title: Would an antimatter beam create multiple matter-antimatter explosions?

text: To expand upon the question - in an atmosphere, would a beam of pure antimatter (disregarding technical difficulties creating such a beam or passing it through a medium) interact with the matter in the atmosphere to annihilate each other and release energy in the form of a blast? Just had a thought that if the counterparts of antimatter and matter - say hydrogen and anti-hydrogen or proton-antiproton met in such a way, would it create such an outcome? Obviously, if its a beam, then the density of particles in the beam matters a lot for the annihilation rate and so does the density and composition of the atmosphere.


title: weight of fully loaded RAM

text: I had read a post Is a hard drive heavier when it is full?   It states, there is some difference in the weight of fully loaded hard drive and an empty hard drive due to different orientation of magnetic dipoles. Now, does a RAM or any other volatile storage device, completely filled with binary numbers weight more than an empty one??


title: Moment of Inertia (triangular plate)

text: I want to generalize the formula for the MOI of a triangular plate (sides $a,b,c$) about an axis passing through mid point of one sides and perpendicular to it's plane . ![enter image description here](http://i.stack.imgur.com/11Qmy.png) The mass of plate $M$ is uniformly distributed on it's area. I can use parallel axis theorem if I know the MOI about an axis passing through COM of the plate, but I don't even know that. Also there is not symmetry in this, so taking small strips of masses to integrate will also not help.So, I was unable to start Please Help. If you solve by integration then please help me get the integration term too.


title: Hot Air Balloon and Buoyancy

text: This is a conceptual question in a solution I am trying to understand. **Problem statement:** I have a balloon with a volume of V $m^3$. The outside air temp is $K$ kelvin and mass to lift is $m$ kg. I am to find the temperature inside the balloon to barely lift the given mass. The formula used is $F_{b}=ρgV$ **Solution:** I know that in order for the balloon to lift $F_b > F_g$ (subscript $b$ = buoyancy and subscript $g$ is gravity force) Using $F_{b}=ρgV$ I tried taking the differences of the two air densities and setting it equal to the mass I need to lift. As seen here. $(ρ_{air}-ρ_{balloon})Vg=mg$ **Concern:** Since $ρVg=F_{b}$ I understand this as there are THREE acting forces; One inside the balloon and one outside the balloon(which is in the NEGATIVE y-direction(drag?)) and mg. I find this strange since the balloon is not moving.


title: Young Tableau Projectors: Does the order of symmetric and anti-symmetric projectors matter?

text: Given a Young Tableau we find the irreducible basis of an arbitrary tensor by projecting, The projectors are usually defined as first symmetrise over the row entries and then anti-symmetrise over the column entries. What happens if we interchange this order that is anti-symmetrise first and symmetrise after. I understand that these projectors do not commute, however, do they correspond for a given young tableau, to the same invariant subspace, albeit with different basis?


title: Angular Velocity after a frictional impulse

text: I am modelling 2D physics collision into simulations. In Physics for Game Programmers, Grant Palmer book, ![enter image description here](http://i.stack.imgur.com/cB7RS.png) the velocity Vn1 after collision is mentioned to be independent of the friction coeff. between the surface. ![enter image description here](http://i.stack.imgur.com/Fpg7E.png) for a sphere Vn1=5/7Vn0. But this holds to true only when the sphere is assumed to be in pure rolling after the impact. How can i determine the velocity of the ball for a sliding case as well. I have to use it in a physics simulation and the duration of impulse is not known. Is there any other way to determine it?


title: Relationship Between Stream Function and Velocity Potential

text: My understanding is that a velocity potential is a scalar function whose gradient yields the x and y components of a flow's velocity vector. Velocity potential can be graphed, where lines of equivalent velocity potential denote equipotential lines. Equipotential lines, when graphed, resemble contour lines on a map which specifies altitude. Therefore, the gradient of any given equipotential line denotes 'slope', or the x and y components of velocity. I have heard that the stream function, when graphed, is perpendicular to the velocity function; also, it can only be found for 2-D flows. What is the motivation for having a stream function? What does it represent? And does it relate to the aforementioned picture of velocity potential? If so, how?


title: "Weakness" of gravitational force

text: I often hear that gravitational force is much "weaker" than electroweak and strong forces. But how can you compare the strength of interactions without the parameters like mass, charge on which it depends? Surely the gravitational force can be made as (practically) strong as one wants by increasing the mass?


title: Maxwell's equations and wave equations

text: As can be proven, Maxwell's equations don't work for spherical or cylindrical waves, but these waves satisfy the wave equation. Are there any other examples of this type that satisfy the wave equation but not Maxwell's equations?


title: Anti-particle problem for Dirac sea

text: According to the Dirac hole theory we know that Dirac sea is completely filled with negative energy, called vacuum. We will need $2mc^2$ or greater to get electron and a positron by incident photon. Now my question is, **why can't an electron fall into the see**? Is it the reason angular momentum symmetry, and if a electron falls on the Dirac sea then conservation will be violated?


title: Harmonic Oscillator - Energy quantisation

text: The one-dimensional quantum HO can be solved in Schrodinger representation by getting Hermite Differential Equation $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ with solutions $$ y(x) = \mathcal{H_n(x)} $$ which is true for integer values of $\lambda$. In the case of non-integer values of $\lambda$ the solutions are given by hyper- geometric functions $$ y(x) = c_1 H_{\frac{\lambda }{2}}(x)+c_2 \, _1F_1\left(-\frac{\lambda }{4};\frac{1}{2};x^2\right) $$ and it seems like these functions are not square integrable and hence not valid/physically acceptable wavefunctions. In this perspective, I tend to understand that quantisation comes up only as a feature of restricting the solutions to integer values, which alone are square integrable. I am looking forward to explanations concerning this aspect, since there are so many cases where the polynomial solutions occur (which are solutions for some integer value of the eigen-value in differential equation). (For instance, the solutions of Hydrogen atom). PS : In the case of infinite square well potential, the quantisation of the energy seems to come from the boundary conditions. So what is an analogy to that in the case of Harmonic Oscillator and Hydrogen atoms cases.


title: Integrating Radial Vector Fields

text: Given a integral $$\int_vd^3{r} \;\vec{r}\;\rho(r)$$ and How do you convert it to spherical coordinate system, noting that $\rho(r)$ is indeed as it is without vector, i.e. it is spherically symmetric $\rho(\vec{r})=\rho(r)$. $$\int_0^{\infty}\int_0^{2\pi}\int_o^\pi \dots \rho(r)r^2\sin(\phi)d\phi d\theta dr$$ I guess all I have to do is convert $\vec{r}$ from radial vector coordinate system to spherical coordinate system. But, I am stuck. **ADDED** There should not be $v$. $$\int d^3{r} \;\vec{r}\;\rho(r)$$ Actually, I was asked to calculate dipole moment in this case.


title: Unitary quantum field theory

text: What do physicists mean when they refer to a quantum field theory being unitary? Does this mean that all the symmetry groups of the theory act via unitary representations? I would appreciate if one could provide some references where the definition of a unitary QFT could be found. Especially in the case where there might not be a special direction singled out as "time".


title: Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

text: I have come across the equation which comes out of the nothing in Zettili's book _Quantum mechanics concepts and applications_ p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ How do I know that I get _function of position and time_ if i take an inner product of a state vector with an vector $\vec{r}$? Where is the proof?


title: Why are Only Real Things Measurable?

text: Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with operators in quantum mechanics, why must measurable quantities be Hermitian, and consequently real?


title: Adjoint of a Wave Function

text: Why is the adjoint of a function simply it's complex conjugate? Normally with a vector we consider the adjoint to be the transpose (And the conjugate? I don't know why), so does this concept carry through to these functions? Should I imagine the conjugate of a wave function to actually be a column vector? Further what does it mean for a function to be a row vector, versus a column vector. Do they live in a completely separate space? Are our operators always square?


title: Classical mechanical problem

text: I have two planes, one characterized by equation $$\phi_1=f(x)-z=0$$ and another $$\phi_2=\alpha y-z=0$$ where $\alpha$ is arbitrary. In their line of intersection(we assume it exist and is continous) is moving a body with mass $m$, that was dropped from height $h$ and moving because of gr. force $F_g=(0,0,-mg)$. I have to solve lagrange's equations of the first kind and find an equation of movement of the body. Here is what i've done: $$m\ddot{\vec{r}}=\vec{F}-\lambda\nabla\phi$$ that gives $$m\ddot{x}=-\lambda_1f'(x)$$ $$m\ddot{y}=-\lambda_2\alpha$$ $$m\ddot{z}=-mg+\lambda_1+\lambda_2$$ and from derivation of the planes, we get $$\dot{\phi}_1=f'(x)\dot{x}-\dot{z}=0\implies f'(x)=\dot{z}/\dot{x}$$ $$\dot{\phi}_2=\alpha\dot{y}-\dot{z}=0$$ $$\ddot{\phi}_1=f''(x)\dot{x}^2+f'(x)\ddot{x}-\ddot{z}$$ $$\ddot{\phi}_2=\alpha\ddot{y}-\ddot{z}$$ that, with combination with first equation is: $$m\ddot{x}=-\lambda_1\frac{\dot{z}}{\dot{x}}$$ $$m\ddot{x}\dot{x}=-\lambda_1\dot{z}$$ $$m\dot{((1/2)\dot{x}^2)}=-\lambda_1\dot{z}$$ Now, I've multiplied all other equations, summed them, and get $$1/2(m(\dot{x}^2+\dot{y}^2+\dot{z}^2\dot{)}=\dot{z}(\lambda_2(1-\alpha)-mg)$$ integrated it and happily used the energy conservation(that I extracted from the equations) and was very surprised, when I got $$\lambda_2(z-\alpha y)=mgh$$ but $(z-\alpha y)=0$ from $\phi_2$. I realized, I can't integrate it, if $\lambda_i$ is a function of spatial coordinates(that are function of time) and now I am stuck. Am I missing some trick I should use?


title: Special Relativity & Mirror Reflection

text: If you move at $5$ $ms^-$$^1$ towards a plane mirror, your reflection moves $10$ $ms^-$$^1$ towards you. But what happens if you're moving much faster, say $0.8c$? Would your reflection move at $1.6c$, since it's not a physical object? Or is it still confined to the speed of light and you have to apply the Lorentz factor? Or, does some strange light-reflecting thing occur since you're moving so fast at a mirror?


title: Calculating minimum object distance for a lens

text: I am trying to figure out the Minimum Object Distance based on a simulation available here. In case you can't open it (it's a Java applet...) I'll describe it briefly. It shows an object emitting light rays which go through 2 positive thin lenses. You can move around both the object and the lens that is further away from it. There are some questions that the user is supposed to answer. It's in French, by the way. The second question there asks what condition has to be met for the object to remain focused. I thought that, for starters, the object should be at least at the Minimum Object Distance from the lens, so I tried to calculate it like this: First, I calculate the focal length: $$ \frac{1}{b} + \frac{1}{g} = \frac{1}{f} $$ where b is the distance to the image, g is the distance to the object and f is the focal length. I am assuming that b is the distance between the rightmost lens (the one further away from the object, the one through which rays exit) and the point where the rays converge. I am also assuming that g is the distance between the leftmost lens (the one closer to the object, the one through which rays enter) and the object itself. Is that right? After that, I calculate the MOD like this: $$ MOD = \frac{f\cdot b} {b-f} $$ The thing is that, no matter how I change the values, MOD is always equal to g. This does not make sense at all. Shouldn't the MOD be always the same regardless of the position of the object? I am completely new to this, so I'm sure that I must be making an obvious mistake.


