title: Matrix and exponential term problem

text: We know the Schrodinger equation for free Hamiltonian is : $$ i\hbar\frac{\partial\psi}{\partial t} = H_f \psi $$ the wave function could be written as $$ \psi(x,t)=\hat{S}(t) \psi(x,0) $$ $$ \hat{S}(t)=\exp\left(-\frac{i}{ \hbar }H_ft\right ) .....(1) $$ I want to know that how exponential term arisen in the equation (1) and how can we write the equation below . Please show me the matrix manipulation for the below equation . $$ \hat{S}(t) = \tau_3 S^\dagger \tau_3 $$ Thanks .


title: Could general relativity and gauge theories in principle be covered in one course?

text: It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the gauge group and the gauge bosons "$A_{\mu}\times T^i_{\ j}$" are much simpler than the relation betreen the diffeomorphisms/isometries and the Christoffel symbols $\Gamma_{\mu\nu}^\rho$. At the very least, at first glance, I don't see the $SO(3)$ matrix lingering in $\Gamma$. I don't really know the technicalities of the Erlangen program and I specifically don't know how that "generating a manifold from a single point using the isometry group" (as I understand it) really works. I'd like to know if it is really directly possible to view both, that is general relativity and the classical gauge theories, just as special cases of the general Cartan connection framework, or if there is a fundamental difficulty to (I guess) general relativty which makes this not work. Especially if the answer is no, then what is it that makes the metric theory so different? (At first I'd guess it's the fact that the GR manifold looks different in each patch due to $g$, but $A$ can also be difficult and both are related to what else is _in_ space. So I suppose it's the interplay between the isometry group and the bigger diffeomorphism group.)


title: Can wireless power transfer harm organic life by anyway?

text: Now, humanity can beam 50-60 watts power through 10 centimeter-thick concrete block. Can such wireless power transfer harm us by anyways?


title: Which topics & books should I read to understand whirlpool, tornado, hurricane etc.?

text: I want to be expert to understand physics behind whirlpool, tornado, hurricane. Which topics are needed for this?


title: Normalizing a Wave Function

text: How would I normalize the wave function: $ \psi (x)$ = $Ce^{i\rho_0x/\hbar}e^{-|x|/2\Delta x}$? I squared it, which got rid of the imaginary part-Then I considered breaking up the absolute value-but this didn't seem to be integrable. I considered also using one of the exponential integrals from negative infinity to infinity, but I didn't seem to have one that fit the form, and I was uncomfortable using an integral I haven't shown to be true.


title: Can entropy be equal to zero?

text: I've searched for it but I only found contradicting answers from "scientists": Dr. David Balson, Ph.D. states: "entropy in a system can never be equal to zero". Sam Bowen does not refutes the following affirmation: "It is know[n] that entropy is zero when a pure crystalline substance is at absolute zero". He says there's residual motion in particles, but he doesn't say clearly that 0 entropy isn't possible. I this link is a long discussion about that, it isn't really conclusive, but most people seem to agree that entropy can be zero. I'm not very receptive to the idea of zero entropy.


title: Gravitation force- Attraction and repulsion

text: Gravitation force is always attractive. Now assume(not a practical one): I took our Earth in my hand and started shaking up and down. This will create a disturbance in space-time warp and it moves like a wave outward direction. So when the crest part of this disturbance comes in touch with another mass, it should carry that mass with it and start moving away from Earth. It was visible because I used a very massive object like Earth. Though it is not visible it should be true with low mass objects too? So is gravitation field is always attractive?


title: Thermal imaging camera

text: I'm making a tea bag experiment (Make a hollow cylindrical tube from light paper e.g. from an empty tea bag. When the top end of the cylinder is lit, it takes off.) I need to know how hot air is moving so please help me if you have thermal imaging camera or with out it write your opinion


title: Is the superposition principle universal?

text: In David J. Griffiths' _Introduction to Electrodynamics_ , he claims that the superposition principle is not obvious but has always been found to be consistent with the experiments. So I was wondering have we found some physics quantities which do not follow superposition principle? If we have not till now why can't we generalize and make it into a law? More specifically: Griffiths was talking about electromagnetic force. My question is about the existence of something like mass or charge and which doesn't follow this superposition principle.


title: Heat transfer from hot water through a copper pipe to oil in a tank

text: I want to build a geyser attached to a pump with copper tubing running into an oil tank. This is done because the oil solidifies at a temperature below 15 degrees Celsius. I have done a few calculations to determine the amount of time it would take to heat the oil, but i am not sure if they are correct. Pumping velocity = $3m/s$ Water temperature = $60C$ oil temp =$ 5C$ Desired Temp $20C$ 15000 liter oil tank 20mm diameter copper tubing was used 6m of tubing is submerged into the oil Calculating the heat transfer coefficient: $$Re=(C.d)/Vf$$ $$ =(3x0.02)/0.475x10^{-6}$$ $$=126 315$$ thus the flow would be turbulant $$Nu=0.023Re^{0.8}xPr^{0.33} (0.33 for cooling)$$ $$Nu=397.98$$ $$Nu=(α.d)/λ$$ $$α=(Nu.λ)/d$$ $$=(397.98x654.3x10^{-6})/0.02$$ $$α=13.02kW/m^2.K$$ I then used the following equation for the heat transfer inside the tank: $$Q^*=αA(deltaT)$$ $$Q^*=13020xπx0.02x6(60-40)$$ $$= 31.248kW$$ Energy required to heat the oil: $$Q=mcpT$$ $$Q=15000x1800x(20-5)$$ $$Q=405000J$$ Heating time: $$t=405000/31.248$$ $$t=3.6 Hours$$ Can someone please tell me how you would do the calculations. Also please assume any information not provided. Thank you.


title: Is battery capacity proportional to Ampere-hour in every case?

text: Suppose I have two batteries one inside the laptop with 2.5 AH giving a backup for around 2 hours. The battery is external inside a UPS system with 7.5 AH. Since it has thrice the capacity of the previous one, can I conclude it will give backup for 6 hours?


title: Dominos vs. Conservation of Energy

text: In this video a single flick of a finger tips 116000 dominos. Domino video I understand the work that needs to be done to move 116000 pieces (at least 100 kilos) of plastic is greater then that expended by flick of a finger. How does the energy conservation law apply here?


title: Formula of Gauss' Law of Gravitation

text: Gauss's law for Gravitation: $$\int g\cdot \mathrm{d}S=4\pi GM$$ where $g$ is the gravitational field and $S$ is the surface area. Am I correct?


title: Milky Way Voyeurism

text: When and where on earth do we get the best view of the Milky Way? I believe it helps   * if it is night   * if it is winter   * to look in the direction of the galaxy center I think that this information from wikipedia: > The Galactic plane is inclined by about 60 degrees to the ecliptic (the > plane of the Earth's orbit). is needed, but I can still not figure it out. I am hoping to get a general sense of what to expect _when and where_ , so that I can make sure to enjoy the view on future travels. **Edit:** From the answer of @Johannes and corresponding comment of @WetSavannaAnimal aka Rod Vance, I believe that a good time and place is December to February on the southern hemisphere. So what is the optimum? New years eve on the South pole?


title: how must i understand this 2-loop integral?

text: let be the 2-loop integral... $$ \int d^{d}l\int d^{d}k \frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \int d^{d}l\frac{1}{(k+l)^{2}}=G(k,d)$$ this is the regularizñation of the subdivergence assumed to exist by dimensional rgularizaton then the final step would be $$ \int d^{d}k \frac{G(k,d)}{k^{4}(k+p)^{2}}$$ where $ G(k,d) $ must be taken in finte part to eliminate the pole when $ d=4-\epsilon $


title: Integration over $S^2$ in electrostatics

text: I'm studying for a test in electrostatics and I'm always failing on putting up the correct integrals. In one problem I have the surface of a sphere with radius $a$ and an opening angle of $2\theta$. There is a total charge $Q$ evenly distributed on the surface. I want to calculate the potential. In order to calculate the potential I first have to calculate the surface charge distribution, $\sigma(r)$, and this is where I always fail. I've done like this: $Q=\sigma A=\sigma 4\pi r^2$ Then I tried to convert this into spherical coordinates by changing $r$ into $r=a \sin\theta \cos\phi +a \sin\theta \sin\phi +a \cos\theta$ The correct integral should be $Q=\sigma a^2\int_0^\theta \sin \theta d\theta \int_0^{2\pi}\\!d\phi$ Can someone explain how to put up integrals like these?


title: Are the axial electric field lines of a dipole the only ones that extend to infinity?

text: Consider an electric dipole and its electric field lines. There will be many field lines that do not extend to or originate from infinity, but rather begin at the positive charge and loop back around to terminate at the negative charge. However, the field lines exactly on the dipole axis will extend out to infinity (or originate from infinity). But are these the only field lines that extend to infinity, or are there other off-axial lines that also go to infinity? How can one prove this? I have tried writing out the differential equations for the field lines, but they seem impossible to solve. No heuristic argument has satisfied me either. One can assert that all off-axial electric field lines must curve away from the axis as one goes to infinity, but does this curvature ultimately cause them to curve all the way back or do they reach some sort of linear asymptote?


title: Radiation and absorption

text: Can thermal radiation from a cooler object (B, which emits longer wavelength radiation) ever ADD to the overall thermal energy level of a warmer object (A, which emits shorter wavelength radiation)? Subsidiary question: What exactly happens - at the molecular level - to the longer wavelength radiation from B as it arrives at A?


title: Parity of proton is 1?

text: I have found from Wikipedia that "a parity transformation is the flip in the sign of spatial coordinates". Now when we operate parity operator, does that mean we are taking any physical entity at ${\bf x}$ to $-{\bf x}$. Or we are just reverting axes of the co- ordinate system? However if we take parity transformation as active transformation then what does it mean that parity of proton is 1? Doesn't it anyway depend on the **origin** of the co-ordinate system? Please elucidate the meaning of the parity. I get too much confused whenever I hear "parity"!


title: PDE from dispersion relation?

text: Suppose I have knowledge of a system's dispersion relation $f(\omega,k)$. Is it possible to recover the underlying PDE describing the system? Can I simply use the replacement $k=-i\nabla$, $\omega=i\frac{d}{dt}$ to go back? I came across one source which claimed that the original PDE could only be recovered to a certain extent. An example given was the Dirac and Klein Gordon equations which both satisfy the same dispersion relation. But I didn't quite follow. Example: I have a polynomial dispersion relation of the form $\omega^2+c^2k^2=1$. Can I automatically say that the underlying PDE is $\frac{d^2u(x,t)}{dt^2}+c^2\nabla^2u(x,t)+u(x,t)=0$? Or is there a subtlety I'm missing? Thanks


title: Vertex operator - state mapping in Polchinski's book

text: In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure 2.6(b), so operators $\alpha_m^\mu$ with $m>=0$ vanishes. I am a bit confused about why $\partial X^\mu$ has no pole inside the contour. Before this section $\partial X^\mu$ always has the singularity part ($1/z^m$). Therefore would it be possible for you to give a more mathematical argument what condition requires $\partial X^\mu$ having no poles in this case? Thanks a lot for your time!


