or substructure; in other words, > it is generally thought to be an elementary particle._ My confusion is: Why is atom being called as basic unit of matter, when it consists of electrons, protons etc, which have masses. These particles electrons should be called as basic unit of matters, instead.?


title: Homogenous vector bundles

text: In the definition of homogenous vector bundles, an equivalence class is defined. Briefly: G is a lie group and H a (lie) subgroup. Define $$ \rho : H \rightarrow GL(V) $$ where V is a vector space. The equivalence class is defined as $$ (g_1, v_1) = (g_1h, \rho(h^{-1})v_1) $$ The question is why the above definition and not $$ (g_1, v_1) = (g_1h, \rho(h)v_1) ? $$


title: Would approaching a distant star at near the speed of light unfold its entire history in "fast-forward"?

text: The light we detect today in our telescopes from distant stars is really old. If we could travel fast (lets say, 90% the speed of light) towards a star a million lightyears away, all the time looking straight at it, would we see that star's entire history unfolds in **_fast-forward_** ▶▶ until we reach it? (if it's still there, of course). It seems reasonable that "newer" photons emitted from that star would hit our eyes at a faster rate while we approach it. Thanks.


title: Optimizing water boiling

text: I'm about to prepare some spaghettis and to do so I need 2 liter of boiling water. For getting the water to certain temperature (lets say 100 Celsius degrees) from a starting temperature of 15 Celsius degrees, I might opt by pouring the 2 liters of water in one single action or might choose by adding in 4 times (1/2 liter each time) so I would start with 1/2 liter, when it starts boiling add another 1/2 liters and so on until I have the 2 liters boiling. What requires less energy for getting the 2 liters? Assumptions:   1. There is only one pot size.   2. The time required for pouring additional water is 0s.   3. Pouring water despite it takes 0 seconds, does not spit any water and then there is no additional loose of temperature other than the one coming from mixing the water.   4. The surface of the pot distributes the temperature homogeneously.


title: Sun energy production as a function of depth

text: I've heard a few times that the Sun's energy output per unit volume is less than that of a human, with the Sun's much higher power output being a result of its much greater size. In a blog post, Phil Plait runs through the numbers and concludes that humans put out 60,000 times more energy per unit volume than the Sun. However, he also makes the counterargument that one shouldn't assume the energy generation density is uniform inside the Sun, since most of it takes place in the core. I would like to know how heat generation is distributed inside the Sun. Is it more or less uniform inside the core? Is there any part of the Sun that generates substantially _more_ energy per unit volume than a human? Phil Plait seems to say that there is, but I think he makes an error in his reasoning: he seems to imply that because the temperature of the inside of the Sun is millions of Kelvin, it must be generating millions of times as much heat. But of course, the temperature isn't the same as the rate of heat generation. If the core of the Sun were thermally insulated, it could stay at millions of Kelvin without generating any heat at all. (Of course I know this isn't really the case.) So I would like to know the figure for the amount of heat generated per unit volume in each part of the Sun.


title: Can a harmonic drive reducer/strain wave gear be used to gear up instead of down?

text: I wish to be able to gear up from a rotating bicycle hub with a revolutions per minute range of 150 to 850 to a ratio of 200:1. Can I do this by simply reversing the direction of a harmonic drive reducer? To clarify could I apply torque through the flex spline and use what is normally the torque in put shaft as an output shaft?


title: Why does grinding steel create sparks but aluminium doesn't?

text: Grinding steel produces sparks. Grinding aluminium does not. Why? I found this article about why grinding steel produces sparks but would like to confirm if it is true. I would also like to know why does it behave so on a molecular level if anyone can explain that.


title: Saturation of an iron core?

text: Lets assume we have a 1 kg iron core, and a 100 kg iron core. Now saturation is defined as how much that core can absorb the magnetic field, since they are different sizes, don't they saturate at different fields? Lets assume the 1kg core saturates at 1 Tesla, shouldn't the larger core saturate at a higher magnetic field?


title: 1+1D Bosonization on a line segment or a compact ring

text: I have been informed that **1+1D Bosonization/Fermionization on a line segment** or **1+1D Bosonization/Fermionization a compact ring** are different - Although I know that Bosonization can rewrite fermions in the non-local expression of bosons. But: > bosons and fermions are fundamentally different for the case of on a 1D > compact ring. Is this true? How is the Bosonization/Fermionization different on **a line segment** or **a compact ring**? Does it matter whether the **line segment** is finite $x\in[a,b]$ or infinite $x\in(-\infty,\infty)$? Why? Can someone explain it physically? Thanks!


