title: Differences of waves in water and air

text: When one drops a pebble in a body of water, one can observe multiple waves emanating from the point the pebble came in contact with water. Be it because the water "jumps" up and comes back down, creating more waves or because the water does "up and down" motion, there are definitely many waves. But when one snaps ones fingers, thus creating a wave in the air, there is only one clear sound, no multiple, diminishing snaps (unless there is echoes, but this is not a concern), unlike one would expect just by observing waves in water. Why is this?


title: 3 Axis Gryroscope with forced Precession and Limits of Motion

text: I am working a problem concerning a 3 axis gryoscope, the spinning mass is a magnet (dipole). This is part of a optical sensing device. The inner gimbal is for pitch rotation, and the outer gimbal is for yaw precession. The outer gimbal is fixed in the sensor casing. The whole assembly is well balanced and not affected by gravity. The casing contains 4 electrical coils located at 90 degree intervals. These coils are pulsed to make the magnetic rotor spin as in a motor. There is another coil that is wrapped around the casing and it is pulsed to precess the gyro in pitch and yaw. The spinning mass is on a bearing through which extends a non spinning axis (forward). On the end of that axis is an optical sensor (which does not spin). The outer casing clearly will limit the range of pitch and yaw motion. And technically, it is desirable to not come into contact with it. The part that would come into contact is the optical sensor. There are other sensors that keep up with spin speed, pitch and yaw gimbal angle, and dipole location. This is needed to be able to send proper spin and precession commands. I have built a very good simulink model using Euler's torque equations, and all of the primary suff works. I even have a good idea of how to model the impact of the rotor precessing INTO the casing wall (reflect the pitch and yaw rotor velocities). But I cannot seem to figure out how to do the other case: Casing rotating into the rotor, where rotor precessions are essentially zero. How do I trasfer the casing momentum to the rotor and still keep the solution physically bounded by the casing itself? Since the casing is mounted in a moving platform with 6 DOF, it can clearly rotate into the rotor. But remember, only the non-spinning part of the sensor will touch the casing, so the angular momentum of the rotor is not DIRECTLY involved with contact. Thank you, and sorry for the long description, MikeO Here is a graphic, certainly not scale. And the design is such that the optical sensor would contact the encasement before the rotor could. Two spin coils are not depicted. This drawing is purely conceptual. In the actual device, the gimbals are quite small and hidden inside a hollow spot within the center of the rotating gyro mass. You cannot see the gimbals. ![Precession Commanded Gyro](http://i.stack.imgur.com/6sYwH.jpg) The inertia tensors for the case and the rotor are known. The gimbals are taken as inconsequential. The intertia tensor of the case is much larger than that of the gyro and can be seen to be unaffected by the gyro. Whether the case rotates about the rotor, or the rotor rotates does not matter (by rotate, I mean pitch and yaw, not spin). The only parts of the entire assembly that can come into contact are the non-rotating optical sensor assembly and the case wall. Therefore the rotor will not touch the case. I am looking to write a conservation of angular momentum equation that would describe the initial rotations (pitch and yaw) of the rotor assuming the rotor was not pitching or yawing at the time of contact, in other words, the case rotated into the optical sensor which in turn transmits torques about the gimbal axes. In the equations v1=u1*(m1-m2)/(m1+m2) + u2*(2*m2/(m1+m2)) and v2=u2*(m2-m1)/(m1+m2) + u1*(2*m1/(m1+m2)) May I subsitute the inertia tensors for mass and the body rates for velocities? ![enter image description here](http://i.stack.imgur.com/yHh2X.jpg) ![enter image description here](http://i.stack.imgur.com/zY9jK.jpg) ![enter image description here](http://i.stack.imgur.com/ULXOk.jpg) The problem I am trying to solve is how to compute the reactions at the the boundry (the cylinder that blocks motion). I think I have solved the problem of what to do when the gyro is driven into the wall... I am reflecting the precession values (pitch and yaw). But the other problem is harder... what to do when the cylinder is driven into the gyro. Remember, the rotating mass never comes into contact with the wall, just the non-rotating mass at the end of the axle (spin axis). At first glance, it might seem that the answer is the same, but it does not work. I need to be able to calculate the "imparted torque" from the collision, and confine computed location within the cylindrical boundary.


title: Euler-Lagrange equations and friction forces

text: We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i (\mathbf{F}^{(a)}_i+\mathbf{f}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=0$$ Where $\mathbf{f}_i$ are the constrainded forces and are supposed to do no work, which it's true in most cases. Quoting Goldstein: > [The principle of virtual work] is no longer true if sliding friction forces > are present [in the tally of constraint forces], ... So I understand that we should **exclude friction forces** of our treatmeant. After some manipulations we arrive to: $$\frac{d}{dt}\frac {\partial T}{\partial \dot q_i}-\frac{\partial T}{\partial q_i}=Q_i$$ Further in the book, the Rayleigh dissipation function is introduced to **include friction forces**. So given that $Q_i=-\frac {\partial \mathcal{F}}{\partial \dot q_i}$ and $L=T-U$, we get: $$\frac{d}{dt}\frac {\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}+\frac {\partial \mathcal{F}}{\partial \dot q_i}=0$$ **Question:** Isn't this an inconsistency of our proof, how do we know the equation holds? Or is it just an educated guess which turns out to be true?


title: What advantages does action-at-a-distance description have over the field view of forces?

text: It is written in Jackson (page 3) : > In fact, though there are **recurring attempts** to eliminate explicit > reference to the fields in favor of action-at-a-distance descriptions of the > interaction of charged particles, the concept of the electromagnetic field > is one of the most fruitful ideas of physics, both classically and quantum > mechanically. What does he mean? **What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?**


title: Derivation of Scattering Equation 9.88 in Thornton & Marion

text: I am confused as to how a particular equation in Thornton & Marion's 'Classical Dynamics of Particles and Systems' was derived. It is equation 9.88, on page 354 of the fifth edition.   An incoming particle, $m_1$, is fired with kinetic energy $T_0$ at a stationary particle, $m_2$. After the collision, the incident particle $m_1$ deflects at an angle $\psi$ with kinetic energy $T_1$, while the target particle $m_2$ deflects at an angle $\zeta$ with kinetic energy $T_2$. All quantities are measured in the lab frame.   The equation in question is $$ \frac{T_2}{T_0} = 1 - \frac{T_1}{T_0} = \frac{4m_1m_2}{(m_1+m_2)^2}\cos^2(\zeta). $$ The only seemingly relevant equation given immediately before this equation is the following: $$ \frac{T_1}{T_0} = \frac{m_1^2}{(m_1+m_2)^2} \left[ \cos{\psi} \pm \sqrt{ (\frac{m_2}{m_1})^2 - \sin^2{\psi}} \right]^2. $$ I am sure that the second equation must somehow lead to the first, but I don't see how. If I could find a relation between $\zeta$ and $\psi$, I am sure that would help.


title: Gas pressure and centrifugal force

text: I think about a rotating torus (simplified tire) filled with ideal gas. Mass of gas is $m$ and molar mass is $M$. Pressure in non rotating torus is $p_0$. Temperature is constant $T$. Inner radius of torus is $r$ and outer radius is $R$. Then the cylinder begins to rotate with angular velocity $\omega$ How can I derive difference of pressures near the wall between inner wall and outer wall? Do I need more data (for a simple model)? If someone knows how to make it numerically, It would be also interesting for me. I would also appreciate if someone could tell which book should I read to know more about solving such problems.


title: Magnetization force on domains after alignment?

text: When a domain is aligned to an exterior uniform magnetic field. There are no forces acting on it? Magnetization force = 0, and all other force?


title: What is radiation(in simple words)?

text: I am a student who is interested in scientific facts. I want a simple definition for radiation. Neither too scientific, nor too simple.


title: Physics of every-day life: rotating bag of tea

text: Whilst studying for my physics courses, I like to drink tea. Today, I noticed that if you pull a bag of tea out of a hot cup of water, it gradually starts to rotate, picking up speed as time progresses and reaching a asymptotic speed after a while. My first question is: why does it start rotating? Of course it has something to do with the liquid water entering the gas phase just above the cup of tea, but how does that give the bag of tea a directed motion to one side? I presume that the momentum of the gaseous water is quite randomly directed vertically, so I don't see a reason for a horizontal rotation. Second question, how is the direction determined? I haven't tried, but might the direction of rotation be equally probable to be left-handed and right-handed? Third question: might the shape of a tea bag have an influence on what we observe? If we take a pyramid tea bag, for example, does the behavior change? I will perform some of these experiments while making my cup of teas and report back to you :)


title: Numerical simulation of sound propagation in air

text: How does one go about accurately modeling sound propagation in a room (with reflections, absorption, and diffusion characteristics) from the motion of a loud speaker? More specifically what are the governing equations that are needed? Obviously Naiver-Stokes but this is too general? Is there an easier way?


title: Vector $\vec{z}$ and its conjugate transpose $\overline{\vec{v}^\top}$ - is it the same as $\left|z\right\rangle$ and $\left\langle z \right|$

text: Lets say we have a complex vector $\vec{z} \\!=\\!(1\\!+\\!2i~~2\\!+\\!3i~~3\\!+\\!4i)^T$. Its scalar product $\vec{z}^T\\!\\! \cdot \vec{z}$ with itself will be a complex number, but if we conjugate the transposed vector we get $\overline{\vec{z}^T}\\!\\! \cdot \vec{z}$ _(this is a inner product right?)_ and a positive real number as a result: \begin{align*} \overline{\vec{z}\,^{T}}\\! \cdot \vec{z}&=\begin{pmatrix}1-2i&2-3i&3-4i\end{pmatrix} \begin{pmatrix}1+2i\\\2+3i\\\3+4i\end{pmatrix} =\\\ &=\begin{pmatrix}(1-2i)(1+2i) + (2-3i)(2+3i) + (3-4i)(3+4i)\end{pmatrix} = \\\ &= \begin{pmatrix}(1-2i+2i+4) + (4+6i-6i+9) + (9-12i+12i+16)\end{pmatrix} = \\\ &= (5 + 13 + 25) = 43 \end{align*} * * * **1st question:** I know that ket $\left|z\right\rangle$ is a vector of a Hilbert space and i know that $\vec{z}$ is the same as $\left|z\right\rangle$. But what about $\overline{\vec{z}^T}$? Is it equal to a bra $\left\langle z\right|$ ? * * * **2nd question** Notation $\overline{\vec{z}^T}$ means we have to conjugate & transposethe a vector $\vec{z}$. Can this notation be swapped with a dagger $\dagger$ _(afterall this is an operation namedconjugate transpose)_? * * * **3rd question:** From all of the above it seems logical to ask if this equality holds $\left|z\right\rangle^\dagger = \left \langle z \right|$ ?


title: de Sitter versus Minkowski QFT and cosmological constant

text: WMAP/Planck results confirm than we live in a de Sitter-like phase, i.e., a Universe with positive acceleration or positive cosmological constant! Therefore, I believe that a way to solve the cosmological constant problem could be to extend QFT from Minkowski to de Sitter space. I would not be surprised if our "mistake" to get the theoretical "wrong" value of the cosmological constant is associated to the fact of calculating the vacuum energy in a Minkowskian QFT setting. My question is: what are the problems that a QFT on de Sitter space faces in order to explain the observed vacuum energy? Could it work? PS: Does someone know good references about QFT in de Sitter space?


