title: Complex energy eigenstates of the harmonic oscillator

text: Given the Hamiltonian for the the harmonic oscillator (HO) as $$ \hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,, $$ the Schroedinger equation can be reduced to: $$ \left[ \frac{d^2}{dz^2}-\left(\frac{z^2}{4}+a\right)\right]\Psi=0~, $$ where $a=-\frac{E}{\hbar\omega}$, $z=\sqrt{\frac{2m\omega}{\hbar}}$. Now, the two independent solutions to this equation are the Wittaker's functions (Abramowitz section 19.3., or Gradshteyn at the beginning, where he defines the Wittaker's functions) $D_{-a-1/2}(z)$ and $D_{-a-1/2}(-z)$. Apparently, there is no constraint on the values for $a$. In Abramowitz, especially, there is written "both variable $z$ and $a$ can take on general complex values". Therefore my first question is: Let us fix $a=i$ and let us therefore take the Wittaker's function $D_{-i-1/2}(z)$. This functions is solution of the time independent Schroedinger equation, and, therefore, is an eigenfunction of the ho hamiltonian. Since its value for the parameter $a$ is $i$, it follows that its eigenvalue $E$ must be $E=-i\hbar\omega$. However, this result is contradictory, since the hamiltonian must have only real eigenvalues, since it is hermitian. What do I do wrong? My second question is: Since the functions $D_n(z)$ form a complete set for $n$ positive integer with zero, I can expand my function $D_{-i-1/2}(z)$ onto the basis set $D_n(z)$. $$ D_{-i-1/2}(z)=\sum_n C_n D_n(z)~. $$ But, evidently, if $D_{-i-1/2}(z)$ is itself an eigenfunction with a different eigenvalue with respect to any of the $D_n(z)$, the expansion above does not make sense. This question is somewhat correlated to the previous one. So, I believe I do something wrong which is in common to both of them.


title: How does an electron "move" in an s-orbital?

text: I have read multiple answers on StackExchange about this question, but I wasn't able to find a concrete answer. Like other questions, the reason I ask about the s-orbital is because it has a zero orbital angular momentum. But, the implication of having a zero angular momentum is unclear? Some answers discuss probability distributions, but the question is how can there be multiple places for an electron, if it can't move. I read another answer that says that the electron passes through the nucleus or curve around it as a wave. I would appreciate it if anyone could provide a resolution to this question.


title: Writing wave functions with spin of a system of particles

text: Suppose I have 2 fermions in a potential $V(x)$. Both particles are moving in one dimension: the $x$ axis. Then, neglecting the interaction between the particles, the spatial wave function of the system would be of the form $$\psi_{n_{1}}(x_{1})\psi_{n_{2}}(x_{2}) $$ Now, if I'm considering particles with spin 1/2, the notation $\alpha(1)$ indicates that the particle 1 has spin up, and $\beta(2)$ denotes the particle 2 having spin down. Now, I want to write the complete wave function, a function of the form $$\psi_{n_{1}n_{2}s_{1}s_{2}}(x_{1},x_{2},s_{1},s_{2})=\psi_{n_{1}}(x_{1})\psi_{n_{2}}(x_{2})F(\alpha,\beta)$$ where $F(\alpha,\beta)$ is a function of the spin of the system. To this end, I have that the only physically possible functions $F(\alpha,\beta)$ are: Symmetric: $\chi_{\alpha}:=\alpha(1)\alpha(2),\quad\chi_{\beta}:=\beta(1)\beta(2),\quad\chi_{+}:=\frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2)+\alpha(2)\beta(1)\right]$ Antisymmetric: $\chi_{-}:=\frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2)-\alpha(2)\beta(1)\right]$ In order to write down the complete wave function with spin, I understand I have to consider the energy levels. For example the ground state: $\psi_{1}(x_{1})\psi_{1}(x_{2})$. If $\psi_{1}(x_{1})\psi_{1}(x_{2})$ is symmetric ( _as I understand it is_ ), then I must multiply this function times the antisymmetric function $\chi_{-}$ (in order to get an antisymmetric wave function, for two fermions). If $\psi_{1}(x_{1})\psi_{1}(x_{2})$ is antisymmetric (and I understand this is impossible, since the ground state is not degenerate), then I'd have 3 wave functions, obtained by multiplying $\psi_{1}(x_{1})\psi_{1}(x_{2})$ times $\chi_{\alpha}$, $\chi_{\beta}$ and $\chi_{+}$. Now, for the 1st excited level, say $\psi_{2}(x_{1})\psi_{1}(x_{2})$, **my question is** , what happens when this function is not symmetric neither antisymmetric? I mean, I could built a symmetric $$f_S=\frac{1}{\sqrt{2}}\left[\psi_{2}(x_{1})\psi_{1}(x_{2})+\psi_{2}(x_{2})\psi_{1}(x_{1})\right]$$ or an antisymmetric $$f_A=\frac{1}{\sqrt{2}}\left[\psi_{2}(x_{1})\psi_{1}(x_{2})-\psi_{2}(x_{2})\psi_{1}(x_{1})\right]$$ wave function. But which one of these must I choose? Or, must I calculate the resultant complete wave functions with both? Then, when I count the states with the 1st excited energy, I'd have 4 instead of 1 or 3.


title: Which way does the scale tip?

text: I found the problem described in the attached picture on the internet. In the comment sections there were two opposing solutions. So it made me wonder which of those would be the actual solution. So basically the question would be the following. Assume we would have two identical beakers, filled with the same amount of the same liquid, lets say water. In the left beaker a ping pong ball would be attached to the bottom of the beaker with a string and above the right beaker a steel ball of the same size (volume) as the ping pong ball would be hung by a string, submerging the steel ball in the water as shown in the picture. If both beakers would be put on to a scale, what side would tip? According to the internet either of the following answers was believed to be the solution.   1. The left side would tip down, because the ping pong ball and the cord add mass to the left side, since they are actually connected to the system.   2. The right side would tip down, because of buoyancy of the water on the steel ball pushing the steel ball up and the scale down. Now what would the solution be according to physics? ![enter image description here](http://i.stack.imgur.com/Mj3Y0.jpg)


title: Why does the thermal conductivity of water decrease with increasing salinity?

text: Intuitively I would expect the thermal and electric conductivity to be positively related, and since electric conductivity increases with salinity, so should thermal. But according to this table (p.10) it decreases. Why is this? Related: is there such a thing as the Wiedemann-Franz law for liquids like water? There's a paper that has theoretical derivations about it, but it's nowhere to be found:   `Predvoditelev, A. S., "Some invariant Quantities in the Theories of Heat Conductance and the Viscosity of Liquids," Russian Journal of Physical Chemistry, Vol. 22, p. 339 (1948)`


How many different types of (hypothetical) multiverse theory exist? For example, the obvious Quantum MWI, eternal inflation, cyclic universe etc


title: Neutral wire capacity

text: I have three DC circuits in my solar array. I have a three lead + ground (bare copper) tec armoured cable (#6). Why not run one circuit using the bare ground as negative? In fact, why not run three circuits and use the the bare copper as the common negative?


title: The universe appears to have a lower bound in the time dimension, why not an upper bound?

text: The Big Bang looks like a lower bound to the "size" of the universe in the time dimension. Could it also have an upper bound, some furthest point in time from the Big Bang?


title: How do moving charges produce magnetic fields?

text: I'm tutoring high school students. I've always taught them that: > A charged particle moving without acceleration produces an electric _as well > as a magnetic field_. It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?


title: Why are magnetic fields only produced by moving charges?

text: Why do charged particles only produce magnetic fields while in motion?


title: Is a magnetic field just a moving charge?

text: I am trying to understand electromagnetism. I would like to pare myself down to the minimum number of necessary concepts. From my physics education (high school only), I understand that a moving charge **creates** a magnetic field. The implication is that magnetism is a discrete concept that I also need to understand. However, is this view actually confusing: is it the case that a magnetic field **is** a moving charge? Either way, a link to a primer accessible to someone with a engineering degree level maths would be wonderful. If it is the case magnetism **is** a moving charge, then an introduction following this approach would be wonderful. Thank you!


title: Why don't stationary electric charges possess a magnetic field?

text: Why don't stationary electric charges posses magnetic field, while moving charges do?


title: No magnetic field from a static charge - Is there a simple physical argument to show why?

text: For a charge moving in an electric field $\vec E$, its equation of motion is given by the electric part of the Lorentz force $$\frac d {dt}\gamma m \vec v = e\vec E$$This comes from the conservation of relativistic energy in a static electric field. But a magnetic field would still make this conservation law true since the magnetic force is always orthogonal to the velocity of the charge and therefore doesn't change its energy. Is there a simply physical argument that shows why a static charge doesn't create a magnetic field?


title: Rotating frame of reference

text: Can you help me to do this: Two frames of references $S$ and $S'$ have a common origin $O$ and $S'$ rotates with constant angular velocity $\omega$ with respect to $S$.   A square hoop $ABCD$ is made of fine smooth wire and has side length $2a$. The hoop is horizontal and rotating with constant angular speed $\omega$ about a vertical axis through $A$. A small bead which can slide on the wire is initially at rest at the midpoint of the side $BC$. Choose axes relative to the hoop and let $y$ be the distance of the bead from the vertex $B$ on the side $BC$. Write down the position vector of the bead in your rotating frame. Show that   $\ddot y-\omega^2 y=0$ using the expression for the acceleration. Hence find the time which the bead takes to reach a vertex $C$. I showed that $\frac{d^2\vec r}{dt^2}=(\frac{d^2\vec r}{dt^2})'+2\vec\omega\times(\frac{d\vec r}{dt})'+\vec\omega\times(\vec\omega\times\vec r)$ where $'$ indicates that it's done in rotating frame. $\vec r$ is position vector of a point $P$ measured from the origin. I got that   $\vec r=r\cos\theta\vec i+y\vec j$   $\vec r'=(\dot rcos\theta-r\dot\theta \sin\theta)\vec i+\dot y\vec j$   $\vec r''=(\ddot rcos\theta-\dot r\dot \theta sin\theta-\dot r\dot\theta\sin\theta-r\ddot\theta\sin\theta-r\dot\theta^2cos\theta)\vec i+\ddot y\vec j$   $\omega\times\vec r'=-\omega\dot y\vec i+(\omega\dot r\cos\theta-\omega r\dot\theta\sin\theta)\vec j$   $\vec\omega\times (\vec\omega\times\vec r)=-\omega^2 r\cos\theta\vec i-\omega^2 y\vec j$ I suppose I have to write Newton's second law now, but I don't know which forces do I have in this motion.


title: Is there spatial data loss in light passing through transparent crystal?