title: Quadrupling Power in a Circuit

text: If I have a heating wire with resistance $R$ to be connected across a constant potential difference $V$, it would seem like cutting the wire into two (thus each having half the resistance) and connecting both across the voltage gap would quadruple the power output according to $P = \frac{V^2}{R}$. Why is this not such a good idea? I thought that it was because the current going through the wires would double, but that doesn't seem too bad. What actually makes this not such a good idea?


title: What does the electric field caused by a charged cylinder look like?

text: I am talking about a cylinder with a large enough radius:length ration that it cannot just be treated as a line. I would think that the electric field lines come out the curved and flat surfaces orthogonally, and they remain straight. Is this correct?


title: Estimating elevation by camera angle

text: I'm reading this tutorial on kalman filters, Section 7.2, pg 85-86. It reads as follows: > You spotted an enemy mortar installation about thirty kilometers away, on a > hill that looks about 0.5 kilometers higher than your own position. You want > to track incoming projectiles with a Kalman filter so you can aim your guns > accurately. You do not knowthe initial velocity of the projectiles, so you > just guess some values: 0.6 kilometers/second for the horizontal component, > 0.1 kilometers/second for the vertical component... Letting $d(t)$ and $z(t)$ be the horizontal and vertical components of the position of the mortar (relative to the observer) at any given time t, respectively, the author further states: > All you have [available] to track the shells is a camera pointed at the > mortar that will rotate so as to keep the projectile at the center of the > image, where you see a blob that increases in size as the projectile gets > closer. Thus, the aiming angle of the camera gives you elevation information > about the projectile’s position, and the size of the blob tells you > something about the distance, given that you know the actual size of the > projectiles used and all the camera parameters. The projectile’s elevation > is $$e=1000\frac{d}{z}$$ I recognize $\frac{d}{z}$ as the tangent of the angle made between the observer's horizon and the straight line connecting the observer and mortar. But I can't seem to understand where this factor of $1000$ is coming from. Why is this factor of $1000$ in this equation? Am I missing something very obvious? Or is this just a given equation for demonstrative purposes only with no real physical explanation or reason?


How do you derive Noether's theorem when the action combines chiral, antichiral, and full superspace?


title: Interpretation of the Instanton in SUSY QM

text: This is a loose follow up to this question: Interpreting Argyres' spectrum of spontaneously broken SUSY QM. In SUSY QM, the Hamiltonian can be cast as a 2x2 matrix $$ H = \frac{1}{2}p^2 + \frac{1}{2}W(q)^2 + \frac{1}{2}W'(q)\sigma_3= \begin{pmatrix}H_+&0\\\0&H_-\end{pmatrix} $$ where $H_\pm = \frac{1}{2}p^2 + V_\pm$ are the spin-up and spin-down Hamiltonians and $V_\pm=\frac{1}{2}W(q)^2 \pm \frac{1}{2}W'(q)$ are their potentials ( **do I understand this correctly?** ). Eigenfunctions of this system is always of the form $\begin{pmatrix}\phi_+\\\0\end{pmatrix}$ and $\begin{pmatrix}0\\\\\phi_-\end{pmatrix}$. **Question: In light of statements below, Is this always true?** On page 87 of Mariño's notes, he illustrates SUSY breaking with the superpotential $$W(q) = \lambda q^2 -\mu^2\,,$$ and the corresponding potentials are plotted: ![Potentials plus and minus](http://i.stack.imgur.com/BYm9i.png) Now, it is claimed that instantons allow transitions from the left vacuum to the right vacuum: $$\langle q_+, \uparrow, T=-\infty |\hat\psi_+(t)| q_-, \downarrow, T=+\infty \rangle = \text{non-zero}$$ We need a $\psi_+$ operator to soak up a fermionic zero mode, and flips the spin from $+$ to $-$ or vice-versa. The mathematical computations is absolutely clear to me. But now I'm running into a frustrating **contradiction** : If the Hamiltonian is diagonal, how can the system possibly _flip_ the spin as it tunnels from one vacuum to the other? Remember, stationary states can always be put in the form $\phi_+$ _or_ $\phi_-$ of definite spin. Flipping spin can only happen if the Hamiltonian has an off-diagonal term as in $\begin{pmatrix}H_+&H_x\\\H_x&H_-\end{pmatrix}$. But in SUSY-QM it doesn't! Is there some _effective_ Hamiltonian that allows this?!, (with the off- diagonal elements acting as some kind of 't Hooft effective vertex?)? If so, standard treatments (e.g. numerical ones) to solve the QM system as an eigenvalue problem would totally miss this effect because they would start from a diagonal Hamiltonian! What's the resolution to this contradiction?


