Projectile Motion Question involving a ball and a ramp inclined at an angle

The question is to finde the initial horizontal velocity of the ball at end of the ramp, where it is released. I know how to do this using gravitational potential energy and kinetic energy ($v=\sqrt{2gh}$), assuming all potential energy is converted into kinetic energy but the question is asking me to find the error in the experiment. It gives me the values on $y$ as the ramp is lifted up (it is lifted up about 10 cm each time) and the corresponding values of $x^2$. I drew a graph of $x^2$ vs $y$ and found the gradient. How can I use this to find the experimental initial horizontal velocity? This sketch shows what the experiment looks like: ![Experiment](http://i.stack.imgur.com/RfGvy.png)


Colors of light

White light is always said to contain all the different wavelengths of light. Why, then, can we 'make' new colors simply by adding wavelengths? Is it just a matter of our perception, that, when two colors are added together, they appear to be the wavelength of another color. Or are the waves actually forming into another, differently 'waving' wave, with a different wavelength?


Temperature below absolute zero?

I saw this Nature article today, which cites e.g. arXiv:1211.0545. And it makes no sense to me. The temperature of a collection of particles is the average kinetic energy of those particles. Kinetic energy cannot be less than zero (as far as I'm aware), so I don't understand what this article is trying to say, unless they're playing around with the conventional definition of "temperature". The only thing I can thing of is if you have something like: $$\frac{1}{kT} ~=~ \left(\frac{\partial S}{\partial E}\right)_{N,V}$$ And they've created a situation where entropy decreases with increasing energy.


Less than absolute zero possible?

> **Possible Duplicate:**   >  Temperature below absolute zero? According to this article http://www.sciencemag.org/content/339/6115/52 (preprint: http://arxiv.org/abs/1211.0545) it is. What do you think. Is this team of scientists a faker team or you could actually go under absolute zero? Could this be considered a way to dark energy something which is less than absolute zero?


Quantum phyics project for a high schooler

> **Possible Duplicate:**   >  Study Quantum Physics I am a high schooler who is interested in physics and mathematics, and I have a kind of 'high-school thesis' coming up in a year and a half or so. I want mine to be about Quantum Physics, and I have already prepared by self-studying Linear Algebra (and I'm planning on starting with Differential Equations too). I just have a couple of obstacles:   * If I know Linear Algebra and (Ordinary & Partial) Differential Equations, what parts of Quantum Physics will be in my reach?   * What other fields of mathematics would you recommend me studying to gain a grasp of rudimentary quantum physics and to make me able to make a thesis on quantum physics?   * What would you guys consider interesting for a thesis on quantum physics? What are fascinating experiments, concepts, results, etc. of quantum physics that would be worth making a thesis about?


Why is pressure in a liquid the same in all directions?

I'd like answers both in the more intuitive side an on the more precise side. Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ? I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?


use of Mobile phone on Petrol pump

I was trying to find out the cause behind the Explosion at petrol pump due to the Mobile phones but eventually it turns out it is not Radio Frequency which may cause the explosion rather it is totally the static electricity which may cause serious explosion and ignition of vapors, The other reason I found is, due to the spark by the mobile phone battery in case the mobile has fallen out of your pocket and it seems more legit. The second scenario, Its been said that Electromagnetic Waves absorb in liquid across a wide wavelength , is there any possibility that the absorption of radio Frequency in Petrol may cause any explosion ? My question is, anything can produce Static electricity then why mobile phones are prohibited ? Do they produce more Static Electricity than other equipments ?


Physical reason for annihilation?

What is the fundamental reason as to why matter and antimatter annihilate? Is it because both particles and antiparticles are excitations of quantum fields, and the annihilation process corresponds to a transition to the _ground state_?


Casimir force between plate and sphere

In our quantum optics lecture we derived the formula for the Casimir force between two plates (or at least the force per unit surface, since for two infinite plates we would have an infinite force). It was then said, that in experiments one often uses a sphere and a plate to measure the force, where the small distance between the sphere and the plate is varied by a piezo. I'm not particularly interested in how exactly the force is measured but rather how to theoretically calculate it in this case. Our professor said, the measured forces agree with the theory up to 1% so I guess, there is a way to calculate it. Let's say I have a sphere of radius $R$ in a distance $d$ from a plate where $R\gg d$. How can I theoretically calculate the Casimir force between a plate and a sphere?


How do we know we've unified two interactions?

1. What is the precise definition of unification of fields (in classical and quantum mechanics)?   2. In general, does unification of a field mean that we can write both of them at both sides of an equation (like Maxwell's laws)? Or does it mean that one of them can produce the other (like $E$ and $B$)?   3. Is there any intuitive explanation of how electroweak unification works? Like an electric charge will feel a weak field or that a flavoured particle will produce a weak field?


Density Matrix Characterization

I am working a two dimensional Hilbert space with basis {$|0\rangle, |1\rangle$} and I am trying to show that the density matrix is characterized by 3 real numbers and show that these three numbers are the expected values of the Pauli matrices. I understand that a density matrix is given by $M_{ij} = \langle i|\rho|j\rangle$ where $|i\rangle$ is an orthonormal basis of $H$ and $\rho$ is the density operator. How do I go about showing this?


What is the difference between QFT and elementary particle physics?

I'm a little unclear as to how QFT differs from Elementary particle physics. They both use pictorials of Feynman graphs, is it that Elementary particle physics assumes the point particle perspective, while QFT treats them as fields?


Recommendation: Advanced topics in quantum field theory

I have read Srednicki's Quantum Field Theory book. I want to learn more about advanced topics in field theory, such as geometry and topology in field theory, topology defect, anomaly, soliton, instanton, renormalization，advanced application in condensed matter physics and so on. Does anyone have anything recommended? Thanks


Top spun up with string under tension problem

> **Possible Duplicate:**   >  Homework about spinning top I have a top with an unknown mass. It has a moment of inertia of 4.00 * 10^-7 kgm^2 a string is wrapped around the top and pulls it so that its tension is kept at 5.57 N for a distance of .8 m. Could somebody help me derive some equations to help with this? Or to get me in the right direction? I have been trying to derive some sort of equations from KEr = 1/2 * I * w^2 but I cant get anywhere without ending up at radius = radius or mass = mass.


Diffusion in the standard map

Consider the standard map (also known as Chirikov map): $$ p_{n+1} = p_n + K \sin(\theta_n) \\\ \theta_{n+1} = \theta_n + p_{n+1} $$ I know that the diffusion coefficient according to the Einstein relation is defined through the relation: $$ \sigma^2 = 2Dt $$ being $\sigma$ the variance of the position in the case of the $1$-D random walker. Can you figure out why the diffusion coefficient for the standard map is given by: $$ D = \lim_{n \rightarrow \infty} \frac{<(p_n - p_0)^2>}{n} \quad ? $$ I mean, the $n$ makes sense, but what about the variance, why this is given just in terms of $p$, what happens with the $x$.


Question about gravity felt at various radii within a massive sphere

Imagine there's a sphere of radius $R$ which has constant density $\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it. Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\mathbb R^4$ so that we're on $S^4$. What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?