text: There is perfect parallelepipedal bar made of transparent crystal with cubic lattice floating in vacuum. Faces of parallelepiped are parallel to lattice axis. There is image, forming checkerboard pattern, each cell alternating between two distinct colors. This image is projected in perpendicular on the one face of crystal bar, so projection has spatial resolution (e.g. density of outgoing photons) such high, so every single molecule in the first layer of crystal bar gets several photons of different wavelength. Questions:   1. How the output image at the opposite side of crystal bar compares to original image projected?   2. Will it correlate to thickness of bar?   3. How are multiple photons hitting same molecule handled?   4. What is the limit of resolution of incoming image, to perfectly match output image? Is it one-to-one correspondence of incoming photons to molecules in the first layer?   5. Will there be scattering of photons on the molecules of the last layer? If any, will interference between scattered photons cancel non-perpendicular directions?


title: Why isn't light scattered through transparency?

text: I'm asking a question that has bothered me for years and years. First of all, let me give some context. I'm a layman in physics (college educated, math major). I've read Feynman's QED cover to cover, and watched his messenger lectures. My hope is to get an answer at the "Feynman level", ideally in the crystal clear terms that feynman himself uses :) Okay, so with that out of the way, my question is about transparency. My understanding from Feynman and other materials is that when a photon hits an atom, the photon can either be absorbed, or absorbed and released. What I don't understand is how transparency could arise from this. If the light were absorbed and then a new photon emitted, it seems incredible that the new photon would have the exact trajectory as the old one. Why wouldn't it go off randomly in a random direction? Here's some of the homework I've done trying to find out the answer on my own: 1) Transparency of materials (one commentor alludes to scattering not being explained here) 2) http://alemassociates.com/mambo/content/view/33/1/ (this seems to address it, but I will admit it's mostly over my head) 3) http://www.av8n.com/physics/white.htm (seems to talk about it, but doesn't explain _why_ the photon would ever come out the same direction as it came in) 4) Discover magazine posted this: "when a packet of light energy, or photon, hits a solid object, three things can happen. Light can disappear: If the photon has the same vibrational frequency as the electrons in the material it strikes, those electrons absorb its energy, changing the photon from light into heat. Light can also be scattered: the surface electrons can grab the photon's energy and then eject a photon of the same wavelength which is how you see pretty much everything that doesn't emit light on its own. But if the photon doesn't have the right vibrational energy for absorption, and if the atoms in the material are arranged in patterns that discourage reflection (such as the random jumble of molecules in glass or air) **then the photon's energy passes from atom to atom, emerging on the other side still bright and shiny.** Then you have transparency." I'm posting here for help. I realize that this forum is geared for graduate level and beyond physics, but I'm hoping that this transgression will be forgiven. I would really, truly, like to understand why anything is transparent at all. From my understanding, nothing should ever be transparent! At best we might get translucency (a bunch of photons coming out in random directions), but I cannot understand transparency. Why would a photon interacting with an atom come out the exact way it came from? In other words, as per that Discover magazine article, why would the photon be passed from atom to atom always at the exact same vector that it came from? Thank you! edit: If possible, I'd like to keep it in terms of photons. What's going on at the photon level, and avoid the wave framework. Let's talk about photons and probability amplitudes a-la Feynman.


title: Why doesn't mass of bob affect time period?

text: Please correct me if I'm going wrong -   By the gravitation formula: $F = \frac{G m_1 m_2}{r^2} $, So if the mass of a bob is greater then the torque on it should increase because the Force increased now by a very small magnitude. So it should go faster and thus the Time period should be lesser. But my Physics book says that Time period is only affected by effective length and $g$ Why doesn't mass of bob affect it?


title: Shape of the rainbow

text: I have watched Walter Lewin's lecture(http://www.youtube.com/watch?v=6QVbE_tU2sA) which was about the rainbows. But there is still a question bothering me. I understood the first part of the lecture which he talked about axial symmetry that holds in the drop of water because the light beam is coming from a preferred direction. But I cannot understand that why he relates the semicircular shape of the rainbow to axial symmetry.   To make my question clearer, I say an example. Considering the red part of the rainbow for instance, we found that this color of rainbow is seen if we look about 42 degrees above the reference line(from the sun). But I think that for looking at other red parts of the bow, one should rotate his head and other red parts have different angles than 42.I know that all red parts should be at 42 degrees but I am confused and cannot understand it. I have another problem and that is why we emphasize on $\varphi_{max}$. For example,at the angle of 40 degrees, we have both blue and red color but they always use 40 degrees just for blue but we can have a mixture of red and blue. I will be thankful if you can help me with these.


What is meant by a local Lagrangian density? How will a non-local Lagrangian look like and what is the problem that we do not consider such Lagrangian densities?


title: What physics considerations direct the charging of cars with wireless?

text: Is there any frequency at which cars can be charged with using wireless? Surely, wireless transmission can be safely assumed to be a form of energy transfer, and there can be charging of cars without physical electric outlets. Is there a frequency at which cars can be charged? One can see the problem of energy transmission as a form of optimization problem in which the parameter energy (E) is to the transmitted with maximal effect(I could be nebulous here!) Is there such a frequency for wireless car charging?


title: If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

text: I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in spacetime. Then, it is claimed that $I^+(r)\subset I^+(p)$. But why? Let me first try to prove this conclusion. I notice there is a theorem: Let a subset $S\subset M$ ($M$ the spacetime manifold) and set $B=\partial I^+[S]$. Then, if $x\in B-\bar S$, there exists a null geodesic $\eta\subset B$ with future endpoint $x$ and which is either past-endless or has a past endpoint on $\bar S$. So we can set $S=\\{p\\}=\bar S$. Since $r\in\partial I^+(p)$ and $r\ne p$, $r\in B-\\{p\\}$ with $B=\partial I^+(p)$. Therefore, there is a null geodesics $\eta$ lying on $B$ and passing through $p$. Is this correct? Thank you!


title: How can earthquakes shift the earth's axis?

text: One often comes across news articles that claim that an earthquake shifted the earth's axis. > > http://news.google.com/?q=earthquake%20shifted%20OR%20shifts%20earth%27s%20axis If you ignore the influence of other celestial bodies, an internal event like an earthquake surely can't change the direction of the angular momentum of the Earth (unless stuff is ejected out of Earth), since angular momentum has to be conserved in the absence of an external torque. So the axis has to remain fixed. Am I missing something? Or are geologists trying to say that the resulting movement of tectonic plates causes a change in the point of intersection of the axis (which remains the same) and the plates that include the poles, so that it _seems_ as if the axis has shifted? **EDIT** Some articles mention the value of the shift in the axis and also the change in the length of the day. If, as Ted Bunn's answer indicates below, the shift in the axis isn't actually real but is because of the movement of tectonic plates with respect to the axis, shouldn't the shift be different at the north and south poles? How are the shifts and the change in day-length calculated?


title: Does Quantum Mechanics Allow Macroscopic Anomalies?

text: I've read a few other posts, and none seem to give me an answer that satisfies my curiosity. Thus far I've only been studying time independent QM, so I'm not even sure how wave functions evolve over time for microscopic things, let alone macroscopic things. However, what mandates that events that expect to happen actually happen? Why does it make sense that when I put something in a box, that it should remain in the box, rather than tunneling elsewhere? Or why should a pot ever come to a boil? Certainly these questions are being caused from a tragic misunderstanding of quantum theory, can someone clarify for me? At the end of the day, there is a nonzero probability that macroscopic anomalies will occur, right?


title: Finding height of floating iceblock above seawater

text: This may be a very simple question but somehow I can't seems to solve it. For the sake of completness, I will use the same original question I am working on. **Question:** The density of ice is $917 kg/m^3 $ and the density of seawater is $1020 kg/m^3$. If a piece of ice with dimensions $ 1m $ x $1m $ x $0.65m $ is left in seawater, what height of the ice will be above sea water? ![enter image description here](http://i.stack.imgur.com/rGScS.png) This is all it was given. **My Attempt:** This is how I understand this problem. There will be two forces acting on this ice block   * The upthrust force ( which is equivalent to the weight of the seawater being displaced)   * The weight of the ice block itself Since the ice block is floating, I assume that the upthrust is either equal or greater than the weight of the ice block. This is the simple sketch I come up with ![enter image description here](http://i.stack.imgur.com/7bc8z.png) but now I am confused because according to my sketch: $ upthrust = \rho_{seawater} vg $ (v being the volume of water being displaced by the ice block) $ weight\,of\, ice\, block = mg = \rho_{ice} vg$ If i subtract these two equations, the volumes will cancel out each other so there will be no height information left. I supposed I have to somehow retrieve the `height` inforation from `volume` using $v = Ah$ but I am not sure how. How do I find the height of ice block above sea water?


title: Coriolis effect on Tsunami

text: The Japanese tsunami, moving at about 700 km/h, affected areas as distant as Chile's coast, 20 hours after the earthquake. How does the Coriolis force affect tsunami? Also, I saw an image of a boat caught within a large whirlpool. Is the whirlpool's rotation due to Coriolis force?


title: Stringy corrections to Friedmann equation

text: Does anyone know a reference or a paper which discusses string theory correction to Friedmann equations?


title: Why does the Debye-Scherrer procedure work (powder diffractometer)

text: In the Debye-Scherrer procedure a sample of crystalline powder is hit by a beam of monochromatic photons. The diffracted photons are measured with a detector. We have constructive interference of the photons if the Bragg conditions are met. In my lecture notes it says that there are always crystalline planes oriented in a way that the Bragg conditions are fulfilled, thus leading to the typical rings observed with powder diffractometer. ![Graphic taken from Wikipedia](http://i.stack.imgur.com/f4Tes.png) (Graphic taken from Wikipedia) I have a hard time seeing why there are discrete rings, because from the reason above I would expect that for every angle there is constructive interference and hence we would have a continuous intensity distribution for all angles. **Question** : Why are there discrete rings and not a continuous distribution?


