Order of magnitude estimation for some intriguing questions

The physics TA showed us a few examples in which one can estimate many things from first principles and sound logic based and scaling arguments. This led usually to understanding of why some numbers have the values they do. I asked myself a few questions and after struggling with them, I could not get a clear answer: why our body temperature is about 30 C, is there a way to understand where that number come from? why pregnancy period in humans is about a year, not 10 years, not a month? why our life time is about 100 years, not 1000, not a year? Any help will be really appreciated


Are locality and separability two distinct notions?

Is there any difference between locality and separability in quantum mechanics, or do they mean the same thing? It seems authors do not always agree.


Problem in the Ladder paradox in relativity?

The ladder paradox consists of a ladder rushing towards a garage with two open doors. In the actual paradox, proper length of the ladder is greater than that of the garage, but in this case lets consider them to be of the _same proper length_. In the frame of the garage, the ladder is Lorentz contracted, and hence it is possible to "trap" the ladder inside the garage. By "trap" I mean that both doors of the garage must be closed while the ladder is completely inside the garage. But in the frame of the ladder, the garage is Lorentz contracted, and hence it is never possible to "trap" the ladder inside the garage. If I am able to trap the ladder in one frame, I should be able to do it in the other frame too right? Why then is there this paradox?


Rotating frames

A bird of mass $m$ is on a merry-go-round of radius $a$ which rotates at constant angular velocity $-\omega_b$ in the $y$ direction. A woman of mass $M$ is on a second merry-go-around of radius $b$ which rotates at constant angular velocity $\omega_s$ in the $z$ direction around the same origin. At $t=0$ the bird is at $(a,0,0)$ and the woman at $(b,0,0)$. What is the force of friction acting on the bird as seen by the woman?


Smallest view-able object by radio wave or UV magnification with metamaterial?

**What is the theoretical and in practice microscopic maximum/best resolution of a meta material radio wave XOR UV lens (assuming diffraction or beyond diffraction techniques)?** I'd also like to use diffraction or beyond diffraction as described in this paper: A three-dimensional self-supporting low loss microwave lens with a negative refractive index. Given the Theory of Optical instruments defines the maximum resolving power of a light based microscope as: ![enter image description here](http://i.stack.imgur.com/vRmVh.png) ![enter image description here](http://i.stack.imgur.com/2cLag.png) Diffraction based analysis allows for an increased resolution but I have not found those formulas. If a meta material lens and diffraction alone do not result in a sub cellular wave what additional tools could be used to generate the tiny subwave? Would UV metameterials and defection techniques result in a small enough wave? I also have not found a radio wave scanning device consistent with an A4 scanner. These are the numbers I know: 11Ghz or wavelen 27.27mm, the focal length between 6 and 10cm, a refraction index of -1, a lens of 255mm by 255mm which is 40mm thick.


Why do rockets need a cosmic ray detector?

I was watching the video _Video camera installed on rocket that reaches 121,000 ft._ , and this rocket has a cosmic ray detector. Why is this needed in a rocket?


Textbook Problem: Free energy expression

> In Klotz, Introduction to Chemical Thermodynamics, Ex. 8.2 requires me to > derive $$ dG = V \left( \frac{\partial p}{\partial V} \right)_V dV + \left[ > V \left( \frac{\partial p}{\partial T} \right)_V - S \right]dT, $$ where $G$ > is the Gibbs free energy, $S$ is the entropy and all the other variables > have their ordinary common-sense meaning. I'd appreciate feedback on my solution, which I find just very simple, maybe too simple to be right. First of all, when recognizing that $G$ is a function of $V$ and $T$, we can write the total differential as $$ dG = \left( \frac{\partial G}{\partial V} \right)_T dV + \left( \frac{\partial G}{\partial T} \right)_V dT $$ In Eq. 8.19, we learn that $$ dG = V dp - S dT $$ and so, when multiplying both sides by $1/\partial V$ at constant temperature, we find $$ \left( \frac{\partial G}{\partial V} \right)_T = V \left( \frac{\partial p}{\partial V} \right)_T - S \left( \frac{\partial T}{\partial V} \right)_T $$ where $(\partial T /\partial V)_T$ is zero, because we're at constant $T$. A similar procedure yields the bracket expression, this time instead we just multiply both sides with $1/\partial T$ and using constant $V$, we arrive at $$ \left( \frac{\partial G}{\partial T} \right)_V = V \left( \frac{\partial p}{\partial T} \right)_V - S \left( \frac{\partial T}{\partial T} \right)_V = V \left( \frac{\partial p}{\partial T} \right)_V - S $$ and from comparison with the total differential expression, one immediately recognizes the above very first expression. Is this a valid solution/Ansatz?


Missing radio source coordinates in G/T measurement technique ITU-R S.733.2

Section 3 ‘Suitable radio stars’ of this recommendation lists the different stars/nebula and their corresponding flux-densities that can be used in the G/T measurement. There is no reference to the coordinates these flux densities were measured at. Some of the Nebula have large angular extension and I’m not sure where to point my antenna. > Is there a list of celestial coordinates of the radio stars used in this > recommendation? http://www.itu.int/dms_pubrec/itu-r/rec/s/R-REC-S.733-2-200001-I!!PDF-E.pdf


Maximum efficiency for a counter-current heat exchanger (double flux controlled motorized ventilation)

I am not sure if I can explain the question correctly because I don't know the name of this mechanism in English. This is my explanation attempt: In a house, a tube is expelling the air from the inside to the outside, and a tube is aspiring the air from the outside to the inside. The 2 tubes are interlaced in order to transmit the heat between their contents. The goal is to have a good thermal isolation between inside the house and outside:   * When the house is more hot than the outside, the outgoing air is warming the incoming air.   * When the house is more cold than the outside, the outgoing air is cooling the incoming air. Question: **In theory** , what is the maximum efficiency of such thermal exchange? Could it be possible to reach something close to 1 or the upper bound of the efficiency is well under?


Determining if a particle is bounded or unbounded with motion in polar coordinates

![a](http://gyazo.com/5be8000bab6937ba21923ffa75cf0997.png) Firstly, I determined the equation for the particles path for $k <-mV^2a^2$ as $u = {Ae^{\sqrt{\alpha}\theta}+Be^{-\sqrt{\alpha}\theta}}$ where $ \alpha = 1 + \dfrac{k}{mV^2a^2},$ $u = 1/r$ I have two questions; (1) Could I solve for the constants $A,B$ with the information given? (It is not given that at r = a, $\theta = 0$, so I figured no) (2) I'm not sure how to determine if the motion is bounded or unbounded in polar coordinates - For instance, if we were dealing with $r(t)$ then I'd take $t \to \infty$ and see what happens, but we have theta here, how can we deal with theta? In this case with $k <-mV^2a^2$ it is bounded (if we take $\theta \to \infty$)


What are the good introductory resources for M-theory towards AdS/CFT?

I see a list here with a section titled M-theory - http://www.superstringtheory.com/links/reviews.html In there these two look promising, http://arxiv.org/abs/hep-th/9607201 and http://arxiv.org/abs/hep-th/9802051 I would like to know if people have opinions of good starting points towards understanding M-branes and how they are used for constructing exact examples of AdS/CFT. [...I guess all exact AdS/CFT scenarios are M-theory constructions..]


Principle of locality and forces

I have a silly confusion about the statement written in the link Einstein and Locality _''external influence on A has no direct influence on B; this is known as the Principle of Local Action.''_ Since GTR and Electromagnetism obey principle of locality. And in both cases change in A position affect the B and vice versa. Then why this contradiction come. Please correct me.


What does it mean for electrons to be "diffusive"?

I'm reading this paper and it has the line (end of 3rd paragraph, page 2): > It turns out that the simple fact that electrons are diffusive instead of > freely propagating leads to a profound modification of the traditional view > based on the Fermi-liquid theory of metals. So it kind of suggests it means something that's _not_ freely propagating, like bound I guess?


Why does a larger sprocket need more power to accelerate at the same rate?

Why does a large drive sprocket require more power to accelerate at the same rate as a smaller drive sprocket if the overall gear ratio is the same (with all other components such as the chain the same between systems)? As an example, we could compare 11/33 to 13/39 (same 1/3 ratio). Why does the 13 tooth sprocket need more power to accelerate at the same rate as the 11 tooth setup? ![enter image description here](http://i.stack.imgur.com/QmYrH.jpg) Can we agree that the top sprocket will need more torque to accelerate an atv at the same rate as the same atv with the bottom sprocket? And assuming the overall gear ratio is the same. The moment of inertia differences of the sprockets are insignificant. Assume the moments of inertia are equal.


How do I calculate integral analytically for small $k$?

In a Heisenberg antiferromagnet, the dispersion relation is \begin{equation} \omega_{\mathbf{k}} =JSz\sqrt{ 1-\gamma_{\mathbf{k}}^2} \end{equation} where $\gamma_{\mathbf{k}}=\frac{1}{d}\sum_{i=1}^d \cos k_ia$, where $d$ is the dimensionality of the cubic lattice. In staggered magnetisation calculation, we get an integral which is given by \begin{equation} \delta m =\int_{BZ} \frac{d^dk}{(2\pi)^d}\frac{1}{\sqrt{ 1-\gamma_{\mathbf{k}}^2}}. \end{equation} This integral is dominated by small $k$, since we have Goldstone modes at $k=0,(\pi,\pi,\dots)$. Using this property, we can extend the limits of this integral from Brillouin zone to infinite $k$-space and we replace $\gamma_{\mathbf{k}}$ by its small $k$ limit: \begin{eqnarray} \delta m=\frac{\sqrt{d}}{a}\int_0^\infty \frac{d^dk}{(2\pi)^d}\frac{1}{k} \end{eqnarray} In 1D, its gives the logarithmic divergence which is expected. But, in 2D and 3D we get \begin{eqnarray} \delta m &\sim& [k]_0^\infty, \quad \text{in}\, 2D\\\ &\sim&[k^2]_0^\infty, \quad \text{in}\, 3D \end{eqnarray} which still show divergence. This is not correct as if we solve the Brillouin zone integral we get some finite value. I do not understand what is wrong with the infinite k space integral? Could anybody please tell me how to solve such integral analytically for small $k$?


How is graphene a major breakthrough?

What is graphene from a physics standpoint? Why do I keep hearing that graphene is considered to be such a major breakthrough? How is graphene going to transform the world?


Quantum Regime of Particles in Solids

On my midterm today, I read that when the deBroglie wavelength of a particle exceeds the spacing between the particles in a solid or liquid, the particles begin to behave quantum dynamically. Why is this? I thought a larger deBroglie wavelength implied a less quantum mechanical behavior.