Understanding the Selection Rules of a Spin-Forbidden, Magnetic Dipole Transition in Molecular Oxygen

I am studying the transition from the second excited electronic state of molecular oxygen, $b^1\Sigma_g^+$ , to the ground state, $X^3\Sigma_g^-$. I know that the ground state has total angular momentum $J=1$, total spin $S=1$, and three spin sublevels $(m_s=-1,0,+1)$. The upper state has $J=0$, $S=0$, and one sublevel with $m_s=0$. I am specifically interested in the transition from the upper state to the ground state $m=\pm1$ level and I will refer to it as the $b-X,1$ transition. I would like to understand the the selection rules that govern this transition and the language that is used to describe this transition. Here is what I know so far about this transition.   * Brecha, Pedrotti, Krause, "Magnetic rotation spectroscopy of molecular oxygen with a diode laser" JOSA 1998: They describe this transition as "doubly weak" and I quote, "First, it is a magnetic dipole transition and thus is roughly 5 orders of magnitude weaker than a normal electric dipole transition. Second, the transition is a singlet-triplet intercombination band, making it 3 orders of magnitude weaker still."    * Minaev, Agren, 1997: They describe this transition as "magnetic dipolar" and "spin-forbidden." They also say that it is spin-orbit coupling that accounts for this "doubly weak" transition being as large as it is.    * Wikipedia-Selection Rules: Provides the rules for Magnetic Dipole (M1) transitions and discusses Spin-Obrit (LS) Coupling but I do not understand where this came from.    * Sannigrahi, "Derivation of Selection Rules for Magnetic Dipole Transitions," 1982: This is pretty clear but just seems to say that $\Delta m_s=\pm1$. This is true for the $b-X,1$ transition but how do I tie in $L$ and $J$ into the selection rules? Most of what I have found on the internet and in textbooks relating to selection rules is directed to electric dipole transitions. Where I have found something discussing magnetic dipole transitions, it is either a summary of the rules or I do not have the required background to understand it.   * What does it mean for a transition to be spin-forbidden?   * What is a singlet-triplet intercombination band? Does this change the selection rules? How is this related to being spin-forbidden?   * What does spin-orbit coupling have with this? I look forward to any feedback on this topic. # Follow up Questions ## Meaning of Spin-Forbidden Does 'spin-forbidden' just mean that the transition from a $J=1$ to $J=0$ state is not allowed because the selection rules for magnetic dipole transitions say that $J$ cannot change? I expected 'spin-forbidden' to imply something about the change of the spin between the initial and final states. For example, suppose I have a time-dependent perturbation like $V_{md}(t) = \frac{e}{m} \vec{S}\cdot \vec{B}(t)$ and I am interested in the transition rate between the initial state $\left| s m_s \right\rangle$ and the final state $\left| s' m_s' \right\rangle$ with the quantization axis in the z direction. As you pointed out, the transition rate will be proportional to $\left\langle s' m_s'\right| \vec{S} \cdot \vec{B}(t) \left| s m_s \right \rangle$. Now if $\vec{B}(t)$ is circularly polarized, the rules for an allowed transition will be $s'-s=0$ and $m_s'-m_s = \pm1$. For $\vec{B}(t)$ polarized in the z direction, $s'-s=0$ and $m_s'-m_s = 0$ so there is not a transition to other states. I would think that if $s'-s=0$ and $m_s'-m_s = \pm1$ are **not** true (like if $s'=1$ and $s=0$), then the magnetic dipole transition between $\left| s m_s \right\rangle$ and $\left| s' m_s' \right\rangle$ would be called 'spin-forbidden.' The same arguments could be made for $\vec{L}$ or $\vec{J}$ as I did with $\vec{S}$. Would you also call a magnetic dipole transition between $\left| L=0 \right\rangle$ and $\left| L=1 \right\rangle$ 'spin-forbidden?' ## Spin-Orbit Coupling Now the difference in the energies of the singlet $(b)$ and triplet, ground state $(X)$ is $1.63\text{ eV}$. That seems too large to be due to spin-orbit coupling breaking a degeneracy. If I was pretending this was a hydrogen atom, I would say this was like a transition where the principle quantum number $n$ changed. I am not sure how to talk about this in a molecule. This isn't key to the rest of your explanation but I did want to clarify to make sure we were on the page. ## Singlet-Triplet Intercombination Band Do you know what the expression 'singlet-triplet intercombination band' is referring to? After your explanation, it seems to refer to the mixing of the singlet and triplet states due to spin-orbit coupling (SOC). Is this true? ## Mixing of unperturbed states How did you know that the perturbed upper state could be written as a combination of the unperturbed upper state and the $M_s=0$ ground state? It makes sense that the states would get mixed up by SOC but I don't know how. If this is a messy explanation, don't worry about it. **I would like to reiterate that your (George G's) explanation has been incredibly helpful. Thank you.**


Is there some quantum potential producing exponential eigenvalues?

Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" potentials which have only a single discrete eigenvalue. I was wondering, what kind of potential do we need for producing an exponential $e^n$ set of discrete eigenvalues?


OPE and 4-point correlation function in CFT_d

I'm reading this paper where to determine the coefficient $C^{\phi\phi O}(x_{12},\partial_2)$ of the OPE (p.10) $$\phi^\alpha (x_1)\phi^\beta (x_2)=C_\phi \frac{\delta^{\alpha\beta}}{x_{12}^{2\eta}}+C^{\phi\phi O}(x_{12},\partial_2)O(x_2)\delta^{\alpha\beta}$$ they require consistency of the OPE with the 3-point correlation function, i.e. comparing the fixed form of the 3-point correlation function to what happens when we apply the OPE to two channels in the 3-point function and then use the form of the 2-point function. Anyway, to solve the form of the coefficient that satisfies eq. 2.29 they use the integral representation $$\frac{1}{(ab)^\rho}=\frac{1}{B(\rho,\rho)}\int_0^1 dt \frac{[t(1-t)]^{\rho-1}}{[at+(1-t)b]^{2\rho}}$$ which comes from the book Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik. However, looking at that book (I only have access to the 7th ed.) I only found the relation $$B(\mu,\nu)\frac{1}{(a+c)^\mu(b+c)^\nu}=\int_0^1 dx x^{\mu-1}(1-x)^{\nu-1}\frac{1}{[ax+(1-x)b+c]^{\mu+\nu}}$$ which reduces to the desired relation when $c=0$ however, in the book it says that the relation is valid for $c>0$, so how can they use that relation?


Why infrared absorption is a nonlinear technique?

I am looking for a good explanation and/or reference quotation explaining why infrared absorption technique is essentially nonlinear (eg. for carbon monoxide quantification). When using UV/visible/near-IR absorption technique, Beer-Lambert Law often is valid and then you have quite straight way to convert transmittance into absorbance which linearly respond to concentration (provided BLL limitations are not met). Above technique relies on electronic transitions, when IR absorption technique relies on vibrational and rotational transitions for molecules which has permanent or transient dipole moments. I would like to figure out why such technique is essentially nonlinear.


How fast do molecules move in objects?

I guess it depends on the heat or the type of the material but can you give some examples or formulas to calculate it ? The best example would be the average speed of the air molecules (all types in the air) at room temperature or water molecules at human body temperature.


Are the $10^{500}$ different string theories being whittled down?

An example of a test: Ask each variant whether its estimate of the electron mass lies within $\pm\,x\%$ of the known value. This surely can't take long per theory. Although $10^{500}$ is huge, whittling them down could be essentially a background task for a few thousand computers. But is it being done, and if not, why not?


What equation of state is needed for liquid states?

I'm familiar with the ideal gas law $$PV=nRT$$ but I don't think it applies to liquids like water. If I'm wrong, please correct me! If I'm right, then what equation of state applies to liquids such as water?


Can a wavefunction be solved to any arbitrary precision, given enough computer time?

I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only approximations can be obtained. Well, I'm not sure I like the vague wording with "impossible". Basically, a hydrogen atom can be solved in terms of elementary functions. Elementary functions, as far as I know, are just arbitrarily defined (Is $\sqrt{}$ one? What about $\text{erfc}$?). For instance, I could say that $\sin{x}$ cannot be solved exactly except in a few special cases (x = some rational multiples of $\pi$). To analogize, a general quintic equation cannot be "solved" exactly using elementary functions -- that is, unless you define a Bring radical as a new elementary function. To me, it's really arbitrary and all of these solutions come down to numerical, iterative refinement. Solved exactly vs solved approximately seems to be an ill-defined concept. So what I am asking -- given enough computer time -- can I solve the Schrodinger equation of, say, Lithium, to whatever precision I like? Much as how I can solve $\sin{x}$ to however many decimal places I want? Or is there some _other_ limit to such a calculation?