Isn't this statement regarding projectile motion wrong? > _If a body is thrown at an angle to the horizontal with initial velocity > $u$, then displacement of body as a function of time is > $\vec{s}=\vec{u}t+\frac12\vec{g}t^2$. (Air drag is neglected)_ How can it be correct? Gravity acts in downward direction, so wouldn't the displacement be $\sqrt{x^2+y^2}$?


title: Which component shows spin squeezing under twisting Hamiltonian?

text: Given a many body spin system, a collection of N spin-1/2 particles, under the interaction of the twisting Hamiltonian: $$H_{int} = \sum_{i,j=1}^Na_{i,j}\sigma_{z,i}\sigma_{z,j}= A J_{z}^{2}$$ assume all $a_{i,j}$ are equal and define: $$\mathbf{J} = \sum_{n=1}^{N} \mathbf{\sigma}_{n}$$ the collective spin operator, $\mathbf{\sigma}_{n}$ is the pauli spin operator for the $n$th spin, and $A$ characterizes the strength. Which component of the spin system will display reduced variance/will be squeezed? What assumptions does this require regarding the initial state? The context of the question is the notion of spin squeezed states, as originally put forward by (1) Wineland et al. and (2) Kitagawa & Ueda EDIT In particular in figure 2 of (2) the evolution of the coherent spin state $|\pi/2,0\rangle$ (pointing along $\hat{x}$) is described, they show that squeezing occurs along the y- and z-axes. As I see it these variances seem to oscillate out of phase. What would be helpful is if someone could explain how to visualize this so- called twisting dynamics. So far i thought that, starting with an initial (Q-)distribution in phase space, the distribution evolves with precession frequency proportional to $J_{z}$. But from there I do not see how the variance along $\hat{z}$ would change... Also as a side note: to my knowledge these type of quadratic in angular momentum terms are not very common, but also appear in nuclear physics as the Lipkin model.


title: How strong is the spacetime curvature at distance $d$ for a nonmoving point mass?

text: Consider a point mass $A$ with mass $m$ in empty space. The point mass $A$ does not have a velocity and does not rotate. Since gravity is symmetric for nonmoving objects, the spacetime curvature around $A$ is also symmetric. So at a distance $d$ from the point mass $A$ how strong is the curvature $C$ ? $$ C = f(d,m) $$ $$f = ???$$


title: How can we get the mass of an uncharged proton?

text: How can we get the mass of an uncharged proton, i.e. how varies the mass of the charged proton if i remove the electric charge? For the isotopic spin theory neutron and proton have the same mass and it is possible to distinguish that particles only for the different values of the third component of the isotopic spin. This is an approximated symmetry because the masses are different and the charges too. In a ideal world we can remove the electric charge from the proton and we get a "uncharged" proton. The question is how the mass changes. You can interpret that question as what it is the contribution to the mass due the charge, not only for proton but for all particles.


title: Help with momentum question involving vectors/trig

text: The problem: A car with a mass of 1200 kg and speed 10 m/s runs into a traffic barrier at an angle of 45 degrees, and is thrown outwards at an angle of 45 degrees relative to the barrier with a speed of 5.0 m/s How big was the change in momentum of the car during the collision? I tried decomposing the momentum, but failed to get the right answer. Can someone explain to me how I can solve this question without using the fact that Pinitial and Pfinal make a right triangle? Additionally, someone made the following solution to this problem: $$\vec{p_e} = 1200 kg \cdot 5 m/s [{1 \over \sqrt{2}} , {1 \over \sqrt{2}}] \\\ \vec{\Delta p} = \vec{p_e} - \vec{p_f} = 1200 [-{5 \over \sqrt{2}}, {15 \over \sqrt{2}}] kg m/s \\\ |\vec{\Delta p}| = 1200 \sqrt{({5 \over \sqrt{2}})^2 + ({15 \over \sqrt{2}})^2} kg m/s = 1,3 \cdot 10^4 kgm/s$$ Can someone please explain to me how vectors and trig are used to solve this? I especially don't understand what the following means, and where it came from: $$[{1 \over \sqrt{2}} , {1 \over \sqrt{2}}] \\\$$


title: Finding the tension in rope tied to ladder using the principle of virtual work

text: A ladder $AB$ of mass $m$ has its ends on a smooth wall and floor (see figure). The foot of the ladder is tied by an inextensible rope of negligible mass to the base $C$ of the wall so the ladder makes an angle $\alpha$ with the floor. Using the principle of virtual work, find the magnitude of the tension in the rope. My equations: I assume length of ladder = $L$ $x$-direction: $$N_2 = mg$$ $y$-direction: $$N_1 = T$$ torque equation about $B$: $$ T_{about B}=T_{mg}+T_{A} = \frac{L}{2}(mg)cos\alpha-LN_1 sin\alpha=0$$ Therefore, $$\frac{L}{2}(mg)cos\alpha-LT sin\alpha=0$$ And we get $$T= \frac{1}{2}(mg)cot\alpha$$ which is the answer. However, I am supposed to find this using the principle of virtual work. To do this, I guess I would have to displace the ladder downward at $A$ and rightward at $B$, in which case the normal forces $N_1$ and $N_2$ do no work. I then conclude that the work done by the gravitational force $mg$ along the downward displacement of the center of mass of the ladder counterbalances the tension times the horizontal displacement at $B$. However, finding the displacement of the center of the ladder as the ladder slides downwards doesn't seem to be an easy task, so I must be on the wrong track. The final equation I have to deal with seems to be $$mg.\delta y_{com}-T.\delta x_{at B}=0$$ Any inputs would be appreciated. ![enter image description here](http://i.stack.imgur.com/yhIPX.jpg)


title: Why is the crust of my meat pie only slightly warm?

text: I just cooked a meat pie in the oven. Almost immediately after pulling it out of the oven, I felt that the crust was only slightly warm, but when I cut it open the filling felt very warm. I can't understand how the crust could be at a lower temperature. My understanding was that the oven heats the air, then the air heats the crust and the crust heats the filling inside. As long as the air is hotter than the crust (should be always), there is no way for the crust to lose heat, except by heating the filling and there is no way for the filling to gain heat, without being heated by the crust. According to this logic, the filling can't be hotter than the crust. Given this situation, why does the filling feel hotter than the crust?


title: Can one make a ball rotate around a vertical axis using only a combination of horizontal axis rotations?

text: This is a nice problem that I would like to share. **Problem:** In a public garden, there a statue consisting of a spherical stone and a stone cup. The ball is 1 meter in diameter and weighs at least a ton. The cup is an upside-down hemispherical shell, and the ball sits in this shell and fits it almost exactly. Water is pumped into the bottom of shell so that a thin film exists between shell and the ball. The result is a ball that is free to rotate with negligible friction. You only have access to the ball near the top, so while you can push it to make it turn around any horizontal axis, you can't get enough of a grip to make it turn around the vertical axis. Can you impart a net angular momentum around the vertical axis anyway, so that the balls spins around the vertical axis? **Source:** _Vector Calculus, Linear Algebra, and Differential Forms_ by Hubbard and Hubbard.


title: Can a free hydrogen atom constitute ionizing radiation?

text: Radiation is basically just particles flying around, right? Are free hydrogen atoms just typically not moving fast enough to be considered "radiation"?


title: Best Sets of Physics Lecture Notes and Articles

text: This post is inspired by this math.se post. Let me start by apologizing if there is another thread on phys.se that subsumes this. I often find that I learn best from sets of lecture notes and short articles. There are three particular reasons that make me feel this way.   1. Lecture notes and articles often times take on a very delightful informal approach. They generally take time to bring to the reader's attention some interesting side fact that would normally be left out of a standard textbook (lest it be too big). Lecture notes and articles are where one generally picks up on historical context, overarching themes (the "birds eye view"), and neat interrelations between subjects.   2. It is the informality that often allows writers of lecture notes or expository articles to mention some "trivial fact" that every textbook leaves out. Whenever I have one of those moments where a definition just doesn't make sense, or a theorem just doesn't seem right it's invariably a set of lecture notes that sets everything straight for me. People tend to be more honest in lecture notes, to admit that a certain definition or idea confused them when they first learned it, and to take the time to help you understand what finally enabled them to make the jump.   3. Often times books are very outdated. It takes a long time to write a book, to polish it to the point where it is ready for publication. Notes often times are closer to the heart of research, closer to how things are learned in the modern sense. It is because of reasons like this that I find myself more and more carrying around a big thick manila folder full of stapled together articles and why I keep making trips to Staples to get the latest set of notes bound. So, if anyone knows of any set of lecture notes, or any expository articles that fit the above criteria, please do share! I'll start: **People/Places who have a huge array of fantastic notes:** 't Hooft John Baez    **The _Feynman_ lectures on Physics:**   * The Feynman Lectures on Physics, Volume I: mainly mechanics, radiation, and heat.  * The Feynman Lectures on Physics, Volume II: mainly electromagnetism and matter. * The Feynman Lectures on Physics, Volume III: quantum mechanics.    **Introductory Courses**        * **Introductory Classical Mechanics**   1. http://farside.ph.utexas.edu/teaching/301/301.pdf   2. http://space.wccnet.edu/~gkapp/   3. http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/   4. http://academics.smcvt.edu/abrizard/Classical_Mechanics/Notes_070707.pdf   5. http://www.maths.tcd.ie/~dleen/mathsoc/pdf/Notes.pdf      * **Optics and Thermodynamics &  Electromagnetism**   1. http://farside.ph.utexas.edu/teaching/316/316.pdf   2. http://seagull.ukzn.ac.za/~mukaror/   3. http://www.sicyon.com/resources/library/pdf/optics.pdf      * **Waves and  Oscillations **   1. http://farside.ph.utexas.edu/teaching/315/Waves.pdf   2. http://www.ma.hw.ac.uk/~bernd/F12MS3/   3. http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/OscWavesIndex.htm      * **Statistical Mechanics and Thermodynamics**   1. http://farside.ph.utexas.edu/teaching/sm1/statmech.pdf   2. http://www.spms.ntu.edu.sg/PAP/courseware/statmech.pdf   3. http://www.physics.umd.edu/courses/Phys603/kelly/   4. http://stp.clarku.edu/notes/      * **Electronics**   1. http://openbookproject.net/electricCircuits/   2. http://ocw.mit.edu/courses/music-and-theater-arts/21m-380-music-and-technology-live-electronics-performance-practices-spring-2011/lecture-notes/   3. http://zebu.uoregon.edu/~rayfrey/432/DigitalNotes.pdf   4. http://courseware.ee.calpoly.edu/~jzhang/EE112/   5. http://www-inst.eecs.berkeley.edu/~ee130/sp07/lecture.html   6. http://www.engr.sjsu.edu/kghadiri/EE122/Class_notes.htm   7. http://www.cramster.com/course-introduction-lecture-note-r30-6638.aspx      * **Computational Physics**   1. http://farside.ph.utexas.edu/teaching/329/329.pdf   2. http://www.physics.umd.edu/courses/CourseWare/EssentialMathematica/   3. http://www.cmth.ph.ic.ac.uk/people/a.mackinnon/Lectures/compphys/   4. http://math.fullerton.edu/mathews/numerical.html      * **Introductory Quantum Mechanics**   1. http://farside.ph.utexas.edu/teaching/qmech/qmech.pdf   2. http://galileo.phys.virginia.edu/classes/252/home.html   3. http://walet.phy.umist.ac.uk/QM/QM.pdf   4. Video:  http://physicsstream.ucsd.edu/courses/spring2003/physics130a/   5. http://quantummechanics.ucsd.edu/ph130a/130_notes.pdf   6. http://www.lecture-notes.co.uk/susskind/quantum-entanglements/      * **Classical And Quantum Optics**   1. http://people.seas.harvard.edu/~jones/ap216/lectures/lectures.html   2. http://atomoptics.uoregon.edu/~dsteck/teaching/optics/          (from my Blog http://quantizd.blogspot.com) So my goal is to ameliorate this list by adding more resources. Thank you.


