title: Does electron in wave form have mass?

text: I heard from my lecturer that electron has dual nature. For that instance in young's double slit experiment electron exhibits as a particle at ends but it acts as a wave in between the ends. It under goes diffraction and bends. But we don't see a rise in energy. It has to produce 500kev of energy (please correct if my approximation is wrong) according to mass energy equivalence relation. But wave is a form of pure energy and doesn't show properties of having mass as of experimental diffraction. So where is the mass gone?


title: Derivation of the Lorentz transformation

text: I have been reading Young's book called: _University physics with modern physics_ and on page 1284 author states that we can derive Lorenz time transformation by eliminating two equations for Lorentz space transformation along the vector of relative speed $u$. First we eliminate them for $x$ and then for $x'$. The equations to eliminate are: \begin{equation} x=\gamma(x'+ut') \end{equation} \begin{equation} x'=\gamma(x+ut) \end{equation} After elimination we are supposed to get the following Lorentz transformations: \begin{equation} t'=\gamma(t+ux/c^2) \end{equation} \begin{equation} t =\gamma(t-ux/c^2) \end{equation} There is also an article about this here. But neither in this article, neither in the Young's book the solution is shown. I have been trying to derive it myself, but failed and am now puting it as a topic on this forum.


title: Monopoles in non-abelian semi-simple gauge groups

text: Relative to the following > Indeed, the modern point of view is that the operator of electric charge is > the generator of a U(1) group. The charge quantization condition arises in > models of unification if the electromagnetic subgroup is embedded into a > semi-simple non-Abelian gauge group of higher rank. In this case, the > electric charge generator forms nontrivial commutation relations with all > other generators of the gauge group. I have a few questions: Could someone explain me why it is important that the gauge group $U(1)$ embedded in the a large gauge group should have non-trivial commutation relations to guarantee charge quantization. Isn't it enough that it should be compactly embedded? Why does the group has to be semi-simple? Aren't the main Grand Unified Theories that are now considered _simple_ group, like $SO(5)$ or $SO(10)$ ? And finally: the group has to be non-abelian to embed the groups $SU(2)$ and $SU(3)$ describing the other forces? 'Magnetic Monopoles', Yakov M. Shnir Googlebooks: http://books.google.be/books/about/Magnetic_Monopoles.html?id=g3L8SWx8ulkC&redir_esc=y


title: Should water cool faster if is inserted metal canister with ice inside either mix only with ice?

text: Let's say that we have two canister first bigger (metal canister 2l) with 1l of water at 100C, and second smaller (metal canister 1l) with 1l of ice. And we want to cool down boiled water to 50C. So we will insert ice (only ice not hole canister) into boiled water and after some time get wanted water (50C). **But what if we insert into boiled canister - small canister (hole canister with ice inside), should boiled water be cooled down faster?** (cause smaller canister with **metal** surface **have bigger thermal conductivity than pure water** )


title: Including special relativistic effects in momentum in Heisenberg's Uncertainty Principle

text: I've been told that an electron is somewhere within the space of $10^{-10}m$ and am supposed to find the uncertainty in its velocity. Simply applying $m\Delta x \Delta v \geq \frac{h}{4\pi}$ results in $\Delta v \geq 5.79\times10^{10} m/s$. Now, this velocity is not approaching the speed of light, but I wanted to know if I could use the Lorentz factor to make this calculation more accurate (of course in this situation it would not make much of a difference. So I redefined momentum uncertainty as $\Delta p = \gamma m_0 \Delta v$ and solved the inequality... but my final answer states $\Delta v \leq 3\times10^8$... this relation is obvious... but why has it popped out of this? What if the space was even smaller when special relativistic effects actually would come into play, how would I then tackle the question?


title: Infinitely many points of space-time?

text: Please can someone provided me with academic literature (Journals/Books, titles & links) which discuss the current view on space time i.e. that there is not Infinitely many points of space-time?


title: Function to fit solar radiation data

text: I have ground-level radiation data of solar incoming radiation from a radiometer (cosine collector) measured along the day. In the following plot you can see PAR irradiance (ie visible light) in Watts per square meter versus time of day (local time). AS usual for this type of sensor, radiation intensity varies with the Sun's angle with respect to the ground level. ![PAR vs local time](http://i.stack.imgur.com/ivFcD.jpg) As you can see it doesn't look perfectly smooth, rather it has some 'chopped' sections (most notably after maximum at noon) due to the presence of transient, passing-by clouds. I would like to use these data points to fit a model and obtain the ideal radiation curve as if it was a clear-sky day, ie perfect, continuous curve. As you can see it is not a gaussian curve... I've heard before that the appropriate model is the Rayleigh distribution, but I'm not sure... What do you think? is that the correct one or should I use another distribution? I don't want just to fit any equation, rather I want to use the _appropriate_ one which is suitable for investigating ant testing the _corresponding parameters_. Thanks!


title: What comes after string theory/M-theory?

text: Classical mechanics is a good approximation to special relativity, which is a good approximation to general relativity etc. I have heard that if string theory/M-theory is right, then it is not just an approximation to a more accurate theory, but represents the end of the line as a TOE, why is this so? Can this be proved in string theory/M-theory?


title: Complex Conjugate of Wave Function's Derivative

text: I am reading Griffiths QM textbook and I got confused by the following identity: How to prove from $$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$ to $$\frac{\partial \Psi^*}{\partial t} = -\frac{i\hbar}{2m} \frac{\partial^2 \Psi^*}{\partial x^2} + \frac{i}{\hbar}V\Psi^*.$$ Can anyone give me a mathematical proof of this complex conjugate identity?


title: What is the criterion of stability of thick-walled spherical shell?

text: Is there the formula (if someone already has discovered it) or what is the algorithm (if a particular formula was not deduced), to calculate the critical pressure of thick-walled spherical shell $−$ the pressure which provokes buckling?


title: Dimensional reduction of Yang-Mills to m(atrix) theory

text: The Yang-Mills action are usually given by $$S= \int\text{d}^{10}\sigma\,\text{Tr}\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\theta^{T}\gamma^{\mu}D_{\mu}\theta\right)$$ with the field strength defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig\left[A_{\mu},A_{\nu}\right]$ , $A_{\mu}$ being a U(N) Hermitian gauge field in the adjoint representation, $\theta$ being a $16\times1$ Majorana-Weyl spinor of $SO(9)$ in the adjoint representation and $\mu=0,\dots,9$ . The covariant derivative is given by $D_{\mu}\theta=\partial_{t}\theta-ig\left[A_{\mu},\theta\right]$. We are using a metric with mostly positive signs.      We re-scale the fields by $A_{\mu}\to\frac{i}{g}A_{\mu}$ and let $g^{2}\to\lambda$ which gives us $$S=\int\text{d}^{10}\sigma\,\text{Tr}\left(\frac{1}{4\lambda}F_{\mu\nu}F^{\mu\nu}-\theta^{T}\gamma^{\mu}D_{\mu}\theta\right)$$ with the field strength defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\left[A_{\mu},A_{\nu}\right]$ and the covariant derivative $D_{\mu}\theta=\partial_{t}\theta+\left[A_{\mu},\theta\right]$.      Now we perform a dimensional reduction from $9+1$ to $0+1$ , so that all the fields only depend on time, thous all spatial derivatives vanish i.e. $\partial_{a}(\text{Anything})=0$ . The $10$ -dimensional vector field decomposes into $9$ scalar fields $A_{a}$ which we rename $X^{a}$ and one gauge field $A_{0}$ which we rename $A$ . This gives (note that $\gamma^{t}=\mathbb{I}$ and that $\gamma^{a}=\gamma_{a}$. $$F_{0a}= \partial_{t}X^{a}+\left[A,X^{a}\right],\quad F_{ab}=+\left[X^{a},X^{b}\right] \gamma^{t}D_{t}\theta= \partial_{t}\theta+\left[A,\theta\right],\quad\gamma^{a}D_{a}\theta=\gamma_{a}\left[X^{a},\theta\right]$$ The action for this theory is then $$S=\int\text{d}t\,\text{Tr}\left(\frac{1}{2\lambda}\bigg\\{-\left(D_{t}X^{a}\right)^{2}+\frac{1}{2}\left[X^{a},X^{b}\right]^{2}\bigg\\}-\theta^{T}D_{t}\theta-\theta^{T}\gamma_{a}\left[X^{a},\theta\right]\right)$$ with the covariant derivative defined as $D_{t}X^{a}=\partial_{t}X^{a}+\left[A,X^{a}\right]$ and $D_{t}\theta=\partial_{t}\theta+\left[A,\theta\right]$      Now to the question. I need the potential energy $V=+\frac{1}{2}\left[X^{a},X^{b}\right]^{2}$ to be negative, not positive.      Taylor has a discussion on this in his paper "Lectures on D-branes, Gauge Theory and M(atrices)" (http://arxiv.org/abs/hep-th/9801182) on page 10, where he writes:   "Because the metric we are using has a mostly positive signature, the kinetic terms have a single raised 0 index corresponding to a change of sign, so the kinetic terms indeed have the correct sign. The commutator term $\left[X^{a},X^{b}\right]^{2}$ which acts as a potential term is actually negative deﬁnite. This follows from the fact that $\left[X^{a},X^{b}\right]^{\dagger}=\left[X^{b},X^{a}\right]=-\left[X^{a},X^{b}\right]$. Thus, as expected, kinetic terms in the action are positive while potential terms are negative."      But I don't understand where the Hermitian conjugate $^\dagger$ comes from, to me this term is just: $$\left[X^{a},X^{b}\right]^{2}=\left[X^{a},X^{b}\right]\left[X_{a},X_{b}\right]$$      Note that Taylor uses a little different conventions when he re-scales, instead of $A_{\mu}\to\frac{i}{g}A_{\mu}$ he uses $A_{\mu}\to\frac{1}{g}A_{\mu}$ and $\theta \to\frac{1}{g}\theta$. But this should not cause any troubles I think.