Can I levitate soon?

Whereas I have seen frogs fly in high magnetic fields and Sumo ringers stand on high-temperature superconductors it seems that these are not the only forms of levitation. Recently a group of physicicsts managed to levitate coffee without a magnetic field (PNAS link, only abstract is free though). So while I am still trying to grasp how this is even possible my question is a bit more far fetched: Can I levitate using this approach?


Is the conjectured noncommutative heavy scalar "brother" of the already detected Higgs boson is a pseudo scalar?

This is a technical (may be trivial?) question about this sigma scalar field advertised by Chamseddine and Connes to improve the electroweak vacuum stability involved by the weak mass of the already found Higgs-like particle. **Is it a scalar or a pseudo scalar field?** (motivation: I was looking for a possible connection with Majoron models)


About the stability of the ground state of the bosonic string

In Polchinski's string theory vol 1, p. 23, it is said "It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known." The book was published on 1998. What is the current answer of this question? Is bosonic string a realistic theory nowadays? In addition, the ground state here is $$|0;k \rangle $$, not $$| \mathrm{vacuum} \rangle$$. Is that because the latter one is absolute nothing so we don't study?


Gauss law in classical U(1) gauge theory

I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture. Let $A$ be the space of all field configurations and let $G$ be the group of gauge transformations. Then, the actual configuration space $C=A/G$ where we consider $A$ modulo the action of gauge transformations. So, first of all, why don't we do our variational calculus to obtain field equations on configuration space? It has always been taught to me that I should consider variation of "paths" in configuration space only while applying Hamilton's principle. Instead, we first obtain field equations regarding $A$ as our configuration space and then "project" them perpendicular to gauge orbits using Gauss law. Are these two methods equivalent? Moreover, I don't understand how Gauss law really projects my fields orthogonal to gauge orbits. According to Manton's book (see the relevant pages scanned below) on topological solitons, he starts with a variation of the fields $\phi$ and $a$ in a small interval of time. Then, he claims that a "naive contribution to the kinetic energy" would be some expression involving these variations (see below) which is non-zero for infinitesimal gauge transformations also. But, when I look at the expression of kinetic energy (which is $T=\int(\frac{1}{2}e_{i}e_{i}+\frac{1}{2}\bar{D_{0}\phi}D_{0}\phi)d^{d}x$) in terms of fields, then I find that the expression is gauge invariant. I don't see why the particular expression $\frac{1}{2}\int\frac{1}{(\delta t)^{2}}(\delta\vec{a}.\delta\vec{a}+\bar{\delta\phi}\delta\phi)d^{d}x$ be the change in kinetic energy. In short, I want to understand this geometric insight into Gauss law whose details are attached below. Also, I wonder if this Gauss law has a relation to the Gauss law I learned while doing high school electrodynamics. Any information or resource on this would be very helpful. ![enter image description here](http://i.stack.imgur.com/3rP4Y.jpg) ![enter image description here](http://i.stack.imgur.com/nRlWL.jpg)


From where do the permanent magnets get energy from?

I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently. Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy?


What is the typical orbital life of an artificial satellite?

The orbit of satellites around Earth eventually decays, or so I read. This is typically caused either by atmospheric drag, or by tides. I would assume most satellites have a limited service life in orbit. Hence the question - What is the typical orbital life of a satellite? How long before it's successor must be launched into orbit ? EDIT: For instance, a weather satellite


How do I calculate the position on the Bloch sphere of a quantum gate with a given diagonal matrix?

In quantum computation there are several principal quantum gates that have corresponding matrix representations. One of these is the Z gate, whose matrix is $\left[\begin{smallmatrix} 1 & 0 \\\ 0 & -1 \end{smallmatrix}\right]$. ... anyway, I've found the eigenvalues (equal to +1, -1) using the characteristic equation, and used them to derive the corresponding eigenvectors, which come together quite nicely in a 2×2 matrix $\left[\begin{smallmatrix} 1 & 0 \\\ 0 & 1\end{smallmatrix}\right]$, equal to the identity. So, in diagonalizing this matrix, I find that the diagonal matrix $D$ is the same matrix as the one for gate $Z$. ... the next step and where I'm stuck is to find the corresponding point on the Bloch sphere for this gate. In order to do that, I need to compute how to take the diagonalized matrix call it $D_z$ and derive two things: **(a)**  its diagonal representation $| 0 \rangle \langle 0 | - | 1 \rangle \langle 1 |$, and **(b)**  the normalized eigenvalues $a, b$ for $Z$, where $Z = a|0\rangle + b|1\rangle$ and must be orthonormal i.e. $a^2 + b^2 = 1$. The $a$ and $b$ terms correspond to the probabilities of measuring 0 or 1 for the state, respectively (I think). After I have the values for $a$ and $b$, I'll be able to locate the gate on the Bloch sphere because the calculation of its coordinates on the sphere is straightforward: $a = \cos(\theta / 2)$, and $b = e^{i\phi}\sin(\theta/2)$.


Does the anthropic principle entail the existence of a huge multiverse?

As many cosmologists have pointed out, the universe is extremely fine-tuned for life in so many respects, and life, especially intelligent life, is extremely unlikely to emerge. The strong anthropic principle tries to explain it statistically by an extremely huge multiverse ensemble. However, are there alternative explanations? One is an intelligent designer, but given how hard it is to create intelligent life, It would probably have to engage in significant trial-and-error by simulating many prototype universes before stumbling upon one that works. This is probably true no matter how intelligent It is. So, the multiverse reemerges through the back door. This still applies even if the Designer is a blind naturalistic algorithm. Can you think up of some mechanism for intelligent life to emerge which doesn't require a huge multiverse ensemble?


Modal analysis with aerodynamic damping

I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the aerodynamic drag force as it moves through the air is very significant. I have the mass and stiffness matrices for the beam ($\boldsymbol{[M]}$ and $\boldsymbol{[K]}$) and so I am able to obtain the eigenvectors and eigenvalues ($\boldsymbol{[V]}$ and $\boldsymbol{[D]}$). The eigenvalues yields a vector containing the natural frequencies of the system (the square root of the diagonal entries of $\boldsymbol{[D]}$, I'll call it $\boldsymbol{\omega}$. The system is excited by a force vector in the global reference frame $\boldsymbol{F_0}$ at an excitation frequency $\omega_F$. There is a rotation matrix which rotates the beams from the local to the global reference frame $\boldsymbol{[R]}$. I have successfully used the procedure to calculate the steady state response without aerodynamic damping:   * Get eigenvalues and eigenvectors $\boldsymbol{[V]}$ and $\boldsymbol{[D]}$   * Get generalised force vector $\boldsymbol{Q}=\boldsymbol{[V]}^T \boldsymbol{F_0}$   * Calculate modal participation factor vector $\boldsymbol{q}$. The individual elements are calculated as follows: $$q_i=\frac{Q_i}{{\omega_i}^2}\frac{1}{\sqrt{[(1-(\omega_F/\omega_i)^2)^2+(2\zeta_i\omega_F/\omega_i)^2]}}$$   * Calculate response from $\boldsymbol{x}=\boldsymbol{[V]q}$ Of course, this drastically overpredicts the amplitude of the oscillations because there is no aerodynamic damping. I've found a few examples for single d.o.f. systems in which the energy lost from the system is calculated (by integrating the drag force x the velocity). The integration results in the following equation for energy loss: $$\Delta E=\frac{8}{3}\frac{1}{2}\rho A C_D \omega^2 X^3$$ This can then be converted to an equivalent damping coefficient as follows: $$C_{eq}=\frac{\Delta E}{\pi \omega X^2}$$ Which can be converted to a damping coefficient by doing: $$\zeta=\frac{C_{eq}}{2 \omega_n}$$ I've modelled a simple spring-mass system with air resistance and it works fine (I have to iterate to find the final position because the amplitude calculation requires knowledge of the damping which requires knowledge of the amplitude so there is a circular reference, but that's no biggy!) Could someone please explain the equivalent for a multi-d.o.f. system? My initial forays have been unsuccessful, with much higher drag coefficients than would really occur.