title: Principle of Least Action

text: Is the principle of least action actually a principle of least action or just one of stationary action? I think I read in Landau/Lifschitz that there are some examples where the action of an actual physical example is not minimal but only stationary. Does anybody of know such a physical example?


title: What makes a Lagrangian a Lagrangian?

text: I just wanted to know what the characteristic property of a Lagrangian is? How do you see without referring to Newtonian Mechanics that it has to be $L=T-V$? People constructed a Lagrangian in Special Relativity and General Relativity. But is there a general recipe to find a Lagrangian for a theory or is a Lagrangian just chosen so that it works out? Of course there are some symmetry and invariance properties that play an important role there, but is it possible to put the theory of Lagrangian to a more abstract and general level so that we can say a priori how a Lagrangian for a given theory must look like? Which information are necessary to construct a Lagrangian for a physical theory?


title: Observables with transcendental eigenvalues

text: Are there any "natural" physical observables which have non-empty point spectrum which consists of numbers which are not algebraic numbers?


title: Infinitesimal volume using differentials

text: I don't understand why in some texts they put that infinitesimal volume $dV = dx dy dz$. If $ V = V(x,y, z)$ infinitesimal volume should be $$dV = \frac{\partial V}{\partial x} dx +\frac{\partial V}{\partial y} dy + \frac{\partial V}{\partial z} dz$$


title: Is proper time renormalization gauge invariant?

text: Proper Time Renormalization is achieved by putting: $$ \int_0^\infty e^{iat} dt = {1\over ia} $$ Is it true that this is the only kind of normalization that is gauge invariant? If so, why do famous books ignore it even though it has such an important property (gauge invariance)? Does the reason behind the name have anything to do with this?


title: Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field

text: I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 + q\phi $$ and also $$H=\frac{1}{2m}(\vec{p}-\frac{q}{c}\vec{A})^2 + q\phi $$ with $\vec{p}$ the momentum, $q$ the electric charge, $\vec{A}$ the vector potential, $\phi$ the scalar potential and $c$ the speed of light. So basically the difference is in the term $1/c$ multiplying $\vec{A}$, present in the second form (which I use in my lectures) but not in the first one (used by Griffiths in _Introduction to quantum dynamics_ to treat the Aharonov-Bohm effect). Why does this difference exist and what does it mean? And how does the term $1/c$ affect the dimensional analysis (the units) of the problem?


title: What does the graviton-matter Feynman scattering diagram look like?

text: The question is in the title really. I know gravity + QFT is on shaky ground but I'm not looking for Feynman rules, just a schematic, if that is possible. Edit 1: added possible diagram for clarification. ![Possible diagram](http://i.stack.imgur.com/isxw2.png) **Edit 2: Actually, isn't _this_ diagram the most important?** ![enter image description here](http://i.stack.imgur.com/ObveI.png) G represents a graviton and m represents some matter, e.g. proton, or quark.


title: Why is the contribution of a path in Feynmans path integral formalism $\sim e^{(i/\hbar)S[x(t)]}$

text: In Feynman's book "Quantum Mechanics and Path Integrals" Feynman states that > the probability $P(b,a)$ to go from point $x_a$ at time $t_a$ to the point > $x_b$ at the time $t_b$ is $P(b,a) = \|K(b,a)\|^2$ of an amplitude $K(b,a)$ > to go from $a$ to $b$. This amplitude is the sum of contributions > $\phi[x(t)]$ from each path. $$ K(b,a) = \sum_{\text{paths from $a$ to $b$}} > \phi[x(t)]$$ The contributions of a path has a phase proportional to the > action $S$: $$ \phi[x(t)] = \text{const}\ e^{(i/\hbar)S[x(t)]}$$ Why must the contribution of a path be $\sim e^{(i/\hbar)S[x(t)]}$? Can this be somehow derived or explained? Why can't the contribution of a path be something else e.g. $\sim \frac{S}{\hbar}$, $\sim \cos(S/\hbar)$, $\log(S/\hbar)$ or $e^{- (S[x(t)]/\hbar)^2}$ ? **Edit:** I have to admit that in the first version of this question, I didn't exclude the possibility to derive the contribution of a path directly from Schrödinger's equation. So answers along this line are valid although not so interesting. I think when Feynman developed his formalism his goal was to find a way to quantize systems, which cannot be treated by Schrödinger's equation, because they cannot be described in terms of a Hamiltonian (e.g. the Wheeler- Feynman absorber theory). So I think a good answer would explain Feynman's Ansatz without referring to Schrödinger's equation, because I think Schrödinger's equation can only handle a specific subset of all the systems that can be treated by Feynman's more general principle.


title: Do the properties of two streams flowing into an engine add together when they mix?

text: Let's say I have two stream supplys that have separate mass flow rates, enthalpies, and velocities that flow into an engine. Upstream of the engine's inlet, the two streams mix to form one stream that flows into the engine inlet. Do the properties of the streams add together when they mix like the mass flow rate adds together?


title: Is there any know weakness with GNUplot fits? (especially with gauss)

text: my faculty push pretty hard on us using GNUplot but it gives me somewhat unconvincing results while attempting to fit a gauss-like function. Is there any known weakness with GNUplot fits or should I trust it? Should I try and learn ROOT (also heavily pushed by my faculty)? Thanks (btw I'm studying physics at Padova University, Italy)


title: Book against a wall and forces

text: If you take a book with mass of 1kg and push it against the wall. With how much force do you have to push the book so it does not fall? The problem is I know how to calculate this problem, you say $F_{friction}$=$F_{gravitational}$ and $F_{wall / normal}$ = $F_{human / push}$. The problem is solvable if you say that $F_{gravitational}$ = $F_{normal / wall}$, but why is this true? How would you calculate this problem if you didn't know that $F_{normal}$ = $F_{gravitational}$, how would you prove this statement? For me the problem is that $F_{normal}$ = $F_{gravitational}$ * $\cos(\alpha)$, but cos(angle) is 0, I don't understand the relationship between $F_{gravitational}$ and $F_{normal}$ in this scenario.


title: Drum head coupled with an RLC circuit

text: I'm thinking about the possibility of an electroacoustic drum that would be tuned with some kind of RLC circuit. The drum head would have an electromagnet attached to its center, which would be directly above a small permanent magnet mounted firmly to the sides of the drum. The electromagnet would be the inductor in an RLC circuit. If I tuned the circuit to some physical mode of the drum head/electromagnet, would I get any coupling between the circuit and the physical drum? Would it be enough to to alter the sound of the drum?


title: Name of the guy that Feynman mentioned during a lecture: the diagram is of a chain hanging over a triangle

text: In a Feynman book, he talks about a man (I believe he lived 400-500 years ago) that discovered something about the dimensions of triangles (I think)by hanging a chain around the triangle. I've searched in "Six Not So Easy Pieces" but haven't been able to find it. I'm trying to find the name of this man; it would be great if someone remembers this (or know whom I'm talking about off the top of their head).


title: A hot object exposed to low temperature in a vacuum doesn't lose heat?

text: I heard somewhere that if the human body were exposed to outer space where the temperature is extremely low, the human won't actually feel cold because in a vacuum, the heat energy doesn't have another physical object to travel too. So my question is, i thought heat is just a form of radiation, and that radiation can travel through a vacuum. So then, why doesn't a body lose heat in a vacuum the same way our sun emits a whole bunch of UV radiation? In which case, the body should be losing energy and feeling cold.


title: Even-branes in IIA and odd-branes in IIB

text: The R-R sector of IIA and IIB are respectively given as, $8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$ $8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$ Now looking at this Polchinski in his lecture notes states about developing the vertex operators for the R-R states as, "..This will involve a product of spin fields $e^{-\frac{\phi}{2} - \frac{\bar{\phi} }{2}} S_\alpha \tilde{S}_\beta$. This again decomposes into anti-symmetric tensors of $SO(9,1)$ as, $V = e^{-\frac{\phi}{2} - \frac{\bar{\phi} }{2}} S_\alpha \tilde{S}_\beta (\Gamma^{[\mu_1}...\Gamma^{\mu_n]} C)_{\alpha \beta}G_{[\mu_1 ... \mu_n]}(X) $ with $C$ as the charge conjugation matrix. In IIA theory it is $16\otimes16'$ giving even $n$ ($n \sim 10-n$) and in IIB it is $16\otimes16$ giving odd $n$..." I have absolutely no clue as what happened here in the above argument! What are these spin-fields from nowhere? What is this $G$? How is this $V$ conceived? Where did these $16$ and $16'$ representations come from?   * Can someone kindly split this up into conceptual chunks and may be reference/explain what happened here? How does this lead to the conclusion about there being only $0,2,4,6,8$-branes in IIA and $-1,1,3,5,7,9$ branes in IIB?


title: Motion of objects (beginner)

text: I have begun physics and we are seeing systems of reference as an introduction.Now I have a small exercise which I'm not sure of, I would need you to check if it's correct or not. There are two cyclists A and B. Suppose that you take place on a sidewalk and look at them pass. They move to the right and to a constant speed. What trajectory do you associate to the cyclists ? Rectilinear (so, one "x" axis.) What answer would the cyclist A give if we asked her to determine the trajectory of cyclist B ? Rectilinear also. Represent the trajectory of the reflector placed in the front wheel of cyclist B, such that you would see her from the sidewalk. ![enter image description here](http://i.stack.imgur.com/eEEOb.jpg) The curves are halfs of a circle and I considered that the reflector began at half the height of the wheel.( so there's the highest point, middle point and the case where the reflector "touches" the ground.)


title: Relativistic space-time geometry

text: What subject (suggest book titles, etc.) should I study to get a clear grasping of hypersurfaces, 2-surfaces, and integration on them, mostly in special relativity (I'm not messing with general relativity yet).


title: What are constraints on a "purity" operator in quantum mechanics?

text: Consider the normalized state, written in some orthonormal basis as: $$\psi = A |0\rangle + B |1\rangle$$ Let's define a "purity operator" for a basis as any operator whose _expectation value_ gives 1 for a pure state in this basis, and 0 for the most mixed state in this basis. Inbetween states should give between 0 and 1, although the specific value doesn't matter. One possible example (please note my question is on the general case though, this specific example is just to aid discussion), is $$\langle \mathcal{O} \rangle = 1 - 4 \frac{|A||B|}{|A|+|B|}$$ What mathematically prevents such measurements in quantum mechanics?