title: Resolution of vectors

text: What is the fundamental basis of resolution of vector. Suppose we have a vector $\vec{mg}$, now we resolve it into two components, horizontal and vertical. My question is what is the basis for telling that this is the horizontal and this is the vertical component since the Cartesian plane can be drawn on any direction.


title: Wave equations for two intervals at Potential step

text: Lets say we have a potential step as in the picture: ![enter image description here](http://i.stack.imgur.com/HB3XI.png) In the region **I** there is a free particle with a wavefunction $\psi_I$ while in the region **II** the wave function will be $\psi_{II}$. Let me take now the schrödinger equation and try to derive $\psi_I$: \begin{align} &~~W \psi = -\frac{\hbar^2}{2m}\, \frac{d^2 \Psi}{d\, x^2} + W_p \psi ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber \\\ &~~W \psi = -\frac{\hbar^2}{2m}\, \frac{d^2 \Psi}{d\, x^2}\nonumber \\\ &\frac{d^2 \Psi}{d\, x^2} = -\frac{2m W}{\hbar^2}\,\psi \nonumber\\\ {\scriptsize \text{DE: }} &\boxed{\frac{d^2 \Psi}{d\, x^2} = -\mathcal L\,\psi}~\boxed{\mathcal{L} \equiv \sqrt{\tfrac{2mW}{\hbar^2}}} \nonumber\\\ &~\phantom{\line(1,0){18.3}}\Downarrow \nonumber\\\ {\scriptsize \text{general solution of DE: }} &\boxed{\psi_{I} = C \sin\left(\mathcal{L}\, x \right) + D \cos \left(\mathcal{L}\, x \right)}\nonumber \end{align} I got the general solution for the interval **I** , but this is nothing like the solution they use in all the books: $\psi_{I} = C e^{i\mathcal L x} + D e^{-i \mathcal L x} $ where $\mathcal L \equiv \sqrt{{\scriptsize 2mW/\hbar^2}}$. I have a personal issue with this because if $x= -\infty$ part $De^{-i \mathcal L x}$ would become infinite and this is impossible for a wavefunction! I know that i would get exponential form if i defined constant $\mathcal L$ a bit differently as i did above: \begin{align} {\scriptsize \text{DE: }} &\boxed{\frac{d^2 \Psi}{d\, x^2} = \mathcal L\,\psi}~\boxed{\mathcal{L} \equiv -\sqrt{\tfrac{2mW}{\hbar^2}}} \nonumber\\\ &~\phantom{\line(1,0){18.3}}\Downarrow \nonumber\\\ {\scriptsize \text{general solution of DE: }} &\boxed{ \psi_{I} = C e^{\mathcal L x } + D^{-\mathcal L x} }\nonumber \end{align} This general solution looks more like the one they use in the books but it lacks an imaginary $i$ and $\mathcal L$ is defined with a - while in all the books it is positive. Could anyone tell me what am i missing here so i could understand this?


title: Force of an impact on water

text: Today, at the Brazilian news, I heard that a men fell from a cruise into the sea, from a height of 50m, and decided to see if he could have survived. I took his weight for being 80kg. I did $V^2=V0^2 + 2 G H$, this resulted $V = 31.3 m/s$ or $112.69 km/h$. Also $F = m a=784 N$. My question is: How to relate this to the surface tension of water to get the actual strength of the impact? Assuming he fell standing and the area of contact was $700cm^2$, two foot side by side.


title: Why does the sky look black in pictures taken from the summit of everest?

text: In pictures taken from the summit of Mount Everest (such as this one), the colour of the sky is a very dark blue or even black in some pictures. I remember from my own experiences of hiking in the Himalayas that the sky appeared to be darker blue as I climbed higher. In shots looking outward from the summit there's a very interesting effect of having a black sky above with a blue sky lower down near the horizon. (one example here) I believed that this happens because there is less atmosphere above the climbers to scatter the blue wavelengths. However, this same black sky effect does not occur when looking out the window of a plane. As the plane climbs, the sky does not appear to "change colour" at all as the plane ascends. The cruising altitude of commercial airliners on an international flight is slightly higher than Everest, so one would expect the sky to appear quite similar. What explains these two observations?


title: 30kHz solenoid and power amplifier design

text: I have some difficulties designing my experimental apparatus: The goal is creating a Solenoid in air (or iron powdered core) able to produce at least 50G of peak B field on his axis at a distance of 1". I would like to drive this solenoid in such a way that the B field varies sinusoidally from 0G to 50G (hopefully even higher) with a frequency ranging from DC to 30kHz. Probably the easiest way is to use a function generator as an input for a power amplifier (I have trouble choosing it, I don't know if I need a four quadrant amplifier or a traditional one) Since I need a fast varying B field I suppose that I have to try to keep inductance as low as possible -> so maybe I'll put not so many turns but more current..inductance will probably also create problem in the correct impedance matching with the power amplifier (I don't have a strong background on this topic.)


title: NASA Theoretical Faster Than Light Space Travel (IXS-Enterprise)

text: I was recently reading about NASA Faster than Light proposed spaceship, it supposedly will bend spacetime behind the spacecraft, so technically instead of bringing the ship to the destination, "it will bring the destination to you." now, I have an understanding of calculus-based physics, and a basic understanding of relativity. From my lackluster knowledge of relativity, by conventional means achieving the speed of light is nearly impossible if not impossible, according to Special Relativity. if I am not mistaken if we try using a spaceship, as we approach the speed of light because of relativity, wouldn't the mass of the spaceship go to infinity? and if we have infinite mass doesn't it mean we need an infinite amount of energy to push it? Even so, if we travel to say alpha centauri, which is 4.4 light-years away it will probably take around a month to go there at the speed of light, which is not bad. It will take a month for the traveler to go there, what about those outside? to put it simpler, if someone goes to Alpha Centauri at the speed of light and he comes back, how much time would have passed in Earth? (I would have used, calculations to illustrate better, but I do not know how to use equations in Stack-Exchange yet.)


title: Finding radioactive nucleus given table of values

text: **Question** Measuring the number of decays per minute $N(t)$ of a radioactive source every four days we have that: $N(t=0):=N_0=200$, $N(t=4)=141$, $N(t=8)=100$, $N(t=12)=71$, where $t$ is measured in days ($d$). ¿Which radioactive isotope is it? **Answer** From the decay law we have that $N(t)=N_{0}e^{-\lambda t}$, then $\lambda=\frac{1}{t}\ln\frac{N_{0}}{N}$, and $$T_{1/2}=\frac{\ln2}{\lambda}=\frac{t\,\ln2}{\ln\left(N_{0}/N\right)}$$ Inserting the values for $N(t=4)$ , I find that $$T_{1/2}=7.93171\ d$$ Inserting the values for $N(t=8)$ , I find that $$T_{1/2}=8.00000\ d$$ And inserting the values for $N(t=12)$ , I find that $$T_{1/2}=8.03154\ d$$ ** > **My question is** which one of these half-lives is the correct? And knowing > the right value for $T_{1/2}$, how can I know the isotope? **


title: Spherical Shaped Electrical Leads for Thermal Conductivity Mitigation?

text: I am designing a thermionic converter and one of the biggest inefficiencies is the thermal conductivity and limited electrical conductivity of the leads. I'm trying to mitigate the loss of heat besides finding other materials that conduct heat poorer and electricity better than Tungsten. One option open is the shape of the lead. Although I don't know if changing the shape of the lead will make any appreciable difference. I am considering a spherical shaped lead in between two flat electrodes. The point of contact of the sphere should be flattened somewhat to allow a better connection. I don't have the mathematical skill to apply Fourier's law to a semi-sphere. I also need to know its overall resistance and I don't have the skills for this either. If you think it would be worth calculating please share your answer and formulas. Or if not, please explain your reasoning. update 6/29/14 It seems to me that if you had a sphere with 2 opposing contact areas (Ca) it would not conduct heat any faster or slower than a straight cylinder with the same contact area (Ca). The cooling part and the heating part are the same size in area. Calculating the resistance was fairly easy to do in slices in javascript. And the resistance reduction from a straight cylinder with contact areas (Ca) on either side to a sphere with a limited contact areas (Ca) was considerable. How far is my reasoning now?


title: $R = dV/dI$ for varying temperature

text: I'm trying to do my prelab for an E&M course, and am asked if, for plotting $V$ vs $I$ with a varying temperature, I should expect a linear slope. I know that both $V$ and $I$ depend on $R$, and since $R$ is proportional to $T$ that I shouldn't. Trying to show this, I'm having some problems -- probably because my differential equations skills are a bit rusty. Simplely, if $V = I \,R(T)$, how do I take $\mathrm{d}V/\mathrm{d}I$? I can find $\mathrm{d}V/\mathrm{d}T$ by chain rule, but i'm just drawing a blank with $\mathrm{d}V/\mathrm{d}I$ since $I = V/R(T)$ Excuse my ineptitude, and thanks in advanced for helping me out.


title: Fermion as a mixture of particle and antiparticle

text: The solution to the Dirac equation (in the Dirac basis) are 4 coupled fields. The first 2 of them represent a particle (spin up/down), the other 2 fields are the antiparticle (spin up/down). When the particle is observed from its rest reference frame, the antiparticle solutions are zero. However once the particle is moving, all the 4 fields become coupled. Does it mean that a moving electron is a little bit of a positron at the same time (in the given reference frame)?


title: What invariant counting process derives the Minkowski metric?

text: Starting with Euclidean space, suppose I make a number of copies of a coordinate system all coincident at the origin, together with copies of a standard unit length. For any space interval common to all the coordinate systems, all the observers will count the same minimum number of times their unit length can be laid end to end to one another between this interval. And from this invariant counting process we derive the Euclidean metric in terms of its orthogonal components. What invariant counting process do we carry out to derive the Minkowski metric in terms of its orthogonal components?


title: Integrating the generator of the infinitesimal special conformal transformation

text: (c.f Di Francesco, Conformal Field Theory chapters 2 and 4). The expression for the full generator, $G_a$, of a transformation is $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} \Phi - \frac{\delta F}{\delta \omega_a}$$ For an infinitesimal special conformal transformation (SCT), the coordinates transform like $$x'^{\mu} = x^{\mu} + 2(x \cdot b)x^{\mu} - b^{\mu}x^2$$ If we now suppose the field transforms trivially under a SCT across the entire space, then $\delta F/\delta \omega_a = 0$. Geometrically, a SCT comprises of a inversion, translation and then a further inversion. An inversion of a point in space just looks like a translation of the point. So the constant vector $b^{\mu}$ parametrises the SCT. Then $$\frac{\delta x^{\mu}}{\delta b^{\nu}} = \frac{\delta x^{\mu}}{\delta (x^{\rho}b_{\rho})} \frac{\delta (x^{\gamma}b_{\gamma})}{\delta b^{\nu}} = 2 x^{\mu}x_{\nu} - x^2 \delta_{\nu}^{\mu}.$$ Now moving on to my question: Di Francesco makes a point of not showing how the finite transformation of the SCT comes from but just states it. $$x'^{\mu} = \frac{x^{\mu} - b^{\mu}x^2}{1-2x\cdot b + b^2 x^2}$$ I was wondering if somebody could point me to a link or explain the derivation. Is the reason for its non appearance due to complication or by being tedious? I am also wondering how, from either of the infinitesimal or finite forms, we may express the SCT as $$\frac{x'^{\mu}}{x'^2} = \frac{x^{\mu}}{x^2} - b^{\mu},$$ which is to say the SCT is an inversion $(1/x^2)$ a translation $-b^{\mu}$ and then a further inversion $(1/x'^2)$ which then gives $x'^{\mu}$, i.e the transformed coordinate.


