title: What is off-diagonal long range order in superfluid?

text: From Wikipedia: > _[...]Off-diagonal long-range order (ODLRO) [...] exists whenever there is a > macroscopically large factored component (eigenvalue) in a reduced density > matrix of any order._ How to understand the ODLRO in superfluidity?


title: Rate of effusion in kinetic molecular theory?

text: According to the kinetic molecular theory obeying Maxwell-Boltzmann distribution of speeds, the rate of effusion through a pinhole of area $A$ is   $$R=\frac{PA}{\sqrt{2\pi M R T}}$$ where $M$ is the molecular weight, $R$ is the gas constant and $T$ the absolute temperature. To derive this, I consider the collision frequency on any small area ($A$):-   using $$v_{avg}=\sqrt{\frac{8RT}{\pi M}}$$ I get the result, that the atoms in volume $v_{avg}A$ can hit the area. The number of particles in this volume is $nv_{avg}A$ ($n$=number density). But the derivation includes a factor of $\frac14$ before this term to find the actual number of atoms in this volume hitting the wall. I want to know how that factor of $\frac14$ came into picture to make the collision frequency per unit area as $\frac14nv_{avg}=\frac{P}{\sqrt{2\pi M R T}}$. I know the origins of a factor of $\frac12$ before the pressure term while calculating it, considering change in momentum of a molecule on collision with the wall, which is to account for the fact that $<v_x^2>$ includes both the $v_x$ terms, going towards and going away from the wall(positive and negative directions) but the ones colliding are just the half of these $\frac12<v_x^2>$(going in any one direction).


title: How is momentum conserved when a magnet attracts a metal?

text: Suppose your have any magnetic object and no external force acts upon it, and the object comes near a metal which causes an impulse (think that will happen). However, the magnetic force is internal to the object, and momentum should be conserved, so where have I gotten it wrong?


title: Your favorite Physics/Astrophysics blogs?

text: What are the Physics/Astrophysics blogs you regularly read? I'm looking to beef up my RSS feeds to catch up with during my long commutes. I'd like to discover lesser-known gems (e.g. not well known blogs such as Cosmic Variance), possibly that are updated regularly and recently. Since this is a list, give one entry per answer please. Feel free to post more than one answer.


title: Can Deuterium undergo a nuclear fusion without the presence of Tritium?

text: I've been reading about fusion fuels for a while now, and I understand that in Lithium-Deuterium fuel, the neutrons from the fission reaction bombard the Lithium to produce Tritium and the D-T reaction occurs and we get the energy. So what about Deuterium-Deuterium fusion ? Is it possible without the presence of Tritium at all ? If so, then does it release more or less energy than D-T fusion ?


title: How does fire create thrust in rocket?

text: All big rockets are burning either gas or fluid to create thrust. While this is so, I have filled up a plastic bottle with air at high pressure, and it can go long distances by blowing the pressurised air at reverse direction. If my bottle can do this without using any fire, why don't rockets just use air? How is the effect of combustion in thrust?


title: Why are "heavier" particles harder to detect than "lighter" ones?

text: Something I have read multiple times that I've never intuitively understood is that "heavier" particles are harder to detect than "lighter" ones... For example, I quote from Stephen Hawking's "The Grand Design" in relation to supersymmetry: > But various calculations that physicists have performed indicate that the > partner particles corresponding to the particles we observe ought to be a > thousand times as massive as a proton, if not even heavier. That is too > heavy for such particles to have been seen in any experiments to date, but > there is hope that such particles will eventually be created in the Large > Hadron Collider in Geneva. Could someone please explain, in simple terms, why heavy particles are harder to detect? Intuitively (to me, a non-physicist) it would seem that it should be the other way around, because a particle with more mass should interact more strongly with other matter.


title: Capacitor in series?

text: Say you have two charged capacitors in series. Zoom in on one capacitor. For this specific capacitor, the charge on the two plates will be the same in magnitude, according to my textbook. My teacher said that the charge on the two plates won't be the same if the gap between the the plates is large. In fact, the charge on each plate in a capacitor is _never_ the same. They're only approximated to be the same if the gap is small. Why is the charge not the same on both plates on a capacitor in series if the gap is large? ![enter image description here](http://i.stack.imgur.com/u9x6F.png)


title: How come gas molecules don't settle down?

text: If the earth's gravity exerts a net downward gravitational force on all air molecules, how come the molecules don't eventually lose their momentum and all settle down? How is the atmosphere is still miles thick after billions of years?


title: friction on a solid body due to another solid body

text: Suppose we have a block A and is moving on a rough horizontal surface with a velocity 60 m/s and another block B of same mass moving with velocity 5 m/s on same horizontal surface and suppose they move same distance . The block A due to friction gets more tear on its bottom surface than block B so can we conclude that friction is varying with velocity? If not then why it happens ? does it happens or not ?? I know friction depends on normal reaction in solids.


title: How long for a frictionless top to fall over?

text: We've previously discussed why it is that spinning tops do not fall over, see: Why don't spinning tops fall over? However, as the highest rated answer notes, the angular momentum of the spinning top is "quite high". On the other hand, I know that if the angular momentum of the top is zero it will, in fact, fall over. This suggests that increasing the angular momentum cannot stop the top from falling over, but instead can only increase the time (but this turns out not to be the case). So suppose a top with mass concentrated in a spherical body M of radius S with center situated a distance R to a point a horizontal table at an angle $\theta$ while rotating at angular speed $\omega$. How long does it take for the top to fall over? Some simplifying assumptions: (1) Assume the point of the top is fixed on the table. (2) Assume that the time at which the top falls over is when $\theta = 0$. ![Drawing](http://i.stack.imgur.com/Ch8dP.jpg)


As I read in The Road to Reality by Roger Penrose, the Joukowsky transform $$w(z) = \frac12\left( z + \frac1z \right)$$ after Nikolai Zhukovsky (transcribed in several versions from Никола́й Его́рович Жуко́вский) can be used to **calculate the flow** of a non-viscious, incompressible and irrotational flow **around an airfoil**. This can be done since the solution of a potential flow around a cylinder is known in full analyticity and the given transform conformally maps a circle on an airfoil-like geometry. I don't understand this argumentation, so: > ### How is the Joukowsky Transform used to calculate the Flow of an Airfoil? An **example** of such a transformation is given in the mentioned Wikipedia article: ![RotatingDisc](http://upload.wikimedia.org/wikipedia/commons/d/d5/Jouktrans.png) Thank you in advance.


title: What is the wavefunction of the observer himself?

text: I am aware about different interpretations of quantum mechanics out there but would mostly like to see an answer from the perspective of Copenhagen interpretation (or relative quantum mechanics if you wish). Let an observer being a man with brain consisting of molecules and atoms. According the basic principles of quantum mechanics each of these particles has a wave function. The question is: is there a combined wave function of all those particles which constitute the observer? Can such wavefunction be (in theory) determined by the observer himself? Since the observer cannot isolate himself from his own brain, this would mean that the wave function, at least the part which determines his thoughts is permanently collapsed (i.e. the measurement happens instantly once the state changes). Does this "permanently collapsed" wave function imply special physical properties of the observer's own brain? Does knowing his own thoughts constitute a measurement? Which moment should be counted as the moment of the collapse of wave function when making measurements on own brain? Pretend an observer tries to measure the wave function of his own brain by a means of an X-ray apparatus or other machinery and read his own thoughts. Would not his own knowledge of that measurement or its results invalidate the results thus making the whole measurement impossible? Does the behavior of particles which constitute the observer's brain differ statistically (acoording his measurements) from the behavior of particles which constitute the brains of other people? Is there a connection with quantum immortality here?


title: What determines the frequency output of a Barkhausen-Kurz tube?

text: I looked at the wikipedia entry and it doesn't give a very detailed or thorough description of the principles of operation of Barkhausen-Kurz oscillator triodes. But I do wonder if it is physically possible to design such a self-oscillating thermionic valve(DC input) that can oscillate in the audio frequency range(20Hz - 20 KHz).


title: A bumpy ride on a bike?

text: While riding a bike at a higher velocity, I go over the speed breakers (Hump) I get jumped over the bike for a certain height. How do I calculate this height of jump with respect to the seat of the bike?


title: Hot Big Bang vs. Big Bang

text: This should hopefully be a quick one. Is there any difference between the Big Bang Theory and the Hot Big Bang Theory? Around Cambridge I hear everyone using "Hot Big Bang Theory", for example ther first line here of the pages of the Centre for Theoretical Cosmology. In my undergraduate university it was referred to as the "Big Bang Theory". I have tried searching online, and there does not seem to be much more mention of a "Hot Big Bang Theory" theory anywhere else on the internet. For example, wikipedia returns no results for the term. I am almost certain that they are one and the same theory, and that this is simply a case of nomenclature, with Cambridge trying to be different as usual, but I just want to make sure. It's terribely unnecessiarly confusing! So can someone confirm that there is no difference between these two theories for me please?


title: Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

text: For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ **isn't** a square matrix. $x_i$ is standard Cartesian coordinates, and $x^\prime_j$ is a point in the new system. I have to mention that the new system's axes **are not linearly independent**. So the last relation can be written as $$\left(\matrix{x_0^\prime\\\x_1^\prime\\\x_2^\prime\\\x_3^\prime}\right)=\left( \matrix{T_{11} & T_{12} & T_{13} \\\ T_{21} & T_{22} & T_{23} \\\ T_{31} & T_{32} & T_{33} \\\ T_{41} & T_{42} & T_{43} } \right)\cdot \left(\matrix{x\\\y\\\z}\right)$$ The matrix $T_{ij}$ is well defined. What I need is a rotation operator that will transform a point in the primed system, as the standard rotation operator does. So say I have the standard rotation matrix in Cartesian coordinates around the z-axis: $$R_{ij}= \left( \matrix{\cos{\theta}&-\sin{\theta}&0\\\ \sin{\theta}&\cos{\theta}&0\\\0&0&1} \right)$$ So to rotate a point in Cartesian coordinates, we use the standard operator formula: $$P^\prime_i=R_{ij}P_j$$ where $P_j$ is the point before rotation, and $P^\prime_i$ is the point after rotation. How can I write this rotation formula for a point in the new coordinates system that uses 4 points? How will the rotation matrix look like? I expect a rotation matrix that is $4\times4$, but I don't know how to derive it. Please help in that.