title: Why are large batteries described by their power?

text: Reading about renewable energy sources, I came across the idea to use huge batteries as a temporary storage to be able to deal with the fluctuations that e.g. solar panels might produce. The most important parameter for these batteries seemed to be their power in Watts (or MW) i.e. how much energy they can take up/release per unit time, not the amount of energy they can store. Why is that? Why is, in the context of renewable energies, power more important than storage capacity? Thank you for your answers!


title: Definition of the Effective Particle

text: We define the effective particle creation and annihilation operators which are collectively and commonly denoted by $\hat{q}_s$: $$\hat{q}_s := \hat{U}_s \, \hat{q}_0 \, \hat{U}^\dagger_s $$ where $s$ is a renormalization group parameter, $0< s < \infty$, $\hat{U}_s$ is an unitary operator and $\hat{q}_0$ collectively and commonly denotes the creation and annihilation operators in our familiar canonical theory. Due to the values of $s$, for each bare particle/particles state we get a family of effective particle/particles states. The particles associated with the effective particle/particles states are called the effective particles. My question: Is $\hat{U}_s$ is a function of $\hat{q}_0$; i.e., $\hat{U}_s (\hat{q}_0)$ ?


title: Why does a transformation to a rotating reference frame NOT break temporal scale invariance?

text: Naively, I thought that transforming a scale invariant equation (such as the Navier-Stokes equations for example) to a rotating reference frame (for example the rotating earth) would break the temporal scaling symmetry due to the introduction of the period of rotation as an external time scale. Now I was told that such a transformation to a rotating reference frame does not break the temporal scale invariance, its form would just look different (more complicated ?) in such a case. I dont understand why this is and what it means exactly, as explained in the first paragraph it seems counterintuitive to me ... My naive idea to verify that the transformation to a rotating reference frame does indeed not spoil the temporal scaling symmetry would be to show that the corresponding transformations commute. I mean I would start with 1) writing down the analogue of the equation (1) $\qquad \exp(-ia^{\mu}\hat{P}_{\mu})\phi(x) = \phi(x+\delta x)$ for a translation in space(time) for both transformations, 2) Taylor expand the r.h.s of (1), 3) insert the coordinate transformations that correspond to the scaling transformations and the transformation to a rotating reference frame, 4) and approximate the exponential on the l.h.s of (1) to first order. 5) Comparing the l.h.s with the r.h.s should then provide the differential operators for the two transformations such that the commutator can be calculated. **My question now is:** Is testing if the two transformations commute the right procedure to show that a transformation to a rotating reference frame does not break temporal scale invariance? If this is not the case how can it be shown otherwise (mathematically or intuitively)? (In addition to "direct help" I would appreciate being pointed to some literature that deals with this issue too)


title: How many anti-particles hit the ground?

text: I am curious to know the amount of flux of anti-particles that arrive to the ground in the cosmic rays. The reason is that I thought it should be very improbable that an anti-particle traveling through the atmosphere to the ground will survive being annihilated during its journey. Nevertheless, the positron was discovered on the ground, without waiting much for it. Therefore I want to evaluate the probabilities. I have two questions regarding this issue: 1- How much is the flux of anti-particles hitting the atmosphere from space. 2- Can someone show me how to calculate the probability for a positron or an anti-proton to be annihilated before hitting the ground ?


title: Fluid mechanics Grashoff number

text: When I derived the Grashof number (for natural convection at a vertical plate) starting with the ratio of buoyancy forces to viscous forces, I realized that I had set the Reynolds number to 1 (ie I had to set velocity*length/kinematic viscosity = 1) to get the accepted equation. Why is that?


title: If there were no space between nuclei, how big would the Earth be?

text: I have some people telling me it would be the size of baseball. I am quite doubtful on this. If this is true, then the gaps must be so incredibly huge that everything should be transparent. I am not a physics student, just wondering.