title: Has a photon or electron ever been observed in a state of superposition?

text: Has subatomic particles ever been seen in a state of superposition or do we just detect information like qubits about the state of the particle? So is actual matter in superposition or is it just information about matter that's in a state of superposition?


title: Strömgren Sphere of Sun

text: I have a homework problem: > The Sun emits $ \sim5 x 10^{23}$ photons per second with $hν > 13.6$ $eV$. > If the density of hydrogen atoms in interplanetary space is $n =$ $109 > m^{-3}$, what is the size of the Stromgren sphere? Assume a recombination > coefficient $α = 2.6 x 10^{-19} m^3s^{-1}$. From Wikipedia, I was able to get to $$ R_S = \left( \frac{3S_*}{4\pi n^2 \beta_2} \right)^\frac{1}{3} $$ And I know $\beta_2 = \frac{\alpha}{T}$, but I have no idea what the $S_*$ is. Wikipedia describes it as a source of flux, which is obviously the sun, but I cannot figure out anything else about it. I'm actually quite sure that the professor would have given us a different equation, but he never went over it, so I don't know. Anyone know how to solve this?


title: Components of the Weyl spinor field

text: In the Weyl basis we can separate the spinor field into 2 components: the right-chiral spinor and the left-chiral spinor. Each of these fields has again 2 components which are coupled. What is the physical interpretation of these 2 components that make up the left-chiral (or right-chiral) field? In the Dirac basis the interpretation of the 4 components is:   1\. Electron spin-up   2\. Electron spin-down   3\. Positron spin-up   4\. Positron spin-down   So my question is what is the corresponding interpretation in the Weyl basis (in the massless case). Is it like this?   1\. Left-chiral electron $\psi_{4}$   2\. Left-chiral positron $\psi_{3}$   3\. Right-chiral electron $\psi_{2}$   4\. Righ-chiral positron $\psi_{1}$   If this is the case than I don't understand why the left-chiral electron $\psi_{4}$ couples to left-chiral positron $\psi_{3}$ as can be seen in the equations: $$ \partial_{t} \psi_{4} + \partial_{x} \psi_{4} - i\partial_{y} \psi_{4} + \partial_{z} \psi_{3} = 0 $$ $$ \partial_{t} \psi_{3} + \partial_{x} \psi_{3} + i\partial_{y} \psi_{3} - \partial_{z} \psi_{4} = 0 $$ $$ \partial_{t} \psi_{2} - \partial_{x} \psi_{2} + i\partial_{y} \psi_{2} - \partial_{z} \psi_{1} = 0 $$ $$ \partial_{t} \psi_{1} - \partial_{x} \psi_{1} - i\partial_{y} \psi_{1} + \partial_{z} \psi_{2} = 0 $$


title: Curie temperature and magnetization?

text: If an ferromagnetic object is heated and reaches Tc the magnetization gradually drops as we get closer to Tc or it's a instant drop? Can I assume as I heat the object, the magnetization is weakening gradually? Likewas as it cools?


title: ideal gas law modelization in a vacuum chamber

text: I am trying to model a pressure sensor in COMSOL. The basic work is that there is current flowing inside it, if the pressure of the gas around it drops down, the temperature goes down as well (ideal gas law) , hence there is a variation in the conductivity (described by the temperature coefficient of resistivity) so variation of current. I am figuring out that i have a vacuum chamber with a specific amount of gas (let 's say air) and then fixed volume. I then start to pumping out the gas, which means reduction of pressure and reduction of the number of molecules. I tried to use the built-in COMSOL modules (version 4.3a) but i was not able to properly describe the physics. Then I tried to use the gas law to "manually" calculate the temperature of the gas in the chamber and i have: $PV=nRT\Rightarrow T=\frac{PV}{nR}$ but then i have two unknown parameters. I mean, i can decide to sweep the pressure (for instance simulating a drop of pressure from 760 torr up to 1E-4 torr) but i will have problem with the n (how many molecules have been pumped out from my vacuum chamber?) and viceversa. Is there a way to describe the pressure inside the chamber as a function of the number of molecules?


title: Tale of two magnets

text: Please help. My example: You take a sheet of steel and suspend it in the air. Then you take a permanent magnet (i.e. a neodymium magnet) and attach it to the sheet of steel and from that you suspend a 1kg weight. The weight and magnet will stay put providing that the room doesn’t change and that gravity is still acting on the objects. Now you take a separate piece of steel (which is not magnetized) and try attach that and the 1kg weight to the suspended sheet of steel. You notice that without an external magnetic force you are unable to attach the two pieces together. Now here is where my confusion lies. You then attach a electromagnet to the 2nd piece of steel and notice that it stick but the battery you have used to power the device drains. But all this time the permanent magnet with the same load and in the same conditions hasn’t why??


title: Forward-scattering for a single impurity in an infinite system

text: I'm slightly confused with the following situation: Suppose you have an electron in a tight-binding model, and let's say we are in one dimension with $N$ lattice sites. Add to this a single impurity at a specific site $i$. The pure system is translationally invariant and has as eigenfunctions the plain waves. As a consequence, the probability to find the (free) electron on the impurity site should scale as $1/N$ and I'd imagine that the full forward- scattering Green function for the problem has some form $$G(k,\omega) = G_0(k,\omega) + \frac{1}{N} T(k,\omega) G_0(k,\omega)^2$$ where $T(k,\omega)$ would be some function independent of $N$. As a consequence, in the limit of an infinite system $N \rightarrow \infty$ I would expect to not see the impurity effect at all as a consequence of that $1/N$ factor. However, if I imagine that the impurity is a temperature effect (an example would be a flipped local spin in a ferromagnetic ground state, or an excited phonon) then the impurity could occur on any of the $N$ sites of the system. If I confine myself to very low temperatures where I assume that the entire system contains at most one impurity, the temperature averaged propagator would then look like $$G_T(k,\omega) = 1/Z \times \left[ \left\langle \phi_0 | \hat G(k,\omega) |\phi_0 \right\rangle + \sum_i e^{-\beta E_I} \left\langle i | \hat G(k,\omega + E_I) | i \right\rangle \right]$$ where $|\phi_0\rangle$ is the free system's ground state and $| i \rangle$ is the state with an impurity at site $i$ and $E_I$ is the energy cost of the impurity. $Z$ is the partition function and in our simple example just gives $1 + Ne^{-\beta E_I}$. If that's all correct, then the finite temperature Green's function should just be $$G(T,k,\omega) = \frac{G_0(k,\omega) + Ne^{-\beta E_I} \left( G_0 + G_0^2 1/N T_{kk}(\omega)\right)}{1 + Ne^{-\beta E_I}}$$ which can be simplified to $$G_0(k,\omega) + \frac{e^{-\beta E_I} G_0^2 T_{kk}}{1 + Ne^{-\beta E_I}}$$ What confuses me here is that if we let $N$ go to infinity, the impurity term will again vanish at small temperature and thus seems to play no role at all in determining the Green function or the spectral function, but clearly that cannot be correct, since we know that impurities can, for example, lead to bound states. Am I missing something really obvious here? EDIT: With your answers so far, let me just add what I think I could do right now. At low temperature, I'd assume a low density of thermally excited impurities, so I will do an expansion to first order in that density. The partition function is $$Z = (1 + e^{-\beta E_I})^N = \sum_{n}^N \begin{pmatrix}N\\\n\end{pmatrix} e^{-\beta n E_I}.$$ The numerator for $\langle G \rangle$ becomes approximately, only accounting for interactions with _one_ impurity at a time, justified by the low density: $$\sum_n e^{-\beta n E_I} \begin{pmatrix}N \\\ n \end{pmatrix} \left[ G_0(\omega) + n T(\omega) G_0^2(\omega)\right]$$ Now the $G_0$ don't depend on anything we're summing over, so we can pull them out of the sum and obtain $$\langle G \rangle = G_0 + \langle n \rangle T G_0^2$$ where I omit the momentum and frequency arguments because they're the same everywhere anyway. I like this answer, it makes intuitive sense: The extra term due to impurity scattering is, to first order, linear in the expected number of thermally excited impurities. I understand that this gets the physics of disordered systems wrong since it can't account for interference due to multi-impurity- scattering and thus for example won't ever see Anderson localization. But just to get started on this, I think it's at least the correct way to start thinking about things.


title: To what extent are quantities fundamental?

text: Arguably the most well-known and used system of units is the SI-system. It assigns seven units to seven ‘fundamental’ quantities (or dimensions). However, there are other possible options, such as Gaussian units or Planck units. Until recently, I thought that these different systems differed only in _scale_ , e.g. inches and metres are different units, but they both measure _length_. Recently though, I discovered that it is not simply a matter of scale. In the Gaussian system for example, charge has dimensions of $[mass]^{1/2} [length]^{3/2} [time]^{−1}$, whereas in the SI-system it has dimensions of $[current] [time]$. Also, I have always found it a bit strange that mass and energy have different units even though they are equivalent, but I find it hard to grasp that a quantity can be ‘fundamental’ in one system, and not in an other system. Does this mean that all ‘fundamental’ quantities are in fact arbitrary? Would it be possible to declare a derived SI-unit fundamental, and build a consistent system with more base units? What is the physical meaning of this?


title: What is probability to find electron at certain distance from nucleus

text: Given for example, Hydrogen electron in ground state. What is probability to find that electron at certain distance (not interval of distances) from center of nucleus, for example at radial coordinate $r=0.5 \cdot 10^{-10}$ (and any angle)?


title: Clouds in closed hydrosphere

text: Is possible estimate the needed size of an geodesic dome (like in the Eden project) for creating an real hydrosphere - especially clouds (and rain)? With other words, under what circumstances can happen clouds formation in an closed space -isolated from the outer atmosphere?


title: Modification of de Donder gauge

text: The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \bar g_{ab} + \gamma_{ab}$, then the de Donder gauge reads   $\nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma) = 0$. The partial differential equation for the gauge transformation vector $v^a$ is $ \nabla^b\nabla_b v_a + R_a^b v_b = \nabla^a(\gamma_{ab} - \frac{1}{2}\bar g_{ab}\gamma)$. In chapter 7.5 of Wald, I read that this equation can always be solved because it is of the form $g^{ab}\nabla_a\nabla_b \phi_i + \sum_j (A_{ij})^a\nabla_a \phi_j + \sum_j B_{ij}\phi_j + C_i$.   Theorem 10.1.2 of Wald says that in a globally hyperbolic spacetime this equation has a well posed initial value formulation on any spacelike Cauchy surface. In stead of de Donder gauge, I want to use a similar gauge:   $\nabla^a(\gamma_{ab} - n \bar g_{ab}\gamma) = 0$.   The partial differential equation changes to   $ \nabla^b\nabla_b v_a + (1 -2n)\nabla_a\nabla_b v^b + R_a^b v_b = \nabla^a(\gamma_{ab} - n\bar g_{ab}\gamma)$. This equation is not covered by theorem 10.1.2 of Wald. My question is: is the existence of a solution for this equation guaranteed in an AdS background when $n=1$?