Does sound cancel itself out?

If there are two 10 x 2 x 1 foot rectangles in space and they are lined up so if they hit each other there will be no spots that are not hit in the front of the rectangle. Then they are pushed forward with the same exact same force, (this is space and there is no resistance) and they hit, will the tiny shockwaves of sound moving through them hit each other (they are exactly the same kind of sound wave), will the sound hit it's opposite would they cancel themselves into heat on contact? I know that the sound does not cancel itself out. You cannot say the wave of atoms hitting each other collides with a wave that is exactly the same and stops both of them in a word. Can you?


Magnetic core, and a DC solenoid

A DC solenoid creates a magnetic field B. A highly permeability magnetic core is added to solenoid to increase B. Now, there is change in flux, due to the increase in magnetic field. What would be the result for the circuit? Would more power drawn support the new stronger field? Or P would be the same before/after?


What does a subatomic charge actually mean?

I was recently reading a popular science book called The Canon - The Beautiful Basics of Science by Natalie Angier, and it talks about subatomic particles like protons, neutrons and electrons in chapter 3. I came across this section on subatomic charges that made me wonder about the nature of the positive and negative charges that we associate with protons and electrons respectively. > When you talk about a fully charged battery, you probably have in mind a > battery loaded with a stored source of energy that you can slip into the > compartment of your digital camera to take many exciting closeups of > flowers. In saying that the proton and electron are charged particles while > the neutron is not, however, doesn't mean that the proton and electron are > little batteries of energy compared to the neutron. A particles's charge is > not a measure of the particles's energy content. Instead, the definition is > almost circular. **A particle is deemed charged by its capacity to attract > or repel other charged particles.** I found this definition/description a bit lacking, and I still don't grasp the nature of a "subatomic charge", or what do physicists mean when they say that a proton is positively charged and electron is negatively charged?


Lagrangian and conservation of energy

If Lagrangian of the motion is $$\mathcal{L}=\frac{1}{2}m\left(a^2\dot\phi^2+a^2\dot\theta^2\sin^2\phi\right)+mga\cos\phi,$$ how can I show that total mechanical energy is conserved? I've read this: > _If the time $t$, does not appear [explicitly] in Lagrangian $\mathcal{L}$, > then the Hamiltonian $\mathcal{H}$ is conserved. This is the energy > conservation unless the potential energy depends on velocity._ Potential energy of this motion doesn't depend on velocity. Also, $t$ does not appear explicitly in Lagrangian. Is this enough to say that total mechanical energy is conserved?


Why and how exactly is electric motor torque limited?

Inspired by this question and specifically this answer to it. From my experience there's always some very specific limit to how much torque an electric motor can output. For example, an electric drill will often have a manually switched mechanical transmission - if one needs to drill some relatively weak materials (like wood) he will use the setting that outputs lets torque at higher RPM and if one want to drill steel or mix cement mortar he will use the settings for more torque at lower RPM. The relation between RPM and torque is more or less clear if one imagines a set of two gears of different diameters and thinks that their radiuses are lever arms - three times more RPM automatically induces three times lower torque and vice versa. But where does the limit to any given electric motor torque come from? Say I have some specific motor right now in front of me and it can output 40 Newton- meters at 500 RPM. Why exactly 40 and not more?


How to explain the different forms of the Hamilton-Jacobi equation?

In Arnold's _Mathematical Methods of Classical Mechanics,_ he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, q, t \right) ~=~ K(Q, t). $$ However, this is different from what I've seen in other physics texts. For example, Goldstein uses the generating function $S_2(q, P, t)$ to get the equation $$ H\left(\frac{\partial S_2(q, P, t)}{\partial q}, q, t\right) ~=~ - \frac{\partial S_2(q, P, t)}{\partial t}. $$ Why is there this difference? Are the two equations saying the same thing?


What's the size of a 'locally' flat space-time patch in a black hole background?

A black hole collapses and emits Hawking radiation with average wavelength 1/M. The observer at I^+ sees it, so do the local observers (infalling or hovering). None of these cases should violate the equivalence principle. What is the size of the 'local' patch associated with the observer, the size beyond which he is allowed to see deviations from flat space because curvature effects become measureable? I'm confused because on the one hand I'd have thought this size goes to infinity if the observer moves towards I^+, but on the other it seems to me it should stay at 1/M so that the (average) Hawking radiation has wavelengths the size of the locally flat patch. What am I getting wrong? (Forget about firewalls, just thermal radiation.)


Did Newton discover gravity waves without realizing it?

Newton's theory explained the changes in sea level by the effect of gravity (moon/sun). Now we are trying to detect gravitational waves from distant cosmic surces. But, apart from the differences in "signal frequency" (extremely small in case of sea effects (10-4 / 10-5 Hz or so), maybe higher in case of "signals" from cosmic sources (in the order of 1 Hz?) what is the difference in principle? At least in principle, can we say that the change in sea level is an effect of gravity waves?


Is there a physical reason why the detection pattern of microphones and antennas is a cardioid?

I was wondering if there's an underlying physical reason why detection in microphones and antennas is a cardioid, or if it's just that a cardioid happens to be the mathematical object that best approximates the shape of the detection pattern. In other words, is there an "ab initio" reason, or it's an empirical fact?


Was Einsteins work with relativity necessary for successful space travel?

So I know that Einstein and general relativity had huge impacts on the way we view the world, but how crucial were these scientific advancements to the success of our space programs? Would Newtonian physics sufficed, or would using those formulas and methodologies when travelling off-earth and to the moon resulted in catastrophic failures?


Can one predict mass distribution in a solar system given the mass of the star?

I'm trying to write a program to procedurally generate solar systems at a reasonably high level (I'm not interested in doing an n-body dust particle formation simulation). Is there some sort of distribution that defines where I am most likely to find mass inside a solar system? i.e. should I expect heavier planets further out... or is it completely random?


Wigner's friend and quantum Zeno effect

Suppose Winger's friend is placed into a black box, thoroughly isolated from the outside world. He constantly observes an atom with a delay of some microseconds. According to Zeno effect, atom's lifetime will increase. He writes his observations to a log. After a day passed, the box is opened and Wigner's friend asked when the atom decayed. Will the increased atom lifetime be seen by the observers who were outside of the box?


Where does the minus sign appear from in the metric tensor?

Trying to understand Schutz's AFCIGR, where does the minus sign appear from in the metric tensor? ![metric](http://i.stack.imgur.com/jXilu.gif) I understand that this expresses the invariance of the spacetime interval. Schutz says (I think) that the metric is a (0,2) tensor. I assume that means it is the product of two one-forms, so presumably one of these one-forms has a -ve time component. What does that mean? Why don't both one-forms have a -ve time component? Looking at a Minkowski diagram what is a straightforward way to understand/visualise those one-forms? At my level, two things have been multiplied together to give a 4x4 matrix which has -1 in the top left corner. What are those two things and why do they give a -1 time component?


Reference for the predictability of rigid body dynamics

I'm looking for a reference, journal article, paper, etc. that supports the idea that classical mechanics, in particular rigid body dynamics, is largely predictable. A view coming from the background of computer physics simulations would be an added bonus.