title: Covariant Description of Light Scattering at a fastly rotating Cylinder

text: Let us consider the following Gedankenexperiment: A cylinder **rotates** symmetric around the $z$ axis with angular velocity $\Omega$ and a plane wave with $\mathbf{E}\text{, }\mathbf{B} \propto e^{\mathrm{i}\left(kx - \omega t \right)} $ gets scattered by it. We assume to know the _isotropic_ permittivity $\epsilon(\omega)$ and permeability $\mu(\omega)$ of the cylinder's material _at rest_. Furthermore, the cylinder is infinitely long in $z$-direction. The static problem ($\Omega = 0$) can be treated in terms of Mie Theory \- here, however, one will need a **covariant** description of the system for very fast rotations (which are assumed to be possible) causing nontrivial transformations of $\epsilon$ and $\mu$. Hence my question: > ### What is the scattering response to a plane wave on a fastly rotating > cylinder? ![RotatingDisc](http://www.personal.uni- jena.de/~p3firo/PhysicsSE/RotatingDisc.png) Thank you in advance


title: Witten Index and letter partition function

text: I haven't seen any reference which explains these things and I am not sure of all the steps of the argument or the equations. I am trying to reproduce here a sequence of arguments that I have mostly picked up from discussions and I would like to know of references for the background details and explanations.   * It seems that in general the Witten Index (${\cal I}$) can be written as $ Tr (-1)^F \prod_i x_{i}^{C_i}$ where $x_is$ are like the fugacities of the conserved quantity $C_is$ and $C_is$ form a complete set of simultaneously measurable operators.  I would like to know the motivation for the above and especially about calling the above an ``index" even if it depends on the fugacities which doesn't seem to be specified by the Lagrangian of the theory or by the underlying manifold? Even with this dependence does it reproduce some intrinsic property of either the underlying manifold or the theory?   * Say $\phi$ and $\psi$ are the bosonic and fermionic component fields of a superfield and ${\cal D}$ be the superderivative. Now when the above tracing is done over operators of the kind $Tr({\cal D}^n\phi)$ and $Tr({\cal D}^n\psi)$ it is probably called a "single trace letter partition function (STLP)" and when it also includes things like products of the above kind of stuff it is called a ``multi-trace letter partition function (MTLP)"  I would like to know about the exact/general definitions/references for the above terminologies and the motivations behind them.   * Probably for adjoint fields, if $f(x)$ is a STLP then apparently in some limit ("large N"?) one has, MTLP = $\prod _{n=1} ^{\infty} \frac{1}{1-f(x^n)}$    * One also defines something called the ``full STLP (FSTLP)" where one probably includes terms also of the kind $Tr({\cal D}^n \phi ^m {\cal D}^p \psi ^q)$ and then the MTLP can be gotten from that as MTLP = $exp [ \sum _{m=1} ^ {\infty} \frac{1}{m} FSTLP]$   * The upshot all this is probably to show that MTLP is the same as Witten Index (in some limit?).  I don't understand most of the above argument and I would be happy to know of explanations and expository references which explain the above concepts (hopefully beginner friendly!). (I see these most often come up in the context of superconformal theories and hence references along that might be helpful especially about the representations of the superconformal group.)


title: Why beauty is a good guide in physics?

text: Dirac once said that he was mainly guided by mathematical beauty more than anything else in his discovery of the famous Dirac equation. Most of the deepest equations of physics are also the most beautiful ones e.g. Maxwell's equations of classical electrodynamics, Einstein's equations of general relativity. Beauty is always considered as an important guide in physics. My question is, can/should anyone trust mathematical aesthetics so much that even without experimental verification, one can be fairly confident of its validity? (Like Einstein once believed to have said - when asked what could have been his reaction if experiments showed GR was wrong - Then I would have felt sorry for the dear Lord)


title: Peskin and Schroeder "Particle Creation by a Classical Source"

text: In page 32 of Peskin & Schroeder, it is written: > But if $j(x)$ is turned on for only a finite time, it is easiest to solve > the problem using the field equation directly. I am wondering:   1. How is it possible to "solve" the field equation at all? I thought that _once we make the quantization_ , the field equation is meaningless. It is only classically that it gives a field varying in time and space and quantum mechanically it would be what exactly? A differential equation for an operator?   2. If I insisted to find the spectrum of the system using the methods used before, how would I go about it? I tried to compute the Hamiltonian by the definition: $$ H=\int d^3x[\pi(x)\partial_0\phi(x)-\mathcal{L(\phi)}] $$ where $\pi(x)=\partial^0\phi(x)$ and $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2+j\phi$. Thus: $$ H=\int d^3x[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}m^2\phi^2-j\phi] $$ If we denote by $H_0$ the free Hamiltonian, then we get: $$ H = H_0 -\int d^3xj(x)\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}(a_p e^{-i p_\alpha x^\alpha} +a_p^\dagger e^{i p_\alpha x^\alpha})$$ which clearly does not lead to the same expression between equations 2.64 and 2.65 (for one, the integration over $x$ is only 3-dimensional and we would need a 4-dimensional integral to get to $\tilde{j}(p)$.)


title: Degrees of freedom in m(atrix) theory

text: The Hamiltonian for m(atrix) theory is given by $$H=\frac{1}{2\lambda}\text{Tr}\left(P^{a}P_{a}+\frac{1}{2}\left[X^{a},X^{b}\right]^{2}+\theta^{T}\gamma_{a}\left[X^{a},\theta\right]\right).$$ Where $X^a$ are the $9$ bosonic $N\times N$ matrices and $\theta$ are a $16$-component matrix-valued spinor of $SO(9)$. (Given by Taylor on page 8 in "M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory" at http://arxiv.org/abs/hep-th/0101126). This can be derived from an action looking like $$S=\frac{1}{2\lambda}\int\text{d}t\,\text{Tr}\left(\partial_{t}X^{a}\partial_{t}X_{a}-\frac{1}{2}\left[X^{a},X^{b}\right]^{2}+i\theta^{T}\partial_{t}\theta-\theta^{T}\gamma_{a}\left[X^{a},\theta\right]\right)$$.      Since this is a super-symmetric theory the bosonic and fermionic degrees of freedom must match. The equations of motion for the fermions halves the it's degrees if freedom and we are left with only 8. But what about the bosonic degrees, we should have 8 not 9. There is however also a Gauss-constraint given by $\left\\{ \partial_{0}X^{I},X_{I}\right\\} +\left\\{ \theta^{T},\theta\right\\} =0$ (which we must impose if we are to have a gauge invariant theory, here the usual gauge field $A$ has been put to zero $A=0$), but this involves both the fermions and bosons and can not only restrict a degree of freedom for the bosonic matrices?   How do I get the bosonic and fermionic degrees of freedom to match?


title: What's the relationship between quantum entanglement and the relativity of time?

text: Apologies in advance for what may be a stupid question from a layman. In reading recently about quantum entanglement, I understood there to be a direct link between entangled particles, even at arbitrarily large distances. I'm wondering how time plays into all of this.   1. Is there any data on how variable passage of time impacts this linkage?    2. For example, if one entangled particle is in a large gravitational field, while another is not (making the passage of time different for the two), is the link the same?    3. If spin or position is one of the observable traits of these entangled particles, do these traits operate on any measurable cycle?    4. If so, would these cycles be necessarily synched in order for the observed linkage to be present?    5. And if so, would asynchronous passage of time ever be able to break the synchronization and/or entanglement?  I feel like this unleashes a whole host of related questions, but hopefully you see where I'm going.


title: Visualising scalar and vector fields

text: As I have been studying electromagnetic theory, I have always been lead to confusion when trying to visualise the fields. Fortunately, the electric and magnetic fields are vector fields and also along the propagation direction they provide a 3-Dimensional Coordinate system. However, in general to visualise a N-dimensional field (I mean field depending on N coordinates), N+1 dimensions are necessary. For instance, in the case of 1D field $\phi(x)$, we can plot a graph $x$ vs $\phi(x)$. But I am wondering if it is also possible to describe the fields using contours. To elaborate, if I have 2D field $\lambda(x,y)$ will it possible to visualise the field in 2D itself using contours (without actually plotting the variation of $\lambda(x,y)$ vs $x$ and $y$).


title: Are all electrons identical?

text: Why should two sub-atomic (or elementary particle) - say electrons need to have identical static properties - identical mass, identical charge? Why can't they differ between each other by a very slight degree? Is there a theory which proves that? Imagine an alien of size of order of Milky-way galaxy examining our solar system with a probe of size of 10's of solar system dimension and concludes that all planets are identical.


title: How can Excel be used to model the flight of a rocket?

text: Firstly, sorry for any silly mistakes or improper use of this website - this is my first time using it! I am trying to use Excel to model the flight of the Apollo 11 Saturn V rocket using the real values from the mission evaluation report. However, I can't seem to get the right acceleration. It is always too low. My initial acceleration at t=0.1s is 2.06 m/s^2 while the values in the evaluation report show the acceleration to be close to 10 m/s^2. My final goal is to calculate the acceleration at the S-IC OECO event (at t=161s). My spreadsheet showed 28m/s^2 while the real value is 39m/s^2. Obviously I have done something terribly wrong since the difference between the values is quite large, but I've been looking over my spreadsheet for days now and I just can't find my problem. What _should_ the columns be? This is what I have so far:   * Range time (s) - in increments of 0.1s down the column   * Mass (kg) - the mass at that time calculated by previous mass - mass flow rate*0.1s   * g (m/s^2) - the acceleration due to gravity at that altitude   * Weight (N) - mass*g   * Thrust (N) - the thrust at that time using real values from the graphs in the evaluation report   * Drag (N) - using the drag equation and estimated coefficients    * Total force (N) - weight + thrust + drag    * Acceleration (m/s^2) - using F=ma so I did total force/mass    * Velocity (m/s) - previous velocity + acceleration*0.1s    * Altitude (m) - previous altitude + velocity*0.1s I've taken into account changing thrust, mass flow rate, acceleration due to gravity and drag as well. The values in the report are given as 'inertial acceleration' but I haven't learnt this yet (still in high school). So is inertial acceleration different to the acceleration I have calculated using F=ma? Have I misused Newton's second law? Was I wrong to consider the weight of the propellants in this case? I'll appreciate any and all helpful feedback and I can provide more information if needed. Thank you so much! :)


title: capacitor in ac and dc circuits

text: In a dc circuit work done by battery=QV while energy stored in capacitor =(QV)/2 loss in energy=(QV)/2 While a capacitor in ac circuit has a no power loss.why is it so? Shouldn't heat be lost in ac circuit capacitor too while charging.


title: Find drop-off rate of magnetic interference from a mass of pure iron on a magnetic compass

text: How can I find the magnetic interference a stationary 35000 kg block of 100% pure iron would have on a magnetic compass and what the drop off rate of the interference would be. So if said 35000 kg block of iron was 1 meter away from the compass, 100 meters away, or 1000 kilometers way I would like to calculate the rate of drop off of the interference. * * * This may seem absurd, but it is very important for a conceptual project I am working on. For the context of this question, assume everything is perfect, and that we are basically operating in a vacuum and there is no interference from anything else and that all instruments are 100% accurate and infinitely precise. And that I have only a very very basic understanding of physics, mathematics and magnetism.