title: Ampere's Circuital Law

text: I'm having trouble understanding the surface used for Ampere's Circuital Law. In classes, we've been using simple circular or rectangular loops. Is the surface supposed to be area of the circle or the rectangle? Or is it a simple case of the law? A example would be the surface used for displacement current. This one is just lost on me. The surface looks like a pot enclosing the wire and the capacitor. Does the law specify a closed loop of a surface? Is the line integral taken on the mouth of the pot-like surface or the bottom ($S_1$ below)? And is the enclosed current in the smaller mouth or the larger bottom surface ($S_2$)? ![enter image description here](http://i.stack.imgur.com/5Rlor.jpg)


title: Difference between RPA and generalized RPA

text: The random phase approximation (RPA) is an approximation method in condensed matter physics and in nuclear physics. What is the difference between RPA and generalized RPA?


title: $\cos^{2}(\phi)$ in the kinetic energy term of the Lagrangian is one?

text: I'm doing some homework in Classical Mechanics, and is about to write out the Lagrangian of a system. But, when I check the answer from my teacher, something is missing. The kinetic energy I'm using is given by: $$T =\frac{1}{2}m\dot{x_{m}}^{2},$$ where $$\dot{x_{m}} = \dot{x}+L \cos(\phi)\dot{\phi}$$ Now, when I want to write out the entire Lagrangian I need to expand the squared $\dot{x_{m}}$, which in my calculations give me the following: $$\frac{1}{2}m(\dot{x}^{2}+2\dot{x}L \cos(\phi)\dot{\phi}+L^{2} \cos^{2}(\phi)\dot{\phi}^{2})$$ But apparently, according to the solution, the $\cos^{2}(\phi)$ term becomes $1$, and only $L^{2}\dot{\phi}^{2}$ is left. So my question is: Why is that? I've been checking around, but nothing I find makes it one. I know that $\cos^{2}(\phi) = \frac{1}{2}(1+\cos(2\phi))$, but that doesn't seem to help me either.


title: Boundary effect on the Beer-Lambert for F-center/photon interaction

text: The Beer-Lambert law is: $ I=I_{0}e^{-n\sigma(\lambda) x} $, where $ n $ is the volume concentration and $ \sigma $ the "efficient" cross-section. For interaction with F-centers (kind of crystal defects: electrons trapped in negative ions lakes), what is the boundary effect on the exponential parameter: $ n\sigma(\lambda) $ ? Thank you.


title: How to determine "timelike"-ness without using a coordinate system?

text: It has been stated here that: > we can say, without introducing a coordinate system, that the interval > associated with two events is _timelike_ , _lightlike_ , or _spacelike_ . This assertion appears at variance with   * the definition of " _time-like (or light-like, or space-like) intervals_ " as defined here: http://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals explicitly in terms of " _differences of the space and time coordinates_ ", and   * the definition of " _curves_ " as " _chronological (or timelike)_ ", " _null_ " (or "lightlike"), or " _spacelike_ ", and thus correspondingly of pairs of " _points_ " of any such " _curve_ " as " _timelike (or lightlike, or spacelike)_ related to each other" of http://en.wikipedia.org/wiki/Causal_structure#Curves which explicitly requires the notion of Lorentzian manifold and thus according to http://en.wikipedia.org/wiki/Manifold#Mathematical_definition uses coordinates as subsets of $\mathbb R^n$ and their topological relations (as " _coordinate system_ ").  Therefore I'd like to know:   How can be determined whether the interval associated with two events (which are given or characterized and distinguished by naming, for either event, the distinct individual participants which had been coincident at that event) is for instance " _timelike_ ", **without** using any coordinates and coordinate system? Is it correct that the interval associated with two events (given as described above) is " _timelike_ " if and only if there exists at least one participant who took part in both of these events?


title: QM 2D Gaussian wave packet translation

text: I've been reading a lot but cannot find an example of 2D Gaussian wave packet moving in a particular direction. I've done some of the math myself, in a 1D case, and then kind of guessed the generalization to 2D. I'd be thrilled if someone could confirm or correct the expression. This is supposed to model a free particle moving in the x axis with a fairly definite momentum: $\Psi (x,y,t) = \frac{1}{\sqrt{\alpha + i\beta t}} \exp \left[-\dfrac{(x-vt)^2+y^2}{4(\alpha + i\beta t)} + i(kx-\omega t) \right] $ The wave function is not normalized and the constants $\alpha$ and $\beta$ are not relevant in my opinion, I could add how they relate with mass and $\hbar$ if someone thinks they're necessary. Also I would like to add to the question what's the relation between $k$, $\omega$ and $v$ and what does each one represent. My guess is that $v$ is the group velocity, and you'd get the average momentum from the fourier transform of that wave function, but I'm not sure what's $k$ then. Is there any need at all to add the phasor $e^{i(kx-wt)}$, does it change anything?? The packet is going to move anyway and the amplitude is not going to change unless you add other wave functions. Thank you very much!


title: String Theory and Fourier Analysis

text: Me and my friend, both many years from learning string theory, had a recent debate about it anyway. He said he already partially discounts it because after learning waves, he believes any function, and thus any kind of physical process can be created through a superposition of an infinite sum of waves. Since string theory to our understanding is made up of primarily oscillating strings, he argues that it is a mathematical necessity for string theory to create equations for the physical world, regardless of whether the theory actually is "true" or not. My understanding is that Fourier analysis can only create arbitrary periodic functions, can it in principal create any real world field or particle, etc.? Besides this, I argued that this might be a pointless question anyway, as if we can create a perfect model of the universe with string theory, then the question of whether the universe is really made of oscillating strings is more of a philosophical question that may not mean anything at all. Through reading Brian Green's books I know that M-theory at least incorporates more than just strings, but I have no clue as to how things such as branes work, can his argument even apply to these other fundamental objects in string theory? So my question is, is Fourier Analysis essentially what String Theory is? And if so, does that make it any less of a true physical theory of the universe?


title: Which electromagnetic radiation is faster in water, microwaves or light?

text: Well I've been asked this question, but I haven't been able to come with an answer yet using books and some web searches. The point is as the title says, to answer the question with the whole phenomenon explanation if possible.


title: How can helicity be conserved but chirality not?

text: I read in a book that for $\beta$-decay the electrons have always been found to have an expectation value for their helicity of $h=-v/c$. Then ist is said in the book, that it follows from this fact that such electrons are in a left-handed chiral state which is characteristic for the weak interaction. In another article I read that the chiral state of an electron is not conserved in time. The electron will soon evolve a component with a right- handed chiral state and it will be a mixture of right- and lef-handed chiral states. Suppose after the decay one electron moves like a free particle.   When it evolves a right-handed chiral component in addition to the left-handed component it starts off with, how can its helicity be conserved?


title: Relation between p+ip wave Superconductor and Moore-Read State

text: I am quite interested in the understanding of the relation between p_ip wave superconductor(SC) and the Moore-Read(MR) state. They share many similar properties, for example, p+ip SC has majorana as vortex excitation, MR has nonabelian quasi hole excitation. More interestingly, their wavefuncions are similar to some extent. All of them can be found in Read and Green's paper: http://arxiv.org/pdf/cond- mat/9906453v3.pdf. (where you can find the wavefunction for p+ip SC in eqn.12 and above, MR wavefunction in eqn.5.1 in http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf). These two wave functions are similar only when electrons are separated far from each other, as shown in eqn 16 in Read Green's paper. Does anyone know how to explain the relation between these two states? Why they share similar property at long distance? Thanks in advance.


title: Why can't we perfectly focus light-abberations aside

text: I don't understand why there is necessarily a diffraction limitation on optical systems. Where does this limitation in focusing light come from?


title: Can mu-metal reduce the negative effect of metal close to low frequency RFID antennas?

text: I have an 134.2 khz RFID antenna for sport events which is basically a 1x1 meter square sheet of rubber with 3 loops of antenna cable embedded. The problem is that the transponder recognition range is drastically reduced when the ground below the antenna contains metal (e.g. reinforced concrete). Would it be possible to solve this problem by attaching a layer of mu-metal or permalloy to the lower side of the antenna? I read that mu-metal draws magnetic field lines into itself and thus redirects them around what's behind (in this case the metal in the ground). Would this make sure that the magnetic field on the upper side of the antenna is the same as if there was no metal in the ground? (and that the transponder recognition range is as good as without the metal) What thickness of the material would be required, would a foil be sufficient?


title: Is it possible to have a floating bullet in the air?

text: I've been asking myself a question for quite some time :   * say that a bullet gets out of a gun at 900 km/h (I'm european, hence the metric system).   * say a train could go in a straight line at 900 km/h   * say you're at the back of that train, and you'd shoot a bullet backwards. The bullet exits your gun at 900km/h and flies away from the train at the same speed. However, what would happen for an earth observer? You'll agree that if you shoot in an open field, the bullet will stay at the same altitude for several seconds before falling to the ground (with a field large enough). So from that train, same thing applies. So for the earth observer, the bullet will not have an horizontal speed, and will stay at the same altitude for several seconds before falling to the ground. So basically it'll float up in the air without falling. That'd be strange for the earth observer, right? But the horizontal speed will decrease of course. So for the earth observer, the bullet will, for a few seconds, stay still at the same position, and then after several seconds will start to fall down a little bit, and will start to move towards the train, because its horizontal speed towards the train will decrease. Strange again! OK last one: what would happen if the earth observer tried to catch the bullet with his bare hands? Nothing right? Just maybe a very hot bullet because of the explosion but it'd be basically as easy to catch as a stone floating up in the air? And don't tell me that the bullet will slow down due to air friction, because remember the bullet is not moving. Or very slowly.


title: What is the relationship between consistent histories and path integrals?

text: As can for example be learned from chapter I.2 of Anthony Zee's _Quantum field theory in a nutshell_ , path integrals can be used to to calculate the amplitude for a system to transition from one state to another by considering all possible paths between the two states. This article gives a nice technical introduction to consistent histories and explains that this method can be used to calculate the answer to questions about alternative histories in a consistent set. For two histories to be consistent, their probabilities must be mutually exclusive such that their probabilities are "additive". Looking at these two methods to calculate answers to meaningful questions in quantum mechanics (and how the relevant formulas are derived), they seem very similar to me. So my question is: What is the exact relationship between the method of path integral and consistent histories? Could one say that the consistent histories are some kind of the "classical limit" of the path integrals? Or does coarse graining (by an appropriate method?) Feynman's path integrals lead to consistent histories as a limit (IR fixed point)? (In the article about the consistent histories it is mentioned, that if one considers too fine grained histories, they are due to the uncertainty principle no longer consistent and would resemble Feynman's path integrals. But I'd like to see an extended technical / mathematical explanation of the relationship between the two things to really understand it.)


