title: Placatory motion of more than two bodies

text: Recently in a documentary I heard that Newton's law of gravitation very well explains motion of two bodies such as sun and earth. And when applied to three bodies the answer is chaotic and not stable. Then how do physicists explain the motion of higher number of bodies such as the solar system or just the three bodies of sun, earth and moon?


title: CP violation from the Electroweak SU(2)$_{weak,flavor}$ by $\int \theta F \wedge F $

text: **Question: Why there is NO Charge-Parity (CP) violation from a potential Theta term in the electroweak SU(2)$_{weak,flavor}$ sector by $\theta_{electroweak} \int F \wedge F$?** **(ps. an explicit calculation is required.)** * * * **Background:** We know for a non-Abelian gauge theory, the $F \wedge F $ term is nontrivial and breaks $CP$ symmetry (thus break $T$ symmetry by $CPT$ theorem), which is this term: $$ \int F \wedge F $$ with a field strength $F=dA+A\wedge A$. $\bullet$ **SU(3)$_{strong,color}$ QCD** : To describe strong interactions of gluons (which couple quarks), we use QCD with gauge fields of non-Abelian SU(3)$_{color}$ symmetry. This extra term in the QCD Lagrangian: $$ \theta_{QCD} \int G \wedge G =\theta_{QCD} \int d^4x G_{\mu\nu}^a \wedge \tilde{G}^{\mu\nu,a} $$ which any nonzero $\theta_{QCD}$ breaks $CP$ symmetry. (p.s. and there we have the strong CP problem). $\bullet$ **Compare the strong interactions $\theta_{QCD,strong}$ to U(1)$_{em}$ $\theta_{QED}$** : For U(1) electromagnetism, even if we have $\theta_{QED} \int F \wedge F$, we can rotate this term and absorb this into the fermion (which couple to U(1)$_{em}$) masses(?). For SU(3) QCD, unlike U(1) electromagnetism, if the quarks are not massless, this term of $\theta_{QCD}$ cannot be rotated away(?) as a trivial $\theta_{QCD}=0$. $\bullet$ **SU(2)$_{weak,flavor}$ electro-weak** : To describe electroweak interactions, we again have gauge fields of non- Abelian SU(2)$_{weak,flavor}$symmetry. Potentially this extra term in the electroweak Lagrangian can break $CP$ symmetry (thus break $T$ symmetry by $CPT$ theorem): $$ \theta_{electroweak} \int F \wedge F =\theta_{electroweak} \int d^4x F_{\mu\nu}^a \wedge \tilde{F}^{\mu\nu,a} $$ here the three components gauge fields $A$ under SU(2) are: ($W^{1}$,$W^{2}$,$W^{3}$) or ($W^{+}$,$W^{-}$,$Z^{0}$) of W and Z bosons. **Question [again as the beginning]** : We have only heard of **CKM matrix** in the weak SU(2) sector to break $CP$ symmetry. Why there is NO CP violation from a potential Theta term of an electroweak SU(2)$_{weak,flavor}$ sector $\theta_{electroweak} \int F \wedge F$? **Hint: In other words, how should we rotate the $\theta_{electroweak}$ to be trivial $\theta_{electroweak}=0$?** ps. I foresee a reason already, but I wish **an explicit calculation is carried out**. Thanks a lot!


title: How can a Higgs decay to heavier products than its mass?

text: How is it possible that a higgs at ~125 GeV can decay into 2 W bosons @ ~ 80 GeV a piece (for example)? Shouldn't a particle only be allowed to decay to lighter particles + energy? Diagram copied from this question ![enter link description here](http://i.stack.imgur.com/eljaD.jpg)


title: What is the interstellar density of mm sized "dust" particles?

text: If you assume an interstellar spacecraft of 100 m in diameter traveling to the nearest star - that ship would plow through 3 x 10^20 m^3 of interstellar space. At 0.5 c the energy release of such a spaceship encountering a piece of space "dust" weighing (~0.01g) becomes significant. Only one such impact would be enough to cause severe spacecraft damage given the kinetic energy involved (150,000 Joules). This collision would explode like a pinpoint bomb (30 g of TNT - one half the charge in a hand grenade). So, what are the chances of such a space ship encountering such a interstellar grain of dust in traveling to the nearest star?


title: Did the singularity really existed in the begining of the universe?

text: I read in Hawking's brief history of time the following: > The final result was a joint paper by Penrose and myself in 1970, which at > last proved that there must have been a big bang singularity provided only > that general relativity is correct and the universe contains as much matter > as we observe. There was a lot of opposition to our work, partly from the > Russians because of their Marxist belief in scientific determinism, and > partly from people who felt that the whole idea of singularities was > repugnant and spoiled the beauty of Einstein’s theory. However, one cannot > really argue with a mathematical theorem. So in the end our work became > generally accepted and nowadays nearly everyone assumes that the universe > started with a big bang singularity. It is perhaps ironic that, having > changed my mind, I am now trying to convince other physicists that there was > in fact no singularity at the beginning of the universe \- as we shall see > later, it can disappear once quantum effects are taken into account. So, according to our current understanding of physics and black holes, what does support the existence of singularities and what does not?


title: Can the two electromagnetic field tensors be combined into a more general tensor?

text: Given the electromagnetic field tensor $$\begin{align} F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\\ E_{x} & 0 & B_{z} & -B_{y} \\\ E_{y} & -B_{z} & 0 & B_{x} \\\ E_{z} & B_{y} & -B_{x} & 0 \end{pmatrix} \end{align}$$ and its dual $$\begin{align} G_{\mu\nu} = \begin{pmatrix} 0 & B_{x} & B_{y} & B_{z} \\\ -B_{x} & 0 & E_{z} & -E_{y} \\\ -B_{y} & -E_{z} & 0 & E_{x} \\\ -B_{z} & E_{y} & -E_{x} & 0 \end{pmatrix} \,, \end{align}$$ does there exist a general tensor $H_{\mu\nu}$ that combines $F_{\mu\nu}$ and $G_{\mu\nu}$ into a more general tensor in Minikowski space?


title: Glass Hardness and Pressure

text: How is glass hardness defined? I understand that ordinary kitchen knives cannot scratch most toughened glass, such as the ones found on cell phones. However (I once tried) - with enough pressure, one can actually scratch the glass. So when testing the hardness of glass, is pressure standardized?


title: Acceleration of the Expansion of Universe

text: Is the acceleration of the expansion of the universe (AEU) uniform in direction? Meaning that if a particular direction in the sky is chosen and the AEU is measured and then other directions are chosen and the AEU are measured, do all these measurements agree?


title: What are the linear maps which preserve the time-like cone?

text: I'm looking at the set of time-like vectors: $\mathcal{T}_+ = \\{ x \in \mathbb{R}^4 \mbox{ s.t. } x^T \eta x \geq 0 \:, x^0\geq 0\\} $, where $\eta = \mbox{diag}(1, -1, -1, -1)$. I want to be able to characterize the set of linear operators that preserve this set: $$\mathcal{V} = \\{L\in M(4,\mathbb{R}) \mbox{ s.t. } L x \in \cal T_+\,, \forall x \in \cal T_+\\}$$ Clearly $L$ must satisfy $$x^T L^T \eta L x \geq 0\:, (Lx)^0\geq 0 \:\:\forall x \in \mathcal{T}_+\:.$$ Is $\mathcal{V}$ convex? It's clear that the orthochronous Lorentz transformations preserve $\mathcal{T}_+$, since $x^T \Lambda \eta \Lambda^T x = x^T \eta x$. Do these form the boundary of $\mathcal{V}$? A clue: I know that $\mathcal{T}_+$ is convex. Maybe there's a good reference on linear maps preserving convex cones... edit: changed $GL(4, \mathbb{R})$ to $M(4,\mathbb{R})$.


title: Why do powers of lenses get added?

text: In our physics class, we were told that a power of a lens is defined as the reciprocal of its focal length. Also, the powers of multiple lenses get added up. But I didn't understand why this is. I mean consider a series of two convex lenses with focal lengths $f_1, f_2$. Clearly, the resultant image distance is going to depend upon the distance between the lenses (that is how a compound microscope works; by adjusting the distance between the objective and the eyepiece different magnifications can be obtained). So, I'm guessing, the resultant focal length of the system (whatever that is), will also depend upon the distance between the lenses. How does the power of the lens fit into all this? Also, what do they mean by the 'resultant power of the lenses' in the first place? The resultant system of lenses is not going to be like these lenses which have negligible thickness. Where are you going to measure the focal length from? The system of lenses does not have a fixed pole.


title: Quantum Mechanical States

text: What can be the precise answer to the question that > Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they belong to the Hilbert space, why they are infinite dimensional, is this because of the projection operator?


title: What should I look for at SAR levels for cell phones?

text: I've been trying to wrap my head around SAR levels when comparing cell phones. Considering that I spent most of my time not talking on the phone but carrying it around near my body, would a phone **lower body** value but **higher head** value be preferable one with the reverse combination? Or, suppose one phone has a value that is 2x of another. How should I think of it? Is it significantly worse or slightly worse?


title: Regularization and renomalization in the lightcone quantization of bosonic string

text: This question relates to this link. But I still don't understand it >_< In Polchinski's string theory vol I, p. 22, there is a divergence term (when $\epsilon \rightarrow 0$) in the zero point energy of open-string (1.3.34), $$ \frac{D-2}{2} \frac{ 2l p^+ \alpha'}{ \epsilon^2 \pi} $$ It is said > The cutoff-dependent first term is propotional to the length $l$ of the > string and can be canceled by a counterterm in the action proportional to > $\int d^2 \sigma (-\gamma)^{1/2}$. In fact, Weyl invariance requires that it > be cancelled. I have a couple of questions related to this statement and I don't know whether it is good to split into several threads.   1. About the underlying logic of regularization and renomalization. In quantum field theory, we meet a couple of divergences. One is in quantizing the scalar field. There is an infinity, we argue that is the zero-point energy. We throw it away since energy is a relative quantity. And we left the cosmological constant problem. If the divergence in (1.3.34) is in this sense, string-theory is a quantum gravity theory. We cannot simply throw it away like QFT. Later in QFT, we meet other divergences in the loop calculations. The reason is, the presented QFT is a low-energy theory. It has certain scale that the theory is not applicaple. Therefore we met the divergence. We regularalize it and renormalize it, where we met the terminology "counterterm". But, string theory is regarded as a final theory(?!), if I follow the logic of QFT, why there is still divergence in (1.3.34)? What is the scale string theory is not applicable?   2. How the counterterm works in $\int d^2 \sigma (-\gamma)^{1/2}$?   3. Why "Weyl invariance requires that it be cancelled"?


title: Why is binding energy $\Delta mc^2$?

text: As we know the mass-energy equivalence relation $E=mc^2$ originally came from special relativity. And the binding energy is $\Delta mc^2$. How do we know that the extra mass coming from theoretical calculation is associated with the binding energy? What is special relativity inside the nucleus?


title: Physical meaning of imaginary part of Electric field?