A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free state. In particular, he argued the integral $$\tag{1} \int d\alpha\frac{e^{-iE_{\alpha}t}g(\alpha)T_{\beta\alpha}^+\Phi_\beta} {E_\alpha- E_\beta+i\epsilon}$$ would vanish, where $d\alpha=d^3\mathbf{p}$ (also involves discrete indices like spin, but of no relevance here). In his argument, he used a contour integration in the complex $E_{\alpha}$ plane, in which the integral of central interest is the integration along real line $$\tag{2} \int_{-\infty}^\infty dE_\alpha\frac{e^{-iE_{\alpha}t}g(\alpha)T_{\beta\alpha}^+\Phi_\beta} {E_\alpha-E_\beta+i\epsilon}$$ I don't see how to obtain (2) from (1), since the lower bound of energy is the rest mass, in the best case I could get something like $\int_{m}^\infty dE_\alpha\cdots$, but how could one extend this onto the whole real line.


Rotating sphere and circular trajectory: minimum speed

I have a sphere (mass = 3 kg), constrained to a fixed length rope, rotating (radius = 5 m) on a vertical plane. My textbook ask me about the minimum speed in the highest point in order to keep the circular trajectory. Now, I know that in the highest point (v=speed): $$F_c=mg+T$$ $$\frac{mv^2}{r}=mg+T $$ $$ \frac{3}{5}v^2=29,43+T$$ I know that with a low speed I have a low Tension, so let's put T=0 and go on: $$v^2=49,05 \rightarrow v=7$$ The result is correct, but I have a little doubt: if the Tension is 0, why the sphere doesn't go along the tangent or start falling down? Exactly what is forcing the sphere to preserve the circular trajectory? In my ignorant opinion, the minimum speed is a little more than 7.00 m/s, because the tension mustn't be 0, even a very small number, but not null..


How to obtain stabilizer's generators of a QEC code

The theory of QEC with stabilizer codes defines an alternative way to represent a quantum state in terms of operators. To understand better what I am concerning about, let's consider the 7-qubit Steane code: $$ \left|0\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|0000000\right\rangle + \left|1010101\right\rangle + \left|0110011\right\rangle + \left|1100110\right\rangle + \left|0001111\right\rangle + \left|1011010\right\rangle + \left|0111100\right\rangle + \left|1101001\right\rangle) $$ $$ \left|1\right\rangle_L \equiv \frac{1}{\sqrt{8}}(\left|1111111\right\rangle + \left|0101010\right\rangle + \left|1001100\right\rangle + \left|0011001\right\rangle + \left|1110000\right\rangle + \left|0100101\right\rangle + \left|1000011\right\rangle + \left|0010110\right\rangle) $$ Papers and books directly define the relative generators as: $$ K^1 = IIIXXXX $$ $$ K^2 = XIXIXIX $$ $$ K^3 = IXXIIXX $$ $$ K^4 = IIIZZZZ $$ $$ K^5 = ZIZIZIZ $$ $$ K^6 = IZZIIZZ $$ where I is identity matrix and X,Z Pauli's matrices. Therefore, my questions are: how these generators have been calculated? How can I calculate the generators for a generic superposition of n qubits? Thank you


Why has the ether been disregarded as a valid medium through which light can propagate?

> **Possible Duplicate:**   >  Ether theory acceptance Although the Einstein's theory of relativity seemed to make the concept of an ether obsolete, did it necessarily invalidate it? Are there any experiments which have truly invalidated the ether, with undeniable proof? I'm very curious as I haven't found any, besides of course the Michelson–Morley experiment, which I don't think offers irrefutable evidence that contradicts the ether's existence. May someone explain why this has been the definitive conclusion (or offer some new insight)?


Can we make a change of variables (for example to polar coordinates) into a divergent integral?

I know that if the integral is convergent we can always make a change of variable to make it better, however what happens with DIVERGENT integrals? can we make a change of variable into a divergent integral after having it regularized? I mean we insert the regulator $(q+a)^{-s}$ or similar on each variable and then make the change of variable to polar, cilindrical or other


Electric field of not-grounded conducting plate with a given potential?

I have been trying to find an equation (or some solution) of how to calculate the electric field strength (in N/C) of a conducting rectangular (nearly flat) plate which has non-zero potential to it, say the plate has a potential of 5V. I have searched over the internet (and read Griffiths book on the method of images section), and it seems one would need to apply the method of images to solve the problem. However, since almost all examples regarding method of images is about "grounded" plates at 0V, it is very difficult for me to understand how to use method of images to solve for the electric field of a plate at 5V? Also, I would like to mention that within the electric field of the 5V conducting plate, there is a small freely-moving positively charged particle in the vicinity (I assume because the charge on this small particle is the same as the charge on an electron/proton-- it can be regraded as a test charge and therefore, modification to the combined electric field between the conducting plate and the charge is not necessary?-please correct me here if I am wrong.)


How to interpret the factor $\frac{(\vec v\cdot \hat n)}{|\vec v| 4\pi r}$?

There is a small area element of size $da$ and normal vector $\hat n$. I understand that the particles with speed $|\vec v|$ that hit this area element in the time interval $(t,t+\delta t)$ lie in a spherical shell with radius $r=|\vec v|t$ at $t=0$. Let the number of particles with energy $E(|\vec v|)$ within a solid angle $d\Omega$ be $dN$. Why then is the total number of particles hitting $\hat n da$ with velocity $|\vec v|$ (assuming isotropy of the velocity distribution) equal to $$\frac{(\vec v\cdot \hat n)}{|\vec v| 4\pi r}da \,\,dN$$? I don't quite understand how to get the factor $\frac{(\vec v\cdot \hat n)}{|\vec v| 4\pi r}$. Thanks.


Can Massless Particles Have Dimensions?

We have been taught that anything occupying space/volume has mass. Light has both a particle nature-photon and wave nature. Since It has an momentum of h/λ, energy of hc/λ. Since a photon carries energy and experiments have put an upper limit to its mass of 1e-18 eV/c^2, does it possess dimensions, a shape ?


Mass loss in Red Giants via dusty-winds and chromosphere activity

I'm reading some literature on mass loss in the RGB/AGB branches and so far I'm getting a lot of information regarding mass loss via dusty- winds/pulsations but almost no explanation of mass loss by 'chromosphere activity'. I asked my mentor about it and he couldn't come up with anything so I'm turning to whomever might be able to throw a few references or explanations my way.


vertical wind gradients in the atmospheric boundary layer

I'm reading the following paper: > Intercomparison of Bulk Aerodynamic Algorithms for the Computation of Sea > Surface Fluxes Using TOGA COARE and TAO Data and am having trouble working through some of the equations, I have copied these equations below: ![enter image description here](http://i.stack.imgur.com/ymp8m.png) Would someone be able to show me the process of how integrating (1) to (6) gives (7)? Any help would be appreciated.


Is a 3 LED flashlight brighter than a single LED one? (same LEDs, same power through each LED)

Say I have one LED in a flashlight, it produces some light. Now, if I have 3 of those LEDs in the same flashlight, _each_ LED receiving the same amount of energy than the single LED was receiving in the model described above (thus producing, individually, the same amount of light), will I get (perceive) more light? Will it be brighter? (considering the 3 LEDs very close to each other, same direction, etc.) **Quickly said** : does the perceived light "sum up"?


Anti-matter repelled by gravity - is it a serious hypothesis?

> **Possible Duplicate:**   >  Why would Antimatter behave differently via Gravity? Regarding the following statement in this article: > Most important of these is whether ordinary gravity attracts or repels > antimatter. In other words, does antihydrogen fall up or down? Is this a seriously considered hypothesis? What would be the consequences on general relativity? If this is seriously studied, can you point to some not-too-cryptic studies on the (anti ;-)matter?


Complete annihilation of matter-antimatter

> **Possible Duplicate:**   >  More on matter and anti-matter Everyone knows that when matter and anti-matter come in contact, they result in pure energy.   I want to know that if atoms and anti-atoms of two different elements are brought in contact, and one having much more charge and mass than the other..will it still result in pure energy because in this case equal amount of opposite charge is not present to counter each other.   Will it result in leftovers ?