Are there any non magnetic materials that attract to each other as if they were magnetic? This is an argument I am having with a friend. Thanks,


title: Finding the time of convergence for this system of particles

text: > Point A moves uniformly with velocity $v$ so that the vector $\vec{v}$ is > continually "aimed" at point B which in turn moves rectilinearly and > uniformly with velocity $u < v$. At the initial moment $\vec{v} \perp > \vec{u}$ and the points are separated by a distance $l$. How soon will the > points converge? This seemed a very daunting problem at first, but then I tried to break it down with some calculus: Consider a coordinate system such that $A$ moves along positive $y$ axis and $B$ along positive $x$.Let the particles meet at time $T$. At an aribitrary time $t = t_0$, the velocity of the particle is inclined at an angle $\theta$ to the horizontal, so the horizontal component of the velocity is $v\cos\theta$. Displacement in a small time interval $dt$ at $t_0$ along the $x$ axis (along direction of motion of $B$) is: $$v\cos\theta\ dt$$ Summing from $0$ to $T$. The displacement along $x$ axis of particle $A$ equals $uT$. $$\int_0^T v\cos\theta\ dt= uT \implies \int_0^T\cos\theta \ dt = \frac{uT}{v} \tag1$$ Similarly, for the displacement along the $y$-axis: $$\int_0^T\sin\theta\ dt = \frac{l}{v} \tag2$$ I'm not sure how to solve these equations, and I'm not sure if the solution is possible either using my method. How do I go about the problem? I went over the solutions that have been written on other websites. Most of them first establish the same thing I did in the first step $$\int_0^T \cos\theta \ dt = \frac{uT}{v}$$ Its the second step I don't understand. At time $t$, ![enter image description here](http://i.stack.imgur.com/RsPU5.png) It is claimed that at time $t$: $$\frac{dr}{dt} = v - u\cos\theta$$ where $r$ is the distance between the two particles. This is the rate at which $B$ moves away from $A$ at time $t$. They then integrate the above and set it equal to $l$, which I don't understand: $$\int_0^T(v - u\cos\theta)dt = l \tag3$$ How is this integral equal to $l$? If we consider the time when $A$ is about to meet with $B$, in a small time interval, the displacement of $A$ as given by $(3)$ would be along the line of motion of $B$ rather than perpendicular to it. How is this possible? **EDIT** : This problem is from General Problems in Physics by IE Irodov


title: Boltzmann–Gibbs-distribution as resulting from a limiting density of states?

text: I'm interested in the relation between the probability distribution $p_i$ over states of a system on the one side and the density of states $\rho(\eta)$ of its environment. (Meaning, $\int_{\eta_a}^{\eta_b} \rho(\eta) ~ \mathrm{d} \eta$ is the number of environment states with energies in the interval $[\eta_a, \eta_b]$.) If the whole (system + environment) is energetically closed ("isolated") with a total energy $E = e + \eta$, but system and environment are in thermal equilibrium (i.e. the whole is described by the microcanonical ensemble), then it holds $$ p_i = \frac{ \rho(E - e_i) }{ \sum_i \rho(E - e_i) }. $$ This means, the probability distribution over states of the system is determined by a) something that only characterizes the energetic _structure_ of the system (the $e_i$s), b) something that only characterizes the energetic _structure_ of the environment ($\rho(\eta)$) and c) the total energy $E$. This relation holds generally, for arbitrary small or large systems and/or environments. Please note that we have not yet taken any limits! If we now consider the thermodynamic limit, i.e. an environment composed of an infinite number of subsystems, the probability distribution $p_i$ over states of the system becomes the Boltzmann–Gibbs-distribution (aka canonical ensemble) $$ p_i = \frac{ \exp(- \beta e_i) }{ \sum_i \exp(- \beta e_i) }. $$ Using the first relation above, this distribution could now be interpreted as corresponding to a _limiting density of states of the environment_ of the form $$ \rho(\eta) \propto \exp( \beta \eta ) $$ which characterizes the "infinite environment". However, the expression refers to the parameter $\beta$ of the Boltzmann–Gibbs distribution, which represents the temperature and depends on the total energy $E$ (per subsystem). Whereas in the finite case $E$ only serves to connect $\rho(\eta)$ and $p_i$, it here defines $\rho(\eta)$ itself. To me this suggests that it does not make sense to characterize an infinite environment by a density of states – but maybe there's some way around this? Or is there a mistake in the derivation somewhere else?


title: Interpreting Physics Results Regarding Circuitry Analysis (Highschool Level Physics)

text: I conducted an experiment which aimed to detail and record the voltage across series resistors in series circuits. I noticed that:   * The Final Voltage, V3, stayed the same.   * The resistors with the highest resistance had the highest voltage drop. How do I put my observation in words?


title: Twist of null Killing fields

text: I have a (hopefully) quick question: is it possible to have a null Killing field $\xi ^ \mu$ such that the twist 1-form $\omega_{\mu} = \epsilon_{\mu\nu\alpha\beta}\xi^\nu \nabla^\alpha \xi^\beta \neq 0$ but the exterior derivative $(d \omega)_{\mu\nu} = 2\nabla_{[\mu}\omega_{\nu]} = 0$? Or does $(d \omega)_{\mu\nu} = 0$ always imply $\omega_{\mu} = 0$ for a null Killing field?


title: Photoelasticity and existence of stress

text: Shall we consider photoelasticity a non-mathematical, or purely visual, proof of the existence of stresses in mechanical structures undergoing external loading?


title: Can we introduce the mass of a quantum field as an interaction?

text: At a free massless Lagrangian \begin{equation} L_0 = \frac 1 2 ( \partial \psi)^2 ,\end{equation} add an interaction term \begin{equation} L_I = \frac 1 2 m^2 \psi^2\end{equation} where m is small enough to converge its perturbation calculation. From these setup, is it possible to gain the same results as of the free massive Lagrangian? Please point me a book description or some internet site. Thanks.


title: If it takes less than a year to accelerate to the speed of light at 1g why will it take the Voyager 10,000 years to reach Alpha Centauri?

text: Today I was doing my physics homework and there was a problem involving a space ship falling at 9.8 m/(s^2) to simulate gravity, and it asked how long would it take for the ship to reach to speed of light, and I calculated it to be just under a year. So if it takes less than a year to go to the speed of light at 9.8m/(s^2) to reach the speed of light and proximus centari is one light year away even with the consequences of relativity (which I understand in the abstract) why does it not take about six years to get to alpha centari, and why will it take Voyager (travelling at a much faster acceleration) so long, especially in space where there is no friction? Thank you.


title: simple question about collimating lens

text: Let me preface this by saying that I have a very limited knowledge of optics -- basically, I know enough to be dangerous. So I have a square Fresnel lens that measures about 10.5 inches on a side. The focal length is about 11.5 inches. I'm trying to find a presumably-smaller glass collimating lens that will take the light from the Fresnel lens and convert it to a beam with an indefinite focal length (or at least, a focal length of a few feet). I don't care about image preservation, I just mainly want to transfer some focused light from point A to point B with a minimum of loss. I basically need to know what kind of lens to buy and where to place it. I've tried single and double convex and concave lenses and seemed to have the most luck with single convex, but wasn't able to get much of a beam past a couple of inches out. The attached picture shows kind of what I am looking for (lens on left represents Fresnel lens, lens on right represents collimating lens). Any help you could give me with this would be great. ![enter image description here](http://i.stack.imgur.com/fWk1M.png)


title: Glashow-Weinberg-Salam mass terms

text: At the end of spontaneous symmetry breaking I get these mass terms: $$W_{\mu}^{\pm}=\frac{1}{\sqrt{2}}\bigl(W_{\mu}^{1} \mp i W_{\mu}^{2} \bigr )$$ $$\mathcal{L}_{mass}=\frac{1}{2} g^2 \frac{v^2}{4} W_{\mu}^{+}{W^{\mu}}^{-} + \frac{1}{2} g^2 \frac{v^2}{4} W_{\mu}^{-}{W^{\mu}}^{+}$$ So I have $$M_{W^+}=g \frac{v}{2} \quad M_{W^-}=g \frac{v}{2} $$ Is it right? Or there are too many terms and it is enough: $$\mathcal{L}_{mass}= \frac{1}{2} g^2 \frac{v^2}{4} W_{\mu}^{-}{W^{\mu}}^{+} $$


title: Examples of symmetric field collapse to produce asymmetry

text: It would really help me with an idea I have if I could see how something such as a symmetric field could collapse to something asymmetric. I know that if $x$ occures before $y$ is symmetric, then $y$ occurs before $x$, which doesn't make sense if we take time to be asymmetric (only flowing in one direction). Is there a relationship between energy in the universe being symmetric, and the production of anti-matter asymmetric? I am mostly interested in the collapse of the wavefunction, and a search to find if this collapse is asymmetric I am left with the thought that I might have to specify what the symmetry/asymmetry would be in respect to, that being said, is there some sort of "field" whether physical or purely abstract which is symmetric and if a "disturbance" (observation?) happens, there is a collapse to an asymmetric field such as the collapse of the wavefunction? Maybe I have it all wrong?   * Digging deeper, what I am looking for examples of, especially of quantum systems is that of "symmetry breaking". I also found the answered question concerning spontaneous symmetry breaking in Quantum mechanics here: spontaneous symmetry breaking in Quantum Systems, and while I have some background in physics and pure math, I hope to get a much more canonical answer so as to understand more the foundations. \


title: How do geocentric theories explain seasons?

text: Did the geocentric theorists also use the fact that the earth had an axis tilt? Or was their explanation of some other kind?


title: Info request on studying QIT/QIS or QM with a Computer Science background

text: I've been considering a career change for a long time and recently discovered the Two-Slit Experiment, which, to put it frankly, blew my mind. I then started some hefty reading and investigation into all things Quantum (Bell, Campbell, entanglement, QIT), which has led me here. I have 20 years experience in IT working as a programmer, having graduated with a Bachelor in Computer Science back in the early 90s. As mentioned, I have been considering a career change for some time (having become quite burned out in this industry), and have found something that has piqued my interest more than anything else (tho I did consider Astrophysics a couple of years ago). I am not yet sure which area of "Quantum" I will be most drawn to, possibly Quantum Information Theory/Science. I am aware that there is a significant amount of Maths and Physics pre- requisites involved. It seems likely, given that I have not done any Maths or Physics study probably since I was doing my CS undergrad, that I will need to start over with an undergrad in Science/Physics in order to get the fundamentals. So my questions are:   1. What is the ideal path of education to get to QIT/QIS or QM?   2. Is the path to QIT/QIS also via the Physics route? Can I leverage my CS background?   3. Resources (books, online courses, etc) that would help with the transition from CS-type thinking to Physics/Maths-type thinking. Apologies if this encompasses elements too broad or off-topic, I'm trying to get a better understanding on what is to come following this path. Thanks in advance.


