How does an atomizer work?

Will someone please explain how a basic pump atomizer works? I have a feeling it has something to do with pressure and the Venturi effect, but I'd really like to know precisely how they work as completely as possible. If you know, please feel free to use this image as a reference in explaining. EDIT: Please note that I'd specifically like to know how atomizers of the type pictured in the link I provided, the kind with a spring-based finger pump, work. I'm not talking about e-cigarette atomizers or atomizers with the bulb design.


In various formulae for black-body radiation where $c$ appears, there is an implicit index of refraction dependence, since $c=c_0/n$, where $c$ is the speed of light, $c_0$ is the speed of light in vacuum, and $n$ is the index of refraction into which the blac kbody is radiating. This comes up, for example, in Planck's law, the Stefan-Boltzmann law, and Wien's displacement law.$^1$ I find this observation surprising, because it implies that black bodies radiating into $n\neq 1$ materials emit a factor of $n^2$ more power than otherwise identical bodies emitting into vacuum. To me, it seems that this would violate the laws of thermodynamics. Consider the situation where two black bodies of equal area face one another. Suppose that the gap between the two black bodies is partly filled an index-$n$ material, with one black body in contact with the index-$n$ material, and the other surrounded by vacuum. Then, according to the argument of the preceding paragraph, one of the black bodies will radiate $n^2$ more power than the other. If the two black bodies are thermally isolated from the rest of the universe, then they will equilibrate to different temperatures. As I understand it, this violates the zeroth law of thermodynamics. There must be something wrong with the above argument. None of my books mention an $n$-dependence, so at first I thought that perhaps the $c$ in Planck's law actually indicated what I call $c_0$, the speed on light in vacuum. However, this does not seem to be the case; I've found at least one reference that explicitly mentions this $n$-dependence.: "For radiation into a _medium_ within which the speed of light is not close to $c_0$, [Planck's law] must be modified by including an index-of-refraction multiplying factor." How is the index of refraction dependence in Planck's law compatible with thermodynamics? * * * $^1$ The $n$ dependencies in the Stefan-Boltzmann law and Wien's displacement law follow from the one in Planck's law, whereas the two former can be derived from the latter.


Calculating vertical velocity component of a particle with mass, given the hit point of parabolic motion

Consider the following situation: ![enter image description here](http://i.stack.imgur.com/ZhUhk.png) I have a particle with a given mass that at a given instant of time (let's say $t_{0}$) is placed at the system origin. The particle has a constant velocity component $V_{0x}$. I have to calculate the value of the vertical component ($V_{0y}$) to apply to that particle at time $t_{0}$, in order to provoke a parabolic motion that makes the particle hit the point $P$. The data I know are the following:   * $P$ coordinates   * $dx$   * $V_{0x}$   * $t$ (that can be computed from $dx$ and $V_{0x}$)   * $m$ (the mass of the particle) From the data above, is possible to calculate the value of $V_{0y}$? If yes then give some hints to solve it?


What is displacement? Position relative to a reference point or change of position

What is the "official" or most useful definition of displacement in the context of kinematics? There are two common ones:   1. Displacement is the length and direction of a line from a fixed reference point. (Basically position).   2. Displacement is the change in position. Textbooks using the first definition frequently define velocity as $v=\Delta s/\Delta t$ (where $s$ is displacement) but then for acceleration give equations such as $s=ut+1/2at^2$ (if they were consistent, they would have to use $\Delta s$). What throws me off is that are the better textbooks. A similar confusion comes through in Wikipedia: A displacement vector is the straight path between the initial and the final position. But velocity is defined as $v=\Delta d/\Delta t$. Background: I am writing on a learning tool for students but different textbooks require them to learn contradicting definitions and I would appreciate your help on which is the right definition to learn.


Dropping condition

Imagine opening a water tap in order to have a smooth and cylindrical outflow and then slowly decrease the flow by adjusting the knob. At a certain moment, the side profile of the flow will become uneven, and even if you decrease the scale of the faucet, the water shall start to drip. It is possible to determine, based on the thermodynamic conditions of the system (pressure, temperature, and surface tension of water) the dropping condition?


Behavior of shock waves at relativistic speeds

Suppose I am in a spaceship traveling inertially at a velocity $v$ that is of the same order as $c$. As I pass by a metal bar that is oriented parallel to $v$, someone hits it with another metal bar, causing it to vibrate. The vibration waves travel down the bar with a velocity of $u$. Observing the bar from my spaceship, I see the waves travelling with velocity $u+v\over 1+uv$.   1. Suppose that the waves are longitudinal, so that $v$ and $u$ are parallel. The atoms in the bar are individually moving back and forth, with a maximum velocity $w$ which is quite different from $u$. $w$ and $u$ are related in a way that depends on the material properties of the bar; I think on its elasticity its density. By comparing $u$ and $w$ I can calculate these properties of the bar. If I observe the motion of the individual atoms, I will see them moving back and forth with a maximum velocity of $v+w\over 1+uv$ and by comparing this with the observed velocity of the travelling wave I will obtain a measurement of the stiffness and density of the bar. But this value will be different from the value that an observer would calculate at rest. What explains this discrepancy? Is it due to relativistic dilation of the bar's mass and volume, and hence its density? Is there some other effect?   2. Now consider transverse waves. Again $v$ and $u$ are parallel. But this time the individual atoms are moving back and forth _perpendicular_ to $u$ so the observed dilation of their velocity $w$ is different than it was for longitudinal motion. If I do the same calculations as in (1) I should get different results for the physical properties of the bar: I see the transverse and longitudinal waves moving at the same speeds in the bar, but when I look closely at the motion of atoms, and try to calculate material properties of the bar I get two different sets of results, because the atomic motion causing the longitudinal wave is relativistically dilated to a different degree than is the atomic motion causing the transverse wave. What is going on here? What does it really look like? Presumably there is no discrepancy and I see a consistent picture regardless of $v$. How do all these apparently conflicting measurements iron out in special relativity?


Angular momentum operator and expectation values

I was reading some notes and it says that $\langle L_z^2\rangle=\langle L^2\rangle$ IFF the system is radially symmetric. I can see that in order that the LHS of the statement implies that $\langle L_x^2\rangle=0=\langle L_y^2\rangle$. But I am not sure how to rigorously prove to myself that in order to make those 2 vanish we must have an isotropic system. Could someone please kindly explain this? Many thanks.


How to interpret the continuity conditions in the PDEs (for example, Maxwell equations) originated in physics?

I am currently working on _PDEs_ in physics, mostly Maxwell equations. I am a mathematics graduate student, and this question has been haunting me for years. In PDE theory, or more specifically the _Sobolev space_ theory, the solution to the _Galerkin-type_ weak formulation of a PDE normally satisfies some continuity conditions across any interface in the domain of interest. From a mathematical point of view, the continuity conditions guarantees that the weak derivative of the function is well-defined. That is, whenever we do integration by parts in each subdomain and then sum them up, the results should be the same with doing the integration by parts on the whole domain. * * * For example, the Maxwell equations like the following I was working on: $\newcommand{\vect}[1]{\boldsymbol{#1}}$ $\newcommand{\vH}{\vect{H}}$ $\newcommand{\vE}{\vect{E}}$ $\newcommand{\vB}{\vect{B}}$ $\newcommand{\vD}{\vect{D}}$ $\newcommand{\vJ}{\vect{J}}$ $\newcommand{\vn}{\vect{n}}$ $$ \left\\{ \begin{aligned} \nabla \times \vE &=- \frac{\partial \vB}{\partial t} \quad &\text{ in } \Omega\times (0,T) \\\ \nabla\times \vH &= \frac{\partial \vD}{\partial t} + \vJ \quad &\text{ in } \Omega\times (0,T) \\\ \nabla \cdot \vD &= q \quad &\text{ in } \Omega\times (0,T) \\\ \nabla \cdot \vB &= 0 \quad &\text{ in } \Omega\times (0,T) \end{aligned} \right. $$ If we have two different mediums, being labeled as $a$ and $b$, as in the following picture (shamelessly ripped off from the book Electromagnetic Theory and Computation: A Topological Approach): ![Continuity condition](http://i.stack.imgur.com/NGDAV.png) the continuity conditions of the electric and magnetic fields/fluxes are given as: $$ \begin{aligned} \vn \times (\vE^a - \vE^b) &= 0 \\\ \vn \cdot (\vB^a - \vB^b) &= 0 \\\ \vn \times(\vH^a - \vH^b) &= \vJ_S \\\ \vn \cdot (\vD^a - \vD^b) &= q_S \end{aligned} $$ where the superscripts denote the limiting value of the vector fields on the interface being approached from each side, and $\vJ_S$ and $q_S$ are the surface current and surface charge respectively. Here $\vE$ and $\vB$ are continuous in the tangential and normal direction on the surface respectively. And these conditions play a pretty important role in the numerical simulation for Maxwell equations, for example, the use of Nédélec elements in computing $\mathbf{curl}$-$\mathbf{curl}$ type equation derived from time-harmonic Maxwell equations. (The introduction ends here.) * * *   * How do we interpret the physical meaning of these tangential/normal continuity conditions in real life (analogous to the interpretation of Stokes theorem as the conservation of a certain quantity)? Are the real electric fields continuous in the tangential direction of a surface? Or are these continuity conditions governed by certain physical laws?   * From my understanding, most PDEs from physics are derived from conservation like Ampère's law and many others, and in the integral form of Maxwell equations no differentiabilities are assumed, hence no continuity conditions are enforced. Then are these continuity conditions artificial just because we would like to get some PDEs?


Why the electric force does no work on the charge when perpendicular to the field?

I was reading my physics textbook and came across this sentence: > _When a charge moves in an electric field, unless its displacement is always > perpendicular to the field, the electric force does work on the charge._ So it is saying that if the displacement is perpendicular to the field, the electric force does no work on the charge.   1. Why is it so?    2. By perpendicular to the field does it mean perpendicular to certain field line?    3. Is it implying that if the displacement is perpendicular to certain field line, some other force does work on it (but not the electric force)?    4. But how to be "perpendicular to certain field line" when field lines are normally curved?    5. Let's say it means literally the field but not field lines, then how to be "perpendicular to the field" when the field is not even a straight line but a 3 dimensional region in which electric force is effective regardless of the presence or absence of a material medium?