text: As far as I know (or I thought I knew), if we have an electric field $$\mathbf{E}=\mathbf{E_0}\cos(\omega t - kx),$$ we can define it as the real part of $$\mathbf{E}=Re(\mathbf{E_0}e^{i(\omega t - kx)}).$$ Introducing imaginary components to the electric field is only a matter of making the maths easier, in particular in the case of complex exponentials which are eigenfunctions of the differential operators. Now. In a Ph.D. thesis I was reading, about lasers and cavity misalignment (which doesn't really matter, my question does not require any knowledge of this), I came across something along the lines of: The electric field to first order misalignment can be written as $$\mid E\rangle = | 00 \rangle + i | 01 \rangle, $$ where the two kets on the LHS are the fundamental and 1st excited Hermite-Gaussian modes. But the physical part of the electric field is surely its real part? What purpose can the imaginary component above serve?


title: Work done against gravity

text: The work done against gravity is $mgh$, well at least that's what my textbook says. I have a question: I can apply a force say 50N, so total work done = $mgh + mah$. Where $ma$ = Force. But the truth is irrespective of the force applied, the work done against gravity is always $mgh$. Why? For example, when I move an object with a force, the work done is more, so work depends on the Force. But in case of gravity it is always depends upon the weight


title: What is a virtual state?

text: In quantum mechanics / Raman spectroscopy, what is a virtual state? What is the difference between a virtual state and a superposition of states? Can you simply think of the virtual state as a superposition of eigenstates? If you have the complete set of eigenstates for a system, then you can represent any configuration of the system as a linear combination of the eigenstates, so it seems that you should be able to represent the virtual state that way. Is there something more to the virtual state than that?


title: Quantum mechanics for macroscopic charges?

text: OK first off tell me if my understanding of the following is correct: In a Hydrogen atom, one would expect that the opposite charges (electron and nucleus) to attract each other, according to classical electromagnetism, but this is not the case. Quantum mechanics will tell us what happens in such a system, and indeed by solving the Schrodinger equation we find the probabilities distribution functions for the electron cloud. Long story short - for subatomic problems classical electromagnetism cannot give all the answers, we must trust what quantum mechanics says. Solving for the energies of the Hydrogen Atom, we find that they are : $$E_n = -\frac{\alpha^2}{2}\mu c^2 \frac{1}{n^2} $$ where $\mu$ is the reduced mass of the electron-proton system. What if, instead of an electron and a proton, we had two macroscopic charges $Q_1$ and $Q_2$? Would they still form a bound system? Is this formula still applicable?


title: Effect of expansion of space on CMB

text: Is it true that the expansion of space time cause the CMB to become microwaves from a shorter wavelength. If it is has the amplitude been increased? Seeing as the amplitude has decreased; why hasn't it increased (/"stretched") in the same way the wavelength has?


title: Scaled energy output of a hydrogen atom

text: I'm trying to get a grasp of the power within a single hydrogen atom (e.g. the power released when whatever process happens in a hydrogen bomb)... If we could enlarge a hydrogen atom up to the size of your fist, how much energy would it give off?


title: Value of weak force coupling constant

text: I'm trying to get my head around the weak force coupling constant $\alpha_w$ but getting confused by different resources. Hyperphysics suggests it is tiny compared with the strong force coupling, ~$10^-6$ times the size, but Griffiths Introduction to Elementary Particles claims it is ~$1/30$, and so five times larger than $\alpha_{EM}$, which hyperphysics gives as the usual $1/137$. Not sure what's going wrong here, but can anyone explain it to me? Thank you! http://hyperphysics.phy-astr.gsu.edu/hbase/forces/couple.html#c4 http://en.wikipedia.org/wiki/Weak_interaction#Properties


title: Why can't a gas be liquified by pressure above its critical temperature?

text: What is the cause behind a gas being difficult to liquify above its critical temperature no matter how much pressure is applied on it?


title: Different cross section for W and Z boson

text: At hadron colliders the cross section for **W** production is about ten times larger than the production cross section for **Z** bosons (e.g. Figure 2 in this review article). I guess the dominant contribution is from Drell-Yan production so let's only consider this. A factor of two probably comes from the fact, that there is a positive and a negative **W**. But what about the remaining difference of factor 5?


title: Where did the universe get its initial momentum?

text: If, according to Newton's third law, forces come in pairs then what about the big bang? where did the universe get that initial push/momentum?


title: Why is a Electron negative not both?

text: I was pondering on the question of magnetic dipoles but then I had an thought that electron and other particles are either positive or negative or have no magnetic charge but then I thought of particles like being small spherical magnets (in shape of pellets) but then I realized both have a negative and positive poles so what makes an electron negative or is it an mono-pole? If it were like my description, then surely we can never call it negative as it would be relative to the observer. I'm only in Middle-School so please correct me if any of my knowledge is wrong.


title: How far will water squirt out from a hole in a can?

text: Say you have a bottle of water filled up to a height H. A small hole is drilled in its side at a height d, so that water squirts out. The squirting water travels in an arc as it falls, covering some horizontal distance S away from the bottle before it hits the table top that the bottle sits on. Multi- part question to try to understand this completely:   1. At what height d should the hole be placed so that its horizontal travel distance S is maximized?    2. Will S or the optimum value of d depend on the hole diameter?   3. Does the answer change if you assume that the hole presents zero impedance to the water flow? This might not make sense to ask, or it might be equivalent to asking what happens in the limit of a very large-sized hole (see #2). Not sure. Thanks to anyone who can help!


title: Do photons interact with each other, or with themselves only?

text: Even if the double slit experiment gives interesting (weird) results, it only concludes that each photon interacts with itself after passing the two slits. I have been thinking about a different experimental setup, where you have two well defined light sources (with specific wave lengths and phase) but no slits. And now to my questions: Has anyone ever done such an experiment, and will there be an interference pattern on the wall? If the answer to the second question is "no", light can not be a true wave - it only has some wavelike properties. But if it is "yes", things become much more interesting. If there is an interference pattern on the wall, there has to be an interference pattern even if both light sources are emitting single photons at random, but as seldom as, say, once per minute. That in turn would mean that the photons know about each other, even if they are separated in time with several seconds, and the light sources are independent (not entangled).


title: How are heat transfer rate and heat capacity constant of a material related?

text: How are heat transfer rate and heat capacity of a given material is related? When I was a kid I imagined it like "Capacity is like a container or bucket of some sort, if it gets filled easily it can give away heat easily. Metaphorically overflowing". Or is it related at all?


title: Can the study of the quantum information structure in QFT with holographic duals be relevant to string theory?

text: I'm interested in characterizing the behaviour of measures of quantum information in strongly correlated quantum field theories which admit a gravity dual description, e.g through AdS/CFT duality. In a more broad sense, my question points towards the following: in my limited experience, I use to accept that a duality means that we don't have any right to give a physical meaning to one part of the duality and use the other part as a mere computational tool. I would like to be illuminated by some experts.


title: What's inside the slit in double slit experiment?

text: If double slit experiment is done in a environment with air, then slits could also contain air made up of (approx. 80% nitrogen, 20% oxygen), then there is not empty space inside the slit. How can it affect the experiment? Has it been taken in account? What are specific requirement to define the slit? If it were not that way. Why it doesn't affect the measurement?


title: Alignment of a triaxial accelerometer

text: I was wondering if someone could explain the alignment issues associated with a tri-axial accelerometer. It measures acceleration in terms of $g$. I have read some literature but I am unsure about the alignment issues involved.


title: Thomson's jumping ring experiment

text: For the jumping ring experiment (I think it is also called Thomson ring experiment) why does the ring float and not move up and down the ring as would be expected, I think by Lenz's law, since would the ring when the current thru the coil is decreased (due to it been ac) not fall back down to increase the magnetic flux thru it that has been lowered due to the smaller current??


title: Would dark matter absorb gravitational waves?

text: Would the vast and seemingly diffuse clouds of dark matter floating around our galaxy (and most others) absorb gravitational waves? Is this perhaps why we haven't detected any yet?


title: A better conceptual model for cooper pairs in a superconductor

text: The conceptual model I have been introduced to for cooper pairs in a bulk superconductor is what I would call the "wake" model, where one electron deforms the positively charged lattice, changing the potential energy landscape and causing the second electron to be drawn closer. This seems like a terrible conceptual picture both because the electrons in a superconductor should have a well defined localized wave function and because the ground state of the BCS theory has zero net momentum, which would suggest they orbit around each other or something... If anyone can explain how I should be thinking about this I would be very grateful.


title: How does my air-cooling unit work?

text: During the winter, my office gets a little cold. I have a unit (essentially a space heater) that heats the area around it to help keep me warm. Now that it's summer, I realize the unit also has a cool setting. I don't understand how this can work. It takes energy from the wall, and removes energy from the air. That energy has to go somewhere, doesn't it? Where does that energy go if not to the surrounding air? Here's a picture of the unit. It's model number is 02044. Extensive investigation (i.e. shaking and listening) show no signs of water inside. Apparently it was recalled in 2008 for starting fires. Hopefully this is a newer model? :) ![enter image description here](http://i.stack.imgur.com/BDej5.jpg)


title: Uses and interpretation of the 'Bowen Ratio' ($B_o=SH/LE$)

text: The Bowen Ratio is the ratio of sensible heat flux to latent heat flux, so presumably it gives some information about the relative importance of these processes. But it is not clear how this information can be used to make inference about a system (e.g. the land-air interface of a corn field). What are some uses of the Bowen Ratio? What are typical values, and what is the range of meaningful values (e.g. under what conditions does $B_o\to\pm \infty$?)


title: How does the resistance force on a rolling ball depend on the ball radius?

text: A billiard ball set gently rolling on a billiard table slows and stops, because it is decelerated by resistance forces at the contact between the ball and table. I expect the magnitude of the resistance force will depend on the ball radius, but what is the exponent on the radius in the equation that describes the force? The exponent is ~3 if determined by the ball volume, ~2 if determined by the surface of the ball or contact area, and ~1 if determined by the ball perimeter. It would be ~0 if the resistance is independent of ball radius. There may be an empirical constant of proportionality in the equation, but what I would like to know is how I could use physical reasoning to fix the exponent on the radius to an integer, and then fit this model to my experiment. ADDENDUM: After reading some of the references suggested in the comments, I saw that my question was more naive than I realized. The subject of rolling resistance is large, complex, and with significant engineering applications. My requirements for an answer are likely more modest than what I now think might be offered here. Really, what I would like is a bit more understanding of the primary cause of rolling resistance, the underlying physics, and enough mathematics to be able to relate this to what I currently know about mechanics. Clearly experimentation is necessary to really understand this accurately, but some theoretical background would be useful for experimental design or explaining it to students.


title: For a particle to have physical mass, is it always necessary to have a mass term in the lagrangian?

text: Since the self-energy adds to the bare mass defined in the Lagrangian, is it possible to create a physical particle mass from the self-energy alone, with no mass terms occuring in the Lagrangian? On a possibly related note, Wikipedia says: "The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass."


title: Why does Snell's law not factor in wavelength?

text: If dispersion is caused due to different wavelengths bending with different angles, how is Snell's law right in generalizing it as $\sin(i) / \sin(r)$? Or am I missing something?


title: What size will the Sun become once it is a red giant?

text: How big will the Sun be once it becomes a red giant? How much of the solar system will it engulf?