title: The 4D Physics of a Superstring Background

text: Suppose I have a compactification of string theory to four Minkowski dimensions of the form $M^{1,3}\times X$, where the internal CFT on $X$ is a $c=9$, $(2,2)$ SCFT. For example, let the SCFT on $X$ be described by the IR fixed point of an LG model with the superpotential $W(\Phi)=\Phi^p$. Can someone explain to me (hopefully in details) how can we extract the space-time physics in $M^{1,3}$ from the data of the internal CFT. What is the relation between the world-sheet superpotential $W(\Phi)=\Phi^p$ and the space-time superpotential? What is the space-time interpretation of the chiral ring of $W(\Phi)$? How does the non-perturbative corrections to the space-time superpotential are related to the correction to $W(\Phi)$. I know this is a basic question in string theory and it should be easy to find an answer in Polchinski but I hope someone will clarify things further.


title: Would a fish in a sealed ball swim normally?

text: This question led me to wonder whether swimming would be the same experience for a fish in a full, sealed ball as it is normally. If the fish is about 7cm from the walls of the tank, a pressure wave can propagate from the fish to the wall and back in .0001 seconds, while the time scale on which a fish wiggles is tenths of a second. So unlike the open ocean, the water surrounding the fish can all communicate with itself on the time scale that the fish wiggles, and unlike a normal fish tank, the water has nowhere to go and so can't change its shape. Would the fish notice any hydrodynamic effects in a full, sealed tank compared to normal swimming?


title: More on matter and anti-matter

text: 1.) Does every particle that has rest mass also have an anti-particle with which it would annihilate? 2.) Does annihilation only occur between like particles? For example what happens if a antineutron (anti u, anti d, anti d) collides with a proton (uud)? What happens if a positron collides with a proton? 3.) Since the Tevatron accelerates antiprotons is this more difficult to handle and dump? 4.) I've read about WIMP annihilation detection. Why would one assume there is any different proportion of WIMPS to anti-WIMPS than as is for non-WIMP matter (where there is far more matter than anti-matter).


title: Units of the Stokes-Einstein rotational diffusion coefficient

text: The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as: $$D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi \eta)(r)^3}$$ Where $k_B$ is Boltzmann's constant, $T$ is the temperature in Kelvin, $\zeta_f \approx (8 \pi \eta)(r)^3$ is the friction, $\eta$ is the viscosity of the medium (e.g. $\approx 1 \space cP$ in pure water), and $r$ is the radius of the sphere. For an example calculation of $D_r \approx 0.2 \space rad^2/s$ (or $Hz$) of a $\approx 1 \space \mu m$ radius sphere in pure water at room temperature, please see this site (this shortened URL is going to WolframAlpha). However, as we can directly see in the calculation, the units for $D_r$ are clearly expressible as rad/s or Hz. Why are the units for $D_r$ always reported as rad$^2$/s in the literature?


title: Physics of scaling up an animal: the neck

text: Consider an animal like a horse. Now scale its neck longer and longer. How can a giraffe, or even worse a huge dinosaur, raise its neck without the tendons snapping? The dinosaur case in particular seems ridiculous. Is there a "physics trick" the animals use to make this more manageable? Or does the tendon tension not scale as badly as my intuitiion is claiming?


title: $\frac{1}{\sqrt{2}}$ (|Independent particle Model⟩+ |Strong Interaction Model⟩)?

text: What is an adequate way to understand this simultaneously. One has the underlying assumption that matter is saturated and has the merit of being able to come up with an accurate formula for the Binding Energy (the SEMF/Bethe- Weizsäcker formula) and the other can explain the magic numbers by building up energy levels in a shell structure similar to atomic orbitals. However, the underlying assumptions appear to be completely contradictory. I am looking for a better perspective on how to understand these two simultaneously.


title: Tangential acceleration in circular motion?

text: A lot of my problems have objects moving in circular paths with tangential and normal components of acceleration. If the tangential component is non-zero though, the speed is changing so the radius of its path would also be changing. So how can its path be considered circular? The size of of its radius of curvature of constantly changing.


title: Translation Operator for Position on Momentum

text: Consider the translation operator $$\hat T=\exp[-ic\hat p/\hbar], $$ which acts on the position operator in the following way: $$\hat T^\dagger \hat q\hat T = \hat q+c.$$ If I take $\hat T ^\dagger \hat p \hat T $, will this just spit back p? I tried doing out the math explicitly and found that the commutator $[\hat p,\hat T]=[\hat p,e^{\hat p}]=0$ which means that it does not actually translate it. My confusion is if this will spit back p, or if this quantity should be evaluated to be 0.


title: Local Charts in General Relativity

text: We may consider a "local" region in curved spacetime (local in respect of the spatial and the temporal coordinates). A "local inertial frame" may be constructed by some transformation that produces flat spacetime locally. This transformation produces the diagonal [1,-1,-1,-1] in an approximate manner. A physical point: You are performing some experiment in a small laboratory room (stationary w.r.t. the Earth) for a small period of time. Are you in a local inertial frame? If there was a freely falling lift in front of you, it would have been a better approximation to the inertial frame concept. So your laboratory does not qualify to be an inertial frame when compared with the falling lift. (The difference becomes even more conspicuous if you imagine "gravity" to be a 100 or a thousand times stronger.) Now, let's move into the problem: We take a small region of curved spacetime and use some suitable transformation to produce a "local inertial" frame. For this transformation we are _disregarding_ all the anisotropies and inhomogeneities of the surrounding space in the original manifold.Incidentally the objective behind creating local inertial frames is to apply SR. The associated Lorentz transformations are supposed to hold good in the ideal conditions of isotropy and homogeneity of space Our transformation transforms a small region of curved spacetime to a small region of flat spacetime. To this small/finite flat spacetime, we add the rest of it, ignoring all the anisotropies and inhomogeneities of the surroundings in the original curved spacetime [original manifold]. In effect we are _assuming_ a "Global Transformation" without working it out! (And we expect correct results when we go back to the original space by the reverse transformations) We are in effect _assuming_ a "Global Transformation" from curved spacetime to flat spacetime. Is it not as sacrilegious as "Flattening a Sphere"? How much unreliability is the aforesaid error going to cast on calculations performed in the local inertial frames?To what extent will this error affect the Principle of Equivalence?


title: Is it possible to have a singularity with zero mass?

text: A singularity, by the definition I know, is a point in space with infinite of a property such as density. Density is Mass/Volume. Since the volume of a singularity is 0, then the density will thus become infinite because Mass/0 = undefined However, is it possible to have a singularity with a mass of 0? 0/0 is indeterminate, but would it be possible for a singularity to exist even if its mass and volume were zero?


title: Predicting stellar evolution and life cycle of a star

text: Is there a way to model/predict the evolution and life cycle of a star based on certain initial conditions? That is, whether it will become a red giant, brown dwarf, etc. I'm basically looking for equations defining the progress of different star-types and how varying initial conditions may affect that progress.


title: Is there a critical order of the Abelian gauge theory in (2+1)D

text: In (2+1)D spacetime, it is known that the $U(1)$ gauge theory is always confined (according to Polyakov), while the $\mathbb{Z}_2$ gauge theory can support a deconfined phase. Now consider a generic Abelian gauge theory, i.e. the $\mathbb{Z}_n$ gauge theory (with $n$ being the order of the gauge group), the question is that does the $\mathbb{Z}_n$ theory in (2+1)D spacetime always support a deconfined phase, or there exist a critical $n_c$ such that for $n>n_c$ the $\mathbb{Z}_n$ theory is always confined? The motivation of this question is that I believe in the $n\to\infty$ limit, the $\mathbb{Z}_n$ theory should behave the same as a $U(1)$ theory (let me know if it is not the case), while when $n=2$, it must reduce to the $\mathbb{Z}_2$ theory. But the $\mathbb{Z}_2$ theory is deconfined and the $U(1)$ theory is confined, so I am wondering if there exist a critical order $n_c$ such that beyond $n_c$ the Abelian gauge theory is always confined. Also the gauge theory has a string correspondence (in terms of the dynamics of flux tubes, or in terms of the dynamics of membranes covering the Wilson loops). I would like to know if the above question could also be explained in the string theoretical language (in terms of the string tension, string renormalization ect.).


title: What direction does the force vector point in regards to Earth's E and B fields?

text: In regards to the right hand rule, given Earth's electric and magnetic field, in which "direction" would a particle go?


title: Why is there no massless spin 1/2 particles with charge? Is there any underlying theory?

text: Whether any massless spin 1/2 particles cannot have charge? If so, why?


title: Are there (or could there be) electrically charged particles that move at the speed $c$?

text: A photon, a neutrino (if it has zero rest mass) move at $c$ but what about charged particles? If the answer is no, is there a fundamental reason or just because of the radiation it emits?


title: How might a resonant antenna and black body radiation interact?

text: How does an antenna behave when it is cooled so that its black-body radiation is emitting energy at its resonant frequency? Edit: To clarify, its not how they're related in general, but how might thermal radiation and resonance interact with each other when their spectra are aligned well? Edit: Also, I'm sure that the thermal radiation spectra that have a significant peaks are associated with incredibly high temperatures, and peak at incredibly small wavelengths, rendering such an antenna completely impractical to build. Still, I'm still interested in the theoretical concept.


title: Pseudoscalar action in classical field theory

text: I was reading Landau and Lifschitz's "Classical Field Theory" and came across a comment that the action for electromagnetism must be a scalar, not a pseudoscalar (footnote in section 27). So I was wondering, is it possible/interesting to construct a classical field theory with a pseudoscalar action? If not, why not? (Note: I was motivated to look at this because of a question from Sean Carroll's "Spacetime and Geometry", which asks us to show that adding a pseudoscalar term ($\vec{E} \cdot \vec{B}$) to the Lagrangian doesn't change Maxwell's equations.)


title: Would the ionic wind power fence actually work?

text: Described here and in the references (patent, patent), the idea is to use a large fence as a grid electrode. Spray water droplets near a high voltage to charge them, and then the wind blows them away, converting wind pressure into electric potential?, then they hit another electrode or the Earth to complete the circuit. ![enter image description here](http://i.stack.imgur.com/sCvNH.png) > The scientific basis for this is shown in FIG. 9 which shows charged > particles 16, 16' removed toward infinity from a single isolated conducting > sphere 17 of radius r0 According to the principles of electrical physics, > work W=QV is done in removing the charges to infinity. Would such a device actually work? If so, why isn't it built or used? Can the energy output be estimated and compared to a traditional wind turbine? Some simple calculations: According to the patent, the pylons to support the fence in the image are 1 km apart and 150 m high, and a 100,000 m² section of this size would produce 45 MW. This would be 450 W/m². The largest turbine in the world is the German RePower turbine producing 5 MW while sweeping an area of 12,000 m² = 401 W/m², so not much of an improvement?


title: How does the Dyson air multiplier fan create negative pressure?

text: I'm curious as to how the Dyson air multiplier fan works. So far I understand that it creates negative air pressure that causes air behind and around the device to be sucked into the air stream and blown out. My question is how is the negative air pressure created? I haven't found an explanation for that yet.