What would be the speed of a hypothethetical object created during the big bang, and that has been moving around in the universe and that is now close to the earth, so the time elapsed (from the hypothetical object's perspective) is around 6,500 years, that is the age of the universe according to some people?.


What is the furthest object from which fermion rays were detected?

What is the furthest object from which non-electromagnetic cosmic rays were detected?


How are these balls reflected after they hit each other?

> **Possible Duplicate:**   >  Physics of simple collisions I have 2 photos of the balls, one before the collision and one after the collision. They do a elastic collision. I want to know how is the reflected value calculated. I know it uses dot product and such stuff. But it all is very vague to me. Can anybody show me a working calculation and explain me based on this figures? I have been trying to understand this for past 2 days but I can't get how it all works. This is the image before collision: ![enter image description here](http://i.stack.imgur.com/nNpCv.jpg) This is the image after collision: ![enter image description here](http://i.stack.imgur.com/vlq45.jpg) Thanks in advance :)


Physics of simple collisions

I'm building a physics simulator for a graphics course, and so far I have it implementing gravitational and Coulomb forces. I want to add collisions next, but I'm not exactly sure how to go about doing it. A quick summary of how this is working so far is: All objects are spheres of a set radius, mass and charge. The mass's and charges of the spheres are treated as point charge/mass for the calculation. Every step in time (about 1/50th of a second) the forces acting on each object are calculated in a nice big nested for loop that figures out the coloumb and gravitational force between 2 objects, for every set of 2 objects, and then they are summed together. I use this net force to determine the acceleration, and the rest is fairly obvious from there. What I want to add in is collisions. I can deteremine pretty easily if a collision is happening (if the distance between them <= radius of one + radius of other), what I am not so certain of is how I should add in the collision force (and would need to do it component wise). I want elastic collisions, though for now I'd just be happy with getting conservation of momentum The information I have easily available are: the velocity of each object (and I mean velocity, speed and direction), the position of each object, the mass/charge (charge obviously not needed here) and the Net force calculated so far for each object for the next step in time. I dont need exact formulas (would prefer if they arent exact), just need a nudge in the right direction.


Collision of two balls of finite radius

This is your typical elastic collision problem except the balls have finite radius. To be clear:   * two billiard balls in the plane each with radius $\sigma$   * move at constant velocities $v_1, v_2$   * starting at positions $x_1, x_2$ What is the criterion for there to be any collision at all? When does it happen? And at what angle to they collide? I need it to write a computer simulation of the hard sphere model. ![enter image description here](http://i.stack.imgur.com/Lkndc.png)


The physics of two circles colliding

> **Possible Duplicate:**   >  Physics of simple collisions I have looked around the internet and gradually built up a better understanding. However I'm still not 100% clear on the physics. If I have 2 identical circles (in a 2D world, so they are on the same plane) and they collide, from any 2 directions, how do I calculate which direction they will travel in after the collision. My initial though was to get the tangent between the 2 circles, and each circle would bounce off the tangent in the same a ball bounces off a flat object (angle into the collision equals the angle out, but reversed around the normal): (where the red lines are parallel, and the line between the center of the circles is perpendicular to the red lines. And the angles x1 and x2 are equal in value, as are y1 and y2, but x1 and y1 are not [necessarily] equal) ![](https://docs.google.com/drawings/pub?id=13Kw1gxNjd_7Pd181QdVFzoOAtWXw4LO1mJerdnFrVj4&w=521&h=497) But after reading articles on the internet, it doesn't appear this is right? (NOTE: This is not homework, but rather learning by example)


Bouncing back of a ball

> **Possible Duplicate:**   >  Physics of simple collisions Let the unit vector along the positive x- axis be i and that along the Y-axis be j. Let us consider a rigid wall with the normal to the wall being along Y-axis. Let a ball with velocity V1 i - v2j move towards the wall. After hitting the wall will it bounce back with a velocity v1i +v2j or v2j-v1i?


Is accelerating particles through a chain of accelerators a continuous or batch process?

At the Advanced Photon Source, they use two accelerators before injecting the electrons into the large storage ring. Is the addition of particles to the storage ring done in "batches" (however small/short they may be), or continuously? By a "batch", I'm thinking where the LINAC would throw however many electrons into the booster ring, which would then accelerate them all together, then dump them all into the storage ring when they're at the appropriate energy.


If I lift a box vertically, why is the work I do equal to the distance I lift it times the force of gravity on the box?

I have problems fully understanding the concept of work, so please forgive me if this is simple. If I take a box of mass $ m $, and lift it a distance $ d $ vertically, **why is the work I have done equal to $ gmd $, where $ g $ is the force gravity exerts on the box?** I understand that work is equal to force times distance--so I'm not asking about the definition of work--but if I exert an upward force equal in magnitude to gravity's, won't the box remain motionless, i.e., net zero force, in which case the velocity is constant, and displacement and work done will be equal to zero? Edit: To be clear, **what I'm asking _is not a duplicate_ of "Why does holding something up cost energy while no work is being done?", because I'm not asking about work done on an object with zero displacement, nor is it a duplicate of "What exactly is F in W=∫baFdx?", because I'm not asking about the distinction between the work done by an individual force and net force.**


Keep an object at a certain hight, does it require energy?

If i have an object with a defined mass, like a stone, and put it on the shelf it will sit there forever (lets forget all external influences). But if I have to hold up the stone I will eventually get tired, feel pain, and give up. Obviously I am using up energy to keep the stone where it is. But, if it takes energy to keep an object at a certain hight it could not sit on the shelf forever because it would need infinite energy. What am I missing?


So there are 6 quarks, what are anti-quarks considered then?

I just recently got into particle physics and the quantum world and I love it. So my first big question is. I watch all these videos and people explain the quarks (up, down, top, bottom, strange, charm). And they all say there are 6 quarks. But every so often someone speaks of an anti-quark. What is this anti- quark if there are only 6 quarks? Is it anti-matter? Is it still a quark? (if so that means there are 12 quarks?) Second question, more for clarification. So there are force carriers and particles. Force carriers are bosons, they carry the strong weak and electromagnetic force and the gravity force carrier is still a mystery as to what carries it (mystery as in we just have not observed the gravity force carrying particle; i.e. higgs/gravitron)? Non-force carriers are leptons and are comprised of only quarks? And quarks can then make more massive particles like protons (uud)?


S-Wave for minimally coupled scalar field

This question is in reference to the paper here (Equation 3).The extremal 3-brane metic in $D=10$ can be written as: \begin{equation*} ds^2 = A^{-1/2}(-dt^2 +dx_1^2 +dx^2+ dx^3) + A^{1/2}(dr^2 +r^2 d\Omega_5^2) \end{equation*} where \begin{equation*} A = 1+ \frac{R^4}{r^4} \end{equation*} In this background the $s$-wave of a minimally coupled massless scalar satisfies: \begin{equation*} \left[\rho^{-5}\frac{d}{d\rho}\rho^{5}\frac{d}{d\rho}+ \frac{(\omega R)^{4}}{\rho^{4}}+1\right]\phi(\rho) =0 \end{equation*} How do I derive this result?


Is a soundproofed wall really only as strong as its weakest area?