title: A calculation problem on Conformal Field Theory

text: This problem is on Di Francesco's book I. It's exercise 7.1: Calculate the norm of the following vector, where $\lvert h\rangle$ is the state of highest weight. $$L_{-1}^n\lvert h\rangle$$ I have just tried to use the commutation relations of the $L$ operators and the fact that $L_1$ acts on $\lvert h\rangle$ is $0$. But as the calculation goes on, things began to be troublesome. I just found them too complicated. Commutation relations: $$[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}\delta_{n+m,0}n(n^2-1) $$   It is in fact asking us to calculate: $\langle h|L_1^nL_{-1}^n|h\rangle $. And we have the following relations: $$\langle h\rvert L_{-1}=0 \quad\mbox{and}\quad L_1\lvert h\rangle=0. $$


title: The Euler-Poincare equation

text: Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone derive either equation from the other? Finally I would be jolly grateful if anyone had a good BASIC source of information at an introductory level to the subject.


title: Is polarization of a wave just a description of its motion in three dimensions?

text: Since a polarization of the wave is described by complex numbers, we can try to give that mathematical formalism geometrical meaning. With having two different axes, one imaginary and other real, it is possible to represent the motion of the wave in three dimensions as the motion of 2 waves in two planes. These two planes are being observed in three-dimensional space from the point in the third plane which is perpendicular to the two other planes. This third plane is actually a complex plane, from whose perspective these two planes are viewed as lines. This also means that the notion of a complex plane just refers to the plane which is an auxiliary construction, on which coordinates of a point in two other planes are projected. The coordinates in each plane correspond to their motion in two dimensions, parametrically described, where one of two parameters is time. This can be mathematically interpreted as a function where y=y(x). These two equations with two variables for each one, or the lines as viewed geometrically are real and imaginary part of a complex number. Each part is showing the two-dimensional projection of a curve in three dimensions. If we can observe these two projections as independent of each other, it is possible to describe this curve in three dimensions. This is done by introduction of mathematical abstraction, a complex number, where imaginary and real part are independent of each other. If our attention is focused only on single point in this three-dimensional world, we can just determine its location from three spatial coordinates. In other words, there are two equations sharing the connection with same parameter. Uniqueness of that parameter arises from setting the same beginning of both planar coordinates system at the place where complex plane intersects two other planes. Therefore, the solution of the equation are the coordinates of the point in three-dimensional space (x, y, z). But if we bring in another dimension to an observed object, i.e. if we make that point a line (given algebraically as the function with two variables), we get a new, fourth variable or fourth dimension. That dimension is time, from the viewpoint of a physicist. Is this geometrical interpretation of complex number description of polarization correct?


title: Can't virtual images be put on a screen?

text: Apparently the plane mirror makes virtual images, and virtual images cannot be projected onto a screen. If you look below, it is? ![Virtual Image projection](http://i.stack.imgur.com/vg05N.png) If the eye is basically a screen of rods and cones, then everything is a real image? I'm guessing I'm wrong, but how? Thanks


title: Current in an RC circuit inside in a magnetic field

text: Does my question make any sense? * * * **Context** : The capacitor is charged with charge $Q_0$. There is a magnetic field $B=B_0 \hat{z}$ perpendicular to the monitor's screen. The length of the (AB) wire is $\ell$. There is no friction. ![enter image description here](http://i.stack.imgur.com/kfsS5.png) **Goal** : I am trying to calculate the velocity of the (AB) wire. From the 2nd law of Newton, we have: $$m \frac{du}{dt}=F_\text{Laplace} = B\ell I(t)$$ In my lecture notes I see that $I(t)$ is replaced by the current $I_c(t)$ which corresponds to the current due to the discharge of the capacitor, **as if there was no magnetic field, i.e. as if $\mathcal{E}=0$**. So it goes like: \begin{align} V_c(t)&=V_0 \exp(-t/RC) \Rightarrow\\\dV_c/dt &= -V_0\left(\frac{1}{RC}\right) \exp(-t/RC)=-I_c(t)/C \Rightarrow\\\I_c(t) &= -C dV_c/dt \end{align} And then substitutes $-C dV_c/dt$ in the place of $I(t)$ at the Newton's law: \begin{align} m \frac{du}{dt} &= B \ell \left( -C dV_c / dt \right) \Rightarrow\\\ du &= -\frac{B C\ell}{m} dV_c \Rightarrow\\\ u(t) - u(0) &= -\frac{BC\ell}{m} (V_c(t) - V_0)\Rightarrow\\\ u(t) &= \frac{BC\ell}{m} \left(V_0 - V_c(t)\right) \end{align} **Questions** : Why use $I_c(t)$ instead of $I(t)$? We do have an induced electromotive force so it should be: $$I(t) = \frac{V_c(t)-\mathcal{E}(t)}{R}$$ What am I missing here? * * * After going through all the calculations he ends with the following differential equation: $$ I(t) = \frac{V_c(t)-\mathcal{E}}{R} = \frac{V_c(t) - u(t)B\ell}{R} \Rightarrow\\\ \frac{dV_c}{dt} + \underbrace{\left(\frac{\ell^2 B^2}{m R} + \frac{1}{RC}\right)}_ {k_1}V_c(t) = \underbrace{\frac{\ell^2 B^2}{m R}}_{k_2}V_0 $$ With the following solution: $$ V_c(t) = V_0 \left(1 - \frac{k_2}{k_1}\right)e^{-k_1 t} + \frac{k_2}{k_1}V_0 $$ And after some calculations the velocity $u(t)$ is given by: $$ u(t) = \frac{BC\ell}{m}V_0 \left(1 - \frac{k_2}{k_1}\right)(1-e^{-k_1 t}) $$ with $k_1,k_2$ being constants depending on $B,C,m,R$, etc. This equation looks weird because there is a combination that leads to zero velocity ? ($k_1=k_2$) I'm at a loss here!


title: Geometric mechanics - Symplecticity

text: I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the topic. I'm sure this is a common question but I would like help with a couple of specific points:   * What would be the main points to take away from the "symplectic structure of phase space". Specifically what does knowing its symplectic do for us? We quote it all the time when talking about Hamiltonian mechanics/Liouville equation/Poisson brackets... etc ... In other words what would I be saying in differential geometry terms when I say " a canonical transformation preserves the symplectic structure?" I understand this may have been answered before but so far having seen the previous question/answers I'm still struggling. I'm specifically after answers relating to classical mechanics linear/non-linear alike, but not relativity if it can be avoided. So to summarise: what are the salient points of a geometrical understanding of classical mechanics and what does this do that a basic understanding doesn't!


title: How do objects heat up?

text: If every body emits radiation at a given frequency and temperature exactly as well as it absorbs the same radiation, how do objects heat up?


title: Mandelstam variables 1 positive 2 negative

text: The three Mandelstam-variables are defined as: $$s=(p_A+p_B)^2=(p_C+p_D)^2,$$$$t=(p_A-p_C)^2=(p_B-p_D)^2$$$$u=(p_A- p_D)^2=(p_B-p_C)^2.$$ Where A and B are the incoming particles and C and D are the outgoing particles, and the second equation follows from the conservation of 4-momentum. The Mandelstam-variable $s$ gives the center-of-mass energy and is thus always positive, now for the Mandelstam-variables $t$ and $u$ the demands can be found by looking at them in the laboratory-frame, after an easy calculation it can be shown that:$$s\geq \max[m_A^2+m_B^2,m_C^2+m_D^2],$$ $$t\leq \min[m_A^2+m_C^2,m_B^2+m_D^2],$$ $$u\leq \min[m_A^2+m_D^2,m_B^2+m_C^2].$$ This means that there should exist a situation in which the three Mandelstam- variables are positive. I was wondering is such a situation exists, or that I am simply overlooking some facts and that indeed only ONE Mandelstam-variable can be positive? > To elaborate more on the calculation which lead me to this conclusion, I've > calculated these variables in the frame in which particle A is at rest > (since $s$, $t$ and $u$ are invariant I can do this). This yields for the > momenta: > $$p_A=(m_A,0),$$$$p_B=(E_B,\vec{p}_B),$$$$p_C=(E_C,\vec{p}_C),$$$$p_D=(E_D,\vec{p}_D).$$ > So for my Mandelstam variables I would get that: > $$s=(p_A+p_B)^2=p_A^2+p_B^2+2p_A\cdot p_B = m_A^2+m_B^2+2m_AE_B,$$ $$t=(p_A- > p_C)^2=p_A^2+p_C^2-2p_A\cdot p_C = m_A^2+m_C^2-2m_AE_C,$$ $$t=(p_A- > p_D)^2=p_A^2+p_D^2-2p_A\cdot p_D = m_A^2+m_D^2-2m_AE_D,$$ where I used the > first equality in the definition of the Mandelstam variables. I could do the > same calculation with the second equality, if I would then discard the terms > of the form $2m_AE$ I would get the inequalities above. $\,$ > > Edit: I know that the second equality doesn't give simple terms $2mE$, but > terms of the form > $$E_CE_D-\vec{p_C}\cdot\vec{p_D}=E_CE_D-|\vec{p_C}||\vec{p_D}|\cos(\theta),$$ > and since $E^2=m^2+|\vec{p}|^2$ we have that $E>|\vec{p}|$, so these terms > are positive. $\,$ > > Edit 2: I've calculated these for the non-elastic proces > $e^-e^+\rightarrow\mu^-\mu^+$ (since inelastic problems are the source of > the problem) and I found that: $$s=2m(m+E_B),$$ $$t=m^2+M^2-2mE_C,$$ > $$u=m^2+M^2-2mE_D,$$ where $m$ is the mass of the electron and $M$ is the > mass of the muon, particle $A$ (either an electron or positron) is standing > still and for particle $B$ (electron or positron) and $C$ and $D$ (the 2 > muon particles) the energies are $E_B$, $E_C$ and $E_D$. The variable $s$ is > obviously positive, to check if $t$ of $u$ are positive I summed them and > used the conservation of energy: $m+E_B=E_C+E_D$, which gives: > $$t+u=2(m^2+M^2)-2m(m+E_B)=2(m^2+M^2)-s,$$ now in order to be able to create > 2 muons the energy should be high enough, hence $s=(2M)^2$, filling this in > yields: $$t+u=2(m^2-M^2).$$ Which brings me back to the question: ''Should > only one of the three variables be positive (just as most books claim) or > are there special cases''? In my derivation above I used the conservation > laws, but as you can see in this example the possibility of $t$ and $u$ > being positive (for small enough $E_B$) vanishes by demanding that de center > of mass energy $\sqrt{s}$ is large enough, will this always be the case ?