title: Examples of manifolds and fluxes coming from generalized complex geometry

text: The paramount object in generalized gomplex geometry is the Courant algebroid $TM\oplus T^\star M$, where the manifold $M$ is called _background geometry_ I think (I am not sure). More generally this Courant algebroid can be twisted by a closed 3-form $H$ on $M$, which I suspect is called a _flux_ (I am not sure too). In fact P. Severa shown that every exact Courant algebroid arise as such a twist of the standard Courant algebroid $TM\oplus T^\star M$. I think these objects arise in Physics when one tries to make compactifications with fluxes in String Theory, which are more realistic ones. They also appear in the AdS-/CFT correspondence (see Generalized Complex Geometry and Theoretical Physics). Please correct me if I am saying something wrong. Could somebody sum up the way these forms $H$ appear in Physics in mathematical terms? However my true question is the following: could somebody provide concrete examples of both manifolds $M$ and forms $H$ such that, when combined together into the twisted Courant algebroid $(TM\oplus T^\star M,H)$ they are relevant to String Theory? Thank you!


title: Is Earth's orbit altered by recoil from take-off/launch/recovery of aero/space vehicles?

text: Just what the title states. Pretty much all movement on Earth is by pushing against the much greater mass of Earth. Given there are easily thousands of aircraft taking flight/landing, and a lesser number of rockets (Satellite launch etc). Understandably the effect of aircraft would be reduced because an aircraft (usually) comes to ground eventually but with a reduced mass by virtue of the reduced fuel mass ... How much is Earth's orbit affected by such reaction if observed/calculated over an extended period - say, of the order of millenia or longer?


title: 1MeV ray hits a band gap of 2eV : trying to understand X-rays

text: I am trying to understand why radio waves pass though the human body, light does not, but X-rays still pass. In my mind I imagine that radio waves don't supply enough energy and don't excite any interesting modes of the material that they hit. This is in contrast to the photon excitation and associated scattering I imagine when a light beam hits a material. I am confused as to why this doesn't happen for X-rays? I mean they come out the other end, so I assume that they have a pretty good transmission though soft tissue. Certainly, the 1MeV is a enough to excite the band-gaps in the human body which, in turn, should cause all the X-rays to be absorbed?


title: Why is there a search for an exchange particle for gravity?

text: If I understand correctly, according to Einstein's General Theory of Relativity, mass results in a distortion in space-time. In turn, the motion of the mass is affected by the distortion. A result of the interplay between mass and space-time is that the 'force' of gravity may be explained away. Masses are not subject to a force, but are merely following a 4-dimensional space- time geodesic; gravity is just geometry. And yet physicists are searching for exchange particles for the force of gravity, and are trying to unify quantum mechanics with relativity, or to unify the weak/strong/electromagnetic forces with that of gravity.   1. What have I missed? Are these different communities of physicists? Does relativity explain only part of the story of masses acting under gravity?   2. Is gravity a force or not? Is it only an apparent force or not?   3. Can such an apparent force 'generate' exchange particles? Are the exchange particle and geometric models both different views of the same underlying truth?   4. A side question might be: why can't the other forces be explained away similarly? Or is that what is happening with all this talk of small extra dimensions? I'd appreciate any illumination on this matter, or suggested reading (preferably at the 'popular science' or undergraduate level).


title: What are the alternatives to the Higgs mechanism?

text: Can someone summarize, with references if possible, all of the alternatives to the simplest model (that requires only a single scalar Higgs field with the Mexican Hat potential) of spontaneous electroweak symmetry breaking?


title: Impulse - distance question

text: What impulse should be applied to an object of mass m, having a known coefficient of friction u to get to a distance d ? Thanks! Update 14.02.2011: I still wasn't able to find an answer to this. Could you please share the solution? I need an "Idiot's solution" for this. I'm not experienced enough to find a formula by myself starting from your clues. I've searched the forums but they take into account other parameters. Thank you ! Update 13.02.2011: Here is a representation of what I need: ![http://img412.imageshack.us/img412/2192/20110213112259.png](http://i.stack.imgur.com/Pc7qa.png) It's been a long time since I studied the principles of physics, sorry if I did not make myself clear in the first post. I've tried formulas like: 1)m * a = u * m * g => a = u * g 2)a = v / t 3)d = v * t So from these three equations, distance = u * g * t^2 The impulse p = m*v , so I get p= m * u * g * t However, the time is not known and it is not important how long it takes for the object to get to target. Does this formula (p= m * u * g * t) seem right? The mass, u, g are known, but what about time?


title: Is anyone studying how the general topology of spacetime arises from more fundamental notions?

text: Stephen Wolfram in his book _A New Kind of Science_ touches on a model of space itself based on automata theory. That it, he makes some suggestions about modelling not only the behaviour of matter through space, but the space itself in terms of state machines (a notion from computing). Here, the general topology of space arises from a small-scale connection lattice. I wondered whether any theoretical work is being undertaken along these lines within the physics community. The reason for my interest in this regards one of the mysteries of quantum mechanics, that of quantum entanglement and action at distance. I wondered whether, if space is imagined as having a topology that arises from a notion of neighbourhood at a fine level, then quantum entanglement might be a result of a 'short circuit' in the connection lattice. That is two points at a distance through 'normal' space might also still be neighbours at a fundamental level; there might be a short strand of connectivity in addition the all the long strands relating the two. (I think Richard Feynman also alluded to this sort of model with his take on quantum electro-dynamics.)


title: Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

text: or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$


title: Is there a sound that explodes gunpowder?

text: Is there a certain frequency of sound/noise that will cause gunpowder to explode?


title: Electric and Water Flow Analogy

text: I’ve been studying electricity lately and it has been quite hard to understand the terminology, and picturing it in my mind. After struggling to understand what each property of electric currents meant, I found an analogy that related electric current to water flow. I am not sure if this analogy is 100% accurate, but it sure has helped me understand the complex (at least for young, starter scientist like me) topic of electricity. This far I have managed to relate in the analogy that: \-----Electricity-------------------------Water------------ Current (Amps)(A)-------Flow rate (Volume/Time) Resistance (Ohms)(Ω)----pipe size (Inches) Voltage (Volts)(V)------Water pressure (psi) But I am still missing properties like charge. What properties am I missing? How could it be related to water flow? If it can’t be related, can you please give me a short definition and explanation to better grasp the concept? Please include units in both sides of the analogy. Also, please keep the answer short and simple for us, the inexpert scientist (including me) that struggle to grasp the concept. **Thanks a bunch for your expert help** It is kind of discussed in these articles: Flow of water and flow of electrons, how this analogy works? Power in hydraulic analogy But I couldn’t find (or understand) the answer provided in them.


title: Are two thin blankets significantly warmer than a single thick blanket?

text: Almost every source I can find online maintains that two 0.5 cm blankets are significantly warmer than a single 1cm blanket due to air trapped between the thin blankets. However, the thermal conductivities of air and wool are roughly comparable according to this website. That would seem to mean that air only aids the insulation significantly if the thickness of the air layer is significant compared to the thickness of the blankets, which I would guess it is not. Further, in order for heat transfer through the air to be dominated by conduction, the air needs to be extremely still, which it seems it would not be around a living, breathing, human. Otherwise the air would mix and not provide significant insulation. Is there any empirical evidence that two thin blankets are superior? If so, why is that, given the above considerations?


title: Forces acting on a body in equilibrium

text: The resultant force acting on a body in equilibrium is 0: $$\iiint_R \rho {\bf b}\ dV + \iint_S {\bf t}^{(n)} ds = 0,$$ in which $R$ is a region inside the body, $\rho {\bf b}$ the body force per unit volume, $S$ the surface of the region and ${\bf t}^{(n)}$ the surface traction. I am not quite sure what to make of all these parts... for example, let $R$ be a solid cube made from uniform material, resting on a table. In this case $S$ consists of the 6 faces of the cube, and $\rho$ is its (constant) density. But what are ${\bf t}^{(n)}$ and $\bf b$? It seems ${\bf t}^{(n)}$ should point upwards (against gravity) on every point on the bottom face of the cube, and be 0 on the other five faces. Also, I think $\bf b$ is the gravitational acceleration $g$... is this correct?


title: What decides the velocity of a bullet coming out from the gun?

text: I was concerned about what determines the speed of the bullet coming out from the gun. Is it determined by the strength of the explosive or how fast does it burn up? For example if we take the same amount of gun powder and a plastic explosive (probably TNT) in a bullet shell which bullet will go further the one with the plastic explosive or the one with the gun powder


title: Why does it seem that the potential difference dependence of capacitance and total energy stored in a parallel-plate capacitor are contradictory?

text: Consider a parallel-plate capacitor. Charge is stored physically on electrodes ("plates") which are flat and parallel to one another. If one electrode has charge $+Q$ and the other electrode has charge $-Q$, and $V$ is the potential difference between the electrodes, then the capacitance $C$ is $$C = \frac{Q}{V}$$ (This definition of $C$ is given in, for example, _Introduction to Electrodynamics_ by David J. Griffiths.) But, now, let's think about the energy stored in the electric field between the electrodes of this parallel-plate capacitor. As stated in Griffiths on page 105, "How much work $W$ does it take to charge the capacitor up to a final amount $Q$?" It turns out that $W$ is $$W = \frac{1}{2} CV^2$$ So: (i) the capacitor's capacitance $C$ goes like $\frac{1}{V}$; and (ii) the energy $W$ stored in the electric field goes like $V^2$. **Are statements (i) and (ii) at odds with one another?** I am sure that they _cannot_ be. But conceptually I am having difficulty. We desire high capacitance -- we want to put as much charge on the electrodes as possible, because if we accomplish this, then I think that will increase the energy density of the system. But is what I just said true? If we manage to increase $Q$, then by $V = \frac{Q}{C}$, the potential difference $V$ between the plates will also increase. This, I think, is why capacitor electrodes are separated by a material (such as a polarizable dielectric material like a slab of plastic); otherwise $V$ will become too large and the breakdown voltage will be reached, generating a spark. But, now, the equation $W = \frac{1}{2} CV^2$ (where I think that $W$ can be conceptualized as the energy stored in the electric field between the electrodes) seems to say that as $V$ increases, so does the energy $W$, quadratically. So, my question is, **do we want a capacitor to have a _large_ potential difference $V$ or a _small_ potential difference $V$?** If $V$ is large, then $W$ is large (which we want), but $C$ is small (which we do **_not_** want). Am I somehow thinking of two different potential differences $V$ and confusing them?