Is "Egyptian Year" the same as a modern sidereal year?

Copernicus uses the term "Egyptian Year" throughout his discussions of the movements of the Earth, and of his and other models of the movements of the planets; but is unclear from his text, or from the general definitions I've found, what this corresponds to in modern astronomical terms. What, precisely, is an "Egyptian Year"? Is it identical with a modern sidereal year; if not, what are the correct conversions between the two?


Does the second law of thermodynamics tell me how the entropy changes?

In thermodynamics I can e.g. compute the properties of ideal gases with certain energies $U_1,U_2$ in boxes with certain volumes $V_1$ and $V_2$. Say I have two such boxes and they have some specific parameter values, like say I know their temperatures $T_1,T_2$. Now I _put them together_ , the gases can interact and I thereby construct a new box with volume $V_1+V_2$ and the energy is $U_1+U_2$. Using the laws of thermodynamics, I can compute everything else again now. Say I find a new temperature $T_{\text{new}}$. One says that the temperatures of the gases in the systems _changed_ by putting the boxes together. However, on the computational side what I did was just considering a system of new specifications. To get the new result, I didn't have to enter the real world staring conditions except for the variables which also were necessary to compute their respecive properties - effectively, the theory didn't have to tell me how the system changed, just what the restrictions are - and I basically entered into a new system with these values. For example, it's not relevant which gas had with specifications before. I just did a little trivial algebra and computed what has to be. When then talk about a _change_ of entropy and how the gas behaves, but that seems to be only decorative. If I say that the second law of thermodynamics tells me that heat flows from a system with high temperatures to a system of low temperature I'll be able to derive rules for the entropy say, how the combined system has to look like if I state their variables and insist on extensivity of certain variables and so on - but it's not about change in the sense that I compute how the system developes from one point to the other in details. Rather I just compute how the end configuration has to be. My question is > Is the starting configuration of a thermodynamics system ever relavant? And secondly, > In thermodynamics, if I compute "the change a system" in the sense of the > above example, do I always induce the necessarity for doing such a thing by > stating "now we bring system one and system two in contact"?


When one can make telescope that can distuingish 1 mas?

When one can make telescope that can distuingish(angular resolution) 1 mas object by angular diameter? If already possible, tell the name of telescope/observatory. How to calculate angular resolution $\theta$ of Telescope from the diameter $D$ of its mirror? Just in case I'm understood wrong; is it through Rayleigh equation; $\theta \approx 1,22 \lambda/D$?


Interference, photon's phase, and the Hilbert space

Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one. Let $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ be a two-photon state. Let's consider interference between the two photons. If one "rotates" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\rangle$ and $|2\rangle$ should be different than the interference between $|1\rangle$ and $e^{i\varphi}|2\rangle$, and that between $e^{i\varphi}|1\rangle$ and $|2\rangle$ (where $\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter. Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\frac 1 {\sqrt 2}\left(|1\rangle|2\rangle + |2\rangle|1\rangle\right)$ is equivalent to $\frac 1 {\sqrt 2}\left(|1\rangle|2'\rangle + |2'\rangle|1\rangle\right)$, where $|2'\rangle=e^{i\varphi}|2\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon. Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\rangle,|2\rangle$ we obtain different interference than in terms of $|1\rangle,e^{i\varphi}|2\rangle$, respectively $e^{i\varphi}|1\rangle,|2\rangle$. So, either the Hilbert space does not contain all the needed information, or I am missing something (which is by far more likely). Could you please explain me where I am wrong, and what is the correct description of interference using the two-photon state? _P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what._


Einsteins equation : Black hole solution

Let Einstein's equations satisfy $ R_{\mu \nu } = 0 $. Suppose we solve it numerically with the aid of a computer. Can we know from the numerical solution if there is a black HOle in the solutions? For example, how can you know when you solve Einstein's equation if your solution will be a black hole or other particular non-smooth solution?


Would Titan moon blow up if you light a match?

Given that the atmosphere of Titan is mostly methane, and from a comment on this TED talk... Would it blow up from a lighted match-stick? Seems like it's at _least_ plausible. But, wouldn't the lack of oxygen hinder this event? Or perhaps I'm confusing the _blow up_ with **combustion**?


Is strictly harmonic 2D lattice made of Hooke springs possible?

If we connect a set of point masses in a 1D lattice with Hooke springs and consider longitudinal oscillations, we'll have a strictly harmonic system, for which there exist eigenmodes and the frequency of oscillations doesn't depend on amplitude. If we choose transverse oscillations, they cease to be harmonic: the potential for small oscillations appears to be $\sim \Delta x^4$. Now I'd like to generalize the first case, where the oscillations are linear, to 2 dimensions. However, it appears that if we make, e.g. a rectangular lattice with nearest neighbours connected by Hooke springs, then any displacement of a lattice site leads to longitudinal deformation of two springs _and_ transverse deformation of two others, thus making the returning force nonlinear function of displacement. Arranging the masses into a triangular lattice doesn't help to compensate the nonlinearity. So, the question is: is it possible to make a _strictly harmonic_ 2D lattice using just point masses connected by Hooke springs? What would be an example of such an arrangement?


What does this quote about the four dimensional divergence of an antisymmetric tensor mean?

> _In the beginning, God said that the four dimensional divergence of an > antisymmetric second rank tensor equals zero and there was light._ Can someone explain what is the meaning of this quote by Michio Kaku?


Can the zeroth-component of a 4-velocity be negative?

Is it allowed to have the zeroth-component of a four-velocity be negative? I presume the answer is yes, but I just want to make sure. Many thanks. * * * This is referring to $V^0$ for a curved space metric with signature $-+++$.


Other application of Liouville's theorem besides thermodynamics

Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in Thermodynamics?


What Is Energy? Where did it come from?

The simplistic undergrad explanation aside. I've never really understood what energy really is. I've been told that it's _something_ when converted from one kind of something to another kind does some "work", as defined by us, but what is that something? Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learnt from thermodynamics where it _goes_ , but where does it come from? I know this sounds as something trivially simple, but there is so much going on over here and I just can't grasp what it is. Of course, I lack the mathematical understanding to grasp the subtle things the universe is doing, but what is it doing and how do I get to point of understanding what it's doing? ( **Note:** What prompted me to ask this was this answer. I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.)


What is energy?

What exactly is energy (in terms a non-physicist/non-mathematician can understand)? I've been researching this topic, and it seems that energy is not a real physical state, i.e. it is only a scalar that is assigned to the state of physical systems for mathematical abstraction. In other words, energy appears to be a trait of mathematics used to describe traits of matter and non-matter fields. How accurate is this assumption? If energy is "nothing," how can the breaking of a chemical bond catalyze another reaction (e.g. formation of a bond)? That energy that is transferred, what exactly is it?


Why is electric charge conserved?

We have long been taught that electric charges are neither created nor destroyed. But somehow it is okay to destroy two oppositely charged particles at once ! Why is that so? Let's just take a look at electron-positron annihilation; Two equally oppositely charged particle are bombarded into each other, both are instantly destroyed (lose their existence) and gamma ray photons emerge. It is certainly convenient to say that at $t$ second the net charge in system was 0 and at $t+1$ second it is also zero. But certainly it also raises the question what if you observe only a positron or an electron, in observed system certainly charge will not be conserved however overall it maybe. It jumps out and say that you cannot destroy individual charges, but if you get 2 equally and oppositely charged particles we can destroy both ! (kind of Romeo-Juliet) Then why is the fundamental postulate of conservation of charge that /charge can neither be created nor destroyed"? Since, clearly it can be destroyed, just in pairs (also created in pairs)


calculate force between two magnets

I have used a couple of online calculators to estimate the attractive force between two cylindrical bar magnets. Using d=1"; t=1"; x=0.1"; Neo35 (Br=12,300 Gauss): source 1 gives 63.5 lbf   source 2 gives 29.6 lbf source 2 explains data is based on measured data, so this accounts for the difference from source 1. But in general, I think this is the right ballpark number. I then tried to reproduce these numbers myself, using a formula I found on Wikipedia. I could not get close to the right answer - my answer is about 4000 lbf. **Could someone please explain what I did wrong in my calculation?** ![formula](http://i.stack.imgur.com/boNho.png) ![enter image description here](http://i.stack.imgur.com/WUd0K.png) **EDIT** Tough crowd. Let me try and be more specific. The problem itself is pretty straightforward. I did not really mean to imply I need someone to check the numbers. I mean more on if I am applying the right approach to the problem:   1. Is the formula above valid when trying to replicate results in the calculators I linked? (i.e. "Calculate the Pull and Repulsion Forces Between Two Disc Magnets")   2. Perhaps "real life" is significantly different than theory for this application? In which case, the engineering calculators would apply a de-rating factor that is not represented in the Wikipedia formula. I did find any references to this, but perhaps someone with more knowledge could advise?   3. Assuming the formula is valid here, I think Magnetic Flux Density (B_0) is the most likely source of my error. It is my understanding that this is a physical property of the material. Is that correct? The value I found for Neo35 is Br = 12,300 Gauss (CGS units). I converted this to SI 1.23 Tesla using a 10^-4 conversion factor. Is this the correct interpretation of B_0?


