title: What are the units of these virial coefficients?

text: I'm reading some papers for calculating the vapor pressure of alkali metals as a function of temperature, and I've come across some familiar-looking virial expansions, but when I tried to work out the unit analysis, it was very much unclear how to make everything work out. Here's the first equation (where they give the form in terms of the physical quantities): $$ \mathrm{ln}\left(p\right) = I_{c} - \frac{\Delta H^{\circ}_{v}}{RT} + \frac{\left(C^{g}_{p} - C_{p}^{l}\right)}{R}\mathrm{ln}\left(T\right) $$ This is equation 4 of the following paper: > (1) Alcock, C. B.; Itkin, V. P.; Horrigan, M. K. Canadian Metallurgical > Quarterly 1984, 23, 309–313., (doi: 10.1179/000844384795483058 From the definitions section, $\Delta H_{v}^{\circ}$ is the molar enthalpy of vaporization in $\frac{BTU}{lb\cdot mol}$, $R$ is the gas constant. No units are given for this, but given the other values, we'll call it $\frac{BTU}{lb\cdot mol ^{\circ}R}$. $C_{p}^g$ and $C_{P}^l$ are the isobaric (constant pressure) specific heats, given in $\frac{BTU}{lb\cdot mol^{\circ}\\!R}$. So working this out, I'm going to assume that this is supposed to be a unitless quantity somehow. I'm assuming that $I_{C} = \mathrm{ln}\left(p_{0}\right)$, so we get: $$ \mathrm{ln}\left(\frac{p}{p_{0}}\right) = -\frac{\Delta H_{v}^{\circ}}{RT} + \frac{\left(C_{p}^{g}-C_{p}^{l}\right)}{R}\mathrm{ln}\left(T\right) $$ Everything there is unitless except for $\mathrm{ln}\left(T\right)$. (Alternately, maybe he is secretly assuming that $p = \frac{p}{torr}$ - the unit analysis is basically the same). What is going on there? The other form I see for this is: $$ \mathrm{log}\left(p\right) = A_{c} - \frac{B_{c}}{T} + C_{c}\mathrm{log}\left(T\right) $$ Again, there's a logarithmic temperature in there! And, bizarrely, he defines the following quantities (A is nowhere to be found on this list): $$B = \text{second virial coefficient, }\frac{ft^3}{mol}$$$$ C = \text{third virial coefficient, }\frac{ft^6}{mol^2} $$ According to the paper, $_{C}$ is the indicator for "empirical constant". I'm guessing $A$ is supposed to be unitless (don't know how to scale it to change the pressure units, but one thing at a time). Trying to find a second opinion, I went to this paper: > (1) Ewing, C. T.; Chang, D.; Stone, J. P.; Spann, J. R.; Miller, R. R. > Journal of Chemical & Engineering Data 1969, 14, 210–214. doi: > http://dx.doi.org/10.1021/worsp). But they have listed _no_ units, and suffer from the same problems. Here is their version of the expansion: $$ A + \frac{B}{T} + C\cdot\mathrm{log}\left(T\right) + D\cdot T\cdot10^{-3} $$ They plug in some values for this: $$ \mathrm{log}\left(p(atm)\right) = 7.55339-\frac{27007.7}{T} + 0.19699\mathrm{log}(T) - \frac{0.171188\cdot T}{1000} $$ Is temperature a unitless quantity now? What am I missing here?


title: What really is the Magnetic Force on a wire?

text: I have a doubt regarding the significance of a force on a wire. Well, first of all, I know that if I have a particle and if there are several forces acting over it, then we can compute one total force $F$ that gives the same effect as the combination of the several forces, and this force is just the vector sum. Well, this is pretty good: a vector _depends_ on the point it's being applied (since it's an element of the tangent space at the point), so that since all the forces are on the same point (the same tangent space) we can take their sum and get another thing acting on the same point. This is pretty clear and simple to understand. My doubt is when we have a wire for example. For a wire normally we use the relationship: $$F=\int_\gamma i \ \gamma'(t)\times B(\gamma(t)) \ dt$$ In other words, we parametrize the wire with some curve $\gamma$, and we integrate $i \gamma' \times B$ over the wire to get the "force on the wire". But now this is confusing me, we are summing vectors at different points, and getting a vector that I don't know where's located. What I mean is: while when working with particles it makes sense to add the forces and use the total force on the same particle, with a wire this is kind of confusing, because it has length, so what should really mean "a force on a wire" since it's not just a point? Thanks in advance for the help.


title: Frequency of small oscillation of particle under gravity constrained to move in curve $y=ax^4$

text: How to find the frequency of small oscillation of a particle under gravity that moves along curve $y = a x^4$ where $y$ is vertical height and $(a>0)$ is constant? I tried comparing $V(x) = \frac 1 2 V''(0) x^2 + \mathscr O(x^3) = \frac 1 2 kx^2$ (assuming $V(0)$ is ground state and $V'(0)$(slope) remains is horizontal at extrememum, but unfortunately $V''(0) = 0$. I am pretty much clueless. Thanks for your help. **ADDED** :: Letting $m=1$ the Lagrangian is $L = \frac 1 2 (\dot x^2 + \dot y^2) - gax^4 = \frac 1 2 (\dot x^2 + (4 ax^3 \dot x)^2)-gax^4 = \frac 1 2 \dot x^2(1 + 16a^2x^6)-gax^4$ The above Lagrangian gives the equation of motion as $$\ddot x(1+16a^2x^6)+\dot x^2 96 a^2x^5 + 4 gax^3 = 0$$ Since we are considering the system a small oscillation whose potential is of order 4, $\mathscr{O}(x^{k>4})$ can be ignored which reduces into $\ddot x = -4agx^3$. To solve this, $$\frac 1 2 \frac{d}{dt}(\dot x^2) = -\frac{d}{dt}(agx^4)$$ which gives $\dot x = \sqrt{k - 2agx^4}$, Assuming the system begins from $t=0$ at $x=x_0$ with $\dot x = 0$, $k = 2agx_0$, which turns the integral into $$\int_0^{T/4} dt= \int_0^{x_0} \frac 1 {\sqrt{2agx_0^4-2agx^4}} dx=\frac{1}{\sqrt{2ag}x_0}\int_0^1\frac{1}{\sqrt{1-y^4}}dy$$ Which gives $$T = 2 \sqrt 2 \sqrt{\frac{\pi}{ag}}\cdot \frac{\Gamma(5/4)}{\Gamma(3/4)}\cdot \frac 1 {x_0}$$ which is a dubious result. Please someone verify it. Any other methods are welcome.


title: Hemisphere irradiance

text: How do I calculate sky irradiance from radiance (L) from a hemisphere above a surface which is tilted relative to the normal (x=0,y=0,z=1). I have L as a function of zenith (0 to 180deg) and azimuth (0 to 360 deg)? Any known MAtlab functions doing this kind of thing?


title: Is it better to call the doppler effect a change in wavelength or frequency?

text: Why is it preferable to say that the doppler effect causes a shift in frequency rather than a shift in wavelength? I often read on websites that they define the doppler effect as a change in frequency.


title: Why is $ \vec{S}^{(A)} \otimes \vec{S}^{(B)} = \frac{\hbar^2}{4}(\sigma_x \otimes \sigma_x + \sigma_y \otimes\sigma_y + \sigma_z \otimes \sigma_z)$?

text: I haven't been taught tensor product in class but they have taught us addition of spin. I looked up online in this link->http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf (pg 148, pg 10 in the pdf) and found an explanation. I think I understand most of it except this step: $$ \vec{S}^{(A)} \otimes \vec{S}^{(B)} = \frac{\hbar^2}{4}(\sigma_x \otimes \sigma_x + \sigma_y \otimes\sigma_y + \sigma_z \otimes \sigma_z). $$ I know that: $$ S^{2} = \frac{\hbar^2}{4}((\sigma_x)^2 + (\sigma_y)^2 + (\sigma_z)^2), $$ but I don't see the connection.


title: Interpretation of de Broglie wave

text: Until what point can the de Broglie wave be thought as a real wave? I mean, is it made of something? What amplitude does it have? Is it a sine wave? How can it be related to the wavefunction of the particle?


title: Derivative of $\epsilon$ in Lennard-Jones equation

text: I just want to calculate the derivative of epsilon for the following Lennard- Jones equation: LJ = $4\sqrt{\epsilon_{ii} \epsilon_{jj}}$ $[(\sigma/ r)^{12}-(\sigma/r)^{6}]$ with respect to $\epsilon_{ij}$ I am very glad if anyone explains this.


title: Making a cut trough a center of mass, can the masses of the pieces be equal?

text: Let's say point $P$ is the center of mass of an irregularly shaped object.   1. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same mass? This would be possible if the centers of mass of each piece was the same distance from $P$, but is this possible with an irregularly shaped object of uniform density?    2. What about non-uniform density?  This is just a question I thought about.


title: Commutator as a time-ordered product

text: I'm reading through Seiberg and Witten's paper "String Theory and Noncommutative Geometry," and one part in $\S$2.1 isn't quite clear to me. (Sorry, in advance, for the length.) My question is about equation (2.7) on page 9, where they say: > The coefficient $\theta^{ij}$ in the propagator also has a simple intuitive > interpretation, suggested in [15]. In conformal field theory, one can > compute commutators of operators from the short distance behavior of > operator products by interpreting time ordering as operator ordering. > Interpreting $\tau$ as time, we see that > > $$ > [x^i(\tau),x^j(\tau)]=T(x^i(\tau)x^j(\tau^-)-x^i(\tau)x^j(\tau^+))=i\theta^{ij}. > \tag{2.7} $$ ${}$ **Question: How did they get the first equality?** I'm sure that there is probably some really simple argument I'm not seeing, but, as this equation is the crucial step in which noncommutativity of coordinates appears, I figure I should probably understand it. Any help? * * * The context is: consider a flat space (with metric $g_{ij}$) in the presence of a constant NS $B$-field with $Dp$-branes. Further, for simplicity, they assume that $B_{0i}=0$, $B_{ij}\neq 0$ for $i,j=1,\cdots,r$, and $g_{ij}=0$ for $i=1,\cdots,r$ and $j\neq 1,\cdots,r$, where $r$ is the rank of the matrix $B_{ij}$. Then one can obtain the action $$ S=\frac{1}{4\pi\alpha'}\int_{\Sigma}\big(g_{ij}\partial_ax^i\partial^ax^j-2\pi i\alpha'B_{ij}\epsilon^{ab}\partial_ax^i\partial_bx^j\big) $$ (with $\Sigma$: the string worldsheet), which leads to the boundary conditions (along the $Dp$-branes): $g_{ij}\partial_nx^j+2\pi i\alpha'B_{ij}\partial_tx^j\big|_{\partial\Sigma}=0$ (where $n$ is the normal to $\partial\Sigma$). Taking $\Sigma$ to be a disc, these become: $$ g_{ij}(\partial-\bar{\partial})x^j+2\pi\alpha' B_{ij}(\partial+\bar{\partial})x^j\big|_{z=\bar{z}}=0. $$ Then, it was shown elsewhere that the propagator can be written: $$ \langle x^i(z)x^j(z') \rangle = -\alpha'\bigg[g^{ij}\log|z-z'|-g^{ij}\log|z-\bar{z}'|+G^{ij}\log|z-\bar{z}'|^2 $$ $$ \hspace{3.4in}+\frac{1}{2\pi\alpha'}\theta^{ij}\log\frac{z-\bar{z}'}{\bar{z}-z'}+D^{ij}\bigg], $$ where $G_{ij}$ is the open string metric given by $$ G_{ij}=g_{ij}-(2\pi\alpha')^2(Bg^{-1}B)_{ij} $$ so that $$ G^{ij}= \bigg( \frac{1}{g+2\pi\alpha'B}g\frac{1}{g-2\pi\alpha'B} \bigg)^{ij}, $$ the constants $D^{ij}$ depend only on $B$ and "play no essential role and can be set to a convenient value," and $$ \theta^{ij}=-(2\pi\alpha')^2\bigg( \frac{1}{g+2\pi\alpha'B}B\frac{1}{g-2\pi\alpha'B} \bigg)^{ij}. $$ Finally, they focus on open string vertex operators, restrict to real $z$ and $z'$ (which they then denote by $\tau$ and $\tau'$), and make a convenient choice of $D^{ij}$ so that the above propagator reduces to $$ \langle x^i(\tau)x^j(\tau')\rangle = -\alpha'G^{ij}\log(\tau-\tau')^2+\frac{i}{2}\theta^{ij}\epsilon(\tau-\tau'). $$ Then the above claim about the commutator is made.