title: Why there are no uuu and ddd baryons with spin 1/2?

text: What is preventing $Δ^{++}$ and $Δ^-$ baryons from going to a lower-energy state with spin 1/2 similar to that of protons and neutrons? I don't think the Pauli exclusion principle can prevent it because the quarks have different colors. The whole purpose of the quark color is to allow more than one quark to be in the same state. What's so special about protons and neutrons? What allows them to have lower energy compared to $Δ^+$ and $Δ^0$?


title: Tear drop shape

text: i heard that a tear drop shap is the most aerodynamic shape possible or the best is this true? If this isn't true what is since i need to make a fast ROV? Also since i need to have a propeller to propel it should it be on the front or back to produce the most thrust and the best hydrodynamcics?


title: Integral ambiguity

text: I'm a bit confused with some notation I encounter in physics calculus. Consider this: ![enter image description here](http://i.stack.imgur.com/rAssQ.png) Taken from here. Integration operates on functions, correct? What does it mean to integrate $\frac{d{\bf p}}{dt} dt$? Could someone point me to some reading about the manipulation of Leibniz notation?


title: What is the density and energy of a photon?

text: As I understand, photons are considered mass-less, which is a necessary condition for moving at the speed of light. However, does that mean their density is 0, as they will occupy some volume. If their density is zero, that means there is no matter inside a photon. Thus, shouldn't a photon be able to pass through matter instead of colliding with it? As $E = mc^2$, shouldn't a photon have zero energy, as it has zero mass?


title: Hidden momentum

text: I'm trying to learn about hidden momentum. After reading what I could find with a google search, I understand that it is equal to the momentum carried by radiation, calculated with the Poynting vector. I can't really understand how the momentum conservation laws apply to the situations in which there is a magnet(current through the loop) and in which this magnet is turned off. How should i write these laws?   I think I somehow understand the origin of hidden momentum, but I don't see how everything sums up in both situations. Could you please help me understand?


title: Clarification of Tensor, Jacobian

text: Is this correct? Tensors are linear mappings between two coordinate systems on a manifold. The elements of that mapping (which include the different changes of bases at each point of the manifold) are governed by the components of the Jacobian. The Jacobian, the ratio of the volume elements of the two states – is itself a tensor. (I expect this is elementary for those who know – I’d like to reach that state myself, so please feel free to expand).


title: Titanic Question

text: I was reading this question: Was Titanic's captain's decision correct? @oakad commented about a Russian vessel keeping itself afloat by going in reverse, using hydrodynamic pressure to keep its damaged bow high enough to prevent fatal flooding. This got me wondering... My main question(s): The Titanic stopped not long after it collided with the iceberg. Could she have stayed afloat a bit longer if she kept moving, either in forward or reverse? How much longer (or shorter, as the case may be?) Would moving have had a net beneficial effect? For example, would the benefits of moving be outweighed by an increased rate of flooding due to the flowing water further widening the gash, or a reduced rate of pumping because less steam was available to be diverted to the pumps? What are all of the phenomena that would have an effect on this scenario?


title: Geometric Interpretation of Rotated basis of Hamiltonian and collective Dicke states

text: Suppose I start with a basis of states for a two spin-1/2 particle system, namely, $\\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, \left|\uparrow\downarrow\right\rangle, \left|\downarrow\uparrow\right\rangle \\} $, and then I apply a Hamiltonian that gives me the new basis, $\\{\left|\uparrow\uparrow\right\rangle, \left|\downarrow\downarrow\right\rangle, \cos(\gamma)\left|\uparrow\downarrow\right\rangle +\sin(\gamma)\left|\downarrow\uparrow\right\rangle, \sin(\gamma)\left|\uparrow\downarrow\right\rangle -\cos(\gamma)\left|\downarrow\uparrow\right\rangle \\} $ Which gives us a transformation matrix linking the original basis kets to the new basis kets which corresponds to an inversion and rotation of the last two kets in the original basis. A few questions: Is there any physical significance to the inversion, or is it just an "arbitrary phase"? If not, what is the most convenient way to see a geometric interpretation of the state? What would be an arbitrary collective Bloch vector for this space? I have heard of so-called Dicke states, in which a system of N spin-1/2 particles can be thought of as a single spin-N/2 particle, in state $\left|N/2,M\right\rangle $, where $M= -N/2,-N/2+1,...,N/2-1,N/2$. Does this entail the use of a collective Bloch vector for all possible states solely excluding the singlet state $\left|0,0\right\rangle$ ?


title: Imaginary voltage in simple RC circuit

text: As a homework assignment, we have to find $U_a(\omega)$, that is the voltage that drops over the right resistor in relation to the frequency $\omega$ of the input AC voltage $U_e$. ![](http://wstaw.org/m/2012/07/01/blindleistung3.png) For the two extreme cases, $\omega \to 0$ and $\omega \to \infty$ I expect $U_a = 0$ since in the $\omega \to 0$ case, the capacitors will block the whole current. In the $\omega \to \infty$ case, the left capacitor will short the whole thing so that the right resistor does not get any current through it. To get the voltage $U_a(\omega)$ I tried these steps:   * Calculate the total impedance of the circuit $Z$: $$ Z(\omega) = R + \left( \frac 1{Z_C(\omega)} + \frac 1{Z_C(\omega) + R} \right)^{-1}$$ The impedance of the capacitor with capacitance $C$ is: $$ Z_C(\omega) = \frac{1}{i \omega C}$$   * Calculate the total current: $$ I(\omega) = \frac{U_e(\omega)}{Z(\omega)} $$   * Use the current divider rule to get the current that flows through the right branch (with capacitor and resistor). The current should be just this: $$ I_r(\omega) = I(\omega) \frac{Z_C(\omega)}{2 Z_C(\omega) + R} $$   * Multiply that current with $R$ and get the result: $$ U_a(\omega) = I_r(\omega) R $$ When I plot real and imaginary parts of $U_a$ against $\omega$, I get the following: (The top line is the real part, the bottom line the imaginary part.) ![](http://wstaw.org/m/2012/07/01/3.png) When I plot the absolute value of $U_a$, I get the following plot: ![](http://wstaw.org/m/2012/07/01/4.png) Which one is the current that is really measured? The real part of the absolute value? I have to find out the value of $\omega$ for which the output voltage is the highest. I tried $\frac{\mathrm d U_a(\omega)}{\mathrm d \omega} = 0$, but I only get $\omega = \pm \frac{i}{RC}$ as solutions, which do not really make sense to me. How can I find the actual maximum?


title: Does anti-lock braking system (ABS) perform reverse rotation sometimes?

text: In cartoons we can see, how vehicles brake by rotating wheels reverse. Will this be helpful sometimes in real situations? I know that the main tactics of ABS is to maintain tractive contact. But may be it will be better sometimes, at some speeds, friction and so on -- to rotate back? Does current ABS systems do something like that? Were any experiments on this performed?


title: Power and magnetism

text: If we have a solenoid and a magnetic field passes through it, a DC voltage will be produced in the wire. If we want to calculate the power, we find out the current using Ohm's law. I know there is power loss due to the resistance (joule effect). But what about the magnetic field due to current (lenz's law), does it contribute in the power loss?


title: Solution to the Maxwell's equation

text: Question goes as > Show that the solutions to the Maxwell's equations $$ \nabla \times \vec H = > \frac 1 c \frac{\partial \vec E}{\partial t}+\frac {4\pi} c \vec J, \hspace{ > 2 mm} \nabla \times \vec E = -\frac 1 c \frac{\partial \vec H}{\partial t}, > \hspace{ 2 mm} \nabla \cdot \vec H = 0, \hspace{ 2 mm} \nabla \cdot \vec E = > 4 \pi \rho $$ are given by $$ \vec E = -\nabla \phi - \frac 1 c > \frac{\partial \vec A}{\partial t}, \hspace{2mm} \vec H = \nabla \times \vec > A,$$ where $ \phi$ and $\vec A$ are scalar potential and vector potential > respectively which satisfy the equations $$ 1) \nabla \cdot \vec A + \frac > 1c \frac{\partial \phi}{\partial t} = 0 $$ $$2) \nabla^2 \phi - \frac 1 > {c^2} \frac{\partial ^2 \phi}{\partial t^2} = -4\pi \rho $$ $$3) \nabla^2 > \vec A - \frac{1}{c^2}\frac{\partial ^2 \vec A}{\partial t^2}= -\frac {4\pi} > c \vec J$$ Hint requested thank you!!   Also what kind of regions is described by $$ \nabla \cdot \vec E = 4 \pi \rho?$$ It is uniformly charged region?


title: Dirac field and stress-energy tensor density

text: I read somewhere that stress-energy tensor density is a symmetric tensor. But if I take the Dirac Field tensor: $$T^{\mu \nu}=i \psi^\dagger \gamma^0 \gamma^\mu \partial^\nu \psi $$ How could I demonstrate this property?


title: What is Supersymmetry (SuSy)?

text: In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles...etc.   * what is symmetry breaking?   * What is supersymmetry (SUSY)?   * What is spontaneous symmetry breaking?


title: How many bytes can the observable universe store?

text: Is the number of states in the Universe countable? What framework could be used to answer the question in the title?


title: Do individual photons increase entropy as they travel?

text: If I draw a control volume around a single light wave traveling through empty space, is entropy increasing as it goes? How is the degradation of the quality of energy manifested?


title: How to connect the dimension of perturbation constant with renormalizability

text: Let's have the Lagrangian $$ L = L_{0} + \lambda V , \qquad (1) $$ where $\lambda$ is constant which is small in the next senses: if $\lambda$ is dimensionless, it means that $\lambda < 1$; if it has dimension $[\lambda ] = l^{-n}$, it means that it is small in compare with characteristic energy of the free states: $\frac{\lambda}{ E^{n}} < 1$. How to connect the dimension of $\lambda$ with renormalizability of theory which is given by $(1)$? Is a dimensionless of $\lambda$ the necessary condition for renormalizability of the theory?


title: Deriving entanglement entropy from Renyi entropy

text: My questions are based on this paper - http://arxiv.org/abs/0905.4013   * Firstly I want to know as to whether some assumptions are needed about the relationship between the systems $A$ and $B$ for the Hilbert space to factor as tensor products as ("assumed"?) on page 3?  I mean assume the more common reverse scenario - if you are given a system C and you decide to call some part of it as $A$ and the rest as $B$ then does it automatically mean that the Hilbert space of C factors between A and B? (...that doesn't intuitively feel to be true...then what exactly is the assumption being made here?..)   * Secondly given the definition of $S_A$ and $S^{(n)}_A$ as in equations 2 and 3 how does this claimed equality follow that, $S_A = \lim _{n \rightarrow 1} S^{(n)}_A = - \lim_{n \rightarrow 1} \frac{\partial \rho^n_A }{\partial n }$ I am unable to see the proof of the above 2 equalities. It would be great if someone could help.