Simulation of fluid flow using Euler equation

I have been looking on Euler's equations for a while and can't grasp one thing. Suppose we have initial system state with volumes of fluid "hanging" in air (time is frozen and equal to zero), each of them has its initial (x, y) coordinates and velocity vector (Vx, Vy). The space is divided in stationary volumes in which measurement takes place (Euler measurement model). ![2 dimension representation of segmented space and restrictions](http://i.stack.imgur.com/AQBYW.jpg) As I understand Euler's equations impose mutual restrictions between functions of velocity ( u(x, y, t) ), mass density ( m(x, y, t) ), and pressure ( p(x, y, t) ) at any instance of time for stationary volume of space with coordinates (x, y) that contains volume of fluid (in that instance). So why each function has time parameter when equations must be solved for any instance of time (iteration process) ? We must have new set of these functions for any moment of time that satisfy equations. And it's unclear whether volume of space contains volume of fluid or not, how do we take this in consideration ? And the second question is how state of each volume of fluid (its coordinates and velocity vector) is related to these functions ? I need a system that has some initial state (array of fluid volumes with their coordinates and velocities) and uses Euler's equations to compute next state, which becomes initial for next iteration and so on. Something like this: ![scheme](http://i.stack.imgur.com/DRe1Q.jpg) I don't need math details, I just want to grasp basic idea in context of computer simulation (how method of simulation can be bounded to standard analytic equations). Most papers are very complicated, with additional physical factors and user interaction, and they don't cover this "link". I want to consider the easiest case and how to start iteration process (just idea and how it is usually done). Sorry if some things sound ridiculous, please adjust my statements if they are wrong. If my model of understanding is completely wrong just point me in right direction, thanks.


In what ratio does the charge distribute if a charge and uncharged body touch each other?

Lets say a large charged body has $n$ elementary charges and is brought in contact with a small uncharged body. When the charge distributes, does it distribute in the ratio of number of atoms or each body gets half of the charge?


Significant figures vs. absolute error

On NIST Avogadro's Number, $N_A = 6.022\;141\;29 \times 10^{23} \text{ mol}^{-1}$ has 9 significant figures and a standard uncertainty of $0.000\;000\;27 \times 10^{23} \text{ mol}^{-1}$. First, can it be written as the following? $$ N_A = 6.022\;141\;29 \times 10^{23} \pm 0.000\;000\;27 \times 10^{23} \text{ mol}^{-1} $$ If yes, consider the following quote from Rice's lab guide on error analysis (emphasis added): > Whenever you make a measurement, the number of meaningful digits that you > write down implies the error in the measurement. For example if you say that > the length of an object is 0.428 m, you imply an uncertainty of about 0.001 > m. To record this measurement as either 0.4 or 0.42819667 would imply that > you only know it to 0.1 m in the first case or to 0.00000001 m in the > second. You should only report as many significant figures as are consistent > with the estimated error. The quantity 0.428 m is said to have three > significant figures, that is, three digits that make sense in terms of the > measurement. Notice that this has nothing to do with the "number of decimal > places". The same measurement in centimeters would be 42.8 cm and still be a > three significant figure number. **The accepted convention is that only one > uncertain digit is to be reported for a measurement. In the example if the > estimated error is 0.02 m you would report a result of 0.43 ± 0.02 m, not > 0.428 ± 0.02 m.** and the following quote from A Project of Joliet Junior College (IL), Lee College (TX) and the National Science Foundation (emphasis added): > The absolute error places bounds telling you that the actual value is > expected to fall between x - D x and x + D x. The absolute error is an easy > way to report the error and makes it easy to check that you are not > reporting too many significant figures. **For example, suppose you had a > measurement that was reported to be 10.02 ± 0.3 m. Because the reported > value is uncertain in the tenths place, it is wrong to report it to the > hundredths place. It should be reported to the same decimal place as the > absolute error: 10.0 ± 0.3 m.** Another point is that the absolute error and > the reported value must have the same units. If there is such a convention (cf. Rice's lab guide) and it is wrong to report more significant figures than is warranted by the absolute error (cf. the joint project's guidelines), and NIST should have adhered to such convention and striven to be correct, is it true that NIST adheres and is correct because the standard uncertainty has two least significant digits (i.e., $27$)? Had the standard uncertainty been $0.000\;000\;2\text{ mol}^{-1}$, would NIST have reported Avogadro's Number as $N_A = 6.022\;141\;3\times 10^{23}$ with 8 significant figures?


What is the smallest increment of space and the smallest increment of time?

I believe that every point in space is a surface point. And if it was possible to break through or tear a point in space, it will create a black hole. I believe that's why a black hole can exist anywhere in space. Since every point in space is a surface edge then punching through a point in space will causes matter to get suck into a very powerful and vast vacuum (outside of our universe). Does any of this hold water? And what about the smallest increment of time and space? Where can I find supporting Theory and equations, if any?


Mixing generators of different dimensionality

Reading a paper about compactified manifolds used in Kaluza Klein theories the author discusses in which ways you can get $SU(2)\times{}U(1)\times{}U(1)$ as a subgroup of $SU(3)\times{}SU(2)\times{}U(1)$. At a certain point, it starts considering ways of getting $U(1)$. It says you can do it using the $\lambda_8$ Gell-Mann's matrix as a generator of $U(1)$. The original $U(1)$ group as well can be, of course, used, whose generator he calls $Y$, and finally any of the $U(1)$ generators of $SU(2)$, which he calls $T_3$. Then he combines the three generators in a linear combination $p\lambda_8+qT_3+rY$ Which is supposed to be a generator (via complex exponentiation) of a $U(1)$ group. How is this supposed to be understood? Being all three generators matrices of different dimensionality, how do we add them?


Electrial Conductivity of Thin Metal Films

What is the best way to find specific/electric conductivity which is dependent of very thin film thickness?


Is there a simple, accurate formula for calculating transit times from rise and set times?

I have access to "rise" and "set" times for astronomical objects, and want to determine the corresponding times of culmination. Is there a reliable and accurate way to do this? It isn't clear to me from the sources of the data I have whether the times I have are for the geometric centers of the objects, or take into account refraction, or the angular extents of the objects, so I'm hoping there is an approach that is insensitive to these factors. In particular, I'm interested in transit times for the Sun.


constraint on scaling dimension

How can we show that for any scalar operator $\Delta\geq1$ (where $\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?


What is a bulk phase transition?

I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of what exactly a bulk phase transition is. Could someone please help me with a definition of _bulk phase transition_? I would very much appreciate it.


What is the most efficient way to destroy the universe?

Don't worry... I won't actually do it, just would like to know how.


Mnemonics to remember various properties of materials

I'm trying to figure out how to remember that   * hardness: how resistant it is to deformation   * toughness: how resistant it is to brittle failures   * stress: force on a surface area   * strength: ability to withstand stress without failure   * strain: measurement of deformation of a material Does anyone know of a mnemonic or easy way? I only know these from googling them, and I'm finding it tricky to remember them.


Is quantum entanglement mediated by an interaction?

You can get two photons entangled, and send them off in different directions; this is what happens in EPR experiments. Is the entanglement then somehow affected if one puts a thick slab of EM shielding material between the entangled photons? Have such experiments been made? According to EPR experiments measurements of the entangled states are at odds with SR, so based on that I'd assume the answer is "no"/"don't know", but any citations would be appreciated!


Time derivative to energy equates $0$

Following up on a related question. This implies that, From the reference we can write \begin{align*} E(\lambda) &= I_F [g_\lambda ] + I_K [f_\lambda , g_\lambda ] + I_V [f_\lambda ] \\\ & = \lambda^{4-d} I_F [\bar A] + \lambda^{2-d} I_K [\bar \phi,\bar A]+ \lambda^{-d} I_V [\bar \phi]. \end{align*} This must be stationary at $\lambda = 1$, therefore we must have, \begin{equation} 0= (4-d) I_F [\bar A] + (d-2) I_K [\bar \phi,\bar A]+ d I_V [\bar \phi].\tag{1} \end{equation} I think the equation (1) equates the time derivative to zero. Is it right? If this is true, then why are we taking this derivative and why does it need to be zero? * * * Source: Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43)


Different invariant gauge groups (IGG) on different lattices with the same form mean-filed Hamiltonian?

Suppose that we use the Schwinger-fermion ($\mathbf{S_i}=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$) mean-field theory to study the Heisenberg model on 2D lattices, and now we arrive at the mean-field Hamiltonian of the form $H_{MF}=\sum_{<ij>}(\psi_i^\dagger u_{ij}\psi_j+H.c.)$ with $u_{ij}=t\sigma_z$($t>0$), where $\psi_i=(f_{i\uparrow},f_{i\downarrow}^\dagger)^T$, and $\sigma_z$ is the third Pauli matrix. Now let's find the IGG of $H_{MF}$, by definition, the pure gauge transformations in IGG should satisfy $G_iu_{ij}G_j^\dagger=u_{ij}\Rightarrow G_j=\sigma_zG_i\sigma_z $ on link $<ij>$—(1) . Specifically, consider the IGGs on the following different 2D lattices: (a)Square and honeycomb lattices(unfrustrated): These two lattices can be both viewed as constituted by 2 sublattices denoted as $A$ and $B$. Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ in **the same sublattice** are _site-independent_ while those in different sublattices differ by $G_A=\sigma_zG_B\sigma_z$ and $IGG=SU(2)$. (b)Triangular and Kagome lattices(frustrated):Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ are _global_ (site-independent) and $G_i=\bigl(\begin{smallmatrix} e^{i\theta }& 0\\\ 0& e^{-i\theta } \end{smallmatrix}\bigr)$ which means that $IGG=U(1)$. So my question is: The same _form_ mean-field Hamiltonian $H_{MF}$ may has _different_ $IGGs$ on _different_ lattices?


How do I calculate the time it would take for a space craft launching from earth to reach another planet?

I am making a matlab program the my final project for my programming class in college. I have chosen this problem, but I am bashing my head against the wall trying to figure out the equations and how they interact with each other. I have the user play with the thrust and the payload to carry to the other planet. The user also decides what planet to go to and what type of orbit to use (elliptical or parabolic). I know I have to usde the rocket equation, escape velocities have to be claculated, and time formula, but I still haven't figured out what to do.


Why does Gauss's law work for a charge off center in a spherical surface?

CASE 1: Consider an enclosed spherical surface with a charge $q$ at its centre. From Gauss' law we can say that the flux through this sphere is $q/\epsilon_0$. CASE 2: The charge is inside but _off centre_. In this case the electric field will not be normal to the surface. So how can we use the formula $q/\epsilon_0$? It turned out that the answer is same for both cases. This is my confusion.


Why do degenerate normal modes appear as complex conjugate pairs?