title: Temporal coherence in Computational ghost imaging

text: In pseudo-thermal ghost imaging using a laser beam passing through a rotating ground glass plate, the temporal coherence is important since the intensities in both scanning and bucket detectors have to be measured within the coherence time of the beam. However, in computational ghost imaging this requirement is obviously not enforced (since the scanning detector intensity is calculated using the phase added by the spatial light modulator). I wanted to know what is the basic difference between these two approaches? Why is it that the phase fluctuations of laser (giving rise to finite coherence time) are important in conventional pseudo-thermal ghost imaging but is not important in computational ghost imaging. Any help would be wonderful. Thanks.


title: Mistake in the Feynman Lectures Volume 1 Ch. 18-2 - Rotation of a rigrid body

text: I just read http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-S2 In my opinion, in this chapter the equations 18.6 and 18.7 are wrong. Have a look at the Picture http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-F1 . > If $OP$ is called $r$, then the length $PQ$ is $rΔθ$, because of the way > angles are defined. The change in $x$, then, is simply the projection of > $rΔθ$ in the $x$-direction: $$\Delta > x=-PQ\sin\theta=-r\,\Delta\theta\cdot(y/r)=-y\,\Delta\theta$$ That is equation 18.6. _It is wrong because $\Delta\theta$ doesn't mean that $\theta_{Q}-\theta_{P}$ is infinitesimally small and equation 18.6 is only valid for infinitesimally small $\Delta\theta$._ An example is: I take $P$ to be $(1|0)$ and $Q$ to be $(0|1)$ than $\Delta x$ is $-1$ but calculated using equation 18.6 it should have been $-y\Delta\theta=(-1)(\pi /2)\approx -1,57$, which is wrong. In the equations $d\theta$ should have be used instead of $\Delta\theta$. **Is my reasoning correct?**


there is a relation between time and space in special theory of relativity: $$t^2c^2-L^2=\tau^2.c^2$$ what is relation between time and space in general relativity?


title: Physically implementing quantum measurement of energy

text: If there were a particle in a box, how could one measure its energy? I understand the theory behind quantum measurements: the Hamiltonian operator represents the energy observable, so you perform an "energy measurement" that returns an eigenvalue of the operator, and that eigenvalue is the energy. But how would you actually physically do this?


title: Proof for torque=force*perpendicular distance to line of action of force

text: I want to derive from first principles the proof for magnitude of torque about a point O = perpendicular distance from O to the line of action of the force * magnitude of the force. I want to derive this result for a general 3-D case for any arbitrary point O and force **_F_** . I need to prove this mathematically without knowing the cross-product magnitude in terms of the angle between the vectors..... that is given any force F = Fx i + Fy j + Fz k and any point O and using torque= r cross F & using the determinant expansion for the same & also the expressions for magnitude of a vector, I need to prove this.... In other words, I need to prove that the magnitude of the cross- product is indeed F*r*sin(theta


title: Projection of Quantum State onto Bell State

text: I am very interested by this paper on entanglement swapping and timelike entanglement. The one thing I get really tripped up with is the whole idea of a projection onto a bell basis. I understand the whole idea of projecting into different basises, but I am not sure what the authors are doing. From the article:               When the two photons of time  (photons 2 and 3) are     projected onto any Bell state, the first and last photons     (1 and 4) collapse also into the same state and entanglement     is swapped. The first and last photons, that did not share     between them any correlations, become entangled.      I am trying to see how that is the case from Equation 3. I am missing the way to mathematically take the projection and then see what is happening with photons 1 and 4. Any help in understanding how to project onto the bell basis would be very much appreciated!


title: Is a semi-Euclidean space possible?

text: Does exists a geometry (3d for example) which is Euclidean in 2 dimensions (x and y coordinates) and non-Euclidean when the third dimension (z) is taken into account? In other words a space where it is possible to construct a _square_ but not a _cube_?


title: Finesse of an Fabry-Pérot interferometer

text: during an undergrad experiment we have to estimate the finesse of an Fabry- Pérot interferometer. The reflectance $R$ is given so the theoretical value can be calculated by $\mathcal{F} = \frac{\pi\sqrt{R}}{1-R}$. For an estimation of the practical value we can use the definition $\mathcal{F} = \frac{\Delta\lambda}{\delta\lambda}$, where $\delta\lambda$ is the full-width half-maximum of an transmission band and $\Delta\lambda$ is the free spectral range (FSR). In our experiment we had an expandet, monochromatic source in an magnetic field, so due to the Zeeman effect we had in fact an splitted line. How can we get the FSR? Or is there another way to calculate the finesse?


title: What is quantum entanglement?

text: **What is quantum entanglement?** Please be pedagogical. **Edit:** I have updated my background under my profile.


title: Why do things that are far away seem smaller?

text: As you see things that are far smaller, a funny question about this: Imagine there are many people in a row (all are same height)                   A            Other Guy B             C             D         |                             you |  distance     |        distance        distance          | <-------->    |        <------->   |  <------->            |               |                    |             |         |               |                    |             |         ------------------------------------------------------  floor         From your point of View you see D much smaller than you, then.. Remember, if this is an "effect" then D is not smaller than you, it's just far away (and this is not earth curvature, because D is smaller in all sense not only in height) so.. Imagine you have a laser pointer and want to point your partner's face to annoy him so how do you target it? do you align the laser with the floor at the height of your own face believing in geometry? or do you point directly at the "little person" D ?                   A               B                    C             D         |. . . . Laser path . . . . .  . >. . . . . . >. . . . . . >. . .  . >         |  ^            |                                you |  |parallel    |                    |         |  |            |                    |             |              |  v            |                    |             |         ------------------------------------------------------ floor         Then you don't hit him, and the "visual effect" became so real that you start to think D is for Dwarf. There is something Wrong? Physics opinions are welcome! I think the answer is that floor itself will seem not aligned eigther so targeting the Dwarf you will be really aligned with floor because floor itself will have perspective effect.                   A               B                    C             D         |. . . . Laser path . . .            |               |        ''''''''''''......     you |               |                    |      ......>|         |               |                    |      _______|___floor            |               |         ___________|------                                 --------------------------      Anyway its weird


title: What is the maximum range of a bullet flying through the sky?

text: **How far would my bullet fly with the following characteristics?**   * 1.2km/second initial velocity   * 45 degree angle into the sky   * 'Normal' atmospheric conditions   * No wind   * Typical high-powered rifles have 5-10 gram bullets I've tried various online calculators but they only do vacuum, not normal atmospheric conditions, so the range is 102km.


title: What is the speed of a thrown ball

text: I would like to know how much speed is loss when a baseball is pitched. When a pitcher hurls the ball, it must lose speed before the catcher catches it because of the air resistance. So, if the catcher catches a 90 mph fastball, what speed was it going when it left the pitchers hand? The drag coefficient for a ball is 0.47 and the distance traveled is 60.5 feet. Let's make it easier by ignoring gravity. How do I figure this out?


title: Is the Hilbert-Felber model of repulsive gravity correct?

text: This theory states that at relative velocities exceeding 3^-0.5 c, gravitational repulsion ensues. The relevant papers are on arXiv by Franklin Felber. Here's one of them http://arxiv.org/abs/0910.1084 I have seen two interesting critiques thus far. The first is that the effect is an artifact of the choice of coordinates: http://arxiv.org/abs/1102.2870 and rebutted by Felber here http://arxiv.org/abs/1111.6564 This answer also seems germane http://physics.stackexchange.com/a/83175/11633 The second critique involves the feasibility of the proposed LHC experiment: http://arxiv.org/abs/0912.1323v1


title: Poincaré symmetry and linearized gravity

text: When working with linearized gravity, is Poincaré symetry assumed to be the symmetry of space-time?


title: Continuity and Blood Flow

text: Consider a tube with two different diameters (one section wider than the other). Because what comes in must go out (conservation of mass), the flow rate in the wider section must be the same as the flow rate in the smaller section (you can't loses some or what your put in or gain some of what you've not put in). Also, the flow of a liquid follows the pressure gradient (liquid flows from high pressure to low pressure in the tube). The left end of the tube is at a high pressure and the right end is at a lower pressure (consider the fluid moving from left to right). According to continuity, $\Phi_1$ = flow rate 1 $= \Phi_2$ = flow rate 2 or $A_1v_1 = A_2v_2$ where $A$ and $v$ are the cross sectional area and velocity respectively. Now consider the a disease/condition where the blood vessels are contracted (think of a normal blood vessel as the bigger diameter tube and the contracted blood vessel as the smaller diameter tube above). I forgot what this disease is called but that's beside the point. The main thing is that blood flow is reduced in the bad section of the blood vessel (small diameter section), requiring surgery to change out the bad section for a normal one. Does this not contradict the continuity equation, which states that the flow rate should be the same in each section? My response: After talking to a professor, he gave the response (I didn't follow but goes something like) the continuity equation does not take into account the amount of pressure in the bigger diameter section and the amount of pressure in the smaller diameter section (it just takes into account the difference in pressure between the two ends). Can anyone help shed light on this? Please discuss what the continuity equation does and does not do as well as what's going on in the blood vessel please. Final note: $\phi = \frac{\pi (P_A -P_B)}{8\eta L}R^4$ This equation describes flow rate taking into account the viscous drag force. It was derived considering a cylindrical tube. The derivation starts (I'm giving you this because even if you don't know what the equation is, I'm giving you context. I don't understand this equation either. It considers a tube all with a single diameter. $A = \pi r^2$, $SA$ stands for surface area. Note, velocity is greatest at the center and smallest along the tube walls due to friction) (driving force) - ($F_{viscous}$) = 0 (at equilibrium) $\Delta PA$ - $\eta (SA)\frac{dv}{dr}$ = 0 Anyways, the final equation for flow rate shows that $\phi \propto R^4$. I thought flow rate was constant? In summary, why is blood flow reduced in the contracted blood vessel and said to be the same in a pipe with different cross sectional areas?