title: momentum conservation question involving a rocket and a spaceship

text: With the engines off a space ship is cruising at a velocity of 230m.s It fires a rocket straight ahead at the enememy vessel. The mass of the rocket is 1300kg and the mas of the ship (not including the rocket) is $4\times10^6$kg. The firing of the spaceship brings the spaceship to a halt. What is the velocity of the rocket.


title: How do we know that C14 decay is exponential and not linear?

text: In my previous question I asked Please explain C14 half-life The OP mentioned that I was thinking of linear decay and C14 was measured in exponential decay. As I understand it, C14 is always in a state of decay. If we know the exact rate of decay then shouldn't it be linear? How do we know that C14 decay's exponentially compared to linear and have there been any studies to verify this?


title: When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion?

text: Im trying to get my head around SMH out of curiosity because it seems simple yet I'm not getting the concept behind some ideas. For a SMH equation : $$ x=a \sin(\omega t+\phi) $$   * Under what conditions will each of the two ( velocity and acceleration ) be at maximum ? we know that : $$ x'=-\omega \cos(\omega t + \phi) $$ $$ x''=-\omega^2 \sin(\omega t + \phi) $$ Also :   * how would it be different for $x = a \cos(\omega t - \phi)$ ?


title: What formulations of QM are there?

text: It is usually said that there are different formulations of QM, for example historically there was Schrodinger's (wave mechanics), and Heisenberg's (matrix mechanics), then Dirac's (which showed they are equivalent) Since they are all physically equivalent I have a few questions: 1-Is Dirac's formulations considered more fundamental in the sense that it can be reduced to one of the 1st two? 2-I also hear about the path integral and density matrices, are they another formulations? 3-Are there more formulations less known to undergraduate students but known by researchers because they are technically advanced?


title: Does path integral and loop integral in a Feynman diagram violate special relativity?

text: 1. Consider a correlation function between two points $A(x_1,t_1)$ and $B(x_2,t_2)$, we need to integrate over paths which could be infinite long. But the time length $(t_1-t_2)$ is finite, so if $A$ and $B$ are the coordinates of one single particle, then all of the paths from $A$ to $B$ should be time-like curves, the maximum length should be $c\times(t_1-t_2)$, which is not infinitely long. It seems special relativity could be violated here.   2. Consider any loop integration of higher order correction to a Feynman diagram calculation, we have infinitely many off-shell processes and "internal virtual particles", which could be created and annihilated without taking any time. Does this violate special relativity? Is there any better reason than simply saying these are "internal virtual processes"? They do affect our final real physical observations! for the second case, one might argue that no particles here but only fluctuating fields? But when, where, and how are those fields created? This picture is not that physically clear to me. Any better insight?


title: Clarification needed in the concept of apparent depth & real depth

text: I understood the concept of apparent depth from here: ![enter image description here](http://i.stack.imgur.com/1VzQx.png) ![enter image description here](http://i.stack.imgur.com/GjVyc.png) But one thing I didn't understand is, will there be difference in the real depth and apparent depth when we are looking not at an angle as shown above but vertically downward (along the normal) as shown in the figure below : ![enter image description here](http://i.stack.imgur.com/QGc9i.png) According to me real depth and apparent depth should be same because the light rays coming out of the object is not undergoing any refraction. If I'm right. Then the answer to the question below must be 6 m/s but Its 8m/s. How come? > A bird is flying 3 m above the surface of water. If the bird is diving > vertically down with speed = 6 m/s, his apparent velocity as seen by a > stationary fish underwater is : (A) 8 m/s (B) 6 m/s (C) 12 m/s (D) 4 m/s


title: Air pressure relative to a force on a bag?

text: Assume an airtight bag occupied by air such that the pressure inside the bag is equal to the atmospheric pressure. Assume the surface tension of the bag is negligible. What is the change in air pressure inside the bag relative to a weight placed on it? Is it possible to approximate the weight of the object on top of the bag by the air pressure inside the bag, without knowing the volume of the bag or the area of the object?


title: How specifically do emulsifiers work?

text: I'd like to understand better how emulsifiers prevent droplet coalescence. There must be something more they do than just lower the surface tension between the droplet and the ambient substance. I tried to look up the answer but there's a lot of qualitative semi-explanations floating around that seem to choke out any precise explanation. If anyone can give a detailed explanation, that would be great!


title: How would natural (resonant) frequencies affect amplitudes?

text: I read $y=A\sin(2\pi ft)$, where $A$=Amplitude, $f$=Frequency, $t$=Time and $y$=$Y$ position of the wave. Since natural frequencies only take the most effect when they are close to the frequency. How would one natural frequency and several natural frequencies affect the equation? Would I be correct in thinking it's something to effect of: `y=Y_Position*NaturalFrequency` where Y_Postion is the first equation and NaturalFrequency is similar to the first equation but with a low amplitude?


title: How to find speed from Newton's law?

text: I have a question about Newton's law. The question says > block A(mass 2.25kg) rests on a tabletop. It is connected by a horizontal > cord passing over a light, friction less pulley to hanging block B(mass > 1.30kg). The coefficient of kinetic friction between block A and the > tabletop is 0.450. After the blocks are released from rest, find (a) the > speed of each block after moving 3.00 cm and (b) the tension in the cord. > Include the free-body diagram or diagrams you used to determine the answer. I tried to solve using following formula $$f_{k} = \mu_{k}N$$ $$T-w_{B} = m_{b}a$$ $$T = m_{A}a$$ First time, I tried to get $a$ Since $$f_{k}=\mu_{k}N$$ $$m_{A}a = \mu_{k}N$$ $$m_{A}a = \mu_{k}m_{A}g$$ From given $m_{A} = 0.45, \delta x=0.03(m), m_{A}=2.25$ I cancelled out $m_{A}$ both side then I got $$a = \mu_{k}g$$ which is $$a = 0.45*9.8 \implies a = 4.41\ \mathrm{m/s^2}$$ Part a asked for its speed so I used $\delta x = \frac{V^2-V_{0}^2}{2a}$ I got $V = 0.897\text{ m/s}$ but the speed I got was not right speed. So I think I made some mistake at some point. What did I do wrong?


title: Hamiltonian of the charged current in SM (related to the lorentz invariance)

text: recently when I was studying the scatterings which involves a vector boson (like W boson) as an intermediate particle, I saw that the propagator is not Lorentz invariant, I read that there is another source of Lorentz non- invariance in the Hamiltonian of the charged current. I mean the Hamiltonian is not simply the minus of the Lagrangian in this case h=-l. the extra term is not Lorentz invariant and, these two source of the Lorentz non-invariance will cancel each other eventually in the S-Matrix. (you can see :The standard Model, A primer written by Burgess chapters 4,5 and problem 5.4). I don't understand this completely, I'll be grateful if anyone can help me and explain it or introducing a reference for it. I apologize for my bad English writing. Thanks


title: In what situations do I use the characteristic length of a fin to find the surface area?

text: So I'm learning about fins in heat transfer and it seems that there are two separate formulas for the surface area of a rectangular fin of length L, width w and thickness t. The fin is attached to a wall at one of its L x t face. The formula for the surface area for a normal rectangular fin would be: $$A_{f}=2Lw+2Lt+tw$$ Now there is also a formula using the characteristic length $L_{c}$ $$A_{f}=2wL_{c}$$ Now my question is **Under what conditions do I use the characteristic length to determine the surface area of the fin?**


title: Physics for taffy pulling?

text: I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea. Is there a general equation for an objects threshold and pulling it apart?


title: Could entropy explain dark energy?

text: This was 3rd beer idea, so please bear with me. What if the universe was not actually expanding but the speed of light was slowing? Wouldn't that be indistinguishable to our observations? Either way we should see an increase in the redshift the further a light source is from us? Suppose we're in a giant extremely advanced compute cluster, and the speed of light has to do with how long it takes data to travel between compute resources on the network/bus. As more data travels across this bus (entropy increases in the universe) it takes longer to reach its destination (speed of light slows.) This seems vaguely reasonable considering fundamental limitations past and present on computers of varying types, there's always a trade off between size of data and speed of access that has it's roots in physics. So I know it's a bit wacky, but it sounds easily falsifiable to me, simply plot entropy over time and dark energy over time and see how well they correlate (or don't most likely.) Can anyone help me test my armchair theory? I think that would be a fun exercise. I know dark energy has been plotted over time, but what about entropy, is that possible?


title: Status of experimental searches for tachyons?

text: Now that the dust has settled on the 2011 superluminal neutrino debacle at OPERA, I'm interested in understanding the current status of experimental searches for neutrinos. Although the OPERA claim was a fun puzzle for theorists trying to find explanations, the claimed scenario wasn't anything like our current best guess as to what to look for in a search for tachyons. E.g., we do expect tachyons to be electrically neutral ( _pace_ Baldo 1970), but we also really expect them to have spin 0. The only particle-physics test I've been able to locate information about is Alvager 1968. They searched for production by photons on lead and put an upper limit on the cross-section. The experiment would only have been sensitive to charged tachyons. This is extremely old, predates the modern understanding of how tachyons fit into QFT, and is of less interest than searches for neutral tachyons, which are what we actually expect to exist. There is a big bibliography here, but it mostly focuses on the OPERA debacle. Recami 2009 is a recent review article, but I don't have access to the conference proceedings, and the article doesn't seem to be available online. Tachyons are expected to exist as waves that are either localized or superluminal, but not both (Shay 1977, Baez). So it seems like superluminal propagation of a wave packet, as in the OPERA result, is not really the right experimental signature to look for. What is the right signature? Non- electromagnetic Cerenkov radiation? If they're expected to be nonlocalized, does it make more sense to look for cosmological evidence, as in Davies 2012? It seems tough to define an appropriate way to search for tachyons, since QFT basically says they shouldn't be able to exist in the naive classical sense envisioned 50 years ago. On the other hand, the willingness of theorists to try to explain the OPERA results suggests that there is wiggle room. If you look at a freshman text like Tipler 2003, he's still referencing Alvager 1968 as the null experimental result on tachyons. This seems unsatisfactory. I can't believe that there haven't been better direct searches in the last 45 years. This question is similar to, but much more specific than, a previous question: Tachyon and Photons Alvager and Kreisler, "Quest for Faster-Than-Light Particles," Phys. Rev. 171 (1968) 1357, http://adsabs.harvard.edu/abs/1968PhRv..171.1357A Baez - http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html M.Baldo, G.Fonte & E.Recami: “About charged tachyons”, Lett. Nuovo Cim. (first series) 4 (1970) 341-247. http://dinamico2.unibg.it/recami/erasmo%20docs P. C. W. Davies, Ian G. Moss, "Cosmological bounds on tachyonic neutrinos" http://arxiv.org/abs/1201.3284 Recami, "Superluminal waves and objects: An overview of the relevant experiments," Journal of Physics: Conference Series, Volume 196, Issue 1, article id. 012020, 14 pp. (2009) Shay and Miller, "Causal and noncausal propagation of both tardyon and tachyon wave functions," Nuovo Cimento A 38 (1977) 490, http://link.springer.com/article/10.1007%2FBF02730018 Tipler, Modern Physics, 2003 [EDIT] I did succeed in finding some references more recent than 1968. I don't know if these represent the state of the art, but to avoid wasting other people's time, I'll list them here: Clay, A search for tachyons in cosmic ray showers, http://adsabs.harvard.edu/full/1988AuJPh..41...93C Australian Journal of Physics (ISSN 0004-9506), vol. 41, no. 1, 1988, p. 93-99. -- cosmic rays, time of flight Baltay, C., G. Feinberg, N. Yeh, and R. Linsker, 1970: Search for uncharged faster-than-light particles. Phys. Rev. D , 1, 759-770, doi:10.1103/PhysRevD.1.759. -- accelerator experiment The following are not experimental papers, but do give helpful surveys of methods of searching for tachyons: Recami, "Superluminal Waves and Objects: Theory and Experiments. A Panoramic Introduction," ebooks.iospress.nl/Download/Pdf/28897 Sudarshan, "The nature of faster-than-light particles and their interactions" http://wildcard.ph.utexas.edu/~sudarshan/pub/1969_005.pdf


title: Liquid benzene magnetic susceptibility

text: In a solid state physics problem, I'm asked to make a rough estimate of the contribution to the diamagnetic susceptibility of the outer electron of each carbon atom. The wavefunction of these electrons extend round the whole ring of atoms. The only data I'm given is the benzene hexagon side length, 1 A; the benzene density, and the molecular weight. Attempt of a solution: I think that I have to consider the effect of an applied magnetic field B perpendicular to the plane of the hexagonal molecule. That will make the outer electrons move along the ring, generating a current that generates a magnetic field opposing B. Am I right? If yes, how to derive the magnetic susceptibility following this line of reasoning? Thank you in advance,