Can anyone prove this? THE DETAILS Suppose we have a system with n components, (i.e. $|~{\psi}(t)\rangle=\sum \langle x_i|~\psi(t)\rangle ~|~x_i\rangle$) where our equation of motion is described by $$\frac{d^2}{dt^2}|~\psi(t)\rangle=\hat{M}^{-1}\hat{K}~|~\psi(t)\rangle$$ And say we are looking for solutions $|~\omega\rangle$ such that $\frac{d}{dt}|~\omega\rangle=\omega|~\omega\rangle$, and thus our equation simplifies to $$\hat{M}^{-1}\hat{K}~|~\omega\rangle=\omega^2|~\omega\rangle\tag{1}$$ (Note that $\hat{M}^{-1}\hat{K}$ is Hermitian). Is there any way to prove that if $|~\omega_1\rangle,|~\omega_2\rangle$ are linearly independent and both satisfy (1), then $|~\omega_1\rangle=|~\omega_2\rangle^*$, i.e. the components of one vector are the complex conjugates of the components of the other vector? I ask this because this keeps on popping up in our class study of coupled oscillators and I want to know why this is happening Thanks!


Is there an idealist rather than realist interpretation of QM?

The many-worlds interpretation of QM is a realist explanation as it makes the wave function of the universe real. That is it makes the probabilities of outcomes real outcomes. One could argue that this view is biased that realist ideas of reality are better by definition. But philosophically speaking idealist ideas of realities have been mooted, for example by Berkely & the Cambridge Platonists, (or Brahman in Indian Philosophy). Is there is an idealist explanation which explains the probabilities inherent in the wave function as epistemological? That is states of knowledge of observers. Except here by observer I do not mean necessarily some one that is conscious, but simply a system that reacts to the information context around it.


Why does evenly heating soup with a microwave take so long?

My anecdotal observations (which could be incorrect, they're totally unscientific) indicate that it takes almost as long to evenly heat a big bowl of soup in a microwave as it does to heat it on the stove - 4-5 minutes in my most recent example. Yet a plate of meat or vegetables is reheated in 30 seconds or less.


Presence of planets in Milky Way and other galaxies

Are there only planets in the Milky Way galaxy, or are there other planets in other galaxies? If planets are only in the Milky Way, why aren't there planets in others?


Has anyone actually produced any braided anyons (quasi-particles) yet? (quantum computer)

Quasiparticles theoretically can form braids on a 2D surface. The braids apparently are quite stable for reasonably long periods, allowing a superposition state more time before it decoheres through environmental interaction. This would be a big step in quantum computing. Has this actually been achieved or is it theory only at this point?


Thermodynamics for Dummies: Entropy and temperature

I do not study physics and I have never had a course in thermodynamics. I have no idea what it is about, but I am currently taking a course where we had something about entropy. Would be great if you could help me in easy words.. Wikipedia and other Websites did not help. So: We are looking at a box full of gas with N particles in it (see Maxwell's demon). The question is now to look at two different states. The first state ist when the whole box is used for the gas. The second state is when only one half is used. Now we consider the Transition. 1) -> 2): The second state has less states, hence less entropy. So we lose entropy. This Forces us to do work, to invest energy, otherwise this would not happen (or only with small probability), as the 2nd law of thermodynamics states. So what does this mean for the temperature in this box? Are we considering the temperature in this box or around this box, or both? Does it get warmer or colder? Where? 2) -> 1): This is more probable, and energy is set free. We have more entropy then. But also, this has to do something with temperature, so what are the consequences? I know how to calculate the energy that has to be invested to enforce 1)->2), but what I do not know is the consequences for the temperatures. So we talk about $\Delta S \geq - N k_B ln 2$ (why $\geq$ and not $=$), the difference in entropy, so the temperature difference is $\Delta Q = T \Delta S$.. Sometimes I have read that more entropy means hotter. As if particles are hotter, they move faster, so there's more uncertainty. But then, I also read that if we go 2) -> 1) that the temperature gets colder. But the entropy grows. So I have no idea. My guess is, that sometimes one talks about the temperature in the gas, and sometimes one talks about the temperature outside this box. Thank you for your help. Hope I am right here.


Electron photon interaction potential in old fashioned perturbation theory (OFPT)

In this PDF on old fashioned perturbation theory (OFPT) we find from equation (14) the potential describing the interaction between the electron and photon: $$ V = \frac{1}{2}e \int \mathrm{d}^3x\, \psi_e(x)\phi(x)\psi_e(x) \tag{14}$$ where $\phi=photon. $ My question is why $V$ looks as it does and where exactly does it come from?


Macroscopic Forces from QED

In QED the carrier for electromagnetic interaction is a photon, while macroscopic forces are due to electromagnetic interaction (by macroscopic forces I mean: normal force, object collision, friction etc..). But for example, in the collision of two billiard balls, if 1 kgm/s of momentum is exchanged between the balls, the amount of Energy required in the form of photon virtual particles would be enormous (3E8 J). And while they do not necessarily obey the mass-shell relation, still the farther off shell interactions are, the less they should contribute to the actual phenomena. For least action to be satisfied it seems like some other type of exchange interaction would occur. Is there some kind of larger scale decoherence type interaction going on?


Can a nearly-extremal black hole be stable against Schwinger vacuum breakdown?

I was doing some basic algebra to estimate the range of possible masses $M$ and electric charge $Q$ for a nearly extremal Reissner-Noström black hole. I want to see if the logic is correct the electric field near the event horizon of such a black hole would be $$ E= \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0} \frac{q c^4}{G^2 M^2} = 10^{66} \frac{\text{Volt Kg}^2}{\text{m Coul}} \frac{q}{M^2}$$ since Schwinger-pair vacuum breakdown occurs at $10^{18} \frac{\text{Volt}}{\text{m}}$, it implies that $$ \frac{q}{M^2} \lt 10^{-48} \frac{\text{Coul}}{\text{Kg}^{2}}$$ Would be a bound that guarantees electromagnetic vacuum stability **Question:** can a charged black hole that is stable against this breakdown still be nearly extremal? well, it seems so, because the extremality condition: $$ r_Q = \frac{r_s}{2}$$ Implies that $$ Q \sqrt{ \frac{G}{4 \pi \epsilon_0 c^4}} = \frac{GM}{c^2}$$ $$ \frac{Q}{M} \approx 10^{-16} \frac{Coul}{Kg}$$ So the above two relations would imply that a black hole with a mass $$ M \gt 10^{32} Kg \approx 100 \text{ Solar masses}$$ Can be both stable (against the specific Schwinger-pair decay) and extremal. In particular, if the mass is $10^{32}$ kilograms, a charge around $10^{16}$ Coulomb would make the black hole arbitrarily close to extremal as one might want, and is not at all clear that it would decay from the extremality state But probably i'm overlooking other ways than the extremality of the black hole can decay (in particular, i'm ignorant about how Hawking radiation will contribute in here, specially if the temperature is below 511 KeV). Any insights into this would be greatly appreciated


Determining the moment of force

A force $F = 3i + 2j$ passes through through a point $P$ with respect to an origin $O$. How do I determine the moment of the force at the origin.


How does Newton's 2-prism experiment help to explain why light does not get dispersed into 7-colors in a parallel glass slab?

![enter image description here](http://i.stack.imgur.com/Tze2p.png) In a real parallel glass slide(with two prisms imagined to be touching each other to form a parallel glass slide),   The ray of light should pass through the Z in between without any dispersion or change in direction because the density in both of the prisms is the same(and they are touching each other). So Newton's experiment won't help in explaining this because in it, there is a vaccum(or air) in between and hence the change in direction of light. I hope that you are getting me, if not I'm ready to elaborate more. _[In case anyone is wondering that I did post a Q about dispersion earlier too, then that is true - but I did not get the answer of the question that I really meant to ask and hence this, I hope that you guys will help me better understand this.]_


Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very strange to me. Second, I don't understand why applying the raising operator and then the lowering operator is different from applying the lowering operator and then the raising operator. The two are explained in terms of moving up and down energy levels, so shouldn't going up then down, or down then up, be the same operation? What is asymmetric about harmonic oscillators that causes this asymmetry?


Quantum of Time

I am not a String Theorist, but I just want to ask if there is anything which specifies a quantum of time, as per M-theory? Can a quantity such as $$ t_p = \left(\frac{\ell_p}{c}\right),$$ where $\ell_p$ is Planck length and $c$ speed of light in vacuum, be called as time quantum?


Why do mirages only appear on hot days?

A previous question asked why the road sometimes appears wet on hot days. The reason is that when there's a temperature gradient in the air, it causes a gradient in the index of refraction, causing the light to bend. This diagram (from Lagerbaer's answer to the previous question) is ![enter image description here](http://i.stack.imgur.com/8eZCm.png) I have two questions about this. First, Wikipedia says that in order for the effect to appear, there must be a temperature gradient on the order of a few degrees per hundred meters above the asphalt. To investigate this claim, I assumed the air near the surface of the asphalt is at constant pressure, so the density is inverse-proportional to the temperature (i.e. I assumed the density variation due to the weight of the air is small compared to the density variation induced by temperature gradient. Otherwise we would see mirages all the time.) Then I assumed the difference of index of refraction of air from one is proportional to the density, i.e. $n = 1 + a\frac{\rho}{\rho_0}$ with $n$ the index of refraction of air, $a$ some dimensionless constant, and $\rho_0$ some reference density. Looking up the index of refraction of air online, I guessed $a = 0.0003$. Then I assumed the temperature above the Earth is modeled by $T = T_0 + gy$ with $y$ the height and $g$ a temperature gradient in ${}^{\circ}\mathrm{C}/\mathrm{m}$. This completes the model. Fermat's principle gives a variational problem to solve for the path of the light. However, the resulting differential equation was hard to work with, so to first order I approximated that the light starts at a horizontal distance $x = -L$ at height $y =h$, slopes down towards the ground with some slope $m$ until it gets to $x=0$, then slopes back up at the same slope $m$ until it gets to $x = L$ and $y = h$ again. Then I chose $m$ to minimize the travel time of this path. To first order for small $h$, I found $$m = \frac{agL}{2T_0}$$ The problem is that when I plug in $g = 5^{\circ}\mathrm{C}/100\mathrm{m}$, $a = 0.0003$, $T_0 = 300 \mathrm{K}$, and $L = 100 \mathrm{m}$ I get $m = 2.5*10^{-6}$, which is too small to explain the mirage. It would only allow light to dip a quarter millimeter in the path from a car $200 \mathrm{m}$ away to me. In the Wikipedia image of a mirage, the light clearly dips down at least 1000 times that much (check out the blue car). ![enter image description here](http://i.stack.imgur.com/eb2iN.jpg) So my first question is: what's wrong? Why are there mirages when this model predicts that there are not? My second is: why do we see them only when it's hot? This model depends only weakly on absolute temperature, and hot absolute temperatures actually decrease the effect. Evidently, there are only high temperature gradients on hot days, by why should that be? A temperature gradient over the asphalt comes from sun heating the asphalt directly more than it heats the air. Shouldn't that happen on cold days as well as on hot days?