title: Maximum electron-phonon coupling SC temperature

text: In many articles and web pages I found people claiming that the maximum critical temperature for superconductivity in the BCS framework is about 40K. Sometime, more accurate writers report that the above temperature is the maximum critical temperature achievable with electron-phonon coupling. In both cases I never sow a reliable proof of these statements, so I'm asking if they are correct and where I can find a reference (a paper or simply few physical considerations) leading to this result.


title: Why do ferromagnetic thermocouples change their behaviour near the Curie point?

text: Having read Wikipedia's current explanation that certain types of thermocouple (specifically Type K thermocouples) experience a step-change in their potential when they reach the Curie point of the constituent iron, it does not explain the reason, nor why the Curie point is important for the thermoelectric properties of ferromagnets. There seems to be no more of a hint on Wikipedia's page about the thermoelectric effect, nor for the page on the Curie temperature. Searching the internet more generally for information only seems to find similar assertions without explanations, or abstracts from academic papers hidden behind paywalls. Can anyone explain this connection in terms a layman reading Wikipedia would understand, or failing that an undergraduate physics student?


title: What are the benefits of quantum information "teleportation"?

text: I read occasionally popular science articles and from time to time encounter issues about quantum information teleportation. (this one for example http://www.physorg.com/news193551675.html) So far I have the following understanding of basic principles:   1. after measurement entangled particles have identical random state (i.e. we can not predict the result, but we know that it will be similar for both particles).   2. "decoding key" should be sent by traditional channel As I understand the underlying process is the following:   1. Some amount of entangled particles is sent to A and B.   2. Particles are measured at A and B and results R (equal in both places) are recorded.    3. Knowing the results A computes a "decoding key" (some function F, that F(R) has some valuable meaning) which is sent by traditional means to B.    4. B applies key to it's copy of R, and thus receives information. The benefit is that amount of information that needs to be transmitted by traditional means is small. In case everything stated before is correct, I have the question: how is this "quantum information teleportation" process differs from just sending two identical sets of random data to both destinations by traditional means? I could be wrong about understanding of principles of quantum information teleportation, in this case, please, correct me.


title: Optics, What is a Foil?

text: I was using a program for simulating optical elements; mirrors, gratings, foils, crystals, slits, and zoneplates are what it mentioned. Does anyone know what a foil is in terms of optics? I know what aluminum foil is, obviously, but in terms of an optical system I don't know what a foil, or foil, would be, and what it's unique optical properties are.


title: How does the viscosity (besides density) of a fluid affects sinking

text: If water, air of whatever fluid had a different viscosity, but the same density, would things fall/sink differently?


title: Nuclear fuel pellets - when does the chain reaction start?

text: Questions about nuclear fuel: Does nuclear chain reaction start in fuel pellets even before they being installed in reactor? If not, why not? My understanding is that since the fuel pellets are uranium enriched, they should be enough to sustain chain reaction. It seems I am wrong (found no reference support me), but I don't know why. Also, for fuel pellets and rods are radioactive, how do we transport them? There are plenty of articles about spent fuel rod transport, but I found almost zero about "fresh" fuel rod transport. According to Nuclear fuel cycle page of wikipedia: _`In the case of some materials, such as fresh uranium fuel assemblies, the radiation levels are negligible and no shielding is required.`_ Erruh. This is against my understanding of the first question. Can anyone help relieve my headache?


title: Entropy / Structure Relations

text: I want to check on the validity of the following objective definition of order. Is it correct? Is there a more rigorous statement of this concept? The further a system is from its maximum thermodynamic entropy, the more ordered it is and the lower its entropy. Specifically, if a system’s microstates are not of equal probability, which is usually the case because of interactions or correlations between the microstates, it is more ordered when it spends time in the less likely states. If the microstates are of equal probability, it can be constrained to remain in one of the states a disproportionate amount of time by imposing constraints on the system’s random fluctuations. Thanks very much for the input.


title: Is there no fourth dimension (time) in the universe?

text: In this article (and numerous others) http://phys.org/news/2011-04-scientists- spacetime-dimension.html and http://www.rebelscience.org/Crackpots/notorious.htm they suggest that there is no time dimension and even in the last one that there is no motion in space- time. Do you believe these theories, are they true?


title: Why doesn't breath condense around poop of flies on the window's glass?

text: One day it was raining outside and I was breathing on the glass of my window. I noticed that around the spots of fly poop there is a circle where vapor does not condense. With time the circle becomes larger and larger as the condensation slowly disappears. Why does this happen? Edit: Note that the poop is there from days ago, so it's not warmer than the glass. If the effect has something to do with the poop being warmer, please give me a motivation. There is an image here which is also shown below. Sorry that it is out of focus; my phone's autofocus couldn't pick out the foreground image. ![enter image description here](http://i.stack.imgur.com/M3kw3.jpg)


title: Transformation law for spinor functions multiplication

text: Let's have Dirac spinor $\Psi (x)$, which formally corresponds to $$ \left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0 \right) $$ representation of the Lorentz group. What representation is true for $\Psi (x) \Psi^{+}(x')$? I expect something like $$ \left[\left(\frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)\right]\otimes \left[\left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0\right)\right] = $$ $$ =\left(\frac{1}{2}, 0 \right) \otimes \left(\frac{1}{2}, 0 \right) \oplus \left(\frac{1}{2}, 0 \right) \otimes \left( 0 , \frac{1}{2}\right) \oplus \left( 0, \frac{1}{2} \right) \otimes \left(0, \frac{1}{2} \right) = $$ $$ \tag 1 =\left[\left( 0, 0\right)\oplus (1, 0) \right]\oplus \left( \frac{1}{2} , \frac{1}{2}\right) \oplus \left[\left( 0, 0\right)\oplus (0, 1) \right], $$ but I'm not sure. Also I know that $$ \tag 2 [\Psi (x), \Psi^{+}(y)]_{+} = i\left( i\gamma^{\mu}\partial_{\mu} + m\right)\gamma_{0}D_{m}(x - y), $$ where $D_{m}(x - y)$ is a lorentz scalar function, so formally $(2)$ doesn't coinside with $(1)$. How to compare it with $(1)$?


title: Can the apparent equal size of sun and moon be explained or is this a coincidence?

text: Is there a possible explanation for the apparent equal size of sun and moon or is this a coincidence? (An explanation can involve something like tide-lock effects or the anthropic principle.)


title: How can the Earth's day increase and its rotation slow down at the same time?

text: I heard that the Earth rotation is slowing down, but I also heard the Earth's length of day is increasing. Does the two theories go together or conflict with each other?


title: What age to start kids with astronomy

text: My son is getting interested in astronomy. I was thinking about getting him a scope when he gets a bit older, he is 4 now. At what age have other people gotten their kids telescopes? (I have a 10" dob that we use together) Also at what age is taking a kid on a dark sky camping trip start to make sense.


title: Does the wing-propeller plane or jet-plane struggle up at the high altitudes because of low pressure?

text: We all know that wing-propeller planes rely on air to create a thrust. They suck the air in and push them back hard that it's opposite reaction pushes the plane (Newton's law). Does that mean wing-propeller struggles up at high altitude because of the low pressure of the air ? Same question for jet engine.


title: Wave function - Working towards Orthonormal solution

text: As you can see below, the two pages attached describe how to obtain the final solution of orthonormality. I do not completely follow the discussion but I will talk you through what I DO understand and then hopefully you could fill in the gaps ? So, we consider a wavefunction $\psi$ which when measured has discrete eigenvalues $f_n$. When we obtain $f_n$ we then have $\psi$=$\psi_n$ We assume the eigenfunction ($\psi_n$) is normalise as per eqn. 3.1. Applyin the principle of superposition : Since we have multiple eigenfunctions which satisfy the wavefunction $\psi$ we must also be able to state these eigenfunction as a linear combination which in turn is also a solution. Hence, we obtain 3.2. **First problem :** In 3.2 where did $a_n$ come from ? This completes the first picture discussion. Now moving onto the second picture : **Second problem :** The first paragraph talks about bilinear expressions and what not. I do not follow the discussion in this section or how 3.3 was obtained. If these two issues are explained to me I think I can work out the rest. :) ![enter image description here](http://i.stack.imgur.com/kdHBL.jpg) ![enter image description here](http://i.stack.imgur.com/nOS5l.jpg)


title: Could CP violation be exploited to violate the second law?

text: I don't understand well CP-violation, but since it introduces a microscopic irreversible process, couldn't it be somehow exploited to decrease entropy? For instance, if we could somehow use it to bias average particle movement in a particular direction I suppose it could be used to extract work.