title: Solving time dependent Schrodinger equation in matrix form

text: If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$ With Hamiltonian $H$ given by $$H=\hbar\omega \begin{pmatrix} 1 & 2 & 0 \\\ 2 & 0 & 2 \\\ 0 & 2 & -1 \end{pmatrix}$$ With $$\psi_t(0)=(1,0,0)^T$$ So I proceeded to find the stationary states of $H$ by finding it's eigenvectors and eigenvalues. $H$ has eigenvalues and eigenvectors: $$3\hbar\omega,0,-3\hbar\omega$$ $$\psi_+=\frac{1}{3}(2,2,1)^T,\psi_0=\frac{1}{3}(2,-1,-2)^T,\psi_-=\frac{1}{3}(1,-2,2)^T$$ Respectively. Could anyone explain to me how to go from this to a general time dependent solution, and compute probabilities of location? I have only ever encountered $\Psi=\Psi(x,y,z,t)$ before, so I am extremely confused by this matrix format. I would be extremely grateful for any help!


title: What is the hierarchy problem?

text: **BACKGROUND** So far I understood that the hierarchy problem was the large difference between the gravitational scale, $M_{pl}\sim 10^{18}\; [GeV]$, compared with the electroweak scale, $M_{ew}\sim 10^3\;[GeV]$. However, I heard that the hierarchy problem is due to the existence of quadratic divergences in the scalar sector of the Standard Model. **QUESTION** Can someone explain with ease the hierarchy problem? Additionally, Is it possible to relate both of the above points of view?


title: How to find the center of mass?

text: A massless rod of length $L$ attached to mass $m$ and with axle to cart of mass $M$. The cart has a shape of equilateral triangle (edge $L$). the cart is at rest and its center of mass is above $x=0$ and a rod is perpendicular to the ground. the cart is free to move without friction. at time t=0 the mass $m$ is released and starts to fall to the left. at time $t=\tau$ the rod is parallel to the ground. Where is the center of mass at $x$ axis at time $t=\tau$ of the cart? ![enter image description here](http://i.stack.imgur.com/8WWPf.jpg) Why does the answer say: $mx_m+Mx_M=m(x_M-L)+Mx_M=0$? How does $x_m=x_M-L$? And isn't $x_M=mL/(M+m)$ is the center of mass of the cart with the mass $m$ and not just the cart as the question asks?


title: Origin of Higgs ghosts

text: In _M. Veltman's Diagrammatica_ , appendix E, one can find the full Standard Model lagrangian. Some sectors (e.g fermion-Higgs and weak sectors) contain so-called Higgs ghosts $\phi^+,\phi^-$ and $\phi^0$. > Are Higgs ghosts Faddeev-Popov ghosts? If so, > why does the Higgs field, not being a gauge field, yield Faddeev-Popov > ghosts? If not, > in which sense are they _ghosts_?


title: Projectile motion along the earth

text: Suppose a projectile is launched from the Earth's surface with initial velocity $v_0$ well below speed of light and initial angle $\theta_0$ with respect to the vertical line perpendicular to the Earth's surface. Omitting Earth's rotation, but knowing that Earth is not flat (as in the real world), what is the maximum height of the projectile with respect to center of the Earth? (Suppose air resistance is negligible.) Is it possible to solve this problem using only the material in the first $8$ chapters of Fundamentals of Physics; D. Holliday, R. Resnick & J. Walker? I'm eager to see a various number of solutions!


title: Negative sign of acceleration

text: This is the problem from our Physics textbook : > _A player throws a ball upwards with an initial speed of 29.4 m s–1. (a) > Choose the x = 0 m and t = 0 s to be the location and time of the ball at > its highest point, vertically downward direction to be the positive > direction of x-axis, and give the signs of position, velocity and > acceleration of the ball during its upward, and downward motion._ Answer is : > _x > 0 (upward and downward motion); v < 0 (upward), v > 0 (downward), a > 0 > throughout;_ a > 0 is what is bothering me and I have done hours of searching and trying to understand. Kindly explain all the parts of the answer. Thanks in advance.


title: How does a receiving antenna get an induced electric current?

text: From this **question**, I've noted that an electromagnetic field carries _no electric charge_ but it has two components:   * Electric field   * Magnetic field Now what I failed to understand is how does the receiving antenna get an induced small voltage. Is it due to the electric or the magnetic field component of EMF? and How?


title: Electric field of a capacitor in dielectric medium with weird size

text: I have been learning gauss's law in capacitor recently, recently I come up with this problem that I couldn't solve myself. If we have a capacitor,and a dielectric medium with half the volume between the plates, the permittivity of the dielectric is epsilon.Find the electric field inside the dielectric. The second one is when the dielectric only covers half the area. First picture (Symbols are E1,E2,Epsilon for clarification) Again,What is the Electric field in the dielectric?Express In terms of Q A(area) epsilon0 and epsilon . How do we go about solving this problem?Is there anything to do with boundary conditions? ![a](http://i.stack.imgur.com/IX0Vu.png) ![enter image description here](http://i.stack.imgur.com/KJmls.png)


title: What is the lifetime of an induced magnetization in a para/diamagnetic material?

text: To the best of my knowledge thermal fluctuations are responsible for washing out any effective magnetization, once the external field is switched off. Since thermal fluctuations need some time to drive the system back to random orientation, what would be the time scale for this process? Is it of the order of 200 ps?


title: Does a moving escalator make it easier to walk up the steps?

text: I was discussing with my colleagues why it feels easier to walk up an escalator when it is moving. My natural assumption was that the movement of the escalator imparts some extra acceleration on the rider that helps to move up the stairs. But my colleagues insisted that this was nonsense, and that the affect is purely psychological (i.e. it just _seems_ easier). We actually came up with three contradictory hypotheses, and I'm not sure which is right: 1\. The escalator is constantly accelerating the rider since without constant acceleration the body wouldn't be able to counteract the force of gravity (i.e. my theory). 2\. The rider is not accelerating since no acceleration is needed to maintain a constant velocity. 3\. The acceleration of the escalator actually makes it harder to get to the next step since it pushes the rider against the current step. Which of these is correct?


title: Speed of a falling pencil

text: If you balance a pencil of length $d$ on its tip, and let it fall, how do you compute the final velocity of its other end just before it touches the ground? (Assume the pencil is a uniform one dimensional rod)


When I learned classical Markov process, I noticed some similarity of quantum process and Markov process, and the only difference of them is between probability and probability amplitude whose modulus square represents a probability. I show the some similarity here. The Schordinger's equation gives us the relationship between present state and next state.$$\frac{d}{dt}\lvert\psi\rangle=\hat{H}\lvert\psi\rangle$$ Here, we ignore the complex number and Planck constant.Its discrete time version is $$\lvert\psi\rangle_{n+1}-\lvert\psi\rangle_{n}=\hat{H}\lvert\psi\rangle_{n}$$ I found it very similar to Markov process which express as $V_{n+1}=MV_{n}$ or $$V_{n+1}-V_{n}=(M-1)V_{n}$$ where $V$ is the vector in the state space. If we ingore the difference between probability and probability amplitude, the matrix $(M-1)$ is something very like Hamiltonian. In the Schordinger picture, the similarity is more obviously. $$\lvert\psi(t)\rangle=\hat{O}(t,t_0)\lvert\psi(t_0)\rangle$$ Actually, my question is about Markov process. I want to know how to deal with a Markov process whose state space has infinite dimensions. I think maybe quantum mechanics could give me some help. Can I treat a quantum process as a Markov process? Is it true that the difference between them is just the difference between probability and probability amplitude? How to deal with the case of infinite dimensions? Can I introduce some thing like probability amplitude to help me solve this question?


title: Optics - Derivation of Linear Magnification of Lens without Sign Convention

text: In some posts in this forum I've read that sign convention in optics is useful for making formula memorizing easier, and that we have to use sign convention again while applying the formulae to neutralize the effect of taking sign convention... I tried to use geometry and saw that indeed without sign convention we get two formulae for lens maker's formula and so on... so in that way sign convention is useful... But i could not derive the formula for linear magnification by geometry - without sign convention the derivation does not match the formula given in the text books (which says m must be negative for convex lens when image is real). Can somebody please explain how to get it, and if not why don't we get it? The Sign Convention used is:   1. The direction of the incident ray is positive.   2. Incident ray is always incident from the left side on the refracting surface.   3. Perpendicular distance of an object is positive when measured upwards from principal axis.   4. Distance is always taken FROM the pole of the refracting surface.


title: What does the differential of $d_s\sin(\theta) = m\lambda$ help us see, with respect to waves through diffraction gratings?

text: With respect to waves traveling through a diffraction grating, we have an equation like this one: $$d_s\sin(\theta) = m\lambda.$$ Where $d_s$ is the distance between slits in the grating, $\theta$ is an approximate angle at which the waves bend through each slit of the grating, $\lambda$ is the wavelength of the waves passing through the gradient, and $m$ is the number of wavelengths by which distances traveled by one wave from one slit differ from an adjacent slit. $d_s$ and $m$ are usually given a remain constant in the scenarios I'm working with. My physics book says that the _differential of the above mentioned equation_ is $$d_s \cos(\theta)d\theta = md\lambda$$ (without confusing the single $d_s$ (distance) with the ones in $d\theta$ and $d\lambda$). What does it mean to call the second equation the "differential" of the first? I am trying to understand the concept behind the differentials more so that I may later make sense of the physics. EDIT: In user6786's question, user6786 states that "according to the formula $dy=f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential)". I'm trying to see how that works.


title: A proton is trapped in an infinitely deep well of L meters. The proton is in the first excited state. How does the excited state change the question.

text: The real question was: **A proton is trapped in an infinitely deep well of 1*10^-14m.** I suppose that is unimportant as that should only help us decided the limits of our integration. What I'm worried about is the second part of the question. **"The proton is in the first excited state."** Does "The proton being in the first excited state" have some effect on the wavefunction or the question? Or for any probability required should I just find the integral of the wavefunction of a free particle over what ever limits they ask for? For examples. **Calculate the probability of finding the proton within 0.25*10^-14m of the left hand wall.** Which would give us an integral between say (0... and 0.25*...) or to be more specific (0.25*... and 1*10...). The question also asked for the wavelength of the photon emitted if the proton returns to the ground state. [ **these are past paper questions** I'm working on in preparation for a final **so I have no solutions or help * _:(_ ***]   * ]


title: How long until "final totality"?

text: It is given that the angular size of the Sun as viewed from Earth is $0.533^\circ$, the distance of the Moon from Earth at perigee is $3.633\times 10^5$ km, and the mean radius of the Moon is $1737.1$ km. Suppose that the perigee of the Moon is receding from Earth at a uniform rate of $4$ cm/year. Estimate how long it will take to reach "final totality" (i.e. when we can no longer observe total solar eclipse on Earth).


title: What are some interesting calculus of variation problems?

text: That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)


title: units and nature

text: I am wondering whether the five$^1$ units of the natural unit system really is dictated by nature, or invented to satisfy the limited mind of man? Is the number of linearly independent units a property of the nature, or can we use any number we like? If it truly is a property of nature, what is the number? -five? can we prove that there not is more? In the answers please do not consider cgs units, as extensions really is needed to cover all phenomenon. - or atomic units where units only disappear out of convenience. - or SI units where e.g. the mole for substance amount is just as crazy as say some invented unit to measure amount of money. \-- $^1$length, time, mass, electric charge, and temperature (or/and other linear independent units spanning the same space).


title: The difference between EMR and lone Electric and Magnetic fields

text: I fail to understand the true difference between EMR and electric and magnetic fields. When current flows, there is an electric field due to the electron flow and a magnetic field, however no EMR (Save blackbody radiation from the wire) is emitted. Then how can radio broadcasts function if magnetic fields only are emitted from wires? Obviously I am missing something major here.