title: Mechanics Problem

text: I'm trying to follow Feynman's lecture. Unfortunately I'm a bit stuck on a small piece, so if you could show me what I'm doing wrong then I'd greatly appreciate your help. I want to exactly know how Feynman came up with $W_{ac} = \int^c_a \bf{F} \cdot d \bf{s} = F s \cos \theta$ when this implies $\bf{F} = F \cos \theta$. I have not been able to convince myself that this is true using triangles and trigonometry. The relevant section says, > Let us magnify one of the triangles, as shown in Fig. 13-4. Is the work done > in going from $a$ to $b$ and $b$ to $c$ on a triangle the same as the work > in going directly from $a$ to $c$? Suppose that the force is acting in a > certain direction; let us take the triangle such that the side $bc$ is in > this direction, just as example. We suppose that the triangle is so small > that the force is essentially constant over the entire triangle. What is the > work done in going from $t$ to $c$? It is $W_{ac} = \int^c_a \bf{F} \cdot d > \bf{s} = F s \cos \theta$ ![](http://i.imgur.com/BdcB6.png) I'm stuck getting the $F s \cos \theta$. Below you'll see the figure I'm imagining while trying to solve this. The triangle ABC is the original in Feynman's figure; the triangle AED is the triangle representing the forces, where the line segment AD is the "F" in the diagram above. I define line segment AE as b, line segment ED as a, and line segment AD as c. The law of cosines says that $a^2 = b^2 + c^2 - 2bc \cos \theta $   If we let $a=b$ then,   $b^2 = b^2 + c^2 - 2bc \cos \theta$   iff $c^2 = 2 b c \cos \theta$   iff $c=2b \cos \theta$   which implies   $\frac{c}{2 \cos \theta} = b$ $c$ in this case is in the direction of the applied force but what we need to determine is the work by the actual force along $ae$ which should depend on $c$; yet my answer appears incorrect because according to Feynman the answer is $b=c \cos \theta$. Can you tell me where I went wrong? Here is where my triangle is located: ![](http://i.imgur.com/JiWwW.png)


title: Yutsis graphs in quantum theory

text: Cubic Yutsis graphs appear in the context of the quantum theory of angular momenta. The recognition of these graphs is NP-complete. Is there any implication to quantum physics if recognition of Yutsis graphs can be done in polynomial time? Is there any insight from quantum mechanics into why recognizing such graphs must be intractable?


title: Wave function of Hydrogen Atom

text: Wavefunction of a Hydrogen atom is expressed in eigenfunctions as: $$\psi(\boldsymbol r,t=0)=1/\sqrt{14}(2\psi_{100}(\boldsymbol r)-3\psi_{200}(\boldsymbol r)+\psi_{322}(\boldsymbol r) ).$$   1. Is this an eigenfunction of parity operator?   2. What is probability of finding system in state (100),(200),(322)?    3. In another energy eigenstate?    4. What is the expectation value of energy, $L^2$,$L_z$?  I know some algebra but I don't know what the subscript of eigenfunction means ._. The only thing I know is about parity operator and its eigenfunctions. Others are just like foreign language.


title: How does electromagnetic induction happen?

text: Electric and magnetic fields are different from each other(i think i am correct).. How does changing magnetic field induce electric current??? ![enter image description here](http://i.stack.imgur.com/ZpE91.gif)


title: Poynting vector of a wire between capacitors

text: A capacitor made of two circular plates of radius $L$ separated by $d$, initially the plates carry $\pm$ Q charge. Then a wire of resistance R is placed between them, how do I go about deriving a time-dependent expressiion


title: A question concerning the act of observing distant galaxies

text: The comoving radius of the observable universe is currently put at $46$ to $47$ billion light years. Source wikipedia. When we observe galaxies at great distance, such as the Hubble Deep Field at $13$ billion light years, how do we know we are observing very young galaxies in the very early universe, rather than galaxies that have entered our observable universe at a time when the comoving radius of the observable universe was $13$ billion light years and are therefore older than we might otherwise expect.


title: The Deflecting System in a Hot Cathode Ray Tube

text: In an HCR-Tube, the deflecting system used to deviate the _electron_ beam is made of _positively_ charged plates. How is this justified? If, due to some malfunction, the electron beam deflects from its original path before it is actually supposed to, won't it stick to the plates? And why is it not wiser to use _negatively_ charged plates? If this is done, then the electrons of the beam will return to their stream even if they are accidentally deflected, and when the actual deflection is required positive plates can be introduced.   So why only positive plates?


title: Hamiltonian Operator for nonrenormalizable Effective Field Theories?

text: Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form $\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - \frac{1}{2} m \phi^2 - \frac{\lambda}{4!}\phi^4 + \text{higher dimensional terms}$. with arbitrary higherdimensional and nonrenormalizable terms such as $\partial_\mu\partial_\nu\phi ~ \partial^\mu\partial^\nu\phi$ etc. N-point greens functions can be defined through the pathintegral: $\langle \mathcal{T}\lbrace \phi(x_1) \dots \phi(x_n) \rbrace \rangle = \int \mathcal{D}\phi \ \phi(x_1) \dots \phi(x_n) e^{i\int d^4x \mathcal{L}_{eff}[\phi]}$ which may be evaluated perturbatively. My Question is: **How can we obtain the Hamiltonian-Operator (as the generator of time- evolution $U(t, t_0) = e^{iH (t - t_0)} $) from the Lagrangian that is used in the Pathintegral ?** $\left \langle \phi_b \right| e^{-iH_{eff}T} \left|\phi_a \right\rangle = \int \mathcal{D}\phi \ e^{i\int_0^T dt \int d^3x \ \mathcal{L}_{eff}[\phi]}$ I would expect the Hamiltonian to be of the form $H_{eff} = \int d^3x \frac{1}{2}\dot\phi^2 + \frac{1}{2}(\nabla\phi)^2 + \frac{1}{2}m\phi^2 + \frac{\lambda}{4!}\phi^4 + \text{higher dimensional terms} $, as a sum of arbitrary higher dimensional Operators (but finitely many at each perturbative order maybe ?). EDIT: If a Legendre Transformation should be the answer: **What would the justification be for doing a Legendre Transformation ?** In contrast to the case where time derivatives appear quadratic in the Lagrangian we have arbitrary higher power of time derivatives, terms in the Lagrangian containing $\frac{d^n}{d^n t}\phi$. The usual way of casting the Pathintegral from the Lagrangian form into the Hamiltonian form: $\int\mathcal{D}\phi e^{i\int \ \mathcal{L}[\phi]} \rightarrow \int\mathcal{D}\phi\mathcal{D}\pi\ e^{i\int (\pi\dot\phi - H[\phi, \pi])}$, and deriving the Hamilton Operator from it or vice versa doesn't work !


title: Problem finding the average velocity

text: Problem: A car travels along a straight line for first half time with speed of 40 km/h and second half time with speed 60 km/h. The mean speed of the car is__________. My answer: 48 km/h Why? ![enter image description here](http://i.stack.imgur.com/MrxUd.png) However, the answer given in the book is 50 km/h, why? Any help will be really appreciated, thanks. :)


title: What are Wightman fields/functions

text: Simple question: What are Wightman fields? What are Wightman functions? What are their uses? For example can I use them in operator product expansions? How about in scattering theory?


title: Dealing with dirac notation with regards to different basis'

text: So this should be a pretty simple question. So we say that $\langle x | \psi \rangle = \psi(x)$. In other words $\psi(x)$ is the ket $|\psi\rangle$ expressed in terms of the $x$ basis. Now suppose I wanted to get $\psi(x+a)$. How would I act on $|\psi\rangle$ to get it? I think I could just do $$ \langle x' | \psi \rangle = \psi(x+a)$$ Where $x' = x+a$. But I don't really know what this means in terms of how $\langle x' |$ and $\langle x |$ are related, or the formal justification for this. Perhaps my trouble is I don't thoroughly understand what the $x$-basis is or what $\langle x | \psi \rangle$ is really doing.


title: Lorentz transformations for sine wave in random direction

text: Hello i have been struggling to derive Lorentz transformations for a sine wave, which is traveling at random direction. I started by prooving that phase $\phi$ is invariant for relativity and that equation $\phi = \phi'$ holds. By using the above equation i am now trying to derive Lorentz transformations for angular frequency $\omega$, and all three components of the wave vector $k$, which are $k_x$, $k_y$ and $k_z$. This is my attempt: \begin{align}\phi' &= \phi\\\ \omega'\Delta t' + k' \Delta r' &= \omega \Delta t + k \Delta r\\\ \omega'\Delta t' + [{k_x}' , {k_y}' , {k_z}'] [\Delta x' , \Delta y' , \Delta z'] &= \omega \Delta t + [k_x , k_y , k_z][\Delta x , \Delta y , \Delta z]\\\ \omega' \Delta t' + {k_x}'\Delta x' + {k_y}' \Delta y' + {k_z}' \Delta z' &= \omega \Delta t + k_x \Delta x + k_y \Delta y + k_z \Delta z \end{align} Now with just the left side: \begin{gather} \omega' \gamma \left(\Delta t - \Delta x \frac{u}{c^2}\right) + {k_x}' \gamma \Bigl(\Delta x - u\Delta t \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \gamma \left(\omega' \Delta t - \omega' \Delta x \frac{u}{c^2}\right) + \gamma\Bigl({k_x}' \Delta x - {k_x}' u \Delta t \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \gamma \left(\omega' \Delta t - {k_x}'c\, \, c \Delta t \, \frac{u}{c^ 2}\right) + \gamma \Bigl({k_x}' \Delta x - \frac{\omega'}{c} u\frac{\Delta x}{c} \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z\\\ \Delta t \, \gamma \Bigl(\omega' - {k_x}' u \Bigl) + \Delta x \, \gamma \Bigl({k_x}' - \omega' \frac{u}{c^2} \Bigl) + {k_y}' \Delta y + {k_z}' \Delta z \end{gather} From this I can write down the Lorentz transformations. \begin{equation} \begin{split} \gamma\Bigl(\omega' - {k_x}' u \Bigl) &= \omega\\\ \gamma \Bigl({k_x}' - \omega' \frac{u}{c^2} \Bigl) &= k_x\\\ {k_y}' &= k_y\\\ {k_z}' &= k_z\\\ \end{split} \end{equation} My professor said that my signs are wrong, but what am i doing wrong?


title: Entanglement entropy and area law

text: I am currently reading a review "Area law for the entanglement entropy" by Eisert, Cramer and Plenio (2010). From what I understand:   1. In one dimension, for local gapped models, we have an area law for entanglement entropy.   2. In one dimension, some models with long range interactions + all critical systems obey a log(N) law, where N is the size of the subregion of the 1D space, the pre-factor depends for CFT models on the central charge.   3. For higher dimensions, local gapped (equivalently quasi-free) models obey an area law.   4. For higher dimensions, critical bosonic models obey an area law, fermionic ones obey a divergent log law with pre-factor dependent on the topology of the Fermi surface (which is also probably related to the central charge of the CFT, but no exact results are known). However I am a bit confused about the treatement of the Klein-Gordon field. The massive Klein-Gordon field is a local gapped model, hence not critical. Papers by Bombelli and Srednicki show that entanglement entropy obeys an area law for dimension greater than 2, in agreement with point 3. However in the review they seem to say that this is a critical system. Thanks to point 4 there is still no contradiction with the area law result. However, this becomes problematic for the 1D case: a massive or massless KG system obeys a divergent log law, as shown here: http://arxiv.org/pdf/hep- th/9401072.pdf and in the review. To me this is only true if the field is massless. A massive KG field is a local gapped system, so not critical and hence it should follow an area law according to point 1, not a divergent log law. So it seems to me that the problem is that these papers consider the KG field (massive or massless) as critical, whereas to me only a massless KG field is critical. What did I misunderstand?