Truck Mass & "4th power law"

Is it true that the damage caused to a road surface is proportional to the 4th power of vehicle axle weight, and if it is, what factors cause it to be related by a power of 4, or is it just the results of some measurements? I'm chiefly looking for the answer to "why?", although it'd be handy to know if it is indeed the 4th power, and if not the 4th power, then what?


Einstein Equation at the Singularity. Understanding the Dr. Michio Kaku's explanation

I'm trying to understand what Dr. Michio Kaku is exaplaining in this video: http://www.youtube.com/watch?v=hydDhUNvva8 I'm just able to rewrite this two equations. I think I've probably made some mistake in the process of copying. Otherwise, I don't understand the meaning of the two $\mathrm O$ and the $I^{'}$ in the second equation. In the video the Dr. Michio Kaku is solving the first equation right? But I saw that is slightly different from the classical field equation, why? What's the difference? $As^2$ is the solution of the first equation? Is possible understand something from the video or is it too cutted? \begin{equation} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu} \end{equation} \begin{equation} As^2=\left(_{\mathrm O}\left(1-\frac{2MG}{r}\right)^{I^{'}\mathrm O}\right) \end{equation}


Is the minimum radius of a positronium system of the order of compton wavelength or less than that?

Since from electron-positron annihilation energy and uncertainty principle,the minimum radius of positronium comes out as half of the compton radius.


Units in cgs system

How do I find the dimensions of this quantity (in $cgs$)... $$\frac{4\pi me^2}{h^2n_o^{1/3}}$$ where $m$ is the mass of electron $e$ is the magnitude of electronic charge $h$ is the Planck's constant & $n_o$ is equilibrium number density of electrons


Determining distance from object in concave lens

so I was studying for my Yr11 Physics test tomorrow and I came across a question that I got the wrong answer on, all of my friends are getting the same answer. Question: You wish to project the image of a lamp, magnified three times, onto a screen 5.0m from the lamp. How far from the lamp should the mirror be placed? I though that we would just use $M=\frac{-d_i}{d_o}$ where $M=3$ and $d_i=5$, which will work out that $d_o\approx -1.7$ which has a magnitude of 1.7m from the lamp However, our answers book says that the answer is 2.5m What have I done wrong? Also, please keep in mind that I am in year 11 and may not understand anything beyond that level


Why does letting air into a box of wine help the flow?

I've notied that when I get down to the last part of a box of wine, the flow slows to a trickle. If I open the nozzle, tilt the box back (letting air gurgle into the bladder) and then tilt it back forward the flow is greatly increased. Why does letting air into the bladder help increase the flow?


Falling into a black hole emitter vs observer

Let's say we are working with the Schwarzschild metric and we have an emitter of light falling into a Schwarzschild black hole. Suppose we define the quantity $$u=t- v$$ where $$dv/dr= 1/(1-r_{s}/r)$$ where $r_s$ is the Schwarzschild radius. What is the $u$ as observed by the emitter? I just need a _definition of $u_e$_. I have problems identifying the quantities as measured by an observer at large $r$ and that of the emitter. Would I be right at least to say that $$t_{e}=\tau$$ the proper time? Many thanks. * * * In fact, I've been told that $$du_o/d\tau=du_e/d\tau$$ Why is it? * * * My apologies for leaving out a subscript $e$ in the question. It has been added.


Apart from Alfvenic turbulence, what is the most important unsolved problems in plasma physics?

**Apart fromAlfvénic turbulence**, what is the most important unsolved problems in plasma physics? Is there a review article describing some important unsolved problems in plasma physics, I repeat, apart from turbulence, which is an unsolved matter in all areas of physics.


Poincare recurrence and the multiverse

In this paper Susskind claims that a stable de Sitter universe is problematic (among other things) due to the existence of Poincare recurrence, which happen because of finite entropy. I disagree that this is a problem for reasons similar to those given here. In any case, he also proposes a variant of eternal inflation driven by false vacuum decays. These decays, as far as I know, are driven by Coleman - de Luccia instantons and are probabilistic in nature, that is to say there is always some probability of instanton creation via quantum fluctuation. My questions are:   1. Since false vacuum decay is probabilistic is it possible that in this bubble cosmology setup there are casual patches which live beyond their Poincare recurrence time?    2. Is there any possibility in string theory that our Universe, assuming it has finite entropy, will eventually be stabilized for an infinite amount of time or is this question settled beyond all reasonable doubt?  Thanks!


Velocity and acceleration with displacement

Find the velocity and the acceleration for a moving object with displacement $= te^{-t}$. The $e$ is the exponential function.


In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In the Dyson-Maleev representation, $(S^+)^\dagger\neq S^-$, so we have a non-unitary transformation. Are eigenvalues and matrix elements invariant under such a transformation? In matrix form, what exactly are the two transformations that will give rise to the two representations, say for spin $S=3/2$?


Time derivative of angular velocity in rotating reference frame

I am going through a section in a textbook regarding the Newton Euler equations for a system of rigid bodies (robotics text). There is a particular line in the derivation I don't understand, I've attached the image to make it clear: ![enter image description here](http://i.stack.imgur.com/scZIL.png) I think the derivative of the second term on the right-hand side of 3.28 is using the product rule to get to 3.32, but I don't understand where the cross- product comes from. Could someone explain this particular part of the derivation? Thanks.


What is the weakest magnetic field that can practically be measured?

I would like to be able to measure magnetic fields that are generated by nerve impulses in extremities like in fingers. I know that they are very weak: about 100 mV electric potential would give us about current 10 mA if resistance = 1/10 Ohm, but interested in measuring magnetic fields. **How many rounds should you have in a coil to measure 10 mA current?** The jumping of electrons from one level to another may become a problem in this case. However, I am not sure where and how. Assume that there is a steady current in the situation so Biot-Savart law. The radius of the nerve is r = 0.003m.


Virtual particles contributing to universal expansion.

> **Possible Duplicate:**   >  Virtual particle production in space-time. Does virtual particle production contribute to observed universal expansion?


Find magnetic scalar potential for superconducting sphere

In regions where $J = 0$, the curl of the magnetic field $B$ is necessarily zero (since $\nabla \times B = \mu_0 J$). Therefore $B$ can be written as $B = -\nabla V_m$, where $V_m$ is a scalar function of position.   A superconducting ball of radius $a$ is placed in an otherwise uniform magnetic field $B = B_0 \hat {z}$. Since the $B$ field is zero inside the ball, it must hold that $B_{\perp} = 0$ just outside the surface of the ball (by magnetic boundary conditions).   (a) Find the magnetic _scalar_ potential $V_m (r, \theta)$ everywhere.   Far from the sphere the field is $B_0 \hat {z}$, and hence $V_m \rightarrow -B_0 z$. So in spherical coordinates, $V_m \rightarrow -B_0 r \cos \theta$ for $r \gg a$. Is the following true? $$ \frac {\partial V_{\text{above}}}{\partial n} = \mu_0 K $$ Since $V_{\text{below}} = 0$ due to the sphere being a superconductor. Using separation of variables, the first condition yields $$ \sum_{l=0}^{\infty} A_l r^l P_l (\cos \theta) = -B_0 r \cos \theta, $$ So, only one therm is present, $l = 1$. Then $A_1 = -B_0$ with all other $A_l$'s zero. From the second condition, $$ \frac {\partial V}{\partial n} = A_1 \cos \theta - \sum_{l =0}^{\infty} (l+1)\frac {B_l}{R^{l+2}} P_l (\cos \theta) = \mu_0 K. $$ Now I'm stuck as to how to proceed and remove some of the $B_l$ terms. Perhaps my boundary conditions are not correct.


Find work done by force along a path - is parameterization the only way?

$F = x^3y^4 \hat i + x^4y^3 \hat j$ from $(0,0)$ to $(1,1)$. I am given different paths. For example, "ﬁrst along x axis and then along the y axis" is one of the paths. Is this problem solvable only using parameterization or is there another way to handle/integrate it?


Kinematics - moving body

If I have an inclined plane, which has acceleration $\alpha$, will a block placed on the inclined plane slip back with the same acceleration? (considering that the surface is frictionless)


Why is the pressure inside a soap bubble higher than outside?

Apparently, the air inside a soap bubble is under higher pressure than the surrounding air. This is for instance apparent in the sound bubbles make when they burst. Why is the pressure inside the bubble higher in the first place?


Gauge fields and strings: Loop equations

I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) = \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, {\rm d} s $$ where the non-abelian phase factor around a closed loop $C$ is defined as $$ \Psi(C) = {\rm P}\exp \left(\oint A_\mu dx^\mu \right) = {\rm P}\exp \left(\int_{0}^{2\pi} A_\mu \dot{x}_\mu\, {\rm d}s \right)$$ It seems that he is using the relation given on p. 116: $$ \delta \, {\rm P} \exp \int_{0}^{2\pi} M(\tau) {\rm d}\tau = \int_{0}^{2\pi}{\rm d}t\,{\rm P} \left(\delta M(t) \exp \int_{0}^{2\pi}M(\tau){\rm d}\tau\right)$$ Matching with (7.25) I find $\delta A_\nu = F_{\mu\nu} \delta x_\mu$. This relation seems to be saying that if I change the position of the loop at the parameter $s$ by $\delta x_\mu(s)$ then the vector potential changes by $\delta A_\nu(x(s)) = F_{\mu\nu}(x(s)) \delta x_\mu(s)$. I don't know how to derive this relation. Is it legitimate?


Will a spinning object come to rest?

Will a sphere spinning on its own axis come to rest given enough time, provided no other forces act upon it? I know that if you have _two_ spinning spheres in the depths of space and orbiting each other they'll eventually stop spinning -- they'll tidally lock each other. The internal friction of the shifting tidal bulge eventually overcomes the rotation. But I don't see what forces would stop a sphere from rotating around its own axis, in absence of other forces. And yet, each point on the sphere (except points along the axis) is experiencing accelerated motion, which implies a force. My intuition says that something must maintain the acceleration of each point, which implies that the acceleration causes internal stress. I know this to be true macroscopically (the sphere will have an equatorial bulge), but what forces maintain the acceleration of each particle, and will the stresses ever overcome the rotation of the sphere? **Clarification** : I am _not_ considering infinitely rigid spheres or point particles. I'm wondering about actual non-rigid macroscopic spheres, such as planets.


Could this suggest that there is a wavelength smaller than Planck's?