title: Alternate young double slit experiment

text: What will happen in young's double slit experiment, when instead of screen I put a black screen with a hole and a second screen behind the black one. Will it still form interference pattern albeit a faint one or no interference pattern will be observed?


title: Confusion about wavefunction separability

text: A wavefunction is inherently a multi-particle function. If you have a container that is perfectly isolated from the external universe (not possible, but just imagine it) and filled with $n$ particles, then the information necessary to perfectly describe everything there is to know about that system would be contained in $\Psi(\vec{r_1}, \vec{r_2}, ..., \vec{r_n}, t)$. The probabilities of all obtainable measurements could be derived from that function. However, I frequently read about wavefunctions of individual particles within a system such that the assumption $$\Psi(\vec{r_1}, \vec{r_2}, ..., \vec{r_n}, t) = \Psi_1(\vec{r_1}, t) \Psi_2(\vec{r_2}, t) ... \Psi_n(\vec{r_n}, t)$$ is made. One of my professors has said that individual particle wavefunctions are a more fundamental description of reality; however, I did not think this was correct because it would completely exclude the effects of quantum entanglement (then again, this was a graduate chemical engineering course). Where I'm going with this is: in what situations is it okay to separate the _complete_ wavefunction like that? How do you know that an isolated electron in an experiment isn't influenced by the rest of the particles in the universe? It's frequently said that when a measurement of a system is made, the wavefunction collapses into one of its eigenstates. But how is this possible if there is no such thing as a single-particle (or few-particle) wavefunction? Wouldn't it instead be the case that the overall $$\Psi\left(\text{particle in experiment} + \text{measurement apparatus} + \text{all other particles}, t\right)$$ has evolved in time? I realize my question is somewhat vague, but any kind of clarification on this issue would be appreciated.


title: Would superluminal signalling imply the violation of the No Cloning Theorem and unitarity?

text: The no-cloning theorem implies that we cannot use entanglement to send signals faster-than-light. Has anyone proved the contrapositive? That is, if we are given a system for superluminal signalling can we use it to violate the no- cloning theorem?


title: Hydrogen extraction efficiency of newish nano particle AA battery

text: The internet is ablaze with the new nano particle based extraction method of converting water to hydrogen. In 2002/2003 when I was 16 there was a similar craze about a lawn mower motor which supposedly ran on water. Indeed there were even conversion kits on the internet for vehicles. They didn't work because of a fundamental inefficiency of the conversion. Will this new nanotechnology improve the process so that cars can run on converted water or does there still exist a limit?


title: why some elements of same group cannot be used for doping semiconductors?

text: 15th group elements like As, Sb, P can be used for doping whereas N and Bi cannot be used for doping even though they too belong belong to 15th group. Why ?


title: What do you exactly mean when you say that momentum is conserved?

text: I am taking for granted that when we say that something is conserved it is understood ' _in its full integrity'._ **Energy** is represented by a number (of J, or other) and is usually conserved. But **momentum** is the product of a number (of Kg) by velocity which is a _vector_ , and therefore has a direction: p = m * v. In circular motion we say that speed is constant but velocity is constantly changing. If an **elastic ball** is thrown against a wall energy is conserved, the value of the speed is unchanged, but the direction of the vector is not conserved. In an **elastic collision** if a ball A (m =1) with v = +8 in the x axis hits a ball B at rest (M =3) , B will proceed ( v = 4 *3 = +12) and A will bounce in the opposite direction (p = v = -4). We conclude that: +12 -4 = +8, momentum is conserved. I am asking: the momentum of what is conserved? if it is the momentum of the system this is not true in the first case, if it is the momentum of the single balls this is not true in the second case. Language of science is and must be precise, I hope you will not consider this as hair-splitting, but I ask is there a general definition of 'conserved' valid for all instances? why do you apply such a broad/permissive interpretation of the principle in the case of momentum? Or, most likely, where did I go wrong? EDIT: What puzzles me is that in an elastic collision with the wall the ball takes all energy when it rebounds and there is 0 energy available for the wall and therefore the wall **must have momentum = 0** , ball A will have p = **-8** and W = **0** , and not +16


title: How to prove when electron spin is perpendicular to linear momentum, orbital angular momentum can't be 0

text: And when they are paralleled, orbital angular momentum is 0.


title: What is Work? What does the quantity suggest intuitively?

text: The mathematical formula for work says that work is force into displacement, but what is the philosophy behind it? I mean what does the quantity suggest?


title: The context of definition of work

text: It is unclear whether the work is defined considering opposite force on the same body or in ideal case where there is a single force acting on a body.If we consider ideal case where only a single force is present on the body, the displacement will not attain a specefic value. It will go on increasing and we cannot fix the value of displacement, I mean when do we say a force has worked on a body? Do we say this when a certain force has to' work' against another opposite force to cause displacement? I think the work is defined considering 2nd case since only then we can have a fix value of displacement because when one force is removed the other force will eventually stop the body.


title: Definition of Ampere

text: On Wikipedia it says: > This force is used in the formal definition of the ampere, which states that > it is "the constant current that will produce an attractive force of $2 × > 10–7$ newton per metre of length between two straight, parallel conductors > of infinite length and negligible circular cross section placed one metre > apart in a vacuum." In reference to the definition of an Ampere, why was $2 × 10–7$ chosen?


title: If I bury a cylinder (top end sealed) in sand, how much force do I need to pull it out?

text: Let's say   * the cylinder has a diameter equal to its height of $25\;\mathrm{cm}$   * the cylinder is sealed at the top and filled with sand   * the cylinder is also buried in the sand at sea at a depth of $10\;\mathrm{m}$ with only its top exposed   * the top has a handle for you to pull on   * the cylinder's weight is negligible ($0.0\;\mathrm{kg}$) How much force is needed to pull it out (excluding friction)? I just need the formulas to figure out how big of a cylinder I need to anchor a buoy. Precision is up to you. Please correct my computations if they are wrong. Assuming the density of wet sand is $1920\;\mathrm{kg\;m^{-3}}$:   * Volume of cylinder $= \pi r^2 h = 3.1416 \times 0.015625\;\mathrm{m^2} \times 0.25\;\mathrm{m} \approx 0.012272\;\mathrm{m^3}$   * Mass of sand in cylinder $= 0.012272\;\mathrm{m^3} \times 1920\;\mathrm{kg\;m^{-3}} = 23.56\;\mathrm{kg}$   * Weight of sand in cylinder $= 23.56\;\mathrm{kg} \times 9.8\;\mathrm{m\;s^{-2}} = 231\;\mathrm{N}$ * * *   * Water pressure at $10\;\mathrm{m}$ depth $= 2\;\mathrm{atm} = 200000\;\mathrm{N\;m^{-2}}$   * Surface area of top of cylinder $= \pi r^2 = 3.1416 \times (0.125\;\mathrm{m})^2 = 0.049\;\mathrm{m^2}$   * Pressure force on top of cylinder $= 0.049\;\mathrm{m^2} \times 200000\;\mathrm{N\;m^{-2}} = 9800\;\mathrm{N}$ * * *   * Total force $= 231\;\mathrm{N} + 9800\;\mathrm{N} = 10031\;\mathrm{N}$ Edit: Converted to newtons ($\mathrm{N}$). Thanks @KvdLingen! Note: This is not homework. I'm a freediver. I am literally thinking of burying a plastic cylinder into a patch of sand at a $10\;\mathrm{m}$ depth. At the top of the cylinder, a one way check valve will be attached to let water get out when burying and a release valve so I can pull it out. I will attach to the "handle" a rope and a buoy to the other end of the rope. The cylinder will serve as a temporary anchor. A real anchor can be easily pulled up by vertical forces when there are waves acting on the buoy. An anchor can also damage corals in dive sites--thus having a need to secure it in a patch of sand. I've seen this technique used in construction of underwater tunnels and oil rigs to fix the structure in place. It is called a vacuum-anchor (e.g. Troll A platform). I want to apply it in a smaller scale.


title: If the universe didn't expand faster than light, would our nights brighter like day?

text: There's a common QnA which has amused and inspired many kids: > **There are billions of Stars in the sky. If we combine lights received from > all stars, wouldn't it beat Sun? Why is night dark really?**   >  It's because speed of light is finite and lights from most of stars are in > the way to the Earth. My Question: If universe didn't expand faster than light in any point of time, would our nights brighter like day? Common sense says, Yes. But, unfortunately, universe is full of common sense busters. Is there anything else which can affect this? Note: I am talking about visible light only. I know, night is already bright in other wavelengths. **Update:**   I am clarifying the behavior setup of universe in If clause:   * Universe expansion speed is calculated classically with Super Cluster Red Shift. No GR globally separated reference frame glitch here.   * Universe is expanding today, but not accelerating. Similar to original Hubble's discovery except Inflation shouldn't accelerate universe to faster than light. My common sense reasoning:   * After universe became transparent to light, all luminous objects should continuously illuminate possible position of Earth until their death. As universe is young, death probability should be low.   * Till the time of birth of Earth and till today, light from most of slightly younger stars should reach Earth for continuous illumination. This is the reasoning of my common sense. If I am still not clear, let me know.


title: Dangerous levels of ozone from combined UVB lamps?

text: I have a reptil UVB lamp in my room, and a de-humidifier that uses a UV lamp to clean the air passing through it. Honestly since I got it working (the lamp was dead at first) my room is mold free. I just read that UV air purifiers were banned, and although the dehumidifier model is not in the ban notice, I wonder whether the addition of both the lamps could be too much ozone in my room. I got the lamp specs and the gov page stating a safe level of ozone should be under 0.050 ppm, but I don't know how to calculate the ozone generation by those devices. Could anyone calculate it for me? Reptisun 5.0 UVB lamp: http://zoomed.com/Library//ProductDBFiles/EPREP1109_ReptileLighting.pdf Gov page about ozone generators ban: http://www.arb.ca.gov/research/indoor/o3g-list.htm Extra link about required/safe levels, if you wonder (please fix the link. I couldn't post more than two) www [remove this space] landesbioscience [dot] com/pdf/01ReichrathHolick.pdf The dehumidifier description is www [dot] sunpentown [dot] com/sdmidewitiuv.html Thanks.


title: What is the reasoning behind hole carriers being able to carry heat?

text: In the Peltier effect, we consider charge carriers being able to carry heat. As for electrons or ions, this attitude makes sense, since external electric potential drives particles with mass in a direction and effectively transfers heat from one point of material to another. But for holes, this situation is only virtual, holes moving in a direction is only reformulation of the fact that the electron making up the hole environment are traveling opposite direction. I just cannot grasp the concept that the holes can indeed carry heat. I know one can argue that the holes have effective mass, but effective mass is only a measure of how much can you accelerate / decelerate the electron (or hole) in a crystal field. Can you help me out please?