title: What is the geometry of our universe?

text: I have some questions about the notion of space in current cosmological theories. I have been trying to decipher a few papers written on this topic such as this one, but I keep running into some trouble. Most of the papers I have been skimming talk about anti-de Sitter space (AdS) and about how the best definition of nothing is > _the limit of anti-de Sitter space in which the curvature length goes to > zero._ This is great and all, but I have read on Wikipedia that AdS implies a negative cosmological constant, which would imply that dark energy should be attractive. Is my reasoning here incorrect? If not, could someone explain why cosmologists spend so much time devoted to studying AdS when the universe is clearly expanding? I have read somewhere about local geometry vs. global geometry, but I wasn't sure if that is what is going on here. The papers are so complex.


title: Quantum Entanglement - Measuring Twice

text: In the answer here and on the wiki article and many other articles it is mentioned that if one of 2 entangled particles is measured their state collapses according to the Copenhagen interpretation. Lets take the example from the EPR paradox article, which mentions a positron and an electron occupying quantum states and becoming entangled. There are two observers, Alice and Bob. > In state I, the electron has spin pointing upward along the z-axis (+z) and > the positron has spin pointing downward along the z-axis (-z). In state II they are opposite. > Alice now measures the spin along the z-axis. She can obtain one of two > possible outcomes: +z or -z. Suppose she gets +z. According to the > Copenhagen interpretation of quantum mechanics, the quantum state of the > system collapses into state I. Meaning if Bob measures the spin, he will get -z. My question is, what happens if Bob or Alice measure the spin along the z-axis again, will it remain z+ for Alice and z- for Bob or can it change between measurements?


title: What happens to the wave function after applying the D'Alembert operator?

text: Is the result of applying the D'Alembert operator on a wave function always zero? Or are there exceptions?


title: Do photoelectrons move fast enough to use special relativity? (+ homework)

text: What are the speeds of the photoelectrons which are knocked out of the metal during photoeffect? Do i have to use special relativity? I am having trouble solving a homework using relativity. But if i do it in a nonrelativistic way my result is wrong... The homework: > We expose a same metal to a light with $\lambda_1=350nm$ and a light with > $\lambda_2=540nm$. What is the work function $A_0$ of this metal if the $v_1 > = 2 v_2$ relation holds for the maximum speeds - the speeds with which > electrons leave the metal. * * * **NONRELATIVISTIC APPROACH:** It was easy doing this in a nonrelativistic maner when kinetic energy is of a form $W_k=\tfrac{1}{2}mv^2$: \begin{align} W_1 - A_0 = W_{k1} &= \tfrac{1}{2}m{v_1}^2 = 2m{v_2}^2\\\ W_2 - A_0 = W_{k2} &= \tfrac{1}{2}m{v_2}^2 \end{align} The workfunction is the same in both cases so: \begin{align} A_0 &= W_1 - 2m{v_2}^2 \xleftarrow{~ ~ \text{i insert for $v_2$}~ ~ } {v_2}^2 = \tfrac{2(W_2 - A_0)}{m}\\\ A_0 &= W_1 - 4(W_2-A_0)\\\ A_0 &= W_1 - 4W_2 + 4A_0\\\ A_0 &= \frac{4W_2 - W_1}{3}\\\ A_0 &= \frac{4\tfrac{hc}{\lambda_2} - \tfrac{hc}{\lambda_1}}{3}\\\ A_0 &= \tfrac{hc}{3}\left(\tfrac{4}{\lambda_2} - \tfrac{1}{\lambda_1}\right)\\\ A_0 &= \tfrac{6.626\cdot 10^{-34}Js~~~ \cdot ~~~2.99\cdot10^{8}\tfrac{m}{s}}{3} \left(\tfrac{4}{350\cdot 10^{-9}m} - \tfrac{1}{540\cdot 10^{-9}m}\right)\\\ A_0 &= 6.32\cdot 10^{-19}J\\\ A_0 &= 3.95 eV \end{align} The result is wrong as it should be $A_0 = 1.87eV$. * * * **RELATIVISTIC APPROACH:** I can't seem to solve this problem in a relativistic way where $W_k = mc^2[\gamma(v)-1]$. Here is what i did do: \begin{align} W_1 - A_0 & = W_{k1} = mc^2\gamma(v_1) - mc^2\\\ W_2 - A_0 & = W_{k1} = mc^2\gamma(v_2) - mc^2 \end{align} I solve second eq. for $mc^2$ and insert it in the first one to get: \begin{align} W_1 - A_0 &= mc^2 \gamma(v_1) - \left(mc^2\gamma(v_2) - W_2 + A_0\right)\\\ W_1 - A_0 &= mc^2 \gamma(v_1) - mc^2\gamma(v_2) + W_2 - A_0\\\ W_1 - W_2 &= mc^2 \gamma(v_1) - mc^2\gamma(v_2)\\\ W_1 - W_2 &= mc^2\underbrace{\left[\gamma(v_1) - \gamma(v_2)\right]}_{\llap{\text{how can i get this relation out of a $v_1=2v_2$?}}} \end{align} This is where i am stucked. Can anyone help me to solve this?


title: How to derive a physical model of temperature above the earth surface?

text: At a point on the surface of the Earth, if I go straight "up" a distance $z$ (radially outward from the Earth), I'd observe a temperature profile $T(z)$. The conventional wisdom from the meteorology community (that I can find at least) is that this temperature decreases linearly up to about 11km as $T(z)=T_0-\gamma z$, where $\gamma$ is a constant known as the lapse rate. However, I have measurements of the temperature over several heights within the first few meters of the earth that clearly are not linear in height. I suspect the constant lapse rate is a simplification of the physics that is applicable after the first few meters up to 11km. My question is, what are the governing dynamical equations for air temperature above the surface of the earth? I wish to set up and solve such an equation, and use the solution form as a fitting function to my measured data. My guess is that it would be some combination of the heat equation with solar irradiance as a kind of forcing term, coupled with a fluid dynamics equation that accounts for the thermal expansion and motion of the air. The model would have to account for the observed non-linear profile in the near-surface environment, and probably become linear at higher ranges. Any insight or references are appreciated. **Edit** To clarify, I'm okay with something that can't work over the whole earth surface. Suppose I know local parameters of the surface, and let's say that is homogeneous over a large enough swath that you can neglect some lateral effects. You can even assume it's flat if you want. I don't know what the reasonable assumptions are, nor do I know what are workable ones that lead to simple solutions. I'm just trying to get a handle on the physics of temerature and air in this environment.


title: How to deal with non fixed springs?

text: In every physics book they explain how to compute the force that a spring exert if one of its ends is fixed to a wall (or equivalent) and the other end is compressed or stretched. But how to deal with the problem if both ends are 'free'. For example, suppose you have two blocks attached with a spring, and someone apply an horizontal force of, say, 50N. (Lets suppose that there is no friction with the surface, the blocks are originally at rest, the spring is originally in its relaxed state, and the spring constant is, say, 300N/m). How you would compute the acceleration of each block? ![enter image description here](http://i.stack.imgur.com/4JqAR.png)


title: Lie algebra and BPS stats

text: i would know what is a charges Lattice and the relationship between it and roots lattice ? and if there is relationship between mutation of quiver and weyl group? and how the PBS stats correspond to weight in in a given representation?


title: Divergence theorem in complex coordinates

text: This question is related to Stokes' theorem in complex coordinates (CFT) but, I still don't understand :( Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in Polchinski's string theory $$\int_R d^2 z (\partial_z v^z + \partial_{\bar{z}} v^{\bar{z}})= i \oint_{\partial R} (v^z d \bar{z} - v^{\bar{z}} dz ) (1) $$ I may try $$ \int_R dx dy ( \partial_x F_y - \partial_y F_x) = \oint_{\partial R} (F_x dx + F_y dy)(2) $$, but what kind of substitution I should use to get Eq. (1)?


title: As the universe expands, the wavelengths of photons are stretched, and energy is lost. What about electrons?

text: Will electrons, and other particles, also loose energy as they travel through the cosmos? They have wavelengths. Do they get "stretched"? My guess is that the EM force, somehow, counteracts this effect. What about neutrinos? Reference found: http://books.google.com/books?id=AmlEt6TJ6jAC&pg=PA96&lpg=PA96&dq=redshift+of+de+broglie+wavelength&source=bl&ots=oAW0q9bmaj&sig=nAwE- ohARZ3YTCZrHDBBjZ_IYfQ&hl=en&sa=X&ei=0lQDVJjtLMKEjAL80IHQCw&ved=0CB4Q6AEwAA#v=onepage&q=redshift%20of%20de%20broglie%20wavelength&f=false


title: Is there a way to directly observe the spin texture of the surface states of topological insulators?

text: Is there a way to directly, here I means in real space, observe the interaction of the surface states of 3D topological insulators with defects (dopings and adatoms)? How to observe the spin texture of these surface states? Is there any experimental proposal for this?


title: What determines the apparent radius of the rainbow?

text: Let's say I know how to compute the apparent radius of a rainbow from the viewpoint of the observer: take a photo of the scene, measure the distance to a known reference object, and its dimensions. Using triangle similarity, I can extrapolate the radius of the rainbow. But my question is: which physical phenomenon determines the radius?


title: Does the potential energy for a given photon increase or decrease in quanta?

text: As a photon leaves a strong gravitational field, it loses energy and redshifts. Is the exchange in potential energy of a photon characterized by energy quanta?


title: Why do particles move?

text: Is all particles moving and forever?   Can the movement of particles be stoped ?


title: How reliable is Radiometric dating? Are there limitations?

text: Young earth creationists dismiss radiometric dating as unreliable, whats the truth?


title: velocity of a ball after sliding down a inclined plane?

text: I am making kind of a game that involves inclined plane and physics simulation. In the Game a M kg ball at the constant speed of V is sliding up an inclined plane(inclined angle is θ) and the COF(Coefficient Of Friction) of inclined plane is μ. I need the velocity of the ball in then second position in the picture.(after the ball has climbed the inclined plane all the way down .)![enter image description here](http://i.stack.imgur.com/MBxap.png)


title: Differences between absorption, transparency, reflection, and emission

text: Can someone help me conceptualize the differences between a photon's involvement with absorption, transparency, reflection, and emission? To be more specific, my current understanding of the matter is that when a photon interacts with an atom containing electrons (not a free electron), if the frequency is high enough, it can be _absorbed_ , and the electron moves to a less-stable, yet higher energy level. In this sense, all other photons that were not absorbed are then _reflected_ back outwards (giving the object its corresponding color). But if the frequency is too low to make the energy gap, the photon passes through the electron cloud and the atoms are _transparent_ in the visual spectrum (as in glass, or air)... Given this, what is the main ingredient that causes a photon to either be reflected off of the object versus passing straight through it (as in transparency)? As well, when does an _emission_ of the electron get involved (as in, if the electron absorbs a certain frequency, when is a photon emitted versus an electron, and does the electron itself get emitted if it's a valence, or does a separate electron in the sea of electrons elsewhere get emitted to make up for the balance?) Sorry if it's a bit of a mess; please let me know if my assumptions above are inconsistent, and need tweaking as well before moving forward.