Suppose the earth receives a photon with a wavelength $\gamma_1$. Since spacetime is expanding, we know that this photon had an original wavelength $\gamma_2$, such that $\gamma_2\lt\gamma_1$. This is phenomenon is known as redshift. Nothing special. Now, here's the thing, what if the earth receives from far away a photon whose wavelength is equal to Planck's length $\ell_p$. This means that the photon before traveling all this distance a wavelength smaller than $\ell_p$. But this is impossible, since $\ell_p$ is the smallest possible wavelength. So would this suggest that there is a wavelength smaller than Planck's?


Deuterium density in seawater

I heard today that deuterium for nuclear fusion could be extracted from the Oceans, in which it is present in "heavy water." I heard this claim: > _"Deuterium is distributed uniformly with Ocean depth."_ This contradicted my naive expectation that deuterium density would _increase_ with depth, on account of it being in "heavy water." For example, salinity increases with depth. I asked, and received two contradictory explanations. The first was that because powerful waves and Ocean currents mix Ocean water, gravity is negligible. This surprised me; I thought that Ocean's were stratified, with predominantly tangential, rather than vertical, currents. The second explanation was that my view of seawater is naive - that I should think of seawater not as heavy and regular water molecules, but as a soup of weakly interacting deuterium, the normal hydrogen isotope and oxygen, forming and breaking water and heavy water molecules with a short time-scale. If so, apparently, deuterium density and the normal hydrogen isotope would be distributed evenly with depth. This did not help me, because, I then naturally wondered why _deuterium's_ density doesn't increase with depth, since it is heavier than hydrogen-1? Is the claim correct? and if so, what is really going on? I have the feeling that Ocean behaviour with depth might be non-trivial...


If LHC is the most powerful microscope in the world, what does it magnify?

Some experts (and non experts) have called LHC the most powerful microscope in the world. I am wondering, what does the LHC magnify? Or is calling it a microscope just wrong and misleading?


Embrace Physics

Does anyone have any suggestions as to what is a good topic for a short talk on theoretical physics to a bunch of Math and Physics undergrads that might make them "embrace" theoretical physics? Thanks. (Brain storm: String theory, quantum tunneling, etc...?) Also it would be great if someone could suggest some resources.


How does water falling down a waterfall gain energy?

I'm kind of a noob in the physics area. But I know that energy cannot be created, it can only be transferred. So how and what "type" of energy does water get when free-falling?


Finding out the Potential difference

> _Find the potential inside and outside a uniformly charged solid sphere > whose radius is $R$ and whose total charge is $q$. Use infinity as your > reference point. Compute the gradient of $V$ in each region, and check that > it yields the correct result._ The above one is a question cropped from Griffiths page no 82. When I try to solve the problem they used a potential equation inside the sphere ![enter image description here](http://i.stack.imgur.com/ctSoI.png) My problem is in the right part. Why do we need to consider the next part? we have been asked to solve potential inside a charged sphere. so can't I use only the limit $\infty $ to $ R$? I am saying this because, i have considered the infinity as the reference point and R as the point where I will find the potential. > Is it because we need to add the potential outside of the sphere and inside?


Conversion of energy to mass with radioactive decay

Consider the following problem: _The nuclei of Am-241 decay by the emission of $\alpha$-particles with a kinetic speed of $8.8 \cdot 10^{-19} J$. In a certain source of Am-241 there are $ 4.0 \cdot 10^3$ decaying nuclei per second. Caculate the mass this source loses per year._ So what the correction sheet tells me is that you basically just have to find the mass of an $\alpha$-particle in $kg$ and then just multiply that number until you get the amount of decaying $\alpha$-particles in a year. However, I think this is wrong, since the $8.8 \cdot 10^{-19} J$ isn't accounted for. I'd say you have to take the mass of the $\alpha$-particle, then add to that the $8.8 \cdot 10^{-19} J$ after you've converted it to $kg$ by using $E = mc^2$ and THEN multiplying that number in order to get the amount of decaying $\alpha$-particles in a year. Which is correct, or perhaps better: which is less wrong?


Question about a finite time interval step in the derivation of the Feynman path integral in Sakurai

This may be a possible errata but Sakurai (pp 126 in the 2nd Edition) states that starting with $$S = \int dt \,\,\scr{L_{\mathrm{classical}}}$$ Looking at a finite-time-interval of the action: \begin{equation} S(n,n-1) = \int\limits_{t_{n-1}}^{t_n} dt \left( \frac{m \dot{x}^2}{2} - V(x)\right)\,\,\,\,\,\,(1) \end{equation} Now using a straight-line approximation for points $(x_{n-1},t_{n-1})$ and $(x_n,t_n)$, which implies $$\dot{x} = \frac{x_n-x_{n-1}}{t_n-t_{n-1}} = \frac{x_n-x_{n-1}}{\Delta t}$$ Sakurai then has Eq (1) become: $$S(n,n-1) = \Delta t\left( \left(\frac{m}{2}\right) \left(\frac{x_n-x_{n-1}}{\Delta t}\right)^2 - V\left(\frac{x_n+x_{n-1}}{2}\right)\right)$$ My question is why is the potential now dependent on $ V\left(\frac{x_n + x_{n-1}}{2}\right) \mathrm{\,\,instead \,\,\,of\,\,\,} V\left(x_n- x_{n-1}\right)$? I checked the most recent errata (4/5/13) posted by J. Napolitano ( http://homepages.rpi.edu/~napolj/ErrataMQM.pdf ) and there is none for this page. Could anyone clarify this step for me? Thanks.


How can laser communication be faster than microwave?

My textbook states that "Laser communication is much faster than microwave communication." But, how can that be? Both are electromagnetic waves with different frequencies but, how can the speed be different? Or Is the statement referring to any kind of processing speeds involved?


About Dirac equation in curved spacetime (spherical)

I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta d\phi^{2},$$ where $\alpha$ is a positive constant. The Dirac equation(a part of) is given by $$i \gamma^{\mu}(x)(\partial_\mu+\Gamma_{\mu}(x))\psi(x)=0$$ where $\Gamma_{\mu}(x)$ is spinor affine connections and $\gamma^{\mu}$ are the general Dirac matrices. If we define the tetrad base as $$e^{\mu}_{(a)}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\\ 0 & \sin\theta \cos\phi &\sin\theta \sin\phi & \cos\theta \\\ 0 & \frac{\cos\theta \cos\phi }{r}&\frac{ \cos\theta \sin\phi}{r} & -\frac{\sin \theta}{r} \\\ 0 & -\frac{\sin\phi}{\alpha r \sin\theta} & \frac{\cos\phi}{\alpha r \sin\theta}& 0\\\ \end{array} \right) ,$$ using the tetrad base we obtain(But I cannot!): > Blockquote $\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2r}(1-\frac{1}{\alpha})\csc\theta(\gamma^{(1)}\cos\phi+\gamma^{(2)}\sin\phi)$ > Blockquote . Here, $\gamma^{(a)}$ are flat Dirac matrices. Moreover, $$\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2}\gamma^{(a)}[(\partial_{\mu}e_{(a)}^{\mu}(x)+e_{(a)}^{\rho}(x)\Gamma^{\mu}_{\rho \mu})+i\gamma^{(5)}\varepsilon_{(a)(b)(c)(d)}e^{(b)\mu}e^{(c)\nu}\partial_{\mu}e^{(d)}_{\nu}]$$ Now my question is: how one can obtain $\gamma^{\mu}(x)\Gamma^{\mu}(x)$ expansion above, which is $$\gamma^{\mu}(x)\Gamma^{\mu}(x)=\frac{1}{2r}(1-\frac{1}{\alpha})\csc\theta(\gamma^{(1)}\cos\phi+\gamma^{(2)}\sin\phi),$$ because I couldn't obtain it. I'm not used to this notation :(


To what degree does guitar construction affects the vibration of the strings?

There's an old debate going on in the guitar community about how much does wood choice and body shape affect the sound of an electric guitar. No one denies that there's a difference acoustically (how the guitar sounds unplugged) because in this situation it's the wood and body shape that amplify the sound made by string vibrations, but when we're talking about the sound as it comes from the pickups, things get much more uncertain because there are a lot of variations even in supposedly identical guitar parts, and accounting for all of them for the purpose of doing an experiment where the only difference between two guitars is the wood is difficult (I certainly haven't heard of an experiment that was satisfying enough, but feel free to prove me wrong). I'm interested in a way to circumvent these practical difficulties using a theoretical explanation: Since the pickup only sees the string's vibrations, the question basically becomes "does body shape and wood make enough difference in the way a string vibrates that it changes the sound in a noticeable way", and this sounds like something that may be possible to figure out mathematically, or at the very least should be much easier to test because there are less variables involved. My question is: Is what I'm describing possible to calculate/test, and has this been done before?


Could velocity be taken as fundamental instead of time?

In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as fundamental and then understand time by dimensional analysis in terms of l/v (length divided by velocity)? If not, why not?


about wavefunction and vector entries

I am beginer of physics and I am studying some very fundamental idea of quantum mechanics by myself. In the introducing book I am reading, there is an example to show a particle diffraced by a slit or something small then there show diffraction pattern. So if the wavefunction of the particle is represented in momentum space such that it has n entries in the wavevector so integer n stands for the nth diffraction order. So what's the significance of the square of the nth entry? Does it mean the probabilty of finding the particle with momentum diffracted into the nth order of the diffraction pattern? Let's say I have a source of particles, each particle emitted at different speed towards the slit so they will be diffracted independently. How does the intial speed of each atom affect the diffraction pattern? If we look at the nth diffraction order, how will the speed of the particle change the probablity of finding the particle in the nth order? Thanks a lot


Row of pivoted magnets and energy scale

This question is about a system involving a horizontal row of length **L** of equally spaced pivotable magnets, each with a pole at either end. These magnets will often be referred to as units. So each unit when rotated causes it's neighbours to rotate in the opposite direction to itself. When the first unit is quickly rotated 1/4 of a turn and fixed in place rotationally, each will likely turn less than the previous at first, taking longer to complete the full 1/4 of a turn. They will all eventually rotate the full 1/4 as the far end of the line is still not fixed. . > Turning the first magnet will require energy. For a line of 1000, while this > will be more energy than that for a line of for example 2, it will be less > than 1000 times the energy required for a line of 1. This is because: > >>   1. Momentum means that every unit doesn't need to be rotated the full distance to turn the first unit, it is not as if they are connected together by rods, the first can be turned all the way before the others have had time to rotate much at all. >> >>   2. More distant magnets contribute less force, so most units are contributing very little force to the first unit directly. Another way to think about this is that the line of separated magnets forms a compound magnet that will result in more force than each component magnet has individually, but not as much as 1000 times more, for example if metal becomes attached to an end it will take less than 1000 times the force of an individual magnet to remove. >> >> . Coils around each unit slow rotation but produce **N** electricity after 1/4 of a turn, however long the 1/4 turn takes. The amount of electricity produced appears to be L * N, for an input power that is increment by a diminishing amount for the growth of L, an example of this sort of growth could be sqrt(L). . > **Question:** This has the potential not to add up energy wise for long > lines, what's going on?