How are Monte Carlo simulations used in experimental high energy physics? Particularly in studying detectors limitations (efficiencies?) and data analysis. I will appreciate giving a simple example to clarify how MC is used if the question is too technical to have a simplified answer


title: Is anything actually 1 meter long (or 1kg of weight)?

text: I believe that no real objects are actually ( _exactly_ ) 1 meter long, since for something to be 1.00000000... meters long, we would have to have the ability to measure with infinite precision. Obviously, this can be extended to any units of measurement. Am I wrong?


title: Numerical computation of the Rayleigh-Lamb curves

text: The Rayleigh-Lamb equations: $$\frac{\tan (pd)}{\tan (qd)}=-\left[\frac{4k^2pq}{\left(k^2-q^2\right)^2}\right]^{\pm 1}$$ (two equations, one with the +1 exponent and the other with the -1 exponent) where $$p^2=\frac{\omega ^2}{c_L^2}-k^2$$ and $$q^2=\frac{\omega ^2}{c_T^2}-k^2$$ show up in physical considerations of the elastic oscillations of solid plates. Here, $d$ is the thickness of a elastic plate, $c_L$ the velocity of longitudinal waves, and $c_T$ the velocity of transverse waves. These equations determine for each positive value of $\omega$ a discrete set of real "eigenvalues" for $k$. My problem is the numerical computation of these eigenvalues and, in particular, to obtain curves displaying these eigenvalues. What sort of numerical method can I use with this problem? Thanks. Edit: Using the numerical values $d=1$, $c_L=1.98$, $c_T=1$, the plots should look something like this (black curves correspond to the -1 exponent, blue curves to the +1 exponent; the horizontal axis is $\omega$ and the vertical axis is $k$): ![The curves are supposed to look like this.](http://i.stack.imgur.com/zIRvb.png)


title: Which angle should it be?

text: in the formula $$dB = \frac{\mu_0l ~|dl \times r|}{4 \pi r^3} $$ and the image   ![enter image description here](http://i.stack.imgur.com/NVNYK.png) where dl is in y-z plane and dB is in x-y plane. the ring conductor is in y-z plane carrying current I in direction as mentioned   **EDIT** : also point p can move in the circular ring **EDIT 2** :To clear the confusion...The dl vector is having (L alphabet) and current is I (i alphabet). I want to know that is the angle between dl and r is 'Theta' ? how?


title: Finding stopping time when only given initial angular velocity and an expression for angular acceleration?

text: Question: > A wheel starts is spinning at $27\text{ rad/s}$ but is slowing with an > angular acceleration that has a magnitude given by $\alpha(t) = > (3.0\;\mathrm{rad/s^4})t^2$. It stops in a time of: $\qquad$ ? I happened to guess $3.0\text{ s}$ and it turned out to be right. But I want to understand why. I tried using a kinematics equation and plugging $a$ in for $\alpha$, but it gave me the wrong answer. Am I missing something simple here?


title: How does the renormalization scale $\mu$ cancel in all finite observables?

text: In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} \end{equation} This is done to ensure that the action will have the correct dimensions. Apparently, $\mu$ will not affect any observables. But I don't understand how this can be proven. Edit: I am aware that we do not want to "true" theories to depends on the cut- off $\Lambda \rightarrow \infty$ or $\epsilon=4-d \rightarrow 0$, but this doesn't tell me what will happen with $\mu$. Edit 2: Ok, maybe I understand it. We introduce the renormalization scale $\mu$ and this is an _unphysical_ parameter. Therefore, we _demand_ that that physical observable are not dependent on $\mu$. This procedure results in the renormalization group equations. Is this correct?


title: "String" infinity paradox

text: The smallest particles of the universe (or the smallest part of the smallest particles of the universe) must have infinity density. But if it have infinity density, you can calculate mass of one of these particles as infinity - so if any thing would composed of these particles, would it have infinity mass. How is it possible?


title: Why can't a dislocation terminate in the bulk?

text: We are told that they can only terminate on surfaces, grain boundaries or other dislocations but we are not told why they can't terminate inside the crystal.


title: Definition of angular velocity

text: I'm trying to follow the Feynman lectures and I'm stuck on a particular piece. Let me frame it for you before I ask my question. Nobody does it better than the author himself, > we must find out how a particular particle moves when the angular velocity > is such and such. To do this, let us take a certain particle which is > located at a distance $r$ from the axis and say it is in a certain location > $P(x,y)$ at a given instant, in the usual manner. If at a moment $\Delta t$ > later the angle of the whole object has turned through $\Delta \theta$, then > this particle is carried with it. It is at the same radius away from $O$ as > it was before, but is carried to $Q$. The first thing we would like to know > is how much the distance $x$ changes and how much the distance $y$ changes. > If $OP$ is called $r$, then the length $PQ$ is $r \Delta \theta$, because of > the way angles are defined. The change in $x$, then, is simply the > projection of $r \Delta \theta$ in the $x$-direction: $$\Delta x = -PQ \sin > \theta = -r \Delta \theta \dot (y/r) = -y \Delta \theta$$ Similarly, > $$\Delta y = +x \Delta \theta$$ ![Angular velocity problem](http://i.stack.imgur.com/cXuy7.png) _Source: Feynman Lecturs on Physics_ Now I will try to reproduce this in a way I find logically consistent: we can imagine an isosceles triangle formed by $ABC$ where the angle $ACB$ is $\gamma$ and the line segment $AC$ is $b$, the line segment $CB$ is $a$ and the line segment of $AB$ is $c$. The law of cosines says that, $c^2 = a^2 + b^2 - 2ab$. In the case of an isosceles triangle we can simplify this by saying $a=b=r$ ($r$ because that is what we have in our example), therefore our law simplifies to: $c^2 = r^2 + r^2 - 2r^2 \cos \gamma = 2r^2(1 - \cos \Delta \theta)$. If we suppose that $c$ is the length of the line segment $PQ$ then it should be: $$c = \sqrt{ 2r^2(1 - cos \Delta \theta)}$$ Instead, Feynman says that this should be $r\Delta \theta$. Okay, maybe it is only an implicit approximation for infinitesimal angles, so we can look and see that $\cos \Delta \theta \approx 1$ for very small angles which immediately implies that $c \approx 0$. So why and how does Feynman find $c=r \Delta \theta$? Where did I go wrong?


title: If Electricity and magnetism are the same viewed from a different reference frame then is electricity medium dependant?

text: If Electricity and magnetism are the same viewed from a different reference frame (they are the same force as unified by maxwell) then is electricity medium dependant? I came to this question when trying some experiment with two wires close to each other where I place between them non-conductor mediums such as: paper, wood, plastic... And I see some difference. So is the electrostatic force medium dependant? That is, if I have a charge Q1, and Q2, they will attract/repulse each other (F=k.Q1.Q2/r²) based on the medium where they are separated? If so, does it have to do with density, what is the general formula?


title: Should particles in Smoothed Particle Hydrodynamics (SPH) always move during a simulation?

text: Or can they just be used as an interpolation points and use some other "transported property" which are just evolved and propagated from boundary conditions like for eg. heat conduction through a solid metal bar. It seems to be possible, but isn't clearly elaborated. The force computation step is always used for fluid flow to move particles forward according to Newton's laws, but what if heat is transported purely by conduction in a solid? Here, nothing moves but still, heat flows and destroys temperature gradients. What equations are to be considered here (apart from the heat diffusion equation) to adapt this problem for this kind of discretization?


title: What is the Moduli Space, and why do we care about it?

text: 1. What is the moduli space of a QFT?    2. What does it mean exactly that there are different inequivalent vacua?    3. Can someone give a precise definition of the moduli space, and some easy examples?    4. And why is it so important and studied nowadays?


title: Does a diatomic molecule falling into a black hole dissociate?

text: I've just answered Dipping a Dyson Ring below the event horizon, and while I'm confident my answer is correct I'm less certain about the exact consequences. To simplify the situation consider a diatomic molecule falling into a Schwarzschild black hole with its long axis in a radial direction. ![Molecule](http://i.stack.imgur.com/KZea4.gif) The inner atom cannot interact with the outer atom because no radially outwards motion is possible, not even for electromagnetic fields travelling at the speed of light. However I find I'm uncertain exactly what the outer atom would experience. We know that the outer atom feels the gravitational force of the black hole even though gravitational waves cannot propagate outwards from the singularity. That's because it's experiencing the curvature _left behind_ as the black hole collapsed. Would the same be true for the interaction of the outer atom with the inner atom? Would it still feel an electromagnetic interaction with the inner atom because it's interacting with the field (or virtual photons if that's a better description) left behind by the inner atom? Or would the inner atom effectively have disappeared? If the latter, presumably the fanciful accounts of observers falling into the black hole (large enough to avoid tidal destruction) are indeed fancy since it's hard to see how any large scale organisation could persist inside the event horizon. **Later:** I realise I didn't ask what I originally intended to. In the above question my molecule is freely falling and the question arose from a situation where the object within the event horizon is attempting to resist the inwards motion. I'll have to go away and re-think my question, but thanks to Dmitry for answering what I asked even if it wasn't what I meant :-)