title: Connection between Kolmogorov entropy and Boltzmann entropy

text: http://math.stackexchange.com/questions/527384/what-is-the-connectivity- between-boltzmanns-entropy-expression-and-shannons-en mentions a relationship between Shannon entropy and Boltzmann entropy. Is there a relationship between Kolmogorov–Sinai entropy and Boltzmann entropy? And Kolmogorov entropy and Shannon entropy?


title: Explicit definition of the energy operator in the Ising model

text: I've simulated a few 2d Ising models at critical temperature on triangular lattice and I'm now trying to check that the correlation functions are right. I alraedy did it for the spin operator ($\sigma$) and I'd like to do the same for the local energy operator that should satisfy the following space dependence: $$\langle\epsilon (x) \epsilon (y) \rangle \propto \frac{1}{(x-y)^2}$$ (scaling weight of 1) but I'm not sure of the definition of this operator, I would guess something as: $$\epsilon(x)=-J\,\sigma(x)\sum_{\text{neighbours}}\sigma$$ where J is the critical coupling ($=\frac{1}{T_{\text{crit}}}$) Is that right? Edit: new research led me to **reject my first proposition** : This energy operator should be odd under the $\text{Z}_2$ symmetry of the system $\sigma \leftrightarrow - \sigma$ which is clearly not the case of my previous guess.


title: A condition for relative gas

text: Consider an ideal gas in a box. we know (no matter what, right?) that each massive particle's energy holds $E^2=c^2p^2+m^2c^4$ . how come that the condition for that gas to be relativistic is $E\approx mc^2$ ? isn't means that the particle's speed is low enough to neglect the kinematic energy, i.e the exact opposite?


title: Does a current-carrying wire running through the centre of a solenoid experience force?

text: Imagine looking at a solenoid from above. Current passing is through it in a clockwise direction. The direction of the field lines therefore is towards the bottom of the solenoid. Now pass a straight current carrying wire through the central axis of the solenoid. The current in the wire is travelling in a downward direction so the field around it is going in a clockwise direction if viewed from above. My question now is how do the field lines interact, and is there any resultant force on the wire? My attempt at thinking this through says that the field lines of the solenoid and wire will become downward spiralling field lines. Using this I believe that the wire will experience force in the upward direction. Is this correct?


title: Existence of negative temperatures and the definition of entropy

text: How negative temperatures can be possible has been treated on StackExchange before (several times in fact), but in light of some recent academic discussion, most of these answers seem to be possibly wrong or incomplete. The literature I am referring to is Dunkel & Hilbert, _Nature Physics_ **10** , 67 (2014) arXiv.1304.2066, where as I understand it, it is shown that negative temperatures are an artefact of choosing an incorrect definition of entropy. The Wikipedia article on the matter has also been amended to reflect this. This was later challenged by similarly well-known scientists, in arXiv:1403.4299 where it was, among other things, pointed out that this argument is actually decades old (citing Berdichevsky _et al._ , _Phys. Rev. A_ **43** , 2050 (1991)). The original authors quickly countered the arguments made in the comment, by what seems to be a rigorous treatment of the matter, arXiv:1403.6058. The first arXiv comment (arXiv:1403.4299) has been updated since and it still reads that "Obviously severe points of disagreement remain". What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. I would also be interested in hearing whether changing the definition of entropy from that of Boltzmann to that due to Gibbs might potentially change any other results. Might for example the Wang-Landau algorithm be affected seeing that it does use the density of states and that you can never simulate infinite systems (although as I understand it, even in the present context with finite scaling you should be able to get consistent results)?


title: Negative sound rooms

text: Well it is pretty well known that rooms with sound less than zero decibels, $\approx$ -15 decibels. How is it possible to create a room which is quieter than soundless? And it is claimed that just staying in the room for a mere 30 - 45 minutes, depending on your source, could drive you crazy. Why is this?


title: Intermolecular Hydrogen Bonding

text: I learnt about intermolecular hydrogen bonding today, which occurs between molecules such as ortho-nitro-phenol. What I was told is that in case of intermolecular bonding, the molecules separate from each other, opposite of what happens during intramolecular hydrogen bonding. I don't understand why is this. Why would attraction within a molecule cause separation of the molecules? There is obviously some intramolecular hydrogen bonding present in this case, but I understand that this might be negligible. Still, why should the molecules spread away from each other?


title: Are the elements of physics different on the upper deck, of a double-decker bus, than those on the lower deck?

text: In this question, I hypothesise an event that affects the entire bus; I'm not asking about menace unique to upper floor passengers, such as an overly high upper floor or hanging objects (because this trivially affirms the question). My friends claim that they feel more 'wobbly' or 'unstable' while seated on the upper deck, than on the lower. Also, they allege that in a car crash accident, sitters on upper floors are more likely to be ejected out through the windshield. Would someone please vindicate or refute, using physics?


title: Derivation of (2.45) in Peskin and Schroeder

text: I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int d^{3}y ~(-i\delta^{(3)}(\vec{x}-\vec{y})(-\nabla^{2} +m^{2})\phi (\vec{y},t)) $$ I've tried using the relations $$[\phi (\vec{x},t), \pi (\vec{y},t)] = i\delta^{(3)}(\vec{x}-\vec{y})$$ and $$[A,BC] = [A,B]C + B[A,C], $$ but run into $$[\pi (\vec{x},t), (-\nabla^{2} +m^{2})\phi (\vec{y},t)] ,$$ which I don't know how to evaluate. Any help would be appreciated.


title: Was the Higgs mass correctly predicted by asymptotic safety of gravity?

text: This paper was published in Phys Lett B in 2009, and predicted the Higgs mass to be 126 GeV based on the asymptotic safety of gravity. Is this prediction taken seriously by the theory community, or is it considered only to be a lucky guess? EDIT: Thanks to Heidar for this fantastic compilation of Higgs mass predictions. This makes the point that it was probably a lucky guess, but nonetheless it could still be interpreted as (weak) evidence.


title: Why can we use the energy of a pendulum to calculate its frequency?

text: The question might sound rather vague; to calculate the frequency using the energy we simply use that the total energy is constant, set the derivative to zero and solve the equation of motion that follows. But my question is more about the why; how does the fact that the total energy is constant imply that we are able to calculate the frequency of the system? It seems to me as if this implies some sort of deeper connection, other than simply because the math allows us to, or the fact that both the frequency and the energy are constants.


title: Does a four-divergence extra term in a Lagrangian density matter to the field equations?

text: Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: > ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu > A^\mu-\frac{1}{2}\partial_\mu A^\mu\partial_\nu A^\nu $   >  $\space\space\space\space=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu+\frac{1}{2}\partial_\mu[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) > A^\mu]$ > > The last term is a four-divergence which has no influence on the field > equations. Thus the dynamics of the electromagnetic field (in the Lorentz > gauge) can be described by the simple Lagrangian > > ${\mathcal L}^{\prime\prime}=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu > A^\nu$ Yes, **if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field** , indeed according to the variational principle this four-divergence will not influence the field equation. And actually I calculated the time-derivative dependence of 0-component of $[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$, in which only $[A_0(\partial^0 A^0)-(\partial_0 A^0) A^0]$ could possibly contain time- derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case. But how can he seem to claim that **it holds for a general four-divergence term** ,`The last term is a four-divergence which has no influence on the field equations`? EDIT:   I only assumed the boundary condition to be $A^\mu=0$ at spatial infinity, not at time infinity. And the variation of the action $S = \int_{t_1}^{t_2}L \, dt$ is due to the variation of fields which vanish at time, $\delta A^\mu(\mathbf x,t_1)=\delta A^\mu(\mathbf x,t_2)=0$, not having the knowledge of $\delta \dot A^\mu(\mathbf x,t_1)$ and $\delta \dot A^\mu(\mathbf x,t_2)$, which don't vanish generally, so the four-divergence term will in general contribute to the action, $$\delta S_j=\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \partial_\mu j(A(x),\nabla A(x),\dot A(x))^\mu =\delta \int_{t_1}^{t_2}dt\int d^3\mathbf x \dot j^0 =\int d^3\mathbf x [\delta j(\mathbf x, t_2)^0-\delta j(\mathbf x, t_1)^0]$$ which does not vanish in general!


title: Why does the $\pi$-flux state have time-reversal symmetry?

text: It's known that the $\pi$-flux state  of the antiferromagnetic Heisenberg model on the square lattice is an important concept. The $\pi$-flux state is described by the (simplified) mean-field Hamiltonian $$H=t_1f_{1\sigma}^\dagger f_{2\sigma}+t_2f_{2\sigma}^\dagger f_{3\sigma}+t_3f_{3\sigma}^\dagger f_{4\sigma}+t_4f_{4\sigma}^\dagger f_{1\sigma}+H.c.$$, where $t_i=\left | t \right |e^{i\frac{\pi}{4}}(i=1,2,3,4)$, and the spin-1/2 operator is $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$. It's obvious that the mean-field Hamiltonian $H$ is _not_ invariant under time-reversal operation( $T$ ), say $H\neq THT^{-1}$, and $H$ is also not $SU(2)$ gauge equivalent to the time-reversal transformed Hamiltonian $THT^{-1}$. So due to what reason, the projected spin-state $\psi_{spin}=\hat{P}\psi_{MF}$ is time-reversal invariant? Where $\psi_{MF}$ is the ground state of the mean-field Hamiltonian $H$ and $\hat{P}=\prod (2\hat{n}_i-\hat{n}_i^2)$ is the projection to the spin subspace. **Remarks:** Here the effect of translation(with _one_ lattice spacing along the $\hat{x}$ or $\hat{y}$ direction) on the Hamiltonian $H$ is the same as the effect of time-reversal $T$. Thus, if the spin-state $\psi_{spin}$ has $T$ symmetry, it must also have the translation symmetry. Thanks in advance.


title: How is extra information encoded in digital radio?

text: Earlier today, I was listening to my digital radio, and was intrigued as to how they are able to encode the information displayed on the LCD screen (which was basically a short string of text giving a brief description of the currently playing record/discussion topic), alongside the audio transmission. My question is: > How is extra information (such as "currently playing song"), transmitted > alongside the audio in digital radio encoding? I am somewhat familiar with the workings of a radio, along with some basic understanding of RF-transmission, but nothing particularly advanced. And I was intrigued as to how they can transmit information within the same bandwidth of the audio transmission without causing interference. My first thought was that they could polarize both text and audio transmissions so their planes of oscillation are orthogonal to each other, and thus no interference would occur, but I didn't know how plausible that would be. Thanks in advance, apologies if this a stupid question.


title: bond energies database

text: I’m looking for a public database of bond energies of biochemical compounds. Do you know about any? Otherwise, which is the standard computational methodology to calculate them? Any references would be very appreciated. Many thanks


title: Can we construct Axiomatic system of physical laws?

text: If we construct axiomatic system of physical laws that are independent one another as in axioms in mathematics, what should they be? Can there be such a finite system of physical laws that can explain every physical phenomenon? Or is it impossible to have such finite axiomatic system in physical laws?