In mathematics, the Chern number is defined in terms of the Chern class of a manifold. What is the exact definition of Chern number in condensed matter physics, i.e. quantum hall system?


Relationship of multiple particles under collision

Consider 3 particles. All 3 particles travel along the x-axis.   * The 1st particle possesses some mass, m, and its initial position is somewhere on the negative x-axis. It contains some (positive) velocity v.   * The 2nd particle possesses some mass, A*m, and its initial position is at the origin of the x-axis (0). It contains no velocity (at rest).   * The 3rd particle possesses some mass, B*m, and is initially situated somewhere on the positive x-axis. Like particle 2, it contains no velocity (at rest). Find the relation between A and B, under which the 1st and 2nd particles will collide with each other more than once. note: it is assumed the collisions are elastic as well as there being no external forces considered. also the collisions are such that all particles remain on the x-axis.


Chiral anomaly and decay of the pion

I am told that if all classical symmetries were reflected as quantum symmetries, the decay of the neutral pion $$\pi^0 ~\longrightarrow~ \gamma\gamma$$ would not happen. Why would the conservation of the axial current in QED prevent the decay of the pion? What is the non-conserved charge in this decay?


Does my model aircraft propeller beat the sound barrier?

I fly model aircraft. Recently I changed from a "1300kv" motor to a "1500kv" motor. The difference is in speed; the 1300kv can go about 16,400rpm full- throttle and with a fully charged battery and the 1500kv can go nearly 19,000rpm. (This is no load speed, but the prop adds very little load when the aircraft is not flying.) I noticed that when I increased the throttle beyond about 75% the motor and propeller became incredibly loud. Below that they were loud, but no where near as much. So I did some ballpark calculations. I'm using a 7x3.5" propeller - which means it has a 7" diameter and 3.5" pitch. In this case we are only interested in diameter. 7" diameter means 3.5" radius, or 8.86 cm. Doing 19,000 rpm means than I'm doing about 317 rps. According to 2 * pi * r the circumfrence of the prop's tip is 55.64 cm. Plug this into 317 rps and we get 17,638 cm/s or 176.4 m/s. Which is about half the speed of sound. Although this is not actually the speed of sound it's very close to it, and much higher than I expected. So what does that mean? I figure that although the propeller itself isn't traveling faster than the speed of sound, the air it pushes out of the way is. My physics teacher showed that if you take a metre stick and swing it through the air (avoiding hitting anyone) it makes a wooshing sound - the air is moving faster than the ruler was. So the prop is pushing the air away faster than this and causing this very loud noise. Is my reasoning correct? Is my prop breaking the sound barrier?!


Is the Keppe Motor "real"?

Its makers say that it's a highly efficient motor because it resonates with (what they call) "Essential Energy", which is free energy that is everywhere in space. (I think that’s just a different name for Dark Energy which somehow they are managing to disturb with the coils of the motor, and when it disturbs back, a bit of power is generated in the coil). I’ve listened to a radio interview of the makers, and they sad that a small motor powered by a 9v battery ran for about 2 hours with enough torque that they couldn’t stop it with their own hands. Is that what is really happening? Is it revolutionary? Or just bull… just a motor that consumes less energy since its coils are turned off half of the time? Also, strange things happen on the proximities of a running motor (a similar motor will start running even though it has no power, but it will run slower than the first one). Caused by the same thing that wireless cell phone chargers use to work?


Speed of ejection of liquid from revolving tank

Consider a cylindrical tank filled with a liquid. Suppose it has an orifice somewhere at the bottom and the liquid is spurting out of this hole at a certain velocity. Now, if you rotated this cylinder about an axis passing through its centre, will the speed of ejection of fluid change? I feel it should, because of a centrifugal force, but how do I prove this mathematically?


How is space ship's acceleration perceived if the acceleration is perpendicular to the velocity?

Spacecraft in orbit around the Earth are constantly accelerated by the gravitational field of Earth. That's why the spacecraft ($m \ll M$) is in an (elliptical) orbit around the centre of gravity of the Earth, accelerated by $ \mathbf{g}=-{G M \over r^2}\mathbf{\hat{r}} \, .$ Plugging in the numbers for a spacecraft orbiting at roughly 420 km (such as the International Space Station), this gives: $ \mathbf{g} = {- 3.986 \cdot 10^{14} \over (6371 + 420) \cdot 10^3 m} = - 8.6 \, \mathrm{m}/\mathrm{s}^2$ So, an astronaut on-board the ISS is in a reference frame that is constantly accelerating at an acceleration of $8.6 \, \mathrm{g}/\mathrm{m}^2$. Yet unlike the astronauts featured in this related question, they do in fact feel no gravitational acceleration at all; at most they may feel some centrifugal pseudoforce, but this is considerably less than the gravitational acceleration, and at most at microgravity levels. Is acceleration only perceived when it changes the magnitude of the velocity, as opposed to the direction? What is the fundamental reason for this? Or am I misunderstanding something?


Finding the electric field

The charge per unit length on a long, straight filament is -91.9 µC/m. > (a) Find the electric field 10.0 cm from the filament, where distances are > measured perpendicular to the length of the filament. (Take radially inward > toward the filament as the positive direction.) MN/C > > (b) Find the electric field 50.0 cm from the filament, where distances are > measured perpendicular to the length of the filament. MN/C > > (c) Find the electric field 150 cm from the filament, where distances are > measured perpendicular to the length of the filament. $λ_q=\frac{dq}{dl}→dq=λ_qdl$ $\vec{E} =k_e\frac{q}{r^2}\hat{r}$ $||\vec{E}||=\frac{λ_qdl}{r^2}$ How can I solve this without knowing the length of the filament? Supposedly this is the suggested way of solving this problem--and I'd like to understand this method very much. Could someone help me? Also, in addition to solving it by this expedient, is it possible to solve this problem by employing a Gaussian surface?


Capacitor Charging and Discharging when connected to the ground

When we charge a capacitor using a battery and then remove the battery, the plates of capacitor becomes charged. One holds positive charge and the other one gets equal negative charge. o. k. ? Now if we attach a wire to the positive plate and connect it to the ground , will the electrons from ground climb on the positive plate and make it neutral ? No. But if we connect positive plate to the negative plate then the capacitor will get discharged. Now consider a situation when we connect 4 capacitors A,B,C,D of equal capacitance in series and connect them to a 10 Volt battery. Now the P. D. between positive and negative plate of capacitor A will be (10- 7.5) i.e. 2.5 . For B it will be (7.5 -5 ) i.e. 2.5 , For C it will be (5- 2.5 ) i.e. 2.5, For D it will be (2.5-0) i. e. 2.5. So potential at positive plate of A is 10 and potential at negative plate of D is 0 and the Potential Difference is 10 volts, which is the potential difference of the battery. Now connect the wire joining C and D capacitor to ground and now record the potential difference at A, you will find it 7.5 and at positive plate of D it will be 0, and at negative plate of D it will be -2.5. This happens because negative charge from ground climbs on the positive plate of capacitor D and makes it neutral. My question is why in this case negative charge climbs on this positive plate of D and makes its potential zero ? But such thing does not happen when we connect positive plate of a charged capacitor to the ground.


Pellet of lithium in a vacuum

What would happen to a grain of sand sized pellet of lithium in a vacuum? Because there is no pressure, would it become more like a sticky liquid?


Is infrared light diffused on walls?

Lamps that produce visible light can light up a whole room because the ceiling and the walls diffuse light. This way, when there is a single lamp in a room, something that is in the shadows is not completely dark. If I set up an infrared lamp in a room, will the whole room be lit by infrared light like with a regular lamp, or will it be absorbed by the walls ? If it is not, will the wavelengths be the same as the ones emitted by the lamp ? Thanks !


What is the $\delta$ in fluctuation model

I found this formula : $G-G_0=\frac{-nkT}2\delta^2$ on this Site and I'd like to ask what does the $\delta$ stand for?


Why must the excitation of closed strings in String Theory be spin-2?

In String Theory it is predicted that as a result of the closed strings we have spin-2 gravitons. 1) How do we know there must be an excitation of spin-2 particles? 2) Why does a spin-2 particle HAVE to be a graviton?


Firing machine question

Suppose we have a firing machine on a frictionless surface at point $x=0$. It fires a bullet of mass $m$ every $T$ seconds. Each bullet has the same constant velocity $v_0$. There's a body of mass $pm$ ($p$ times larger than the bullet's mass) at point $x=x_0$. I know, according to the law of momentum conservation (inelastic collision) that the velocity of the body as a function of bullets inside of it will be: $v(n)=\frac{v_0 n}{n+p}$. How can I know the elapsed time $\Delta t$ from the $n-1$ bullet hit to $n$ bullet hit? Is there any way to solve this without a recursion? What I mean by recursion is this (the answer to the question): $ \left\\{ \begin{array}{l} a_1=\frac{x_0}{v_0}\\\ a_n=a_{n-1}+ T \cdot \frac{{p-1+N}}{p} \end{array} \right. $ This series represent the time when nth bullet hits the object. I do not quite understand why $a_n-a_{n-1}= T \cdot \frac{{p-1+N}}{p}$.


How is it possible to equate the internal energy at constant volume with the internal energy of an adiabatic process?

I hope my question makes sense. My problem is that, I have read through numerous textbooks that nC(cons. volume)dT = -PdV when deriving the relationship between T and V for an adiabatic process, though I can't seem to understand why this relation has been made. Surely, at constant volume, work done by the system should be zero? I am lead to believe that, at constant volume, no work is done by the system, but for an adiabatic process, work is only done by the system, not on it. So why is it possible to equate the two internal energies together when they describe two different systems?


Why uniform ensemble and microcanonical ensemble are the same in thermodynamic limit?

So uniform ensemble is the ensemble of all possible states with energy less than E, and since entropy derived from both are equivalent with each other in the thermodynamic limit, many functions derived from them are the same, however, I am wondering why that is the case since apparently the uniform ensemble contains more states and thus more information than the microcanonical ensemble.