st. venant shallow water equations for pipe flow

As far as I know from the fluid dynamics class, St. Venant equations (the shallow water equations) are derived by depth-integrating the Navier-Stokes equations. This depth-integrating is done with the assumption that the horizontal scale is much larger than the vertical scale, hence it would be used for overland flow or surface runoff modeling. But I was wondering if the St. Venant equations can also be used in open channel pipe flow, say to model the flow inside the drainage pipe. I imagine the flow would more or less be similar as that in a river channel, but in this case the horizontal dan vertical scale would be the same at some point of water depth, given the geometry of the pipe. So my question is, can St. Venant equations be applied correctly for modeling open channel pipe flow? If they can, will there be any constraint from the water depth condition? For example, if the water depth is 5cm in a 1m diameter pipe then perhaps SWE can be applied, but when the water depth is more than half of the pipe then wouldn't the horizontal to vertical scale ratio be the same?


James Joule's Conservation of Energy

Is there a manifest or documentation of the original Conservation of Energy as stated by James Joule word for word? **Edit 1.1:** I found Joule's Memoir on google, skimming through to find any conclusive evidence.


Does electricity flow on the surface of a wire or in the interior?

I was having a conversation with my father and father-in-law, both of whom are in electric related work, and we came to a point where none of us knew how to proceed. I was under the impression that electricity travels on the surface while they thought it traveled through the interior. I said that traveling over the surface would make the fact that they regularly use stranded wire instead of a single large wire to transport electricity make sense. If anyone could please explain this for some non-physics but electricly incline people, I would be very appreciated.


Physical position eigenfunction normalisation

We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ Our professor told us that for any value $a>0$, the physical position eigenfunction is $$\psi_{x_0}(x)\cong N_1\delta_a(x-x_0).$$ How can I show that $\psi_{x_0}$ is a physical position eigenfunction?


M1 vs M2 on a frictionless surface

I was wondering, if I had a frictionless surface and I had an M1 that was 100kg and an M2 that was 1kg and I pushed each separately with an equal force F would they move at the same speed or am I assuming incorrectly?


Why does string theory (in case it is true ) have NO divergencies?

Why is string theory in 10 or 26 dimension not divergent? Due to the high number of spacetime dimension (10 or 26) it should have a lot of UV divergencies of the form $ \int k^{n}dk $ and gravity within the approach of the string theory should be non-renormalizable too, or shouldn't it?


What is the theoretical/intutive meaning of $x(t) = x_0 \cos ( \omega t + f )$?

What is the meaning of $x(t)=x_0 \cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ the phase constant? I know how to solve the mathematical problems which use this expression, but what I don't understand is how sine and cosine functions are used to express AC waves. I know the math but I do not understand the theory behind it. I do not have an intuitive understanding of these expressions. If you cannot explain at least guide me to a resource that can. i want to understand the math in a theoretical manner like are we going to take the sine/cosine of the wt or the solved value of it, since sine/cos is a ratio so basically are we multiplying the since/cos fraction value to the variables, oh i am so confused please explain .


Equation of state of a rubber band

I have the following question that I attached in png format. ![enter image description here](http://i.stack.imgur.com/v5MpZ.png) I have done part (a), but I am having difficulties in part (b) when I proceed according to the book. I have non zero tension at equilibrium length. I reached that conclusion by taking the derivate of entropy with respect to L and n at the same time. What is the physical principle underlying this?


Difference between 1PI effective action and Wilsonian effective action

What is the simplest ay to describe the difference between these two concepts, that often go by the same name?


Where does the excess energy emitted by a microwave go?

If there is nothing in the microwave, where does the excess radiation go? Why doesn't the radiation accumulate and blow it up? Should I cook two pieces of Canadian Bacon twice as long as I cook one?


Where do the high order rainbows appear?

This site shows locations of the high order rainbows up to sixth order. My thought is that, There should be even higher order bows like 7th, 8th, etc. (Though they would be practically impossible to observe.) So, Where would they appear on sky?


Shoot object into the Sun using minimal energy

Say I want to shoot a cannonball into the Sun with minimal energy (minimal initial velocity relative to Earth).   In which direction do I shoot it? Let's neglect Earth's gravity, if that would make things very complicated.


Help identifying an expression for the action

I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from: $$ S = \int_{0}^{N} \left [ \frac{1}{4}\eta_{\mu\nu}{\mathrm{d} X^\mu \over \mathrm{d} s}{\mathrm{d} X^\nu \over \mathrm{d} s} -m^2c^2 \right]\mathrm{d}s$$ I'm not even sure what some of the symbols mean; no idea what $N$ is for example. Big note-taking fail of my part! Any help is appreciated.


Solar System and the Plane of our Galaxy

Why do all of the space probes launched travel within the plane of the solar system? Can we direct a probe perpendicular to the plane of the solar system and galaxy.


Filter out polarized light

Unpolarized light enters the polarizer and gets polarized at one certain angle. If we place an analyzer behind the polarizer and align them, we observe that all light is transmitted. If we rotate the analyzer by an angle $\theta$, then the intensity of light leaving the analyzer would fall as $\cos^2(\theta)$. What I am asking is: is there some device that when placed instead of the analyzer will transmit light as a delta function: max intensity for $\theta = 0$ (perfect alignment) and zero intensity for all other angles ($\theta$ not equal to zero)?


Rolling a deformed ring

Consider a ring rolling without slipping along a horizontal surface. Regardless of the speed of the ring, it is continuously in contact with the surface.   Let's deform the ring slightly so that it becomes an ellipse with small eccentricity of $e\rightarrow 0$. Now, consider the deformed ring rolling without slipping along the surface. At low speed, the ring still keeps contact with the surface continuously. However, at some critical speed the ring will jump. I am interested in how to find this speed?


Pendulum axes confusion

Suppose I have a simple pendulum and I want to calculate its acceleration when the bob reaches the maximal angle. I usually choose my axes such that the y-axis will be parallel to the rope. Then the acceleration would be $g \sin\alpha$ (Because: $mg\sin\alpha=ma$). However, if I choose the y-axis to be parallel to the gravitational force ($mg$), the acceleration equals to $g\tan\alpha$ (Because: (1): $T\cos\alpha=mg$ and (2): $T\sin\alpha=ma$ where $T$ is the tension force (centripetal force) and $\alpha$ is the given angle). Obviously, these two accelerations aren't the same. So my question - which one is right? And why the other one is wrong?


How to calculate the resulting velocitys and rotation speed after two concave polygons collide in 2d

so I've been searching google for how to do this, but all anyone seems to care about is collision detection =p, and no results come close to teaching me how to calculate **2d elastic collision between two concave polygons**. I know how to solve elastic collisions between two 2d circles, but I can't find anything more complicated than that. I'm also a very visual person, so it would be great if someone could show me how to do this or point me to a website! Thanks =)


Alpha particle and helium nucleus

The symbol for the alpha particle is α or $α^{2+}$, it can be written as $He^{2+}$. What I want to know is that, are they same? I mean alpha particle and helium nucleus are same or any subtle difference exists?


Where does the energy produced by fusion come from?

Fission, in layman's (or "initiate's") terms, is easy enough to understand; a large atom with a lot of protons and neutrons requires a large amount of force provided by the strong interaction to overcome electromagnetism and keep the nucleus together. The necessary energies were imparted to the atom during its formation, which we can replicate to some degree within a nuclear reactor; a combination of heat and fast-moving free protons/neutrons recaptured by the atoms of the fuel turns uranium into plutonium, even as the actual atom- splitting produces much lighter krypton and barium isotopes (or a host of other possibilities, some more likely than others, as with many other types of reactions). That atom-splitting, by the way, reduces the size of the resulting nuclei, and therefore reduces the amount of strong interaction force required to contain them; the leftover energy is released as a high-frequency gamma photon. Fusion, however, is an odder beast. A fusion reaction requires a large amount of starting energy; enough to strip the electrons off of the deuterium/tritium nuclei and form plasma. That heat is also enough to accelerate the particles fast enough that when they collide, the initial electromagnetic repulsion is overcome and the strong interaction binds them together. Now, that's a lot of energy required; the Sun's inner core where most of the fusion occurs is estimated to be about 15 million Kelvin. The reaction, however, doesn't seem to release any energy, based at least on this simplistic explanation. Now, obviously that's wrong; pretty much all the energy we have available to us right now is, however indirectly, a result of the Sun bathing us in the energy from nuclear fusion. The rest of it, such as from nuclear fission, is also star-based, via creation of superheavy elements in stellar nucleogenesis. But where is the source of this energy on the subatomic level? Surely more strong force is required to contain a helium atom than a hydrogen one, because the helium now actually has two protons and must actually overcome the Coulomb force (as opposed to hydrogen, which only has one proton and so the Coulomb force between the proton and its zero-charge neutrons is lesser). So, what imbalance does fusion induce within the nucleus that is then resolved with the release of a photon?


Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not?

Does the force of kinetic friction increase with the relative speed of the objects involved? I have heard and read that the answer is no. This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting. What phenomena are at work with kinetic friction, and why does it not increase with relative speed?