title: Is Schrodinger's Cat a real conceptual problem or just a problem with approximations?

text: In this thought-experiment a cat is placed in a box set with a bottle poison that will release and kill it depending on whether or not a certain radioactive particle decays. The box is kept closed and it is asked "is the cat alive or dead?" Since the decay of the particle is a quantum mechanical process- it is represented by a state function. This function remains in a "superposition state" until it is observed whether the particle has decayed or not. The cat is presumably and unfortunately entangled with this function- it's own life/death is in a superposition state until it is indeed observed to have dodged the bullet or not. Schrodinger used this thought experiment in an attempt to show the ridiculousness of certain aspects of quantum theory. His motives were clear in devising it and are well documented in his correspondences with Einstein. My question is not about the cat, the particle, or Schrodinger. I want to know about the box. The box that contains this scenario must effectively isolate the outside observer from any information about what is (or could be is perhaps more correct) going inside of it. So what sort of box would qualify? It seems that such a box could not have any interaction at all with the outside universe- no gravitational or thermal effects- anything. Otherwise would not some kind of information about what was going on inside escape without it being opened? And this being said- were such a box constructed would it not be entirely cut off from reality and how could we be sure it even contained a cat still or was even there at all? Is this experiment possible or are the problems I outline above a deterrent to it's physical possibility?


title: Sign Paradox in Berry's Phase

text: Suppose we have normalized states $| n(\vec{R})\rangle$ indexed by continuous variable $\vec{R}$. Then fixing our choice of gauge and ignoring dynamic phase, the phase difference between two states is the Berry's Phase: $$\tag{1} \langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~=~ e^{i\gamma}$$ where, if $C$ is some curve that goes between $\vec{R}_0$ and $\vec{R}_0 + \Delta \vec{r}$, $$\tag{2} \gamma~=~i\int_C \langle n(\vec{R}) | \nabla_{\vec{R}} | n(\vec{R})\rangle \cdot d\vec{R}$$ If $\Delta \vec{r}$ is small, then $$\tag{3} \gamma \approx i\langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r}$$ However, we can directly calculate this as well: $$\langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~\approx~ \langle n(\vec{R}_0) | n(\vec{R}_0) \rangle + \langle n(\vec{R}_0) | \nabla_{\vec{R}}| n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r}$$ $$\tag{4} \approx 1 + \langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r} ~\approx~ \text{exp}\left[ \langle n(\vec{R}_0) | \nabla_{\vec{R}} | n(\vec{R}) \rangle \Big|_{\vec{R}=\vec{R}_0} \cdot \Delta \vec{r} \right]$$ and therefore $$\tag{5} \langle n(\vec{R}_0)| n(\vec{R}_0 +\Delta \vec{r} ) \rangle ~\approx~ e^{-i\gamma} $$ There's a minus sign now! What am I doing wrong here?


title: Is there a normalized form of the Euler equation discretized with finite volumes?

text: I want to calculate a flux on my fpga using the Euler equations with the finite volume method. Unfortunately the values of the state variables differ a lot. For example the pressure has a value of 100000 and the density 1.16. This makes it complicated to calculate on the FPGA. Now I'm wondering if there is a normalized form for the Euler equations with finite volumes, so that the values of the state variables are in the same range. I've tried to set them all to one, but my simulation crashed. I think that's not possible because of the non-linearity of the equations.


title: Why does the counterterm's propagator have inverse units of the propagator? $\phi^4$-theory

text: According to Peskin & Schroeder (page 325), the Feynman rule for the counterterm                ------(x)-----       for $$ \frac12 \delta_Z(\partial_\mu\phi_r)^2-\frac12\delta_m \phi_r^2$$ being $\phi_r$ the renormalized field, is given by $$i(p^2\delta_Z-\delta_m)$$ which resembles rather the (multiplicative) inverse of the propagator for the original Lagrangian (whith physical quantities). Why?


title: Charge carrier injection in heterostructures - help with concept definition

text: I have this report to do on "Charge injection in heterostructures". I have been searching and reading but I still have some trouble with the basics, i.e. defining the concept. As far as I understood a heterostructure is a junction between two or more different semiconductors and the carrier injections comes in the sense of bringing electrons and/or holes to the heterostructure through some kind of contact as a way of making a current or a light emitter by electron-hole recombination. So the first question, is this correct? What is difference between this and conduction through a metal-semiconductor contact? Is there such thing as a metal-heterostructure contact? Does electron-hole pair formation by light counts as charge injection? Can you advise me with some bibliography about this topic? What is super-injection? I'm really having trouble expanding this concept in details. Thanks for the help.


title: Using wavepackets instead of photons in quantum computer

text: Why does a photonic quantum computer require photons? Why wouldn't wave packets work just as well, better in fact since it would get away from the use of fragile single photons? (Article)


title: Could there be a correlation between the earth's magnetic field and quantum superposition

text: I do apologise in advance for what might seem like a stupid question coming from a non-physicist. But I was wondering if the earth’s magnetic field could influence the wave pattern that one observes in a simple double-slit experiment. Meaning, in a sense, something similar to what De Broglie-Bohm suggest where the electron/particle simply rides a 'probability wave' and then takes a position at the point of measurement. Could it be that the wave that the particle/electron is riding is actually simply the earth's magnetic field that is causing these electrons/particles to move in that way? Like a boat on a giant sea. Again, thanks in advance.


title: What is meant by Proton Structure Function?

text: I am going to embark on a project involving deep-inelastic scattering but first I am trying to do some really basic background reading to get me up to task. My only background in particle physics is Griffiths chapters 1-5, and most papers I find on the topic are really advanced (I don't expect papers to be expository in nature anyway). I kept seeing the following terms pop-up but I can't find any "beginners" explanation for them on the web. I was wondering if someone could give me an expository explanation on the following section found on pages 7-8 of this document. In particular:   * What does it mean by Proton Structure Function $F_2$? **In summary** : I have managed to read up on the layman explanation of deep- inelastic scattering and what it is supposed to do. But upon venturing further (i.e. once symbols and jargons appear), I can't find any explanation for the symbols and jargons which they use, or how I should understand the subject. Any exposition/references would be greatly appreciated!


title: Big Bang Question

text: In the early Universe it it's believed that matter and anti-matter annihilated one another till there was but a remaining 1% left. In a matter/anti-matter collision, is there not an explosion of energy left behind? According to basic Newton principals the energy can't be destroyed. My question is: what exactly happened to the energy of this reaction? Can it be a contender for the explanation of the origin of dark matter or dark energy?


According to Srednicki's QFT textbook, page151 "According to our development of LSZ formula in section5, each incoming and outgoing particle should correspond to a single-particle state that is an exact eigenstate of the exact hamiltonian. This is clearly not the case for a particle that can decay." Why an unstable particle is not the exact eigenstate of the exact hamiltonian? Because its eigenvalue of the full hamiltonian is the $m_\phi$. Where is my fault?


title: Would a magnetic field be affected permanently if it interacts with another magnetic field?

text: Would a magnetic field still be changed after it stops interacting with another magnetic field. As far as I know, a magnetic field is generated by spinning electrons. Thus, does a magnetic field affect the spin of the electrons?


I'm an undergraduate physics student currently studying for the Solid State Physics Exam. This is the introductory paragraph of Chapter 1 from "Introduction to Solid State Physics" by Charles Kittel: "The serious study of solid state physics began with the discovery of x-ray diffraction by crystals and the publication of a series of simple calculations of the properties of crystals and of electrons in crystals. Why crystalline solids rather than nonclystalline solids? The important electronic properties of solids are best expressed in crystals. Thus the properties of the most important semiconductors depend on the crystalline structure of the host, essentially because electrons have short wavelength components that respond dramatically to the regular periodic atomic order of the specimen. Noncrystalline materials, notably glasses, are important for optical propagation because light waves have a longer wavelength than electrons and see an average over the order, and not the less regular local order itself." Well, I have problems understanding two sentences: 1)"because electrons have short wavelength components that respond dramatically to the regular periodic atomic order of the specimen." What does it mean "that respond dramatically to the regular periodic atomic order of the specimen" 2)What does it mean "light waves have a longer wavelength than electrons and see an average over the order, and not the less regular local order itself."


title: Radioactive nuclear fuel?

text: I suppose this is a not-too-bright question but, in a nutshell: why are nuclear fuels radioactive? With this I mean, which is the connection between being a fissile (or fertile, for that matter) material and being radioactive? Does the former imply the latter? Does the latter imply the former? Why?   Would it be thinkable for an isotope to be able to capture a neutron, become instable and split without it being radioactive in the first place?


title: Torque of an electric engine

text: So I have some follow up questions from this thread: Electric engine transmission Basically, I am looking for a more precise mathematical statement that makes this true. Why is it that the electric motor can provide torque as necessary and the gas engine can't? What can we do to the gas engine to make it behave like an electric engine? I have only taken 2nd year analytical mechanics so keep that in mind.


title: Oort Cloud Detection

text: The hypothetical Oort cloud is an explaination for long period comets. It presumably is made up of icy bodies orbiting up to 50,000 AU from the Sun. If so, shouldn't the Spitzer IR telescope have detected the radiation generated by solar energy from such a spherical cloud?


title: Why does microwaved food get cold faster

text: My evidence is entirely anectodal and non-scientific, but I've noticed food gets cold faster when it's been heated in a microwave instead of a stove. Is this true? And if it is, why does it happen?


title: What is the electric field between and outside infinite parallel plates?

text: I know that Gauss's law says $$\oint_S {\vec{E} \cdot d\vec{A} = \frac{q_{enc}}{{\epsilon _0 }}}$$ and that because $\vec{E}$ is always parallel to $d\vec{A}$ in this case, and $\vec{E}$ is a constant, it can be rewritten as $$\left | \vec{E} \right |\oint_S {\left | d\vec{A} \right | = \frac{q_{enc}}{{\epsilon _0 }}}$$ which is also equal to the electric flux through a Gaussian surface. I've included a picture to make it easier to ask my question. ![Parallel plates of uniform, opposite charge densities with Gaussian cylinders I - IV](http://i.stack.imgur.com/Mloif.png) What I don't get is how, mathematically, there is no electric field "outside" of the plates and how the electric field between them is determined. I - IV are Gaussian cylinders with one face on a plate. $$\vec{E} = \vec{E_+} + \vec{E_-}$$ Where $\vec{E_+}$ is the electric field from the positive plate and $\vec{E_-}$ is the electric field from the negative plate. For I: $$\left | \vec{E_+} \right | \pi r^2 = \frac{\sigma \pi r^2}{{\epsilon _0 }}$$ $$\left | \vec{E_+} \right | = \frac{\sigma}{\epsilon_0}$$ $$\left | \vec{E_-} \right | \pi r^2 = \frac{0}{{\epsilon _0 }}$$ $$\left | \vec{E_-} \right | = 0$$ This doesn't make sense to me because it's saying that the magnitude of the electric field due to the negative plate is 0, but even if I just assume that's because Gauss's law only works for surfaces that enclose some charge and disregard the 0 I got for the negative electric field, I'm still confused for the following reason: For III: $$\left | \vec{E_+} \right | \pi r^2 = \frac{0}{{\epsilon _0 }}$$ $$\left | \vec{E_+} \right | = 0$$ $$\left | \vec{E_-} \right | \pi r^2 = \frac{-\sigma \pi r^2}{{\epsilon _0 }}$$ $$\left | \vec{E_-} \right | = \frac{-\sigma}{\epsilon_0}$$ Now I have values for $\left | \vec{E_+} \right |$ and $\left | \vec{E_-} \right |$, but when they're going in the same direction (as they are between the plates), they sum to 0, which isn't right. To the left, when you add them going in opposite directions, you get $\frac{2\sigma}{\epsilon_0}$ and to the right you get the same thing. What am I doing wrong?