title: Functional Derivation of Holonomy

text: I would like to know how to take the functional derivative of the holonomy, or Wilson line. I have tried it and I will show what I have done below, but before I wanted to say that I also have seen and done this with the characteristic deifferential equation for the holonomy $$ \frac{\partial U}{\partial s}+\dot{\gamma}^a A_{a} U=0 $$ with $\dot{\gamma}$ a tangent vector to the curve and $A$ the connection. By varying this equation I can find what $\frac{\delta U}{\delta A}$ is, but I would like to know how to do it from the expression for $U$ $$ U=\mathcal{P}\exp \left[ -\int_{\gamma} \dot{\gamma}^a(s) A_a(\gamma(s)) ds \right] $$ with $\dot{\gamma}^a=\frac{dx^a}{ds}$ as before. Now I have tried to directly vary this with respect to $A_b$ $$ \frac{\delta U}{\delta A_b(x)}=\mathcal{P} \exp \left[ -\int_{\gamma} \dot{\gamma}^a A_a ds \right] \cdot \frac{\delta}{\delta A_b}\left[ -\int_{\gamma} \dot{\gamma}^a A_a ds \right] $$ Now if $A_a=A_{a}^{i}\tau^i$ then $$ \frac{\delta}{\delta A_{b}^i }\left[ -\int_{\gamma} \dot{\gamma}^a A_{a}^j \tau^j ds \right]=-\int_{\gamma} \dot{\gamma}^a \delta _{ab}\delta_{ij} \delta^3(\gamma(s)-x) \tau^j ds=-\int_{\gamma}\dot{\gamma}^b \delta^3(\gamma(s)-x) \tau^j ds $$ So I end with $$ \frac{\delta U}{\delta A_{b}^j}=U(\gamma)\left[ -\int_{\gamma}\dot{\gamma}^b \delta^3(\gamma(s)-x) \tau^j ds \right] $$ Which isn't right.. Can someone point me in a better direction, Thanks.


title: Relation between coordinates and frames of reference

text: I always get a little uneasy that all the theories I can think of (at least since Newton) are constructed in a way such that they would be true in heaven and on earth ... but we can never go everywhere and test it out. So here is the question: > Is there some good justification to implement something like the principle > of relativity in scientific theories other than it turned out to work good > so far? Some more motivation: We have an understanding of different places in space (and time) and what different velocities are. Like imagine me and my droogs cruising our skateboards down the neighborhood and there is a truck driving in the other direction. I see a cactus on the roadside and I wonder how the trucker in his ride sees it. ![](http://i.imgur.com/7qhvM.png) Now in the maths, space $\vec x$ and spacetime $t$ represent physical space and physical time. And if I know my coordinates, the form of the plant and its location and orientation in space, I can find out what I see and also what the trucker from his position sees. A _coordinate transformation_ (replacing some letter on a piece of paper with some other letters in a systematic way) is conventually interpreted as taking the data from one "perspective" and transforming it into "another perspecitive". It's supposed to be a fruitful approach to physics to consider only the observable quantities. Maybe I interpret the principle of relativity the wrong way, but I find it funny that a theory tells me there are spacetime events where I can never get to (outside the light cone). And simultaneously I'm guaranteed that if I'm there I would also be able to physics and come to the right conclusions. At the very least, I feel this is somewhat redundant - why not drop it?


title: Coupling of vector gauge and a massive tensor field

text: I was reviewing the paper-Coupling of a vector gauge field to a massive tensor field In the calculation I found the term $ 2\mu^2 \varepsilon^{ijk} \dfrac{\partial_j}{\partial^2}B_k\dot{B}$ which needed to be integrated by parts. How can I do this? Please help me. _Note_ : Here $B_k$ is the space component of an antisymmetric tensor field and $\dot{B}$ is the time derivative of $B_k$. ( _Edit: corrected the index_ )


title: are pinch-off bubbles valid solutions to general relativity?

text: are bubbles of spacetime pinching-off allowed solutions to general relativity? With "pinch-off bubble" i really mean a finite 3D volume of space whose 2D boundary decreases until it reaches zero and it disconnects from the mother spacetime, becoming a compact "child" spacetime I've read this as a way in eternal cosmologies to produce new universes, but i don't know if that is something something allowed in general relativity or is something from a more advanced theory (i.e: string theory) In particular, unlike unitary black holes, this does not seem to conserve information, unless someone believes that complementarity can do strange things as clone the information (including the quantum one). Does that automatically make this an unphysical solution?


title: Why is parasitic capacitance in inductor said to be in parallel?

text: Internal resistance of inductance (or other devices) are said to be in series. But parasitic capacitance is said to be in parallel (in case of an inductor). Why is that so? What determines whether an internal property is in series or parallel?


title: Impostor Higgs?

text: I recently came across this article, published in the respectable _European Physical Journal A_. (Apparently, there isn't any corresponding arXiv article for this, so I'm sorry if everyone isn't able to access the article.) Here's the possibility that the author has suggested: > While looking for the putative Higgs boson of the Standard Model of particle > physics, recently, the CMS and the ATLAS experiments at CERN have found > strong signals of a new particle at about 125 GeV. ... > > Here we show that what they may have found at 125 GeV is the long sought for > and missing ingredient of the strong interaction: the sigma-meson of the > Chiral Sigma Model, within the framework of the Skyrme model with a > topological interpretation of the baryons. Just like a massless gauge boson > is a requirement, and hence a prediction of the local gauge theories, in the > the same manner, a very heavy scalar meson is a requirement and hence a > prediction of the Skyrme model of the hadrons. The 125 GeV particle > discovered by the CMS and the ATLAS groups may be an experimental > confirmation of this unique prediction of the topological Skyrme model. > However the bottom line is that even if the experimentalists finally confirm > that this 125 GeV entity is the expected Higgs boson, then there still > remains to discover another heavier scalar particle as the sigma-meson of > the chiral sigma model/Skyrme model, which remains its unique prediction, as > shown in this paper. I appreciate that the physics community in general must be very confident that they have identified Higgs correctly (they wouldn't have awarded a Nobel prize if there was any doubt remaining). The point is, while hoaxes keep sprouting up all the time, this is a well- sourced and peer-reviewed article. Does this possibility appear irrational for any definite reason? Especially since this $\sigma$ meson is still elusive? Edit : Found another. This one is in Phys. Rev. D and incidentally carries the same name as my question title ! Here are the links - PRD Link, arxiv link


title: Behavior of multiple flat coils in a cylinder

text: I am trying to wrap my head around how this would work. The setup is quite simple - there are multiple flat coils sealed in insulated disks, and an active coil with alternating current underneath. All is placed in a cylinder. The question is, how will the disks behave - all should induce the same magnetic field and thus repel each other. Will this force be stable enough, when the field collapses on each pulse? Thanks!


title: Error estimation in peak location determination by centroid method

text: I am trying to locate peak in a data set by numerically calculating the peak using centroid method. How can I estimate the error associated with this peak determination?


title: Particle Collision with Static System

text: I have a system of particles with equal distance with each other and another at random positions which is moving with time. What I want to know is :   1. The method by which I can reduce the number of particles from the first system, as I know the maximum motion of any particle is X ( **Meaning: Maximum displacement from starting to end of simulation is X)**.   2. How to efficiently calculate the collision between them, I have heard about quadtree and octree but (as per I understood till now) they are for collision of particle from each other. In my problem,second system of particles doesn't collide with each other. Note: Sorry, if it is a very basic question, I am (very) new to this field. **Update:** ![enter image description here](http://i.stack.imgur.com/ZTixa.png) ## **The combined System** ![enter image description here](http://i.stack.imgur.com/uEgwX.png) ## **Movable parts (Same color means same radius and mass)** ![enter image description here](http://i.stack.imgur.com/i3whf.png) ## **Unmovable Grid (Colors means nothing and yes, it's a cube of spheres)**


title: What causes radioactivity? Is it a quantum mechanical effect?

text: I'm just curious what causes radioactivity. I've been told that in the case of alpha decay, since the nucleus is quantum mechanical, there is a probability that the configuration of protons and neutrons is in such a way as to have an alpha particle outside the nucleus, and then the electric repulsion pushes it away. Is this true? Does quantum mechanics give a probability of this sort of thing happening which would be connected to the half life. Also the second part of my question is,why we don't have many other kinds of decay, why alpha particles or beta only (and fission etc artificially)? Thanks a lot in advance, if you can point to articles or books I can read on the subject it would also be a plus. edit: for the configuration , I mean some kind of quantum entanglement or uncertainty principle.


title: What is the angle at which water is splashed when a vehicle tyre rides in water on the road?

text: I would like to know the following:   1. What is the angle at which water gets splashed when I ride my vehicle through a water on the road?   2. How does angle of water varies with speed?   3. What is the relation between the distance the water goes with the speed of the vehicle? Thanks!


title: Why and how is the speed of light in vacuum constant, i.e., independent of reference frame?

text: I was told that the Galilean relative velocity rule does not apply to the speed of light. No matter how fast two objects are moving, the speed of light will remain same for both of them. How and why is this possible? Also, why can't anything travel faster than light?


title: Relativistic addition of velocities of spaceships

text: If Spaceship 1 is traveling at speed $.5c$ relative to Earth, and Spaceship 2 is traveling at speed $.3c$ relative to Earth in the same direction, what does Spaceship 2 see Spaceship 1's speed as? I don't think it is quite as simple as $.5c-.3c = .2c$.


title: Why light can't go faster then 300 000 km/s? What prevents it?

text: Why can't the speed of light be faster then approx 300 000 km/s? What prevents it?


title: What is the speed of light from a star moving towards a observer.?

text: We can say that in vacuum speed of light is constant. But if the star is moving with a certain velocity does it add that velocity to the velocity of photons emitted out of that star?


title: Nothing can travel faster than the speed of light. In what frame of reference?

text: Since frames of reference are arbitrary, I can define a frame of reference that moves backwards with a speed greater than $c$, then any static object in that frame of reference is already traveling forwards faster than $c$!      So what does "nothing can travel faster than light" exactly mean, and where does it apply?


title: The Speed of Light

text: I know that the speed of light is constant for some reason.   1. But why is it that when shining light from a spacecraft that is moving at (almost) the speed of this light, the speed of this light wouldn't seem to travel at (almost) twice the speed of light to an outside observer, or does it?   2. I also would like to know why the speed of light is constant?    3. Finally, I also would like to know why the speed of light considered the fastest speed possible? I read that there is no explanation, it is just a physical law of our universe but is there a possible explanation?


title: Ground state energy $ E_{0} $ and evaluation of physical energies

text: Given the lowest eigenvalue $E_0$ of an Schrödinguer operator, do the other energies $ E_{n} $ for $ n >0 $ depend strongly on the lowest eigenvalue of the system? I mean, if we somehow fixed the eigenvalue $E_{0}$, could we get more or at least better approximations to the other eigenenergies of the system? thanks


title: Was the Universe's entropy equal to zero at the Big Bang? Is zero-entropy state unique?