Can we say that nanomaterials have higher cohesive energies compared to coarse-grained counterparts?


title: Born approximation to Lippman-Schwinger integral equation

text: I am having the following problem understanding the Born approximation in the case of the Lippmann-Schwinger equation. This exercise is for something which is entitled "computational physics lab course" and the following equations are all taken from the lab manual. Following the derivation in Chapter 3 of Weinbergs _The Quantum Theory of Fields_ in a similar fashion one obtains the Lippmann-Schwinger-Equation: $$ |\Psi_\mathbf{k}^\pm\rangle = |\mathbf{k}\rangle + \lim_{\epsilon\rightarrow 0^+} [E_k\pm i\epsilon - H_0 ]^{-1} V |\Psi_\mathbf{k}^+\rangle.$$ From this one can obtain the tranfer matrix $T$ by defining $$T|\mathbf{k}\rangle = V | \Psi_\mathbf{k}^+\rangle.$$ Then following holds: $$T|\mathbf{k}\rangle> = V|\mathbf{k}\rangle + V \lim_{\epsilon\rightarrow 0^+} [E+i\epsilon-H_0]^{-1} T|\mathbf{k}\rangle.$$ From which in turn an expression analogous to eq 3.5.1 in Weinberg can be obtained: $$\langle\mathbf{l}| T |\mathbf{k}\rangle = T(l,k)=\hat{V}(\mathbf{l}-\mathbf{k}) + \lim_{\epsilon\rightarrow 0^+} \frac{1}{(2\pi)^3}\int_{\mathbb{R}^3} \hat{V}(\mathbf{l} -\mathbf{p})\langle \mathbf{p}|[E+i\epsilon-H_0]^{-1} T|\mathbf{k}\rangle d^3p,$$ where $\hat{V}$ is the Fourier transformation of the potential. After a partial-wave decomposition one obtains: $$\hat{V}(\mathbf{k}'-\mathbf{k}) = 4\pi\sum_{l=0}^{\infty} (2l+1)\hat{V}_l(k',k) P_l(\cos\theta).$$ After further algebra we end up with the following expression for $T$: $$T(k',k) = \hat{V}(k',k)+\frac{2}{\pi} \int_0^{\infty} \frac{2m/\hbar^2}{k^2-q^2}\left[ q^2\hat{V}(k',q)T(q,k) - k^2 \hat{V}(k',k)T(k,k)\right]dq - 2i\frac{mk}{\hbar^2} \hat{V}(k',k)T(k,k).$$ Now a real kernel is defined by: $$K(k',k)=\hat{V}(k',k)+\frac{2}{\pi} \int_0^{\infty} \frac{2m/\hbar^2}{k^2-q^2}\left[ q^2\hat{V}(k',q)T(q,k) - k^2 \hat{V}(k',k)T(k,k)\right]dq.$$ My problem lies with the following: Given a potential $$V(r) = V_0 \frac{1}{\mu r} \cdot (e^{-2\mu r} - e^{-\mu r})$$ (where actually $r=||\mathbf{r}||_2$) and using the following expression: $$\hat{V}(k',k) = \sum_i \frac{V_i}{\mu_i}\frac{1}{k'k}Q_l\left( \frac{k^2+k'^2+\mu_i^2}{2kk'}\right),$$ where the sum is in this case over just two components. I have to solve the equation for $K$ for $l=0,1$ in Born approximation. Unfortunately the only expressions for a Born approximation I have are for a state $\Psi_\mathbf{k}^\pm(r)$(Wikipedia says essentially the same) and not for a matrix equation. Now it took me some time and the equation 3.5.3 from Weinberg $$T_{\beta\alpha}^+ = V_{\beta\alpha}+\int\frac{V_{\beta\gamma}V_{\gamma\alpha} }{E_\alpha - E_\gamma +i\epsilon}+\dots$$ to get to the following conclusion: _The Born approximation in physics is the same as one step of aPicard iteration in mathematics._ Where the latter one is an iterative solution of a Cauchy problem $y(0)=y_0, \dot{y}= f(t,y)$ in the form: $$y^{(0)}(t) = y_0, y^{(n+1)}(t) = y_0 + \int_0^t f(t', y^{(n)}(t'))dt'.$$ So in this case I would obtain for $K$: $$K(k',k)=\hat{V}(k',k)+\frac{2}{\pi} \int_0^{\infty} \frac{2m/\hbar^2}{k^2-q^2}\left[ q^2\hat{V}(k',q)\hat{V}(q,k) - k^2 \hat{V}(k',k)\hat{V}(k,k)\right]dq.$$ **Here my question then:** Is this reasoning right or am I missing something fundamental?


title: What prevents this magnetic perpetuum mobile from working?

text: As a child, I imagined this device, which may seem to rotate indefinitely. I have two questions.   1. **Is this perpetual motion machine already known?** If it is, could you please give some references?   2. **What is the exact mechanism that makes it stop?** By this, I mean an explanation, not simply "because it would break energy conservation". Of course energy conservation is true, and of course cannot rotate indefinitely. But for any known (presumed) perpetuum mobile, there was an explanation, usually based on showing that the force generating the motion is balanced by another force. For example, Stevin obtained the laws of the inclined plane from the perpetual motion device ![Magnetic perpetuum mobile](http://i.stack.imgur.com/rhhTv.gif) I would like to make some observations.   * I don't try to convince anyone that it will move forever, because **I don't believe in breaking the energy conservation**. It is true that great physicists like Bohr, Kramers and Slater admitted the possibility, and nowadays some who think that there may be energy exchanges between parallel worlds in MWI believe, but I don't.   * But I don't consider an enough explanation simply to refer to the energy conservation. I am interested in an explanation showing exactly how the magnetic forces making it rotate, are balanced.    * If the forces are balanced, **only then** friction will make it slow down and stop. I don't think that we can explain only by referring to friction, which in principle can be made as small as needed. There has to be a balance of forces.   * Why spending time trying to understand or explain something that admittedly can't work? Well, even though perpetual motion machines cannot actually work, I think they may be interesting as puzzles.


title: What is the third law action pair of buoyancy?

text: Consider a block floating in a fluid. The force due to gravity balances the buoyant force exerted on the block. But there must be a third-law action pair to the buoyant force. If the block exerts a force on the fluid, that would mean that the pressure in fluid just below the block is more than that in other places that are on the same horizontal level. ![enter image description here](http://i.stack.imgur.com/TrHKq.png) I do not believe that to be the case, since it doesn't conform to real world observations. Also, if the block has a $m$ and density $\rho$, and the fluid + the container has a mass $M$. What is the mass of the block + container + fluid (as in the picture), as measured by a weighing machine underneath the container. According to my common sense, since the buoyant force is internal, the measured mass of the system should be $m + M$. Am I right?


title: Building a magnetron powered steamer

text: I am wanting to build a microwave powered steamer for a reverse engineering project in university. I've read up some on this and know that its not as efficient as a heating element. I'm curious about the wavelength changes of trying to heat a container of water directly with the magnetron. Unlike in a microwave which has a larger volume than I want, the water container would be much smaller and therefore microwaves concentrated. I know that there are limitations to the size of the container I use to heat the water that are based on the wavelength of the microwaves (roughly 12.5 cm). Also the water would have to be stirred to have even heating to allow boiling overall. I think my question is really over the possibility of something like this working and things you could see going wrong. Thanks for any and all help :)


title: role of entropy in gibbs free energy intuition

text: I have looked at various derivations of the Gibbs free energy equation and the underlying definition of the gibbs free energy. However I have been unable to attain direct insight/intuition over the significance of entropy in the equation. Specifically, lets take the case of 2 gases reacting to form a solid (the reaction, overall, releases energy as more stable bonds are formed than the ones present before the reaction). Clearly, the entropy change for the system will be negative as the solidification strips the atoms of much of their "freedom". However, what does it mean (in terms of this scenario) that if the temperature were high enough, the reaction would not occur. How is it that the entropy change will determine the spontaneity of the reaction, especially at high temperatures. I really am looking for a verbal answer please (rather than some mathematical derivations which often, at least to the novice (me), lack illuminative powers). Thank you!


title: changes in horizontal velocity and the effect on time airborne (thus overcoming gravity)

text: I would like to know if a object is launched horizontally in the air at a lower speed and a higher speed, why does the higher speed keep it airborne longer. what are the forces and why does gravity suddenly take "longer" And is this relevant to a pole dancer spinning around a pole? Why is it easier to stay high on the pole for longer if more momentum is used?


title: Space travel using pair production

text: If the method for creating electrons and positrons from photons, as described in this article, were proven correct, would it be possible to use it for accelerating space ships – using only an energy source such as a nuclear fission reactor (as opposed to having to use a propellant)?


title: What guarantees the existence of unitary operators implementing Lorentz Transformations?

text: This should be a very basic question. In introductory QFT books, often one of the first things we see is the following claim: for every Lorentz transformation $\Lambda$, we can associate an unitary operator $U(\Lambda)$ such that: $$ U(\Lambda)^{-1} \varphi(x) U(\Lambda)= \varphi(\Lambda^{-1}x)$$ And we also demand it to be a homomorphism, $U(A)U(B)=U(AB)$. Where of course, $\varphi$ is an operator-valued quantum field. I want to know what guarantees the existence of mappings $U$ which satisfy these conditions. It seems like it ought be possible to choose an operator field $\varphi$ such that no such set of operators $U(\Lambda)$ exist. Is the existence of $U$ given by some requirement on quantum fields, or am I missing something else?


title: Propagator in massless scalar field theory

text: Suppose we have the following Lagrangian: $\mathcal{L} = \frac{1}{2} \phi \Box \phi + V(\phi)$, where $\Box = \partial _ {\mu} \partial ^ {\mu}$ and $V$ is the interaction term. We use the $(-+++)$ signature. Is it true that the propagator is $- \frac{i}{k ^ 2 - i \epsilon}$? If we used the $(+---)$ signature, would the propagator be $- \frac{i}{k ^ 2 + i \epsilon}$ or $\frac{i}{k ^ 2 + i \epsilon}$? I am not particularly sure about the sign in each case. Could someone please explain?


title: Neutrino Speed in Supernova

text: I've read that neutrinos in supernova can be affected by "neutrino refraction." Is this analagous to the refraction of light, and if so, is the speed of these neutrinos similarly reduced from their near c speeds via this index of refraction?


title: Why doesn't gravity bend everything equally?

text: If gravity is the curvature of spacetime it should bend everything equally. To clarify my point I would like you to imagine two scenarios. Think of a bird flying in the storm while the wind is blowing sideways. Because the wind is an active force and the bird is fighting against it, the amount by which it's trajectory is bent should depend on it's momentum for a given strength of wind. Qualitatively speaking, more the momentum the less it bends. Now think of a train moving on a curved track. It should always bend by a fixed degree regardless of it's velocity or mass because the path itself is bent by that precise degree. I am aware that in my examples I have only considered curvature in SPACE but the theory of relativity talks about curvature in SPACETIME and I am guessing that every object's trajectory through spacetime is indeed equally bent due to gravity. Can someone explain? **EDIT** : After reading some comments, I have realized that perhaps my choice of words wasn't that great in the original question. I know that different object will follow different geodesic trajectories based on their initial conditions. What I was interested in is the CHANGE in trajectories due to spacetime curvature. The term I should have used in the question is acceleration. For example, the gravitational acceleration an object will experience near the surface of the earth is g regardless of it's mass or initial velocity. So it's natural to expect that every object will experience the same acceleration when they are moving sideways with respect to a gravitational body. In other words the "bending" should be equal. Hope it's a little clearer now.


title: What Do We Get From Having Higher Generations of Particles?