What is the significance of negative frequency in Fourier transform? Why we include the band widths of the negative frequency also while calculating band width of the signal.


title: Covering centeremost slit of a N slit diffraction grating - what happens?

text: For an N-slit diffraction grating, the distance from a maxima to a minima at order p is given by $$\delta \theta = \frac{\lambda}{Np}$$ What happens to this width when the centremost $\frac{N}{2}$ slit is covered? I'm tempted to think that one slit won't make a difference if N is large like 1000 slits.


title: How can a laser pointer have range of several kilometers in atmosphere?

text: Laser pointers manufacturers claim that some pointers have range of several kilometers. Okay, they use a powerful laser, but that powerful laser usually has power less than one watt. Okay, the laser beam is very focused. But what about atmosphere particles like dust and vapor? Why don't these particles diffuse the beam and make it lose energy? How does it happen that a less than one watt laser can have several kilometers range in atmosphere?


title: Is single photon annihination of electron-positron pair prohibited by Feynman diagram analysis?

text: It is obvious that electron-positron pair cannot annihilate to a single photon which will violate the momentum conservation. My question is can we get this knowledge from Feynman diagram or perturbative expansion of QED? If we can't, how can we make sure not to over-count diagrams when we draw Feynman diagram to compute the scattering amplitude?


title: Will two trains running along the equator in opposite direction experience same wear out?

text: Two identical trains, at the equator start travelling round the world in opposite directions. They start together, run at the same speed and are on different tracks. Which train will wear out its wheel treads first? Will their weight change? Answer is given as : the train travelling against the spin of the earth. This train will wear out its wheels more quickly because the centrifugal force is less on this train. How do the forces change when the frame of reference is same for both trains?


title: limits of diamond anvils for high pressure research

text: in this wikipedia article regarding diamond anvils, it mentions that the pressure peaks roughly at 300 GPa. My question is why is this so? is the diamond crystal structure collapsing if higher pressures are applied (like 500-600 Gpa, where metallic hydrogen is expected to form? and if such collapse happens, what sort of phase does the diamond collapses into?


title: Basic question on experimental plots

text: On the following Higgs $\rightarrow$ Tau Tau plot, since we are plotting the ratio of $\frac{\sigma}{\sigma_{SM}}$ on the y axis, shouldn't the expected for this be 1? i.e., shouldn't the expected 68% and 95% be centered at a dotted line at 1? Anything else seems to imply we are expecting something other than the Standard Model... ![Higgs ->Tau Tau](http://i.stack.imgur.com/dffNF.png)


Is Invisibility possible according to physics? Is there any backing theory to prove it true or false?


title: $(\mu,P,T)$ pseudo-ensemble: why is it not a proper thermodynamic ensemble?

text: While teaching statistical mechanics, and describing the common thermodynamic ensembles (microcanonical, canonical, grand canonical), I usually give a line on why there can be no $(\mu, P, T)$ thermodynamic ensemble ($\mu$ being the chemical potential, $P$ being the pressure, and $T$ being the temperature). My usual explanation is a handwaving that all of the control parameters would be intensive, which leaves all the extensive conjugated parameters unbounded, and you can't write your sums and integrals anymore. However, I have always felt uneasy because:   1. I never was able to properly convince myself where exactly the problem arose in a formal derivation of the ensemble (I am a chemist and as such, my learning of statistical mechanics didn't dwell on formal derivations).   2. I know that the $(\mu, P, T)$ can actually be used for example for numerical simulations, if you rely on limited sampling to keep you out of harm's way (see, for example, F. A. Escobedo, _J. Chem. Phys._ , 1998). (Be sure to call it a _pseudo-ensemble_ if you want to publish it, though.) So, I would like to ask how to address point #1 above: **how can you properly demonstrate how chaos would arise from the equations of statistical mechanisms if one defines a $(\mu, P, T)$ ensemble?**


title: Thermal expansion of Sphere

text: How would one go about writing an expression of the expansion of the volume of a sphere of a given material? I noticed a few sources give it as $\Delta V= 3\gamma V\Delta T $   where **V** is the initial volume; $\gamma$ is the expansivity coefficient and $\Delta T$ is change in temperature of sphere. Other texts leave out the 3, but with everything else the same. Any suggestions?


title: What is on the AdS side in AdS/CFT supergravity or string theory?

text: What really is on the AdS side in AdS/CFT, does it always have to be string theory or is sometimes supergravity "enough" or better suited to do calculations? From the answers to my earlier question, I have learned that one can calculate CFT/QFT correlation functions on the boundary from the quantum gravitational partition function valid inside the AdS space by taking the bondery value $$ <O(x_1)O(x_2)...O(x_n)> \sim \frac{\partial^n Z}{\partial \Phi_0(X_1)\partial \Phi_0(X_2)...\partial \Phi_0(X_n)} $$ Does the action that appears in the partition function on the AdS side $$ Z = e^{−S(\Phi)} $$ have to come from supergravity or string theory? When reading about AdS/CFT I have seen it defined with both possibilities and this confuses me. So when and why does it make a difference, of one assumes strings or supergravity to calculate the partition function on the AdS side? Are there cases when one or the other is more appropriate, simpler, useful, etc?


title: Time required for water to freeze

text: recently I was wondering if there is any specific formula in order to calculate how long it takes for certain liquids to freeze (especially water). I know this depends on:   1. the volume of the liquid,   2. the temperature surrounding the liquid,   3. the surface of the liquid container, In order to not make it too complicated let's assume the liquid is surrounded by air and the container has no isolation. This is just out of pure curiosity, so I haven't done any calculations.


title: Electron in Magnetic Field can lead to non-Hermitian Hamiltonian?

text: Consider a charged quantum particle in a magnetic field. The Hamiltonian can be written using minimal coupling: $$ H = \frac{1}{2m} \left( \mathbf{p} - \frac{e}{c} \mathbf{A}(\mathbf{x}) \right)^2 $$ where $\mathbf{A}(\mathbf{x})$ is the vector potential. If one expands this, one gets: $$ H = \frac{1}{2m} \left\\{ \mathbf{p}^2 + \left(\frac{e}{c}\mathbf{A}(\mathbf{x})\right)^2 - \frac{2 e}{c} \mathbf{A}(\mathbf{x}) ~\mathbf{p} - \frac{e}{c} \underbrace{\left[ \mathbf{p}, \mathbf{A}(\mathbf{x}) \right]}_{\textrm{commutator}} \right\\} $$ Now, the commutator becomes $$[\mathbf{p}, \mathbf{A}(\mathbf{x})] = (\nabla \cdot \mathbf{A}(\mathbf{x}))~~ [\mathbf{p}, \mathbf{x}] = -i (\nabla \cdot \mathbf{A}(\mathbf{x}))$$ which is imaginary and independent of momentum. Now, this term drops out when one is in the Coulomb gauge. But if one picks a different gauge, say $\mathbf{A} = ( xy, 0, 0)$ for a gradient magnetic field, **doesn't such a term make the Hamiltonian non-Hermitian?** Which would in turn lead to complex energy eigenvalues? Or, else, do we have to insist on using the Coulomb gauge for non-relativistic problems?


title: Is a permutation of coordinates or labels really equivalent?