title: Energy transfer in a coupled pendulum

text: If you take a look at this video you will see what kind of a coupled pendulum I'm talking about. So I made a similar one in my high school's physics lab, using light metal bobs(much lighter than the Easter eggs he's used in the video) and thin copper wire. **Some Data About My Experiment:** My pendulum did work the same way, i.e. one would stop as the other would swing with max amplitude, which continues periodically without much damping. But the frequency of the combined periodic motion in my case was much slower than that in the video. His pendulum stops after a few oscillations while mine took 10-15 oscillations to completely transfer the energy(I think this is because I used a lighter weight) **My Question:** > In the video the person swings one of the pendulums _perpendicular_ to the > wire. I tried swinging it _parallel_ to the wire and nothing happened. The > pendulum I initially provided energy to kept swinging with the same > amplitude for a long time.   >  **Why doesn't energy transfer take place when I swing the one of the > pendulums _parallel_ to the above wire?** It should because the energy is > being transferred by waves through the string. This motion should also > produce waves. **Possible Reasons:**   * Maybe the energy transfer takes a lot of time, hours perhaps? I hardly observed it for ten minutes.   * There's too much damping, although in which case the first pendulum should stop quickly too.   * The energy transfer is happening in a way not compatible with the direction the bob is swinging in. I really can't seem to explain any of the above reasons. Maybe it's an entirely different reason altogether. Any help would be appreciated!


title: It appears that stationary states aren't so stationary

text: $$ \DeclareMathOperator{\dif}{d \\!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \newcommand{\bracket}[3]{\langle{#1}|{#2}|{#3}\rangle} \newcommand{\linop}[1]{\hat{#1}} \newcommand{\dpd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dod}[2]{\frac{\dif{#1}}{\dif{#2}}} $$ Using the Schrödinger equation and the definition of the expectation value it can be shown that the time dependence of the expectation value of an observable $A$ for a system in arbitrary state $\ket{\Psi(t)}$ is given by $$ \dod{\langle A \rangle}{t} = \frac{\ramuno}{\hbar} \langle [\linop{H}, \linop{A}] \rangle + \big\langle \dpd{\linop{A}}{t} \big\rangle \, , \tag{1} $$ and this equation shows that, in general, if an operator $\linop{A}$ commutes with the Hamiltonian operator $\linop{H}$ and does not have an explicit time dependence, then the expectation value of the corresponding observable $A$ is time independent. For stationary states $\ket{\Psi(t)} = \exponent^{-\ramuno E_{k} t / \hbar} \ket{E_{k}}$ the first term in the expression for the time dependence of the expectation value of an observable vanishes $$ \langle [\linop{H}, \linop{A}] \rangle = \bracket{ \Psi(t) }{ \linop{H} \linop{A} }{ \Psi(t) } - \bracket{ \Psi(t) }{ \linop{A} \linop{H} }{ \Psi(t) } = \bracket{ E_{k} }{ \linop{H} \linop{A} }{ E_{k} } - \bracket{ E_{k} }{ \linop{A} \linop{H} }{ E_{k} } = E_{k} \bracket{ E_{k} }{ \linop{A} }{ E_{k} } - E_{k} \bracket{ E_{k} }{ \linop{A} }{ E_{k} } = 0 \, , $$ and so the time dependence of the expectation value is simply given by $$ \dod{\langle A \rangle}{t} = \big\langle \dpd{\linop{A}}{t} \big\rangle \, . \tag{2} $$ Nevertheless, the statement like the following one > A stationary state is called stationary because the system remains in the > same state as time elapses, in every observable way. Wikipedia is found in many books. The thing that troubled me is the word _every_ , since from (2) it appears that if an operator $\linop{A}$ carries some explicit time dependence, then the expectation value of the corresponding observable $A$ changes in time. So stationary states are, in fact, not so stationary. I have the feeling that I am missing something. And our discussion with Bubble helped clarify what's bothering me. As far as I know operators in the Schrödinger picture _usually_ do not carry an explicit time dependence. Again, _usually_ , but not always. In many books (see, for instance, Griffiths, D.J., Introduction to quantum mechanics, 2nd ed.) one can find that > Operators that depend explicitly on $t$ are quite rare, so _almost always_ > $\dpd{Q}{t} = 0$. And, yet the author claims that > Every expectation value is constant in time. **I feel like there is a gap between operators being _almost always_ explicitly independent of time and _every_ expectation value being constant.**


title: Is it possible to study solid state from kittel after taking only one course in quantum mechanics?

text: I have taken only QM I, which is the 1st half of Griffiths including the chapter on identical particles, will that be enough to understand Kittel's solid state? Should one have also taken a course in statistical mechanics before studying kittel? I have seen in some universities in the states that the prerequisite of an introductory course in solid state physics sometimes are QM I, II, SM, and other times are QM I only (or with SM and QM II are corequisite)


title: The Energy involved in the work done here?

text: When a wire that has current $I$ flowing within it and its in a magnetic field, the wire experience the Lorentz force, and that force moved the wire over a certain distance $x$(no matter how small), can we state that work is done by the Lorentz force on the wire? If so... What kind of energy is transferred here? And, what potential energy was converted for this wire to move? What is the source of energy?


title: Error conversion

text: $\alpha_L = \frac{1}{a}\frac{da}{dT}$ I know error in $a$, i.e., $da$ I need to find out $d\alpha$ from data of $da$. $d\alpha_L = -\frac{1}{a^2}\frac{da}{dT}da$ Is this correct? Note: $\alpha_L$ = Linear thermal expansivity $a$ = Lattice constant


title: Orientability of spacetime

text: In many theoretical setups it is implicitly assumed that the underlying manifold (i.e. spacetime) is orientable. Then our analysis depends on this implicit assumption. For example, Stokes' theorem assumes orientability of the chain on which we integrate. However, we accept that time "always" points forwardly. My question is: doesn't a 1-directional arrow of time provides a hint that spacetime should be modeled, afterall, by a non-orientable manifold?


title: Work-Energy-Power Relations

text: > A wind-powered generator converts wind energy into electrical energy. Assume > that the generator converts a fixed fraction of the wind energy intercepted > by its blades into electrical energy. For wind speed $v$, show that the > electrical power output will be proportional to $v^3$. I tried hard but I get power output is proportional to $v^2$.


title: Can a magnetic field alone be used to move a metallic object within a contained tube?

text: Is it possible to make a metallic object move in a circular contained tube around and around in which a magnetic field is setup to cause the metallic object to move through the electric field within the tube through magnetic forces alone? If not, what other forces must be present to make the object move around and around in the circular tube?


title: Potential Step - Choosing Wavefunctions

text: http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4190016/Quantum_Mechanics_1/ch4-virtual- book.pdf On page 2 of the above pdf they describe how they select their wavefunctions. Finding the general solution is easy but they then go on to say that D= 0 because 'this is not physically meaningful' leaving $\psi=Ce^{ik_2x}$. Fine, I can see that if the wave is transmitted through the barrier it is best represented by such a travelling wave moving in the +x direction. Now, my lecturer also supports and uses the above explanation. However, there are a few university websites and a textbook (Introduction to Quantum Mechanics by Bransden) that state that the wavefunction for the section $V(x) = V_0$ cannot be represented by $\psi=Ce^{ik_2x}$ as this tends to infinity at x= infinity. Also, by the postulates of QM the wavefunction must have a real quantifiable integral which such a function is in violation of. What do I follow ? Who is right ?


title: How can area be a vector?

text: My professor told me recently that Area is a vector. A Google search gave me the following definition for a vector: > Noun: A quantity having direction as well as magnitude, esp. as determining > the position of one point in space relative to another. My question is - what is the direction of area? I can relate to the fact that velocity is a vector. The velocity of a moving motorbike for example, has a definite direction as well as a definite magnitude assuming that the bike is moving in a straight line & not accelerating. My friend gave me this explanation for the direction of Area vector. Consider a rectangular plane in space. He argued that the _orientation_ of the plane in space can only be described by considering area as a vector & not a scalar. I still wasn't convinced. Suppose the plane was placed such that its faces were perpendicular to the directions, North & South for example. Now the orientation of the plane is the same _irrespective_ whether the so called vector points to north or to the south. Further what is the direction of a sphere's area? Does considering area as a vector have any real significance? Please explain. Thanks in advance.


title: What does this summation mean in relativity?

text: Equation 1.2 of 't Hooft's Introduction to General Relativity gives the _Lorentz transformations_ : $$ (x^\mu)' = \sum\limits_{\nu = 1}^4 {L^\mu}_\nu x^\nu $$ Is this the sum of four square matrices each one multiplied by a column vector? Or is this a single square matrix multiplied by a single column vector?


title: Calculating the rainbow angle

text: I've read that the angle for viewing a Primary Rainbow is between 40-42 degrees. How would I go about showing this? How would I calculate that angle from scratch? I figured that Snell's Law and the Law of Reflection would be used for light bouncing off and through a (spherical) water droplet. But I only have a vague idea of how to begin. I would imagine that once you know how to calculate the Primary Rainbow angle, calculating the Secondary Rainbow angle would be very similar if not identical process. Is this right?


title: Origins of many-particle interactions

text: The internal potential energy of an $N$ particle system is a general function of the coordinates of the particles: $U(r_1,...,r_N)$. In some approximations and expansions - e.g. virial expansion \- it is assumed that the potential energy is a sum of pairwise interactions: $$U(r_1,...,r_N) = \sum_{ij}u(r_{ij})$$ with $r_{ij} = |r_i - r_j|$. But most of the interactions between particles I am aware of are pairwise _per se_. The only three-particle interaction I know of is a chemical reaction which needs a catalyst. > Which other specific examples of three- or $n$-particle interactions are > there? **Addendum** : I found this in the Wikipedia article on reaction mechanisms: > _In general, reaction steps involving more than three molecular entities do > not occur._


title: A good book for Quantum Cryptography

text: I am interested in Quantum Information and Cryptography in particular. I have gone through Neilson's text and Preskill's notes . Can someone suggest me some good text for Quantum Cryptography ? I thank you for your time.


title: Magnetic force at the edge?

text: When sliding a magnet over a ferromagnetic surface there is no magnetic force at all, however,there is a magnetic force only at the edge, when the magnet is being pulled of I assume the magnetic field is trying to resist. However, a fact it noticeable, the sliding/lateral magnetic force at the edge is a lot less, by a factor of half - ten times less. What could be the reasons that detains the magnitude of the lateral force? Pelase do explain in detail. Thank you.


title: Field inside a wire?

text: This answer gives a great explanation of why the field inside a wire connected to a battery must be equal at all points: Why doesn't the electric field inside a wire in a circuit fall off with distance from the battery? The answer uses the concept of surface charge buildup to show why the field has to be equal at all points and why it must be perpendicular to the wire. However, there are a lot of ways a field could exist in a wire that is not perpendicular to the wire, but would not cause surface charge buildup: For example these fields: ![enter image description here](http://i.stack.imgur.com/Or4yG.png) could exist instead of: ![enter image description here](http://i.stack.imgur.com/iQuaO.png) and there would still be no surface charge build up. So my questions are: 1) Can these "exotic" fields exist"? 2) If not, why not? 3) If these exotic field exist, how would current be defined in the wire? Since the electrons are not flowing perpendicular to the wire, would current be defined as the component of the movement of the electrons perpendicular to the wire or just the entire movement of the electrons? 4) The image below is a resistor. The "lines" represent the current direction and density ("the current density streamlines") . The "gray objects" represent wires through which a voltage difference is applied. If the electric field is _always_ parallel to the surface how can the current lines be at an angle in the first resistor? ![enter image description here](http://i.stack.imgur.com/SKO6R.png) ** ## **Please address all my questions separately. Thank you.** **


title: Tensor product of Hadamard Operators

text: The Hadamard Operator on one qubit is: \begin{align*} H = \tfrac{1}{\sqrt{2}}\left[\,\left(\color{darkgreen}{|0\rangle + |1\rangle}\right)\color{darkblue}{\langle 0|}+\left(\color{darkgreen}{|0\rangle - |1\rangle}\right)\color{darkblue}{\langle 1|}\,\right] \end{align*} Show that: \begin{align*} H^{\otimes n} = \frac{1}{\sqrt{2^n}}\sum_{x,y}(-1)^{x \cdot y}\,\left|x\rangle \langle y\right| \end{align*} I can evaluate things like $H \otimes H$ in practice, but I don't know how to get a general formula for $H^{\otimes n}$. Are there any tricks I could use?