**Update.** As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive. After a long time I came back to try to understand an article on the Ising model. The review article is Percolation and number of phases in the 2D Ising model by Hans-Otto Georgii and Yasunari Higuchi (published in 2000 in the **Journal of Mathematical Physics** as _Percolation and number of phases in the two-dimensional Ising model_ ). I confess that since after the first half of the statement of the first theme got lost. For an expert my doubts about the second half of the proof of the lemma below is certainly primitive. I reproduce below the statement of the lemma and its proof together as some figures to illustrate what I did the first half of the demonstration of the lemma. > _**Lema 2.1(Existence of infinite clusters)** If $\mu\in\mathcal{G}$ is > different from $\mu^-$, there exists with positive probability an infinite > $+$cluster. That is, $\mu(E^+)>0$ when $\mu\ne\mu^-$._ [Here, $\mathcal{G}$ is the set of Gibbs measures of the Ising model of first neighbors in the net $\mathbb{Z}^2$. And $\mu^-$ is the extremal Gibbs measure, which is obtained by thermodynamic limit of finite volume measures whit borders fixed in negative sign. $E^+$ denotes the event que there exists an infinite cluster of spins in state $+$.] _**Proof.** Suppose that $\mu(E^+)=0$. Then any given square $\Delta$ is almost surely surrounded by a $-*$circuit, and with probability close to $1$ such a circuit can already be found within a square $\Lambda\supset\Delta$ provided $\Lambda$ is large enough. If this occurs, we let $\Gamma$ be the largest random subset of $\Lambda$ which is the interior of such a $-*$circuit. (A largest such set exists because the union of such sets is again the interior of a $-*$circuit.) In the alternative case we set $\Gamma=\emptyset$. By maximality, $\Gamma$ is determined from outside [ see figure below]._ ![enter image description here](http://i.stack.imgur.com/qos6X.png) So far I understand the argument. And in any section of this theorem repeats this argument in several demonstrations. But what comes next does not seem intelligible to a non specialist. I have no idea how the properties below to get the desired equality through the instruction below: _**Continuation of proof:** The **strong Markov property** together with the **stochastic monotonicity** $\mu^-_\Gamma\preceq \mu^-$ therefore implies (in the limit $\Lambda\uparrow\mathbb{Z}^2$) that $\mu\preceq\mu^-$ on $\mathcal{F}_\Delta$. Since $\Delta$ was arbitrary and $\mu^-$ is minimal we find that $\mu=\mu^-$, and the lemma is proved._ $\Box$ **Question:** How to use the strong Markov property and the stochastic monotonicity to finish the proof of lemma? Below the properties used as set out in Article. > $\bullet$ the **strong Markov property** of Gibbs measures, stating that > $\mu(\cdot\,|\mathcal{F}_{\Gamma^c})(\omega) =\mu_{\Gamma(\omega)}^\omega$ > for $\mu$-almost all $\omega$ when $\Gamma$ is any finite random subset of > $\mathbb{Z}^2$ which is determined from outside, in that > $\\{\Gamma=\Lambda\\}\in\mathcal{F}_{\Lambda^c}$ for all finite $\Lambda$, > and $\mathcal{F}_{\Gamma^c}$ is the $\sigma$-algebra of all events $A$ > outside $\Gamma$, in the sense that $A\cap > \\{\Gamma=\Lambda\\}\in\mathcal{F}_{\Lambda^c}$ for all finite $\Lambda$. > (Using the conventions $\mu_{\emptyset}^\omega=\delta_\omega$ and > $\mathcal{F}_{\emptyset^c}=\mathcal{F}$ we can in fact allow that $\Gamma$ > takes the value $\emptyset$.) For a proof   >  one simply splits $\Omega$ into the disjoint sets $\\{\Gamma=\Lambda\\}$ > for finite $\Lambda$. > > $\bullet$ the **stochastic monotonicity (or FKG order)** of Gibbs > distributions; writing $\mu\preceq\nu$ when $\mu(f)\leq\nu(f)$ for all > increasing local (or, equivalently, all increasing bounded measurable) real > functions $f$ on $\Omega$, we have > $\mu_{\Lambda}^\omega\preceq\mu_{\Lambda}^{\omega'}$ when > $\omega\leq\omega'$, and   >  $\mu_{\Lambda}^\omega\preceq\mu_{\Delta}^\omega$ when > $\Delta\subset\Lambda$ and $\omega\equiv +1$ on $\Lambda\setminus\Delta$ > (the opposite relation holds when $\omega\equiv -1$ on > $\Lambda\setminus\Delta$).


title: Getting Energies and Probabilities from the Hamiltonian

text: So I need to find the possible energies and the probabilities of these using the eigenvalues of a Hamiltonian. Once I obtain the eigenvalues, are those the energies E_n in and of themselves? Or do they simply give me the n values, i.e. n = 1, 2, 3, that I would then plug in to the equation ![Equation for finding energies](http://i.stack.imgur.com/4QNC5.gif) Or is it both? Do they both yield the same answer? (I am still waiting on the installation of the computer program to use to find the eigenvalues) Finally, I am completely at a loss as to how to go on to find the probabilities of the energies. I am not given a traditional wavefunction to normalize, so how do I find the probability without the normalization constant? EDIT: I normally don't like to put problem specifics from my homework on here, but I suppose it's hard to understand what I mean by "I am not given a traditional wave function." As such, the exact problem is stated: > Models describing electrons on a crystal lattice are very important to > understanding various phenomena in solids. Here we consider a model in which > an electron lives on a one-dimensional lattice of N sites. The sites are > labeled by i=1,2, ....,N. The system looks like >   >   >       o----o----o-- .... --o----o >       1       2       3           N-1       N >   > > The state of the electron is then a vector of dimension N. The Hamiltonian > is given by an N by N matrix whose elements are: >   >   >                     / - 1,  if i and j are near-neighbors; >        H_{ij} = | >                      \ 0  ,  otherwise. >   > > Physically, the electron can be thought of as hopping from site to site > through a near-neighbor hopping. As you see, the Hamiltonian resembles the > one we obtained in class when we discretized the problem of a particle in a > box. Suppose we prepare the electron in a state |a> with equal amplitude for > all N sites, i.e., a(1)=a(2)=....=a(N). To be specific, let's consider N=5. >   >   >     What is the lowest value we can find if we measure the energy of the > electron? With what probability? >     List all the possible (i.e., with non-zero probability) energy values > that we could find in such a measurement. >   So I used Maple to obtain eigenvalues for N=5, and thus have the energies. So I suppose the root of my question is: if I am only given this information, how do I know what state the electron is in? Is it as simple as the corresponding eigenvector for each eigenvalue? Can I assume anything about the wave function (i.e. follows particle in a box method)? Thank you.


title: Morse potential and chaos

text: I have heard that the Morse potential equation $$\tag{1} -\frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)-E_{n}y(x)=0 $$ is related to the two dimensional equation on the Poincare half plane with a constant magnetic field $$\tag{2} -\frac{y^{2}}{2m}( \partial _{x}^{2}+\partial _{y}^{2})f(x,y)+B\partial_{y}f(x,y) = 0$$ by means of a substitution that turns (2) into (1) but i do not know where to find some free avaliable info.


title: What is the importance of electrodynamics and magnetism in physics as a whole?

text: At my university the second half of a year long sequence in basic calculus based physics focuses on electrodynamics and magnetism. I am wondering what is the significance of these topics to physics in general? How does it relate/prepare you to study the more advanced subjects like quantum mechanics and the like?


title: What happens when a piezo crystal is exposed to a vacuum?

text: Application of mechanical stress to a piezo crystal generates a charge. Quoting from wikipedia, a 1 cm3 cube of quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 V. What happens when the crystal is exposed to a vacuum? Is the converse true? If the structural deformation occurs as a result of exposure to vacuum, is a voltage still generated?


title: Collision frequency at surfaces

text: Collision frequency for particles in gases is well known, and collision theory is used to derive chemical reaction rates in gases, (and particles in liquid solutions as well). Using the mean velocity as a function of temperature, one arrives at $Z = N_A \sigma_{AB}\sqrt{\frac{8k_BT}{\pi\mu_{AB}}}$. I need something similar where the collisions/interactions can only occur at the surface between two areas. I understand phenomenologically the collision frequency will be proportional to the area, but I am looking for something a little more rigorous. I have two types of particles, A and B. They occupy separate volumes of space and do not mix. One can assume no mixing because the time window I am interested in is too small compared to the average velocities or because of surface tension. I haven't thought it out completely yet. The only thing I've thought of yet is to define some sort of mixing depth, and then just use the traditional collision frequency, multiplying by the mixing depth to arrive at collision frequency per unit surface area. But then the mixing depth should itself be derived from the average velocities of the particles and it should not really have sharp boundaries. What is the canonical kind of model used for similar problems, such as reactions on liquid/liquid or liquid/gas boundaries?


title: Spectrum of quantum fluctuations in a harmonic oscillator

text: If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = \left(\frac{1}{2}+n\right) \hbar \omega$$ Even if we can bring this system at $T=0$ into it's ground state, there will be zero point motion or quantum fluctuations remaining. Now if we heat a system, depending on its excitations we need Maxwell–Boltzmann, Bose-Einstein or Fermi statistics to calculate the occupation of each state. The resulting spectra are due to thermal excitations from the ground state. Now if we remain at about absolute zero the system still has quantum fluctuations. From Heisenberg's principle the energy is uncertain but that does not tell me which is the current energy or eigenstate. How can one calculate the 'quantum' spectrum of an harmonic oscillator, or bluntly, how can one calculate the probabilities for observing the system at the state $n=1,2,...$ at $T=0$ if we prepare the system to be in $n=0$?


title: State of the Stars

text: What's the state of stars? I'm not pretty sure whether the stars are stationary or rotating. For instance, if they are not rotating what makes them to be stable?


title: Rigid rectangle in Schwarzschild

text: Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial Riemannian metric $$\mathrm{d}s^2=\left(1-\frac{r_s}{r}\right)^{-1}\mathrm{d}r^2+r^2\left(\mathrm{d}\theta^2+\sin^2\theta \,\mathrm{d}\varphi^2 \right)$$ Now I would like to write down the set points of my perfect rectangle. But I need some help. Although I asked about the notion of angles in curved space I still feel unsure to find the set points of the rectangle. Of course the position of the rectangle shall be such that it is easy to solve the problem.


title: Charging a spherical capacitance

text: How could one charge a spherical capacitor with a battery or any other emf source?


title: Reciprocal lattice and Bragg scattering

text: I know that there are two different ways to write the Bragg condition namely $k^2=(k+G)^2$ and $n\lambda=2d\sin \theta$ where $G$ is a reciprocal lattice vector, $\lambda$ is the wavelength and $d$ is the distance between planes. I am also familiar with the equivalence of both and the intuitive derivation of the second form. But is there an intuitive understanding of the condition using reciprocal lattice vectors?