title: Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

text: I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for this statement and trying to verify it myself I did not succeed. If I make a general ansatz with a spherically symmetric Gaussian wave packet with time dependent width and time and space dependent phase $$ \psi(t,\vec x) = (\pi a(t)^2)^{-3/4} \exp\left(-\frac{x^2}{2 a(t)^2} + i \phi(t,\vec x)\right)$$ and insert it into the Schrödinger equation $$ i \dot{\psi}(t,\vec x) = -\frac{1}{2m} \Delta\psi(t,\vec x) + \frac{k}{2} x^2 \psi(t,\vec x) $$ I get relatively complicated differential equations involving first time derivatives of a and $\phi$ as well as first and second order spatial derivatives of $\phi$. I failed to solve those equations or even show that a solution exists. Is there an easy way to show this? Is there any reference where this is shown? Which are the solutions for $a(t)$ and $\phi(t,\vec x)$ for a Gaussian (given initial conditions $a(0) = \sigma$ and $\phi(0,\vec x)=0$)?


title: Why does the tea in the middle of the cup spin faster when I stir it?

text: Try this: Make a cup of tea, add sugar, and stir it. Now add some milk, while it's still spinning. What I noticed is that the milk appears to move around a lot quicker in the middle of the cup of tea, than the outside. This seems to be the opposite of what happens on a vinyl, though it's the same as what happens with planets orbiting the sun. What's the explanation here?


title: Faraday tensor, antisymmetric electromagnetic tensor

text: I want to write $F^{\mu \nu}F_{\mu \nu}$ in terms of $F_{\mu \nu}F^{\mu \nu}$. How to do it?


title: Photon indistinguishability and beam splitters

text: $\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\ket}[1]{\left|#1\right\rangle}$ Suppose I have a beam splitter that will either reflect a photon by 45 degrees, or will allow the photon to pass directly through. If I send a single photon state through, that is, $\ket{1}_{initial}$, we have: $\ket{1}_{initial} = \frac{1}{\sqrt{2}}(\ket{0}_a\ket{1}_b + i\ket{1}_a\ket{0}_b)$ where $a$ and $b$ represent the paths that pass straight through, and the reflecting paths respectively. Now suppose instead of sending through a single photon, the initial state is two photons, that is, $\ket{2}_{initial}$. Classically I would suspect the state to look something like this: $\ket{2}_{initial} \propto (\ket{0}_a\ket{2}_b + \ket{2}_a\ket{0}_b + 2\ket{1}_a\ket{1}_b$) I have the factor of 2 in front of the $\ket{1}_a\ket{1}_b$ state because classically you could have the first photon passing through and the second photon reflecting, or vice-versa. Is this still correct for quantum mechanics, where photons are indistinguishable? Also, could you please show how to derive result that I have simply guessed for two photons (given the 1 photon case in the first equation)?


title: How to calculate explicit form of stress energy tensor in any situation?

text: I know that the components of stress energy tensor are: energy density, energy flux, momentum density and momentum flux. But can I explicitly calculate the form of stress energy tensor in any complicated situation? Or explicit form is known only for few systems? How do I do it?


title: How to calculate the force between line of charges?

text: As far as I know Coulomb's law only works for point charges but what if there are not any point charges? For example, let's imagine there are three rectangles with different sizes. First one is 50 cm, second one is 30 cm and last one ise 10 cm. The distance between first and second triangle is 10 cm and the distance between the second and third rectangle is also 10 cm. If their centers are collinear what is the net force acting on second ractangle? Should I calculate it by think their centers as a point charge? I searched for it but couldn't find a clear answer. Here is the picture of the problem for those who couldn't understand from my explanation. ![Charges](http://i.stack.imgur.com/OGsDC.png)


title: Finding the cosmological redshift of a galaxy in the expanding Universe

text: Firstly, I understand what the Doppler effect is when it comes to sound or light waves. From everything that I've read, we are told that the universe is constantly expanding since the all the radiation we observed is red-shifted. Assuming we are observing a distant galaxy/start that is moving away from us, the EMR waves incident on us from that galaxy is red shifted. My question is: how do we know that the light is red shifted? When you measure its wavelength you are given just ONE value Λ, right? How does one know what its original wavelength was to begin with? Its only after we know both the values (the actual wavelength when it was emitted and the wavelength that we measure hear on earth) that we can claim that the galaxy/star is moving away. I always assumed that one would observe both the wavelength of the photons and the energy and there is some sort of disparity there that tells us the light is red-shifted. But that does not make sense since one usually determines the wavelength based on the energy of the photon......I am confused here. Any answer will be appreciated. Thanks.


title: Magnetic field of a stationary electron

text: As far as I know, a magnetic field can only be produced by a moving electric charge, or from a particle's spin (this is how a permanent magnet works, all the spins are in the same direction) What is strength and direction of the magnetic field of a stationary electron at the origin with spin oriented straight up? I suspect a function of cylindrical coordinates is most convenient.


title: Interaction of solid objects and change of trajectory

text: I have two solid objects. Each of them has an arbitary complex surface, which is discribed by set of vertices. The aim is to describe their interaction, result of which is the change in trajectory of one of them. Example: one object (M1) is stable and fixed. The second object (M2) moves along some certain axis and has, for example, one degree of freedom perpindicular to the motion axis. In some moment M2 touches M1. Because of the degree of freedom, M2 changes its trajectory from line to some curve, the form of which depends on the form of two interacted objects. So, how it can be described in formulas? Of course, there can be up to 6 degrees of freedom.


title: Why doesn't water in water barometer boil?

text: I have read that the pressure in a water barometer at the top of the water column is around 0.5 psi and at such low pressures water should boil at around ~26°C (Room temperature). [1] [2] ![Water barometer](http://www.thermospokenhere.com/wp/02_tsh/B160___water_barometer/water_barometer.png) How come the water in the column near the surface doesn't boil?


title: Can quarks of different colors annihilate?

text: Wikipedia: "Antiparticles have exactly opposite additive quantum numbers from particles, so the sums of all quantum numbers of the original pair are zero." Is it possible to annihilate a quark-antiquark pair if the antiquark has a color different from the quark? Eg.: Will red up and anti-green anti-up annihilate? And if so, what is the product?


title: Why linear wave equation does not have solitonic solutions?

text: As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for solitons in nonlinear wave equation?


title: Relativity addition and signs

text: I have just covered a very brief module on special relativity as a part of my physics course. I have also done some extra reading mostly; Morrin's Classical Mechanics. While I found the book really illuminating in some aspects, I still feel that regardless of how hard I try there is something with relativity that prevents me form doing anything but the simplest questions. I was trying to pinpoint my problem and I think that a big part of it is velocity addition. I understand that the Galilean transformation would predict the $$V_{A}=V_{B}-V_{rel}$$ provided that A and B are two frames of reference. I also understand that we need to use the Lorentz transformation to get the velocity transformation in relativity; $$ \begin{pmatrix} c \Delta T_A \\\ \Delta x_A\\\ \end{pmatrix} \begin{pmatrix} \gamma & \gamma \beta \\\ \gamma \beta & \gamma\\\ \end{pmatrix} = \begin{pmatrix} c \Delta T_B \\\ \Delta x_B\\\ \end{pmatrix} $$ Transforming the velocity u measured in frame to frame B; $$u = \dfrac{\Delta x_A}{\Delta t_A} = \dfrac{v_B + u_{rel}}{1+\dfrac{v_B u_{rel}}{c^2}}$$ But as far as I understand we could equally reverse the frames A and B and simply transform the other way around which means we need the inverse of the transformation matrix; \begin{pmatrix} \gamma & \- \gamma \beta \\\ \- \gamma \beta & \gamma\\\ \end{pmatrix} This will yield the formula; $$u = \dfrac{\Delta x_B}{\Delta t_B} = \dfrac{v_A - u_{rel}}{1-\dfrac{v_A u_{rel}}{c^2}}$$. However since the naming of frames is arbitrary, how do I know which of the two formula to use, the one with the all plus and the all minus signs. I have tried to look on the internet for the explanation of this, but I could not find anything. Also provided that I know which equation to use how, do I use it what is the sign convention for the velocities? Thank you very much for all the help and sorry for the long post P.S. I would be also very grateful if someone could point me to some good and simple resources on relativistic dynamics especially collisions. Thanks again.


title: Question regarding a two block system and a viscous force

text: I'm just looking to set up a free body diagram for this scenario and I'm having a bit of trouble understanding what's going on. The problem states, "A rectangular block of mass $m_1$ lies on top of a (larger) rectangular block of mass $m_2$, which lies on a horizontal plane without friction. Between the two blocks there is a viscous force with magnitude $F=kv$, where v is their relative speed, which opposes their relative motion. At time t = 0 the larger block is at rest, while the smaller block moves in the horizontal x direction with initial speed $v_0$." My interpretation is that in between the blocks there is some fluid that when the smaller block moves, that fluid opposes the block's motion. Also, since the lower block is on top of a frictionless surface it will move as well and eventually reach the same velocity as the smaller block. Am I correct in that interpretation? So then the forces acting on the smaller block are $F_{net}=F-kv$. I'm not sure how to do the larger block. Any help is appreciated.


title: Faster spinning moon

text: Let's say we somehow speed up the moon's rotation. (I'm thinking about a big asteroid smashing into it, but I'm open to better ideas.) Here are some questions: How fast could we get the moon to spin before gravity stopped holding it together? (An answer in revolutions per minute would be nice.) At half that speed, what would the consequences be on the moon, on space missions to the moon, and here on earth? How many asteroids would it take to get it to spin at that speed?


title: energy required to start and then stop rotating an object, when we have gravity

text: Imagine an arbitrary object with center of mass at the red dot in the picture, rotational inertia from the point of the hanging: **I** , hanging from a fixed point, what is the energy required to start moving at constant angular velocity of $\ ω=ω_0 $ the object within a small amount of time $\ t=t_1$, from position A to Position B (angle=θ Rads), if gravity all this time is against our motion. ![enter image description here](http://i.stack.imgur.com/bmC3S.png)   i believe we need to exert a torque which will result the needed velocity in the given time, and then a torque which will cancel out the torque of gravity so we can keep the constant angular velocity. After these we will need to stop our body which means we should exert the necessary force against the motion, considering the "help of gravity" so we can stop the body in a small amount of time again. So we have a function of Torque, and we will integrate Torque•dΘ from Theta=0 to Theta=θ rads. Is that correct? If yes, can you help with formulation? suppose no air friction and axis friction, the body starts and ends at rest


title: U.S. Standard Atmosphere

text: My textbook states that "Ideally, we would like to have measurements of pressure versus altitude over the specific range for the specific conditions (temperature, reference pressure) for which the pressure is to be determined. However, this type of information is usually not available. Thus, a "standard atmosphere" has been determined that can be used in the design of aircraft..." "The standard atmosphere is an idealized representation of mean conditions in the earth's atmosphere" My question is, why can't we determine how pressure varies with altitude. Say we want to know the pressure at 1000 m. Why can't we measure the pressure at 1000 m for different temperatures?