title: Questions about closed forms and cycles

text: I read the section _closed forms and cycles_ in Arnold's _Mathematical Methods of Classical Mechanics_ (page 196-200), but the problems in this section is too difficult to solve in the way following the hints. What's more, There seems no other materials teach closed forms and cycles in the same way as this book. So, this post is meant to ask for solutions(in the way following the hints) and recommended materials. The problems are following: >   1. Let $X$ be a vector field on $M$ and $\omega$ a differential $k$-form. > We define a $(k-1)$-form $i_X\omega$ (the interior derivative of $\omega$ by > $X$) by the relation $$ (i_X \omega)(\xi_1,\ldots,\xi_{k-1}) = > \omega(X,\xi_1,\ldots,\xi_{k-1}). $$ Prove the homotopy formula $$ i_X d + > di_X=L_X. $$ > >   2. Prove the formula for differentiating a vector product on three > dimensional Euclidean space (or on a Riemannian manifold) : $$ > \mathrm{curl}[a,b] = \\{a,b\\} + a~\mathrm{div}[b] - b~\mathrm{div}[a], $$ > where $\\{a,b\\} = L_ab$ is the Poisson bracket of the vector field. > >


title: Quantum distance ladder

text: There is a lecture on how facts about the cosmos can be derived form rather simple observation. Is there an analogy for the small scale? Is there a technically simple observation that would imply, say, atoms? Anything smaller?


title: Could Gamma ray bursts be caused by matter-antimatter annihilation?

text: As far as I know a collisions of matter and antimatter leads to the complete annihilation of both, whereby 100% of the rest mass of the particles is converted into gamma rays. Could this mechanism be responsible for the Gamma Ray bursts seen by satellites? Could therefore Gamma Ray Bursts tell us something about where all the antimatter is in the universe?


title: Mathematical derivation of $N(\lambda)d\lambda$

text: We all know that in Rayleigh-Jeans law, $$N(f)df ~=~ 8\pi f^2 df/c^3.$$ How do you derive $N(\lambda)d\lambda$? I am sort of confused...


title: Light vs neutrino speed comparison in a real tunnel

text: Given current accuracy of the techniques, is it possible to identify a real, existing tunnel (stright I think) to make the direct comparison of the speed of light and of neutrinos? The hypotetical tunnel from CERN to OPERA would-be too long. If we were able to increase accuracies by two decades, then would a 7.3 Km tunnel suffice ? The 60ns difference would become around 600ps. Suppose there is such a tunnel, suppose it can be availabe to physicists. Apart from all the logistics problems of generating and detecting light and neutrinos at the two ends of such tunnel, are there other difficulties?


title: Can the Heisenberg interpretation or path integrals apply to open quantum systems?

text: It has been claimed by some people that Schrodinger's picture is more misleading compared to the Heisenberg principle or path integrals, and that we would be better off abandoning the Schrodinger picture in favor of either the Heisenberg picture or path integrals. However, when it comes to open quantum systems in a constant interaction with the environment, which isn't fully modeled, it is not at all clear how to apply the Heisenberg picture or path integrals. The Heisenberg picture requires the operators of the system to evolve in such a way in time what it becomes extremely mixed up with the environmental degrees of freedom, and the only way this can be done is to fully model the environment as a whole, or at least all of the part of the environment which ever has a chance of interacting with the system in question. Similarly, how do you even go about adapting path integrals to open systems without including the entire environment, or at least all of those parts which can ever interact with the system? Might it be the case that the Heisenberg picture and path integrals can only be strictly applied to the universe as a whole, or at the very least, causal diamonds?


title: Didactics question ("teams and times")

text: In sports it is commonplace to distinguish   * a " _team_ " (as characterized by the players who took part in a match, playing together against another team), from    * the " _score_ " (such as the final score of two teams having played a match against each other), and from    * the " _ranking_ " (e.g. the ranking of teams within a league at the end of a season).  Similarly, considering a clock it is possible to distinguish   * its " _instants_ " (or " _indications_ ", as characterized for instance by the pointing of clock hands to certain elements of the clock face), along with the fact (or idea) that a clock " _progresses_ " (considering some or all of its indications as a set in a particular order), from   * the " _duration_ " of the clock between any two of its indications, along with  comparisons of durations (in particular determining, durations between which pairs of indications are equal to each other), from   * parametrizations of the indications of a clock by real numbers (often called $t$). Use and comprehension of this entire terminology then allows for instance certain parametrizations $t$ to be qualified as monotonous wrt. the progress of a given clock (while others are not);   and among those, certain parametrizations as " _good_ " (or proportional, or affine) wrt. the duration ratios between indications of the given clock (while other monotonous parametrizations are not). My question:   Is there a didactical disadvantage in projecting much if not all of the described clock-related terminology into one and the same word: " _time_ " ?


title: Lifting frictional force and relative values for kinetic vs static friction

text: My physics textbook says that friction mainly arises due to intermolecular attraction between atoms of the objects in contact and clashes between peaks are only significant for rough surfaces. I was wondering, if friction arises due to intermolecular attraction, why doesn't it oppose lifting a body as well(considering that the atoms are attracted to each other). Why doesn't it oppose rolling? (even when there is no slippage, if atoms of the contact patch are attracted to the atoms in the ground, there should be a force opposing movement) Another related question: Is there an intuitive reason why(for most pairs of surfaces) kinetic friction is lower than static friction?


When differential equation like Schrodinger is separable in some coordinate system? What needs to satisfy the potential ?


title: Toy models of asymptotic safety?

text: Are there some toy model QFTs where the asymptotic safety scenario is realized?


title: Is it possible that black holes are also neutron stars, but so dark that we cannot see them?

text: 1. Since the concept of the singularity in a black hole leads to infinite densities, I wonder if it is really certain that black holes exist?    2. Is there a possibility that massive objects (which are believed to be black holes) are in reality dense stars not emitting light (dark neutron stars)?   3. Is there a possibility to explain the observed facts without the use of black holes?   4. One more question: According to the mainstream, are neutron stars believed to eventually collapse to a black hole, or are they stable objects?


title: Dielectric boundary

text: I am trying to determine why electric field may be confined to a certain region if there is a large difference in the permitivity for example if electric field flows through water and then reaches a water air boundary. I have also been reading about EM waves, is it possible to model electric field as a wave because if so then the transmission T and reflection R coefficients given below in terms of n which is $\propto \epsilon_{r}^{1/2} $ and so if $\epsilon_{1}>>\epsilon_{2}$ then $n_{1}>>n_{2}$ ; $\displaystyle R$ $\textstyle =$ $\displaystyle \left(\frac{n_1-n_2}{n_1+n_2}\right)^2, \rightarrow 1$ $\displaystyle T$ $\textstyle =$ $\displaystyle \frac{n_2}{n_1}\left(\frac{2 n_1}{n_1+n_2}\right)^2 \rightarrow 0.$ and so it is clear that the wave is reflected at the boundary, is this approach valid ? http://farside.ph.utexas.edu/teaching/em/lectures/node103.html


title: Physical meaning of the sign basis in quantum mechanics

text: If we take a hydrogen atom as qubit, let $\lvert0\rangle$ = unexcited state   $\lvert1\rangle$ = excited state then what is the meaning of measuring the qubit value in the sign basis? If the atom may only be in excited or unexcited state, but $\lvert+\rangle$ and $\lvert-\rangle$ are superpositions of those states — then what would the outcome of the measurement be — also a superposition of $\lvert+\rangle$ and $\lvert-\rangle$? Can anyone please help to understand the idea behind the sign basis?


title: Coordinate transformation from earth to solar

text: I am building a 3d model of the solar system and need to figure out the position of the pole stars of each planet in order to tilt the planets in the correct direction the correct amount. I've already found the information of the pole star locations, the problem is that these are in earth relative coordinates. For instance the pole star of Mars is Gamma Cygni at RA 20h 22m 13.70184s, Dec +40° 15' 24.0450". Right Ascension (RA) is hours east from the prime meridian. Declination (Dec) is degrees north (+) or south (-) from the equator. These together define a position on an imaginary celestial sphere on the surface of which one can imagine all the stars. So how do I go from those coordinates to a vector that defines the position of the star somewhere far far away on the celestial sphere? I am using ThreeJS. This question points to a partial solution in terms of defining an origin for a sphere, or at least its texture. I've posted the same question to Stack Overflow here in case some programmer wizard has the answer.


title: Quantum mechanics on Cantor set?

text: Has quantum mechanics been studied on highly singular and/or discrete spaces? The particular space that I have in mind is (usual) Cantor set. What is the right way to formulate QM of a particle on a Cantor set? I can only guess that: i) There will be no momentum operator. ii) Hilbert space will be spanned by position vectors corresponding to points of Cantor set. iii) Corresponding to any two points $x$ and $y$ in Cantor set there will be a unitary operator $P_{xy}$ (analog of the exponential of momentum operator) such that $P_{xy}|x>=|y>$; $P_{xy}=P_{yx}^{-1}$; and $P_{yz}P_{xy}=P_{xz}$ for all $x,y,z$. But I am not able to see what would be correct notion of free particle, and what would be corresponding Hamiltonian.


title: What is the mass of the emergent magnetic monopoles in spin ice and how is the mass of an emergent particle determined?

text: In solid state physics emergent particles are very common.   1. How one determines if they are gap-less excitations?    2. Do the defects in spin ice called magnetic monopoles have mass?    3. What is the mass of emergent particles in the fractional quantum hall effect (FQHE)?  I am not looking for a deep theory, just the general picture.


title: Does launching a rocket in lower density air require more or less fuel?

text: Given two environments that are identical, except for air density (e.g. Cape Canaveral, but at Mount Everest's height), would launching a rocket require more or less fuel at the lower air density?