title: Matrix solution of an equivalent resistance circuit problem

text: Start with a set of points $x_1, x_2, \ldots$ that are connected by wires with some resistance. Represent the resistance by a conductance matrix (conductance being one over the resistance), where $\mathbf{C}_{ij}$ is the conductance between points $i$ and $j$, if the point are connected by a wire, otherwise the $\mathbf{C}_{ij}=0$. Can one solve for the equivalent resistance between two points by some matrix transform of $\mathbf{C}$? **EDIT** The comments bring up some interesting points - and suggest an alternate phrasing: > Can you compute the resistance distance for a graph when the resistances are > not all unit values using matrix operations?


title: How much of current flows through a bird sitting on a power line?

text: I've been googling for hours and went through over a hundred answers. Now, some say the bird doesn't form a closed loop, some say the current is so small that it doesn't kill the bird. From as much as I understand electricity, I'm sure the bird does make a parallel connection to the wire, so there must be some electrons moving through it's body. Could someone please explain which is the right answer and why? And if it's the case that a parallel circuit is formed, how can I calculate the current going through the birds body?


title: $\theta '' = \frac{a}{R}$?

text: ![enter image description here](http://i.stack.imgur.com/0rncK.png)   Assume that this is a general circular motion where $\theta$ is a function of time. I've derived some equations below and found that the angular velocity in the end is zero. Which step of my derivation is wrong? $ \vec{p}(t) = R cos(\theta) \hat{x} + R sin(\theta) \hat{y} ......(1)$   $ \vec{v}(t) = -R sin(\theta) \dot{\theta} \hat{x} + Rcos(\theta) \dot{\theta} \hat{y} ......(2)$   $ \vec{a}(t) = -R [cos(\theta) \dot{\theta}^2 + sin(\theta) \ddot{\theta}] \hat{x} + R[-sin(\theta) \dot{\theta}^2 + cos(\theta) \ddot{\theta}] \hat{y} ......(3)$   $ (2): |\vec{v}| = R \dot{\theta} $   $ a = \frac{dv}{dt} = R \ddot{\theta} ......(4) $   $ (3): |\vec{a}| = R \sqrt{\dot{\theta}^4 + \ddot{\theta}^2} $   $ \Rightarrow \frac{a^2}{R^2} = \dot{\theta}^4 + \ddot{\theta}^2 $   $ (4): \ddot{\theta} = \frac{a}{R} $   $ \Rightarrow \frac{a^2}{R^2} = \dot{\theta}^4 + \frac{a^2}{R^2} $   $ \Rightarrow \dot{\theta}^4 = 0 $


title: Hamiltonian with Dirac Delta function

text: I've to compute this expression $$ \hat{H} ~=~\frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2}) $$$$ \times \left[ \delta(\vec{r})\nabla_{\vec{r}}^2 +\nabla_{\vec{r}}^2\delta(\vec{r}) \right]\Psi(\vec{R}+\frac{\vec{r}}{2}) \Psi(\vec{R}-\frac{\vec{r}}{2}), \tag{15} $$ where $\bar{\Psi}$ is the conjugate of $\Psi$. Using Dirac delta properties, Can I say that $$\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 +\nabla_{\vec{r}}^2\delta(\vec{r}) \right] = 2 \delta(\vec{r})\nabla_{\vec{r}}^2~? $$ If not, how can I calculate this integral? I should obtain $$ \hat{H} = \frac{1}{4}g_2\int d^3R\ \bar{\Psi}(\vec{R})\left[ \nabla^2(\bar{\Psi}(\vec{R})\ \Psi(\vec{R}))\right]\Psi(\vec{R}). \tag{16} $$ A method should be expanding $\Phi = V^{-1/2} \sum_\alpha a_\alpha e^{i\textbf{k}_\alpha\cdot\textbf{r}}$, but I don't have any idea what doing! This integrals (15) comes from the paper **Phys. Rev. A 67 053612** and authors say they do integration for part and then over $\textbf{r}$. Does anyone have ideas how to calculate this integral? **/// Update ///** I tried to calculate the integral using yours suggestions. I'm near the solution! At the last there is an extra term and an extra $1/2$. In the followed images, the conjugate is $\phi^*$ and I indicate $\phi_+ = \Psi(\vec{R}+\frac{\vec{r}}{2})$ and $\phi_- = \Psi(\vec{R}-\frac{\vec{r}}{2})$ The first term is ![firstep](http://i62.tinypic.com/34o3by8.jpg) and the second is ![secondstep](http://i61.tinypic.com/28lukb6.jpg) Summing, $$ \int d^3\vec{R}\int d^3\vec{r}\nabla^2_{\vec{r}}\left( \phi_+^* \phi^*_-\phi_-\phi_+ \right)\delta(\vec{r}) $$ and then, at the last $$ \int d^3\vec{R}\ \frac{1}{2}(\phi^*\phi(\phi\nabla^2\phi^*+\phi^*\nabla^2\phi) - \phi^2|\nabla\phi^*|^2-\phi^{*2}|\nabla\phi|^2) $$ and completing it by adding and subtracting $2\phi\phi^*\nabla\phi^*\cdot\nabla\phi$ $$ \int d^3\vec{R}\ \frac{1}{2}\phi^*\left( \nabla^2(\phi^*\phi)\right)\phi - \int d^3\vec{R}\ \frac{1}{2} (\nabla(\phi\phi^*))^2 $$ I did all calculations by hand and then i checked them with mathematica. is the term $\int d^3\vec{R}\ (\nabla(\phi\phi^*))^2 = 0$ for any reasons? I hope yes. Why is there the constant $1/2$?


title: To estimate the energy of heterogenous vortex for emulsion?

text: I remember seeing in some fluid dynamics course an equation which relates the energy of a nonhomogenous vortex to the frequency. The fluid is here blood (emulsion). **How can you estimate the energy of heterogenous vortex for emulsion?**


title: Plane wave complex notation

text: As far as I know, the function: $$ \vec{E}(\vec{r},t)=\vec{E_0}\cdot e^{i(\vec{k}\cdot \vec{r}-\omega t)} \hspace{2cm}(1) $$ is a mathematical solution of the wave equation: $$ \nabla^2 \vec{E}=\mu\varepsilon\frac{\partial^2 \vec{E}}{\partial t^2} $$ if and only if $\omega$ satisfies the dispersion relation: $$\omega(k)=\frac{k}{\sqrt{\mu\varepsilon}}$$ Previously I wrote "mathematical" because the complex fuction $(1)$ has no physical meaning, if we want to have a physically significative function we have to take the real part of (1) $$ \vec{E}(\vec{r},t)=\vec{E_0}\cdot \cos(\vec{k}\cdot \vec{r}-\omega t) $$ Up to now there should be no problem. The problem arises when I consider a 1D wave packet. **Using the complex notation:** the initial wave packet is given by: $$ E(x,t=0)=e^{-\frac{(x-x_c)^2}{2\sigma^2}}\cdot e^{i k_{\text{c}}x} $$ where the derm $e^{i k_{\text{c}}x}$ derives from the fact that the initial wave packet is a moving wave packet and not a static one. Its temporal evolution can be determined doing a Fourier transform. Let's indicate with $g(k)$ the Fourier transform of the initial wave packet: $$ g(k)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}E(x,t=0)\cdot e^{-ikx}dx $$ So, using the fact that every component of the spectra evolves according to a specific $\omega(k)$, I can determine the temporal evolution: $$ E(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g(k)e^{i(kx-\omega(k) t)}dk \hspace{20mm}(2) $$ Finally, according to what I said previouisly, if I am interested in physical quantities like the temporal evoultion of the amplitude, I have to take the real part of (2): $$ E_{\text{phys}}(x,t)=\Re \left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g(k)e^{i(kx-\omega(k) t)}dk\right]\hspace{20mm}(3) $$ **Reasoning on the physical meaning step by step** The initial condition is a real value wave packet: $$ E_r(x,t=0)=\Re\left[e^{-\frac{(x-x_c)^2}{2\sigma^2}}\cdot e^{i k_{\text{c}}x}\right]=e^{-\frac{(x-x_c)^2}{2\sigma^2}}\cdot\cos(k_{\text{c}}x) $$ so, because of I am interested in determining the temporal evolution of the wave packet, I perform a Fourier transform of the initial wave packet: $$ g_r(k)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}E_r(x,t=0)\cdot e^{-ikx}dx $$ where now $g_r(k)$ is could be a complex value function but this does not hurt me because this function lives in the k-space and I am interested in having real function only in the x-space. The problem is thai if I evolve $g_r(k)$ in the function in the Fourier space multiplying it for $e^{-i\omega(k) t}$ and then I come back in the x space antitrasforming: $$ E_r(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g_r(k)e^{i(kx-\omega(k) t)}dk \hspace{20mm}(4) $$   I obtain that (4) is a function whose amplitude becomes zero in a very rapid time, a very different behaviour from (3). Could anyone give me an explanation of that, or any reference in which the complex notation is well explained?


title: How does light get from one bulb to another in a sequence of light bulb?

text: How does light get from one bulb to another in a sequence of light? Like in the advertisements or a smaller group? Is it the same light?


title: How do I choose my constants?

text: When solutions to Einstein's equations are found (in coordinates $t,r,\theta,\phi$), such as Schwarzschild, Kerr etc. the way they are found seems to be by starting with a generalised metric in terms of functions of the appropriate parameters such as $f(r)$ for Schwarzschild and $f(r,Q)$ for Reissner-Nordstrom and then of course the more complicated issue of cross terms for rotations, and then finding these functions such that Einstein's equations are satisfied. However, when you go about this yourself you realise that the solutions to Einstein's equations are going to result in these functions having constants present simply because these are differential equations we're solving. Of course most people who have studied GR know this, but is it perhaps taken for granted how we set these constants appropriately? How do I know what to set this constant to? Right now I'm assuming we use dimensional analysis and the boundary conditions (i.e. you usually want Minkowski space as $r \to \infty$), but what about choosing dimensionless parameters such as the $2$ in Schwarzschild's solution involving the integration constant $2M$? Then a slightly more complicated issue of $AdS$ space where the boundary conditions are different? Right now, when Mathematica tells me I have a free constant that won't affect Einstein's equations being satisfied, I just set it to $M$, but surely there's more to it?