title: Practical difference between canonical and grand canonical ensembles

text: I'm currently doing some calculations which require evaluating various standard thermal expectation values in the canonical ensemble (both bosons and fermions). Now, in order to make my theoretical machinations easier, I am actually using the grand canonical ensemble, where the chemical potential acts as a Lagrange multiplier enforcing the constraint $\langle \hat{N} \rangle = N$, where $N/V$ is the fixed density of the physical system. The justification for this is that the relative fluctuations in $\langle \hat{N} \rangle$ should vanish in the thermodynamic limit, in which case I expect fixing the average number to be physically equivalent to fixing the number once and for all. (Also, this approach seems to be adopted by a several presumably trustworthy references, see for example Simons & Altland Section 6.3.) This intuition seems reasonable, but I wonder if matters may be more subtle than this argument implies. > Do thermal averages in the thermodynamic limit of the grand canonical and > canonical ensembles coincide? I'm hoping for either a more rigorous justification supporting this procedure, or examples where it can go horribly wrong. Pointers to appropriate references would also be much appreciated.


title: Which function denotes the energy of thermal motion within a system?

text: In thermodynamics, the heat $Q$ is defined as a type of energy in transfer, and is not a state function, which function denotes the energy of thermal motion within a system? 1) $TS$, (there is a contradictory, for an idea gas, $dU=TdS-pdV=C_vdT$) 2) Enthalpy $H$, ($H=U-Yx+pV =TS+pV+\sum_j\mu_jN_j$) 3) Neither of the two, the function has not been well defined, 4) The function cannot be defined. 5) Others.


title: Racing balls question

text: My question is related to simulation of racing ball demonstration. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=142 One ball goes on a straight path, while another one goes on a curved path. On the simulation the second ball going on the blue part down hill which is parabola has a constant component of the velocity in the x direction. Why is that the case when the slope changes?


title: Berry's phase: in which space does the degeneracy appear?

text: This question follows a previous one of mine: Adiabatic theorem and Berry phase. In his original paper [ M. V. Berry, Proc. R. Soc. Lond. A. Math. Phys. Sci. 392, 45 (1984) ], Berry discussed the possibility for an old-forgotten phase factor to arise after one loop is done in a _parameter space_ , using the adiabatic theorem. This became famously known as a Berry's phase, see also Berry's curvature article on Wikipedia. Soon after, [ Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987) ], AA proposed to generalised this adiabatic phase to non-adiabatic -- still cyclic -- evolutions. What confused me is the sentence: > [..] we regard $\beta$ [the AA geometric phase] as a geometric phase > associated with a closed curve in the projective Hilbert space and not the > parameter space [...] I would like to make this sentence a bit more clear, especially regarding the degeneracy point discussed in the previous question Adiabatic theorem and Berry phase. **What is the space in which the degeneracy appears?** Any comment to improve this question is welcome.


title: Speed of approach between image and object

text: A plane mirror image approaches the object at the same rate the object approaches the mirror. So the speed of approach is twice the speed at which the object approaches the mirror. If the object approaches the mirror at a speed greater than half the speed of light (c), does that mean that the speed of approach between object and image will exceed the speed of light?


title: Can I calculate the height of a cliff from weight of falling object and time taken?

text: I'm sure a simple question. I have a video of me jumping off a cliff into a river. I want to calculate how high it is. I know my weight, acceleration due to gravity of course, and I can get the time it took to hit the water. Can I calculate (albeit roughly) how high the cliff was? Cheers, Tim.


Do we take gravity = 9.8 m/s² for all heights when solving problems? Why or why not?


title: No photon interaction in free space

text: How can the Hanbury Brown and Twiss effect (photon bunching) be explained if photons don't interact in free space? To explain it with the influence of the two photons on the two detectors simultaneously would seem strange to me, because of the fact that "if the source consists of a single atom which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible" (same Wiki under the "Quantum interpretation" heading). Comment: The question is not about uniting or dividing two photons, which is possible only on the atomic level. The question is about synchronizing two photons from different sources like the sun.


title: A light so strong it has a shadow

text: I have recently taken an interest in shadows. I know that in order for a shadow to exist that you must have a solid in the way of the light. My hypothesis is that there can be a light so strong, like a laser beam, that it acts like a solid in the sense where it doesn't let light pass through... is this plausible?


title: Thermodynamics of scattering theory

text: I have a couple of conceptual questions regarding the thermodynamics of scattering. Any partial answer or argument will be appreciated. For the sake of discussion, consider the scattering of electromagnetic waves. I understand that the physics would be different for other particles or fields, feel free to base your discussion on those to provide your perspective. The first questions are on the definition of entropy and its governing laws in a scattering process. More specifically: (1) How do we define entropy for electromagnetic waves? I guess, one can always start from the general definition of entropy and perhaps calculate the density matrix of electromagnetic waves (though I've not seen such calculations). While such a first-principles approach might be helpful, I'm more of looking for a phenomenological definition that would capture the physics in relatively simple terms and mathematics. There seems to be arguments saying that electromagnetic waves do not have entropy, but given the fundamental nature of entropy, I think everything has entropy but it may not be well defined (please correct me if I'm wrong). (2) How does entropy change in a scattering event? The second law would be a correct answer to this question, but is there a tighter bound regarding its change? Or, assuming that we now know how to define entropy for the electromagnetic waves, how are the entropies of incoming and outgoing waves related to the properties of the scatterer? (To better illustrate my question, in the case of energy, the difference between energies of incoming and outgoing waves is the dissipation in the scatterer.) My final question is what I'm really after, though it might sound stupid: are the 1st and 2nd laws of thermodynamics independent in such scattering theories? (Can we derive the 2nd law from the 1st law plus something else in the framework of scattering theory?) This is not a random question but has basis in my research. I've been unable to reach clarity in this thought, but feel free to ask me to elaborate on any confusing statement.


title: Am I solving this basic Kirchoff's current law and Ohm's law problem correctly?

text: After moving onto some of the practice textbook exercises in the section covering Kirchhoff's current law and Ohm's law, I came across a problem which evades my best attempts at solving it. ![problem text](http://i.stack.imgur.com/KybEy.png) ![circuit diagram](http://i.stack.imgur.com/goNxE.png) Using Ohm's law, it was relatively easy to determine a equation involving voltage drops around the right loop:$$(1000\,\Omega)I_C+(1000\,\Omega)I_E=V_2\\\\(1000\,\Omega)(I_C+I_E)=V_2$$Similarly, it was simple to write an equation for the node where $I_B,I_E$ meet:$$I_B+I_C+150I_B=I_E\\\151I_B+I_C=I_E$$... but given just $I_B=100\,\mu A=10^{-4}\,A$ it seems impossible to determine a numerical value for $I_C,I_E$. Nevertheless it's straightforward to solve for $I_C,I_E$ in terms of $I_B,V_2$:$$I_C+I_E=\frac1{1000}V_2\\\\-I_C+I_E=151I_B\\\\\implies\begin{align*}2I_E&=151I_B+\frac1{1000}V_2\\\2I_C&=-151I_B+\frac1{1000}V_2\end{align*}$$ Substituting $I_B=10^{-4}\,A$ yields $I_C=\frac1{20000}(10V_2-151),I_E=\frac1{20000}(10V_2+151)$ Am I approaching the problem correctly? Am I justified in ignoring the unlabeled voltage source bridging the two loops along with the labelled elements in the left one (i.e. $V_1,R_1,R_2$)? Am I treating the CCCS properly in writing my nodal equation?


title: When do we see particles to be in a superposition of energy states?

text: I have two doubts: Exactly when does this happen? and If we are in a superposition of states (lets say E1 and E2) and the particle absorbs a photon, what will happen? If E3-E1 = hf, will it go to E3? Thanks!


title: what energies do the wheels of a moving car posses?

text: I saw this question in a test. I would have answered kinectic energy due to rotation and translation. It that correct. Else what is the answer? Oh no, i forgot to mention it was objective type question.There were three other options.I just looked them up 1)kinetic energy due to rotation 2)kinetic energy due to translation 3)none of the above the question looked confusing and the choices looked misleading at that time. Thanks for all your resopnses


title: Water rocket physics

text: According to this page, the current record for greatest altitude achieved by a water and air propelled rocket is 2044 feet (623 meters).   In this connection, i had one question: How the greatest altitude depends of the density of the liquid used in the rocket?   Suppose that we have the opportunity to choose a liquid with an arbitrary density and with the same hydrodynamic properties as water. Then there must be a density at which the greatest altitude has its absolute maximum, i think. To be more specific let's consider a simpler problem: Let a two-liter soda bottle rocket, filled with some amount of a liquid mentioned above and air at the atmospheric pressure, is launched in deep space($g=0$)(say an astronaut is experimenting). What is the absolute maximum of the speed the rocket can achieve? For simplicity let's assume the expansion of the air to be isothermal (which it is not). ![enter image description here](http://i.stack.imgur.com/RVXX7.png)


title: Shear, viscosity and expansion of universe

text: What is the meaning of expansion, shear and viscosity in context of universe? How can we conclude a result after getting a numerical value of above terms?


title: In a low pressure (near vacuum), -20°C vessel, would sublimated moisture rise to the top or fall to the bottom?

text: If I have a strong vessel that I've cooled to say -20°C, and lowered the pressure below 0.006 atmospheres, the water in food placed inside will sublimate and freeze dry the food. But what happens to the water vapor? Does the moisture rise to the top or fall to the bottom of the vessel itself? Or does it just expand to fill the container equally?


title: Figuring out North and South Magnetic Poles for Earth

text: I currently make my living as an electrophysicist, so am a bit embarassed at having not thought this one through, before. Magnetically, the North pole is.... North. We get out our compass, and wait for the North-seeking needle to line up. I've always thought of this end of the needle as the north end. In a current carrying loop, if we use the right-hand rule and point our thumb in the direction of conventional current, the fingers show the direction of the mag field lines around the wire. The direction the mag field lines point "north" when they pass through the interior of the loop. By stacking many such loops we can create a cylindrical electromagnet, for which it is easy to see which end is the "north" end. I resume permanent physical magnets follow the same naming convention: that is, mag field lines emanate from the north end and cycle around to the south. We know that like poles of a magnet repel and opposites attract. So, now, hanging a magnet from a string (or going back and using our trusty compass), if the north pole of the magnet points towards teh magnetic north pole of Earth, doesn't that mean that the pole near the Arctic circle is really magnetically-speaking the south pole? Or conversely, when we say "north-seeking" pole of a magnet, is that really the south pole of the magnet? Which is it?