text: Background: I have written a pop-science book explaining quantum mechanics through imaginary conversations with my dog\-- the dog serves as a sort of reader surrogate, popping in occasionally to ask questions that a non- scientist might ask-- and I am now working on a sequel. In the sequel, I find myself having to talk about particle physics a bit, which is not my field, and I've hit a dog-as-reader question that I don't have a good answer to, which is, basically, "What purpose, if any, do higher-generation particles serve?" To put it in slightly more physics-y terms: The Standard Model contains twelve material particles: six leptons (the electron, muon, and tau, plus associated neutrinos) and six quarks (up-down, strange-charm, top-bottom). The observable universe only uses four, though: every material object we see is made up of electrons and up and down quarks, and electron neutrinos are generated in nuclear reactions that move between different arrangements of electrons and up and down quarks. The other eight turn up only in high-energy physics situation (whether in man-made accelerators, or natural occurances like cosmic ray collisions), and don't stick around for very long before they decay into the four common types. So, to the casual observer, there doesn't seem to be an obvious purpose to the more exotic particles. So why are they there? I'm wondering if there is some good reason why the universe as we know it has to have twelve particles rather than just four. Something like "Without the second and third generations of quarks and leptons, it's impossible to generate enough CP violation to explain the matter-antimatter asymmetry we observe." Only probably not that exact thing, because as far as I know, there isn't any way to explain the matter-antimatter asymmetry we observe within the Standard Model. But something along those lines-- some fundamental feature of our universe that requires the existence of muons and strange quarks and all the rest, and would prevent a universe with only electrons and up and down quarks. The question is not "why do we think there are there three generations rather than two or four?" I've seen the answers to that here and elsewhere. Rather, I'm asking "Why are there three generations rather than only one?" Is there some important process in the universe that requires there to be muons, strange quarks, etc. for things to end up like they are? Is there some reason beyond "we know they exist because they're there," something that would prevent us from making a universe like the one we observe at low energy using only electrons, up and down quarks, and electron neutrinos? Any pointers you can give to an example of some effect that depends on the presence of the higher Standard Model generations would be much appreciated. Having it already in terms that would be comprehensible to a non-scientist would be a bonus.


title: How Does Mass Leave the Body When you Lose Weight?

text: When your body burns calories and you lose weight, obviously mass is leaving your body. In what form does it leave? In other words, what is the physical process by which the body loses weight when it burns its fuel? Someone asked this question on the whiteboard at work. Someone said it leaves the body in the form of heat but I knew this is wrong since heat is simply the internal kinetic energy of a lump of matter and doesn't have anything do with mass. Obviously the chemical reactions going on in the body cause it to produce heat but this alone won't reduce its mass.


title: Do XAFS excitations and subsequent relaxations lead to vibrationally hot molecules?

text: I'm in the process of trying to understand X-Ray Absorption Fine Structure (XAFS) spectroscopy, but I think this question is more fundamental than merely as it pertains to XAFS. I think I don't entirely understand the interplay between electronic, vibrational, and rotational excitations in molecules. I understand that vibrational fine structure can be seen in electronic spectra. But how do we know which vibrational state a transition will end up in? Is it a Boltzmann distribution? So to my main question. I realize that on the energy scale of XAFS, vibrational energy levels are not resolved; I just want to know if these transitions lead to the same type of vibrational state distribution that valence electronic transitions yield. Is that the case? Or if not, why not?


title: Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?

text: Sorry if this is a silly question but I cant get my head around it.


title: Experiments that measure the time a gas takes to reach equilibrium

text: If you take two ideal gases at different temperatures, and allow them to share energy through heat, they'll eventually reach a thermodynamic equilibrium state that has higher entropy than the original. The time evolution of that entropy increase is easy to predict, because we know the time evolution of $T$ and $U$ for each of the gases (assuming we have the necessary constants). This is OK. Now take a system beyond the scope of thermodynamics. A single box containing a gas that is not in thermodynamic equilibrium (doesn't follow the Boltzman distribution). One would expect that gas to quickly thermalise, and reach said equilibrium. Entropy can still be defined using statistical mechanics, and the final state will have higher entropy than the initial state. I'm looking for quantitative experimental evidence of this effect.   **Ideally, I'd like references measuring the time a gas takes to thermalise.** Obviously this time depends on many factors and is not always doable. I wasn't more specific because I don't want to be picky, I'm looking for any experiments that verify it.


title: How do I find average temperature given a temperature distribution?

text: I was told to find the temperature distribution of a wire with a current going through it. So I found $$T(x)=T_{\infty}-\frac{\dot{q}}{km^{2}}[\frac{cosh(mx)}{cosh(mL)}-1]$$ I need to find the average temperature in the wire using this formula. I know I have to use some integral but I can't remember the formula. If someone could just give me the formula then I could probably integrate it myself.


title: What are Low-lying energy levels?

text: I am reading about some canonical transformations of the Hamiltonian (of a system consisting of an electron interacting with an ionic lattice) due to Tomanaga and Lee, Low and Pines. One of the important considerations should be translation invariance (momentum conservation). In a paper by Lee, Low and Pines ("Motion of Slow Electrons in a Polar Crystal", 1953), there is a frequent mention of low-lying energy levels. Intuitively, I would like to think they are referring to energy states close to the ground-state, but I don't think that is quite correct. Do low-lying energy levels have to do with, perhaps, energy of the electron with small momentum? The title of the paper by Lee, Low, and Pines is after all about "Slow Electrons". I'm confused. Some clarification could be helpful.


title: A question about the implication of UV divergence in QFT

text: I have a basic question about the logic of renormalization in quantum field theory (QFT). We met the ultraviolet (UV) divergence in loop corrections. The standard argument is, our current field theory is incorrect at high-energy scale, we regularize and renomalize the theory to make it work. My question is, why do we expect a correct theory will be capable for loop corrections without regularization? Is that possible that nature or saying a correct theory at high energy scale is non-perturbative from free particles? P.S. non-perturbative from free particles mean (i) if one uses perturbative expansion from free particle, the "correct" theory still gives infinity; and (ii) if one uses perturbative expansion from a special interacting solvable system or by certain non-perturbative technique, the results from that theory always agrees with experiments.


title: Why is there water coming out of a car’s tail pipe?

text: I notice yesterday that my neighbor’s car had water coming out of the exhaust pipe in the morning. My first response was since the hot exhaust is hotter than the cold tail pipe, heat is transferred from the hot exhaust through the pipe, and with enough moisture in the exhaust, enough heat leaves such that the humid air is condensed forming water drops.   1. Is my thinking correct here? If not, please correct me.   2. Does the water come from (i) a chemical byproduct of gasoline combustion or (ii) the humidity of the air (i.e. I've only noticed this in the winter, not the summer so I assume that during the summer this would not happen since the tail pipe is “already warm enough.”)


title: Non-trivial components of the stress-energy tensor of the bosonic string ghost action

text: The stress-energy tensor derived from the ghost action of a bosonic string is: $$ T_{\alpha \beta} = \frac{i}{4 \pi} \left ( b_{\alpha \gamma} \nabla_{\beta} c^{\gamma} + b_{\beta \gamma} \nabla_{\alpha} c^{\gamma} - c^\gamma \nabla_\gamma b_{\alpha \beta} - g_{\alpha \beta} b_{\gamma \delta} \nabla_\gamma c^{\delta} \right) $$ My book says, but doesn't give the proof, that in light-cone coordinates, the only two non-trivial components are the diagonal ones, given by: $$ T_{++} = \frac{i}{4 \pi} \left ( 2b_{++} \partial_+ c^+ - c^+ \partial b_{++} \right) $$ and the same thing for $T_{--}$ but replacing all plus signs in the indices with minus signs. But I can't derive this. I know that in light-cone coordinates, $g_{++} = g_{--} = 0$ and $g_{+-} = g_{-+} = -\tfrac{1}{2}$ and the covariant derivative is just the partial derivative. Now for example, consider $T_{+-}$ and focus on the $c^\gamma \nabla_\gamma b_{\alpha \beta}$ term. In light-cone coordinates, it's: $$ c^+ \partial_+ b_{+-} + c^- \partial_- b_{+-} $$ Since these are the only terms involving partial derivatives of $b$, they must kill each other for $T_{+-}$ to be zero. I know that $b$ is symmetric and traceless, and that $b$ and $c$ anti-commute, but that's not enough to produce $$ c^+ \partial_+ b_{+-} + c^- \partial_- b_{+-} = 0 $$ I don't see how it's possible.


title: How does a gauge theory probe a spacetime singularity?

text: Within the framework of string theory, I have read in numerous articles such as the introduction of this this in which it is stated that the gauge theories living on a stack of D-branes can be used to _probe a spacetime singularity_ , such as for example a Calabi-Yau or toric one. What is exactly meant by that?


title: What happens to gravity after matter-antimatter annihilation?

text: Both matter and antimatter have mass and thus gravity, but since energy from annihilation has no mass what happens to the force of gravity that was previously present? What about conservation of energy? Does it include gravity too?


title: Is there a sound theoretical argument against inner-shell induced nuclear chain reactions?

text: There is a claim often made about cold fusion, that it is excluded theoretically. The main theoretical argument is that electronic energies are too low to overcome the Coulomb barrier, since d-d fusion only takes place at KeV energies, while chemistry is at eV energies. This is belied by inner shells, which in Palladium store 3 or 20 KeV of energy per ejected electron, depending on whether the first or second shell is excited. These inner shell vacancies can decay either by x-rays, or by absorbing an electron into the vacancy and simultaneously ejecting a different electron (this second process is electrostatic). The cross section for ejecting a deuteron with tens of KeV's instead of an electron should be larger, since a deuterium is heavier. So I believe deuterated metal with excited inner shells has KeV deuterons running around. If two KeV deuterons do a fusion to an alpha in a dense environment, close to a nucleus or to an electron, I don't know why the process cannot end without a proton or neutron ejected. There are electrostatic matrix elements that allow an unstable alpha-resonance to decay by giving its energy to a charged particle nearby, instead of ejecting a constituent. After a fusion, the resulting alpha leaves an energetic track behind, and charged particles leave behind trails of atoms with ejected inner-shell electrons. So the K-shell holes produce fast deuterons, and fusion in deuterons produces K-shell holes. I don't see why this can't make a chain reaction. I have explained this idea before. I would like to know whether somebody knows a sound theoretical argument which rules it out. Can such a chain reaction in a Pd be excluded theoretically? I am not asking if it is likely, I am asking whether it can be firmly theoretically excluded. Anna v. asks how this process gets started--- it requires a random charged particle to pass through the deuterated material, from spontaneous environmental radioactive decay, or a cosmic ray muon. Charged particles produce K-shell holes. To make my biases clear: I can't exclude it. Regardless of the quality of the experiments, I don't see an argument against cold fusion.