title: Is there a way to obtain an RG flow equation for Quantum spin systems using MERA

text: We restrict ourselves to ground states of translationally invariant 1d quantum systems. I understand that there is the scale invariant MERA(multiscale entanglement renormalization ansatz) which describes quantum critical points in which the tensors ("isometries" and "disentaglers") are the same across different levels. I also understand that away from a quantum critical point, these tensors must vary across different levels (and are same within a level due to translational invariance). Is there a way to obtain an RG flow equation based on the details of how these tensors must vary across adjacent layers. Also, is it possible to go backwards: i.e., to obtain conditions for these tensors across adjacent levels from the correct flow equation. More generally, what are the techniques used to obtain these tensors? A commentary about where I can find answers to these question in the literature will be very helpful. Thank you.


title: Lenz law and magnetic pole

text: When a magnetic north(N) pole is placed close to an coil with out energy source, the coil behaves like an magnet and we find a North pole which opposes the magnet bar which was approximating onto the coil. > My question is, what conditions and mathematcal treatment assure us about > the creation of the pole on the coil. Please write if you have any confusion to get the question.


title: Renormalization of Composite Chiral Superfields

text: On page 24 of these lecture notes http://arxiv.org/abs/hep-th/0309149 it is stated that products of chiral superfields do not suffer from short distance singularities. In other words, if I want to calculate the dimension of a composite operator built out of chiral superfields, I may just add the dimensions of the individual field operators that form the product (in other words, it is sufficient to just renormalize the individual operators). This is definitely not the case in ordinary QFT. Can someone please explain why this is the case? Is the same true for products of vector superfields (the wording in the notes seems ambiguous)?


title: The mighty man and the bridge

text: Let us say we have a mighty man crossing a bridge, carrying 4 bags of concrete, each of which weighs 50 pounds. Let us say, for the sake of the argument, that the mighty man himself weighs 300 pounds. The support capacity of the bridge $x$ is less than the 500 pounds they weigh together - therefore, the mighty man cannot carry the concrete in a stack across the bridge - it will fall. However, the mighty man does have the ability to juggle all the concrete bags so that he is only in contact with one of them at a time. Is there such a value of $x$ possible that he can cross the bridge juggling all four bags of concrete at once, or will the bridge necessarily exceed its support capacity?


title: Is GNSS-based positioning trilateration or multilateration, and why?

text: According to the Wikipedia article on GNSS positioning calculation, GNSS-based positioning needs $t_\text{satellite}−t_\text{receiver}$ for a series of satellites, along with the satellite positions. This is an element of trilateration. However, the clock in a GNSS-receiver is not atomic, and not nearly accurate enough to determine $t_\text{receiver}$ with sufficient accuracy for trilateration. Therefore, it needs a 4th source to determine the time, such as indicated by maptoaster.com or this Yahoo Answers post. That makes sense: it simulatenously determines time, latitude, longitude, and elevation, and therefore it needs four satellites. Why, then, do we still consider this positioning trilateration rather than multilateration? In my understanding, _A GNSS receiver does not need a clock at all_ — after all, the clock that it has, is not nearly good enough. Isn't the implication that it's using the _difference_ between the distances to the satellites for positioning — the principle of multilateration? Yet the latter Wikipedia article states that: > Multilateration should not be confused with trilateration, which uses > distances or absolute measurements of time-of-flight from three or more > sites, or with triangulation, which uses the measurement of absolute angles. > Both of these systems are also commonly used with radio navigation systems; > trilateration is the basis of GPS. Is GNSS-based positioning considered trilateration or multilateration? Why?


title: Refraction of light in medium

text: Given that the plane $y=0$ separates the vacuum ($y>0$) from the optical medium ($y<0$), I would like to calculate the trajectory of a light ray starting at the point $(x_1,y_1)$ and ending in $(x_2,y_2)$ where $x_2>x_1$ and $y_1>0>y_2$. Specifically, I would like to find the point $(x,0)$ at which the light enters the medium. However, using Snell's law, I obtain an equation of fourth degree in $x$. Is there a smarter way of doing this?


title: How to calculate max/min Right Ascension and Declination in telescope field of view

text: I have a telescope with focal length 1200mm. My camera has 5184x3456 pixels with size 4.3um. This combination should give me a resolution of 0.74"/pixel and thus a field of view of 1.06° x 0.71°. Given this FOV and some specified center coordinates for RA and DEC, what equations will give me the max/min right ascension and declination within this area? My first thought was just to add/subtract 1/2 the FOV from each side of the center coordinates, but that doesn't solve the problem at ±90 DEC where the full range of RA should be found in this FOV.


title: Infinite fringes in double-slit interference

text: My professor told us that the fringes formed by interference will always be countable or finite. But I am just wondering, if, it is possible to have infinite fringes? Is there such a case? I am thinking that if the slit is placed too far from the screen, an infinite number of bright fringes will form. Or, in two-slit interference, suppose the screen is placed very far from the slits, will there be an infinite number of bright fringes formed on the screen? Your take on this?


title: Oscillation Question

text: ![enter image description here](http://i.stack.imgur.com/8dRQE.png) Now normally (if it was a block not rotating) all you would have to do is use $w^2 = k/m$ and $E= \frac12k(A\cos(2wt+\theta))^2 + \frac12m(Aw\sin(wt+\theta))^2$ or in other words the translational kinetic and spring potential. But here I don't even know how to start. Please could someone set up what I have to do?


title: Doppler -D'Alembert laws

text: I have found a document refering to the following two equations \begin{align*} &\frac{\partial^2 a\left(x,t\right)}{\partial t^2}+2u\left(x,t\right)\frac{\partial^2 a\left(x,t\right)}{\partial x \partial t}+\left[u^2\left(x,t\right)-a^2\left(x,t\right)\right]\frac{\partial^2 a\left(x,t\right)}{\partial x^2}=0\\\ &\frac{\partial^2 u\left(x,t\right)}{\partial t^2}+2u\left(x,t\right)\frac{\partial^2 u\left(x,t\right)}{\partial x \partial t}+\left[u^2\left(x,t\right)-a^2\left(x,t\right)\right]\frac{\partial^2 u\left(x,t\right)}{\partial x^2}=0 \end{align*} Which can be obtained by combining the mass and momentum conservation equations (As stated in the reference). I kept looking for such Doppler -D'Alembert laws on the internet, but could not find any reference. Does anyone have an idea?


title: Asteroid collision debris calculation

text: I wonder how to determine the directions, in which the collision debris is launched when 2 asteroids collide. I am aware of: m1*v1 + m2*v2 = m*v = m3*v3 + m4*v4 + m5*v5 + ... and this works just fine for the masses and valocity, however I find it difficult to determine the boundaries of the directions and under what circumstances shatter be produced or the asteroids will just "merge". All info is appreciated :)


title: Which types of particles are affected by the wave-particle duality?

text: If we take the double slit experiment as a way of demonstrating the wave- particle duality, which types of particles would show an interference pattern? For example, I know that electrons show such a pattern. But do protons, too? What about atoms? Where is the boundary between "wavey particles" and "classical particles"?


title: Lagrange's Equations for a Tetherball

text: I'm trying to write down the equations of motion for a tetherball moving around a pole while the string is getting shorter. **\--- MAJOR EDIT ---** I started with Lagrange: $$ x(t)=l(t) \sin (\theta) \cos (\phi)\\\ y(t)=l(t) \sin (\theta) \sin (\phi)\\\ z(t)=h(t)+l(t) \cos(\theta)\\\ \\\ T = \frac{1}{2}m(\dot x^2 +\dot y^2+\dot z^2)\\\ U=m g l(t)(1-\cos(\theta)) + mg(S-(h(t)+l(t))) $$ where $l(t)$ is the length of the string. Here only I am assuming the radius of the pole is really small compared to $l$. The polar angle is $\theta(t)$. $h(t)$ is the change in height due to the string wrapping on the pole. $S$ is the length of string when unwrapped. The change in length is given by: $$ \dot l(t) = -\frac{r\dot\phi}{ \sin(\theta)} $$ where $r$ is the radius of the pole. And the sliding pivot point is given by: $$ \dot h(t) = \frac{r\dot\phi}{ \tan(\theta)} $$ After plugging those in $T$ I apply the Lagrange derivative to $L = T-U$ and solve for $\ddot \theta$ and $\ddot \phi$ Now when I simulate the results I get a linear velocity that is some how increasing which is not supposed to happen for a tetherball because no new energy has been introduced to the system and the angular momentum is not conserved.


title: Question about Landau's phase transition theory

text: I have a question about Landau's theory of quantum phase transition. In his model, the free energy is assumed to be \begin{equation} F = f_0 + \alpha (T-T_c) \Delta^2 + \beta \Delta^4 \end{equation} The ground state of the system depends strongly on the sign of $T-T_c$. In this way, we find that the scaling exponent near the critical point is $1/2$, which may be somewhat different from that in experiments -- as a results, we need renormalization group method to understand the discrepancy. This is a theory that has been accepted by this community. OK, now my question is why in the second term the coefficient is $\alpha (T-T_c)$, instead of $\alpha (T-T_c)^\gamma$, where $\gamma$ is a constant, e.g., $\gamma = 3/5$ or $1/3$. This is perhaps a trivial problem, but has never been discussed explicitly in standard textbooks. The answer to this problem is not so straightforward for most of us. A related question maybe like that: how to prove this point in experiments. Thanks very in advance.


title: What Is the Physical Size of a Magnet?

text: With present day materials and technology what is the physical size of a one milli Tesla magnet? How much "power" it has to attract pieces of iron? Please compare it with the objects we have around. What difference in size and "power" it gets each order of magnitude that I go up? What is a weak magnet what is a powerful magnet in this scale?