title: Radial Null Geodesics in Static Maximally Symmetric DeSitter Space

text: Given a DeSitter-space metric from the line element: $$ ds^2=\left(1-\frac{r^2}{R^2}\right)dt^2-\left(1-\frac{r^2}{R^2}\right)^{-1}dr^2-r^2d\Omega^2 $$ Where $R=\sqrt{\frac{3}{\Lambda}}$, and $\Lambda$ is a positive cosmological constant, I am trying to derive the equations for radial null geodesics. I derived the geodesic equations from the definition and Christoffel symbols, but I'm a little suspicious of my solution to these equations. So, the differential equations I derived are ($\lambda$ is an "affine" parameter): $$ \frac{d^2r}{d\lambda^2}-\frac{r}{R^2}\left(1-\frac{r^2}{R^2}\right)\left(\frac{dt}{d\lambda}\right)^2+\frac{r}{R^2-r^2}\left(\frac{dr}{d\lambda}\right)^2=0 $$ $$ \frac{d^2t}{d\lambda^2}+\frac{2r}{r^2-R^2}\frac{dt}{d\lambda}\frac{dr}{d\lambda}=0 $$ Now, I tried to use the identity $\textbf{u}\cdot\textbf{u}=0$ where $\bf u$ is the four velocity (or I guess a vector tangent to the light ray's world line if I'm thinking about this correctly). Thus, because $u^2=u^3=0$: $$g_{\mu\nu}u^{\mu}u^{\nu}=\left(1-\frac{r^2}{R^2}\right)(u^0)^2-\left(1-\frac{r^2}{R^2}\right)^{-1}(u^1)^2=0 $$ $$\implies -\frac{r}{R^2}\left(1-\frac{r^2}{R^2}\right)(u^0)^2+\frac{r}{R^2-r^2}(u^1)^2=0$$ Then, substituting this into the first (radial) geodesic equation, I get: $$\frac{d^2r}{d\lambda^2}=0$$ This is what I am suspicious of. It seems too easy and simple. Do you think this is correct? If I carry on anyways and integrate to find $u^0(r)$ then substitute $r(\lambda)$ I get the following for $t(\lambda)$: $$ r(\lambda)=A\lambda+B $$ $$ t(\lambda)=\frac{D}{AR}\tanh^{-1}\left[\frac{A\lambda+B}{R} \right]+E $$ Where, $A,B,D,E$ are constants of integration. To sum up, is this a viable way to solve these equations or am I missing something? If this is correct, how does one define initial conditions for these solutions. Specifically, the derivative conditions $t'(0)$ and $r'(0)$.


title: Person in a special relativistic frame observes light refract on a passing ship

text: The speed of light is the same in all frames of reference. But imagine a scenario in which you passed by a ship which had (B=.78), while your ship had (B=.94). While somehow being able to look through the window of the ship you were passing, you see someone pointing a green laser into a fish tank, and of course the light refracts. Solving for the new angle is simple,(a bit of linear algebra, using the Lorentz matrix with angular components), but we all know that the speed of that light changes after it was refracted. So, would I measure the same velocity of that light as someone who is in the rest frame of the refraction incidence? I mean it is light, so all observers should measure it moving the same speed, correct? But, my gut tells me that I would clearly have to add the velocities using the velocity component Lorentz matrix. Where is the kicker?


title: Is there an equivalent of a scalar potential for torques?

text: For a given scalar potential $V$, it is known that the corresponding force field $\mathbf{F}$ can be computed from $$ \mathbf{F} = -\nabla V $$ Suppose a rigid body is placed inside this potential. The torque on the body $\mathbf{T}$ exerted by the scalar field will be $$ \begin{align} \mathbf{T} &= \int_M \left( \mathbf{r} \times \mathbf{F}(\mathbf{r}) \right) dm \\\ &= \int_M \left( \mathbf{r} \times -\nabla V(\mathbf{r}) \right) dm \\\ &= \int_M \left( \nabla V(\mathbf{r}) \times \mathbf{r} \right) dm \end{align} $$ with $\mathbf{r}$ the position vector to the mass element $dm$, and the integration carried out over the entire body $M$. So, being not too familiar with rigid body dynamics, I was wondering -- does something like a (vector/scalar) potential $P$ exist, such that the local torque induced by the potential $V$ can be expressed as $$ \matrix{ \mathbf{T} = I \cdot \nabla P & & &\text{(or some similar form)} } $$ with $I$ the moment of inertia tensor of the rigid body? If such a thing exists:   * what is its name?   * where should I start reading?   * What is the proper expression for the torque $\mathbf{T}$?   * how does $P$ relate to $V$? If such a thing _doesn't_ exist:   * why not? :)


title: Extreme heating by Sun on Earth- Can eggs be fried?

text: Will it be possible to fry eggs in Sahara desert by just keeping them under the sun? If so, then will the radiated eggs be any different? Or will we have to use the solar cooking concept to some extent? What is the minimum Intensity required and the temperature to achieve this? (With and without solar cooker)


title: Differences between pure/mixed/entangled/separable/superposed states

text: I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum systems. This is what I have so far:   * superposed: A superposition of two states which a system $A$ can occupy, so $\frac{1}{\sqrt{2}}(|1\rangle_A+|1\rangle_A)$   * seperable: $|1\rangle_A|0\rangle_B$ A state is called separable, if its an element of the (tensor)product basis of system $A$ and $B$ (for all possible choices of bases)   * entangled: $\frac{1}{\sqrt{2}}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)$ is not a state within the product basis (again for all possible bases).   * mixed state: Is a statistical mixture, so for instance $|1\rangle$ with probability $1/2$ and $|0\rangle$ with probability $1/2$   * pure state: Not a mixed state, no statistical mixture I hope that the above examples and classifications are correct. If not it would be great if you could correct me. Or add further cases, if this list is incomplete. On wikipedia I read about quantum entanglement > Another way to say this is that while the von Neumann entropy of the whole > state is zero (as it is for any pure state), the entropy of the subsystems > is greater than zero. which is perfectly fine. However I also read on wikipedia a criterion for mixed states: > Another, equivalent, criterion is that the von Neumann entropy is 0 for a > pure state, and strictly positive for a mixed state. So does this imply that if I look at the subsystems of an entangled state, that they are in a mixed state? Sounds strange... What would be the statistical mixture in that case? Moreover I also wanted to ask, whether you had further illustrative examples for the different states I tried to describe above. Or any dangerous cases, where one might think a state of one kind to be the other?


title: A mirror flips left and right, but not up and down

text: Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?


title: Axis of reflection in a mirror

text: When I look at my reflection in a planar mirror, the image I see is reflected about a vertical axis. Why is it this axis and not the horizontal axis?


title: How far would a radio signal propagate in free space as compared to Earth's atmosphere?

text: A curiosity question. Radio Propagation within Earth's atmosphere is via atmosphere. Broadly speaking a signal loses strength between bouncing off the ionized layer, and absorption by various earthy elements. If an identical signal were to be produced in free space, how far would it travel as compared to it's range within Earth's atmosphere? EDIT: A commercial FM transmitter on Earth probably has a range no more than 25 miles where the listener uses a broadcast receiver. For the purpose of elaboration assume the the receiver being placed in static orbit - how far out could the same signal be generated, and still be intelligible?


title: Rigorous QFT on a Torus

text: The problem description for the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf) says in its "Mathematical Perspective" section that > _Some results are known for Yang-Mills theory on a 4-torus $\mathbb{T}^{4}$ > approximating $\mathbb{R}^{4}$ and, while the construction is not complete, > there is ample indication that known methods could be extended to construct > Yang–Mills theory on $\mathbb{T}^{4}$._ > > _In fact, at present we do not know any non-trivial relativistic ﬁeld theory > that satisﬁes the Wightman (or any other reasonable) axioms in four > dimensions. So even having a detailed mathematical construction of > Yang–Mills theory on a compact space would represent a major breakthrough. > Yet, even if this were accomplished, no present ideas point the direction to > establish the existence of a mass gap that is uniform in the volume. Nor do > present methods suggest how to obtain the existence of the inﬁnite volume > limit $\mathbb{T}^{4}\rightarrow\mathbb{R}^{4}$._ Could someone point me in the direction of a paper that describes the use of compact torus manifolds to construct 4d Quantum Yang-Mills, or else describe some of these attempts? Also, is the difficulty alluded to by Witten and Jaffe solely that a toroidal space is compact whereas a Euclidean space is unbounded, or is there more to the story?


title: How to tell if the collision is elastic or inelastic?

text: I'm a programmer and a game developer, not a mathematician or a physicist. So please go easy on the math :) I know two things:   * How to find the new velocities of two objects after an elastic collision.   * How to find the velocity of the new object formed by two objects in an inelastic collision. However what I need to know is: given the mass, the velocity, and the 'angle' the two objects are going two be when they collide - how can I know if I need to compute an elastic or an inelastic collision? I need this for the 2D game I'm developing, that tries to simulate relatively- realistic physics. Thanks for your help


title: Relation between pressure and particle displacement in an acoustic wave

text: Consider an acoustic wave in some medium, expressed as particle displacement: $$s(t,x) = e^{j(-\omega t + kx)} $$ I understand that pressure must be at its maximum when the particle displacement is zero and vice versa. But there are many functions that satisfy that criterion. How can the pressure be expressed in terms of particle displacement?


title: What would a laser beam's E-M waves actually look like if you zoomed in?

text: I am trying to understand how a real world beam of laser actually reflects the physics description of oscillating electromagnetic waves. So say we are looking side on at a vertically polarized laser beam, and this is section of it propagating through free space: ![enter image description here](http://i.stack.imgur.com/9qqwi.png) Ive cut down the opacity and zoomed in now to illustrate my question on what would the waves look like? Something like... ![enter image description here](http://i.stack.imgur.com/47Ngz.png) But how can this form a Gaussian intensity profile? Maybe there are more of these waves dispersed through out it, and their amplitude denotes the intensity e.g: ![enter image description here](http://i.stack.imgur.com/D4wX5.png) Where the waves closer to the edge of the beam are the same wavelength but smaller amplitude than the main section...? But now we have only considered a horizontal cross section, what would it be like if you looked at it from above? Thanks


title: Propagator of the Klein-Gorden equation

text: ## Does this integral converge? My question is related to this one: Free particle propagation amplitude calculation I am reading the book of Peskin and Schroesder. In the second page of their chapter 2, they estimated the propagator for the Klein-Gorden equation. The final step of the calculation is to calculate an integral of the form $$\int_0^\infty \mathrm{d}p\;p\;\sin(px)e^{-\mathrm{i}t\sqrt{p^2+m^2}}.$$ My question is very simple: does this integral converge in the first place? Note that the integrand is an oscillating function whose amplitude grows without bound. If we take $m=0$, then for sure the integral does not converge. In fact, I can reproduce their result very easily (namely, the $e^{-m\sqrt{x^2-t^2}}$ behavior). But I am not sure whether my calculation makes sense because I don't know whether the integral has a finite value. I know this integral can be converted into a loop integral in the complex plane, and then be estimated using stationary-phase method. But since the amplitude of the integrand does not go to zero uniformly at infinity, I feel such a procedure may not be valid. We may multiply a factor $e^{-\epsilon p}$ ($\epsilon>0$) to the integrand to make it converge, but then we are not calculating what we wanted to calculate in the beginning, right? Then, if the integral does not converge by itself, how should I make sense of the result in the Peskin and Schroesder book? I am quite confused by this. Thanks for any hints!