title: Angular momentums addition in QM

text: We know that the spatial inversion parity for eigenfunctions of $\hat {L}_{z}$ operator (spherical functions) is $(-1)^{l}$, where $l$ refers to angular momentum. So for product of two eigenfunctions with definite summary angular momentum $L = l_{1} + l_{2}$ corresponding wavefunction has parity $(-1)^{L}$. Also full wavefunction (in $\hat {L}_{z}$ representation) of two particles has parity $(-1)^{l_{1} + l_{2}}$ under their interchanging. For example, $l_{1} = l_{2} = l, L = 2l, m_{1} + m_{2} = 2l - 1$ in $\hat {L}_{z}$ it $\hat {L}_{z}$ representation is given as $$ \langle l_{1}m_{1}, l_{2}m_{2}|L = 2l, M = 2l - 1\rangle = \frac{1}{\sqrt{2}}\left(\delta_{m_{1}l}\delta_{m_{2}(l - 1)} + \delta_{m_{1}(l - 1)}\delta_{m_{2}l} \right). $$ The question: is it possible to establish a one-to-one correspondence between spatial inverse parity and interchange parity formally? One can imagine a bit classical mental experiment: let's have two particles with definite angular momentums. Let's use coordinate system with center in a middle of imaginary line which connects the particles. So in this case spatial inversion is equivalent to particles interchanging. I want to formalize it in a quantum case if this correspondence is true.


title: The first $10^{-35}$ seconds

text: I am a rank amateur, so please forgive me if the answer to this is well-known. The following quote can in a weekly update for an EdX course I am following in astrophysics: > "And what a week it's been with the recent discovery of primordial > gravitational waves from the epoch of inflation! This one observation has > pushed back the earliest direct measurements of the Universe from about 1 > second (the epoch of nucleosynthesis) to $10^{-35}$ seconds (the epoch of > inflation)." I would appreciate any comments as to what it will take to close the gap. **EDIT** : Well, initially I asked this question with a few specific alternatives. Perhaps - respectfully and with moderators' approval -I should revert to that (as best as I can remember), in that any one of them could be answered with a "yes" or "no" and a few qualifying, suggestive sentences. \-- Would a viable GUT do the job. \-- Are there currently specifically directed experiments whose intended results will answer the question. \-- Can the gap be closed by correct extrapolation of know experimental and theoretical results. If this is yet deemed unacceptable, I understand, and please feel free to restrict my question as deemed appropriate.


title: Backspin in outer space i.e. in a liquid with no viscosity: not changing direction?

text: No viscosity means no friction between neighbouring particles. I think outer space is such a place. So a golfball or a tennis ball traveling in space with a large spinning such as backspin would not experience the Magnus effect where the ball changes its direction due to spinning. > **Is this correct? So a ball in outer space would not experience Magnus > effect due to a large spinning? So is playing tennis in outer space a very > directional game without spins and curvatures?**


title: Problem understanding sign of volume integral in Minkowski space

text: My professor told me that a 4-dimensional Minkowski - Space Integral I was working on can be written as the product of a metric tensor and a scalar: $\int d^4 k \frac{k^\mu k^\nu}{(k^2-m_1^2)(k^2-m_2^2)} = g^{\mu \nu} B_2$ where $B_2$ is a scalar. I am surprised that apparently for $\mu = \nu = 0$ , the sign of the integral is opposite to the sign for, say, $\mu = \nu = 1$. This seems wrong to me, and in an Eucledian Space it certainly would be. If I had been asked to guess, I would have written the above equation with $ g^\mu_\nu B_2$ on the right hand side instead.. Can someone bring clarity?


title: What's up in this SPAWAR video?

text: Here is a video presentation of infrared recordings of anomalous heating in a deuterium palladium cell: ( youtube video) (see also this presentation if you want more detail, and have time). There are two papers referenced in the link, which give more detail. You can see with your own eyes sporadic localized bursts of energy, which show up as white flashes in the camera (I would like to encourage people to read the linked papers, which include thermal photos with a different color map, and also include piezoelectric detection of bursts) Excluding cold fusion, what could possibly be causing this? Please try to account for the nature of the bursts--- the small radius, the qualitative amount of energy released, etc. Just to be clear, I think the answer is _nothing_ , that it is cold fusion, but I would like to see what explanations people come up with. I will also point out that this research group has been politically shut down recently.


title: Non-Locality and Entanglement

text: Let’s consider a pair of particles [with their signals] comprising an _isolated system_. Any change in some property of either particle is due to the signal/s received from the other. Each particle has the knowledge of the signals emitted by it and the consequences of such signals on the other. This is a “continuous process” which may go simultaneously with increasing separation between the particles. The entanglement of some property ,between the particles, even in the highest level of non-locality seems to be intuitively “natural” _so long the system remains an isolated one_ , comprising the two particles and the signals exchanged between them. Would it be correct to say that the entanglement of properties in the non- local context should not be considered as an injury to commonsense perceptions?


title: How is electric flux related to permittivity?

text: How is Gauss' law related to permittivity? I know that it equals $1/\epsilon_0$ times the magnitude of the charge enclosed. But, I'm unable to understand what it actually means. Can someone intuitively explain it to me?


title: How do we demonstrate that all spectral colors exist in nature?

text: Spectral emission by elements occurs in discrete wavelengths. There are only about 118 elements. My question is whether there are pure colors either disfavored in nature or even disallowed? The sun (via prism) appears to generate a continuum of visible frequencies but of course that would probably appear to be the case, as the eye would mix adjacent spectral colors and the resultant (near-spectral) color would seem to fill in the gap (note: that is a question, not a statement). It is often recited that "black bodies" (like the sun) emit a spectrum of radiation, but how would we notice small discontinuties, if they existed? Red-shift might create visible frequencies that otherwise do not occur. But do we have some basis other than (prism + visual intuition) that all pure visible frequencies do in fact correspond to some possible natural or artificial source? Apologies for the naivete of the question, if that is the case, and the likelihood that the answer may not have much practical significance. There were related question on the site and on Google but nothing directly on point. Thanks.


title: Current scope of Chaos theory and non-linear dynamics?

text: I am a physics undergrad interested in stuff like dynamical systems, chaos theory etc. Is there ongoing research in these fields? I am talking about pure research and not applications to things like weather etc? I hope this question is appropriate for Physics SE. I asked this question, because I browsed through the websites of the physics departments of a few renowned universities (MIT, Princeton, Caltech) etc, but nowhere it mentioned research in these areas.


title: Design of a common table fan

text: A few questions about the design of a common table fan, such as below.   1. Does the round disc (silver in the image) in front of the blades serve any functional purpose?   2. Does the metal casing around the blades serve any functional purpose except protecting the blades?   3. What exactly are the sources of noise in a well-designed fan? Does the metal casing in a well-designed fan have an impact on the noise level?  ![](http://img193.imageshack.us/img193/8081/60933751.gif)


title: Lattice Yang-Mills

text: I have often heard statements like, Continuum limit of Lattice Yang-Mills doesnot exist in dimensions higher then four. Is there a neat way to see this? or could some one point me to some relevant literature where this has been shown. thanks


title: Dirac equation in curved spacetime - found second derivatives of the metric, violation of the principle of equivalence?

text: I am working on the Dirac equation on curved spacetime. A Foldy-Wouthuysen transformation was applied to obtain the semiclassical limit of the equation to study the dynamics of the spin of the electron in curved spacetime. In the semiclassical limit up to order $\hbar$ the Hamiltonian then takes a familiar form: $H =\; ... \,+\, \hbar \, \Sigma \cdot \, \Omega^{(1)}$, where $\Sigma$ is the spin vector and $\Omega$ is the precession frequency due to gravitational effects. If I look closer to $\Omega$, I find that it contains the metric $g_{\mu \nu}(x)$ and its first derivatives $\partial_\lambda g_{\mu\nu}$. As far as I understand, one can always find an appropriate coordinate system in which the first derivatives of the metric can be made vanish at a certain point $x$ (so called normal coordinates). To my knowledge, this is actually good because the principle of equivalence states that gravitational effects (in this case the spin precession of the electron) must be able to be set zero at _one point_. **Q1** : The Dirac equation is certainly a local theory. Therefore, should gravitational effects only appear as fictitious forces? (Should they only contain first derivatives of the metric)? Now I calculated the semiclassical limit to an higher order $\hbar^2$ (I did this by evaluating the commutators in $\Omega^{(1)}$ to a higher precision.). The Hamiltonain looks like this: $H =\; ... \,+\, \hbar \, \Sigma \cdot \, \Omega^{(1)} \; + \hbar^2 \, \Sigma \cdot \, \Omega^{(2)}$ Looking at $\Omega^{(2)}$, I find **_second derivatives of the metric_.** This is a bit more confusing to me, since the Riemann curvature tensor is made of second derivatives and since curvature is the "objective ingredient" in general relativity, there exists no coordinate system in which curvature effects can be "gauged away". **Q2** : Is the appearance of these second derivative terms problematic in any way? Do they mean that the principle of equivalence could be violated here (even if only at the order $\hbar^2$)? This is my ongoing research so any input would be very welcome. _This is my first post. Please excuse me, if my question is not asked optimally._


title: Prove one of the following trajectories is circular

text: In his classical mechanics lecture, Prof Susskind gives a short exercise, which I "feel" is very simple, but don't know where to start with. The question is: > "There is a coordinate system $(x,y)$, which is stationary. Another system, > $(X,Y)$ with origin common to the previous one is rotating with a uniform > velocity, through an angle $\omega{t}$. The trajectories are(ignoring masses > and $t$): > > $$\ddot{X} = {\omega^2}x - \omega\dot{y}$$ > > $$\ddot{Y} = \omega^2{y} + \omega \dot{x}$$ > > Prove that one of the trajectories is circular." However, both the terms will equal to $0$ because the velocity is constant, and hence the acceleration is zero. Now, what should I do with these? Should I prove $\ddot{x}$ = $\ddot{y}$= $0$, or something else? Please explain.


title: Polarizing beam splitter

text: I'm searching for basic literature or papers introducing the action of polarizing beam splitters with single photons. Unfortunately the most books and papers only focus on usual beam splitters. The only paper I found is this one (Zeilinger - Photon statistics at beam splitters) but even there is just a little section about polarizing beam splitters. Above all it is full of typos.


title: Spherical harmonics for two particle?