title: Construction of the $\cal{N}=3$ supersymmetric Yang-Mills Chern-Simons theory in $2+1$ dimensions

text: I am stuck with understanding the following construction. I am breaking it up into segments which I think can be separately answered. This is related to an earlier question of mine. Note that this previous question was edited a lot after the initial comments came in.   * Firstly I would like to know how the following field multiplet is determined,  The gauge multiplet of $\cal{N}=3$ in $2+1$ dimensions is apparently , $A_\mu$, a massive vector field of spin $1$ $\lambda_a$, $3$ Majorana fermions of spin $\frac{1}{2}$ $C_a$, $3$ neutral scalar bosons $\chi$, a Majorana fermion of spin $\frac{1}{2}$ (I guess there should be another scalar field $D_a$ not mentioned in the above list)   * Its not clear to me as to what are the superfields one has in mind with the above multiplet structure that helps construct the lagrangians below.    * Like in the linked earlier question above one had to "know" what exactly is the meaning of doing the dimensional reduction on a given set of supersymmetric transformations, here its not clear to me as to what are the corresponding starting points. (I wonder if I can start from the $\cal{N}=2$ SYM in $3+1$ lagrangian that is given in say equation $27.9.3$ in Weinberg's QFT book)  Then dimensional reduction of $\cal{N}=2$ SYM in $3+1$ apparently gives the following $\cal{N}=3$ SYM in $2+1$, $L_{\cal{N}=3,2+1,SYM} = \frac{1}{g^2}Tr\left [ -\frac{1}{2}F_{\mu \nu} ^2 + (D_\mu C_a)^2 + (D_a)^2 + i \bar{\lambda_a}\gamma^\mu D_\mu \lambda_a + i \bar{\chi}\gamma^\mu D_\mu \chi \+ i\epsilon_{abc}\bar{\lambda_a}[\lambda_b,C_c] -2i\bar{\lambda_a}[\chi,C_a] - \frac{1}{2}[C_a,C_b][C_b,C_a] \right ]$ where the notation is, $D_\mu = \partial_\mu -iA_\mu$, $ab,c=1,2,3$. The gauge group generators in the fundamental representation satisfy $[T^m,T^n]=if^{lmn}T^l$ with normalization as $Tr{T^mT^n}=\frac{1}{2}\delta ^{mn}$ and $f^{kmn}f^{lmn}=c_v\delta^{kl}$. ($c_v$ being the quadratic Casimir) The fields belong to the adjoint representation and $A_\mu = A_\mu^mT^m$. The metric is given by $\eta_{\mu \nu} = diag(1,-1,-1)$. The purely imaginary gamma matrices satisfy, $\gamma^\mu \gamma ^\nu = \eta^{\mu \nu}-i\epsilon^{\mu \nu \rho}\gamma_\rho$.   * It is not clear to me as to how does one argue that the above has $\cal{N}=3$ supersymmetry.    * The dimensional reduction of the $\cal{N}=2$ supersymmetry somehow implies that the theory has the not-so-obvious $\cal{N}=4$ supersymmetry.  * * * One also notes that supersymmetrizing the Chern-Simon's terms one is supposed to get the Chern-Simon's lagrangian as, $L_{\cal{N}=3,2+1,SCS} = kTr\left [ \epsilon^{\mu \nu \rho}(A_\mu \partial_\nu A_\rho - \frac{2}{3}iA_\mu A_\nu A_\rho) - \bar{\lambda_a}\lambda_a + \bar{\chi}\chi \+ 2C_aD_a + \frac{i}{3}\epsilon_{abc}C_a[C_b,C_c] \right ]$   * In all these lagrangians apparently the corresponding $\cal{N}=2$ theories can be obtained by setting $C_1=C_2=D_1=D_2=\lambda_3 = \chi = 0$ and the $\cal{N}=1$ theories can be obtained from the $\cal{N}=2$ theories by further setting $C_3=\lambda_2=0$.  (I am unable to get the equations to wrap around properly!)


title: Calculate the electric field of a moving infinite magnet, without boosting

text: Consider a rectangular slab of permanently magnetized material. The slab's dimensions are $L_x$, $L_y$, and $L_z$, and the slab is uniformly magnetized in the $\hat{x}$-direction. The slab is not accelerating or spinning. Does the slab generate an electric field? In a frame where the magnet is stationary, we know $\mathbf{E}$ is zero everywhere. In a frame where the magnet is moving, there are at least two ways to attack the problem:   1. Drop $d\mathbf{M}/d t$ into Maxwell's equations, solve for $\mathbf{E}$ and $\mathbf{B}$    2. Solve for $\mathbf{B}$ in the slab's rest frame, and use a relativistic boost to transform $\mathbf{B}$ in the slab's frame to $\mathbf{E}$ and $\mathbf{B}$ in the frame where the magnet is moving. Both these methods give the result that $\mathbf{E}$ is nonzero in a frame where the magnet is moving. Now let's consider a long, thin slab ($L_x, L_y \ll L_z$). In a frame where the slab is moving in the $\hat{z}$-direction, is there an electric field (external to the slab) near the 'center' of the magnet? Both the $d\mathbf{M}/d t$ argument and the Lorentz-boost argument seem unchanged. The magnetic field external to the slab does not vanish near the center of the slab, suggesting there is a nonzero electric field. With the backstory laid out, here's my real question: In a frame where the slab is moving in the z-direction, is there still an electric field in the case where $L_z \rightarrow \infty$? The Lorentz-boost argument seems unchanged, and suggests that there is. However, in the $L_z \rightarrow \infty$ case, $d\mathbf{M}/d t = 0$, suggesting no electric field. Can this case be calculated without Lorentz boosts? How do Maxwell's equations account for moving permanent magnets in the case where $d\mathbf{M}/d t = 0$? EDIT: Followup questions: What's a good reference for the electrodynamics of moving media? What is the electric field generated by a spinning magnet?


title: can someone review a 8 page essay for me for accuracy? essay is in cosmology

text: I am a bio-chemist by profession and recently written an essay for a book which is related to the subject of cosmology. This is not my area of expertise. Is there a cosmologist out there who will be kind enough to review this 8 page essay for me to check it for accuracy, or perhaps give me criticism or even suggestions? I am a member of the APS and its division of Biological Physics. Thanks Paul


title: Apple falls for which of these 2 reasons?

text: Needles to say I am a visitor here. I do not belong to the science world;) But I have read both of these things before:   1. Apple falls to the ground because curved spacetime pushes it there (same force as keeps moon in orbit)   2. Apple 'falls' to ground because the ground is rushing up to meet the apple (which is actually suspended in space) because of Earth's acceleration through space. I don't think these can both be true. I'd appreciate any clarification - thank you.


title: Moment of inertia of a hollow sphere wrt the centre?

text: I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method: Consider a uniform hollow sphere of radius $R$ and mass $M$. On the hollow sphere, consider a concentric ring of radius $r$ and thickness $\text{d}x$. The mass of the ring is therefore $\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi r\cdot\text{d}x$. Now, use $r^2 = R^2 - x^2:$ $$\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi \left(R^2 - x^2 \right)^{1/2}\text{d}x$$ and the moment of inertia of a ring wrt the centre is $I = MR^2$, therefore: $$\text{d}I = \text{d}m\cdot r^2 = \frac{M}{4\pi R^2}\cdot 2\pi\left(R^2 - x^2\right)^{3/2}\text{d}x $$ Integrating to get the total moment of inertia: $$I = \int_{-R}^{R} \frac{M}{4\pi R^2} \cdot 2\pi\cdot \left(R^2 - x^2\right)^{3/2}\ \text{d}x = \frac{3MR^2 \pi}{16}$$ which obviously isn't correct as the real moment of inertia wrt the centre is $\frac{2MR^2}{3}$. What was wrong with this method? Was it how I constructed the element? Any help would be appreciated, thanks very much.


title: Fresnel's explanation of rectilinear propagation

text: How did Fresnel explained the phenomenon of rectilinear propagation of light using concept of half period zones?


title: Why don't we need to normalize wavefunction to find probability distribution?

text: > Consider an unormalized wavefunction of a rotor at $t = 0$, a combination of > $n=0$ and $n=2$ states: > > $$\psi(\phi) = 3 - 2 \cos (2\phi).$$ > > Find the probability distribution in angle. The book simply takes probability as $$\lvert\psi(\phi)\rvert^2 = \lvert 3 - 2\cos(2\phi)\rvert^2 = 9 + 4\cos^2(2\phi) - 12\cos(2\phi).$$ I have found the normalized wavefunction to be: $$\psi(\phi) = \frac{3}{\sqrt {11}}\lvert 0\rangle - \frac{2}{\sqrt {22}}\lvert 2\rangle.$$ Do I use probability as $\left\lvert\frac{3}{\sqrt {11}} - \frac{2}{\sqrt {22}}\cos(2\phi)\right\rvert^2$ ? or the one in the book?


title: Thought experiment that seems to involve something growing at twice the speed of light. Is anything wrong?

text: Let foo be some unit of distance and bar be some unit of time which have been chosen so that the speed of light c = 1 foo/bar. Position several observers along a line each separated by one foo, and place light sources some distance apart amongst the observers, initially off. All observers and light sources are stationary with respect to one another. ![diagram](http://i.stack.imgur.com/3GK3n.gif) So in my diagram the tick marks are the observers, and the scale is one foo. For the example, the light sources are ten foo apart at points L and R. Now at time 0 bars, ignite both light sources. In the diagram, at time 4 bars, A has observed source L but not R, C has observed R but not L, and B has not yet observed either light source. At time 5 bars, B observes both light sources, and at time 6 bars all three labeled observers have observed both light sources. At time 7 bar, a total of five observes (A, B, C, and two unlabeled) have observed both sources. Let r be the length of the line segment containing all observers that have observed both light sources. In other words, r is the length of the intersection of the event horizons on the two ignition events (I think I am using that term right, if not, please let me know). It seems that dr/dt = 2 foo/bar; for instance, in the 1 bar interval from 6 to 7, the change in r was 2 foo. As c = 1 foo/ bar, r is growing at twice the speed of light. Since r is not a particle and cannot be used to transmit information amongst the observers, is there any contradiction here? Have I made any logical falicies?


title: Quantum Mechanisms for Isotope Fractionation

text: Are there any quantum properties that would enable isotope fractionation? For example, atoms with odd versus even numbers of neutrons are fermions and bosons, respectively. Has any work been done using the distinct properties of fermions and bosons to isolate isotopes?


title: Lorentz boost matrix in terms of four-velocity

text: As I understand it, the value of a 4-vector $x$ in another reference frame ($x'$) with the same orientation can be derived using the Lorentz boost matrix $\bf{\lambda}$ by $x'=\lambda x$. More explicitly, $$\begin{bmatrix} x'_0\\\ x'_1\\\ x'_2\\\ x'_3\\\ \end{bmatrix} = \begin{bmatrix} \lambda_{00}&\lambda_{01}&\lambda_{02}&\lambda_{03}\\\ \lambda_{10}&\lambda_{11}&\lambda_{12}&\lambda_{13}\\\ \lambda_{20}&\lambda_{21}&\lambda_{22}&\lambda_{23}\\\ \lambda_{30}&\lambda_{31}&\lambda_{32}&\lambda_{33}\\\ \end{bmatrix} \begin{bmatrix} x_0\\\ x_1\\\ x_2\\\ x_3\\\ \end{bmatrix} $$ I have seen examples of these components written in terms of $\beta$ and $\gamma$, which are defined as $$\beta=\frac{v}{c}$$ $$\gamma=\frac{1}{\sqrt{1-\beta\cdot\beta}}$$ where $v$ is the 3-velocity and $c$ is the speed of light. My question is this: How can the components of $\lambda$ be written in terms of the 4-velocity $U$ alone? I know that $U_0=\gamma c$ and $U_i=\gamma v_i=\gamma c\beta_i$ for $i\in\\{1,2,3\\}$, but I'm having trouble deriving the components for $\lambda$ using the matrices based on $\beta$ and $\gamma$. An example of one of these matrices can be found at Wikipedia. How can I rewrite this matrix in terms of $U$ alone?