text: To construct a N-body anti-symmetric wave function some derivations start with the requirement that the N-body wave function should be anti-symmetric under a permutation of coordinates, other derivations start with the requirement that the total wave function should be anti-symmetric under a permutation of labels or states, for example this derivation . I can see the equivalence is trivial if you assume you can freely change the order of the wavefunctions. This is true in the case the wavefunctions are just scalars but what if you are dealing with multicomponent wavefunctions. For example relativistic wavefunctions which are 4x1 matrices. A simple example of my problem is the following, suppose we want to write down a general 2 body wavefunction in terms of single particle wavefunctions, \begin{align} \Psi(x_1,x_2) \propto \varphi_{a}(x_1)\varphi_b(x_2) \end{align} Where $x_i$ denote the spatial,spin,... coordinates, and $a,b$ denote single particle eigenstates. If we permute the coordinates in order to derive a antisymmetric wavefunction we get, \begin{align} \frac{1}{\sqrt{2}} \left(\varphi_{a}(x_1)\varphi_b(x_2) - \varphi_{a}(x_2)\varphi_b(x_1) \right) \end{align} while a permutation in states gives, \begin{align} \frac{1}{\sqrt{2}} \left(\varphi_{a}(x_1)\varphi_b(x_2) - \varphi_{b}(x_1)\varphi_a(x_2) \right) \end{align} Obviously above expressions are identical for scalar wavefunctions. When I try to check the normalisation of the two particle wavefunction (assuming the single particle states are orthormal and general $N \times 1$ matrices) I get for the label permutation, \begin{align} \int | \Psi(x_1,x_2) |^{2} &= \frac{1}{2} \int \varphi_{b}^{\dagger}(x_2) \varphi_{a}^{\dagger}(x_1) \varphi_{a}(x_1)\varphi_b(x_2) - \frac{1}{2} \int \varphi_{b}^{\dagger}(x_2) \varphi_{a}^{\dagger}(x_1)\varphi_{b}(x_1)\varphi_a(x_2) \\\ &\- \frac{1}{2} \int \varphi_{a}^{\dagger}(x_2) \varphi_{b}^{\dagger}(x_1) \varphi_{a}(x_1)\varphi_b(x_2) + \frac{1}{2} \int \varphi_{a}^{\dagger}(x_2) \varphi_{b}^{\dagger}(x_1)\varphi_{b}(x_1)\varphi_a(x_2) \\\ &= \frac{1}{2} - 0 - 0 + \frac{1}{2} = 1 \end{align} Which is the expected result (note that the integral is assumed to integrate over all continuous degrees of freedom, or sum over the discrete ones). For the coordinate permutation however I get, \begin{align} \int | \Psi(x_1,x_2) |^{2} &= \frac{1}{2} \int \varphi_{b}^{\dagger}(x_2) \varphi_{a}^{\dagger}(x_1) \varphi_{a}(x_1)\varphi_b(x_2) - \frac{1}{2}\int \varphi_{b}^{\dagger}(x_2) \varphi_{a}^{\dagger}(x_1)\varphi_{a}(x_2)\varphi_b(x_1) \\\ &\- \frac{1}{2}\int \varphi_{b}^{\dagger}(x_1) \varphi_{a}^{\dagger}(x_2) \varphi_{a}(x_1)\varphi_b(x_2) + \frac{1}{2}\int \varphi_{b}^{\dagger}(x_1) \varphi_{a}^{\dagger}(x_2)\varphi_{a}(x_2)\varphi_b(x_1) \\\ \end{align} Where it is not clear to my why the cross terms, \begin{align} \propto \int \textrm{d}x_1 \textrm{d}x_2 \left[ \varphi_{b}^{\dagger}(x_1) \varphi_b(x_2) \right] \left[ \varphi_{a}^{\dagger}(x_2) \varphi_{a}(x_1) \right] \end{align} should vanish. (Even if they did my main question still stands namely if you can always freely change the order of the wavefunctions). Note that the choice of permutation in coordinates or labels is equivalent to the choice whether to expand to rows or columns in a Slater determinant. EDIT: upon request a more elaborate version. Suppose we want to write down a general wave function for 2 identical fermions $\Psi$. This wavefunction has to be antisymmetric. Suppose the single particle states are given by $$ \varphi_{\alpha_i}(r_i,s_i) \,\,\,\,\,\ i \in \\{1,2\\} $$ Where $\alpha_i$ denotes the quantum numbers of a state, while $r_i,s_i$ denote the spatial and spin coordinates. A way to antisymmeterize $N$-body wavefunction in terms of single particle wavefunctions is often introduced by the concept of a Slater determinant. In the case of two particles a Slater determinant looks like \begin{align} \Psi_{\alpha_1,\alpha_2}(r_1,s_1,r_2,s_2) = \frac{1}{\sqrt{2}} \left| \begin{array}{c c} \varphi_{\alpha_1}(r_1,s_1) & \varphi_{\alpha_2}(r_1,s_1) \\\ \varphi_{\alpha_1}(r_2,s_2) & \varphi_{\alpha_2}(r_2,s_2) \end{array} \right|. \end{align} Now we can calculate this determinant by expanding along a row or along a column. Which is equivalent to the choice whether to permute the quantum numbers of the single particle states or the the coordinates of the single particle states. More explicitly, expanding the matrix along the first row (or equivalently, permuting the label of the single particle states) we get, $$ \frac{1}{\sqrt{2}} \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right). $$ While an expansion along the first column gives (equivalent to a coordinate permutation), $$ \frac{1}{\sqrt{2}} \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right). $$ The two above expression certainly look to be equivalent. However if we consider the wave functions to be $N$-component vectors I get some strange results. Suppose we want to calculate the normalization of the two particle wave function. We would then write the following $$ \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) $$ Note that I have introduced the $\dagger$ instead of the complex conjugate $*$ as the normalization should give a scalar. If we now replace $\Psi$ with the expression for the "label" permutation case we get, \begin{align*} \sum_{s_1} \sum_{s_2} & \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) \\\ &= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right)^{\dagger} \\\ & \times \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \\\ &= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left( \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \right. \\\ & \,\, - \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \\\ & \,\, - \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \\\ & \,\, \left. + \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \\\ &= \frac{1}{2} \sum_{s_1} \int \textrm{d}r_1 \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_1}(r_1,s_1) \sum_{s_2} \int \textrm{d}r_2 \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi_{\alpha_2}(r_2,s_2) \\\ &-\frac{1}{2}\sum_{s_1} \int \textrm{d}r_1 \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_2}(r_1,s_1) \sum_{s_2} \int \textrm{d}r_2 \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi_{\alpha_1}(r_2,s_2) \\\ &-\frac{1}{2}\sum_{s_1} \int \textrm{d}r_1 \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi_{\alpha_1}(r_1,s_1) \sum_{s_2} \int \textrm{d}r_2 \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi_{\alpha_2}(r_2,s_2) \\\ &+\frac{1}{2}\sum_{s_1} \int \textrm{d}r_1 \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi_{\alpha_2}(r_1,s_1) \sum_{s_2} \int \textrm{d}r_2 \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi_{\alpha_1}(r_2,s_2) \end{align*} Note that we have used the general expression $(AB)^{\dagger} = B^{\dagger} A^{\dagger}$ as well as the fact that the product $\varphi^{\dagger} \varphi$ contracts to a scalar and can be safely dragged trough the expressions. Now if one assumes the single particle states are orthogonal we have the following relation, \begin{align} \sum_{s_i}\int \textrm{d} r_i \varphi^{\dagger}_{\alpha_m}(r_i,s_i) \varphi_{\alpha_n}(r_i,s_i) = \delta_{m,n} \end{align} So we get, \begin{align} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 &\Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) \\\ & =\frac{1}{2} \delta_{\alpha_1,\alpha_1} \delta_{\alpha_2,\alpha_2} - \frac{1}{2} \delta_{\alpha_1,\alpha_2} \delta_{\alpha_2,\alpha_1} - \frac{1}{2} \delta_{\alpha_2,\alpha_1} \delta_{\alpha_1,\alpha_2} + \frac{1}{2} \delta_{\alpha_2,\alpha_2} \delta_{\alpha_1,\alpha_1} \\\ &=1 \end{align} Which is to be expected. If we start from the second expression for $\Psi$ however we get the following, \begin{align*} \sum_{s_1} \sum_{s_2} & \int \textrm{d}r_1 \textrm{d}r_2 \Psi^{\dagger}(r_1,s_1,r_2,s_2) \Psi(r_1,s_1,r_2,s_2) \\\ &= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right)^{\dagger} \\\ & \times \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right) \\\ &= \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d}r_1 \textrm{d}r_2 \left( \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \right. \\\ & \,\, - \varphi^{\dagger}_{\alpha_2}(r_2,s_2) \varphi^{\dagger}_{\alpha_1}(r_1,s_1) \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \\\ & \,\, - \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) \\\ & \,\, \left. + \varphi^{\dagger}_{\alpha_2}(r_1,s_1) \varphi^{\dagger}_{\alpha_1}(r_2,s_2) \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right) \\\ \end{align*} It is clear that the first and the fourth term will give $1/2 + 1/2 = 1$ as they are exactly the same as in the previous expression. However I do not see why the cross term should be necessarily zero. So it would seem that starting from the different antisymmeterization expression one gets different results. Hence my phrase "order matters" as the following appears to be true $$ \frac{1}{\sqrt{2}} \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) \neq \frac{1}{\sqrt{2}} \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_1}(r_2,s_2)\varphi_{\alpha_2}(r_1,s_1) \right) $$ A final note about the Dirac notation. If one would start from the Dirac formalism and introduce $ | \Psi \rangle$ as $$ | \Psi \rangle = \frac{1}{\sqrt{2}} \left( | \alpha_1 \rangle | \alpha_2 \rangle - | \alpha_2 \rangle | \alpha_1 \rangle \right) $$ A projection into coordinate space gives the "label" permutation expression, $$ \langle r_1,s_1 ; r_2,s_2 | \Psi \rangle = \frac{1}{\sqrt{2}} \left( \langle r_1,s_1| \alpha_1 \rangle \langle r_2,s_2 | \alpha_2\rangle - \langle r_1,s_1| \alpha_2 \rangle \langle r_2,s_2 | \alpha_1\rangle \right) \\\ = \frac{1}{\sqrt{2}} \left(\varphi_{\alpha_1}(r_1,s_1)\varphi_{\alpha_2}(r_2,s_2) - \varphi_{\alpha_2}(r_1,s_1)\varphi_{\alpha_1}(r_2,s_2) \right) $$ The expression for the normalization starting from the Dirac formalism gives, $$ \langle \Psi| \Psi \rangle = \frac{1}{2} \left( \langle \alpha_1 | \alpha_1 \rangle \langle \alpha_2 | \alpha_2 \rangle - \langle \alpha_1 | \alpha_2 \rangle \langle \alpha_2 | \alpha_1 \rangle - \langle \alpha_2 | \alpha_1 \rangle \langle \alpha_1 | \alpha_2 \rangle + \langle \alpha_2 | \alpha_2 \rangle \langle \alpha_1 | \alpha_1 \rangle \right) $$ Using the unit identities: $1 = \sum_s | s \rangle \langle s |$ and, $1 = \int \textrm{d} r | r \rangle \langle r |$ we get, $$ \langle \Psi| \Psi \rangle = \frac{1}{2} \sum_{s_1} \sum_{s_2} \int \textrm{d} r_1 \int \textrm{d} r_2 \\\ \left( \langle \alpha_1 | r_1,s_1 \rangle \langle r_1,s_1 | \alpha_1 \rangle \langle \alpha_2 | r_2,s_2 \rangle \langle r_2,s_2| \alpha_2 \rangle - \langle \alpha_1 | r_1,s_1 \rangle \langle r_1,s_1 | \alpha_2 \rangle \langle \alpha_2 | r_2,s_2 \rangle \langle r_2,s_2| \alpha_1 \rangle - \langle \alpha_2 | r_1,s_1 \rangle \langle r_1,s_1| \alpha_1 \rangle \langle \alpha_1 | r_2,s_2 \rangle \langle r_2,s_2| \alpha_2 \rangle + \langle \alpha_2 | r_1,s_1 \rangle \langle r_1,s_1| \alpha_2 \rangle \langle \alpha_1 | r_2,s_2 \rangle \langle r_2,s_2| \alpha_1 \rangle \right) $$ Which is when using $ \langle r_i, s_i | \alpha_i \rangle = \varphi_{\alpha_i}(r_i,s_i)$ exactly the same as the normalization expression I get for the label permutation case, which works out nicely to $1$ as expected.


title: How to combine these equations of constraint?