title: Can different species of particles annihilate with other species

text: Obviously electrons annihilate with positrons, but can a muon annihilate with an positron, or can an anti-taon cancel with a muon? similarly for quarks of different species, e.g. u and anti-strange. I think this is possible as long as quantum numbers like charge and spin are conserved, with the excess energy being given off in kinetic energy, but has it ever been observed?


title: What do the four components of Dirac Spinors represent in the Standard Model?

text: I've been trying to get my head around the formalisms used in the Standard Model. From what i've gathered Dirac Spinors are 4 component objects designed to be operated on by Lorentz Transformations much like 4-Vectors are in Special Relativity. However they also incorporate additional information: Spin and "Handedness". Due to the nature of Spin, Spinors also transform differently then vectors. This leaves me with the impression that the 4 components can be classified as: Left Handed and Spin Up, Left Handed and Spin Down, Right Handed and Spin up, Right Handed and Spin Down. My question is if this impression is the right general idea or not?


title: What is the relation between N=2 super Yang-Mills and its twist

text: My question is what is the relation between N=2 super Yang-Mills and its twisted version topological field theory? After twisting N=2 super Yang-Mills, i.e. diagonally embedding $SU(2)'_R$ into $SU(2)_R \times SU(2)_I$, we get a topological field theory. My question is since N=2 SYM and TQFT are different i.e. one is physical and the other is topological. Why can we use TQFT to calculate partition of N=2 SYM? What are the same for these two different theories? **Update** : From the second paper of Trimok, the authors claim that SYM under twist are just redefination. How to understand it?


title: A dimensional regularization identity

text: I had a question on a dimensional regularization identity. A reference or a quick sort of derivation will be greatly appreciated. I looked at some textbooks of QFT, but couldn't find the one I was looking for. I found in http://www.maths.tcd.ie/~cblair/notes/list.pdf, a result for $\int\frac{d^dp(p^2)^a}{(p^2+D)^b}$ (see eq. 3.2 of the above link). I wanted something which is $\int\frac{d^dp(p^2)^a}{(p^2+2pq+D)^b}$ i.e the integrand has a linear power of $p$ too. May be a derivation of the previous equation will help. But anyway, some light on $\int\frac{d^dp(p^2)^a}{(p^2+2pq+D)^b}$ or $\int\frac{d^dp(p_\mu p_\nu..p_\lambda)}{(p^2+2pq+D)^b}$ is what I need. Thanks in advance.


title: Why does water ($\mathrm{H_2O}$) only have two distinct fluid phases?

text: Water (and other substances) can exist in many distinct solid phases (with different crystallic micro-structure), but only in two fluid phases - liquid and gaseous, in which the molecules are oriented randomly (they is no long range order). Is there an explanation in the molecular theory, why there are just two "disordered" phases? Why isn't there just one? Or more than two?


title: What is the sense of introducing generating functional to the summands of expansion of S-matrix?

text: Let's have generating functional $Z(J)$: $$ Z(J) = \langle 0|\hat {T}e^{i \int d^{4}x (L_{Int}(\varphi (x)) + J(x) \varphi (x))}|0 \rangle , \qquad (1) $$ where $J(x)$ is the functional argument (source), $\hat {T}$ is the chronological operator, $\varphi (x)$ - some field. I want to understand the reasons for its introduction for the summands of expansion of S-matrix. As I read in the books, it helps to consider only the vacuum expectation values, forgetting about in- and out-states. But in $(1)$ appear summands like $\int \frac{J(p)dp}{p^2 - m^2 + i0}$ instead of the contributions from external lines. It may refer to the internal lines. So what to do with them and are there some other reasons to introducing $(1)$ except written by me?


title: How can two time theories be compactified to 3+1 without any Kaluza-Klein remnants

text: I have recently been looking into the two-time theories and the implied concepts. For me this seems slightly hard to grasp. How can I see the basic concept in this theory in a fundamental way based on its implied interaction with normal 3+1 dimension? I am interested specifically in how gauge symmetries that effectively reduce 2T-physics in 4+2 dimensions to 1T-physics in 3+1 dimensions without any Kaluza-Klein remnants.


title: CGS Units to SI Units

text: This may be a stupid question but I am having trouble getting the same result in CGS units as I would if I used SI units for a unitless calculation. I have to calculate $E_c=\frac{m_e c\, \omega_c}{eB}$ I have in CGS units that $\frac{e}{m_ec}$ is $(1.76\times10^{7}s^{-1}G^{-1})$ and $B=10^{-4}G=10^{-8}T$ and $\omega_c$ is just some frequency. Essentially, when I do this calculation in SI units and in CGS units I get very different results. From this equation it seems clear to me that $\frac{e}{m_ec}$ in CGS and SI should only be off by a few factors of 10 (namely 4) since $B$ is scaled by $10^{-4}$ from CGS to SI, but this is not the case... What am I doing wrong? Edit: Here are my calculations. (i) CGS: $\frac{(1.52\times10^{18}s^{-1})}{(1.76\times10^7s^{-1}\,G^{-1})(3\times10^{-4}G)}$ (ii) SI: $\frac{(9.11\times10^{-31}\mathrm{kg})(3\times10^8\mathrm{m \, s^{-1}})(1.52\times10^{18}s^{-1})}{(1.6\times10^{-19}C)(3\times10^{-8}T)}$ Unless the value I'm using for $\frac{e}{m_ec}$ in CGS is wrong...


title: On Bell's Inequality (Classical Intuition) and Quantum Mechanical Counter Intuition

text: This posting is directly related to the issue in The System and the Measuring Gadget. The QM expectation is given by: $$\langle\sigma_{1}.\vec{a}{\;}\sigma_{2}.\vec b\rangle=-\vec a.\vec b$$ In the above relation we are considering _the measured value_ of spin which is the outcome between the value of some property of the system itself and the measuring gadget The "classical" formula for evaluating the expectation with the hidden variable is as follows: $$P(\vec a.\vec b)=\int d\lambda\,\rho (\lambda) A(\vec a)B(\vec b)$$ Now some property of the system may depend on the value of $\lambda$ and the probability distribution $\rho (\lambda)$. Is the effect of measurement being fully accounted for by the the hidden variable $\lambda$ and the pdf $\rho (\lambda)$, especially in view of the fact that the process of measurement modifies the wave function itself. Would it be possible remove the contradiction between QM and commonsense intuition,expressed through Bell's Inequality, by considering the above factors? Reference for Bell's Original Paper.


title: Projectile motion equations @ very (!) high starting velocity

text: I've already searched Physics StackExchange for some similar question but I didn't find anything about this. Assumptions:   * Earth is a perfect sphere with it's core (X,Y,Z) -> (0,0,0) as a reference-frame center   * Air resistance can be ignored   * Earth rotation can be ignored   * Moon gravity-effect can be ignored And if I know (projectile starting properties):   * Current Earth GPS-coordinates   * Starting angle   * Starting direction (relative to (0,0,0))   * Starting velocity (can be larger than Earth-escaping speed)   * (mass of the projectile is irrelevant (I guess) when we know starting velocity of the projectile) How could I calculate aprox. coordinates of a projectile landing somewhere around the Earth globe (OR detect that projectile will "leave" the Earth)? EDIT: I've found tons of links on the net about projectile trajectories when starting velocities are quite small (& Earth can be considered like a flat plane), but non about upper situation.


title: More data points or more averages?

text: Perhaps an elementary questions. Given a time limited measurement situation, would it be better for one to measure more averages or more data points? More averages will increase the SNR by $$\sqrt{n}$$ , i.e., making the data point more reliable, but more data points may make the fitting better. Consider the model is A*exp(-Bt) + C*exp(-Dt) which is a difficult model to fit when noise is introduced. Assume due to time limit, one could only do 100 measurements in total. Should one measure 20 data points with 5 averages each or 100 data points, or 5 data points with 20 averages each?


title: What is the capture cross-section of a black hole region for ultra-relativistic particles?

text: What is the capture cross-section of a black hole region for ultra- relativistic particles? I have read that it is $$\sigma ~=~ \frac{27}{4}\pi R^{2}_{s}$$ for a Schwarzschild BH in the geometric optics limit. Where does the coefficient come from? Edit - Sources:   1. Absorption and emission spectra of a Schwarzschild black hole.   2. Fermion absorption cross section of a Schwarzschild black hole.


title: Deriving group velocity

text: At the introduction to quantum mechanic phase $v_p$ and group $v_g$ velocities are often presented. I know how to derive $v_p$ and get equation: $$ \scriptsize v_p=\frac{\omega}{k} $$ What i dont know is how to explain a derivation of a group velocity $v_g$ to myself. Our professor did derive it, but i am having some difficulties with it. * * * 1st he did a superposition of 2 waves with the same amplitude $s_0$: $$ \scriptsize \begin{split} s &= s_0 \sin(\omega_1t-k_1x) + s_0 \sin(\omega_2t - k_2 x)\\\ s &= s_0 \left[ \sin(\omega_1t-k_1x) + \sin(\omega_2t - k_2 x) \right]\\\ s &= 2s_0 \left[ \sin\left(\frac{(\omega_1t-k_1x)+(\omega_2 t -k_2 x)}{2}\right) \cdot \cos\left(\frac{(\omega_1t-k_1x)-(\omega_2t - k_2 x)}{2}\right) \right]\\\ s &= 2s_0 \left[ \sin\left(\frac{\omega_1 + \omega_2}{2} t - \frac{k_1 + k_2}{2} x\right) \cdot \cos\left(\frac{\omega_1 - \omega_2}{2} t - \frac{k_1 - k_2}{2} x\right) \right]\\\ s &= 2s_0 \left[ \sin\left(\overline \omega t - \overline{k} x\right) \cdot \cos\left(\frac{\Delta \omega}{2} t - \frac{\Delta k}{2} x\right) \right]\\\ \end{split} $$ * * * Here $\overline \omega$ is larger than $\Delta \omega$ and this is why:   * $\scriptsize\sin \left(\overline{\omega}t - \overline k x\right)$ is a part which declares an _envelope_ and   * $\scriptsize\cos \left(\frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x\right)$ is a part which declares _phases inside an envelope_. ![enter image description here](http://i.stack.imgur.com/GgLq5.png) * * * Than professor takes only a part which declares an envelopa and says that phase of this part must be **constant** like this: $$ \scriptsize\frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x = const. $$ **QUESTION:** What does this mean? Does a constant phase mean to only look at one point which is allways at the same distance from $x$ axis? Please someone explain this a bit. Well then he derives the group velocity easily from now on like this: $$ \scriptsize \begin{split} \frac{\Delta{\omega}}{2}t - \frac{\Delta k}{2} x &= const.\\\ \frac{\Delta k}{2} x &= \frac{\Delta{\omega}}{2}t - const.\\\ x &= \frac{\Delta{\omega}}{ \Delta k} t - \frac{2}{\Delta k}const.\\\ \end{split} $$ If i partially diferentiate $x$ i finally get group velocity: $$ \scriptsize \begin{split} v_g &= \frac{\partial x}{\partial t} \\\ v_g&= \frac{\Delta{\omega}}{ \Delta k}\\\ v_g&= \frac{\textrm d{\omega}}{ \textrm d k} \end{split} $$


title: How to include Damping in a Simple harmonic oscillator

text: Im designing a model for Kelvin Method. Some of my calculation results are as follows:   1. Radius of the membrane : 50 micron    2. thickness of the membrane : 3.25 micron    3. resonate frequency : 1.32MHz   4. spring constant : 1.81*10^4 N/m   5. mass : 1.045*10^-8 Kg ![enter image description here](http://i.stack.imgur.com/B6rjl.png) Vdc is applied, so that we can get a continuous vibrations of the membrane. Amplitude of vibration is same as the amplitude of Vac when there is no damping. i want to calculate the amplitude when damping(air and support) is included I want to introduce damping into my design. Im not sure of the calculation. How can i calculate the following   1. Damping coefficient (considering the system is surrounded by air)   2. damping ratio    3. Final amplitude with damping


title: How does the shape (form; not cross-section) of a spring impact performance?