title: Formation of atoms question

text: Could you please, explain to me the logic of the folllowing process as you would do to your 8 y/o sister: > Ubiquitousness and stability of atoms relies on their binding energy, which > means that an atom has a lower energy than an unbound system of the nucleus > and electrons. Where the temperature is much higher than ionization > potential, the matter exists in the form of plasma – a gas of positively- > charged ions and electrons. When the temperature drops below the ionization > potential, atoms become statistically favorable. Source: wiki How that that binding energy can be explained in plain English? Is it somewhat intrinsic to atoms, with each kind having its own level of binding energy? I understand that this can be translated in the language of closed systems using terms as Gnatt energy (G), etc. - but how?


title: Can ligthning produce a reverbed sound if it falls on a high-tension line?

text: Last night, a rather strange noise woke me up around 5 am. It was like if a normal thunder sound had been fed to a reverb/delay box. As I was getting back to sleep afterwards, I could hear the sound of rain. As I have been producing acid house tracks the whole day (it's just a hobby), using quite a lot of delays here and there, i'm rather skeptical of myself. Yet it seems I was woken up by a reverbed thunder sound (the reverbed part of the sound didn,t last longer than 5 seconds). As I live under high-tension lines, I'm wondering : if lightning falls on them, can it produce the kind of resonating metallic sound I heard last night ?


title: Lightspeed (invariance) measurement methods

text: I would like to know, how measurements of the speed of light are conducted these days, especially in the context of the invariance of $c$. Do all the methods involve mirrors to redirect the photons back to the place where they were emitted from? If not, could somebody, please, describe the method briefly? Again, what I am chiefly interested in are attempts to test the invariance, like the Michelson-Morley experiment.


title: Chemical potentials for multicomponent solids

text: I have a problem understanding how to deal with the chemical potential when two thermodynamic systems are in contact. For simplicity, let us consider $T=0\ K$. We have a system that is composed of two parts - I and II - where I occupies a space of fixed volume and II occupies the remaining space (essentially going to infinity in all directions). Both parts are initially filled with a crystal that is made up of a finite number of atomic types $i$. This is an equilibrium configuration (1). Now, I apply a long-range displacement field that originates in part I. If the boundary between parts I and II is permeable for the matter, some atoms move across this boundary. Here, part II can be thought of as a reservoir for the matter. The system eventually settles into another equilibrium (2), where the number of atoms $n_i$ in the part I is different from the above, i.e. there is nonzero $$\Delta n_i=n_i^{(2)}-n_i^{(1)}\ .$$ The free energy difference between the two equilibrium states of part I is then $$\Delta F \equiv F^{(2)}-F^{(1)} = V\sum_{ij}\sigma_{ij}\Delta\epsilon_{ij}+\sum_i\mu_i\Delta n_i\ .$$ In each of the two configurations (1) and (2), the parts I and II are in contact equilibrium, which means that $\mu_i^{I(1)}=\mu_i^{II(1)}$ and $\mu_i^{I(2)}=\mu_i^{II(2)}$, where the number in the bracket refers to the configuration. So, we have two chemical potentials for each $i$, one corresponding to configuration (1) and the other (generally different) for the configuration (2). Moreover, the particles move into I from the immediate neighborhood of the boundary I-II, which implies that $\mu_i$ should correspond to the atoms in the part II that are close to this boundary. However, how am I to understand a single $\mu_i$ in the equation above?


title: Quantum Translation Operator

text: > Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that > $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: $$T_\epsilon\Psi(\mathbf r)= e^{i \mathbf{\epsilon} P/ \hbar}\Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(i\epsilon \cdot (-i\hbar \nabla)/\hbar)^n}{n!} \Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(\mathbf \epsilon \cdot \nabla)^n}{n!}\Psi(\mathbf r)= \Psi(\mathbf r) + (\epsilon \cdot \nabla) \Psi(\mathbf r) + \frac{(\epsilon \cdot \nabla)^2 \Psi(\mathbf r)}{2} + \cdots$$ This looks somewhat like a Taylor expansion of $\Psi(\mathbf r)$, but it's different than I've seen before -- I've never seen it in terms of a directional derivative. Can you confirm if this is the Taylor expansion of $\Psi(\mathbf r + \mathbf \epsilon)$? Or if not, what I should be getting when expanding $e^{i \mathbf{\epsilon} P/ \hbar}\Psi(\mathbf r)$?


title: Direction of radial acceleration same in circular motion?

text: I have a big confusion. There is a question in my book which basically says that a ball is tied to a string and rotated. and it asks me to tell whether the following statement is true of false. Direction of radial acceleration MAY remain the same. This statement is true. Please explain to me a case where this is possible. i thought that this is only when an object moves straight and in circular motion radial acceleration is towards the centre.


title: what kind of transformation can be applied to qubits?

text: I had a doubt on what kind of transformations can be applied to qubits. I understand that the transformations need to be reversible , but they also have to preserve the norm thats why the transformations need to be unitary as $U^{\dagger}U=I$.      But then I was reading the paper 'Quantum Mechanics Near closed Timelike Lines , by D.Deutsch'.In it he has applied a transformation give by   $|x\rangle |y\rangle \Rightarrow |x\rangle |y \vee f(x)\rangle$ , where $\vee$ stands for XOR and $x,y \in \\{0,1\\}^{n}$ and $f:\\{0,1\\}^{n} \rightarrow \\{0,1\\}^{n}$      Now I can see that that the transformation is reversible ( apply twice and get the same thing ) but it is not unitary , am I missing something ?


title: What is the idea behind counting the number of excited states and the representation of a group ?

text: While reading Polchinski's Chapter 1, I encountered the following on page 24, "For example, the $(D-1)$ dimensional vector representation of $SO(D-1)$ breaks up into an invariant and a $(D-2)$-vector under the $SO(D-2)$ acting on the transverse directions, $$ \vec{v} = (v^1, 0, 0, \dots) + (0. v^2, v^2, \dots, v^{D-1}) $$ Thus, if a massive particle is in the vector representation of $SO(D-1)$, we will see a scalar and a vector when we look at the transformation properties under $S0(D-2)$. This idea extends to any representation : one can always reconstruct the full $SO(D-1)$ spin representation from the behaviour under $SO(D-2)$. " I can show that the second excited states which is given by, $$ \alpha_{-1}^i \alpha_{-1}^j |0\rangle \bigoplus \alpha_{-2}^i |0\rangle $$ where $i,j$ runs from $\\{2,D-1\\}$ and treating them symmetric, the no. of excited states would be $\dfrac{(D+1)(D-2)}{2}$ which matches with the dimensions of a traceless symmetric irrep of $SO(D-1)$. My question is how can we be sure by just matching numbers, and what does this Physically mean and is there a mechanism to do this consistently ? What does this business of "reconstructing $SO(D-1)$ representation from $SO(D-2)$" mean ? I know a little bit about group theory like, Cartan Matrices, Dynkin Diagrams and Young Tableaux method for SU(N) theory, so it is fine if someone could give me a good reference. A precise answer would ofcourse be great :).


title: Question on envelope-carrier description of traveling wave

text: I'm doing a research internship in attosecond physics right now, and one of the really important things in the field is the description of a propagating laser pulse as the combination of a slowly (or not) varying envelope and a carrier frequency. I was wondering, is this envelope-carrier description completely general, i.e. does it apply to any arbitrary solution of the wave equation? In my waves course we learned briefly about beats and derived the whole envelope-carrier description for a system consisting of 2 sinusoids, but what about in general? One of the reasons I am confused is because if I consider the general solution to the wave equation using a superposition of sines and cosines (i.e. Fourier series), then I can imagine constructing a Gaussian pulse. This obviously won't have a carrier frequency attached to it. So basically, when is this envelope-carrier description valid? If anyone could clear things up, that would be greatly appreciated.


title: How much red, blue and green does white light have?

text: Different kinds of white light have a different spectrum. Light from a white LED will have blue at the peak intensity while white light from a CFL or something else will have a different looking spectrum. I dont understand how this works. Shouldn't pure white light have a unique spectrum no matter what? ![White LED Spectrum](http://i.stack.imgur.com/qwkQ8.png) Thats white LED spectrum


title: Coefficients in V-A interaction

text: To account for parity violation the weak interaction matrix element is written in the form $$ M \propto \sum_{i} C_i (\bar u_p O_i u_n) (\bar u_e O_i (1 + {{C'_i}\over{C_i}}\gamma^5)u_\nu) $$ Why does time reversal invariance imply that the $C_i$ coefficients must be real? I was thinking that maybe it might be related to some kind of hermiticity condition, in order to have $|M_{if}|=|M_{fi}|$. Am I on the right path?


title: Aufbau principle in modern quantum theory

text: What is the rigorous definition of the Aufbau principle and the mathematical model used for its description? From Wikipedia, we have that the principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons. For modeling the postulate, we use the Madelung rule, Pauli exclusion and "if two same energy orbitals are available, choose the less occupied one)". So, $n+\ell$ ordering with constraints. And I also know that the Aufbau principle has its exceptions.


title: How to calculate the upper limit on the number of days weather can be forecast reliably?

text: To put it bluntly, weather is described by the Navier-Stokes equation, which in turn exhibits turbulence, so eventually predictions will become unreliable. I am interested in a derivation of the time-scale where weather predictions become unreliable. Let us call this the critical time-scale for weather on Earth. We could estimate this time-scale if we knew some critical length and velocity scales. Since weather basically lives on an $S^2$ with radius of the Earth we seem to have a natural candidate for the critical length scale. So I assume that the relevant length scale is the radius of the Earth (about 6400km) and the relevant velocity scale some typical speed of wind (say, 25m/s, but frankly, I am taking this number out of, well, thin air). Then I get a typical time-scale of $2.6\cdot 10^5$s, which is roughly three days. The result three days seems not completely unreasonable, but I would like to see an actual derivation. Does anyone know how to obtain a more accurate and reliable estimate of the critical time-scale for weather on Earth?


title: Projectiles problem solving

text: I've only learned about to use kinematics equation when solving projectile problems but today i came across the following equations. where does they come from?   * Distance travelled   * Time of flight   * Angle of reach


title: How to calculate viscous damping coefficient?