title: Calculate impulse based on mass, acceleration and time elapsed

text: I want to move to Sprite Kit framework offered by apple which has physics integrated right into it. The back bone of the physics engine is famous Box2D. Sprite Kit has made it pretty easy by wrapping C++ inside Objective-C but unfortunately some features lack in Sprite Kit. One them is GetReactionForce function, that returns the impulse applied on the body B of a joint. Just disregard the software related terms if you find it difficult to understand. Basically what I need is to create my own function that will calculate impulse. Here are the ingredients I have: Mass, gravity, delta time. P.S. Sorry for the tags, I couldn't find anything more suitable and I have no right to create new ones


title: What would happen to a rocket car in a torus-shaped spaceship that encompasses the earth?

text: Suppose we built a enormous torus shaped space station that encompasses and leaves about 330 kilometers between it and the earth. So it's radius is about 6,371 kilometers . Obviously, this structure will have to spin very quickly to not collapse. So let's say it's going 28,000 kilometers per hour so people in the spaceship will feel weightless. If we now were to build a rail inside this spaceship with a rocket car on it, that holds on to the rail much like a roller coaster holds on to its rail and have the rocket car go in the opposite reaction the space station is spinning. Would it start to exert downwards force on the space station since it moving more slowly relative to the earth? And reach almost 1g when the car is going 28,000 kilometers per hour? The boarder question I guess is, how is the absolute rotation of something determined. What's the reference frame you should use? Is there an absolute reference frame for rotation or should you maybe use the local curvature of spacetime as a reference?


title: 3D ballistic trajectory with quadratic drag. Calculating position and velocity at time $t$

text: A particle starts at the origin and has an initial velocity represented by a 3D vector. The particle experiences gravity and air resistance with quadratic drag (based on velocity^2). What I've been looking for are parametric equations for $x$, $y$, and $z$ position and velocity of the particle at a given time $t$. To keep answers similar the two constants will be g as gravity and a as the air resistance. Also assume $z$ is positive elevation. $v_x(t)$ will be the velocity for the $x$ component at time $t$. ${v_x}_0$ would represent the initial velocity for the $x$ component. At the bottom of this wikipedia article it suggests there's an analytic solution to the 2D problem by a teenager Shouryya Ray. The corresponding Math.SE and Phys.SE links have people discussing solving for a constant of friction, but there's no parametric equations given anywhere. I'm not sure how to go from what they're talking about to the actual parametric equations I need. To cover what I've learned so far. In 3D with gravity and no drag the following parametric equations can be used: $v_x(t) = {v_x}_0$ $v_y(t) = {v_y}_0$ $v_z(t) = {v_z}_0-gt$ $s_x(t) = {v_x}_0t$ $s_y(t) = {v_y}_0t$ $s_z(t) = {v_z}_0t - \frac{1}{2} g * t^2$ With linear drag per the Wikipedia article we'd used: $v_x(t) = {v_x}_0e^{-\frac{a}{m}t}$ $v_y(t) = {v_y}_0e^{-\frac{a}{m}t}$ $v_z(t) = -\frac{mg}{a}+({v_z}_0+\frac{mg}{a})e^{-\frac{a}{m}t}$ $s_x(t) = \frac{m}{a}{v_x}_0(1-e^{-\frac{a}{m}t})$ $s_y(t) = \frac{m}{a}{v_y}_0(1-e^{-\frac{a}{m}t})$ $s_z(t) = -\frac{m * g}{a}t + \frac{m}{a}({v_z}_0 + \frac{m*g}{a})(1 - e^{-\frac{a}{m}t})$ With quadratic drag we have the velocity's length so each component's velocity and position over time depends on the other velocity components. Starting with this post I can write the 3D version for the first part: $v_x' = -av_x*\sqrt{v_x^2 + v_y^2 + v_z^2}$ $v_y' = -av_y*\sqrt{v_x^2 + v_y^2 + v_z^2}$ $v_z' = -av_z*\sqrt{v_x^2 + v_y^2 + v_z^2}-g$ But I lack the mathematical background to continue. It's not even clear to me what the constant of friction they calculate is used for since it's not based on time. I would think any trick that attempts to calculate a constant coefficient would need to know about t. I'm probably thinking of this wrong. Any help would be appreciated as I imagine this is a common problem in physics. edit: Also my problem case is specifically for a constant gravity vector if that helps for specific solutions so the path must be a parabola of sorts. As far as I can tell there should be a function as there's one y value for every x.


title: Simple (but wrong) argument for the generality of positive beta-functions

text: In the introduction (page 5) of _Supersymmetry and String Theory: Beyond the Standard Model_ by Michael Dine (Amazon, Google), he says > (Traditionally it was known that) the interactions of particles typically > became stronger as the energies and momentum transfers grew. This is the > case, for example, in quantum electrodynamics, and a simple quantum > mechanical argument, based on unitarity and relativity, would seem to > suggest it is general. Of course, he then goes on to talk about Yang-Mills theory and the discovery of negative beta-functions and asymptotic freedom. But it is the mention of the _simple but wrong argument_ that caught my attention. So, does anyone know what this simple argument is? And how is it wrong?


title: How to calculate tortuosity of a signal?

text: Not sure which stack exchange forum to put this in. Moderators feel free to migrate it. One definition of the tortuosity $T$ of a curve is: $$T=\frac{C}{L}$$ Where $C$ is the chord length (distance between the ends of the curve) and $L$ is the total length of the curve. I am trying to calculate the tortuosity of a white noise signal. But it is not clear to me what the unit length should be. Here's what I mean: The white noise is a time series $X(t)$ of length $N$ of normally distributed random numbers with mean $0$ and standard deviation of $1$. To calculate the length $L$ of this signal I do the following: $$dy(t)=\left| X(t+1)-X(t) \right |$$ $$L=\sum_{t=1}^{N-1} \sqrt{dy(t)^2+dx^2}$$ Where $dx$ is the unit interval. The question is how to select a suitable value for $dx$. Should it just be $1$? Or perhaps the mean or minimum value of $dy(t)$? The chord length $C$ of this signal is $N/dx$, so it also depends upon the selection of $dx$.


title: Would it help if you jump inside a free falling elevator?

text: Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When?


title: How does lifting an object effect its entropy

text: I have figured out that:   1. When photons leak out from a container, the entropy of the photon collection increases, because each photon has a different escape time.   2. Photons that have leaked out from a container can be collected into a container of the same size, if that container is in a gravity well deep enough.   3. When photons fall some distance in a gravity field, each photon's energy becomes multiplied by some factor, which does not change the entropy.   4. After aforementioned container has been winched up from the gravity well, the entropy of the photons inside must be increased compared to the entropy of the photons inside the first leaky container.  So I suspect that lifting some radiation increases its entropy. Am I on a right track here? Addition1: I have a suggestion how the entropy increases when a cube-shaped mirror walled container of radiation is lifted: A photon moving into just right direction does not touch the floor or the ceiling of the container during the lifting process. The lifting agent does not do any work on that photon. Work done on a photon ranges from zero to X joules, which increases entropy. When gravity does work on photons, every photon's energy increases by K percents, which does not increase entropy. Clarification to point 2: Observer in the gravity well sees the radiation to leak out rapidly. The short time implies small entropy increase. Observer outside the gravity well sees that radiation slows down when the radiation enters the gravity well. The line of photons contracts when photons at the front are slowing down. Oh yes, optically dense material has the same effect as gravity well. And also if a photon catcher device moves rapidly towards the photons, then photons go in rapidly and come out slowly, which allows an arbitrarily large photon gas cloud to be caught into a small volume container, in this process temperature of radiation increases without entropy of radiation increasing.


title: Implications of unbounded operators in quantum mechanics

text: Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\mathcal{H}$. Now I understand a lot of operators employed in quantum mechanics are unbounded operators, in nutshell these operators cannot be defined for all vectors in $\mathcal{H}$. For example according to "Stone - von Neumann", the canonical commutation relation $[P, Q] =-i\hbar I$ has no solution for $P$ and $Q$ bounded ! My basic question is :   * If the state of our system $\psi$ is for example not in the domain of $P$ (because $P$ is unbounded), i.e., if $P\cdot \psi$ does not, mathematically, make sense, what does this mean ? Does it mean we cannot extract any information about $P$ when the system is in state $\psi$ ?


title: Algorithm for distribution of radiation

text: Unstable atomic nuclei will spontaneously decompose to form nuclei with a higher stability. What is the algorithm for deciding what sort of it is? (alpha, beta, gamma, etc. Also, given that alpha and beta emission are often accompanied by gamma emission, what is an algorithm for deciding about the distribution of the radiation?


title: Kinetic energy transfer in matter annihilation?

text: What happens to the kinetic energy of matter when it is annihilated? Is it released in the resultant explosion? In that case shouldn't it be $E=(mc^2 + \frac{1}{2} mv^2)$ ?


title: Correlation between large-scale galaxy structure and CMB fluctuations?

text: During a relatively non-technical astronomy seminar the other day, the speaker displayed the famous WMAP full-sky image as an aid to describing what the CMB is, the scale of its fluctuations, etc. This speaker mentioned that there are correlations between the higher-temperature regions on the map and regions of large-scale galaxy structure seen in deep-sky surveys. I was surprised to hear this. My understanding is that CMB is an image of events currently about 14 billion light years away, while the observed large- scale filaments of galaxies are at approximately half that distance. I wouldn't have expected any density fluctuation 14 billion light years away to share any correlation with a density fluctuation 7 billion light years away. When I asked, the speaker admitted to being "mostly a star guy" and continued with his excellent talk. Is there actually a correlation between the warmer, denser regions of the CMB and the distribution of dense galaxy clusters? Is there a causal reason for these distant objects to be correlated with each other? Is there a lensing effect on the CMB temperature? Or is this "correlation" just an enticing- sounding mistake, slowly working its way into common knowledge?


title: Is the standard model so full of holes it is time to start again?

text: I have read a lot about dark matter and dark energy, the fact that we are spending a small fortune trying to detect their presence, but the only reason they have been postulated it is an effort to protect the standard model. Dark Energy seems to act at a galactic scale, but do we notice it in our solar system? Come on, do we really believe that our solar system is some how special and has no dark energy or dark matter? What else is wrong - Gravity is still absent from the model, so that is 25% of forces not accounted for (and the only force that actually shapes the universe!). Neutrino mass not accounted for, absence of anti-matter, I could go on. From my perspective every barrier we come to we make up more and more unbelievable / incredible solutions in order to try and hold on to a model that is clearly flawed. When the standard model was suggested its simplicity was what made it seem so believable. Do we not need to just get a clean sheet of paper and start again?


title: Formula for residence time/turnover rate with unsteady state

text: I'm not sure this is the correct place to ask my question... but maybe someone could still help me. I'm looking for a way to calculate the residence time/turnover rate. I have the production and consumption rate (they are different, but stable over time). And I have the volume (resp. concentration) change over time. How can I calculate how long a molucule stays in the pool? Or how long it takes until the whole "population" of molecules is renewed? I tried several approaches, but it seems I did not yet find the correct direction (?) 1) close the "inflow" and look how long it takes until the pool is emptied. Then open the inflow, close the outflow and calculate how long it takes to reach the start-concentration. But the problem here is, that it does not take into consideration, that "new" and "old" molecules are mixing and the uptake- change is the same for the new and old molecules. Also it doesn't adress the problem of the changing volume and therefore changing probabilites of the uptake of the "old" molecules. 2) normal residence time. Not usable for me, since I have a different in- outflow and changing volume. I found something from a pulp fabric but this didn't take into consideration, that there is a free mixing of the molecules. 3) next possibility would be in a direction of the "Urn problem". But I do not know how to build it up around the circumstance that there isn't a steady number of balls in there and that I remove "red balls" while put "white ones" into it. Can someone point me into the correct direction?