title: Make a semi transparent mirror with copper

text: The question: How would you make a semi transparent mirror (50% reflection, 50% transmission) with glass with a layer of copper. For light $\lambda$ = 500nm Try to be as realistic as possible What I've done so far. The reflection coefficient of copper for 500nm light is 50%. So that's great. But I'm not sure whether that's enough. (It is an exam question, so I guess not). I'm guessing it'll have to do something with skin depth, but I'm not sure. Any suggestions on how to continue ?


title: How much thrust would be needed to turn a hobbyist weather balloon into a deep space probe?

text: I was reading the article _Weather Balloon Space Probes_ that says you can put your own balloon probe at 65,000 ft temporarily. Is it even remotely possible to raise the probe high enough using balloons, and then put it in orbit or escape the field using an on-board propulsion system? Can it be done with easily purchased materials? How much mass can be added for instrumentation? How much fuel of is required? What kind? I know it's kind of a broad question, but I just want to know a rough estimate.


title: Interchaging boson and fermion on an infinite 1 dimensional line

text: In 1+1 dim bosonization, one introduce the Klein factors, which are Hermitian and satisfies Clifford algebra. (1) In the case of **1 dim space is a 1D ring ($S^1$ circle)** , then one have left-right boson field commutes $$[\phi_L, \phi_R]=0$$ but introduces Klein factor to reproduce the fermionized fermion field anti-commute: $$\\{ \psi_L, \psi_R\\}=0.$$ (2) However, according to this Ref, in page 21, footnote 8, for **1 dim space as an infinite line** , one requires $$[\phi_L, \phi_R]=i \frac{1}{4}$$ to reproduces the fermionized fermion field anti-commute: $$\\{ \psi_L, \psi_R\\}=0.$$ > **How can I see, how can one show that $[\phi_L, \phi_R]=i \frac{1}{4}$ for > 1 dim space as an infinite line**?


title: Magnetic moment with external magnetic field on a lattice?

text: Consider a system in which atoms are located in a regular lattice, each atom having a spin $1/2$ and an associated intrinsic magnetic moment $\mu_0>0$. Assume that each atom interacts only weakly with other atoms so that all the other atoms act as a heat reservior. If the system is placed in an external magnetic field $H_0 \hat{z}$, then the Hamiltonian is given by: $$\hat{H}=-\mu_0H_0\sum_i\sigma^z_i$$ where $\sigma_i^z$ are (the $z$ components of) the Pauli matrices, at the $i$th site. For the magnetic moment $\mu= \pm\mu_0$, the corresponding magnetic energy of the atom is $E=\mp\mu_0 H_0$. What would be the mean magnetic moment?


title: How are the E and B fields inside a coaxial cable related to the current and voltage?

text: Conceptually, I understand that coaxial cables operate in a TEM mode due to the central conductor, and hence the electric field is directed radially outward from the voltage $V_0$ at the centre of the cable, to the supposedly grounded outer cable (the derivation for this involving Poisson's equation and such). Similarly, as the current flows down the cable, the magnetic field around this conductor has direction only in terms of $\phi$ and does not change with $z$. However, I've been asked: _How are the electric and magnetic fields inside a coaxial cable related to the V and I in the wave of the equation:_ $V_+(t-z/v) = Z_0I_+(t-z/v)$ _where the subscript $+$ signifies the voltage and current in the $+z$ direction_ (Note that there is a similar formula for the -ve reflected direction). I've been searching all night, but I have yet to find a solid explanation for this that doesn't involve derivations from first principles. How can I turn the above equation into something involving B and E fields?


title: Why didn't electroweak symmetry breaking happen earlier than it did?

text: I've read that electroweak symmetry breaking occurred "at a picosecond or so" after the Big Bang (in a source I found). I can't help wondering why it took so long to get started. For instance, was something missing up until then, which was needed by the Higgs mechanism? What feature(s) of the Higgs mechanism, or perhaps what other theoretical consideration(s), was/were seemingly responsible for this very brief delay?


title: The problem in Sredniki's textbook: How do I calculate loop corrections for $\phi\phi\to\phi\phi$ with this Lagrangian?

text: The problem in Sredniki's textbook 10.5 : For a free scalar field $\psi$, the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\psi\partial_\mu\psi-\frac{1}{2}m^2\psi^2$$ Here we use the metric $\operatorname{diag}(- + + +)$ If I make $\psi=\phi+\lambda \phi^2$, then the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2-2\lambda\phi\partial^\mu\phi\partial_\mu\phi-\lambda m^2\phi^3-2\lambda^2\phi^2\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}\lambda^2m^2\phi^4$$ For scattering $\phi\phi \to \phi\phi $, how do I calculate the loop correction? Since the Lagrangian is now nonrenormalizable. In loop correction we need to take ghost into consideration.


title: Maxwell's Demon Constant (Information-Energy equivalence)

text: New Scientist article: Summon a 'demon' to turn information into energy The speed of light c converts between space and time and also appears in e=mc^2. Maxwell's Demon can turn information supplied into a system into energy, which suggests there is a constant 'd' such that e=i*f(d) where i is information and f is a function of d. If d appears in the formula relating energy and information like c relates energy and mass, but c is a conversion factor between space and time, then what is d a conversion factor between? What dimensions does d have? Other than "Maxwell's demon constant", is there another name for d? Or is there not a 'd' after all, instead 'c' appears once again in the information-energy equivalence fomula e=i*f(c)? Or is there not a constant here at all and the situation is more complicated? Living things have evolved to exploit all kinds of opportunities - even quantum effects in the form of photosynthesis. Living things have senses to extract information from the environment, but do living cells utilise any demon effects to turn any of this information into energy ?


title: What if the Universe is made of recursions?

text: What do you guys think of Scott Adams' theory that the Universe might be made of recursions, created by observation? I find it fascinating but have no strong opinion yet if it makes sense.


title: Why should the mass of leptons to be near of proton and QCD/chiral scales?

text: The mistery of the mass of the top being in the electroweak scale can be justified by the Higgs mechanism itself; in some sense the top mass is the only "natural" mass, the other masses of fermions being protected by some unknown mechanism so that they are "zero" relative to the electroweak: the typical values of the Yukawas are in the $10^{-3}$ range and even smaller. The masses are still "natural" relative to GUT or Planck Scale cutoffs, because then their corrections are logarithmic and about the same order of magnitude that the mass itself, in fact about a 30%. My question is about the mass of tau and mu. Is there some reason for them to be in the GeV range, where QCD masses -proton and pion, if you wish, or glue and chiral scales- are? I am asking for theories justifying this. For instance, Alejandro Cabo tries to produce first the quark masses from the one of the top quark via QCD, very much as Georgi-Glashow electron-muon in the early seventies, and then he expects that the leptons should have masses similar to the quarks via the electromagnetic/electroweak coupling (albeit some group theoretical reason could be enough). * * * Hierarchically, and from the point of view that any near zero value for a coupling is a hint for a broken symmetry (so that in the limit where this symmetry is unbroken, the coupling is exactly zero), what I would expect is a symmetry protecting all the yukawas except the top, and then still a subgroup protecting the first generation.


title: Distinguishing mechanical systems from general dynamical systems

text: [Remark: I admit that my first attempt on What makes a space a real space? was rather ill-posed and led to some confusion. Sorry for that, but please give me a second try. Part of the confusion arose from the ununsual term "real space". The far better term would have been "physical space" which I hope to be unambiguous.] In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space. Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a mathematical space. **Question 1** > When would you assume that $q$ describes the evolution of a mechanical > system? I would, if I found or was given a function $\mathcal{L}(x,\dot{x},t)$ such that the Lagrange equations hold for all times $t$ when inserting $q(t)$ for $x$ and $\dot{q}(t)$ for $\dot{x}$, and which I can consider physically plausible. In this case I would interpret $\mathcal{M}$ as physical space, maybe distorted by generalized coordinates. I would _not_ , if there is provably no such function. In this case $q$ describes the evolution of a general dynamical system, but not a mechanical one. **Question 2** > Are there mathematically definable conditions on $\mathcal{L}$ for being > physically plausible, or possible, let's say? Besides, maybe, that $\mathcal{L}$ must be of the form $\mathcal{T} - \mathcal{V}$. **Question 3** > Can anyone give an example of a non-mechanical system that _behaves_ like a > mechanical one (in the narrow sense above)? One answer to this question comes immediately in mind: a system that is _designed_ to behave like a mechanical one, e.g. a computer running a simulation. I'd like to exclude such cases if I could, and restrict the question to "natural" systems. **Question 4** > What about possible worlds with a physical space like ours but in which the > Lagrange equations do not hold?


title: Ohms law and multiple resistors

text: In my physics class, we did an experiment in which we supported Ohm's law. We used a simple circuit that includes one resistor. The relationship of I = VR, and variants thereof, became apparent. We mainly looked at the relationship between potential difference and the current in series. What we have not done is see if this relationship is apparent in a series circuit with multiple resistors. I've did further research and read about Non- ohmic resistors, do they have anything to do with this? I can't really make the connection. So my question is, what is the relationship between the potential difference across two resistors in series and the current through them? We are moving towards parallel, but I would like to know this question better.


title: Does the image forced barrier lowering appear wnen the device is in Forward bias?

text: Because of the image forced barrier lowering the barrier potential reduces in reverse bias . What happens to barrier height (potential) in forward bias??


title: Is Palladium an exception?

text: I have been taught in school that atoms cannot have more than 8 electrons in the outer shell. Palladium atom's electron configuration is 2,8,18,18. Why isn't it 2,8,18,17,1 like the case of Platinum 2,8,18,32,17,1 or Nickel 2,8,17,1 and they are all in the same group?


title: Limit on geothermal energy that could be extracted before the earth's magnetic field collapsed?

text: This is more of a theoretical thought-experiment question. Basically, how much geothermal energy can we extract before the loss of the magnetic field makes it a terribly bad idea? Will the geothermal energy inside the earth now, which is lost through "natural processes", last until the end of the Sun (say, another 5billion years?) And if so, how much excess is leftover? a) do we know how much thermal energy there is? Call this: A b) do we know at what point the core would stop producing a magnetic field? (ie: is it when the iron in the core is no longer molten?) Call this: B c) do we know the rate that heat is radiated out naturally? Call the annual rate: C So I guess: A-B - (5.0 * 10^9 * C) is the amount of heat for "safe" geothermal extraction? also: Considering such extreme timescales -- are there any other factors adding heat into the Earth? Maybe tidal interactions with Sun/Moon/etc add anything over billions of years? Thanks!