What is Fermi surface and why is this concept so useful in metals research? Particularly, I can somewhat appreciate the Fermi energy idea - the radius of Fermi surface which is a sphere. But is there any quantitative use of more complicated Fermi surfaces?


title: Cellular structure in solid phase detonations

text: For gaseous and liquid detonations, the detonation propagates by the energy released at the triple-points which form a cellular structure. This structure is traced using a soot foil during experiments. In solid phase explosives such as TNT and C4, the literature is devoid of discussion about the triple-points and soot foils (both experiments and simulations). The primary method of detonation propagation is through hot-spot generation. These hot-spots are formed at imperfections such as voids, grain boundaries, slip boundaries, etc.. Does the cellular detonation structure exist in solid phase explosives? Is the lack of experimental data because of the difficulty in generating the data, or are solid-phase explosives inherently different than gaseous or liquid? Is it possible that the triple-points exist but because of the heterogeneous nature of solid explosives, they are minor or secondary methods of propagation relative to the imperfections in the mixture?


title: Definition of Buoyancy Flux

text: I've seen in some papers the definition of buoyancy flux as follows $$B = g \frac{\rho_a-\rho_0}{\rho_0}Q$$ where $g$ is the gravity acceleration, $\rho_a$ is the density of the ambient fluid, $\rho_0$ is the density of the lighter fluid released into the ambient fluid from a point source, Q is the volumetric flow rate of the lighter fluid. However I'm a bit confused with this definition as it has the units of $m^4/s^3$. But as it is the flux of buoyancy force, I think the formula should at least have mass with the unit in kg there. Can anyone help me out and let me know the accurate physical meaning of buoyancy flux? The definition of the buoyancy flux in question can be found in:   * lboro.ac.uk/departments/cv/staff/docs/227/ch3/ch3.pdf eq.(3-4)   * ctr.stanford.edu/ResBriefs99/basu.pdf eq. 5


title: Is Type-0 phase matching in BBO crystal possible?

text: I know that it has to be possible because it has been observed. I started with following 3 equations (as BBO is a uniaxial crystal).   \begin{equation} \omega_p = \omega_s + \omega_i \ \ \ \ \ \ (1) \\\ \end{equation} \begin{equation} \vec{k}_p = \vec{k}_s + \vec{k}_i \ \ \ \ \ \ (2) \\\ \end{equation} > where, $\vec{k}=\frac{\omega}{c}n(\omega, \theta)(\hat{z} cos\theta +\hat{r} > sin\theta)$   > The type-0 (or V whatever you prefer to call it) is;                Pump  Signal  Idler       o      o       o      where, o -> ordinary light      So the refractive index is just a number and the equations (1) and (2)   can be brought down to, \begin{equation} \omega_s^2 n_s^2 = (\omega_p n_p Z_p - \omega_i n_i Z_i)^2 + (\omega_p n_p \sqrt{1-Z_p^2} - \omega_i n_i \sqrt{1-Z_i^2})^2 \\\ \Rightarrow Z_i^2 -(2 a Z_p) Z_i + (a^2 + Z_p^2 - 1) = 0 \end{equation} This gives, $\Rightarrow Z_i = aZ_p \pm \sqrt{(1-a^2)(1-Z_p^2)} $ > where, $Z=cos\theta$ and $r=\frac{\omega_i}{\omega_p}$ and > $a=f(r)=\frac{n_p^2+r^2 n_i^2-(1-r)^2 n_s^2}{2r n_p n_i}$ If $|a| \le 1$ then $Z_i$ (or $\theta_i$) will be real and physically acceptable.   The same argument can be made for $\theta_s$. For signal photon's case "$a$" is replaced by   \begin{equation} b=f(1-r)=\frac{n_p^2+(1-r)^2 n_s^2-r^2 n_i^2}{2(1-r) n_p n_s} \end{equation} and we can say for all those $\omega_i$ for which $|a| \wedge |b| \le 1$, $\theta_s$ and $\theta_i$ will be real and physically   acceptable. Here is the problem. When I plot "a" and "b" w.r.t I see no overlap region.   Even if the crystal is positive uniaxial it won't work (I checked it by assuming BBO to be positive uniaxial and by $n_o \leftrightarrow n_e$. ![](http://i.stack.imgur.com/47hpu.png) > a vs r ![](http://i.stack.imgur.com/2GLaK.png) > b vs r Hence I see no-way type-0 is ever possible. Somebody please help me out. What am I doing wrong here ? PS: I can't create a new tag for phase-matching or PDC, I think it would help someone can add these tags to this question.