text: The damping of a spring is calculated with: $$[\zeta] = \frac{[c]}{\sqrt{[m][k]}}$$ Where c is the 'viscous damping coefficient' of the spring, according to Wikipedia. m is the mass, k is the spring constant, and zeta is the damping ratio. How is the value of c calculated though? Is it a constant for the air through which the spring is moving or does it depend on the spring itself? What data is required to calculate it and how can it be done? I'm just looking at the oscillation of a spring vertically, and I have data for its decreasing amplitude, and the velocity of the spring at all points. I have the value of the damping ratio, and I'm trying to find the value of 'c' in order to prove the above equation in an investigation. There is almost no information about this online.


title: Compactification of Dimensions

text: As the only way String theory can respect the principles of quantum mechanics and Special theory of relativity, by formulating it in hypothetical nine dimensional space. You could use string compactification for d number of non compact direction, all it requires to have d non compact direction is to choose an appropriate compact space of dimension (9-d). The question arises here, Then Why our Universe is in Three dimensional space?


title: Diffusion of gases in the atmosphere

text: Suppose that the atmosphere is composed of 21% $O_2$ and 78% $Kr$ (instead of $N_2$). Since the density of $Kr$ is greater than the density of $O_2$, the lower atmosphere (where we live) should be deprived of $O_2$. Am I right? I know that diffusion has a role here, but is there a temperature where diffusion can't overcome the fact that $O_2$ should be in the upper atmosphere?


title: Is this the strongest steel?

text: I found this link where it described steel with UTS at 5200 MPa (755kpsi), is this the strongest steel or just a typo because value is seemed to be unreal?


title: Timelike/null generic condition in general relativity

text: My question concerns the following definition > **Definition:** The _timelike_ (resp. null) _generic condition_ in GR is > fulfilled if $$u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne > 0$$ at some point of each timelike (resp. null) geodesic with tangent vector > $\vec u$. ($R_{\rho \mu \nu \sigma}$ is the Riemann curvature tensor.) It is written in many places that this is the right condition to impose if one wants to assume that every freely falling (or light) particle encounters some form of matter or radiation in its history (or something to that effect). But I don't understand, why the particular tensor $u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$ is the right thing to look at in this context. For example, why don't we assume that $R_{\rho\mu\nu\sigma}u^\mu u^\nu \ne 0$ at some point? Or maybe that $R_{\mu \nu}u^\mu u^\nu \ne 0$, or perhaps that $G_{\mu \nu} u^\mu u^\nu \ne 0$? I'm guessing that the last two condition could be too weak to derive the singularity theorems we want, so that something stronger must be assumed. But the expression $u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$ really looks a bit strange to me (i.e. I don't understand its significance). Could someone explain to me why this is the right condition to impose? Thanks!


title: What is Absorption Grating

text: I came across the word "absorption grating" in a review article. I googled it tried to find out what it means but couldn't. Could you explain it to me?


title: Is Schrödinger’s cat misleading? And what would happen if Planck constant is bigger?

text: Schrödinger’s cat, the thought experiment, makes it seem like as if measurement can cause a system to stop being in a superposition of states and become either one of the states (collapsed). So does a system always exist in a superposition? In this sense, do things in the macroscopic world have a wavefunction? Is it because of the size of the everyday object, so it won't behave so much like an electron? Theoretically, if the Planck constant is to be bigger, everyday object would start behaving more like particles in a quantum scale?


title: An issue about the compactness and the existence of CTCs

text: There is a well known fact that a compact spacetime necessarily contains a closed timelike curve (CTC). Proof can be found in several books on GR (e.g. Hawking, Ellis, Proposition 6.4.2), and in essence goes like this: The spacetime $M$ can be covered by open sets of the form $I^+(p)$, chronological future of the point $p \in M$ (note that _a priori_ $p$ is not an element of the set $I^+(p)$, but such situation can happen in a presence of CTCs). Now, suppose that $M$ is compact. Then there is a finite subcover, say $$\\{ I^+(p_1), \dots, I^+(p_n) \\}$$ The point $p_1$ is contained in $I^+(p_{k_1})$ for some $1 \le k_1 \le n$, the point $p_{k_1}$ is contained in $I^+(p_{k_2})$, and so on. Since this subcover is finite, eventually some point $p_{k_r}$ must belong to $I^+(p_{k_s})$, with $s \le r$. Then there is a future directed timelike curve going from $p_{k_r}$ to $p_{k_s}$ (since $s \le r$) and then from $p_{k_s}$ back to $p_{k_r}$ (since $p_{k_r} \in I^+(p_{k_s})$), which gives a closed timelike curve through $p_{k_r}$ (and $p_{k_s}$) in $M$. Q.E.D. * * * The question that bothers me is: what is implicitly assumed about the spacetime $M$, by saying that the family of sets of the form $I^+(p)$ is indeed a covering of $M$? Take for example a flat spacetime with compact space part and finite time direction, e.g. $M = [0,1] \times T^3$, where $T^3$ is a (spacelike) 3-torus. This is a compact manifold (since it is a product of two compact manifolds) and there are no CTCs. The loophole in the argument above seems to be in the fact that the "initial points", $\\{0\\} \times T^3$, are not covered by any set of the form $I^+(p)$. One can easily modify this example by contracting the initial and final spacelike slices to points ("Big Bang" and "Big Crunch"); the resulting spacetime is still compact and contains no CTCs. Do these manifolds fail to be "regular spacetimes" for some reason?


title: About the gauge invariance of Chern-Simons' theory (in local coordinates)

text: I am aware of the differential form language proof of the fact that for arbitrary gauge transformations the Chern-Simons' term shifts by a WZW term (on the boundary). But I am getting confused if in local coordinates I try to prove that under infinitesimal gauge transformations the Chern-Simons' term is invariant (may be upto total derivatives) So I took my infinitesimal gauge transformation as $\delta A^a_\mu = \partial _\mu \epsilon ^a + f^{abc}A_\mu^b \epsilon ^c$ where my structure constants $f^{abc}$ are totally antisymmetric and cyclic. Then I look at the two terms that I have in the CS form, $\epsilon^{\mu \nu \lambda}A^a_\mu \partial_\nu A^a_\lambda$ and $\epsilon^{\mu \nu \lambda}f^{abc}A^a_\mu A^b_\nu A^c_\lambda$ and I vary them infinitesimally to get, $\delta (\epsilon^{\mu \nu \lambda}A^a_\mu \partial_\nu A^a_\lambda ) = 2\epsilon^{\mu \nu \lambda}( \partial_\mu \epsilon ^a \partial _\nu A^a_\lambda + f^{abc}A_\mu ^b \partial _\nu A_\lambda ^a \epsilon ^c)$ $\delta (\epsilon^{\mu \nu \lambda}f^{abc}A^a_\mu A^b_\nu A^c_\lambda) = 3\epsilon^{\mu \nu \lambda}f^{abc} ( \partial _\mu \epsilon^a A_\nu ^b A_\lambda ^c + f^{adf} A_\mu ^d A_\nu ^b A_\lambda ^c \epsilon ^f) $   * Now I can't see how any linear combination of the above two terms can make their combined variation 0 (or a total derivative). I would like to know as to what am I missing here.


title: Can refraction of the atmosphere be so high that the surface of a planet seems concave to its inhabitants?

text: In their novel Inhabited Island by brothers Strugatsky, the inhabitants of a planet think they live on the inner surface of a giant sphere because the refraction of the atmosphere makes the surface appearing concave. At the same time, the atmosphere is suitable for a human to breathe. I wonder whether it is actually possible?


title: Are we big or small?

text: How does the size of humans compare to the size of other objects in the universe? Are we among the relatively large or the relatively small things? My very preliminary research suggests that the smallest thing in the physical universe is the quantum foam which should be about Planck length or 10-35 m. While the universe is about 1026 m. This would seem to bias us toward the big side by about 9 orders of magnitude. I am skeptical of this estimate because it is pretty close to the middle of the scale and I am always skeptical of measures that put humans in the center. So is there another way of thinking about/ quantifying this question? **Edit:** it seems that the smallest thing we have actual evidence of is a neutrino which is 10-24 m, which would put us even closer to the center!


title: How to derive Eq. (1.2.17) in Polchinski?

text: I have a super stupid question about deriving Eq. (1.2.17) in Polchinski's string theory, vol 1. The book seems to derive from $$\tag{1.2.16} h_{ab}=\frac{1}{2} \gamma_{ab} \gamma^{cd} h_{cd} $$ to $$\tag{1.2.17} h_{ab} (-h)^{-1/2} = \gamma_{ab} (-\gamma)^{-1/2}.$$ I cannot figure out how to manipulate the determinants $\gamma$ and $h$. Excuse me, would you provide at least any hint for the derivation?


title: How early might a moon-sized asteroid be detected?

text: If an enormous asteroid, approximately the size of our moon (~2000-mile diameter), was passing close to Earth, how early might we detect something that large? Or alternatively, how close could something that large get without detection? [edited to change term from meteor to asteroid]


title: Does processing for a quantum computer take place in other universes?

text: Apologies in advance if my question seems misinformed. I am a software developer, and neither quantum mechanics nor physics are my specialties. From http://physics.about.com/od/quantumphysics/f/quantumcomp.htm: > The exact physical mechanism at work within the quantum computer is somewhat > theoretically complex and intuitively disturbing. Generally, it is explained > in terms of the multi-world interpretation of quantum physics, wherein the > computer performs calculations not only in our universe but also in other > universes simultaneously, while the various qubits are in a state of quantum > decoherence. And from http://www.cs.man.ac.uk/~toby/writing/PCW/q2.htm: > The amazing power of the quantum computer stems from the fact that if you > have a collection of qubits – a register – in which each qubit is in an > indeterminate state, then the register effectively represents all possible > numbers at once. If you then perform a single computation on the register, > the computation works on every possible number, simultaneously. David > Deutsch explains the process using the idea of parallel universes – although > we see only the single register in our universe, it actually exists in many > other universes too, one for each of its possible states. By operating on > the register in our universe, we kick off computations in all the other > unseen universes, and then magically retrieve the answer. Assuming the multiverse interpretation of quantum mechanics is correct, does this mean that a quantum computer is capable of considering all possible solutions to a problem simultaneously because it delegates the consideration of (calculations for) each possible solution to a different universe?


title: Why isn't a pendulum clock considered a perpetual machine?

text: Why do people say that a perpetual machine is one that runs for eternity? Wouldn't a machine that runs taking no energy from outside but sometimes needs to be restarted, such as pendulum clock, be considered a perpetual machine? It just needs to be restarted one or two times a month. Given that physicists love approximations, wouldn't restarting a machine two times a month be higher order smaller term that we could throw away and call the pendulum clock a perpetual machine?


title: Units kT = 1... why?

text: Why is it convenient to choose units kT=1? It came up in this paper: http://jfi.uchicago.edu/~leop/SciencePapers/scaling%20and%20universality%20in%20statistical%20physics.pdf in page 4, equation (2.1).