text: I want to model a nonholonomic system of an arbitrary rotating disk in 3D, which rolls without slipping, and doesn't have to stay vertical. (think spinning a penny on the table) I want to use the method I just learned of Lagrange multipliers with the Euler-Lagrange equations to solve the system. I can parameterize the system in terms of $(x,y,\theta,\phi,\psi)$, and I can come up with several equations of constraint if I let two (or three, with $(x,y)$ changing as a pair) variables change at a time and hold the others constant. I'm using mathematica so I can afford to have unweildy representations and painful integrals. I wanted $(x,y)$ representing the position of the center of the disk in the horizontal plane, $\theta$ representing the angle from $(x,y)$ to the point where the disk touches the ground, $\phi$ representing the angle from the xy plane to the actual (x,y,z) center of the disk (so, if $\phi=0$ the disk is flat, and if $\phi=\pi/2$ the disk is vertical), and $\psi$ representing the angle of the disk around the axis normal to its face. I wound up on the following linear transformation, which takes a stationary point on the disk's space into world space: $T(x,y,r \sin(\phi)) R_{xy}(\theta)R_{xz}(-\phi) R_{xy}(\phi) \vec{v}$ (where $T$ is a translation, $R_{xy}$ is a rotation in the xy plane etc) This works perfectly and I can actually come up with the kinetic energy in terms of $x,\dot{x}, y,\dot{y},\phi, \dot{\phi}$ etc. * * * So now where the nonholomic part comes in, I need to find the equations of constraint. The only constraint is rolling without slipping. I can find equations with partial derivatives (say, I let x and y vary as I change $\psi$ and hold all other variables constant), but these are just partial constraints and don't represent the true differentials governing the constraints. How can I find the true differentials? My sets of equations are: **1** Rotating the disk normal to its face (exactly like spinning a wheel) $\frac{\partial x}{\partial \psi} =-r \sin (\theta ), \frac{ \partial y}{\partial\psi }=r \cos (\theta )$ **2** Rotating $\theta$, the point where the disk touches the ground, without changing x, y, or $\phi$. $psi$ must change according to: $\frac{\partial\psi}{\partial\theta} =-\cos (\phi )$ **3** Changing the vertical angle of the disk, $\phi$, and having the point of contact stay the same (as well as $\theta$, $\psi$ constant), $x$ and $y$ must change according to: $\frac{\partial x} {\partial\phi} =-r \cos (\theta ) \sin (\phi ), \frac{ \partial y} {\partial\phi} =-r \sin (\theta ) \sin (\phi )$ How can I combine these equations into full differentials for use in Lagrange multipliers with the Euler-Lagrange equations? ## Animated Visualizations Just to show what the parameters mean and what the constraint equations mean in case there's something technically wrong: (the animations seem to freeze. If one isn't moving try dragging it to a new tab) Adjusting the parameters on the transformation equation: ![changing parameters](http://i.stack.imgur.com/qvHLA.gif) Applying partial constraint **1** to visualize rolling without slipping ![constraint 1](http://i.stack.imgur.com/62gJI.gif) Visualizing partial constraint **2** ![constraint 3](http://i.stack.imgur.com/Nb8N6.gif) Visualizing constraint **3** ![constraint 2](http://i.stack.imgur.com/AEaCw.gif) * * * note: I'm pretty new to Lagrangian mechanics, on chapter two of Goldstein classical mechanics, but I don't see a reason why I can't apply everything I've learned (just what I've mentioned) to this problem.


title: What is the evidence for the super-rotation of Earth's inner core?

text: What are the geophysical observations that support (or contradict) the hypothesis that the Earth's inner core rotates at a faster rate than the Earth's mantle? Summary of Answers: 1) Studies of seismic ‘body-waves’ travelling through the Earth’s inner core indicate the inner core is rotating 0.2 to 3 degrees faster than the mantle. These studies are sensitive to heterogeneities in the inner core’s structure. 2) Analysis of the free-oscillations of the Earth following great earthquakes are insensitive to local structure indicate that the differential rotation rate has been within 0.2 degrees over the past 20 years and that the inner core is probably gravitationally locked to the Earth’s mantle. 3) Comparisons of seismic ‘body-waves’ travelling through the Earth’s inner core from earthquake doublets (a pair of earthquakes that occur at the same place but different times,) should also be insensitive to local structure (because the ray-paths would be the same) and can be used to investigate changes over relatively short intervals of time. These studies indicate that the mean rate of rotation of the inner core has been 0.3 to 0.5 degrees per year faster than the mantle, but that the rate has also had decadal fluctuation about the mean of 1 degree per year.


title: Physics the Why vs. How question?

text: I've been told that a physicist never asks why and must always ask how. Is this true? Have all discoveries in physics been due to asking how? or have some been discovered by asking why?


title: Does tea stay hotter with the milk in it?

text: A little thought experiment, similar to this one: Imagine you are making a cup of tea when the door bell rings. You've poured the boiling water into a cup with a teabag in it. As you're just about to pour milk in to the mix, the door bell rings. My question is: what will keep the resulting drink hottest after my chat with the person at the door: quickly pouring the milk in, or leaving it in the carton until after the interruption ends? Being a bit of a mathematician I'm happy to have the questions answered with a formula in terms of $t$ the length of the interruption, $m$, the temperature of the cold milk, $w$ the temperature of the hot water, $a$ the ambient temperature etc. I guess you'll have to use a model of how an insulated body loses/gains heat... feel free to use simplifying assumptions like the environment stays the same temperature etc. My starting point from a physics point of view, is that heat is lost (or gained) proportional to the temperature difference between the object and the environment. So I imagine that would create some sort of exponential curve?


title: Are the ideas of the speed of light and the absolute top speed inside the universe equivalent?

text: The idea of the speed of light being as fast as something can go is pretty well accepted. I'm wondering if the top speed might be (very slightly) higher? Lemme explain. I was reading about particle acceleration, and how accelerating a particle to the speed of light would require infinite energy. According to a quick Google search, the LHC is getting particles up to 0.999999991 c. For someone such as myself, I might as well consider that the speed of light. I mean, it's _so close_ , right? So what if in a similar vein, the speed of light is very, very close to the absolute top speed? Is that possible? _If the speed of light in a vacuum was something like 0.9999999999999999999999999999999999 to the absolute top speed in the universe, would we ever know the difference?_ (this number is just an example to show how the speed of light could be imperceptibly close to the absolute top speed, and yet still not be the fastest speed possible).


title: How is the speed of light an absolute maximum speed in the universe?

text: Under the heading _which came first, the chicken or the egg,_ which came first: the maximum velocity of electrons orbiting the atom, or the speed of light as the maximum velocity limit? Is it possible that when an object reaches the velocity at which its forward velocity equals its electron(s) orbital speed around its nucleus, then its advancing electrons would cease to spin around their orbit, lose their centrifugal force, and collapse into its nucleus, and the retreating electrons would lose their forward speed and the atom would run away from them, and shed them as radiation? Could this limit the maximum velocity at which a mass can travel, and this define the speed of light? If this is true, then at the speed of light its mass would not become infinite, in according to Einstein, but the mass would transmute into pure energy, in other words, become light or some other radiation frequency.


title: Ratio of distance between mirror and person

text: In perspective of a given example, if a man was to stand $2\ m$ away from a mirror which was $0.9\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be if the man was now $6\ m$ away from the mirror and was to maintain a full reflection? Would the mirror be equal to or less than the original height and why? This scenario seems to have caught me out multiple times. So, some reasoning would be much appreciated.


title: Wave frequencies and barrier width?

text: I know a fact that says a wave can go through barriers thinner than its length. This is why for example FM radio can be picked anywhere while antenna TV needs direct sight to the transmitter. Is this correct? If so, what is the explanation? Related, if my wifi is 2.4GHz, then the wave length is 12.5 cm (right?), so I will not go thorough walls thicker than that?


title: What's the relationship between an object's color and it's energy?

text: All else held equal, which has more energy: a violet-painted teacup or a red- painted teacup? I know that violet light has more energy than red light, but does this mean that the inverse holds true for painted things? Does red paint necessarily hold more energy than violet paint? Or is the reflective property not related to energy in any way? I understand that electron energy levels have to do with the color of light emitted, but is there a correlation between the total energy in the material and the color that it reflects?


title: Discrete version of Feynman path integrals

text: I've decided to put a very limited amount of my time into understanding the path integral formulation of quantum mechanics. I'm interested in the mathematical formalism more than the physics, so I'd like to understand how the following abstract and fairly general scenario for a discrete quantum system can be translated into the path integral formalism, assuming it can. A system starts has a time-dependent state that can be represented by an $n\times n$ density matrix $\rho(t)$. It begins in state $\rho_0$ and evolves over time according to $$ i\hbar \frac{\partial \rho}{\partial t} = H\rho - \rho H,$$ where the Hamiltonian $H$ is some constant Hermitian matrix, yielding $$ \rho(t) = e^{-iHt/\hbar}\rho_0 e^{iHt/\hbar}. $$ At time $t'$ I make a complete set of orthogonal measurements, which amounts to (optionally) doing a basis change on $\rho(t')$ and then interpreting its diagonal elements as probabilities. My question is, can someone show me how to reinterpret the above in path integral terms? For example, what is a "path" in a discrete system like this, and, given an arbitrary Hamiltonian, how does one assign each path an action? Alternatively, can anyone recommend a treatment that summarises the path integral formalism with a minimum of physics (by which I mean details about how specific Hamiltonians behave and where they come from, etc.)?


title: Can't seem to reconcile geometric optics and wave optics

text: I was looking at a physics situation involving light, and I _can_ make the correct derivation assuming light is a ray of a given intensity (geometric optics), energy conservation checks out, everything. But when I try to move to a wave model of light to take into account interference, nothing works out anymore - I get completely different results. And I don't know why. It's driving me insane. Here is the situation. Find the total reflected and transmitted light intensity from/across this layer. ![enter image description here](http://i.stack.imgur.com/gHNjZ.png) External medium has refractive index $n_0$, layer $n_1$ and internal medium $n_2$. Using Fresnel equations, call the reflectance of interface $n_i \to n_j$, $\text{R}(n_i \to n_j)$ and call the transmittance $\text{T}(n_i \to n_j)$. The Fresnel properties apply, $\text{R}(n_i \to n_j) = \text{R}(n_j \to n_i)$, $\text{T}(n_i \to n_j) = \text{T}(n_j \to n_i)$, and: $$\text{R}(n_i \to n_j) + \text{T}(n_i \to n_j) = 1$$ * * * I'll just do the transmitted light derivation since its enough to show my problem. The transmitted light is: $$\text{T} = \text{T}(n_0 \to n_1) \cdot \text{T}(n_1 \to n_2) \cdot \sum \left (\text{R}(n_1 \to n_0) \cdot \text{R}(n_1 \to n_2) \right )^k$$ $$\text{T} = \frac{\text{T}(n_0 \to n_1) \cdot \text{T}(n_1 \to n_2)}{1 - \text{R}(n_1 \to n_0) \cdot \text{R}(n_1 \to n_2)}$$ A similar derivation gets the reflected intensity, and they correctly add up to 1, as expected. * * * Now assume light is a wave, the layer has thickness $\delta$ and the incident wave (in red) has wavelength $\lambda$. Now use the Fresnel equations for amplitudes, call them $\text{r}(n_i \to n_j)$ and $\text{t}(n_i \to n_j)$. The phase shift for the $k$th transmitted wave is (assume no phase shift due to reflection): $$\frac{2 k \pi}{\lambda} \left ( 2 n_1 \delta \cos{\theta} \right ) = k \varphi$$ Where $\theta$ is the angle made by the light wave inside the layer (depends on incident angle). Then, the transmitted intensity is: $$\text{T} = \left | \text{t}(n_0 \to n_1) \cdot \text{t}(n_1 \to n_2) \cdot \sum \left (\text{r}(n_1 \to n_0) \cdot \text{r}(n_1 \to n_2) \cdot e^{i \varphi} \right )^k \right |^2$$ $$\text{T} = \frac{| \text{t}(n_0 \to n_1) |^2 \cdot | \text{t}(n_1 \to n_2) |^2}{| \text{r}(n_1 \to n_0) |^2 \cdot | \text{r}(n_1 \to n_2) |^2 - 2 \cdot \text{r}(n_1 \to n_0) \cdot \text{r}(n_1 \to n_2) \cdot \cos(\varphi) + 1}$$ Which isn't even close to the right result! This is completely meaningless. * * * I understand this isn't an elegant way to do electromagnetics and it makes a lot of assumptions (I am not a physics student, this is for computer graphics) but I don't understand why the two models don't give the same answer? I know I'm making a mistake in the second one but I can't find it. I thought this is how you add waves and I feel my sum is correct but the results still don't agree with the geometric optics result. Note I removed the $\omega t$ part from the wave since it cancels out in the end. I guess an equivalent question would be, could someone show me the correct wave optics derivation (which should average out to the geometric optics solution over all possible phase shifts, as noted in comments)? Because I've spent days on this problem and I'm really at my wit's end at this point. * * * EDIT: after putting Mathematica to work on this, it turns out the wave optics average _is_ equal to the geometric optics solution, with one small issue: it's telling me that $|\text{t}|^2$ is equal to the transmittance $\text{T}$, which is apparently wrong according to the Wikipedia Fresnel article, where: $$\text{T}(n_i \to n_j) = \frac{n_j \cos{\theta_j}}{n_i \cos{\theta_i}} |\text{t}(n_i \to n_j)|^2$$ But at least it looks as if I wasn't completely all wrong. So now I just need help on understanding why the transmittance coefficient doesn't work out as it should..