text: Cylindrical compression springs are everywhere and then some applications choose other forms like rectangular or unique polygonal form. What impact does the form of a compression spring have and how do you calculate an optimal solution?


title: Why there is added a partial time derivative in formula for time derivative of potential energy?

text: In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from? ![enter image description here](http://i.stack.imgur.com/xkrx5.png)


title: What is difference between $\frac {dr}{dt}$ and $\frac {\partial r}{\partial t}$?

text: What is difference in **physical meaning** of partial time derivative and ordinary derivative of $r$? $$\frac {\partial r}{\partial t}\quad\text{and}\quad \frac {dr}{dt}.$$ I know that ordinary time derivative is velocity.


title: What is the common difference between partial time derivative and ordinary time derivative?

text: What is difference between partial and ordinary time derivative? for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$? where the $v$ is velocity.


title: Calculating the stability/invertibility of energy maps by multivariate time series model

text: You have the following energy maps for 14 different structures so about 4-5 pixels for one unit. The color density represents energy. ![enter image description here](http://i.stack.imgur.com/4VNj3.png) I am thinking that you can set multivariate time series models from energy maps by `vgxqual` and `vgxset` for instance in Matlab. **How are energy maps related to the measure of stability in quantum physics a) by wave models and b) non-commute dynamics?** I think the latter measure is the right one here. There may be different models too to measure stability however.


title: Two small objects are placed at rest in an empty universe a great distance apart. Given an infinite amount of time will gravity cause them to meet?

text: Let's say the objects are marble size or even single atoms or quarks. They are placed in an otherwise empty universe(expanding or non-expanding) at opposite ends of the universe with an arbitrarily large distance between them. With a combination of great enough distance and small enough mass will the gravitational pull between the two objects ever equal zero or merely approach it? Given an infinite amount of time would they ever meet?


title: Mirrors and light beam divergence technology limits

text: There are many applications for orbital space mirrors in astronomy (better telescopes) and space propulsion (solar power for deep space probes), but this is limited by the minimum beam divergence achievable with current technology So, i'm trying to understand what physical and technological limits exists in our capacity to build mirrors that can keep as small beam divergence as possible. For instance, a sail probe to Saturn would require that the beam doesn't significantly diverge above 300m-600m (the biggest sail we can conceive of building in the inmediate future) at distances as big as 5-6 AU ($10^{11} - 10^{13}$ meters) What is the best beam focusing divergence we can achieve for solar light with mirrors _right now_ , and what limits the improvement of this? technological limits? fundamental physical limits? **Edit** let's assume the concrete case of a wavelength of $10^{-6}$ meters, and a distance of $10^{12}$ meters (Neptune orbit). Can't i, for instance, build a focusing element with a focal length of $10^{12}$ meters that would push the far field beam divergence at farther distances from the focal point? Is this a manufacturing limitation of focusing element engineering (not enough precision to build lens made of atoms with the required focal length), or something more instrinsic, say, a focal point cannot be farther than some finite distance that depends on the wavelength?


title: Entropy exchange of a free fall

text: I have a problem in which the tell me that you drop a bag of 50 kg of sand from 10 meters high, and you have to caltulate the entropy difference of the sand, asuming that the speific heat of the sand is so high that its temperature (298K) doesn't change. My result is 16.5 J/K, and my problem is with the sign. The book doensn't give an explanation but just an answer and it's positive, but I'm thinking that in the impact the heat must flow out of the sand, as it's loosing its kinetic energy, so the entropy would decrease, and the surroundings entropy would increase; what am I missing, or what is really going on here?


title: What is the force on the arms in a pushup?

text: What force do the arms have to generate to do a pushup? Let us look a this simplified model: ![pushup physics](http://i.stack.imgur.com/ueWHS.jpg) The body can be represented by the green plank of mass B. Its angle to the ground is $\theta$. _This question was asked onfitness.stackexchange.com, although not phrased with physics in mind._


title: Misunderstanding of Special Relativity

text: Person A in reference frame A watches person B travel from Star 1 to Star 2 (a distance of d). Of course, from person B's reference frame, he is at rest and is watching Star 2 traveling to him. Now we know from the principle of relativity, each one will measure the other one’s clock as running slower than his own. Let’s say that Person A measures Person B’s speed to be v, and that Person A measures 10 years for person B to make it to Star 2. Let’s also say that person B is moving at the speed so that the Gamma Factor is 2. This means person A observes person’s B’s clock to have elapsed a time of 5 years. Now let’s look at this from Person B’s perspective: Person B observes Star 2 approaching (and Star 1 receding from) him also at speed v. Since the two stars are moving, the distance between them is length contracted (after all, if there were a ruler in between the stars, the moving ruler would be contracted) by a factor of 2. Since person B measures the initial distance to Star 2 to be d/2 and its speed v, he calculates the time to Star 2’s arrival to be 5 years. Since he observes person A’s clock as running slow (since Person A is moving also at speed v), when Star 2 arrives, he measures Person A’s clock to have elapsed a time of 2.5 years. Do you see why I’m confused? Person A measures Person B’s elapsed time to be the same as Person B measures Person B’s elapsed time (both 5 years), but Person B does not measure Person A’s elapsed time to be the same as Person A measures Person A’s elapsed time (Person B get’s a measurement of 2.5 years while Person A measured 10 years). This is asymmetrical, which probably means it is wrong. But I’m not sure what the error is. I suspect if I had done this correctly, each person should measure his own elapsed time to be 10 years and measure the other’s elapsed time to be 5 years. This would be symmetrical and would make the most sense, but again, I can’t seem to justify how person B wouldn’t measure his trip time to be 5 years. What's my mistake?


title: Predict final temperature by taking temperature samples?

text: Is it possible to predict what the final temperature will be by taking temperature samples. For example, an object is 0ºC and moved to a room above 0ºC. I'm taking temperature of the object using a thermometer every second. Can I predict (approximation) on what the final temperature would be after a few samples? I guess the more samples the more accurate it would be. Can I calculate when the final temperature might occur based the rate of the temperature change? Is there any formulas for these kind of calculations?


title: Born-Oppenheimer separation in Dirac bra-ket notation

text: Most derivations I have seen of the Born-Oppenheimer approximation are made using wave-functions. To understand it better, I was trying to write a derivation using Dirac notation, but I am stuck. I am going to post what I have done so far so you guys can help me out. The hamiltonian of the molecule can be written as the sum of two parts, $H_\text{mol} = H_\text{el} + H_\text{nuc}$, where $H_\text{nuc}$ is the hamiltonian of the nuclei by themselves, and $H_\text{el}$ is the hamiltonian of the electrons interacting with the nuclei: $$H_\text{el} = T_\text{el} + V_\text{el-el} + V_\text{el-nuc}$$ $$H_\text{nuc} = T_\text{nuc} + V_\text{nuc-nuc}$$ We want to find the energy levels of the molecule. That is, we want to solve $H_{\text{mol}} |\mathcal{E}\rangle =\mathcal{E} |\mathcal{E}\rangle$, where $\mathcal{E}$ and $|\mathcal{E}\rangle$ are the eigenenergy and corresponding eigenket of the molecule. The state space of the molecule can be separated into electronic and nuclear parts: $\mathcal{S}_\text{mol} = \mathcal{S}_\text{el} \otimes \mathcal{S}_\text{nuc}$. Let $|R\rangle \in \mathcal{S}_\text{nuc}$ be the position basis of the nuclei, where $R$ denotes the coordinates of all the nuclei. Define $H_{\text{el}}(R) = \langle R| H_{\text{el}} |R\rangle$, which is an operator in $\mathcal{S}_{\text{el}}$. Let $E_a(R)$ and $|E_a(R)\rangle$ be the eigenvalues and corresponding eigenkets of $H_{\text{el}}(R)$ in $\mathcal{S}_{\text{el}}$, so that $H_{\text{el}}(R) |E_a(R)\rangle = E_a(R) |E_a(R)\rangle$. For each $R$, the kets $|E_a(R)\rangle$ make a basis for $\mathcal{S}_{\text{el}}$. The set of kets $|E_a(R)\rangle |R\rangle \in \mathcal{S}_{\text{mol}}$ for all $R$ and $a$ then make a basis for $\mathcal{S}_{\text{mol}}$. Using this basis, the state of the molecule can be written: $$|\psi \rangle =\sum _a \int \chi_a(R) |E_a(R)\rangle |R\rangle dR$$ where $\chi_a(R)$ is an amplitude function. Note that: $$H_\text{el} |E_a(R)\rangle |R\rangle = [ H_\text{el}(R) |E_a(R)\rangle ] |R\rangle = E_a(R) |E_a(R)\rangle |R\rangle $$ Therefore, the molecular eigenproblem $H_{\text{mol}} |\psi \rangle =\mathcal{E} |\psi \rangle$ can be written: $$\sum_a \int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |E_a(R)\rangle |R\rangle dR = 0$$ Multiplying by $\langle E_a(R)|$ on the left: $$\int (E_a + T_\text{nuc} + V_\text{nuc-nuc} - \mathcal{E}) \chi_a(R) |R\rangle dR = 0$$ At last, we define a ket $|\chi_a\rangle \in \mathcal{S}_\text{nuc}$ such that $\chi_a(R) = \langle R | \chi_a \rangle$: $$|\chi_a\rangle := \int \chi_a(R) |R\rangle dR$$ Then we can write: $$(T_\text{nuc} + V_\text{nuc-nuc} + E_a - \mathcal{E}) | \chi_a \rangle = 0$$ **HERE ENDS MY DERIVATION SO FAR.** I must have done something wrong somewhere, because the final equation that I obtain is, as far as I can tell, the Born-Oppenheimer _approximation_ , but I am obtaining it here as an _exact_ equation. What did I do wrong? Also, if anyone knows of some reference, textbook or paper, that deals with the Born-Oppenheimer approximation in Dirac notation, please post it.


title: What happens when particle-antiparticle pairs annihilate in MWI?

text: The many-worlds interpretation of quantum physics is built around a configuration space, where the position of a particle is three components of the position of that universe. What happens with particle-antiparticle creation or annihilation? It can't just change the number of dimensions, can it?


title: Why can different batteries with the same  voltage send different currents through the same object?