title: How fast is the Earth-Sun distance changing

text: This is inspired by Evidence that the Solar System is expanding like the Universe?, which referenced an article by G. A. Krasinsky and V. A. Brumberg, "Secular Increase of Astronomical Unit from Analysis of the Major Planet Motions, and its Interpretation," pdf available here. The article seeks to explain the increasing Earth-Sun separation (orbit- averaged separation, as in semi-major axis), and a large part of it is focused on the expansion of the universe. Now Ben Crowell gave a great explanation here as to why the effect of cosmic expansion only comes in with the third time-derivative of the scale factor, noting that the effect is "undetectably small." So then I wonder: **What is the rate at which the Earth-Sun distance _is_ changing?** Krasinsky and Brumberg cite something like $15\ \mathrm{cm/yr}$, but given the number of wrong calculations they do elsewhere, I don't entirely trust this claim. Who has done this analysis, and what do they find? Also, **what methods are used?** I would guess that measuring the precise distance to the Sun's surface would be challenging, due to all the activity it undergoes. * * * Note that discussion of cosmic expansion was background motivation for what sparked this question - I fully understand that any secular change in the Earth's orbit will be dominated by other effects. Also note there is a related question on this site: How is distance between sun and earth calculated? However, that does not go into detail about the method. Moreover, it is not clear that the answers provide work for measuring small changes in the separation. For example is radar really that precise? Perhaps it is, in which case justification of that would serve as an answer here.


title: If I take a bottle of air into space, and open it, where does it go?

text: It seems to me that space doesn't have any/much air, and if my bottle is full of air, when I open it, where does the air go?


I have a boiling pot of water that I am going to cook my Sponge Bob Noodles in. I am very hungry and want the noodles to cook faster, so I turn up the heat from medium to high.( Note, the water was already boiling at medium).What happens to the time it takes for my noodles to cook, and the temperature of the boiling water?


title: Solar stills with Wine

text: Is it possible to obtain more concentrated alcohol by using wine in a solar still instead of seawater? If yes, as I suspected, how does the concentration (of alcohol) of the distillate vary with time? I would expect it to rise to a peak, then fall to the original concentration as the water also get condensed into distillate. Can you give an analysis for this variation?


title: Why is it so hard to mix spaghetti with meatballs?

text: When I cook spaghetti with a sauce that includes meatballs or chopped vegetables, those stay in chunks and won't get uniformly mixed in the pasta. Is there a simple reason for that? Has it been studied? you know some people study piles of sand with various grain sizes?


title: Anomalies and Modification of symmetry algebra

text: This question is motivated by 2-dimensional CFTs where the Classical conformal group (defined by the Witt algebra) is modified to the Virasoro algebra in the quantum theory. In this question, it was pointed out to me that the extra term in the Virasoro algebra is due to the conformal anomaly (for example, when quantizing the theory on a curved surface, the Ricci scalar explicitly introduces a length scale). This led me to wonder about this: Consider a field theory has a classical symmetry with algebra $G$. Is it true that if this symmetry is anomalous in the quantum theory the algebra $G$ is modified? If so, how? For example, Yang-Mills theory with a massless Dirac fermion, has a classical symmetry group $G_{gauge} \times G_{axial}$, where $G$ is the gauge group. In the quantum theory, $G_{axial}$ is anomalous. Does this somehow also imply that the algebra of $G_{axial}$ is modified somehow?


title: J.J. Sakurai book on Q.Mech

text: I just want to ask how recommended is the book on quantum mechanics by J.J. Sakurai. Is it any good as an introductory text? And are there better suggestions (substitutes)? Thanks in advance


title: How can an object with zero acceleration move?