I've seen a few questions about sound waves and sound travel here on Physics SE, so I'm hoping this question is a good fit for this site. During my internet research on soundproofing, I've come across many acoustics gurus who say that, when soundproofing a wall, you have to account for every single little opening such as a wall or ceiling even if the opening is as tiny as the shaft of a screw. If you don't, you may as well not soundproof the wall at all. Since I have approximately zero understanding of the physics of sound waves, my incredulity brings me to doubt those words. I imagine the contact of sound waves upon a wall being much like taking a handful of pebbles and slinging them against the same wall. When the pebbles leave your hand, they spread out. If I were to sling a handful of pebbles at a group of people from a distance, the chances are high that everyone's going to get hit. If you put up a solid wall in front of them, however, no one will feel a thing as the wall will stop 100% of the pebbles. If you cut a hole 2' in diameter and sling the pebbles, perhaps only a few people will be hit since only the pebbles that make it through the hole will have a chance of striking anyone. If I were to apply the "weakest link" argument here, it would suggest that slinging a handful of pebbles at a wall with a small hole in it would be the same thing as slinging the same pebbles as if the wall weren't there. The only way this could be true is if all the pebbles spread out after leaving your hand, magically coalesced enough to fit through the hole, then spread out again to hit the group of people. Now back to sound travel. How could a small, screw-sized hole totally negate the _entire_ soundproofing effort? While I can understand that a little sound would still make it through that hole, I am not seeing how that little leak would make it as if the wall isn't there at all. Can someone explain how a small opening in a well-soundproofed wall affects sound travel?


Snowmaking in the tropics - an estimate of water evaporation

If I set up a snowmaker in the tropics and sprayed water with it how much water would I evaporate? How would I calculate?


Electric potential difference from electric field of isolated spherical conductor

I'm wondering if my train of thought is correct. Say you have Point $A$ which is $A$ distance away from the center of an isolated spherical conductor, and Point $B$ which is $B$ distance away from the center. If I'm to find the electric potential difference, $V_A-V_B$, I know I can use the relationship $V_A-V_B= -\int_B^A \vec{E} \cdot d\vec{s} = -\dfrac{kQ}{r^2} \int_B^A d\vec{s}$ What I'm wondering is if $d\vec{s}$ is equal to $dr$ in this case. I don't really see any other thing it could be. Using that idea, things simplify down to, $\rightarrow = -\dfrac{kQ}{r^2} \cdot r \bigg|_B^A \rightarrow -\dfrac{kQ}{r} \bigg|_B^A$ Is this correct? I would also appreciate an explanation as to why it is or isn't, since I'm a obviously a little unsure of myself.


Inclusive and exclusive searches

Please, I would like to know what means inclusive and exclusive searches in High Energy Physics. Thanks in advance. Fábio.


Translation operator to higher order

IN QM, the space translation operator, or generator of Translations is set to be Ie-ie/h_bar*P up to order e. Now my question is what is the physical justification of only going up to order e and do the higher order terms have any meaning? Is it correct to say(if we are only including up to order e) that we are doing a constant linear translation(equal distances in equal times) and hence assuming there is no change in how fast we do it?


what interactions would take place between a free proton and a dipolariton?

What interactions can be expected to take place between a free proton and a dipolariton, (a) at high energies and (b) at lower energies? A dipolariton is a bosonic quasi-particle mentioned in a recent article in Science. It is a static dipole that consists of an electron bound together with a photon. I suspect the proton and dipolariton would be attracted to one another, but I'm unsure of the specifics.


How close would Earth have to be for us to detect it was habitable, and then inhabited?

Given our current technology (or technology that is near implementation), how close would a clone of our Solar System (and so also Earth) have to be to us in order to detect that the cloned Earth was in habitable, and also how close would we have to be to detect that there is life on the planet (excluding radio signals the cloned humans have broadcast into space)? Basically I'm asking if we assume the worst case scenario where life only exists on Earth-like planets, and that the life is the same as ours (ie is inteligent, builds cities, etc), at what range with our current technology and methods/techniques would we be unable to detect a planet and civilisation the same as our own (seeing as it's the only civilisation we know about)? EDIT: Another way to put this, assume every star system is identical to the Solar System, using our current technology/techniques what is the furthest planet we could "see" that is habitable, and what is the furthest planet we could "see" that is inhabited by a species identical to our own (so the identical Earth 500 lightyears away would actually be in the year 2513, so we'd "see" it in 2013).


Is it possible to witness a circular rainbow?

What conditions would make it possible to see a naturally occurring fully 360° circular rainbow? Would it even be possible?


Do Maxwell's Equations overdetermine the electric and magnetic fields?

Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\vec{E}=(E_x,E_y,E_z)$ and $\vec{B}=(B_x,B_y,B_z)$, there are only 6 unknowns. So we have 8 equations for 6 unknowns. Why isn't this a problem? As far as I know, the answer is basically because the equations aren't actually independent but I've never found a clear explanation. Perhaps the right direction is in this article on _arXiv_. Apologies if this is a repost. I found some discussions on PhysicsForums but no similar question here.


Why does electric field intensity $E$ can be uniquely determined by its divergence and curl?

My question is, the number of following equations $$\nabla\cdot E=\frac{\rho}{\varepsilon}$$ $$\nabla\times E=-\frac{\partial B}{\partial t}$$ is 4 while the number of unknown variables $E=(E_1,E_2,E_3)$ is 3. Intuitively, the equation is overdetermined and the solution may not exist unless four equations are correlated. Is my intuition right?


Lowest temperature possible in the universe?

The third law of thermodynamics states that nothing can reach to absolute zero temperature. What is the lowest possible temperature that can be in the universe? Has any experiment reached to a billionth of a Kelvin? Is there any restriction on how low it can be? Is $10^{-1000}\;\rm K$ possible? Or is there a lowest quantum of temperature?


Why Gravity attracts all objects with the same speed? Is this question was solved? What is the exact answer?


Fitting of data to a model

Imagine that I have some observable value predicted with a theory for some process to be: $1+a x + b x^2$ and observed value is 1.3 with an error 0.2; a and b are some numerical constants. I also have another observable influenced by the same x to be $1 + cx+ dx^2$ and the measured value 1.2 with an error 0.6, c and d are some numerical constants. My question is: How to calculate value of x that explains both measurements the best? I guess this is called benchmark point, but how to get it and how to quantify it?


What affects the damping of a spring?

What variables affect the damping of a spring executing simple harmonic motion? What are the independent variables, and what variables would need to be controlled in an experiment? I'm attempting to complete an investigation where I measure the decrease in amplitude of a damped spring, and to prove the relationship between variables in the motion. Thanks!


How did Einstein come up with his postulates for Special Relativity using thought experiments on simultaneity?

In his popular book on relativity, in chapter IX, "The relativity of simultaneity", Einstein describes an experiment in which a flash happens simultaneously on A and B, as defined by the fact that an observer at the middle point M can see the light coming from A and B at the same moment: > >               --train--> >   > > \---embankment---A-------M-------B---- Then, he proceeds to say that an observer on the train "is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A". So Einstein concludes that simultaneity is not absolute. And an explicit calculation using Special Relativity principles confirms this. The conclusion is that: > So the answer is that the observer on the train sees the lightning strike > the front of the train at $t' = -\gamma\tfrac{vd}{2c^2}$ and the rear of the > train at $t' = \gamma\tfrac{vd}{2c^2}$. The time between the lightning > strikes is $\gamma\tfrac{vd}{c^2}$. By setting $\gamma=1$, we obtain the result in Galilean Relativity (ie: "The time between the lightning strikes is $\tfrac{vd}{c^2}$"), which is the theory of space time before Einstein came out with Special Relativity. The point is that regardless of whether we are calculating using Galilean Relativity or Special Relativity, the conclusion is that different people on different speed will always have different notion of simultaneity. Since this is a conclusion known to everyone before Einstein, how did Einstein use this insight to derive his Special Relativity? Truly, I fail to see how can this thought experiment on simultaneity can give Einstein insights into Special Relativity. Anything I miss?


Why electrons behave as a particle and also as a wave?

Why do electrons (and other very small particles) sometimes behave as particles (i.e. when we are not looking at them) where as other times they behave as waves?