text: According to an answer in this thread on Skeptics: > If you take one of the little 12V garage door opener batteries and short out > (directly connect) the two terminals with a piece of wire or something else. > You'll get a light current flow through the wire or metal. It may get a > little warm. This battery is only capable of supplying a small amount of > current. > > If you take a 12V car battery and short out the two terminals (don't do it, > it's not fun), you will be met with a huge current arc that will likely > leave a burn mark on whatever was used to short it. This is because the car > battery is capable of discharging a large amount of current in a very short > period of time. I'm not sure how this could work given Ohm's law V=IR. If we assume the resistance of toucher is constant, then we'd expect the current to be the same as well.   1. Could it be that a car battery has less internal resistance than a garage battery?   2. Does it have anything to do with contact area? If it does, then how would you model it? Generally resistances add in series, but if I only half touch a contact, then neither the battery nor I change, so you'd expect our resistances to stay the same, but somehow our total resistance changes. So which objects resistance would change - mine or the batteries?


title: Are black holes really that special?

text: ~~Science and science fiction alike~~ Science fiction describes black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time. However, are they really that special? My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark. And, if black holes _must_ have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?


title: recommendation for a physics history/non-fiction book

text: I know that there are a lot theses being published on lives of physicists. Is there a history/non-fiction book that tracks the development of a problem chronologically? Like pieces of a puzzle. I would like it to be mathematical and trying to get into the heads of people trying to solve that problem. Something like a case study.


title: Is it possible for wind to break the sound barrier?

text: I understand that in nature wind would never get high enough, but I am just curious as to whether physics would allow this to occur or not.


title: Entropy $dQ=TdS$ and Work $dW = -pdV$ conditions?

text: What are the conditions in order for the equations: Entropy $dQ = TdS$ and Work $dW = -p dV$ to work? I think for $dQ = T dS$, it must be a reversible process? But for $dW = -p dV$, shouldn't it always hold?


title: Definition of Information in Information Theory

text: I am not sure in which SE site I have to put this question. But since I have learnt Shannon Entropy in the context of Statistical Physics, I am putting this question here. In the case of Shannon Information theory, he defines information $I$ for an $i^{th}$ event as, $$ I_i = -\ln P_i \qquad \qquad \forall i=1,...n. $$ Based on this definition we further define Shannon Entropy as average information, $$ S_\text{Shannon} =\langle I\rangle = -\sum\limits_{i=1}^n P_i\log_2P_i .$$ My question is what is the motivation behind defining entropy as some function that is inversely related to probability? I was told by my professor that lesser the probability of an event more information it possesses, although am still not convinced about this fact. Secondly, what is the reason in choosing the logarithmic function in this definition? Are there places where this definition of information is forfeited?


title: Is there a physics Puzzles site like Project Euler?

text: I am looking for a site that publishing riddles and problems such as Project Euler publishes problems in computer science. Is there any similar site that is hardcore in the area of physics? This site has some good problems set, but it seems like there isn't a community and sharing of solutions/ideas like in Project Euler.


title: Are all classically impossible quantum possibilities entangled?

text: Any entangled state represents a quantum possibility that is classically impossible.   Is the converse true?   That is, are all states that are quantum mechanically possible but classically impossible entangled in some way?   If so, can you give a proof, or a reference to a proof?   If not, can you give a counterexample?


title: Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism

text: I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure). The topic is the Faddeev-Jackiw treatment of Lagrangians which are singular (Hessian vanishes) - similar to what Dirac does, but without the need to differentiate between first and second class constraints. Just looking at classical stuff here, no quantization. Starting with the Maxwell Lagrangian $$\mathcal{L}=F_{\mu\nu}F^{\mu\nu}$$ where $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ we see that it's second order in time derivatives acting on A. We choose to write it in first order form like this $$\mathcal{L}=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})F^{\mu\nu}-{1\over{2}}F_{\mu\nu}F^{\mu\nu}$$ where we're treating $F^{\mu\nu}$ now as an auxilliary, independent variable. Having defined this, Faddeev says "we rewrite (the last equation) as: $$\mathcal{L}=(\partial_{0}A_{k})F^{0k}+A_{0}(\partial_{k}F^{0k})-F^{ik}(\partial_{i}A_{k}-\partial_{k}A_{i})-{1\over{2}}(F^{0k})^2-{1\over{2}}(F^{ik})^2$$" My question is how does he arrive at this from the previous equation ? I don't see how just expanding the indices into time and space values ever gets me to $A_{0}(\partial_{k}F^{0k})$ I can see how there's something special about $A_{0}$, since when I write out the EOM for the first order Lagrangian, $A_{0}$ drops out, which indeed it should do because we'll end up with it being a Lagrange multiplier. I just can't see how you end up with that term, with $A_{0}$ multiplying $\partial_{k}F^{0k}$. It's clearly correct since $A_{0}divE$ is just the Gauss law constraint.


title: Could a solar flare cause the Earth's magnetic poles to reverse?

text: With all the hype of the impending "2012 Mayan doomsday" I was thinking it might be interesting to see what principles of physics prevent the theories of doomsday from occurring. One overarching theory is that on December 21, 2012, the two magnetic poles will reverse as a result of a solar flare equivalent to 100 billion nuclear bombs, enough to reverse the geomagnetic poles. Is there any way a solar flare could lead to reversal of the magnetic poles? If not, what prevents it from happening?


title: Resistive force proportional to velocity

text: Find the displacement and velocity of horizontal motion in a medium in which the retarding force is proportional to the velocity. I kind of understand how to do this problem. We know that the resistive force $F_r \propto v$. Since $F_r$ is the only force present in the x-direction, Newton's second law gives $$F_r=ma=m\frac{dv}{dt}.$$ My book then says that $F_r=-kmv$. So thus we have $$-kv=\frac{dv}{dt},$$ from which it is trivial to find expressions for $v(t)$ and $x(t)$ by using initial conditions and integration. The only part about the problem I don't understand is why $F_r=-kmv$. Why does the retarding force depend on the mass $m$? Since $F_r \propto v$, shouldn't we just stick a proportionality constant $k$ in there and have $F_r=-kv$?


title: What happens to the impinging light if its wavelength is smaller than the absorption band of a semiconductor?

text: Are all wavelengths absorbed in the semiconductor regardless of material's absorption band?


title: Gravitational Potential Energy (GPE) to Kinetic Energy (KE) transfer in Satellite Orbits

text: I am stumped by a mundane A Level Physics question (teacher of physics here obviously a bit short of practice!). My colleagues and I are stumped and were wondering if any one could help us. It concerns the change in KE of a satellite in orbit, if a satellite drops it’s orbit to that of a lower radius, it’s change in GPE is given by the expression $GMm(1/r_1-1/r_2)$. Due to the conservation of energy, I would have thought that the gain in KE would therefore be equal. But I keep seeing in past exam papers that the answer is HALF of this value! However I cannot seem to find a satisfying reason why anywhere.


title: Does the velocity of a falling object increase as the mass of a object increases?

text: I am doing an experiment to show the effect on the velocity of a falling mass due to change in mass. How can i justify my prediction that the velocity of the falling mass will increase as i increase the mass. For my experiment i am dropping masses between 100-800g threw 1000ml of water and timing how long the mass takes to fall the 35cm height. From my results i can see that as the mass increases the time taken for the mass to reach the bottom of the 35cm decreases. Is this due to change in velocity and am i correct in using velocity and mass as my measurements?


title: Resultant vector problem

text: $\newcommand{\v}[1]{\vec #1}\newcommand{\i}{\hat i}\newcommand{\j}{\hat j}$ Problem statement (1,2) > A shopper at the supermarket follows the path indicated by vectors $\v A, \v > B, \v C, \v D$ in the figure. Given that the vectors have magnitudes > $A=51\:\mathrm{ft}, B=45\:\mathrm{ft}, C=35\:\mathrm{ft}, > D=13\:\mathrm{ft}$, find the total displacement of the shopper using (a) the > graphical method and (b) the component method of vector addition. Give the > direction of displacement relative to $\v A$ > > ![enter image description here](http://i.stack.imgur.com/iJbOUm.jpg) My work: ![enter image description here](http://i.stack.imgur.com/RCyp8m.jpg) $$\begin{align}A &=&0&\i + &51&\j \\\B &=&45&\i + &51&\j\\\C &= &10&\i + &-35 &\j\\\ D &= &10&\i + &-13 &\j \\\\\text{Resultant} &=&54&\i+&65&\j\end{align}$$ Essentially I am adding all the components to get the resultant vector but it is not leading me to the right answer. What am I doing wrong here?


title: Fermi Energy Variation

text: 1. What would be a good Internet link that would properly explain Fermi Energy?    2. How does the Fermi Energy of a material vary with external influence, such as doping of the material, and applied electric and magnetic fields for instance? What other factors can effect Fermi Energy?   3. In single walled Carbon nanotubes, how does the Fermi Energy vary with the geometric shape, i.e. length and radius of the nanotube? How does chirality affect the Fermi Energy?


title: Particle coming across a step potential barrier

text: My quantum mechanics textbook says that when a particle (in the classical case) comes across a potential-step barrier of finite height, if it has sufficient energy to surmount the barrier, it will continue on with reduced kinetic energy. I'm finding this hard to understand since force is given by $$F=-\frac{dV(x)}{dx}$$ For a step barrier, this should give an infinite force acting (in the opposite direction) on the particle when it comes in contact with the barrier. The only other thing I can think of is to model the force on the particle by a dirac delta function, so we effectively see it getting an impulse in the opposite direction, which could lower its kinetic energy. Is this reasoning right?


title: Sound intensity and space/things in the way

text: Suppose you have two stationary people, A and B, who are equally effective in terms of hearing ability, and A emits a sound that is heard by B at a certain intensity. If they remain stationary, and B makes the exact same sound, does A hear the sound at the same intensity as B?