text: My physics text has a problem in which it is said that a person moves a block of wood in such a way so that the block moves at a constant velocity. The block, therefore, is in dynamic equilibrium and the vector sum of the forces acting on it is equal to zero: $\sum{F} = ma = m \times 0 = 0$ This is where I get confused. If the person is moving the block in such a way so that the sum of the forces acting on it is equal to zero, how can he be moving it at all? I realize that   1. even if there is no force acting on an object it can still have velocity (Newton's First Law)   2. force causes _acceleration_ (a change in velocity--not velocity itself) but for some reason I don't understand how it is possible for the object to move with constant ~~acceleration~~ velocity. Do I have all the pieces? I really feel like I am missing something. **EDIT** The block was at rest to begin with, and the person is moving the block through the air **EDIT: THE ACTUAL PROBLEM** For the sake of clearing up ambiguity, here is the actual problem. (I thought it was wooden blocks at first, sorry) > Two workers must pick up bricks that like on the ground and place them on a > worktable. They each pick up the same number of bricks and put them on the > same height worktable. They lift the bricks so that the bricks travel upward > at a constant velocity. The first gets the job done in one-half the time > that the second takes. Did one of the workers do more work than the other? > If so, which one? Did one of the workers exert more power than the other? If > so, which one? Basically, **_the intent of this question is to ask if the above problem is entirely hypothetical._** My text does not indicate that it is, which is what I find confusing.


title: What is the frequency of a photon emitted by an accelerated charged particle?

text: When a charged particle is accelerated by a field, it would seem there are many possible photons that could be emitted to obey conservation of momentum. Is it random, or time-dependent, or based on the magnitude of the acceleration?


title: Charge on Sphere due to Contact with Capacitor

text: **Question:** Consider a parallel plate capacitor which is connected to potential difference $V$. Let there be a small spherical conductor, assume that its radius is much smaller than the distance between the plates $R<<d<<\sqrt A \qquad$ ($R$ = radius of sphere, $d$ = distance between plates, $A$ = area of plate). The question is when the sphere is in touch with one of the plate, how much charge approximately(leading order) will be induced on the sphere? ![enter image description here](http://i.stack.imgur.com/FjZ0H.jpg) **Attempt for solution:** Since the sphere is very small we can assume that the electric field everywhere is approximately $E=V/d$ and I will ignore the correction of the plate's charge distribution to counter the sphere's field since it's a higher order effect. Firstly, the sphere will counter the field with a charge distribution $\sigma=3\epsilon_0Ecos\theta$, this charge distribution will produce a uniform electric field that will cancel with $E$ inside the sphere and therefore we get an equipotential surface on the sphere. After that we need to add an extra uniform charge distribution on the sphere so that it is equipotential with the plate in touch. How much charge do we need to add? The problem is the plate's potential itself is not well defined in this case.


title: Does Hestenes Zitterbewegung Explain why complex numbers appear in QM?

text: This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and may deserve its own discussion? It seems to me that in the paper "The zitterbewegung interpretation of Quantum Mechanics " Foundations of Physics , 20 1213-1232 Prof Hestenes argues that spin is an actual precession of something physical and that complex variables exist in Quantum Mechanics because of this. I have also heard that Geometric Algebra, which I think was invented by Prof Hestenes can clearly explain why complex variables are required in QM. If these are both the case (spin is a physical phenomenon and GA explains why complex variables are used in QM) can anyone explain why QM is not taught beginning with the assumptions of Geometric Algebra?


title: Magnets arranged in a sphere

text: If I was to take a bunch of magnets and arrange them in a sphere (And keep them there with glue or plastic or something) so that the north pole faces the outside of the sphere and the south pole faces the inside, would the magnet have the same pole no matter what way I turn it or would the magnet be neutralized or something. I'm envisioning a sphere made of magnets so that no matter what way I turn it it is always the same pole and that a bunch of these repelling each other would be really cool, is this possible? In effect would this create a monopole magnet?


title: What's the relation between virtual photons and electromagnetic potentials?

text: Given that: 1) virtual photons mediate the electric and magnetic force fields 2) the magnetic field is the curl of the magnetic vector potential 3) the electric field is the negative gradient of the electric scalar potential How do we understand the vector and scalar potentials in terms of virtual photons? Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present. For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics. Thanks for any help.


title: Infinite-range 1D Ising model + Hubbard-Stratonovich-Transformation

text: I have a probably quite simple question RE the HST. After some work, I obtain as the partition function for the infinite range 1D Ising model $$Z = \int_{-\infty}^\infty \frac{dy}{\sqrt{2\pi / N\beta J}} e^{-N\beta f(y)}$$ where $$f(y)= \frac{J}{2} y^2 - \frac{1}{\beta} \ln[2\cosh(\beta(h + Jy))]$$ Now I want to evaluate the integral in the thermodynamic limit and thus I will be using the method of steepest descent. I am asked to show that $$Z = \sum_i e^{-\beta N f(y_i)}$$ and have to find the equation satisfied by the $y_i$. So of course my idea was to say that I will get a contribution from each of the local minima of $Z$, which leads to the condition $$\frac{\partial f}{\partial y}|_{y_i} = 0$$ and $$\frac{\partial^2 f}{\partial y^2}_{y_i} > 0$$ However, to obtain the exact form I am asked to show, I'd also need the condition that $$\frac{1}{2} \frac{\partial^2 f}{\partial y^2}|_{y_i} = J$$ because otherwise the prefactors don't come out correct to cancel each other. And now I just don't see how I get this result since the second derivative of $f$ becomes rather messy.