Question on measuring expectation value of spin with time variation

I have a particle with the following wave function: $$\psi(t) = \frac12 |\uparrow \rangle e^{-i(\omega_1+\omega_2)t/\hbar} +\frac12 |\uparrow \rangle e^{-i(\omega_1-\omega_2)t/\hbar} +\frac{1}{\sqrt{2}} |\downarrow \rangle e^{-i(-\omega_1-\omega_2)t/\hbar}$$ I am trying to calculate the expectation value of the spin. How should I go about doing that? My best guess is to calculate $\psi ^* \psi$ to find the time-dependent probability of both spin up and spin down and then multiply each probability by the appropriate eigenvalue ($\pm \hbar/2$) and add them. However, when I did this I got a time-dependent probability for spin up but a time-independent probability for spin down, which seems impossible to me. $$P(\uparrow)=\frac12 + \frac14 e^{i2\omega_2t/\hbar} + \frac14 e^{-i2\omega_2t/\hbar}$$ $$P(\downarrow)=\frac12$$ I don't know if I messed up or if the answer is starting me in the face and I can't see it. Note: the two omega values of the time-dependent terms of the wavefunction come from two different particles; my question is only about the expectation value of spin for the first particle so I have not included other non- necessary information on the second particle.


Question about Atwood Machine and Changing Masses

Say we have an object of X Newtons on the ground called object 1, and it is connected to another object called object 2. If we change the mass of object 2 in intervals until object 1 < object 2, how would the force the the ground exerts on object 1 change and how would it be if object 2 was greater than object 1. I know that the force would be less as object 2's mass increase, but I'm not sure how to display it in my problem with numbers. Would we have to do something with the tension as the mass increases? We can find the acceleration with the two masses but I don't understand conceptually how we could find how the force the ground exerts changes.


If virtual particles have negative mass why do they contribute positive mass to atoms?

According to Lawrence Krauss, atoms containing in our body consists of merely 10% (if I remember correctly) of our total mass. The rest come from virtual particles popping in and out of existence from empty space. So if ~90% of our mass comes from virtual particles, and virtual particles have negative mass, shouldn't virtual particles mass be subtracted to ours? In other words, why we aren't getting lighter due to the effect of those particles?


Physical meaning of some operators formed by $|Q\rangle \langle Q|$

In Dirac's formulation of quantum mechanics, Suppose that $q$ represents position observable. About $|q\rangle \langle q|$: what does this operator mean? I do get that it results in an operator, but unsure of what physical meaning it has. The same with $|\psi \rangle \langle \psi |$. ($\Psi$ representing wavefunction, and the corresponding eigenstate $| \psi \rangle$.)


Is water considered a substance with low friction or high friction?

Like, for example, rubber is a substance with high friction, as an object with neither high or low friction would easily stop on it, and wouldn't skid. Ice is considered an object with low friction, as an object with neither high or low friction wouldn't easily stop on it, and would skid. But, what about water?


Efficiency of reversible engines

I'm a little confused about something. All reversible engines have the same efficiency, or one could drive the other to move more heat in the reverse direction. Also, no engine has an efficiency greater than a Carnot engine. Finally, a Carnot engine is reversible. If all these statements are true, then isn't it true that all reversible engines operate at Carnot efficiency? Where am I misinterpreting the logic?


Is it possible to design a large permanent magnet that creates a multi-Tesla field?

Is it ever possible to create a LARGE magnet (in meters wide/long) that could potentially create a powerful magnetic field?


Why acceleration is not relative in General Relativity?

I was thinking of it, If I say: "I'm moving at a velocity $v_1$ relative to a reference frame $M$ then the acceleration will be the derivative of $v_1$ relative to the reference frame $M$." In other words, from the perspective of my brother at home, I'm travelling at a velocity $v_1$ and I have an acceleration $a_1$. But from my perspective, he is travelling at $v_1$ (and I'm standing still) and thus his acceleration is $a_1$. But General Relativity tell us that acceleration is not relative, so why?


How to prove that a time-oriented spacetime possesses a nowhere vanishing timelike vector field?

Penrose gave a very brief proof to this question. Since the spacetime is paracompact, there exists a positive definite metric called $h_{ab}$. Then, the nowhere vanishing time-like vector field $V^a$ is determined by the following equations: $(g_{ab}-\lambda h_{ab})V^b=0$ with $\lambda>0$, and $h_{ab}V^aV^b=1$. (By his convention, the signature of the Lorentzian metric is (+,-,-,-)) I totally cannot understand the logic. Although $g_{ab}V^aV^b=\lambda h_{ab}V^aV^b=\lambda>0$, so $V^a$ is indeed a time-like vector, how can we be sure there is such kind of vector field satisfying the conditions?


What determines the end of space (universe)?

What determines the end of our universe?   Is it defined by the farthest out physical object or is it defined by the farthest out form of energy like light?


Does the presence of atoms hinder current flow?

When current flows through a material, electric resistance is observed when the electrons collide with the material's vibrating atoms. My textbook says that when we cool the material till the point where the atoms reach a state of rest, the electrons will not collide with the atoms **at all** , and we can achieve **zero** resistance. But how is it possible that we can have absolutely zero collision? The analogy I have always used is to think about walking inside a train. When people are moving around inside the train, I can't move freely as I'll bump into them easily (high resistance). But even when no one is moving, I still can't move totally 100% freely, simply because their very _presence_ is blocking me. Doesn't the _presence_ of the atoms themselves hinder the flow of current, even when the atoms are not vibrating?


Lowest Frequency De-excitation problem

Before the formation of the molecule it is possible for the muonic hydrogen atom to be formed in an excited state and for radiation to be observed from a single transition to the ground state. Calculate the lowest frequency of the radiation that could be observed in this process and estimate the momentum of the recoiling atom. Muonic Hydrogen is 207 heavier than normal Hydrogen To find the highest frequency I would use E=hf and solve for E_2 - E_1=hf , because this is the biggest gap. I don't even have a clue on how to start this question


What determines bubble locations in boiling water?

Something a little different to our usual fare. I was boiling a pan of water for cookery the other day, and got to wondering what caused the location of the bubble streams from the bottom of the pan. The stream of bubbles seems to come from the same spot of the pan, is this a quirk of the fluid flow? Imperfections in the pan? These streams of bubbles appear to continue for several minutes until the water fully boils and it becomes too turbulent to follow anything. Any thoughts?


Why do earphone wires always get tangled up in pocket?

What is the reason? Is it caused by their narrow shape, the soft material, walking vibration or something else?


How does the photon field operator change are understanding of the electromagnetic stress-energy tensor?

Given the following photon field operators \begin{align} \mathbf{A}(\mathbf{r}) &=& \sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}} \left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} + \bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\\ \mathbf{E}(\mathbf{r}) &=& i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar\omega}{2 V\epsilon_0}} \left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} - \bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\\ \mathbf{B}(\mathbf{r}) &=& i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}} \left((\mathbf{k}\times\mathbf{e}^{(\mu)}) a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} - (\mathbf{k}\times\bar{\mathbf{e}}^{(\mu)}) {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right), \\\ \end{align} how do these photon field operators change our understanding of the classical electromagnetic stress-energy tensor \begin{align} T^{\mu\nu} =\begin{bmatrix} \frac{1}{8\pi}(E^2+B^2) && S_x/c && S_y/c && S_z/c \\\ S_x/c && -\sigma_{xx} && -\sigma_{xy} && -\sigma_{xz} \\\ S_y/c && -\sigma_{yx} && -\sigma_{yy} && -\sigma_{yz} \\\ S_z/c && -\sigma_{zx} && -\sigma_{zy} && -\sigma_{zz} \end{bmatrix} ? \end{align}


How is foliation of manifolds' theory useful in General Relativity?

I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying Holomorphic foliations on projective manifolds. I just came across some texts which related the more general Foliation theory to Gen. Rel. Is there research going on that mixes both areas? Can someone direct me to introductory papers and textbooks?


Movement of compressed air inside a rolling tire?

Does it stay still wrt the axis? Does it rotate along with the tire? Is it drawn away from the center of rotation?


Why one-dimensional strings, but not higher-dimensional shells/membranes?

One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this 'generalization', it becomes obvious how to write down the action, not just for a string, but for manifolds of higher dimension as well. In fact, Becker-Becker-Schwarz (the main source I happen to be learning from) actually do this. But (as far as I have read), they merely write down the action and do nothing further with it. My question is: what happens when we proceed along the same lines as string theory, but when replacing a string with a 2-manifold, the simplest example of which would be the 2-sphere, a "shell/membrane"? Assuming 3 spatial dimensions, this is the highest dimensional manifold we can consider (because we don't allow for non-compact manifolds). Furthermore, there is only one compact manifold of dimension 1; however, there are infinitely many compact manifolds of dimension 2, which could potentially make the theory much richer (and probably much more difficult). For example, in string theory, we are stuck with $S^1$ (if you insist upon having no boundaries), but if you allow for 2 dimensions, we could consider the sphere, the torus, etc. Because it seems that this path is not ever presented (in fact, I've never even heard of it), I would presume that something goes wrong. So then, what exactly does go wrong?


How to choose a solution from all possible solutions of general relativity

So there are so many solutions for general relativity - then how does one "choose" the solution that is right one? By checking with observation? (though I also know that it is currently unknown which one is the correct solution.)


Scaling of quantum error correction

I'm having a question regarding quantum error correction. Using a large number of imperfect (but already very good) quantum gates, it is in theory possible to build an equivalent, error-corrected gate. What I don't understand, however, is how it precisely scales when I want to do computation using algorithms with a large input space. To give a precise question: Let's say I can create many individual CNOT gates with an individual probability of success of $\eta=99.99\%$. How many of them do I need to implement Shor's algorithm to factor a $1024$ bits integer with an overall probability of success of $50\%.$ ? Once I reach the point where I can build a single error-corrected gate between two qubits, have I won the fight against decoherence or will it just be harder and harder to correct the errors as the input space scales up ? Thanks.


Heat transfer in infrared

Why does the most of the heat transferred on Earth come from infrared part of electromagnetic spectrum?