A lepton is an elementary particle. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. The heavier muons and taus will rapidly change into electrons through a process of particle decay, Does it happen at high energies (heavier leptons decay)? So, It may be possible that electron decay in very low energies.


title: Definitions of the Normal Ordering Operator in CFTs and QFTs

text: Recall the normal ordering of bosonic operators in QFT is defined by a re- arrangement of operators to put creation operators to the left of annihilation operators in the product. This is designed to avoid accidentally annihilating $|0\rangle$ when looking at an expectation value in relation to the vacuum state. $ : \hat{b}^\dagger\hat{b} : \: =\: \hat{b}^\dagger\hat{b} \\\ : \hat{b}\hat{b}^\dagger: \: = \: \hat{b}^\dagger\hat{b} $ In CFTs, I've seen defined the normal ordering of operators as the zeroth basis field of the Laurent expansion of the radial ordering product. $\mathcal{R}(a(z)b(w)) = \sum_{n = -n_0}^\infty (z-w)^n P_n(w),$ and select $P_0(w) = \: : a(w)b(w) : $ Is there an equivalence between these two definitions? What is the CFT analog of not annihilating the vaccuum/ how do we show this definition has that property?


title: Is there an alternative metric for isentropic efficiency that remains valid when broken up into multiple segments?

text: Say that we have an irreversible expansion process which extracts energy, like a turbine. Isentropic efficiency is commonly defined by the following relation, which applies in a similar fashion for pumps. Here, state 1 refers to the inlet conditions and state 2 refers to the outlet conditions, while the "s" indicates the isentropic pseudo-state. P, h, and s are pressure, enthalpy, and entropy. $$ \eta = \frac{ h_1 - h_2 }{ h_1 - h_{2s} } $$ This has a monumentally huge, glaring problem, that has always bothered me - it can't be applied to sub-segments of a turbine. By that, I mean let's say we keep P1 and P2 the same, but instead of one turbine, we have two in series. Here is the basic situation on a hs diagram: ![Isentropic Process](http://i.stack.imgur.com/OkrAn.png) The general structure, isobaric lines, and 3 labeled states are cannon in the literature. The line should probably be curved slightly. But let's add another point _somewhere in the middle_ of the 1->2 process. Let's label that midpoint "m". If you begin with the assumption that two irreversible expansion processes exist 1->m and m->2(by pressure) with the same isentropic efficiency as 1->2, then I'm saying that it follows that: $$s_1 = s(P_1,h_1) \\\ h_{2s} = h(P_2,s_1) \\\ h_2 = h_1 - \eta ( h_1-h_{2s} ) \\\ \text{Two-Step Process} \\\ h_{ms} = h(P_m,s_1) \\\ h_m = h_1 - \eta (h_1 - h_{ms} ) \\\ h_{2s}' = h(P_2,s(P_m,h_m)) \\\ h_2' = h_m - \eta (h_m - h_{2s}' ) \\\ \text{The Problem:} \\\ h_2' \ne h_2$$ In these equations, I intend for the 2-variable functions to be property lookups. I'm mostly using the steam tables. A more correct way of saying this is that if you assigned values for $\eta_{1m}$ and $\eta_{m2}$ such that you wind up at the intended final point of 2, then both of these values will be different from the original $\eta$. The topic has come up on the site before, considering what the derivative of some expansion line might mean. But that's already jumping the gun, because we don't even have any defined way to draw that line in the first place. I can write down $\frac{\Delta h}{\Delta s}$, but not $\frac{dh}{ds}$. Again, we could replace one turbine with two turbines, and assign them efficiencies _such that_ point 2 has the same properties. But that leaves a free degree of freedom. You could make $\eta_{1m}$ and $\eta_{m2}$ the same, or you could have one a little higher and one a little lower, without point 2 being affected. One simple way to phrase this question would be: how can I calculate P_m and h_m in a physically meaningful way? My main interest is - in doing so, what metric defines the _degree of irreversibility_. Has anyone put forth a definition of a variable which will accomplish this in a truly _differential_ sense along a non-ideal expansion line?


title: Does altitude affect sound pitch?

text: Due to differences in air pressure, temperature, and other factors, the speed of sound varies with altitude on Earth. Does this affect the pitch of the sound in any meaningful way? For example, if I had a tuning fork that vibrates at around 262 Hz, would I hear the same "Middle C" from it while standing on the shores of the Dead Sea as I would while standing at the peak of Mount Everest - an altitude difference of 9,271 meters or 30,417 feet? Would there be any difference at all, and would it be perceptible to the human ear at close range? Would the altitude difference affect how the sound is heard more at a distance than it would nearby? Or, am I not quite understanding properly how sound works?


Does Kepler's law only apply to planets? If so why doesn't it apply to other objects undergoing circular motion? By Kepler's law I'm referring to $T^2 \propto r^3$


title: Branes Collision -> Big Bang

text: ![Branes Collision - Big Bang](http://img84.imageshack.us/img84/4082/collision.jpg) Imagine universe occurred when two parallel branes collided, Momentum of Branes converted to big bang kinetic energy after Collision. Thus, high-energy quanta are high- Vibrating strings. > what cosmological issues can be particularly well explained by such a brane > scenario? what direct or indirect experimental evidence would support it?


title: How is fine tuning of standard model conceptual different than the fine tuning of PI?

text: If I were to try to find pi using a ruler and a compass, I would first try to find out how many rational line segments of the diameter I could fit around the interior circumference and then continue to refine until I began converging on some number. Before the foundations of Non-Euclidean geometry were made, people would not have even questioned that the euclidean pi had a unique nature. Why not think of the standard model parameters in the same light? Just simply as ratios that might change with the geometry of space or some something similar?


title: What is the experimental uncertainty of an ensemble measurement?

text: Let's say you measure the time it takes for 10 oscillations of a mass undergoing simple harmonic motion to within ± 0.01s, what is the uncertainty of the period of one oscillation?


title: Variational approach to search the excitations. What will happen if start from wrong reference state?

text: By 'wrong reference state' I mean a state which cannot be transformed into desired ones via variational ansatz $\left|\Psi\left[\mathbf{n}\right]\right\rangle =e^{i\hat{O}\left[\mathbf{n}\right]}\left|ref\right\rangle \;,\; H\left[\mathbf{n}\right]=\left\langle \Psi\left[\mathbf{n}\right]\right|\hat{H}\left|\Psi\left[\mathbf{n}\right]\right\rangle $ where $\mathbf{n}$ is the variational parameter (field). To be concrete, if we started with the first excitation state in a gapped system, i.e. choosing (making a guess) this first excitation state to be the reference state, then it seems to me that we will never reach the true ground state, as well as the correct excitations. The manifolds are disconnected because of the gap. Are there any concrete examples for my general question here? Maybe I have not properly described this question. Please let me know if you are interested but don't understand. I will try to clarify.


title: Problem in Hamiltonian

text: I need to elaborate the equation ,and need to know what is the physical significance and how matrices will manipulate in the equation $$ \hat{H} = (\hat{\tau_3}+i\hat{\tau_2})\frac{\hat{p}^2}{2m_0}+ \hat{\tau_3}m_0 c^2 = \left| \begin{array}{ccc} 1 & 1 \\\ -1 & -1 \\\ \end{array}\right| \frac{\hat{p}^2}{2m_0} + \left| \begin{array}{ccc} 1 & 0 \\\ 0 & -1 \\\ \end{array}\right| m_0 c^2 $$ Where $$\tau_1 , \tau_2,\tau_3 $$ are Pauli matrices and Hamiltonian comes from "Schrodinger form of the free Klein_Gordon equation And also why did we added Pauli matrices in the free Hamiltonian ?


title: A physical sense of an Inertial frame

text: Definition clarification needed, please: I am hoping to get physical sense of an "inertial frame". Do inertial reference frames all have zero curvature for their spacetime? So is an inertial frame just a flat metric? (Sorry that the question is not too profound. I have read the Wiki article for inertial frame of reference, but I'm just not entirely sure.)


title: Local inertial coordinates

text: It is said that we can introduce local inertial coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do with invertibility or something? Thanks.


title: Loopy lightning

text: What causes lightning to follow the path it does ? ![loopy lightning](http://i.stack.imgur.com/U7fBT.jpg) picture from BBC news: http://news.bbcimg.co.uk/media/images/62891000/jpg/_62891901_untitled-1copy.jpg main page: http://www.bbc.co.uk/news/in-pictures-19597250


title: Can $\langle\Omega|f|\Omega \rangle$ always be reduced to $\langle0|f|0\rangle$?

text: I've not come across any expression involving $\langle\Omega|f|\Omega\rangle$ in Srednicki's QFT book (please correct me if these exist there). On the other hand, they are abound in Chapter 7 of Peskin&Schroeder, in relation to the LSZ reduction formula. Here $|\Omega\rangle$ is the ground state of the interacting theory and $f$ is anything you can imagine that makes sense. Srednicki only uses $\langle0|f|0\rangle$ where $|0\rangle$ is the ground state of the free theory. How come this is so? Can one always go from one expression to the other by rewriting in terms of creation and annihilation operators? Srednicki also derives the LSZ but he only uses the above-mentioned $|0\rangle$.


title: Why we see upright images?

text: Since it's convex lens there in our eyes so image formed on our retina is inverted, so how come that we see upright images?


title: How to calculate time for ball to reach plane while considering mass, gravity and wind resistance?

text: The scenario that I'm having is such that a ball of radius $15mm$ is thrown from a location point $\vec{p}=(2, 5, 2)$ in a direction of $\vec{d}=(3, 0, 4)$. The initial velocity is $30m/s$. There were wind velocity at $15m/s$ in the direction of $\vec{v}=(0, 1, 0)$. The mass of the ball is $2kg$ and the gravity is $9.81m/s^2$. I'm trying to calculate the time for the ball to hit the plane at $z=0$. I started by making the throw direction as an unit vector: $\hat{\vec{d}} = \frac{\begin{bmatrix} 3\\\ 0\\\ 4 \end{bmatrix}}{\left | \begin{bmatrix} 3\\\ 0\\\ 4 \end{bmatrix} \right |}=\begin{bmatrix} \frac{3}{5}\\\ 0\\\ \frac{4}{5} \end{bmatrix}$ This is where the ball will reach at time $t$: $\vec{r(t)} = \vec{p}+t \cdot \hat{\vec{d}}$ $r_z = 2+t \cdot \frac{4}{5}$ Since equation of plane is $z=0$, if $z<15$, ball hits the plane: $r_z < 15 \Rightarrow 2+t \cdot \frac{4}{5} < 15 \Rightarrow t\geq16.25$ This would imply that when $t\geq16.25$ seconds, the ball hits the plane. But I realise this _doesn't_ consider the mass, gravity and wind resistance! So this value is probably invalid from start. I recalled that $F=ma$, so $F= 2kg \cdot 9.81m/s^2$. But $F$ is just a scalar in this case and I don't know how I can use it any where. How can I calculate the time $t$ for when the ball hits the plane with the consideration of the mass, gravity and wind resistance?


title: Infinite Energy of Point Charges (in the context of classical field theories)

text: In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?


How does repulsion and attraction of a magnet work? I have a hypothesis. We all know that repulsion works when people throw balls at each other. This is used as an analogy of how virtual particles are used to explain how magnets repel each other in some cases. The other idea is that virtual particles have momentum and anti-momentum to explain attraction and repulsion. Is it possible to explain attraction by virtual particles leaving the south pole of one magnet at some angle, let's say less than 45 degrees and arriving at the second magnet's south pole at around 90 degrees? As an after thought, the 45 degree angle would be obtained if the north pole particles were repelled by the south pole particles but now I seem t be creating a loop in my thinking.


title: Entalpy and entropy role in freezing-point depression phenomena

text: There's this "atomic" explanation of the freezing-point phenomena on Wikipedia that leaves me really intrigued. > Consider the problem in which the solvent freezes to a very nearly pure > crystal, regardless of the presence of the solute. This typically occurs > simply because the solute molecules do not fit well in the crystal, i.e. > substituting a solute for a solvent molecule in the crystal has high > **enthalpy**. In this case, for low solute concentrations, the freezing > point depression depends solely on the concentration of solute particles, > not on their individual properties. The freezing point depression thus is > called a colligative property. > > The explanation for the freezing point depression is then simply that as > solvent molecules leave the liquid and join the solid, they leave behind a > smaller volume of liquid in which the solute particles can roam. The > resulting reduced **entropy** of the solute particles thus is independent of > their properties. This approximation ceases to hold when the concentration > becomes large enough for solute-solute interactions to become important. In > that regime, the freezing point depression depends on particular properties > of the solute other than its concentration. (source: > http://en.wikipedia.org/wiki/Freezing_point_depression) I'd like to understand better how the enthalpy and entropy "behave" on this process and how they explain it.


title: Does the temperature at which water evaporates influence the amount of chalk left behind?