title: Light emanating from finger tips?

text: My little nephew on two occasions (when he was 4 now 6) said he saw light rays emanating from his fingers. His father said this is true and that there is physics behind it. I have searched the internet but found no satisfying answer yet. Do you guys have any idea about this phenomenon?


title: Name this Mulltivariable Calculus Theorem

text: In Robert Wald's book _General Relativity_ a multivariable calculus theorem is cited on page 16, which states: If $F:\mathbb{R}^n\mapsto \mathbb{R}$ is $C^{\infty}$ then for each $a=(a^1,...,a^n) \in \mathbb{R}^n$ there exist $C^{\infty}$ functions $H_{\mu}$ such that for all $x \in \mathbb{R}^n$ we have $$F(x)=F(a)+\sum_{\mu=1}^n (x^{\mu}-a^{\mu})H_{\mu}(x)$$ (1) What is the name of this theorem? (2) Where can I find a proof?


title: Does being suspended in air allow you to not be affected by Earth's rotation?

text: Let's assume that there was some mechanism by which we could remain suspended in air. By this I mean that our feet is not in contact with the ground. One possible way of doing this would be by means of a jetpack. If we could remain suspended in air for a while in this manner, will we be in a different place when we come down, due to the Earth's rotation? When I thought of this problem in the micro scale using a ball (which rotates), it appears like I'll be in a different place when I come down from my suspended state. Am I overlooking something here?


title: Seismic science questions

text: We just (20mn ago) suffered from another strong aftershock (of the initial March 11th earthquake) in Japan. I have some questions related to seismic science.   1. Why is there so much time between the initial quake and the following strong aftershocks (initial was March 11, the stronger aftershock happened 20mn ago, April 7)?   2. Why does it seem that progress in seismic science (and quakes predictions) is so poor? It seems people have only raw probabilities, based on historical data.   3. Would anybody have any graphical / imagery data of the physical fault move on March 11? Thank you.


Question 1. Can virtual particles, in particular gravitons, interfere? Virtual particles are created and annihilated in a distance too small and a time too short to be measured. Their existence is allowed by Heisenberg's uncertainty principle, which allows a transitory violation of energy conservation in processes that endure a short enough time. The electromagnetic interaction has a single force carrier, the photon. The weak interaction has two force carriers, the W and Z particles (with 2 signs for W). The strong interaction has 8 force carriers, the gluons. The quantum carrier of the gravitational force is the graviton (a massless spin 2 particle). Gravitons act together in such vast numbers that there is a scant hope to seeing evidence for just one or a few. I think that it is established that the less energetic the force carrier, the stronger is the force mediated. If gravitons can interfere, then it is likely that they would interfere destructively (chance dictates that both constructive and destructive interference are present, but destructive interference is more likely, that we can prove). From Planck's relation from quantum mechanics, that means that these gravitons will be less energetic, thus increasing the strength of the force of gravitation, over large scales. I emphasize that the probability of graviton interference increases for large scale systems (galaxies, clusters of galaxies), and it is very small for small scales, due to the ephemeral character of the virtual particles, in this case the graviton. Here is my second question. Question 2. Could this lead to a solution of the dark matter problem (a more accurate analysis than sketched here, of course)? Is it possible that the accumulated tiny effects of graviton interference in the Planck region could lead to an observable deviation from the known laws of gravity at very large scales (galaxies, clusters of galaxies), thus solving the dark matter problem?


title: Simple harmonic oscillator: zero point energy?

text: Today we had a lecture on the simple harmonic oscillator and its quantum mechanical treatment. My teacher derived the equation for it and finally concluded it has some zero point energy.   1. My question is what is the physical significance of this zero point energy?    2. And from where does this energy come from?    3. And yes one more thing. I'd like to know the physical examples where the concept of zero point energy is used.