text: For one particle angular momentum eigenket, we can represent it in position space, such as $\langle\theta,\phi|l,m\rangle=Y_{lm}$ where $Y_{lm}$ means spherical harmonics. But if we have we have two particle with angular momentum $\frac{1}{2}$(of course no angular momentum $\frac{1}{2}$ particle, but for simplicity) , the total angular momentum state $|10\rangle$ can be represented by sum of each angular momentum, $|10\rangle$=$\frac{1}{\sqrt{2}}(|\frac{1}{2},\frac{1}{2}\rangle)+|\frac{1}{2},\frac{1}{2}\rangle$ At this stage, what is $\langle\theta,\phi|1,1\rangle$, when the added angular momentum represened in position space?? First I thought that it was $\langle\theta,\phi|10\rangle$ $= \frac{1}{\sqrt{2}}(\langle\theta,\phi|\frac{1}{2},\frac{1}{2}\rangle)+\langle\theta,\phi|\frac{1}{2},\frac{1}{2}\rangle)\\\=\frac{1}{\sqrt{2}}(Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)+Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi))$ $\langle\theta,\phi|10\rangle$ $= Y_{1,0}(\theta, \phi)$ thus $ Y_{1,0}(\theta, \phi) = \langle\theta,\phi|10\rangle = \frac{1}{\sqrt{2}}(Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)+Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi)Y_{\frac{1}{2},\frac{1}{2}}(\theta,\phi))$ but if we think the bra operator, $\langle\theta,\phi|$ acts on total angular momentum state, it cannot acts on each particle's hilbert space, I think the right side of last equation cannot be hold. so I thought that I have to find the corresponding operator of $\langle\theta,\phi|$, which acts on total angular momentum hilbert space, to each particle's space, but I'm not sure about this idea. Is this right? if then, what is that operator?


title: Is a closed system receiving heat guaranteed to have entropy below maximum?

text: Isolated system will preserve its entropy unchanged (for fully reversible system) or it will increase its entropy (for system with irreversible processes). For a closed system, it can receive energy in form of heat. Does this always imply, that its entropy will be below maximum? Is it possible to add heat in such a way, that entropy will remain unchanged and maximum?


title: Reference paper to support information -- energy relation ($kT \ln2 \rm\frac{J}{bit}$)

text: In answer to Maxwell's Demon Constant (Information-Energy equivalence) there is stated that one bit of information allows to perform $kT \cdot \ln2$ Joules of work. Which paper supports the thesis? (there are many publications on Maxwell daemon, Szilard engine, Landauer's principle).


title: 3D Delta Potential Well

text: The 1D delta potential well $V(x) = -A\delta(x - a)$ always has exactly one bound state. The same is true for the 3D delta potential well $V(\vec{r}) = -A\delta(\vec{r}-\vec{a})$. I can show this for $l = 0$, I don't know how to do the calculations otherwise. So two questions,   1. Can I conclude that there is only one bound state for the 3D potential well for $l \not = 0$? I've seen that the energies of the eigenstates for the hydrogen atom depend only on $n$, but I am wondering whether this is an instance of a more general result?   2. When $\vec{a} = 0$ and $l=0$, there are no normalizable eigenstates. For $l \not = 0$, the effective potential in the radial equation becomes large at the origin, can I use this to conclude that there are no bound states when $\vec{a}=0$?


title: static friction problem without mass

text: I am having trouble with a physics problem. The problem is as follows: A car is traveling at 54.0 mi/h on a horizontal highway. A) If the coefficient of static friction between road and tires on a rainy day is 0.103, what is the minimum distance in which the car will stop? B) What is the stopping distance when the surface is dry and µs = 0.595? This is what I have so far for A)               fs = µs * Fn          Fn = Mg     fs = Ma          Ma = µs * Mg     a = µs * g          V = 54 - a * t     0 = 54 - a * t     t = 53.497 seconds          dist = 54*t - (a/2)*t^2     = 1444.42243 miles     = 2324572.57 meters      This is not the correct answer. Any help would me much appreciated.


title: Where can I find the full derivation of Helfrich's shape equation for closed membranes?

text: I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of variational calculus" to derive the following: $$\Delta p - 2\lambda H + k(2H+c_0)(2H^2-2K-c_0H)+2k\nabla^2H=0$$ But I'm having a lot of trouble tracking down the original derivation. The guy who did it first was Helfrich, and here's his and Ou-yang's paper deriving it: http://prl.aps.org/abstract/PRL/v59/i21/p2486_1 . However, they don't show an actual derivation, instead saying "the derivation will appear in a full paper by the authors" or something like that. Yet everybody cites the paper I just linked for a derivation. Does anybody know a source that can derive this, or can give me some hints to figure it out myself? To be honest I can't even figure out how to find the first variation. Edit: So, after some careful thought and hours and hours of work and learning, I realized that the answer that got the bounty was wrong. The author stopped replying to my messages after I gave him bounty.... thanks guys. That said, I've almost got it all figured out (in intense detail) and will post a pdf of my own notes once I'm done!


title: Why the Principle of Least Action?

text: I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations. Great. But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be _equivalent_ to the Principle of Least Action. After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all: it's the "Principle of Stationary Action". Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional. Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique. But the problems don't stop there. Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$. As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want. From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way. Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory. Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean _period_ ), and in fact, that it is so natural that we would not even dare abandon these ideas?


title: Why does Principle for least action hold for classical fields

text: Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the principle of least action. In case of motion of particles, I know that principle of least action comes from Newton's second laws. But why does the principle of least action also hold for classical fields like EM field and gravitational field? Is there any deep reason why it holds for both EM and gravitational field?


title: How general is the Lagrangian quantization approach to field theory?

text: It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of physics to be preserved. But nevertheless the question about the generality of this approach keeps coming to my mind. My question is how one can be sure that this approach has to be always right and fruitful. Isn't it possible, at least from the mathematical point of view that a future theory of physics does not subscribe to this approach?


title: Calculating torque with gears and wheel-axles

text: For my homework in engineering, I was given a question like this: There is an axle attached to a gear, which is meshed with another gear, which has an axle attached to it. I am given the diameter of both gears, and the torque on the input axle. How would I go about finding the torque on the output axle? I'm ommitting the specific numbers, because I would like to know how to solve such a problem in general. What is the relation between thetorque of the wheel and the torque of the axle? Is this even solveable without axle diameters? We went over the problem in class, but I can't for the life of me remember what we did. Thanks


title: General definition of an event horizon?

text: Horizons are in general observer-dependent. For example, in Minkowski space, an observer who experiences constant proper acceleration has a horizon. Black hole horizons are usually defined as boundaries of regions from which no lightlike curve can reach null infinity $\mathscr{I}^+$. But how can this be interpreted in terms of an event horizon for an observer? Immortal material observers end up at timelike infinity $i^+$, not $\mathscr{I}^+$. Is there some nice way of unifying both cases? In other words, is there a general definition of an event horizon that has both these types of horizons as special cases? [Edited to clarify the question and remove a mistake about the dimensionality of $i^+$ versus $\mathscr{I}^+$.]


title: Degenerate perturbation theory applied to topological degeneracy?

text: Consider a quantum system described by a _gapped_ Hamiltonian $H_0$ with _degenerate_ ground states (GS), adding a perturbation term $V$ to $H_0$, then the low-energy physics can be described by an _effective Hamiltonian_ $H_{eff}$ acting within the GS subspace of $H_0$, where $H_{eff}$ can be calculated from **degenerate perturbation theory**. What if the GS degeneracy of $H_0$ is a **topological degeneracy** ? I learned that the topological degeneracy is robust against ANY local perturbations. Does this imply that: if the above $H_0$ describes a topologically ordered system defined on a torus with a topological GS degeneracy, and $V$ represents ANY local perturbations, then would the resulting _effective Hamiltonian_ $H_{eff}$ be always trivial (i.e., $H_{eff}=$constant number) ?? And in the opposite case, if we find $H_{eff}$ (corresponding to ANY local perturbations $V$) is a constant number _at any finite order_ from a degenerate perturbation theory calculation, then does this imply that $H_0$ describes a topologically ordered system?? Of course, for a finite system, $H_{eff}$ is usually NOT a constant number. All the above we talk about is in the thermodynamic limit. Thanks a lot.


title: Can we make usable energy from subnuclear particles?

text: I understand mass and energy are the same, but in this question I will be talking about mass being turned into usable energy (electricity/heat/etc). We can make our energy through chemical reactions like burning coal. We can also make energy from nuclear decay (and soon the fusion) of atoms. But is there any know way to take subnuclear particles and turn them into energy? How about quarks and other particles that make up protons/neutrons/electrons? More specifically, I have heard (and know little) about Quantum Field Theory. Do we know any way to take the energy form an excited field of, lets say a quark and transfer it into the electron field? I hope this is not a silly question and thanks.


title: Find pressure of liquids without making contact with it

text: Is there any way to find the pressure of a liquid inside a closed container without making contact with it? You can make any assumptions, like transparent container or anything, only rule, the sensing device shouldn't make contact with the liquid.


title: Non-uniform circular motion, computing the angle

text: There is an object moving in circle, with this law: $$ \alpha = -k^2 \theta $$ With $ \alpha = \frac{ d \omega }{dt} $, and $ \omega= \frac{d \theta}{dt} $, $\theta$ = angle, $k$ positive constant. I need to compute $ \theta $ in function of time. My attempt: $$ \frac{d^2 \theta}{dt^2} = -k \theta $$ $$ \frac{d^2 \theta}{\theta} = -k^2 dt^2 $$ $$ \int_{\theta_0}^{\theta} {\frac{d^2 \theta}{\theta}} = - \int{k^2 dt^2} $$ $$ \ln\Big(\frac{\theta}{\theta_0}\Big) d\theta = -k^2 t \cdot dt$$ So I try to get the solution computing: $$ \int_{\theta_0}^{\theta} \ln\Big(\frac{\theta}{\theta_0}\Big) d\theta = -\int k^2 t \cdot dt $$ But it's wrong, the solution is $\theta = \theta_0 \sin \big( kt + \phi \big) $.