text: It is postulated by many cosmologists that at the Big Bang time the universe was in an unusual low entropy state. Does this claim specifically mean that the entropy of the initial universe was zero? Is zero-entropy state unique for given physical laws? Is it possible that entropy was growing always so that only difference in entropy has physical meaning rather than absolute value? Was there ever negative entropy state?


title: Massive photon and gauge invariance of S-matrix amplitude

text: Let's have minimally extended gauge invariant lagrangian (with free kinetic term of EM field): $$ \tag 1 L (\Psi , \partial_{\mu} \Psi) \to L (\Psi , D_{\mu}\Psi ) - \frac{1}{4}F^{\mu \nu}F_{\mu \nu}, \quad D_{\mu} = \partial_{\mu} - iQA_{\mu}. $$ Here $A_{\mu}$ is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ . It is well-known result that for an arbitrary amplitude of EM process (I have cut here photon propagators and polarization vectors; all of operators here is given in Heisenberg picture) $$ \tag 2 M^{\mu_{1}...\mu_{n}}(q_{1}, ..., q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{iq_{1}x_{1} + ...+iq_{n}x_{n}} \langle | \hat{T}\left( \hat{J}^{\mu_{1}}(x_{1})...\hat{J}^{\mu_{n}}(x_{n}) \right)|\rangle , $$ $$ \hat{J}^{\mu} = \hat{\bar{\Psi}}\gamma^{\mu}\hat{\Psi}, \quad \partial_{\mu}\hat{J}^{\mu} = 0, $$ we have identity $$ \tag 3 q_{\mu_{i}}M^{...\mu_{i}...} = 0. $$ Let's have than non-zero photon mass. Then we will get modified photon propagators (so the new force law) and non-invariance of lagrangian, but $\hat{J}_{\mu}$ is still conserved, so it seems that $(2)$ still satisfies identity $(3)$. I have checked this statement for processes $WW^{+} \to \gamma \gamma$ and $e e^{+} \to \gamma \gamma$ at the lowest order. But is this statement true in general? If yes, why we usually say that photon must have strictly zero mass (nonzero but very small mass doesn't contradict the experimental data)? I'm not interested in Higgs mechanism, because I'm not interested in exactly gauge invariance.


title: Why is the axial current said to create particles?

text: In Peskin and Schroeder p. 669 it is argued that the axial current can be parametrized between the vacuum and an on-shell pion state as: $$<0|j^{\mu 5}(x)|\pi^b(p)>=-ip^\mu f_\pi \delta^{ab}e^{-ipx}$$ This is then described as a parametrization of the amplitude for the axial current to create a pion state from the vacuum. This interpretation puzzles me: isn't it the role of the creation operators of a given theory to do this? Does the above expression correspond to an actual physical process or is it just part of an amplitude in a Feynman diagram for an actual process?


title: Interpretation of derivative interaction term in QFT

text: I am trying to understand what a term like $$ \mathcal{L}_{int} = (\partial^{\mu}A )^2 B^2 $$ with $A$ and $B$ being scalar fields for instance means. I understand how to draw an interaction term in Feynman diagrams without the derivative and how to interpret it (connecting external lines, find the correct value for the interaction coupling constant and so on). But if I have a derivative in front of one of the fields, how do I interpret it ? Is it still two $A$ scalar particles interacting with two $B$ scalar particles ? How the derivative changes the interaction ? I search a bit on Internet about that and found some resources : Preskill Notes (see p. 4.33) or Useful Formulae and Feynman Rules (see p. 20) but still... Don't understand.


title: Sinusoidal Wave Displacement Function

text: I am learning about waves (intro course) and as I was studying Wave Functions, I got a little confused. The book claims that the wave function of a sinusoidal wave moving in the $+x$ direction is $y(x,t) = A\cos(kx - wt)$. However, I see a drawing of the wave and they always seem to be $\cos$ graphs. Are sinusoidal waves always cosine graphs? Or can they be sine? If I ever see a sine wave, then does that mean that this is merely a pulse/wave travelling once and not oscillating in a periodic motion? Sorry for the basic beginners question.


title: Why are radians more natural than any other angle unit?

text: I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I _suspect_ that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not). I understand that using radians is useful in calculus involving trigonometric functions because there are no messy factors like $\pi/180$. I also understand that this is because $\sin(x) / x \rightarrow 1$ as $x \rightarrow 0$ when $x$ is in radians. But why does this mean radians are fundamentally more natural? What is mathematically wrong with these messy factors? So maybe it's nice and clean to pick a unit which makes $\frac{d}{dx} \sin x = \cos x$. But why not choose to swap it around, by putting the 'nice and clean' bit at the unit of angle measurement itself? Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light $c = 1$). Sure, you would have messy factors of $2 \pi$ in calculus but what's wrong with this mathematically? I think part of what I'm looking for is an explanation why the radius is the most important part of a circle. Could you not define another angle unit in a similar way to the radian, but with using the diameter instead of the radius? Also, if radians are the fundamentally natural unit, does this mean that not only $\pi \,\textrm{rad} = 180 ^\circ$, but also $\pi = 180 ^\circ$, that is $1\,\textrm{rad}=1$?


title: Space Elevator on Mars with Today's Technology Possible?

text: I know it's not possible with Earth today, but With today's level of material science technology, would it be possible to make cable strong and light enough to make a space elevator system connecting a Mars-synchronous satellite to an anchor on Mars' surface, ignoring any load at all by an elevator car? Or can someone say with a high degree of certainty that the level of strength to weight possible in cables today is still far from possible even with Mars' lower mass compared to Earth?


title: Velocity of Object from electromagnetic field

text: I am wondering if someone could provide me with a formula that would tell me at what velocity a projectile can be launched from something using an electromagnetic field. The idea is much like a rail gun or Gauss rifle, but not exactly. Just looking to find a formula to determine the speed of the projectile. Thank you, Michael Vanderpool


title: Why does the balloon pop?

text: When we pierce a balloon with a sharp needle, it pops and produce a great sound. But, It doesn't pop when we open the mouth of the balloon (through which we have blown air)... So, Why doesn't the gas release slowly when we pierce it with a needle. In fact, it is released slowly when we release the mouth.


title: Find total energy and momentum of an moving electron in a rest frame

text: I have an electron moving with speed $u'$ in a frame $S'$ moving with speed $v'$ relative to a rest frame $S$. How do I find the total energy and momentum of the electron in the rest frame $S$? I thought the equations were: $E_{total} = \gamma \times mc^2$   $p = \gamma \times mv$ But, that doesn't look right... Could someone point out to me what is wrong here?


title: Are the only observables in string theory the S-matrix?

text: Is the S-matrix the only observable in string theory? What about time varying spacetime backgrounds, or thermal states then?


title: what would my path look like using the sun for navigation

text: A theoretical question which thought about instead of writing my C++ code. I know it is poorly defined however I am now intrigued as to the answer! If you got into a boat in the middle of the sea before sunrise, and as the sun cam up sailed directly towards it, and continued to do so throughout the day until you were heading west in the evening at sunset, and then stopped at sun down. Repeat this for say a week/month, then look on a GPS device as to the path you have traveled, what shape would your path/footprint be? Everyone I have asked has come up with a different answer, and short of buying a boat not sure how to work it out.


title: Norton's dome and its equation

text: Norton's dome is the curve $$h(r) = -\frac{2}{3g} r ^{3/2}.$$ Where $h$ is the height and $r$ is radial arc distance along the dome. The top of the dome is at $h = 0$. ![ ](http://i.stack.imgur.com/t7JI7.gif) _ViaNorton's web._ If we put a point mass on top of the dome and let it slide down from the force of gravity (assume no friction, mass won't slide off dome), then we will get the equation of motion $$\frac{d^2r}{dt^2} ~=~ r^{1/2}$$ (Not just me, lots of sources give this answer). But this equation of motion doesn't make sense. Because as $r$ becomes large, the tangential force is also becoming large. The tangential force should always be less than or equal to the drive force from gravity. What am I seeing wrong?


title: An equation that describes massless spin-1 particle

text: Proca action/equation describes massive spin-1 particle, but I was unable to find an equation that describes massless spin-1 particle. Can anyone tell me what the name of this equation is?


title: Confusion about Impact Force

text: Let a person of mass ( m ) = 10kg. then his weight will be 98N. Now, when he jumps from 1m height on the ground, the impact force becomes 9800N ( considering deflection of ground after impact is 1cm ). 100 times. But i am highly confused. 100 times force should simply crush a human! how the force is neutralized? for calculating impact force: http://hyperphysics.phy- astr.gsu.edu/hbase/flobi.html


title: What is pseudo tensor?

text: What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?


title: Heating up a room to 50 °C

text: Suppose I have an empty room of $24 m^3$ and I want to heat it to 50 °C, assuming the current room is well isolated and the current temperature is 20 °C.   * How much power (in Watts) would a single heating element need to produce in order to reach the 50°C?  Having googled around I found this chart, specifically figure 34T seems to be of interest. I was looking at a heating element that provides $3 kW$ and an air displacement of $127 f^3/m$. If I'm reading the figure correctly it specifies for increases of 50 degrees. But I assume more factors are of importance, specifically the room size. > Note: Initially I asked this question on the general chat channel because I > thought it would be too localised and simple but It was not deemed so. > > Edit #1: Removed the time needed subquestion, instead assume the heating > element is powered on for a period of one hour time.


title: Why doesn't Euler's theorem provide an *absolute* energy scale

text: we know that for a one component system $$U= TS - PV + N \mu$$ Now all terms on the right hand side have well defined zeros, but the internal energy is known to be a relative quantity, with no unique zero. What's going on? Similarly, we derive the above from the fact that: $$U(kS, kV, kN)= kU(S,V,N)$$ And it seems to me that taking $k=0$ gives us $U(0,0,0)=0$ - the internal energy of an empty system is the uniquely defined zero of energy.


title: Finding Angular Acceleration of rod given radius and angle

text: > A uniform rod is 2.0 m long. The rod is pivoted about a horizontal, > frictionless pin through one end. The rod is released from rest at an angle > of 30° above the horizontal. What is the angular acceleration of the rod at > the instant it is released? I just used $$sin(30) = \frac{9.8}{a_{centripetal}}$$ Then I related $$a_{rad} = a_{centripetal}$$ Is this right? I am looking at University Physics: $$a_{rad} = v^2 / r$$ Where 9.8 is gravity. But I got none of the answers in the multiple choice ... so I must be doing wrong. Also I haven't used the radius. Any suggestions would be helpful...


title: Why does nuclear fuel not form a critical mass in the course of a meltdown?

text: A BWR reactor core may contain up to 146 tons of uranium. Why does it not form a critical mass when molten? Are there any estimates of the critical mass of the resulting zirconium alloy, steel, concrete and uranium oxide mixture?