title: Obtaining point of application of Ground Reaction Force with use of a hyperstatic load-cell array platform

text: I'm looking for the theory of an experiment that is giving me a hard time to perform. I have an instrument composed of a rigid horizontal square plate rigidly supported under each corner by a load-cell. Considering the origin of the system at the center of the platform, the problem statement is "find the point of application (PosX, PosY, relative to the origin at the center of the platform) of the ground reaction force of an object placed on this platform, given the vertical force measured in each of the load-cells at the corners (Z1, Z2, Z3 and Z4)". I know that the total force vertical component so as the moments around X and Y axis, with the corners numbered clockwise from "upper left" corner as seen from above: $Fz = Z1 + Z2 + Z3 + Z4$ $Mx = -Z1 - Z2 + Z3 + Z4$ $My = -Z1 + Z2 + Z3 - Z1$ and then: $PosX = Mx/Fz$ $PosY = My/Fz$ Unfortunately, though, the system is hyperstatic, and then some papers and texts from manufacturers mention a cross-talk between sensors, thus requiring some sort of calibration matrix that should multiply the measured values to obtain the actual values: $actual \begin{vmatrix}Fz \\\ Mx \\\ My\end{vmatrix} =\begin{vmatrix} C_1 & C_2 & C_3 \\\ C_4 & C_5 & C_6 \\\ C_7 & C_8 & C_9 \end{vmatrix} \times measured\begin{vmatrix}Fz \\\ Mx \\\ My\end{vmatrix}$ So the question proper is: "What's the theoretical basis for this calibration matrix (considering a square plate with hyperstatic support conditions of one support per corner) and how should one be derived from an array of experimental measurements in the form $(Z1, Z2, Z3, Z4) = f(Fz, PosX, PosY)$ ? I'm not sure this question applies here, but it seems to me that it does, since it has a theoretical nature (although derived from a very practical prolem). If it is not the case, then I would appreciate very much any suggestion regarding where I could find help. Thanks for reading!


title: Connected and strongly connected Feynman diagrams

text: 1. Recently I read, that only connected Feynman diagrams give contribution of nonzero values into the scattering amplitude. Why it is so and what is the physical sense of connected diagrams (due to their definition in Wikipedia)?   2. Also, I don't understand why strongly connected (=one-particle irreducible) Feynman diagrams are so important in scattering theory. By the other words, I don't understand why do we cut off one of the internal lines in the Diagram and does it relate to some physical process.


title: Finding the approximate solution for Schrodinger equation by using variational method

text: I need to find the approximate solution of nonlinear Schrodinger equation $$ i\hbar \partial_{t} \Psi + \frac{\hbar^{2}}{2m}\Delta \Psi - g |\Psi|^{2}\Psi - \frac{m\omega^2 (x^2 + y^2 + z^2)}{2}\Psi = 0 $$ by using variational method. $\Psi$ has norm $\int |\Psi|^{2}d^{3}\mathbf r = N$, where N is the number of particles. It was saying to me that I must start from the trial function $$ \Psi (x, y, z) = h e^{-\frac{1}{2\omega^{2}}\left( x^2 + y^2 + z^2\right)}e^{\frac{i}{\hbar}\mu t}, $$ where $\mu$ plays role of chemical potential. So by using the property of $\Psi$ I got $h = \frac{\sqrt{N}}{(\omega \pi )^{\frac{3}{4}}}$. Then, due to the variational method, I must compute functional $J(N) = \langle \Psi | \hat {H}| \Psi\rangle$ and determine N from relation $\frac{\partial J (N)}{\partial N} = 0$. But I don't understand this step, because $N$ already is given in the task and equal to the integer number. So I don't have parameters in the trial function which can minimize functional $J(N)$. Can you help me? Maybe, there is the mistake in the trial function and it must be some parameter $l^2$ instead of $\omega^{2}$?


title: Calculate the acceleration of a moving elevator

text: An elevator is moving down with uniform acceleration. A man inside the elevator who wants to measure the acceleration drops a coin as soon as the elevator starts. The coin is 6ft above the floor of the elevator at the moment it is dropped. It takes 1 second for the coin to reach the floor of the elevator. Calculate the acceleration from all this data. How do I go about solving it ? I have absolutely no idea. Can anyone give at least some hints ?


title: Current Source that produces transverse electromagnetic field

text: Given only $y$ component of Electromagnetic Field $\vec{E}$, $$E_y = \begin{cases} E_0\cos(\omega z/c - \omega t), & \mbox{ z > 0} \\\ E_0\cos(\omega z/c + \omega t), & \mbox{z < 0} \end{cases}$$ I am asked to find the current source that produces this field. So I worked on \begin{array}{lcl} \vec{E} & = & -\vec{\nabla}\phi -\frac{1}{c}\frac{\partial \vec{A}}{\partial t} \\\ \vec{B} & = & \vec{\nabla}\times\vec{A} \end{array} But I couldn't related them to produce meaning results. First things first,   1. How come this field becomes an electromagnetic field? Why don't I see "magnetic field" for both cases?   2. Should I jump from Coulomb gauge to Lorentz gauge?


title: Why supra-conductivity became super-conductivity?

text: The original article by the Kamerlingh Onnes team in Leiden does not give a name to the new effect: > Kamerlingh Onnes, H. _Further experiments with liquid helium. C. On the > change of electric resistance of pure metals at very low temperatures, etc. > V. The disappearance of the resistance of mercury_. KNAW / Comm. Phys. Lab. > Univ. Leiden 14 I, 115–122, Comm. 122b (1911). (note I'm not even sure the title is the orginal one, I'm not sure the paper was in english, since KNAW refers to. I never put a hand on it...) Few years latter, the name _supra_ -conductivity was coined: > Kamerlingh Onnes, H. _Further experiments with liquid helium. IX. The > appearance of galvanic resistance in supraconductor, which are brought into > a magnetic field at a threshold value of the field._ KNAW / Comm. Phys. Lab. > Univ. Leiden 16 II, 987–992, Comm. 139f (1914). And the name stayed for a while, for instance in german articles: > Meissner, W. & Ochsenfeld, R. _Ein neuer Effekt bei Eintritt der > Supraleitfähigkeit._ Naturwissenschaften 21, 787 (1933). > > London, F. & London, H. _Supraleitung und Diamagnetismus_. Physica 2, 341 > (1935). or when it was about discussing this in english > London, F. & London, H. _The electromagnetic equations of the > supraconductor._ Proc. Roy. Soc. Lond. A 149, 71–88 (1935). > > Schrödinger, E. _Phenomenological Theory of Supra-conductivity_. Nature 137, > 824–824 (1936). But it has been changed to _super_ -conductivity at some point, for instance in the London book > London, F. _Superfluids, volume I: Macroscopic theory of superconductivity_. > (Dover Publications, Inc., 1961). I wonder why...


title: Beamsplitter and Dielectric Mirror

text: Among two optical components. One is a Nd:YAG Laser 50/50 Beamsplitter and the second one is a Dielectric Mirror see pictures. Both of them almost looks like the same. If the specification is not mentioned on them. Is there a way we could differentiate them and know which one is a beam splitter and which one is a mirror ? ![enter image description here](http://i.stack.imgur.com/QGpmz.jpg) ![enter image description here](http://i.stack.imgur.com/B0e8a.jpg)


title: Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?

text: I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result. The background for this problem comes from the composition of Brownian motion and studying the densities of the composed process. So if we have a two sided Brownian motion $B_1(t)$ we replace t by an independent Brownian motion $B_2(t)$ and study the density of $B_1(B_2(t))$. If we iterate this composition n times we get the iterated interal in (**) below as an expression for the density of the n times iterated Brownian motion. The result I am interested in is derived in the following paper: The original reference is "Fractional diffusion equations and processes with randomly varying time" Enzo Orsingher, Luisa Beghin http://arxiv.org/abs/1102.4729 Line (3.14) of Orsingher and Beghins paper reads (**) $\lim_{n \rightarrow \infty} 2^{n} \int_{0}^{\infty} \ldots \int_{0}^{\infty} \frac{e^{\frac{-x^2}{2z_1}}}{\sqrt{2 \pi z_1}} \frac{e^{\frac{-{z_1}^2}{2z_2}}}{\sqrt{2 \pi z_2}} \ldots \frac{e^{\frac{-{z_n}^2}{2t}}}{\sqrt{2 \pi t}} \mathrm{d}z_1 \ldots \mathrm{d}z_n = e^{-2 |x|} $   1. How do you prove this result without using probability?   2. Is this a type of path integral (functional integral)? Or is this integrand some kind of kinetic plus potential term arsing in quantum mechanics? Do expressions like (**) ever come up in physics literature? (I tried using the change of variable theorem for Wiener measure to transform (**) into a Wiener integral with respect a specific integrand and have had some success with this.. I think this shows how to compute a Wiener integral with respect to a function depending on a path and not just a finite number of variables but did not see how to take this any further - The change of variable theorem for Wiener Measure was taken from "The Feynman Integral and Feynman's Operational Calculus" by G. W. Johnson and M. L. Lapidus.)   1. I have been studying a slight generalization of ** from the probability side of things and have been trying to use dominated convergence to show the LHS of ** is finite but I am having problems finding a dominating function over the interval $[1,\infty)^n$. Is dominated convergence the best way to just show the LHS of (**) is finite?


title: determination of voltaic cell voltage

text: if i understood this correctly, the determination of voltage for a specific voltaic (gallvanic) cell is determined only by the chemical correlation between the two metals. is this true? for an example , if i use iron instead of zinc in the example from wikipedia, i will get a different voltage?


title: Born-Oppenheimer Approximation equivalent to Tensor-product ?

text: If you have a wave function $\Psi$ of a system consisting of an electron and the vibrational modes of the crystal, THEN we represent the wavefunction $\Psi%$ to be in the Hilbert Space formed by the tensor product of the Hilbert spaces corresponding to the electron with the Hilbert Space corresponding to the vibrational modes if and only if there isn't an instantaneous interaction between the electrons and the vibrational modes; First of all, this is true right? The Born-Oppenheimer Approximation technique tells us that we can write the wavefunction $\Psi$ as a product wavefunction- as a product of the electronic ($\phi$) and vibrational modes' ($\zeta$) wavefunctions. We write $\Psi$= $\phi \zeta$ ? Now my main question: Is the Born-Oppenheimer Approximation technique equivalent to saying that the Hilbert Space representation of the space that $\Psi$ is situated in is the tensor product space of the electronic and vibrational mode wavefunctions?


title: Calculating heat capacity from the equation of state

text: It is known that within thermodynamics alone, given the equation of the state of a system, one cannot explicitly determine the heat capacity. What is the mathematical reason for this? Intuitively, it is clear that the heat capacity contains the information about the degrees of freedom of the system which are not visible to the equation of state and, therefore, we need statistical mechanics to determine them. But how do we see it in the mathematical formalism of thermodynamics?


title: QM and Renormalization (layman)

text: I was reading Michio Kaku's _Beyond Einstein_. In it, I think, he explains that when physicsts treat a particle as a geometric point they end up with infinity when calculating the strength of the particle's field as you approach the particle. First, did I get that part right? Second, how does (in very simple terms) renormalization try to fix this in Quantum Mechanics? If you can include some type of word picture that would be great. Accepted answer will go to the clearest explanation. **Update** I got a little further in the book (lol) and Kaku talks about using symmetry to remove the divergences in the math. I'd appreciate an answer that incorporates this also. Thanks guys!


title: Is there any literature that compares the candle-power to the candela?

text: I am unable to find anything in the literature to back up the claim that appears all over the internet that:               The modern candlepower now equates directly to the number of candelas.      Can anyone please suggest something in the scientific literature to back up this statement? If this is not the correct stack exchange for this question, please suggest a better one.