In a turning plane, will the vector of combined centrifugal plus gravity force experienced by passengers be perpendicular to the floor? In other words, will the passengers experience the feeling of inclination without looking into windows? I think that the reaction of air is perpendicular to the wings and as such, there will be no feeling of inclination. Am I right?


title: Effective Resistance

text: For the combination of the resistors, shown in the figure. Calculate the equivalent resistance between A and B, please help me to find the answer of this complicated question this is a question from an easy part but, I don't know how to solve this. ![For the combination of the resistors, shown in the figure. Calculate the equivalent resistance between A and B, please help me to find the answer of this complicated question this is a question from an easy part but, I don't KNOW TO SOLVE THIS](http://I.stack.imgur.com/3wsWo.jpg)


title: Why are pear-shaped nuclei possible?

text: In a recent question, Ben Crowell raised an observation which really puzzled me. I obtained a partial answer by looking in the literature, but I would like to know if it's on the right track, and a fuller explanation for it. It is a well-known fact in atomic and molecular physics that electronic eigenstates of inversion-symmetric molecules never have electronic dipole moments. This is because the electromagnetic hamiltonian that governs molecular physics is parity invariant: under a reflection the eigenstates must map to themselves, but nonzero vector quantities - like dipole moments - must switch signs. However, it was a fairly big news item earlier this year (see e.g. the University of York press release or the piece in _Nature News_ , 8 May 2013) that atomic nuclei can be 'pear-shaped'. This was predicted in the fifties, such as e.g. > Stability of Pear-Shaped Nuclear Deformations. K. Lee and D. R. Inglis. > _Phys. Rev._ **108** no. 3, pp. 774-778 (1957) and was experimentally confirmed this year in > Studies of pear-shaped nuclei using accelerated radioactive beams. L. P. > Gaffney, P. A. Butler et al. _Nature_ **497** , 199–204 (09 May 2013). > E-print on L.P. Gaffney's L.U. page. A pear-shaped nucleus is one that has a nonzero electric octupolar moment. The pear shape arises from the added contributions of quadrupole and octupole perturbations on a spherical shape, winding up with something like this: ![enter image description here](http://i.stack.imgur.com/8Qp3z.gif) However, this poses an immense problem, because octupole moments have odd parity. If you reflect a pear-shaped nucleus (as opposed to a rugby-ball- shaped quadrupolar one), you get a pear pointing the other way. Having such a nucleus requires a mixing of parity-even and -odd contributions to an energy eigenstate, and this is not allowed for eigenstates of the parity-conserving electromagnetic and strong interactions that (presumably) shape atomic nuclei. To put this another way, having a pear-shaped nucleus requires a way to tell which way the pear will point. The nuclear angular momentum can break isotropy and provide a special axis, but the 'pear' is a vector (pointing from the base to the stem) and one needs parity-violating machinery to turn a pseudovector angular momentum into a vector quantity. Another way to phrase this is by saying that if such an eigenstate were possible for a parity-conserving hamiltonian, then the reflected version should also be a degenerate, inseparable eigenstate. Having a unique such ground state means having a way to lift that degeneracy. I can then pose my question: **why are pear-shaped nuclei possible?** Is my reasoning incorrect? That is, can parity-conserving hamiltonians lead to such parity-mixed eigenstates? Or are there in fact parity-violating interactions that decisively lift the degeneracies and shape these nuclei? If so, what are they?


title: Is it possible to generate electricity using a water tower?

text: Is it possible to build a water tower that will provide enough pressure to run an electricity generator? A water pump can be used to send water up to the tower. The water pump can be powered by solar panels. Alternatively the water pump could also be powered by the electricity produced from the generator. The water tower can hold 20,000 to 30,000 gallons of water. Could this have enough pressure to send the water down a pipe and pass it through an electric generator where it will cause a turbine to rotate and produce electricity? Once the water passes through the generator it can be redirected back to the water pump. The water will then be pumped back to the water tower making it a water circulation system. I would like to find out if it is physically possible to produce more power from the pressure stored in the water tower than it is required to pump it up. If solar panels are used to power the water pump, is it theoretically possible to produce more electric power using this method? I am not familiar with any similar projects but I am looking to see if this setup will have the capacity to produce enough electricity to make it a feasible project.


title: Surface energy as thermodynamic potential

text: Consider free energy of sharp interface $\Gamma$ $$ \int_\Gamma \sigma\;\mathrm{d}S $$ or also free energy of diffuse interface of characteristic width $\epsilon$ given by Cahn-Hilliard/Allen-Cahn functional of phase-field $c\in [0,1]$ $$ \int_\Omega \left[ 12\frac{\sigma}{\epsilon}c^2(1-c)^2 + \frac{3}{4}\sigma\epsilon|\nabla c|^2 \right]\;\mathrm{d}V .$$   1. Is there some reasoning to conclusively deduce if these are Gibbs energies or Helmholtz energies?   2. Or would such a conclusion be equivalent to definition of circumstances under which surface tension/energy $\sigma$ is measured?   3. If latter holds what are these conditions for either Gibbs or Helmholtz energy and which are of practical significance and being typically measured?


title: Does the positive mass conjecture indicate a necessity of interactions in our universe?

text: The positive mass conjecture was proved by Schoen and Yau and later reproved by Witten. Total mass in a gravitating system must be positive except in the case of flat Minkowski space, where energy is zero. Since QG is intended to be a theory of interaction with force particles called gravitons, one may begin to wonder if the interactions are in fact the important defining features of the space in question. So does a theory with interactions also require that space be curved?


title: Formalization (and meaning) of Heisenberg Cut

text: In discussions around a recent  Stack answer which covered axiomatic quantum mechanics, Bell's Theorem and Random Fields (related to axiomatic QFT) the phrase "Heisenberg Cut" was used. The basic question is what does this mean? Having reviewed this discussion I can provide some background to set a context for the question. In (Von Neumann) QM a distinction is made between : OBSERVER [Cut] QUANTUM OBJECT This simplistic distinction has been critized later in that article and elsewhere such that the term "Heisenberg Cut" has fallen into disuse. If one wishes to make distinctions in a QM experiment then here is a larger proposal: ENVIRONMENT [Cut1] QUANTUM OBJECT [Cut2] CLASSICAL INSTRUMENT [Cut3] OBSERVER [Cut4] O-O here O-O is any hypothetical observer-observer as discussed in the "Wigner's Friend" concept. So has the concept of "Heisenberg Cut" been re-introduced in modern axiomatic QM/QFT and if so what is its formalisation (ie how can one make experimental or Bell Theorem-like deductions from it?)


title: Creating a stream of single file atoms

text: How do you go about creating a stream of single file atoms? The aim is to fire a stream of single file atoms (hydrogen, helium) down a vacuum and detect there arrival on the opposite end. Any tips/papers would be a great help.


title: How do we measure mean part or time average of velocity field?

text: How do we measure mean part or time average of a (known) velocity field? In other words, if I know velocity field how can I measure its time average or mean part? * * * What is time average in general?


title: Longitudinal and transverse part of vector field components

text: I was reviewing a paper of coupling to vector field and tensor field. I have got stuck with the term $$A_k \varepsilon^{kmn}\partial_mV_n=V^{T}.(\nabla\times A^{T})-\nabla.(A^{T}\times V^{T})+V^{T}.(\nabla\times A^{L})-\nabla.(A^{L}\times V^{T})$$ Where $A^T$ and $V^T$ are the transverse part of vector $A$ and $V$ respectively. And $A^L$ is the longitudinal part of $A$ . According to the paper, after calculation the only term that survives is this one-$$A_k \varepsilon^{kmn}\partial_mV_n=V^{T}.(\nabla\times A^{T})$$ other vanishes. Well we can use $$\nabla .V^T=0=\nabla \times V^L$$. That vanishes the term $V^T.(\nabla\times A^L)$. But still there are 2 more term left. How can I solve this? Does integration by parts can do any help?


title: How could electromagnetic waves propagate through space although they have no electrons?

text: How could electric fields in these waves propagate through space although in space there's no electrons for the electric field to be formed? is there another type of charged particles that carry the electric field? How could the electric field exist without charges?


title: Spooky action appears to contradict Relativity of time order of multiple events

text: It is well known that in special relativity observers can disagree on the time ordering of two events. It is also well known that entangled particles exhibit so called spooky action at a distance. Today I read in the New Scientist and on the arXiv that although the order of two events can be arbitrary, this is not so for multiple events. As the number of events increase the number of combinations of possible orders increase but not every one of these combinations will actually be possible to always observe. According to the NS article it is possible to entangle three particles a,b and c and then have them in such a way that the collaspe of c cannot be observed to precede the collapse of both a and b by any observer even though c spookily causes the collapse of a and b. Is this an outright contradiction between relativity and quantum mechanics or just a paradox ? What is the explantion ?


title: Find the tangential acceleration

text: > _A racing car driver drives his car on a flat circular track of radius > $\frac{25}3 m$ and a coefficient of friction $0.5$. Find the magnitude of > its tangential acceleration (in $ms^{-2}$) at an instant when car starts > slipping at a speed of $5 ms^{-1}$._ I found the centripetal acceleration to be $3ms^{-2}$ and thought about setting the resultant acceleration (vector sum of centripetal acceleration + tangential acceleration) = $\mu$$g$ where $\mu$ = 0.5 but I don't think it is correct. If it is, then can somebody please explain why? By the way, I am getting the correct answer (=$5ms^{-2}$) by this method.


title: Even and Odd States of a 1D finite potential well

text: Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?


title: potential difference in parallel connecting resistors

text: How could the potential difference be constant across all the resistors of parallel connecting resistors although each resistor has a specific resistance?


title: How to get conserved currents of a theory which are not Noether currents?

text: In the first SuSy lecture last week following theory of two real scalar fields has been considered as a first example: $$\mathcal{L}=(\partial_\mu \phi_1)^2/2+(\partial_\mu \phi_2)^2/2-m^2(\phi_1^2+\phi_2^2)/2$$ which has the usual equation of motion $$(\Box+m^2)\phi_i=0;\quad i=1,2$$ Then the professor has written down following conserved currents for this theory: $$J^{(n)}_{\mu_1\dots\mu_n}=\partial_{\mu_1}\phi_2\partial_{\mu_2}\dots\partial_{\mu_n}\phi_1-\phi_2\partial_{\mu_1}\dots\partial_{\mu_n}\phi_1$$ Using the equation of motion, it is easy to show that these currents are indeed conserved for all $n$. As far as I can see, these are no Noether currents, at least I cannot see how to get something with arbitrary many indices using the standard formula for Noether currents; furthermore I don't see which symmetry of the Lagrangian they could reflect. My questions now are:   1. Are there really no symmetries corresponding to these currents or do I overlook something? If there are no such symmetries and the formula for Noether's currents doesn't apply, is there any way to get such currents or is it rather an educated guess?   2. Is there any way to intuitively visualize such currents with many indices? Can anything be said about their physical meaning (if there is some)?


title: A good example of a nonlinear symplectomorphism?

text: What is a good example of a simple, physically useful _nonlinear_ symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked have been purely mathematical and not connected to any physical problems. Obviously linear examples are well known and useful. Actually, it would suffice if I could see some nice example of a nonlinear diffeomorphism (i.e. change of coordinates) $\varphi: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by $x \mapsto y$, as one can then lift this diffeomorphism to a symplectomorphism by transforming the momenta via $p_y := (d\varphi^{-1})^{*}p_x$. Thanks in advance.


title: Where can I find a good classification for phase transitions?

text: I'm having a hard time to find a good (and modern) classification scheme for phase transitions and related universality classes. Can someone recommend a paper/book/site? Detailed mathematical aspects are very much welcome.


title: When one thinks of a field of operators in QFT, is it reasonable to think of a matrix being associated with each point in space time?

text: Is it correct to visualize operators existing as matrices parameterized by spacetime coordinates in the context of QFT?


title: Is it a goal of modern physical theory to avoid big bang singularities and how do they approach the problem?

text: During discussion on another question, the question of singularties in modern physical theories arose. The big-bang is an obvious singularity in modern conceptions of the cosmos. Is it a goal of modern theories to find a way to remove the big bang singularity from theory, and how is that accomplished in theories being proposed today?