How to name different approaches to relativistic quantum theory

In the introductory chapter of the QFT book by Mark Srednicki the author notes that > [p. 26] So now we have two different approaches to relativistic quantum > theory [...] Which [one of those two] we use is a matter of convenience and > taste. And _quantum field theory_ , the formalism in which position and time > are both labels on operators, is much more convenient and efficient for most > problems. The other approach is described as follows: > [p. 25] Let us discuss the second option [...] to promote time to an > operator. [...] We can use the _proper time_ $\tau$ of the particle [...] as > the time parameter. The coordinate time $T$ [...] is then promoted to an > operator. [...] Relativistic quantum mechanics can indeed be developed along > these lines, but it is surprisingly complicated to do so. Does this " _second option_ " have an own specific name (for distinction from _quantum field theory_ , in Srednicki's sense)?


Internal potential energy and relative distance of the particle

Today, I read a line in Goldstein Classical mechanics and got confused about one line. > _To satisfy the strong law of action and reaction, $V_{ij}$ can be a > function only of the distance between the particles: $V_{ij} = V_{ij}(|{\bf > r}_i-{\bf r}_j|)$._ What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances? I prefer to receive mathematical proof (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome.


Angles between axes after Lorentz transformations

Consider frame $K_1, K_2, K_3$ such that $K_1$ moves along the $y$ axis of $K_2$ with speed $v$ and $K_2$ moves along the $x$ axis of $K_3$ with speed $u$. Find the angle between the $x$ axis of $K_1$ and that of $K_3$. My question is that I can't see intuitively why the $x$ axis of $K_1$ rotates after Lorentz transformations. And moreover, if one calculates the Lorentz transformations connecting $K_1$ and $K_3$, one may have the following: $$t=\gamma\gamma't'+\gamma\beta x'+\gamma\gamma'\beta'y'$$ $$x=\gamma'\gamma\beta t'+\gamma x'+\gamma\gamma'\beta\beta' y'$$ $$y=\gamma'\beta't'+\gamma'y'$$ where the $'$ coordinates are those of $K_1$ and the others are coordinates of $K_3$. However, if we consider transformations of the origin and a point on the $x$ axis, we may get $\Delta y=0$, that is, the x axis does not rotate.


What happens in string theory beyond the Hagedorn temperature?

What happens in string theory when the temperature exceeds the Hagedorn temperature? Is that even possible? If yes, what is the nature of the phase transition and the phase beyond that? What happens to spacetime in the new phase?


Geometric quantization in Kepler problem in hydrogen atom

Why in the usual geometric quantization calculation the dimensions of eigenspaces is wrong (we can see this obstacle for Kepler problem in hydrogen atom). Here is a refference see


Basic energy calculation for N identical spin system

We have a system that has N identical spins $n_i$, and each spin can be in state 1 or 0. The overall energy for the system is $\epsilon\sum_{i=1}^{N}n_i$. My understanding: There is only one distinct way in which we could have 3 spins in state 1, because the spins are identical. Likewise for 4,5 or 6... spins in state 1, there will only ever be one way of choosing. So I think that the multiplicity is $N$, therefore entropy $S(E,N)=k_blog(N)$. I need to calculate the $E(T,N)$, I know $(\frac{\partial S}{\partial E})_{V,N}=1/T$ but my entropy doesn't have an E in it. How can I calculate the energy and is what Ive got so far correct?


Energy density of a quantum mechanical ensemble

How do we determine the energy density of a given system? I have seen that the density operator $$\rho~=~\frac{\exp(-\beta \hat{H})}{\text{tr}(\exp(-\beta \hat{H}))}.$$ What does this mean exactly and how does it relate to the pure and unpolarized states of a system? For example, given a system of relativistic spin-1/2 particles, the completely unpolarized beam density is $\sigma~=~\frac{1}{2} \left| \uparrow\right>\left<\uparrow \right|+\frac{1}{2} \left| \downarrow\right>\left<\downarrow \right|$ or $$\sigma~=~\left( \begin{array}{cc} \frac{1}{2} & 0 \\\ 0 & \frac{1}{2} \end{array} \right)$$ Are the diagonal elements the probabilities that within the ensemble a particular particle will be found to be in that spin state? If so, how does this affect the energy density of the system? I am not entirely sure what the Hamiltonian would be in this case. But I think that spin-up and spin-down are eigenstates with eigenvalues $ \lambda_{\pm}=\pm \frac{\hbar}{2}$, so as a guess, since we are talking about spin-projections, that the energy eigenvalues are going to be proportional to $\lambda_{\pm}$ $$\displaystyle \rho~=~\frac{1}{\text{e}^{-\frac{\beta\hbar}{2}} + \text{e}^{\frac{\beta\hbar}{2}}} \left( \begin{array}{cc} \text{e}^{\frac{\beta\hbar}{2}} & 0 \\\ 0 & \text{e}^{-\frac{\beta\hbar}{2}} \end{array} \right) $$ where $\beta$ is the inverse of temperature given by $\beta=\frac{1}{kT}$, however I am not sure what the constant of proportionality will be since I am assuming that the Hamiltonian will be proportional to $\sigma_z$ and not equal to it. Would it be possible to see an example or have an explanation of what is happening? I am referring to Sakurai's _Modern Quantum Mechanics_ and _Quantum Mechanics_ by Bransden and Joachain.


Difference between photo electron spectrum and photoelectron angular distribution

I am trying to learn the Photoelectron velocity map imaging. While I was going through the article "Chem. Soc. Rev., 2009,38,2169-2177", it is said that the "photoelectron spectrum reflects the energy eigenvalues of the parent atomic or molecular system, while the phtoelectron angular distributions reveal the characteristics of the corresponding wave functions" (in the page 2170 of the said reference). I didn't understand what it is mean by the statement "photoelectron angular distributions reveal the characteristics of the corresponding wave function"!. Can anybody please explain me this?


Work done on a moving particle in electric field

This is one of the "fast answer" exercises I've been given to train (should be answerable in around 6-7 minutes). I can only think of a very long-round way to solve this. The question is as following: > Two point charges 2Q and Q are located at the opposite corners of a square > of length l (2Q at the top right corner). The point A is in the lower left > corner and the point B is located halfway the right side of the square. > Determine the work $W_{AB}$ required to move a particle with charge q from A > to B. There are several methods I can think of, though non come to an answer. My best guess is to calculate the magnitude of the force acting on the particle in A, and in B. And then substract these... However to simply calculate the effective force in position B proves itself to be lengthy task. So there must be a simpler mechanic?


Unknown isotope

Problem description: The most common isotope of a single nucleus stripped of its electrons is accelerated through a potential difference of 1225V and fired horizontally into a B-field directed perpendicularly into the page (flat page lying on desk). The nucleus is observed to travel in a circular path of radius 9.973 cm. Afterward an alpha particle is accelerated through the same potential difference and fired in the same manner into the magnetic field and observed to travel in a circular path with radius 8.980 cm. What element is the unknown nucleus? (For this problem determine the masses of the nuclei--unknown and alpha--by summing the individual masses of the protons and neutrons. Also, include a diagram with your solution showing all aspects of the problem description.) This is what I have so far: ![Valid XHTML](http://i44.tinypic.com/ht6zad.jpg). ![Valid XHTML](http://i40.tinypic.com/qytrba.jpg). This is the point where I am confused as to how to find the mass and charge of the unknown particle, or where exactly I'm supposed to go from this point to solve the problem. (I have heard from other students that there is some guessing involved (from the periodic table)) Help would be appreciated. Thank you.


How To Stop A Heat Ray Gun?

What material would be best suited to create a shield to protect from the new heat ray gun?


How can we know, today, that there's something from 100 light-years from here?

In my understanding, to take a picture of something that is 100 light-years from here, our "camera" would have to travel 100 years at light speed, take the picture, send to us, and 100 years later we would receive it. So, how do we have pictures of things that are 100 light-years from here?


Magnet substitution

I wonder if the two magnet configurations below are equivalent. If they are not - what will be the effects on the flux? ![enter image description here](http://i.stack.imgur.com/Zqble.png)


Why do 'dead' batteries work again after exchanging the places of the batteries in an electronic device?

My camera, which is powered by two AA batteries in series, would not power on. I removed the batteries, exchanged their locations, and the device worked again - for another 15 minutes or so. The temperature of the batteries did not change, and only about few minutes passed before I discovered this. Is there a physical explanation for this phenomenon?


Is relativistic motion equivalent to fluctuating gravitational fields?

The theory of relativity makes very precise predictions about how an object's motion through space-time affects the passage of time for both the object and observers in other frames of reference. I am curious, however, whether this motion is equivalent to changing / fluctuating gravitational fields? What if instead of an object moving through space/time we instead had the same object exposed to shifting gravitational fields, similar to what it would experience if it were actually moving relative to other masses? Can it be shown that the same set of effects would still exist from the object's frame of reference (eg time dialation, length- contraction, etc)?


In GR, how do particles know how to fall in instead of out of a gravitational well?

The geodesic equation (let's suppose that we're talking about massive particles, so I'll parameterize the path by proper time $\tau$) $\frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\rho \sigma}\frac{d x^\rho}{d \tau} \frac{d x^\sigma}{d \tau}=0$ is invariant under $\tau \rightarrow -\tau$. However, falling particles clearly have a direction, they always fall in instead of out. Formally, how does a falling particle 'know' which way to move in $\tau$?


If dark matter is a new type of particle, what does that imply?

My understanding is that dark matter cannot be (or is at least highly unlikely to be) an exotic form of any known particle. On the other hand, articles about particle accelerators seem to say that the Higgs is the last piece missing in the Standard Model jigsaw puzzle. _If_ dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery "stand to the side" of the Standard Model or would it certainly change the foundations? (Forgive my extremely lay understanding and vocabulary -- please feel free to correct my mistakes.)


Do Category Theory and/or Quantum Logic add value in physics?

I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks? More particularly, is it worth investing the time to study these formalisms?


Calculate gravitational force between a sphere and a uniform rod

I want to calculate the gravitational force between a uniform rod of mass 2 kg, length 1 meter and a uniform sphere of mass 0.2 kg. Distance between the sphere and the rod is 0.1 meter ( The sphere is in the same line as the rod I.e. sphere is one meter from the near end of the rod). I thought of integrating $(G*m*M)/x^{2}$ from 0.1 to 1.1 meter which does yield the correct result but the solution in the textbook has a different procedure.