Time dilation factor for the circular orbit at 3/2 Schwarzschild radius

What would the time dilation factor be if a massive (as in rest mass>0) point particle orbiting a Schwarzschild black hole in the photon sphere? If I understand correctly, this is the only possible orbit for photons, but it is also the _closest_ possible orbit for a massive particle. So this is the same thing as asking what the maximum physically possible time dilation for a circular orbit is. That point is 3/2 times the Schwarzschild radius. Any closer than that, and no free-fall path goes to infinity or completes a full orbit. It's also unstable. However, an exactly timed maneuver could put a particle into orbit there for a large number of orbits, so I don't think the instability affects the meaningfulness of the question. One could even realistically transition from r=infinity to the outer edge of this orbit, complete several orbits, and then escape back to r=infinity. Reason for asking: a simplistic application of general relativity circular orbit time dilation tells me that the factor is infinity. In other words, time doesn't pass for a particle in this orbit. I can't even begin to rationalize that. How could the universe be frozen still for such an observer? I don't think that makes any sense.


How do you stop wobbling in a carbon nano tube elevator to space?

If there was a carbon nanotube elevator that went to space how would you keep it from wobbling? As I picture it I see the structure being very strong, but would that prevent wobbling? It would be like a tall, thin building?


What is difference between Acceleration due to gravity and Gravitational field intensity near the Earth Surface?

I have read in so many standard books that Gravitational field intensity near the earth surface and acceleration due to gravity near the earth surface are Two Different physical quantities with same Direction, Magnitude and Units. So if All the units magnitudes and directions are same and also both mean the acceleration produced in a point mass near the earth surface What is difference between them?


What is the amount of EM energy from the sun reaching the earth as a function of wavelength?

The question stems from a desire to know if more light from the sun in the infrared wavelengths has more energy than light at say yellow or violet wavelengths - at the Earth's surface. I'm guessing a simple 2-D graph would illustrate the answer well.


How to prove R value for insulation?

I ran some tests on different types of insulation. I heated water, and placed it into an insulated box. The temperature was recorded every minute, for the water, as well as the air. The insulation which is the best (high R value) will have the temperature drop the least. I have calculated the heat added using: $ Q=mc\Delta T $ Is this right, or should I have down something different? Can the laws of Thermodynamics be used?


What truly is mass, and is there a direct way to measure it?

We know a mass of an object of one kilogram as an object that weighs W = mg = 9.8 N and we reference it to that, (when it should as a fundamental parameter describe weight not the opposite). But if we were to describe mass to an alien civilization on an alien planet we are exchanging knowledge with, by sending them a one kilogram object, according to their gravity they will measure it differently. Also their star could be curving space time in such a way, or their velocity according to SR will cause them to perceive mass on our planet -if they observe from away- differently. If we asked a crew on a space ship moving at a speed close to the speed of light wrt us, or moving in a gravitational field they don't know about, to measure the mass of our planet, they will get different results. On the same principle, we could be measuring the mass of far celestial objects like planets, differently. I perceive space time as full of curves and irregularities. We know the about some of these and we don't about others. I mean we know about the earth's gravity, the sun's but we ignore the effect of the galaxy and the cluster, dark matter, and who knows what else. Besides we change our position all the time with relation to these external factors that bend spacetime. So our measurements to an outside neutral observer, say these aliens could change. I've found definitions of mass like "the quantity of matter in an object" but that seems like the good old: mass = volume x density, but mass is the more fundamental quantity than either of them, with which theses parameters should be described by mass not the opposite, not to mention how relative these other quantities are, considering SR and GR. Or "The resistance of an object to acceleration" but again you have to describe how fast that object is moving and what spacetime it's in according to SR and GR. So what makes us so confident that mass is such a universal value, when we built everything on a concept referenced by our own gravity, and maybe our own reference frame? How do we describe mass to the aliens, who don't know about our (g)? How do we measure the mass of celestial objects say planets, by units like kilograms, and pounds, while they are not subject to the earth's gravity, (I'm well aware of the difference between weight and mass). I mean that's what they will weigh -ignoring their own gravity- if they were in our atmosphere on a huge scale? So what's the method? _**Bottom line_** Is there a direct way to measure mass like we do with other fundamental values like length and temperature, _other than_ using a scale or equations, i.e not depending on other physical parameters to describe it? \--Forgive the length..


Does a cross really have a smaller area moment of inertia than a square of the same area?

I need to calculate the area moment of inertia of a cross for a homework assignment. The cross is a symmetric plus sign:                   _        | |     |==   ==|        | |         -      where the length/width of each arm is 1" (so the total area is 5 square in.). My understanding is that I can just separate the cross into three sections (left, middle, right), so the area moment of inertia would be $$ I = \sum \frac{bh^3}{12} = \frac{1\times 1^3 + 3\times 1^3 + 1\times 1^3}{12} = \frac{5}{12}$$ This is surprising to me, because a 2.2" x 2.2" square would have the same area but a much larger moment of inertia: $$ I = \frac{bh^3}{12} = \frac{(\sqrt{5})^4}{12} = 2.08$$ Since the arms of the cross span 3 inches, I would expect there to be more material further from the center, whereas the square only has a small amount of area in the corners that the cross doesn't have. Are these formulas correct or am I doing something wrong? Thanks in advance.


How to calculate Rest Mass practically with Standard Model?

With relativistic physics, we can apply force to see resistance against acceleration. It'd give us relativistic mass and we have well established formula to get to the Rest Mass as long as we know the velocity. In Standard Model, we have Higgs Mechanism to assign something Rest Mass. I was curious how to apply that practically.


Dark Energy Explanation Found

It's my sincere request to this Physics forum that if i am wrong (which has the least possibility atleast from my side) just reply "wrong" so that i can move on futher. I know personal theories are not allowed but i am not finding any place to get it checked. thanks DARK ENERGY The theory described below does not contain any self made false assumptions about anything but uses only one, most logical assumption about the potential energy of curved space. Let a particle of unit mass curve space according to general relativity.Now, if we suddenly evaporate it (i.e. convert it's mass into radiation), the curvature of space associated with it begins to die out.But what about the potential energy associated with the curved space after the mass is lost? We can better understand this with this example. We stretch a rubberband by applying a constant force (correspond to mg).Now, if we remove this force, the stretched rubberband tends to come to it's mean position.But, the potential energy due to initially stretched rubberband makes it go further from it's mean position making it vibrate along it's mean position. The same happens in space. Bodies cause the curvature of space which induces potential energy in space. So, when a body loses it's mass,the curvature associated with that mass does not just dies out but instead the space starts vibrating along it's mean shape i.e. the potential energy of the curved space associated with that mass makes the space vibrate along it's mean shape producing gravity - anti-gravity fields. But , if there is another body of mass m near it, the magnitude of anti-gravitational field is more in each oscilation as the other body tends to curve space (apply force i.e. mg) along it's direction. This is why the galaxies are moving away from each other.As the stars present in them lose their mass,anti-gravitational fields are produced along all directions making every galaxy move apart from each other. Therefore, the space is not expanding but instead the already curved space produces this effect as it starts vibrating along all directions when a body loses it's mass.


Qubit in Type 1.5 superconductor?

I'm interested in Type 1.5 superconductors, first proposed by Egor Babaev in 2002 and found in the laboratory in 2009 (magnesium dibromide). Such conductors favor small bundles of vortices. The most stable number of vortices in the bundle varies depending on conditions but is typically from 2 to 12. The vortices have fractional quanta. The bundle is sheathed by larger vortex. Lets suppose that we have a bundle of three. Would each vortex be an anyon, and the bundle be a qubit?


Charge inside conductor

I know that the E field inside a conductor is zero. What happens if I put a source of charge inside the conductor? Say the conductor was spherical centered on the origin and there exists a charge at the origin. Then by Gauss's Law, there exists a non zero field inside the conductor. Or is it the case that even though this charge is within the sphere, it will redistribute evenly over the surface and hence the E field inside is zero? Many thanks


Rocket towing an object

I'm not sure if my question makes sense, but I don't know where to start. A crate of mass $m_2$ is connected by a massless rope of constant length $l$ to a rocket of mass $m_1$. We take into account gravity (the crate is initially hanging directly underneath), but the rocket has no vertical acceleration since it's acted upon by an upward force of $(m_1+m_2)*g$. We neglect air drag and treat the rocket and the crate it is towing as blocks of the same size. If a horizontal force $F$ acts upon the rocket, what happens to the crate? It's probably going to lag behind at a certain angle, but how can we find that angle? Is the situation different if there's no gravity? Edit: This link helped me answer my question: http://cnx.org/content/m14061/latest/?collection=col10322/1.83


Would a Nuclear bomb have any effect on the sun that could damage the earth?

I was asked this recently and wasn't sure. The power of the sun is much more powerful than that of a Nuclear explosion but the question was what is the worst that could happen, we are assuming a situation where the nuclear bomb wasn't itself immediately destroyed by proximity of the sun for whatever reason. So, for example, if it was near the surface, could that cause a massive sun spot or similar which could disrupt electronics b.t.w. my background isn't a physics one :-) thanks


Anodized aluminium reflectivity at 10.6 um

I wonder how I can model anodized aluminium reflectivity in ray tracing-based optical simulation. I know that the parts my company is using are anodized to get covered with a ~20 um (as specified) thick corundum layer. I would like to be able to properly model angular dependence of such a surface reflectivity. I am using Zemax, but please do not limit yourself to it. Any hint is welcome. The two options I can think of are 1) Create a model coating in Zemax by specifying reflectivity vs. angle. Can be done if I know the values. I could not find them so far. Could be measured as well but that's a major effort which I'd like to spare. 2) Specify a thin layer of corundum on top of a bulk aluminium part in Zemax. However I could not find the dispersion data for Al2O3 at 10.6 um and I cannot find anything in Zemax' material catalogues. Besides, I assume such a layer of 20 um thickness would act as a thin film resulting in AR or HR properties. But I don't think the anodization process is precise enough in terms of layer thickness so that I can rely on this side effect. Any help or comment is highly appreciated.


What are the arguments towards the Life-and-Entropy relation?