text: I believe the title is straightforward. Evaporating water leaves behind chalk. Assuming you evaporate the same amount of (the same type of) water, does the temperature at which this happens (and so also the time it will take) matter for the amount of chalk that is left behind? Maybe a little side-question: Does the surface or container from which evaporation happens matter?


title: Identifying a critical phenomena?

text: I have a system with a number of measurables (in time). Some measurables are discrete some are continuous (within the measurement accuracy). How can I determine whether my system experiences criticality or not? I am looking for many different ways to (dis)proof criticality. See here for critical phenomena http://en.wikipedia.org/wiki/Critical_phenomena.


title: Three-mass, two springs copled oscillator NOT attached to walls

text: Int he three-mass coupled oscillator problem, we often see it stated that you have three masses, (they can be equal or not, but we'll assume they are equal here) connected by two springs and then another set of springs connecting the masses to the walls on the end. We assume that the spring constants are all the same. (We're assuming one-dimensional motion here as well). So the equations of motion are usually written thus: $$m\ddot x_1+kx_1-k(x_2-x_1)=0$$ $$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$ $$m\ddot x_3+k(x_3-x_2)+k(-x_3)=0$$ (And lord knows tell me if this is wrong and I missed a concept) Well, ok, but what if we assume that the three masses are not connected to anything else? That is, they are connected by two springs but there are no walls? Would we just have to pick a reference frame and go from there, declaring one mas "stationary"? For instance, if we connect the first mass to a wall but leave the other end unconnected, would the EoMs look like: $$m\ddot x_1+kx_1-k(x_2-x_1)=0$$ $$m\ddot x_2+k(x_2-x_1)+k(x_3-x_2)=0$$ $$m\ddot x_3+k(x_1+x_2)+k(x_3)=0$$ Anyhow, I was just curious, because all the problems in this vein I see seem to assume the three masses are attached between two walls. Thanks for your insights, in advance.


title: Time Dependent Position Operator

text: How does one find the time dependent position expectation value for a wave function? I thought we could simple take the time dependent wave and apply the position operator like normal, but this gave me the wrong answer, as the time dependencies multiplied to one and left me with my solution for $t = 0$. Added: The given solution was to use Ehrenfest's Theorem to find the expectation of momentum at a given time, then to use Ehrenfest's again to find the expected position at a time. I still don't understand why I couldn't just apply the momentum operator to the time dependent wave function to get the time dependent momentum expectation value, and do the same for position.


title: Circulate smoke in a closed system

text: Hey i am by no means a scientist but i have a idea for a art work using smoke. Basically what i want to know is whether it is possible to circulate smoke by means of a pump through a transparent box. If the box is completely sealed will the smoke disappear within time? When the smoke is pumped out and back in again, will it be have the same consistency as it had before it was circulated? Thanks in advanced


title: Why must gluinos be spin 1/2 instead of 3/2?

text: Is there some condition in the N=1 SUSY algebra telling that the spin of the superpartners of gauge bosons (either for colour or for electroweak) must be less than the spin of the gauge boson? I am particularly puzzled because sometimes a supermultiplet is got from sugra that contains one spin 2 particle, four spin 3/2, and then some spin 1, 1/2 and 0. If this supermultiplet is to be broken to N=1 it seems clear that the graviton will pair with the gravitino and the rest of spin 3/2 should pair with spin 1, so in this case it seems that a superpair (3/2, 1) is feasible. Why not in gauge supermultiplets?


title: Is there a SQCD gluino string, similar to the gluon string?

text: A gluon string is a particular kind of open string terminated in two particles which are the sources for the field. Is it possible to have a similar arrangement with gluinos? At first glance, it seems to me that such object could not exist, or at least not as an extended object: you need bosons to mediate between the two sources, and you need bosons to build a classical field extended in space. On other hand, the fact that in the world-sheet such structure is just an object with fermionic coordinates --and without bosonic coordinates-- could be telling that you can build it, but it is not extended in space, just a point. But I have never read of such description, so surely intuition is failing here.


title: Flood prediction data and code

text: Is it possible to use the data that is readily available (e.g. of the Shuttle Radar Topography Mission) to predict the flooding of a region? Would it be necessary to use the refined geoid of Gravity Probe B? Is there free software that can do this? I always wondered about this and the current events near Fukushima convinced me to ask this question: http://search.japantimes.co.jp/cgi- bin/nn20110730a5.html


title: Cheat sheet of elementary particles

text: I am trying to teach myself some particle physics. There are too many particles and its too much for me. I hated biology just because of this sort of stuff. Too many names and it was all Greek to me. Is there a good cheat sheet/ reference sheet of elementary particles? It will also be very helpful if you share how you people manage to remember these things. This is the first time I have ever hated studying physics. :(


title: Significant figures in measurement with error

text: Someone can explain me what's the rule behind the correct expression of a quantity $K$ with its error $\Delta K$ as $K \pm \Delta K$? They must have the same number of significant figures? Or the error should have in general 1-2 significant figures? For example, if I have: $$K = 8510.33 \pm 56.97~.$$ This expression is uncorrect? Maybe should be expressed as: $$K= 8510 \pm 57~? $$


title: Why connect cells in parallel?

text: Say there is a circuit with two 1.5V cells, and a 100 ohm resistor. If you connect two cells in series, then the total emf is 3V. And the current will be 3/100 = 0.03 A. (Using V = IR):   ![enter image description here](http://i.stack.imgur.com/qQmhp.png) If you have the cells in parallel, then the total emf is 1.5V, as the terminals of the cells are electrically the same point. So the current will only be 0.015 A:   ![enter image description here](http://i.stack.imgur.com/aHMqv.png) But if you just had one 1.5V cell, so the total emf is again 1.5V. The current will still be 0.015 A.   ![enter image description here](http://i.stack.imgur.com/4TowK.png) So what benefit does adding a second cell have? The emf and the current is the same no matter if you use 2 cells in parallel or just one cell. I'm assuming that all the cells are identical, and internal resistance is negligible.


I am trying to understand Rømer's determination of the speed of light ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in the orbital period of Io. (Jupiter's moon) ![Geometry of the problem](http://i.stack.imgur.com/J5dyj.png) The Earth starts from point A. $r(t)$ is the distance between the Earth and Jupiter. $r_e$ is the radius of the (assumed) circular orbit of the Earth around the Sun, while $r_0$ is the same for Jupiter. $T$ is the period of the Earth's orbit. Under the assumption that the Jupiter-Io system is stationary, $r(t)$ can be expressed as $$r(t) = \sqrt{r_E^2 + r_0^2 -2r_0 r_E \cos \left(\frac{2\pi t}{T}\right)}$$ If we further assume that the period of Io's orbit around Jupiter, $\Delta t$ is much smaller that $T$, then it can be shown that the distance the Earth moves, $\Delta r$ when Io completes one orbit is: $$\Delta r = \frac{2\pi r_E \Delta t}{T} \sin\left( \frac{2\pi t}{T} \right)$$ The point I am stuck is about why is there an apparent fluctuation in Io's orbit as observed on the Earth? And how can we derive the observed delay using these expressions?


title: Surface integral of Poynting vector around static sources

text: Consider fields $\rho \left( \vec{r} \right)$, $\vec{J} \left( \vec{r} \right)$, $\vec{E} \left( \vec{r} \right)$ and $\vec{B} \left( \vec{r} \right)$ in $\mathbb{R}^3$, with their usual meaning as per Electrodynamics. Take any finite volume $V_s$ outside of which $\vec{J}\left(\vec{r}\right)$ and $\rho\left(\vec{r}\right)$ are $0$. Then, we know that $\vec{E}\left(\vec{r}\right) $ and $\vec{B}\left(\vec{r}\right) $ for any $\vec{r}$ are as follows: $$ \vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_0} \iiint_{V_s} \frac{\rho\left(\vec{r}_s\right)}{\left|\vec{r}-\vec{r}_s\right|^3} \left(\vec{r}-\vec{r}_s\right) \space dV\left(\vec{r}_s\right) $$ $$ \vec{B} \left( \vec{r} \right) = \frac{\mu_0}{4\pi} \iiint_{V_s} \frac{\vec{J}\left(\vec{r}_s\right)}{\left|\vec{r}-\vec{r}_s\right|^3} \times \left(\vec{r}-\vec{r}_s\right) \space dV\left(\vec{r}_s\right) $$ Now my question is, is it possible to _mathematically_ prove that the following surface integral will always evaluate to zero? If so, what is the proof? (To clarify, $\partial V_s$ is the bounding surface of the volume $V_s$ mentioned earlier) $$ \frac{1}{\mu_0} \oint_{\partial V_s} \left(\vec{E}\left(\vec{r}\right) \times \vec{B}\left(\vec{r}\right)\right)\cdot d\vec{S}\left(\vec{r}\right) $$ Motivation: I'm looking for proof that time invariant sources (static charges and constant currents confined to a volume) cannot radiate any energy, and I'm trying to do that without invoking the Hertzian dipole and Fourier analysis. Thanks... **Update** As pointed out below, since $\vec{E}$ and $\vec{B}$ are time invariant, applying Poynting's theorem this boils down to proving that the following volume integral is zero: $$ \- \iiint_{V_s} \left( \vec{J}\left(\vec{r}\right) \cdot \vec{E}\left(\vec{r}\right) \right) \space dV\left(\vec{r}\right) $$ So can _this_ be proved, given that $\vec{J}$, $\vec{E}$, $\vec{B}$ and $\rho$ are all time invariant, beyond the trivial cases of $\vec{J} = 0$, $\vec{E} = 0$ or $\vec{E} \perp \vec{J}$ for all $\vec{r} \in V_s$? Or, if this can't be proved, is there a counterexample? Thanks...


title: What factors indicates inelastic collision?

text: I am watching this example from Wikipedia: ![](http://upload.wikimedia.org/wikipedia/commons/c/ca/Collision_carts_inelastic.gif) I am wondering what factors would indicate that the collision of 2 objects will be inelastic (I know macroscopic scale impacts are never perfectly elastic). If these are 2 wagons of the same mass and they have both hard surface, there is no friction and collision takes place (mostly) in 1 dimension shouldn't it be more close to elastic.


title: Cancel out Earth's Magnetic field

text: Is there any tools except helmholtz coil to cancel out earth's magnetic field to calibrate magnetometers in practice.


title: How to measure g using a metre stick and a ball

text: Can I measure the value of g using only a metre stick and a ball? I am not supposed to use a stopwatch and that has been the problem. NOTE: I do not know if a solution exists or not.


title: What's the underlying particle physics of endothermic reactions?

text: I don't just mean reactions that require heat to proceed, storing surplus energy in chemical bonds. I wonder about strongly endothermic reactions that suck heat out of environment. You take some substance A (e.g. Ammonium Nitrate), and some substance B (e.g. water), both at room temperature. You mix them together in a beaker which is room temperature, all performed in room temperature air. As the reaction begins, the two substances binding into substance C, the beaker cools down quite a bit below room temperature. This is what we observe on macroscopic scale. I wonder what happens on the microscopic scale - chemical particles, atoms, elementary particles, their energy. Normally entropy would suggest everything would remain in equilibrium but suddenly we have a higher energy concentration within the substance at cost of energy of the environment. It is not easily reversed. What happens that makes particles "want" to bind so much that they specifically "steal energy" from the environment just so that they can react?