Newtonian Gravity on a Riemannian $3$-Manifold

To solve the Poisson equation for the Newton Potential, say $\phi$, one can use the divergence theorem, such that $$\int_U \nabla^2 \phi \sqrt{g}~ dV= \int_{\partial U} <\nabla \phi,n> \sqrt{g_\Sigma}~ d\Sigma=1,$$ where $\sqrt{g_\Sigma}$ represents the induced metric on the border. Considering that $\phi$ is classic it will only depend on the relative position between the source and the point where we want to compute the potential, so, in order to use the divergence theorem, we need to find the set of points that are at the same distance of a given point (where we consider the source of the field to be) and that will be the border, $\partial U$, of the considered volume that contains the source. Well, if space is the three dimensional Euclidean space then the volume that contains the source is straightforwardly the 3-ball whose center is at the source position and the border $\partial U$ the corresponding 2-sphere. But if space is a different 3 Riemannian Manifold things are a "little" bit harder and finding the set of points that are at the same (geodesic) distance from the source is in general quite difficult or even impossible... But when searching for the geodesic curves on a manifold one can, in some circunstances, instead of searching for the curve that minimizes the distance $$d~=~\int_a^b [g(\dot{c}(s),\dot{c}(s))]^{1/2} ds,$$ search for the curve that minimizes the energy functional $$E~=~\int_a^b [g(\dot{c}(s),\dot{c}(s))] ds,$$ since in those circunstances the curves that minimize the energy are the same as the curves that minimize the distance. Well, finding the set of points that are at the same energy is much easier...I would then ask: what is the significance of the energy defined above and would it make sense, physically, to say that equipotetial surfaces are not the set of points that at the same distance from the source but the set of points that have the same energy. **PS:** Note that in the Euclidean Space, the set of points that are at the same distance is the same as the set of points that are at the same energy.


What physical forces give rise to the peculiar bond angle of hydrogen peroxide?

In one of the Periodic Table videos, Prof. Poliakoff shows a model of a hydrogen peroxide molecule and claims that the H-O bonds will always be at right angles to each other. I have a rudimentary intuition for how molecules arrange themselves based on mutual attraction and repulsion - I've solved some Foldit puzzles - but the H2O2 defies that kind of common sense explanation. I get that there are forces pushing the hydrogens apart and forces pulling them together, but how does that fragile equilibrium of forces result in an _exact_ right angle?


Optical Drive Physics

I have been recently wondering how is data stored on and retrieved from optical devices like CDs, DVDs, and Bluray. What makes these different storage types different from each other?


Is there any way to prove/disprove we are in a computer simulation?

Is there any way to prove/disprove we are in a computer simulation in some transcendental reality? Even if we are "really" not in a simulation, can we ever prove so? Even if some messages/evidence flashing "You are in a simulation by an Overlord" pops up, can we always dismiss it as a naturalistic hallucination? Can quantum mechanics apply to closed systems? MWI and consistent histories are candidates, but run into the preferred basis/choice of consistent histories problem. Copenhagen requires a transcendental observer and a Heisenberg cut, doesn't it? If there is a transcendental observer, doesn't that imply there has to be a transcendental realm? So, if our universe is a subsystem in some transcendental realm which can perform, and only perform, Turing computable computations, isn't that equivalent to saying our universe is a computer simulation in some transcendental realm? This doesn't necessarily imply there is some Overlord who is an Intelligent Designer Programmer. It could be an Overlord who is a "Blind Watchmaker" transcendental observer. This Overlord might be deist or an occasional interventionist. In Copenhagen, in order for something to exist, it has to be observed, right? That means we need an "all-seeing eye in the sky" transcendental Overlord, right? If this Overlord forgets about us, we die, and pop out of existence, right?


Question about laws of conservation in electrodynamics

One way of deriving laws of energy, impulse and angular momentum of electromagnetic field conservation is following:   1. Introduce two values below: $$ \mathbf P = \frac{c}{4 \pi}[\mathbf E \times \mathbf B], \quad W = \frac{1}{8 \pi}(\mathbf E^{2} + \mathbf B^{2}). $$   2. Using Maxwell's equations and time derivative of these values, get the laws. But how to argue an introduce of values from it. 1? Can Noether's theorem help to argue it?


What is a qualitative explanation for order of values of standard entropies of formation for isomers of hexane?

I have literature values for the standard enthalpy and Gibbs energy changes of formation at 298 K, $\Delta H_\text{f}^o$ and $\Delta G_\text{f}^o$, _for product in the ideal gas state_. Since $G=H-TS$ and by definition, the reactions are carried out at constant temperature I can solve for $\Delta S_\text{f}^o$ via: $$\Delta S_\text{f}^o=\frac{\Delta H_\text{f}^o-\Delta G_\text{f}^o}{T}$$ I've done this for three hexanes, n-hexane, 2-methylpentane, and 2,3-dimethylbutane. Below are the results: ![hexanes thermochemistry values](http://i.stack.imgur.com/Fzrd4.jpg) I guess the sign makes sense to me. It says that all three molecules represent more order than their reactants (C and H2). What is strange to me is the order of the standard entropies of formation. Since these are values in the ideal gas state, entropy effects due to intermolecular forces should not be included, correct? Hence, I would have thought that n-hexane would be the most "ordered" of the three molecules. Surely a linear molecule is less likely (requires higher order) than a branched molecule. What have I missed?


What is 'tilt locking' of a laser cavity?

In another question, jasonh mentions a scheme for locking a laser to a resonant cavity called 'tilt locking': > The signal is used for locking the frequency of a laser to a high finesse > cavity. The lock has to be stable enough for intensity and phase > fluctuations resulting from e.g. acoustic vibrations of the cavity to be > strongly suppressed. The error signal is created via the tilt locking > scheme, i.e. it is the difference of intensities on two halves of a split > photo diode. I'm familiar with Pound-Drever-Hall locking of a laser cavity, but I've never heard of 'tilt locking' before. How does it work?


Is vapour pressure based on partial pressures or just total pressure on the liquid?

The explanation for the boiling point of water is that at 100C, the vapour pressure becomes greater than atmospheric pressure. But say you had a a jar of water sealed in argon at 1atm, which is larger than the vapour pressure of water, wouldn't some water evaporate? I'm guessing yes, because there are no water molecules in the argon initially, so wouldn't some randomly leave the liquid, making the partial pressure on average non-zero? Another reason I can think of it being based on partial pressure is that an open jar of water in a windless dark, dry room will still eventually evaporate. So why is it that at 100C there is a sudden dramatic change? Isn't the vapour pressure at e.g. 95C more than the partial pressure of water in the atmosphere as well?


Gauge fixing of an arbitrary field

How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?


Why are there more lunar maria on the near side than on the far side?

It is widely known that the side of the moon that faces us has a particularly large number of dark patches, which resemble seas of liquid, hence, "maria". It is strange, though, that the far side of the moon has a great lack of these features. Why does the far side of the moon have so few maria? If they maria formed due to impacts, is there some reason that the far side would have less collisions?


What objects look best in an O-III filter?

I've heard that an O-III (Oxygen III) filter is great for planetary nebulae. Is this true for all planetary nebulae, or just some or most? What other target types are often improved with an O-III filter?


Ising Hamiltonian for relativistic particles

An Ising system is described by the simple Hamiltonian: $$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units of $\hbar$), the $c_{1i}$ are local magnetic fields and the $c_{2ij}$ are (nearest neighbor) interaction constants. So one usually only keeps constants $c_{2ij}$ if $x_i$ and $x_j$ are located close enough in space; after a certain spatial range the $c_{2ij}$ are supposed to become negligible. Note that (1) can be interpreted as two low-order terms of a general Taylor expansion of a general Hamiltonian H($x_1$,$x_2$,…,$x_n$): $$H = c_0 + \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j + \sum\limits_{i,j,k} c_{3ijk} x_i x_j x_k + … \,\,\,\,\,\,(2)$$   My questions are the following:   Q1) Does (1) also hold if the particles are moving ? Looking at Feynman’s book on stat mech it seems that (1) is then also valid: the form (1) arises according to Feynman due to magnetic dipole interaction or due to a “Coulomb potential combined with Pauli’s exclusion principle”; nothing in the derivation demands that the particles are static.   Q2) Suppose all particles are on a 2D lattice, except two particles (1 and 2) which are emitted from a source in the middle of the lattice and move in opposite direction through the lattice at relativistic speeds. Can we apply (1) here or do we need a corrected version ? (One way to correct (1) if some particles are moving might be to include higher order terms as in (2).)


Effects of space mining on Earth's orbit

I was reading a post about space mining, specially lunar mining. I was thinking about what would change in Earth's orbit if we start bringing tons of rocks to it? I mean, in a huge scale. So, would space mining change Earth's orbit in a way to produce any dangerous effects? If the system in question is not a planet/satellite like Earth/Moon, would that change anything? Would bringing tons of rocks from Mars be any different than Moon?


Can you create mass with $E=mc^2$?

If you use the equation $E=mc^2$ could you make matter by dividing the $c^2$? I'm sorry if this is a really stupid sounding question or if it shouldn't be asked here.


When is many-body perturbation theory valid?

I'm calculating expectation values (thermal, time-independent) using many-body perturbation theory, but I'm unsure how to work out what values the parameter I'm expanding the perturbation series in can take. I read that it's when the matrix elements, $\langle i | H_{pert} | j \rangle$ where $H_{pert}$ is the perturbative term of the Hamiltonian and $| i \rangle$ and $| j \rangle$ are the eigenvectors of the unperturbed Hamiltonian, are much smaller than the energy difference between $i$ and $j$. But I don't really understand what that means, or how it helps me to calculate what values my perturbation parameter can take - Is there a method I can use to figure it out? **Edit:** As requested, to make this concrete, I have a one dimensional fermionic Hubbard model with Hamiltonian \begin{equation} H= -t \sum_{\langle l,m \rangle} (c^\dagger_l c_m + h.c.) + U \sum_l (n_{l\uparrow} - 1/2)(n_{l\downarrow}-1/2) \end{equation} I have a special case where I know that $U$ is very small and I want to use many-body perturbation theory to see its effects on correlation functions. I calculate the correlation functions using the functional integral method (i.e. calculating a functional partition function). For this case, how would I go about finding out how small $U$ has to be in order for perturbation theory to be valid? Secondly, (if this should be a separate question, please let me know!) if instead I have a random $U$, dependent on its position in the lattice, \begin{equation} H= -t \sum_{\langle l,m \rangle} (c^\dagger_l c_m + h.c.) + \sum_l U_l (n_{l\uparrow} - 1/2)(n_{l\downarrow}-1/2) \end{equation} I can then use a similar functional integral technique, but take an average over the functional partition function (e.g. over a Gaussian distribution). This average removes the $U_l$ and leaves $\Delta$, the variance of the distribution we've averaged over. In this case, it is $\Delta$ which the perturbation series is expanded in. How would I go about finding how small $\Delta$ has to be for the perturbation series to be valid? I don't want an answer that's true for any system, I just want to understand how to go about finding it for any system. So if anyone knows of another system where it is shown how small the expansion term has to be, please let me know. Thanks.


Can a harmonic drive reducer/strain wave gear be used to gear up instead of down?