I've heard it from a few people, and I've seen it popup here in the site a couple of times. There seems to be speculation (and studies?) towards this idea, and this is what I've picked up so far:   * Living organisms are low entropy systems.   * The second law of thermodynamics is somehow strongly connected to the origin of life itself. This is a very interesting idea, unfortunately I've never seen anyone elaborate beyond that. I'd like to understand it a little better. Specifically:   * Why are the above statements true (neither is immediately obvious to me)?   * And what are the other arguments in favor of this theory? I'll accept both answers for and against this point of view, as long as it reasonably justifies _why_ the above statements are correct/false. _Note: I believe the thermodynamic definition of entropy is the one commonly used with these arguments. And I'm not sure what definition they use for life. TheWikipedia page is pretty vague on this._


Why cant Electrostatic field lines form closed loops?

My physics textbook says "Electrostatic field lines do not form closed loops. This is a consequence of the conservative nature of electric field." But I cant quite understand. Can anyone elaborate? PS:- This question is answered. Please refer both the accepted answer and also the answer and the comments in Robin Ekman's answer for a complete understanding.


Pseudo scalar mass and Pure scalar mass

Since the only difference between pseudo scalar and a scalar term is just a change of sign under a parity inversion, is it possible that both of them be present in the same field and interact? For example let's say that the vector gauge field is copuling with some other field. And suppose I have a pseudoscalar mass term like $m\varepsilon^{\mu\nu\rho\sigma}\phi_{\mu\nu}\phi_{\rho\sigma}$ and pure scalar mass term $\frac{M}{16} \phi_{\mu\nu}\phi^{\mu\nu} $. In that case is it possible that both of them present in the same time in the field? Or there is any condition that they have to satisfy?


What is Conductance?

What is conductance? I am being said that conductance is inverse of resistance?I do know that resistance is the opposition to the flow of current but do not get conductance? > I do know that:it is ease of flow current but my brain doesn't accept > it,don't know why Ain't it so important!Couldn't even find it on wikipedia.


Is there a point interaction model of the electron? Is there a point interaction model of the electron? I imagine something like $\propto(\bar \psi\psi)^2$ (edited). Is such a thing in use? Since I expect some _but that's not renormalizable!_ -arguments: I think there are theories with scalar photons. Can you integrate out the massless photon completely? And that raises the question more generally: Does going from a renormalizable to an effective description theory change that property? * * * Edit: I just found the Four-fermion interactions. I guess that leaves the question if, or better how, these can result from swallowing the photon. Here is the question in that light: > Coming from QED, is there a point interaction model of the electron? PS: I like the last line of the article. Who would have seen that one coming :P


As an undergrad I was often confused over people's bafflement with Schodinger's cat thought experiment. It seemed obvious to me that the term "observation" referred to the Geiger counter, not the person opening the box. Over time, I have come to realize that the Copenhagen interpretation actually _is_ ambiguous and that "observer" cannot be so easily defined. Nonetheless, an objective collapse theory (which is what I was unknowingly assuming) still seems to me the simplest explanation of wave collapse phenomena. I have read some of the objections cited in the Wikipedia article linked above, but it is still unclear to me why most physicists adopt the Copenhagen interpretation and reject objective collapse. For example, in this question on hidden observers, there was some discussion about the mechanism of wave collapse. It was suggested that perhaps the gravitational pull of a hidden observer would collapse the wave function. In response, it was pointed out that the gravitational pull would be negligible at the scales involved. Okay, then imagine the following: A hermetically sealed (i.e. isolated) box is balanced on a fulcrum. Inside the box is a radioactive isotope, a Geiger counter, and a trigger mechanism connected to a spring loaded with a mass on one side of the box. If the Geiger counter detects a decay, the trigger releases the spring and the mass shifts to the other side of the box. The shift in mass would, under observable conditions tilt the box on the fulcrum. According to the interpretation of Schrodinger's cat that I often hear (the cat is in a superposition) it seems that the box should slowly tilt over as the wave function of the system evolves with the half-life of the isotope. I can't imagine that anyone thinks this is a realistic expectation. I can see that people might object and say "But the contents of the box are interacting gravitationally with the outside system and observer so it is not really isolated!" Well, what of it? The same is true of the cat even if the interaction is less dramatic. The question, then, is: How isolated must a system be for it's wave function to be considered not collapsed?


Massless spin 1/2 particle

Could a massless spin 1/2 particle, or more generally massless half-integer spin particles exist? Does it make sense to say that they could be described for example by the Dirac equation by forgetting the mass term ?


Why do wheels appear to revolve opposite to the direction they are rotating?

When viewing cars that are driving along side of us, sometimes their wheels appear to be turning backwards even though they are traveling in the same direction as our car. Why do they look that way?


Special Relativity Second Postulate

That the speed of light is constant for all inertial frames is the second postulate of special relativity but this does not means that nothing can travel faster than light.   * so is it possible the point that nothing can travel faster than light was wrong?


Minkowski metric -- why does it follow from the constancy of the speed of light

In all the sources I’ve been able to find, the Minkowski metric appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else. Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that.


Understanding the argument that local U(1) leads to coupling of EM and matter

I'm trying to better understand the argument that U(1) local gauge invariance implies a coupling of EM and Dirac fields. I understand the math, but I'm not sure about the chain of logic. You start with $\mathcal{L}_\mathrm{Dirac} = \bar\psi ( i\hbar \partial_\mu - m)\psi $ and the following transformations: $$ \psi \rightarrow \psi' = e^{i\theta}\psi \qquad \left( \bar\psi \rightarrow \bar\psi' = \bar\psi e^{-i\theta} \right) \qquad (1)$$ $$ A_\mu \rightarrow A'_\mu + \partial_\mu \chi \qquad (2)$$ You often by applying (1) to $\mathcal{L}_\mathrm{Dirac}$, and note that to keep the form invariant, you have to change the partial derivative to the covariant derivative: $\partial_\mu \rightsquigarrow D_\mu = \partial_\mu - \frac{i}{e} A_\mu$. You do the calculation again, and then find $$\mathcal{L}' = \bar \psi\left( i \hbar \gamma^\mu (D_\mu - i \partial_\mu\theta + i\partial_\mu\chi) - m\right)\psi$$ Thus, to make $\mathcal{L}$ invariant under the transformations, $\theta = \chi$ (up to factors of $e$ and $\hbar$ that I missed). The phase of the electron wavefunction must in a way absorb the change of the four-potential given by the gauge transformation, or vice versa. I understand (2) comes from classical gauge freedom in electrodynamics. You are free to add a constant potential offset $\Phi_0$ to $\Phi$, or to add a divergence of a scalar field $\vec\nabla \chi$ to $\vec A$. Since $A_\mu = (\Phi, \vec A)$, it follows that (2) leaves the physics invariant. But where does (1) come from? Is this just the regular freedom of phase change you have with any wavefunction? If so, why doesn't the U(1) symmetry show up everywhere? Or does this come from _postulating_ a U(1) symmetry for $\psi$? If so, how is this motivated (other than "oh, it gives the right result")? For example, if I think (2) is better motivated, **is it possible to start with only (2) and possibly the covariant derivative, and get to (1)? In general, what assumptions can I put into the argument, and what can I get out of it?**


Why dimensionality of the Electric Charge varies with the spacetime dimensions?

The point is: We can find via dimensional analysis that the electric charge dimensionality varies with the dimension of space-time. $$[\text{charge}] = eV^{(3-D)/2}$$(You can see below the way I did it) The question is: **Why it occur? What is the meaning of this?** * * * I'm using the Heaviside-Lorentz Units with Natural Units ($\hbar = c = 1$) so that everything has dimensions of energy (eV). $$[x] = [t] = eV^{-1}$$ $$[p] = [E] = eV$$ Using the action from Maxwell Theory $$ S = - \frac{1}{4} \int d^{1+D} x F^{\mu \nu} F_{\mu \nu}$$ And making a dimensional analysis, $$ [S] = [x]^{1+D} [F^{\mu \nu}]^2$$ $$ 1 = eV^{-D-1} [F^{\mu \nu}]^2$$ $$ [F^{\mu \nu}] = eV^{(D+1)/2}$$ Then, using the inomogeneous equation $$ \partial_\nu F^{\mu \nu} = J^\mu$$ Ang making an dimensional analysis $$ [\partial_\nu] [F^{\mu \nu}] = [J^\mu]$$ $$ eV eV^{(D+1)/2} = [J^\mu]$$ $$ [J^\mu] = eV^{(D+3)/2}$$ Now lets look the charg dimension. The relation of the charge with the (1+D) density current is $$[J^\mu] = \frac{[\text{charge}]}{[x]^D}$$ So, $$eV^{(D+3)/2} = eV^{D} [\text{charge}]$$ $$[\text{charge}] = eV^{(3-D)/2}$$ We get that only in 1+3 the charge is dimensionless. If we are at 1+1, and at 1+5 charge is just like time. * * * **Comment 1** At another place I was told that it (dimension of charge) have relation with the renormalizability of QED. And that the fact that charge is dimensionless only in 1+3 is related with Maxwell Equations being conformally invariant at 1+3.


How big would the impact of a nuclear warhead be?

I am writing a post-apocalyptic science-fiction story which includes the detonation of a nuclear warhead. In this context, What scientific information is available about the impact of a modern day nuclear warhead?


How are fictitious forces related to my feeling?

This question arises when I am studying fictitious forces in an undergrad introduction to physics course. Suppose I am standing in an elevator with an acceleration $a$ directed upward. From the ground viewpoint, with $N$ being the normal force on me, we have $N - mg = ma $ and so $N = mg + ma$. From my viewpoint, I feel two force: an upward force of $m(g+a)$ and a total downward force (gravity + fictitious force) of $m(g+a)$ too. If I were standing somewhere on a planet with gravitational acceleration $G=g+a$, then I feel these two forces too. That's why in the elevator I have the feeling of being on the planet with $G = g+a$. OK, so far so good. Now let's say I'm freefalling. From my viewpoint, I feel two force: an upward fictitious force of $mg$ and a downward gravitational force $mg$. The situation is pretty the same as if I were standing on the ground. But clearly I wouldn't feel that normal when freefalling; in fact I feel weightless. Why? Thanks for your help.


Sensitivity of eye

In which color human eye is most sensitive?