title: ultrarelativistic gas

text: Consider the ideal ultrarelativistic gas Hamiltonian $\mathbf{H = }\sum_{i = 1}^N \mathbf{c |\vec{p_{i}}|}$, now if we let molecules to interact with a potential term like $\mathbf{d|\vec{q_{i}}|}$; is it still possible to find density of states $\omega(E)$?


title: Can sound be separated from air?

text: I would like to open up a window in my house, let the air in, but keep the sound out. Could a device be designed to put up on the window, like a screen, and accomplish this?


title: The speed of sound is proportional to the square root of absolute temperature. What happens at extremely high temperatures?

text: The speed cannot increase unboundedly of course, so what happens?


title: Estimate temperature and pressure of steam in a pot of boiling water

text: Suppose we have a pot of known dimensions filled to level L with boiling water on a stove, covered by a lid with a vent of known dimensions. Given the steady-state temperature of the stove (bottom of pot's surface), how to quantitatively estimate the ambient pressure and temperature of steam in the pot, at 1 atm pressure outside the pot? All I can think of is that the pressure will be a function of the vent size (monotonically decreasing) and stove temperature (monotonically increasing). That's because more steam will be produced at higher stove temperature and the velocity of steam escaping through the vent is a monotonically increasing function of pressure; at equilibrium the amount of steam escaping == the amount of steam produced per unit time. But is it only the pressure that's increasing as stove temperature increases? In the limiting case of no water and zero-sized vent, the temperature of steam will be increasing until it reaches the temperature of the stove. Does the presence of 100°C water affect the temperature of steam? One way to attempt to solve this problem might be to consider the amount of energy entering and exiting the pot (the rate of mass lost due to steam escaping is low, so assume no mass is lost). I expect the answer to the whole problem would be a set of simultaneous equations to solve. But what equation to use to find the rate of energy input? It would be to do with the temperature of the stove and the temperature of the water, 100°C, but I can't think of how to find it.


title: change of resistance in semiconductors due to temperature change

text: I have a question that popped up in an old exam I just can't find a way to solve. > If you drop the temperature of this material from room temperature (I guess > around 300K) to 0C (273K) the resistance drops with a factor of 3.5. Is it: silicon, germanium or gallium arsenide ? I've been googling, looking through the books for the last 2h and just can't find any way to solve this question. Any idea?


title: In a positively biased PN junction, where do the injection carriers come from?

text: I am not quite understand i-v character of PN-junction diode. Here is the model in textbook. The PN junction diode can be divided into three regions. They are   1. One depletion region near the PN interface, where no carrier but ionized atoms existed.   2. Two quasineutral(QNR) regions extend from the boundary of depletion region to both ends of the diode. ![This image is take from the post "Pn junction voltage drop?"](http://i.stack.imgur.com/7VaN3.png) When positive voltage applied to this diode, current forms. In order to characterize the current, according to the model, the only thing one need to calculate is the minority diffusion within QNR. But, **what about the majority carrier?** **What is their concentration along the diode? Why don't we count their contribution to the diffusion?** While I guess that majority carriers also contribute diffusion by recombining with minority ones. In order to support diffusion (by maintaining concentration difference), excess carriers of both types must be injected into QNR. Q1 **Where do these injection carriers come from?** Take the n-QNR for example, is it right to say that both electrons and holes are injected from p-QNR? If this is so, then carrier concentration in p-QNR would be lowered, and eventually, equals to zero, which I am sure is not the case. Q2 Here near the interface between depletion region QNR, **do the relationship n*p = ni^2 still hold**? It seemed that it has been broken by both n and p injected. _Q1 and Q2 are quite clearly answered. With the explanation, I am now concerning with the number of carriers in this diode. Take the electrons for example, they are born in n-QNR by thermal excitation, then overcome the potential barrier in depletion region, and finally recombine (after some distance of diffusion) in p-QNR._ For each recombination, however, #electrons and #holes decreases by one. Since current flows continuously, a new electron-hole pair must be generated in n-QNR, otherwise, #electrons that can go through depletion regions would become zero eventually. Q3 **What about the holes being generated in n-QNR, where would they go?** If they go through the depletion region (being accelerated) and become majority in p-QNR, are they risk of being recombined before going through the depletion region (in n-QNR)? Q4 As bias increased, more current flows. Does it imply larger generation rate (in #/time-1 volume-1) of electron-hole-pair. If so, extra energy must be supplied (compare to the T.E case), since the #electrons with high K.E increase but the temperature remains. **Where do these extra energy come from?**


title: When is a flow vortex free?

text: To solve problems in fluid dynamics one states often the assumption that the flow is vortex free _i.e._ $rot(u) = 0$ It is a basic assumption which is needed for potential flow problems etc.. My question is, how do you know that a flow is vortex free? When are you allowed to assume that? In literature $rot(u) = 0$ often occurs, when you are dealing with inviscid fluids but I haven't found a source yet, which makes precise correlation between those two situations like (inviscid fluid => $rot(u) = 0$ or inviscid fluid <= $rot(u) = 0$ or inviscid fluid <=> $rot(u) = 0$).


title: What is the process that gives mass to free relativitic particles?

text: When a free particle move in space with a known momentum and energy then what is the physical process that gives mass to that free (relativistic) particle? What is role does the Higgs field in that process, if any?


title: General Thermodynamic equation of state

text: I heard my professor saying that the equation $$ PV = \frac{2}{3}U $$ is valid for any **non-relativistic** gas, be it Ideal or Real gas(includes quantum ideal gases). Is this true, If it is how can we prove with all generality ?


title: Can a room resonate at a frequency lower than the principal frequency?

text: I'm the bass in our high school a cappella group. Recently we had a performance in our auditorium, and I am almost certain I heard the pitch an octave lower than I was singing. I was singing E3 (164 Hz), and I'm sure I heard E2 (82 Hz). We were singing with no audio equipment, so it couldn't have been any effects processors. Do you folks have any idea what I could have experienced?


title: Why (infra-)red light is not used in diagnostics?

text: I noticed that human body conducts red visible light quite well. A hand placed over a powerful red lamp seems to be semi-transparent while over any other color it does not. Possibly the light conductivity in near infra-red range is even better. So my question is why (infra-)red light is not used for diagnostics instead of harmful X-rays?


title: Spin-statistics theorem on spaces with non-integer dimensions

text: What would be the spin-statistics relation for particles in a space with non- integer dimension, $ 2 \lt D \lt 3 $? In other words (cf. stackexchange questions here and here), what is the first homotopy group for a space with non-integer dimension, $ 2 \lt D \lt 3 $? Although it is not useful for my immediate work, I would also like to know for fractional dimensions $ 0 \lt D \lt 2 $ and $ 3 \lt D \lt \infty $ References/reviews, written for physicists, will also be greatly appreciated!


title: Phasor form of Maxwell's Equations

text: I'm interested in the transformation from the standard Maxwell's equations to their phasor equivalents. From the literature, this means injecting: \begin{equation} E = Re(\boldsymbol{E}e^{j\omega t}) \end{equation} into \begin{equation} \nabla \times E = -\frac{\partial B}{\partial t} \end{equation} to deduce \begin{equation} \nabla \times \boldsymbol{E} = -j\omega\boldsymbol{B} \end{equation} Some demonstrations out there simply jump from the real components equation to the complete equation, ignoring the imaginary components. i.e., from: \begin{equation} Re(\nabla \times \boldsymbol{E} e^{j\omega t}) = Re(-j\omega\boldsymbol{B}e^{j\omega t}) \end{equation} to the expected solution. Example of such demonstration are available at: https://courses.cit.cornell.edu/ece303/Lectures/lecture14.pdf or http://faculty.cua.edu/kilic/EE%20542/Topic2_BasicEMTheory.ppt What assumption allows them to do so? Why is it possible to ignore the imaginary component? Thanks


title: Dynamic equation

text: I'm asked to give the dynamic equations of a charged particle moving on a potential, under relativistic considerations, basically $dp/dt$, with $p$ the 4-momentum vector. The question for the three space coordinates is simply $d\vec p =-q\nabla\Phi$, being $\Phi$ the potential, that is: $$\frac{d(\gamma m\vec v)}{dt}=-q\nabla\Phi$$ For the first term of the four vector, the answer given is this, and I have no idea where this comes from: $$\frac{d(\gamma mc^2)}{dt}=q(\vec E\cdot\vec v)$$, being $\vec E$ the electric field. Where does this come from? IS it even correct, isn't the first term of the four momentum vector just $\gamma mc$ instead of $\gamma mc^2 $


title: How do I derive the Lorenz gauge from the continuity equation?

text: I was reading my old electromagnetics book ( _Elements of Electromagnetics,_ by Sadiku, 3rd edition) and after the author explained what the Lorenz gauge is mathematically and why it is useful in decoupling the scalar electrical potential from the vector magnetic potential to produce the wave equations, the author made a side comment below eq. (9.52) that the Lorenz gauge $$\nabla \cdot \vec A = -\mu\varepsilon \dfrac{\partial V}{\partial t}$$ can be derived from the continuity equation. I don't have a background in relativity. I am an electrical engineering student. So given these limitations, how would I go about deriving the Lorenz gauge from the continuity equation? Note: I am sticking to the conventions of my book in writing equations.