I wish to be able to gear up from a rotating bicycle hub with a revolutions per minute range of 150 to 850 to a ratio of 200:1. Can I do this by simply reversing the direction of a harmonic drive reducer? To clarify could I apply torque through the flex spline and use what is normally the torque in put shaft as an output shaft?


Why does grinding steel create sparks but aluminium doesn't?

Grinding steel produces sparks. Grinding aluminium does not. Why? I found this article about why grinding steel produces sparks but would like to confirm if it is true. I would also like to know why does it behave so on a molecular level if anyone can explain that.


Saturation of an iron core?

Lets assume we have a 1 kg iron core, and a 100 kg iron core. Now saturation is defined as how much that core can absorb the magnetic field, since they are different sizes, don't they saturate at different fields? Lets assume the 1kg core saturates at 1 Tesla, shouldn't the larger core saturate at a higher magnetic field?


Fluid mechanics Grashoff number

When I derived the Grashof number (for natural convection at a vertical plate) starting with the ratio of buoyancy forces to viscous forces, I realized that I had set the Reynolds number to 1 (ie I had to set velocity*length/kinematic viscosity = 1) to get the accepted equation. Why is that?


What is Fermi surface and why is this concept so useful in metals research? Particularly, I can somewhat appreciate the Fermi energy idea - the radius of Fermi surface which is a sphere. But is there any quantitative use of more complicated Fermi surfaces?


I have a boiling pot of water that I am going to cook my Sponge Bob Noodles in. I am very hungry and want the noodles to cook faster, so I turn up the heat from medium to high.( Note, the water was already boiling at medium).What happens to the time it takes for my noodles to cook, and the temperature of the boiling water?


Why is it so hard to mix spaghetti with meatballs?

When I cook spaghetti with a sauce that includes meatballs or chopped vegetables, those stay in chunks and won't get uniformly mixed in the pasta. Is there a simple reason for that? Has it been studied? you know some people study piles of sand with various grain sizes?


How can an object with zero acceleration move?

My physics text has a problem in which it is said that a person moves a block of wood in such a way so that the block moves at a constant velocity. The block, therefore, is in dynamic equilibrium and the vector sum of the forces acting on it is equal to zero: $\sum{F} = ma = m \times 0 = 0$ This is where I get confused. If the person is moving the block in such a way so that the sum of the forces acting on it is equal to zero, how can he be moving it at all? I realize that   1. even if there is no force acting on an object it can still have velocity (Newton's First Law)   2. force causes _acceleration_ (a change in velocity--not velocity itself) but for some reason I don't understand how it is possible for the object to move with constant ~~acceleration~~ velocity. Do I have all the pieces? I really feel like I am missing something. **EDIT** The block was at rest to begin with, and the person is moving the block through the air **EDIT: THE ACTUAL PROBLEM** For the sake of clearing up ambiguity, here is the actual problem. (I thought it was wooden blocks at first, sorry) > Two workers must pick up bricks that like on the ground and place them on a > worktable. They each pick up the same number of bricks and put them on the > same height worktable. They lift the bricks so that the bricks travel upward > at a constant velocity. The first gets the job done in one-half the time > that the second takes. Did one of the workers do more work than the other? > If so, which one? Did one of the workers exert more power than the other? If > so, which one? Basically, **_the intent of this question is to ask if the above problem is entirely hypothetical._** My text does not indicate that it is, which is what I find confusing.


What's the relation between virtual photons and electromagnetic potentials?

Given that: 1) virtual photons mediate the electric and magnetic force fields 2) the magnetic field is the curl of the magnetic vector potential 3) the electric field is the negative gradient of the electric scalar potential How do we understand the vector and scalar potentials in terms of virtual photons? Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present. For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics. Thanks for any help.


Optical Raytracing by using Adiabatic Hamiltonian Method

I'm looking into raytracing a Lüneburg Lens which is a gradient index (GRIN) optical element with a radially varying refractive index: $$ n(\rho)=n_0\sqrt{2-\left(\frac{\rho}{R}\right)^2}, 0<\rho<R $$ The good paper by Hanhong Gao et.al, "Design of thin-film photonic metamaterial Lüneburg lens using analytical approach" describes the theoretical background and applies it to the Lüneburg lens, but does not indicate any code which can be used to carry out the raytracing. I'm considering to try out an implementation in Mathematica, however if there is any comparable example of the application of Hamiltonian optics raytracing this would give me a head start. In the book "Geometric Optics" by Antonio Romano has provided an online site of Mathematica notebooks which were offering some implementation of the Hamiltonian formulations but the Link to these notebooks seems to be broken. Any info about already existing implementations on Optical Raytracing with Hamiltonian Method would thus be most welcome!


Bug in linear thermal expansion, $L_0$ must be $0$

Assume we change the temperature of an object with negligible size in $2^{nd}$ and $3^{rd}$ dimensions from $T_0$ to $T_1$ to $T_2$, with all of them pairwise different. We choose a substance with coefficient $\alpha \ne 0$. $\alpha L_0(T_2 - T_0) = L_2 - L_0 = (L_2 - L_1) + (L_1 - L_0) = \alpha L_1(T_2 - T_1) + \alpha L_0(T_1 - T_0)$   $\Leftrightarrow L_1T_2 - L_1T_1 + L_0T_1 - L_0T_2 = 0$   $\Leftrightarrow (T_2 - T_1)(L_1 - L_0) = 0$   $\Leftrightarrow \alpha L_0(T_2 - T_1)(T_1 - T_0) = 0$   $\Rightarrow L_0 = 0$ Where is my mistake?


Franck and Hertz graph function

I need a function whom graph is the following (the Frank and Hertz experiment graph): > ![](http://i.stack.imgur.com/WifSY.png) Currently I wrote the following function:               function comptuteValue (value) {              var period = 4.9           , low = 6           ;              function addOrSubstract (x) {                  if (x < low) {                 if (x <= period) {                     return Math.pow(x, 3/2);                 } else {                     return Math.pow(period + low - x, 3/2) - x * 1.1;                 }             }                  return addOrSubstract (x - period);         }              return (2 * (Math.floor(value / period + 0.1)) + addOrSubstract(value) * 1.1989).toFixed(2);     }      And the graph looks like: > ![](http://i.stack.imgur.com/gPPca.png) As you can see there are differences between the two graphs. What's the function of the Franck and Hertz experiment graph? $f(x) = ???$ For example: $f (0) = 0$ $f (4.9) = 12$ $f (5) = 6$ And so on.


Why do nucleons feel a repulsive force when less than 1 fm?

My Modern Physics textbook by Taylor states that when nucleons are less than 1 fm apart, there is a strong repulsive force between them. I am fairly certain that it is not the Pauli Exclusion Principle that the book is talking about. So if this is true, what is the mechanism for such a force?


What gonna happen to beer on the moon?

Let's assume that we have bottle of beer on the Moon. What would happen to it in sunlight? Some cases:   1. Beer can be dark or lager.   2. It can be contained in glass bottle (transparent or dark) or aluminium can. Will it freeze or vaporize or what?


Which device amplifies the optical signal in fiber optics?

My teacher ask me one question. How the optical signal or information was transfered in fiber optics communication? Then i explained it and use the word optical amplifier then teacher ask me what is optical amplifier i said which amplifies optical signal then again which device amplifies the optical signal in fibre optics communication? I was dump at that moment i dont have the answer for it and i google it then i read about Doped fiber amplifier(DFA) but i didn't get it so can you give me any suggestion to understand it like any website or explain it.


Orbits of maximally entangled mixed states

It is well known (Please, see for example Geometry of quantum states by Bengtsson and Życzkowski ) that the set of $N-$dimensional density matrices is stratified by the adjoint action of $U(N)$, where each stratum corresponds to orbits with a fixed type of degeneracy structure of the density matrix spectrum. These orbits are flag manifolds, for example the orbit of density matrices of spectrum $(1, 0, 0, 0)$ (pure states) is $\mathbb{C}P^3$ and the orbit of density matrices of spectrum $(0.5, 0.5, 0, 0)$ (and also $(0.4, 0.4, 0.1, 0.1)$) is the complex Grassmannian $Gr(4,2, \mathbb{C})$. These spaces - being coadjoint orbits - are known to be Kahler-Einstein. An observation by Ingemar Bengtsson which was stated and proved in the following article, asserts that in the composite system of two $N-$ state quantum systems $\mathcal{H}^N \otimes \mathcal{H}^N$, whose orbit of pure states is $\mathbb{C}P^{N^2-1}$, the orbit of maximally entangled pure states is $\mathbb{R}P^{N^2-1}$, which is a minimal Lagrangian submanifold.This observation was also mentioned in the first reference. My questions:   1. Does this result generalize to non-pure states, for example is the orbit of maximally entangled states in $Gr(N^2,2, \mathbb{C})$ (the orbit of density matrices of the type $(0.5, 0.5, 0, 0, . . .)$ in $\mathcal{H}^N \otimes \mathcal{H}^N$) isomorphic to $Gr(N^2,2, \mathbb{R})$.   2. Is there a physical interpretation of this result (a question left open by Bengtsson). Update: This is a clarification following Peter Shor's remark. Among the biparticle pure states, the maximally entangled states, have the property that their partial trace with respect to one system is totally mixed (please, see the discussion following equation 22 in the second reference). As a generalization I wish to know the local orbits of the states within a fixed biparticle density matrix orbit whose partial trace has the maximal Von Neumann entropy relative to all other states in the same biparticle orbit. My motivation is that if the local orbits will happen to be the real flag Lagrangian submanifolds of the complex flag manifolds defining the biparticle state orbits (which is the case for the pure states), these manifolds have well known geometries and this can contribute to our understanding of mixed state entanglement.


Complete set and Klein-Gordon equation

In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following: > The solution $\Psi$ can always be expressed as a superposition of free > particle solutions, provided that the latter form a complete set. They from > a complete set only if the negative energy components are retained, so they > cannot be simply discarded. Can anyone explain what this is talking about? First of all, what is a complete set? And why does forming a complete set require negative energy components being retained? Why is the equation (solutions, I guess) need to be a complete set?


Graph Invariants and Statistical Mechanics

Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was looking if similar connection have been found for graph invariants. I am interested to know if any attempts have been made form statistical physics community to solve graph isomorphism problem. Since both knot problem and graph isomorphism are problem about testing structural equivalence there might be connection between them.