title: Entropy of photons

text: Do low energy photons have greater entropy than high energy photons, or is the entropy of all photons the same (and the relationship between high and low entropy radiation is just a function of the number of photons - i.e. a closed system with more photons has higher entropy than the same sized system with fewer photons, regardless of the individual energies of the photons)? Is the entropy related to the number of different energies of a group of photons (i.e. do ten photons with the same wavelength have lower entropy than ten photons with different wavelengths)?


title: Why does glass absorb infrared light?

text: I was under the impression that glass was transparent to light with a wavelength above ultraviolet, but when playing with my latest toy, an infrared thermometer which brought up a question I asked here before, expecting to be able to measure temperature via a mirror, I found that it would essentially just give me room temperature (or mirror temperature), at most slightly more, while via a piece of shiny metal I was able to measure the temperature of the object of interest (at most slightly less). If it were the case that either most infrared radiation were transmitted or reflected I think I would be able to measure the temperature of the object, so I suspect that most heat radiation was absorbed. Why does that happen while it doesn't happen for visible light? I think that for high frequencies electronic energy levels are available to absorb light, and that their unavailability for light in the visible range made glass transparent. Why is it that for infrared energy levels are available (mechanical?) that apparently are not available for higher energy visible light?


title: Physical, intuitive reason for divergence of dielectric constant at electronic percolation transition?

text: Several papers such as this (warning, PDF) and this (PDF again) talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (which, if you read those papers, gives rise to some neat effects). I'm having trouble understanding how, though, aside from just the math. The 2nd paper mentions a more qualitative interpretation: > Near the percolation threshold the metallic clusters are separated by thin > dielectric regions. Each pair of nearest clusters forms a condenser [i.e., > capacitor] whose effective surface tends to infinity near the percolation > threshold. Then the effective capacity of the system diverges, too. Ok, let's assume for simplicity the "dielectric" here is vacuum, and use the form of a parallel plate capacitor, $C=A\epsilon_0/d$. By "effective surface tends to infinity", I'm guessing they mean that $d$ is going to zero (because the distance between clusters is going to 0), and thus $C$ is increasing to infinity. From there, I'm a little confused. My best guess is that the impedance goes as $Z\propto 1/(i\omega C)$, and conductivity $\sigma \propto 1/Z$, and dielectric constant $\epsilon \propto i\omega \sigma$, so when the C increases, Z decreases, the imaginary part of $\sigma$ increases, and thus the real part of $\epsilon$ increases. But I'm not really sure about that, and I'm having trouble reconciling it with what must happen after the percolation transition. Does anyone have a better physical understanding of this? thank you!


title: Estimating volume of moduli space of genus-g Riemann surface with n marked points

text: I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus $g$ and having $n$ marked points. I am reading some old string theory papers which discuss divergences in string perturbation theory. In this respect authors such as Gross, Periwal and Shenker say that the partition function in closed bosonic string theories asymptotically grows as $(2g)!$, where $g$ is the genus and they claim that this estimate can be obtained from the volume of moduli space. It would be great if anyone of you can suggest some reference where I learn about finding out the volume of moduli space of genus-$g$ surfaces with $n$ punctures.


title: For Faraday's law, why does the emf decrease as you increase the area of the loop?

text: I've only recently started learning basic electrodynamics, but I don't understand why a loop of coil with a small area and a magnet falling through will produce a larger emf than a loop of coil with a larger area with the same magnetic field falling through. To clarify, lets say you have a loop of coil and you drop a magnet through the coil. This is will produce an emf according to Faraday's law. Now if you have a loop of coil with a larger radius, and you drop the same magnet through at the same speed, it will produce a _smaller_ emf. Why is this?


title: Effect of Reynolds number on wake region

text: For a flow over a smooth sphere, does the wake region increases with increase in Reynolds number? And if so, then why?


title: Wave/particle duality

text: Apologies if this has been asked before (I did check and I believe it wasn't). I have a question about the particle/wave duality of photons (or other particles). Depending on what and how we measure the photon turns out to be either a wave or a particle. Recently I saw some web page (and I can't remember where) that maybe neither is true. Both the wave perception and the particle perception are just that: perceptions, strengthened by our perhaps limited ability to observe. What if the reality of the photon is something else, something "on top" of our two perceptions?? Could someone direct me to a site that could tell me more about it (or maybe debunk the whole idea?)?


title: Why aren't all photons virtual particles even in the "vacuum" of empty space?

text: I'm thoroughly confused about the nature of electromagnetic radiation. Light is supposed to exhibit both wave and particle characteristics. But does that mean that it is _both_ a wave and a particle or _neither_ a wave nor a particle? If the latter, then it is not a "real" particle at all, but only occasionally behaves as one (hence, a "virtual" particle). And if the former, then it is, depending on the experimental setup being used to detect it, a "real" particle, but one that only exists (as a particle) for a limited time, which again, corresponds to the definition of a _virtual_ particle ("a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time" - Wikipedia). So, if photons invariably appear to exhibit the characteristics of virtual particles, why are they not actually virtual particles (even in a vacuum)?


title: Particles acting like waves

text: Wave–particle duality is kinda bothering me... I read that electrons can act like waves, but I know that electrons are actually particles. The theory says that if you have not observed the particle it acts like a wave and can explore all classically available particle trajectories simultaneously and when you observe it it ends up in just one place. But doesn't that just mean that this particle is a particle the whole time, but you can't know it's location without observing it and the wave just presents a range of possible locations where it might be if you do observe it? And if so why would you call it "Wave–particle duality" if this particle is just a particle doing it's thing and a wave is just a best guess of where it is?


title: Does radio use virtual photons?

text: In radio communication each accelerated electron in the transmitter antenna interacts with an electron in the receiver antenna by exchanging a photon. Is that photon always a virtual photon as described in the diagram below rather than a "real" photon? ![enter image description here](http://i.stack.imgur.com/i4Zqd.gif) As I understand it, Feynman's definition of a virtual photon in his book QED is any exchanged photon that does not appear in the initial or final conditions. Therefore by his definition radio works by using virtual photons.


title: Mass of small fluctuation around vacuum

text: For a potential $V$, how do we define the mass of a small fluctuation around its vacuum? For example I have the potential $$ V_\mathrm{eff}(\phi) = \frac{1}{2} \left(\frac{\rho}{M^2} - \mu^2\right) \phi^2 + \frac{1}{4} \lambda \phi^4. $$ What is the definition of the mass of small fluctuations around the symmetry- breaking vacuum and what is the physical meaning of it?


title: Force on every particle?

text: Say there is a horizontal rod, where a force $P$ is applied to the right hand side of the rod to drag it horizontally to the right. If I drew a force diagram, I'd put some vector with a length of $\| \mathbf{P} \|$ coming out of the right end of the rod. In actuality, if the rod is composed of ten particles, then is $\| \mathbf{P} \|$ distributed evenly? So if I felt like creating a more accurate force diagram, then each particle would have a smaller vector going to the right with magnitude $\| \mathbf{P} \|/10$? Is this true? Is the reason we usually only put 1 vector of length $\| \mathbf{P} \|$ for simplicity? Likewise when we assume the force of gravity $F_g$ on an extended object can be put at the center of gravity of that object as if it was a point mass. But in actuality, there are like a billion really small $F_g$ vectors on each particle?


title: Calories Burned While Hiking

text: I want to find out how many calories are burned on any particular hike by any particular by using a formula. I figure it's probably a simple physics question, but since I never took physics, I don't know the math to do it. My premise is that you will expend (A) energy (calories) when moving (B) weight (C) distance on a level path. And also (X) energy (calories) when moving (Y) weight (Z) distance when moving up against gravity. The formulas I have found will add (A)+(X) to get the totals calories burned. Trouble is, the results vary greatly, and I have no way of judging which one is right. Now, I know there lots of variables, but for this exercise, we'll skip the small ones. Here's what I'm thinking:   1. Obviously there's resistance. At first I was dismissing resistance as being too minor when only traveling 2-4 MPH, but then I realized that without resistance, we'd just keep moving. Now, resistance will be based on the front profile size. Can we simplify things by saying X weight = Y profile?   2. Assuming X weight = Y profile. A backpack would add weight, but not added profile. So we'll say no added weight for the formula.   3. Up & down motion for each step. I figure that makes the formula a lot more complex because we have to deal with the height & stride of each person. So I guess we'll skip that.   4. A fit person burns less calories (more efficient muscles) than an out of shape person doing the same hike. So we'll pick a good average.    5. Downhill is considered the same exercise as level walking.   6. Age & sex may be a factor, but I don't have those numbers, so can be ignored for now. I may later ask on a health forum for those factors.   7. There's probably others I haven't thought about. There are many sites on the internet that calculate this for you, but there's 2 problems with them.   1. The majority want to ask you your hiking speed and time hiked. But most people don't their hiking speed, and some sites are vague with "brisk walk". Then all they do is convert speed with time to get distance.   2. Even the ones that factor weight, distance, & elevation gain, I get widely different answers. So, what I would like to get, if someone is in the mood to figure it out, is a formula for calculating calories burned. Thanks


title: Are cylindrical mirror telescopes possible?

text: Is it possible to use a set of cylindrical mirrors and software correction to build an optical reflector telescope, instead of using a paraboloid reflector? By 'cylindrical' I actually mean a planar mirror curved to a parabolic profile. In other words, rather than focusing light in 360 degrees in one step, is it possible to focus it in two steps -- first along one axis and then using a perpendicular 'cylindrical' mirror to focus it along the other axis? Does this introduce optical distortions, and if so, can software correction be used? The reasoning is that 'cylindrical' reflectors are far easier and cheaper to fabricate (either using sheet metal techniques or linear grinding methods) than traditional ones. They are also easier to stack together, launch into orbit and deploy once it reaches stable orbit (for space telescopes). A Google search revealed only a single document with an attempt to describe such a setup, but it is over a decade old: http://www.bruno.postle.net/2001/telescope/ ![enter image description here](http://i.stack.imgur.com/dc5DA.png)


title: The role of dark matter in black holes and star formation

text: In my understanding, there exists a critical mass for which a star needs to be in order for it to collapse into a black hole. This also applies to a certain critical density of gas in order for stars to form in the first place. However, given a massive star or a massive amount of gas, it must have a relatively large gravitational field and therefore will interact with dark matter, attracting dark matter towards itself. This makes me question, doesn't the critical mass or critical density that we've figured out before our knowledge of dark matter now need to take that into account? OR Shouldn't the critical mass and critical density resemble the following pseudo equations? Critical mass (normal and dark) for a star to turn to black hole = A*normal matter mass + B*dark matter mass, where A and B are fractional amounts Critical density for a cloud of gas (normal and dark) for star formation = C*normal matter mass + D*dark matter mass, where C and D are fractional amounts So if the answer is yes, then what are the fractions, A, B, C, and D? Moreover, What kind of experiment or calculation would allow us to figure out these fractions?