title: What are some examples of infinite state quantum mechanical systems that do not involve free particles?

text: That is, the quanta are in bound states where there are least upper bounds and greatest lower bounds to their energy states but there are at least a countably infinite many energy levels they can assume? I'm particularly interested in examples of entangled systems with this property. Any physical examples that can be created in the laboratory?


title: Is there a way to electrically shock a person who is inside a Faraday cage?

text: I learned last semester that a Faraday cage shields people (among other things) from getting electrically shock, say, from a tesla coil. This was well demonstrated in lecture so I believe it.   1. The person was in a Faraday cage made from chicken wire (i.e. it was full of holes) so why did the Faraday cage "shield" the person?   2. Is this always true that a person is always shield under all circumstances? That is, is there a situation where some low frequency discharge (this implies large skin depth) able to penetrate the Faraday cage and shock a person inside the cage?


title: How long until the sun cannot sustain human life on earth?

text: The sun will last, at its current brightness for 9 billion more years. How long until the sun gets burned down to the point where it cannot sustain life on Earth anymore? **Updated** : I am more concerned with how long until _human_ life cannot be sustained, not how long until the water on Earth vaporizes.


title: Quantum memories: What are they?

text: Searching the literature for the term "quantum memory" seems to bring up results from two different communities. On the one hand there are quantum opticians, who see a quantum memory as something used to absorb a photon and store its quantum state before re- emitting it again. The memory in this case could be something as small as a single atom. More details on these are exponded in a review here. On the other hand, there are those who see a quantum memory as a many-body system in which quantum information is stored. Possibilities for this include error correcting codes, such as the surface code, or the non-Abelian anyons that can arise as collective excitations in certain condensed matter systems. In this case it is not necessarily assumed that the quantum information existed anywhere before it was in the memory, or that it will be transferred elsewhere after. Instead, the state can be initialized, processed and measured all in the same system. An example of such an approach can be found here. These two concepts of quantum memories seem, to me, to be quite disjoint. The two communities use the same term for different things that store a coherent state, but what they refer to is otherwise pretty unrelated. Is this the case, or are they really the same thing? If so, what exactly are the connexions?


title: Can a particle in superposition act as x*(energy?)

text: I can't find an answer anywhere - I even asked my physics teacher, he hasn't a clue. Is superposition an illusion, or can a particle literally act as [x] particles?


title: Quantization of Electron Spin

text: Why is electron spin quantized? I've seen the derivation for the Hydrogen atom's energy levels, but my professor jumped to electrons having spin 1/2 or -1/2 as experimental. Why do electrons obey the same quantization rules for angular momentum as the Hydrogen atom does? Why must the two states be one apart?


title: wave functions of Klein Gordon particles

text: It is known that the Klein Gordon equation does not admit a positive definite conserved probability density. Nonetheless, in Wikipedia (for example), you can read that with the $\textit{appropriate}$ interpretation it can be used to describe a spinless particle (such as the pion I think it says). My question is the following, which is this appropriate interpretration? and how can we get the wave-function of a particle described with the Klein Gordon equation?


title: Why does the Casimir effect require metallic plates?

text: > The typical example of the Casimir effect is of two uncharged metallic > plates in a vacuum, placed a few micrometers apart. Why do the plates need to be metallic?


title: What is the physical size of a black hole?

text: Something that's always confused me. How large is a black hole's physical size - not mass? From descriptions, it would seem that the 'singularity' is a single point, but is it really? Say for arguments sake, a 110 solar mass black hole. Obviously it's not going to be the same massive size as a 110 solar mass star (such as Cygnus OB2-12 at a physical size of 246 R☉), but would it really be a tiny point smaller than say a pin head?


title: Double-spring mass system

text: We just had a lesson about elementary mass-spring systems (SHO), and I thought about a horizontal situation with two springs with the test mass oscillating in between. If we are to manually stretch the mass a distance $\Delta x<\Delta_0$ , we obtain opposing restoring forces, which yields the familiar SHO equation. But do thesee conditions change when we stretch it beyond $\Delta_0$(a little beyond, it's still elastic with Hooke's law), in some intervals the restoring forces point in the same direction and in others don't. Does the system retain its "SHO-ness"? ![System diagrams](http://i.imgur.com/g0rBIf3.png)


what is the difference between constant and changing magnetic and electric fields? How do they occur? How do they form an electromagnetic wave?


title: QED and Newton's 3rd law

text: According to some lecture notes on propagators (see bottom of page) the Feynman particle/anti-particle hypothesis states: > _The emission (absorption) of an antiparticle of 4-momentum $p^\mu$ is > physically equivalent to the absorption (emission) of an particle of > 4-momentum $-p^\mu$._ The antiparticle of a photon is a photon. So if an electron emits a photon with 4-momentum $p^\mu$ that is equivalent to it absorbing a photon with 4-momentum $-p^\mu$. Is that right? So is this the origin of Newton's 3rd law of action and reaction?


title: Where do $L_+$ and $L_-$ live, if not in $\mathfrak{so(3)}$?

text: This question is continuation to the previous post. The lie algebra of $ \mathfrak{so(3)} $ is real Lie-algebra and hence, $ L_{\pm} = L_1 \pm i L_2 $ don't belong to $ \mathfrak{so(3)} $. However, when constructing a representation for $\mathfrak{so(3)} $, one uses these operators and take them to be endomorphisms (operators) defined on some vector space $V$. Let $\left|lm \right> \in V $,then $$ L_3\left|lm \right> = m \left|lm \right> \;\;\;\;\; L_{\pm}\left|lm \right> = C_{\pm}\left|l(m\pm1) \right> $$ Now, how do we justify these two things ? If $L_{\pm} \notin \mathfrak{so(3)}$, then how is this kind of a construction of the representation possible ? I belive similar is the case with $\mathfrak{su(n)}$ algebras, where the group is semi simple and algebra is defined over a real LVS.


title: Does black hole complementarity apply to white holes?

text: If physics is time reversal invariant, there ought to be white hole complimentarity as well. Imagine a white hole so enormous that it is possible for life to evolve entirely within it for billions of years before emerging to tell their story about the interior of the white hole. Meanwhile, an external observer can observe and keep note of everything in the vicinity of the white hole and what comes out of it. Both observers can meet and compare notes. Can the external observer really claim the white hole interior doesn't exist?


title: Is charge density zero in a dielectric material and why?

text: I'm trying to solve a problem involving parallel capacitor. I can't decide whether to use poisson's formula or laplace's formula. ![enter image description here](http://i.stack.imgur.com/hArz0.png) The question is, is there $\rho_v$ in a piece of dielectric and why?


title: Is particle number a problem for formulating statistical physics in a mathematically rigorous manner?

text: Quantities like the chemical potential can be expressed as something like $$\mu=-T\left(\tfrac{\partial S}{\partial N}\right)_{E,V}.$$ Now the entropy is the log some volume, which depends on the particle number $N$. As in this definition, we sum over natural numbers of particles, is there any good way of actually evaluating the derivative? What one practically does, i.e. when dealing with an ideal gas, is computing the quantity $\Gamma_N$, which might turn out to be $\frac{\pi^{N/2}}{(n/2)!}$, and then one will get an expression $S(E,N,V)$ which can of course _be treated as if_ it was a function over $\mathbb{R}^3$. Even if that assumes that one has a closed expression which is a function $N$. In principle I'd be fine with that - if one has a given function (Or at least the bunch of values for all $N$) over a grid and a procedure to introduce more and more grid points to get a finer mash, then there is a notion of convergence to a derivative. But here the N's are clearly always at least 1 value apart - no matter how many N s there are (thermodynamics limit), the mesh doesn't get finer between any two given points. You might define the derivative as computing the average rate of change between two partcle numbers $n$ and $n+d$ and say $\tfrac{\partial S}{\partial N}$ evaluated at $N'$ gives the same value for all $N'$ in one of the $d$-Intervals, but then you would have to postulate how to come up with $d$ in every new situation. This might be overcome in very specific situations in coming up with a "reasonable" fraction of Avogadros number, but this is not quite mathematical and the values of different finite difference approximation schemes are always different. In full generality, I feel there is no categorical understanding of what the fractional dimensional space (phase space in this case) has to be and so the procedure of evaluation of the derivative should be explicitly postulate.


title: power counting and (superficial) non-renormalizability

text: Comment: This stuff is new to me so it doesn't entirely make sense (yet). Question:   As I understand from Peskin and Schroeder chap 10 if you have a theory with interaction terms $\lambda \phi^n$ in the Lagrangian (density) and the dimension of the coupling constant ($\lambda$) is given by $d-n(2-d)/2$. Superficially, if $d\geq0$ the theory is renormalizable. Secondly, I am aware that the Einstein Hilbert action $S\rightarrow C*\int d^4x \sqrt{-g}R$ is superficially non-renormalizable. (The constant $C$ in the above is of course proportional $G^{-1}$.) The constant $C$ ,itself, has mass dimension 2. Why do these dimensional criteria appear to be reversed for this form of action compared to the $\phi^n$ type terms? To rephrase another way: I understand that the dimension for $G$ itself is negative, but why is it important to look for the dimensionality of $G$ here instead of the dimensionality of what happens to actually be constant in front of the Lagrangian term as was the case in the $\lambda \phi^n$ example?


title: Non-zero higher time derivatives of position?

text: My mom told me to use speed control, which would allow the car to remain at constant speed. I told her that its impossible for a car to maintain constant speed, as slight changes in friction on the road cause differences in the acceleration of the car, and hence the velocity. But that got me thinking: Even the differences in acceleration caused by differences in the friction due to the road need not be constant. Thus, the car's acceleration is also always changing, implying that it has non-zero "jerk," defined as the change in acceleration over time. Is this reasoning correct? If so, is there anything keeping the "jerk" constant? How about snap, crackle, and pop (the 4th, 5th and 6th time derivatives of position)? Does it even have to stop there?


title: How to tell local and non-local in QFT?

text: I'm taking QFT course in this term. I'm quite curious that in QFT by which part of the mathematical expression can we tell a quantity or a theory is local or non-local?