Quantum Eraser under Lorentz Boost

Suppose I am conducting the Quantum Eraser experiment. The results of this experiment are easy to understand with the traditional quantum mechanical interpretation of a pair of entangled photons. Specifically, suppose that I am performing the "eraser" part of the experiment in which one photon is diagonally polarized so that when the entangled partner passes through the circular polarizers of the double-slit apparatus it cannot be determined through which slit it passed (thus reproducing an interference pattern). Now suppose (It's like supposition _Inception_! Supposings inside supposings inside supposings.) that I Lorentz boost into a reference frame in which the particle passing through the double-slit apparatus is received at the detector (and consequently has its polarization measured) **before** its entangled partner gets diagonally polarized. It appears that there are two possible outcomes: either (a) I disagree with the results recorded by an observer in the commoving frame of the experiment (i.e. I see no interference pattern.) or (b) I must conclude that the **future** polarization of the entangled partner somehow reached back in time and changed the outcome of my measurement. So which is it? Do I get different results or do I need to embrace a further layer of quantum non-locality? This is a tough one for me to understand because of its complementary experiment: Imagine putting the polarizer much farther away than the double slit apparatus so that there is no interference pattern created (because the entangled partner does not get diagonally polarized until **after** the measurement has already occurred). In that case, I could boost into a frame where an interference pattern _should_ occur because the partner gets polarized first. (Or am I missing something? Will there be - or not be - an interference pattern regardless?) If the answer is that I get the same results as the commoving measurement, then why is the commoving frame preferred? Why is it _its_ results that are maintained in all reference frames and not the (expected) results from my boosted frame (which should be just as valid)? Also, I think that this version of the experiment is fundamentally different from the Delayed-Choice Quantum Eraser experiment and Wheeler's thought experiment since both of those experiments are easily explained by representing the particle in a different eigenbasis (namely, as being in a superposition of the interfering and the non-interfering states - instead of being in a "collapsed" state of another observable). However, I am also interested to know whether I am wrong in this estimation. Are these experiments basically the same? Can conclusions about or explanations of one of them be generalized to the others? **Edit** : Because of the question in the comments, I've added a little additional explanation for why I think the experiments are basically different. Primarily, the difference comes from the fact that both Wheeler's thought experiment and the Delayed-Choice Quantum Eraser can be explained in such a way that apparent "cause" always precedes "effect." Causality is still violated, but at least its not giving us information about the future. For instance, if we didn't know whether or not the entangled partner beam would get polarized a year from now (in the quantum eraser experiment), we could predict whether it does or not **today** by determining whether or not there was an interference pattern at the double-slits. If there exists any boosted frame in which the beam gets polarized first, then that result must carry over to the frames where it gets polarized after. So "future-telling" can happen in the relativistic quantum eraser, but not in the other two experiments. For example, in the Wheeler experiment, we polarize the photon and then _after the fact_ decide at random whether to measure its polarization or not. If we measure the polarization, we get no interference; if we don't measure the polarization, we do get interference. Often this is ascribed to a sort of "predictive" non-locality, but just as easily, we could say that the polarized state of the photon is actually a superposition of states (namely, interfering and non-interfering) and when we measure its polarization we collapse it into the non-interfering state (and if we don't, we collapse it into interference). Likewise for the delayed-choice quantum eraser: instead of thinking that the beam-splitter in the future decided the interference in the past, you could just as easily conclude that the interference (or not) in the past decided the outcome of the beam-split in the future.


Is it possible to enlarge the output angle of a projector with outside lens?

I have a very small projector, with a small output angle (30 degree). In my current demand I want a bigger output angle(90 degree), so that I can get a bigger image within a given distance. Is it possible to get it by outside lens? The projector: ![enter image description here](http://i.stack.imgur.com/fp4XE.jpg) What I expect: ![enter image description here](http://i.stack.imgur.com/LOb0F.jpg) If it's possible, how to design the lens? If not, why? What would I lose? (resolution,brightness,distortion etc)


Path integral with zero energy modes

Consider the field integral for the partition function of a free non- relativistic electron in a condensed matter setting, i.e. $$ Z = ∫D\bar\psi D\psi \exp\left(-\sum_{k,ω} \bar\psi_{k,ω} (-iω + \frac{k^2}{2m} - \mu) \psi_{k,ω}\right) $$ where the action is written in Fourier representation and $\mu$ denotes the chemical potential. Now, this integral is well known to be the determinant $$ Z = \det\left(β\left(-iω+\frac{k^2}{2m} - \mu\right)\right) $$ which is equal to the product of all eigenvalues of the quadratic form in brackets. But here is my problem: > How to calculate a quadratic path integral if the quadratic form has some > _eigenvalues_ that are equal to _zero_? If the chemical potential $\mu$ is positive, then all all momenta with $\frac{k^2}{2m} = \mu$ (the Fermi surface) will represent an eigenvalue equal to zero and would force the determinant to become zero. Of course, I could just drop the problematic eigenvalues from the determinant and call it a day, but unfortunately, I would like to understand quantum anomalies a bit better and > zero energy eigenmodes are important for understanding the axial quantum > anomaly for example of the $1+1D$ Schwinger model. Fujikawa's book on quantum anomalies argues that the axial anomaly comes from an asymmetry of zero modes of the Dirac operator, but I am very confused because a zero mode would make the determinant of the Dirac operator and hence the path integral vanish. How to make sense of this?


Resonance of a tube of air in case of more complex shapes

I've been thinking about posting this question on Music Performance stack, but finally I think it fits more here since I'm interrested in technical details. The subject of resonance of a tube of air is pretty well documented for simple shapes, like cylinders (open of closed), cones, closed rectangular boxes, spheres et cetera. I even found formulas in my high school books. But I am interested in much more complex case. Let's imagine we have a solid tube with a certain shape with holes in it filled with vibrating air of frequency $f_i$. How can we predict the frequency $f_o$ of air that come outside of the tube? For now I can imagine only two ways.   * Decompose the problem into sub-problems, in this case complex shape into much simpler shapes (cylinders, cones, rectangular boxes, spheres etc). But still I have no idea how a whole in the tube would affect the frequency, I suppose we would need to take in count the placement of this whole and its size.   * Run a computer simulation of such system. This would require knowledge how does the frequency propagade through air in such solid tube. To simplify a little bit this complex problem let's forget conditions like air temperature, pressure and humidity. I'm more interrested in second approach, to create a tool that works a little bit like fuild mechanics simulation software, that help for example design heating installations etc cetera, except that this tool would be usefull for acoustic installations, architecture or instrument design. If anyone could share some knowledge usefull to understand this phenomena I would be greatfull. What I have found in school and university books or wikipedia is not enough, or I'm not searching for right words.


Are operators in quantum mechanics linear transformations?

Observables in quantum mechanics correspond to self-adjoint linear operators. If $\psi$ is an eigenvector of $\hat A$, then $\hat A\psi=\alpha\psi$ where $\alpha$ is the eigenvalue of $\psi$. Otherwise, if the initial state $\psi$ is not one of the eigenvectors of $\hat A$, then the terminal state will be one of $\hat A$'s orthonormal eigenbasis with probability determined by the projection.   1. Then what is the physical meaning of $\hat A\psi$ in this situation?    2. Shouldn't observables be more appropriately described by maps from the Hilbert space to the set of probability distributions on that space?


Can an elliptical orbit take the same time as a circular obit?

In the picture below you can see two orbits of potential pbjects. The main aspect of the orbits is that they have a collision point at the maximum of the elliptical orbit. My question is, could the object travelling elliptical orbit take the same amount of time as the object taking the circular orbit? The reason behind this question is that I am doing an analysis on the movie called Gravity. The only information provided in the movie is that the debris cloud will collide every 90 minutes (with the Hubble telescope) and that it is going at 50000 miles per hour (which is much faster that the Hubble telescope). Obviously, since they are going at different speeds, they have to be in different orbits. So this is the only possibility I have come up with where they could possibly collide every 90 minutes, or that the orbit time of the elliptical orbit would take 45 minutes, or 22.5 minutes (any factor of 90). ![enter image description here](http://i.stack.imgur.com/g99H4.png)


asymptotic curvature of the universe and correlation with local curvature

There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions regarding this: 1) Does the fact that a manifold with a) asymptotic (space) curvature zero and b) local inhomogeneities with positive (space) curvature imply that there will be regions with negative (space) curvature? 2) a Region of negative (space) curvature implies dark energy in that region? 3) assuming answer to both 1) and 2) are true: does this represent an independent confirmation of dark energy? or there is somehow an geometric relationship relating asymptotic flatness to accelerated expansion (the traditional reason to introduce dark energy in the first place)? EDITED: to reflect distinction between space and space-time curvatures.


Origins of the principle of least time in classical mechanics

Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least action?


Renormalization and Infinites

Measuring a qubit and ending up with a bit feels a little like tossing out infinities in renormalization. Does neglecting the part of the wave function with a vanishing Hilbert space norm amount to renormalizing of Hilbert space?


Kolmogorov entropy and noise

There are various ways of calculating the Kolmogorov entropy (KE)of dynamical system. According to Pesin's theorum, it is the summation of the lyapunov exponents; frominformation theoretic concet, it is the supremum of the Shannon entropy defines for all the partitions. I am using the formula given here. > (Jaap C. Schouten, Floris Takens, and Cor M. van den Bleek , " Maximum- > likelihood estimation of the entropy of an attractor" Phys. Rev. E 49, 126) The desired signal $x(t)$ whose entropy is to be measured is corrupted with additive white gaussian noise. Say: $$x(t)= ax(t-1) + bx(t-1) + \text{Chaos_Signal}+\text{White noise}$$ I want to calculate the KE of $x$. Assuming, the chaotic signal to be logistic map, my question are (Q1) for different values of the parameters $(a_1,b_1),(a_2,b_2),\ldots$ will the KE measure be same for a particular signal-to-noise ratio (SNR)? (Q2) Will KE decrease with increase of SNR? Intuitively, entropy is the measure of randomness. So, higher the level of noise (low SNR) then greater is the entropy. Again, noise also changes the value of the Lyapunov exponents of the system. I guess, Lyapunov exponent increases. Is there any proof that shows relationship between KE of time series increasing or decreasing with noise?


Particles Associated With Gravitational Waves

I've been reading about linearized GR and the study of gravitational waves, and an odd thought popped into my head. According to wave-particle duality (admittedly, usually used in quantum mechanics!), particles can act like waves, and waves can act like particles. My logic was that, therefore, there should be a particle associated with a gravitational wave. There is one thing, though, that might interfere with this: gravitational waves are plane waves, not longitudinal waves - which is how we describe, for example, photons. But my original question still stands: Is there a particle associated with a gravitational wave? If not, why not? If so, is there a separate mathematical structure associated with it (aside from the metric for the wave)? FYI: Nope, I'm not talking about gravitons. I know those are completely different and unrelated (and, at the moment, hypothetical).