title: A reference request for real world experimental data

text: I always use to wonder how the experimental physicists discover new particles every now and then whose dimensions/properties/mass/charge several order of magnitudes below that of anything that is visible/perceptible. So as engineers do, I guess they also set up a extremely complicated equipment and do some pretty complicated experiments and measure some physical quantity, say (pardon me if my example is poor) an voltage or a magnetic field intensity or could be anything depending on the experimental setup. Now they compare this measured voltage variation with that of what is expected theoretically and then go on to prove the hypothesis. ( This is the hardest thing I could imagine). Now my request is where can i find a set of data (preferable a continuous variation of a physical parameter with respect to another... may be sampled at sufficient sampling frequency) along with the context of experiment (as minimal as possible but sufficient) so that i can carry out some processing of data in my own way so that i can verify the hypothesis or any such a thing. Simply put i need some really cool real world data ( in the form of signal). Is any such thing available on the internet or where could I find one? Please suggest me something which involves signal processing.


title: Mechanical energy problem

text: Question goes: "An anvil hanging vertically from a long rope in a barn is pulled to the side and raised like a pendulum 1.6 m above its equilibrium position. It then swings to its lowermost point where the rope is cut by a sharp blade. The anvil then has a horizontal velocity with which it sails across the barn and hits the floor, 10.0 m below. How far horizontally along the floor will the anvil land?" Now I've been working a while on it and what i got was $PE \ at \ side = 1.6\times 10 (gravitational\ constant) \times m (mass\ of\ anvil).$ $KE\ at\ lowermost\ point = \frac{1}{2} m v^2 = PE$ $16m = \frac{1}{2} m v^2$ $16 = \frac{1}{2} v^2$ $32 = v^2$ $v = \sqrt{32}$ Now to the free-fall problem, $10 = \frac{1}{2} g t^2$ $20 = g t^2$ $2 = t ^ 2$ $t = \sqrt{2}$ So the distance traveled horizontally = $\sqrt{2} \times \sqrt{32} = 1.414$ meters. I checked the model-answers and the answer was 8 meters, would someone please explain this? (The model-answer might be wrong though)


title: Question on particles

text: Is there any theory in which every particle can be further subdivided into any number of particles and the total number of particles any where in the space time are infinity in theory and only due to practical constraints that we are capable of observing the known particles in the current World. Also could you please explain the nature of the two widely accepted theories which explain the small scale and large scale universe which are QP and GR repectively in the current context of the question.


title: Acceleration on an inclined rail

text: I have to use $\vec{P}+\vec{R_N}+\vec{F_f}=m\vec{a}$ with $\|\vec{F_f}\|=\mu\|\vec{R_N}\|$ to express the acceleration as $$a_{x'}=g\sin\alpha(1-\frac{\mu}{\tan\alpha})$$ The configuration is (sorry about the poor image) : ![scheme](http://www.beziaud.org/~work/upload/docs/131112083008.schema.jpg) I'm totally lost. How could I do that?


title: Difference between $K_{\alpha}$, $K_{\beta}$ and binding energies

text: I'm having a small question regarding $K_{\alpha}$ and $K_{\beta}$ emissions. If I'm not mistaken this happens when there is a transition from the L shell to the K shell (Depending on the orbital), in which a photon is emitted with a given energy. My question is: Is there any difference between the energy of the photon that is emitted in $K_{\alpha}$ emission for example, and the binding energy of the electron in the same orbital, or...? Thanks in advance


title: Green functions in QFT

text: What is the sense of Green function $$ \langle | \hat {T}(u_{1}(x_{1})...u_{n}(x_{n})\hat {S})|\rangle , \quad \hat {S} = \hat{T}e^{i\int \hat {L}(x)d^{4}x} ? $$ How is it connected with scattering processes?


title: What's the idea behind the Riemann curvature tensor?

text: The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} \- \partial_l\Gamma^m{}_{kj} \+ \Gamma^m{}_{ki}\Gamma^i{}_{lj} \- \Gamma^m{}_{li}\Gamma^i{}_{kj}$ How did they come up with this? What was the idea? I searched the web but the descriptions I found were too formal, and I was unable to decipher what the author tries to describe. So I'm looking for some thoughts or an easy paper I can start from and derive this formula myself.


title: What does $\text{VAC}$ mean in Weinberg's QFT?

text: Hopefully Weinberg is sufficiently popular that this question isn't too localized. I'm reading Volume II (16.1, p. 63) and he refers to the state $\lvert\text{VAC}\rangle$ quite a lot. Does this mean the vacuum of the free theory or of the interacting one? I think it means the vacuum of the free theory, from how he uses it. But I can't find the page where it's defined (and there's a lot of rather dense pages, so I'd rather not spend my whole afternoon trying to find the right line)!


title: Aggregating daily readings with units of parts per million

text: I have a series of daily readings (NOx emissions) measured in parts per million. I need to aggregate the daily readings into a monthly measure. Clearly a sum operation will be incorrect (parts per million is a ratio). Am I correct that aggregating samples in parts per million can be achieved with a simple mean operation? e.g (if a month had 2 days).               100ppm (day1) + 500ppm (day2) = (100+500)/2 = 300ppm      What's the correct way to get the NOx in ppm over a whole month?


title: What physical sense has following transformation?

text: Let's have an interval expression for Friedmann Universe with 3-metric of a sphere, $$ ds^{2} = c^{2}dt^{2} - c^{2}\frac{ch^{2}(Ht)}{H^{2}}\left( d\rho^{2} + sin^{2}(\rho )(d\theta^{2} + sin^{2}(\theta )d\varphi ^{2})\right), $$ with metric tensor, which is given by $$ \hat {\mathbf g} = \begin{bmatrix} 1 & 0 & 0 & 0 \\\ 0 & -ch^{2}(Ht) & 0 & 0 \\\ 0 & 0 & -ch^{2}(Ht) \sin^2 (\rho ) & 0 \\\ 0 & 0 & 0 & -ch^{2}(Ht) \sin^2 (\rho ) sin^{2} (\theta ) \\\ \end{bmatrix}. $$ What physical sense has transformation $$ \hat {\mathbf T}, \quad |\hat {\mathbf T} = 1|, \quad T_{\alpha \beta } = T_{\beta \alpha}, $$ which is given by $$ \hat {\mathbf T}\hat {\mathbf g} \hat {\mathbf T} = \hat {\mathbf g}? $$ Maybe, the only space part of transformations determines the motion on the sphere, but what means the time part? It's something like Lorentz transformations, but for non-inertial systems. The trivial solution for $\hat {\mathbf T}$ is $\hat {\mathbf T} = diag(1, 1, 1, 1)$. But maybe there exist other solutions, don't they?


title: Velocity change of objects

text: Is it possible for small object (small mass, let's say bullet) to hit large object (big mass, let's say rock) and still move forward (or stop) instead of being reflected (let's say objects don't crush and collision takes place in one dimension)?


title: Testing my understanding of QM - The Double Slit Experiment without the slit

text: First off, sorry to throw in another question from someone who hasn't studied the maths. I'd like to see if I have a correct (if very basic and non-mathematical) understanding of the wave and statistical nature of Quantum Mechanics, and to avoid any complexity I'd like to look at the Double Slit Experiment without the slit. So we have a macro object (which is itself a very complex Quantum object that for the most part can be modeled with "classical physics") that emits electrons and can be switched to emit one electron at a time. Off to one side of the electron emitter is a flat screen that acts as an electron detector. When we tell the emitter to produce a single electron what we get is a (spherical in 3-d space?) wave of some specific energy that travels from the emitter at the speed of light. As the wave intersects with objects (other waves, fields, "classical objects") with which it is capable of interacting with there is a percentage chance (that can be modeled through QM maths) that the wave will interact with it. Not each electron emission will interact with the screen, which is expected, and the wave will continue until it reacts with an object that absorbs its energy and becomes part of the Quantum state of another object. So, is my understanding as presented in the previous paragraph correct?


I am modeling a gas flowing through a liquid. How do I calculate the Reynolds number in multiphase flows? And, at what Reynolds number should I consider the flow to be turbulent? The problem is of a typical drilled wellbore in the oil and gas industry. Fresh cement is poured into an annulus between the steel casing and the drilled formation. Before the cement is set it is vulnerable to formation gas over coming its pore pressure (or hydrostatic head) and flowing through the cement, often times forming channels for more gas to flow through in the future, rendering the cement sheath useless. **EDIT:** I am currently using a Bingham fluid to describe the viscosity of the cement slurry. The yield stress will be in the range of 300 to 700 kPa. The plastic viscosity will be in the order of 0.05-0.10 Pa.s. It is hard for me to estimate gas velocity but in preliminary models I have seen velocities in the range of 5-10 mm/s in preliminary models.


Is force the unique possible relation between particles? It was requested to further specify my question. The followings paragraphs contain some of my thoughts and confusions. **Historical motivation:**   * Since Aristotle the notion of force (dunamis, energeia) was intrinsic to the substance itself. The driving force was that the which caused movement (kinesis). Change from A to B is movement. The cause of this movement is force. However, Aristotle considered motion in a much broader sense than contemporary physics does. For example, he considered the transition between seeing and not-seeing, or sickness and health, as a kind of motion. Since the 17th century the concept of motion was restricted to spatio-temporal movement. Accordingly, the notion of force was changed into that which causes modifications of interrelated moving particles embodied in space-time. So, whereas the notion of force began as something intrinsic to a substance, it became something extrinsic, dependent on the structure of space, relating different particles and their properties. **To go back to my original question:**   * The nature of the formula $F_G = G \frac{m_1 m_2}{r^2}$ contains the spatial distance between the two particles, as well some intrinsic properties of the particles (i.e. their mass). The formula of $F_C = k \frac{Q_1 Q_2}{r^2}$ has a similar form, and also captures some intrinsic properties of the two particles (i.e. their charge). But not only does force "captures" spatio-temporal relationships and some properties of particles, it also expresses the reason of their modification. Let's look at the formula: $F=m \cdot a$. Even though this is a mathematical identity, the equation can be interpreted as a physical difference. On the left-hand-side $F$ can be read as a "cause," whereas $m \cdot a$ on the right-hand-side can be read as an "effect." Force is the cause which brings out an effect (i.e. acceleration). I am aware that this notion of "cause" may be be outdated and that Newton's action-at-distance has been replaced by the notion of a field. But the nature of a field is even more ambiguous to me. Is it merely a mathematical construct? Does it have physical meaning?   * When I look carefully at the fundamental formulas of physics, they all seem to express relations between particles and their respective states. Force is in this sense a relational concept. Moreover, whatever a particle (or a system of particles) undergoes (motion, acceleration, etc.): there was a force which caused this change. However, it's not obvious for me how particles are able to _inter_ -act and can in-fluence each other. My first guess was that forces were responsible for establishing this inter- _connection_. Forces "mediate" "between" particles. Be aware that I don't understand the words "mediate" and "between" here.


title: Why Inox Steel doesn't interact with magnets?

text: My dad has a HUGE magnet on his workshop. I love magnets, and when I saw it, I asked him what it was for. His reply was: "I don't know why, but inox steel bolts don't get attracted to it, so I use it to identify them." Thus I got curious, why a magnet don't attract inox steel bolts? Steel, even if a inox variation is still mostly iron, no?