Logical positivism and black hole interiors

A black hole exists. Eventually, it will completely evaporate away. Alice falls into the hole. Imagine you are Alice. According to logical positivism, the interior of the black hole exists. But you will die at the singularity. You can't measure anything after you're dead, so any event outside after the black hole has vanished doesn't exist. Bob stays outside the hole. Imagine you are Bob. The interior doesn't exist. Enter Quentin. Right now, he holds a qubit in the state ${1\over\sqrt 2} (|0\rangle + |1\rangle)$. He will measure the value of the qubit shortly in the future. If 0, he will jump into the hole. If 1, he will stay out forever. Imagine you are Quentin. Right now, does the interior exist? Curiouser and curiouser.


Why can't my eye see itself in the mirror through polarizing 3D-glasses?

I found a pair of polarizing "3D glasses" lying around, and tried to look at myself in the mirror while wearing them. To my utter confusion, when closing the left eye and only looking through the right eye, I could not see the right eye in the mirror. The light could not pass through the same polarized lens twice. (I could, however, see the closed left eye clearly.) I would expect the opposite to be true, as light going out the right lens with polarization X and coming back in with the same polarization X should pass through unaffected.


Boundary layer theory in fluids learning resources

I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. Not rigorous as in keeping track of all epsilons and deltas, but more rigorous than an heuristic argument. Hope you understand what I mean. Some free resources available on the web would be preferred, but if you can suggest book titles that's also helpful. Thanks. Edit: The applications I have in mind are the calculation of damping in surface waves in basins of various shapes (circular, rectangular, etc).


Questions about Statistical Mechanics

1. For grand partition ensemble, is it true that the introduction of chemical potential allows us to have the sum of number of the particles in each state to be the total number of particles ("On average")? and this fluctuation in the constraint of $N$ will shrink as the number of particles you got is large.   2. Next question is about Boson gas, why would we have to consider the chemical potential to be negative or zero? Is there anything to do with the reference to the ground state energy level?   3. What would it mean if we were to integrate the energy of those states below ground states? NB For last question, I mean when I were to find the total energy of this boson gas system, which I know below a critical temperature Tb, the Bose Einstein condensation becomes important, and we would do the separate sum thing as usual ![enter image description here](http://i.stack.imgur.com/ufZ9N.png)(see first Pic) , Now I wonder why when we do the sum of the energy we integrate the states above ground state , does it mean all the particles at ground state have zero energy?? and we only have to concern those above the ground state? ![enter image description here](http://i.stack.imgur.com/d3ALk.png)


Lorentz force for electrically conducting fluid flow in homogeneous magnetic field

I am mathematician and have paper which models situation when homogeneous magnetic field is applied to moving electrically conducting fluid. There is such Lorentz force formula on which all the work is made: $$F= \sigma (E + V \times B) \times B $$ But I think that the last $\times B$ is not needed. Is this correct formula for this case or the work is not correct?


Is it possible to calculate distance if non-constant acceleration is unknown?

I know that the distance travelled in non constant acceleration is $d=\int_a^b f(x)\mathrm{d}x$, but is it possible to calculate the distance without knowing the value of $f(x)$?


Twins Paradox Paradox

I've recently has special-relativity explained to be in a rather elegant way. All objects travel at the speed of light in space time. Thus, when you travel faster through the three dimensions of space, the speed at which you travel through time decreases. Photons do not experience time because all their velocity is in the spatial directions and none of it is in time. Considering the 4-dimensional universe this invokes, how can I ever interact with anything that has traveled at a different velocity than me unless it changes direction to meet me? In the terms of the twin paradox, how is it possible for the twins to meet? Twin A, stays on earth traveling at a fixed speed in spacetime. Twin B leaves earth traveling faster in space and slower in time but still traveling at c in spacetime. Some basic geometry tells me that if two objects forced to travel at a fixed speed from the same origin in changeable directions (in any number of dimensions) that there is no way their future locations can be the same unless they **both** change direction to provide the possibility of intersection. If Earth is traveling at a fixed speed and Twin B is traveling at a fixed speed in a space (no matter the dimension) then there is no way they could ever be in the same location in spacetime again unless the earth changes direction and meets Twin B in the middle. Wouldn't they be forever doomed to be in different places along the axis of time unless Twin A goes on a relativistic voyage to allow Twin B to catch up? Should we not expect time to behave the same way as the spacial dimensions?


Why the peak of spectrum gets vague when the dimension is lower?

In a many-body system, we can know the spectrum function at a particular temperature from Green function. It means density of states. A peak of spectrum represents one mode. My question is that in the same model why the peak of the spectrum gets broader with lower dimension?


How can the speed of light be a dimensionless constant?

This is a quote from the book _A first course in general relativity_ by Schutz: > What we shall now do is adopt a new unit for time, the meter. One meter of > time is the time it takes light to travel one meter. The speed of light in > these units is $$\begin{align*}\end{align*}$$ > > $$\begin{align*} c &= \frac{ \text{distance light travels in any given time > interval}}{\text{the given time interval}}\\\ &= \frac{ \text{1m}}{\text{the > time it takes light to travel one meter}}\\\ &= \frac{1m}{1m} = 1\\\ > \end{align*}$$ > > So if we consistently measure time in meters, then c is not merely 1, it is > also dimensionless! Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this: The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$ Does Schutz correctly show how $c$ can be a dimensioness constant?


Hemisphere irradiance

How do I calculate sky irradiance from radiance (L) from a hemisphere above a surface which is tilted relative to the normal (x=0,y=0,z=1). I have L as a function of zenith (0 to 180deg) and azimuth (0 to 360 deg)? Any known MAtlab functions doing this kind of thing?


Mistake in the Feynman Lectures Volume 1 Ch. 18-2 - Rotation of a rigrid body

I just read http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-S2 In my opinion, in this chapter the equations 18.6 and 18.7 are wrong. Have a look at the Picture http://www.feynmanlectures.caltech.edu/I_18.html#Ch18-F1 . > If $OP$ is called $r$, then the length $PQ$ is $rΔθ$, because of the way > angles are defined. The change in $x$, then, is simply the projection of > $rΔθ$ in the $x$-direction: $$\Delta > x=-PQ\sin\theta=-r\,\Delta\theta\cdot(y/r)=-y\,\Delta\theta$$ That is equation 18.6. _It is wrong because $\Delta\theta$ doesn't mean that $\theta_{Q}-\theta_{P}$ is infinitesimally small and equation 18.6 is only valid for infinitesimally small $\Delta\theta$._ An example is: I take $P$ to be $(1|0)$ and $Q$ to be $(0|1)$ than $\Delta x$ is $-1$ but calculated using equation 18.6 it should have been $-y\Delta\theta=(-1)(\pi /2)\approx -1,57$, which is wrong. In the equations $d\theta$ should have be used instead of $\Delta\theta$. **Is my reasoning correct?**


Proof for torque=force*perpendicular distance to line of action of force

I want to derive from first principles the proof for magnitude of torque about a point O = perpendicular distance from O to the line of action of the force * magnitude of the force. I want to derive this result for a general 3-D case for any arbitrary point O and force **_F_** . I need to prove this mathematically without knowing the cross-product magnitude in terms of the angle between the vectors..... that is given any force F = Fx i + Fy j + Fz k and any point O and using torque= r cross F & using the determinant expansion for the same & also the expressions for magnitude of a vector, I need to prove this.... In other words, I need to prove that the magnitude of the cross- product is indeed F*r*sin(theta


What is the maximum range of a bullet flying through the sky?

**How far would my bullet fly with the following characteristics?**   * 1.2km/second initial velocity   * 45 degree angle into the sky   * 'Normal' atmospheric conditions   * No wind   * Typical high-powered rifles have 5-10 gram bullets I've tried various online calculators but they only do vacuum, not normal atmospheric conditions, so the range is 102km.


Poincaré symmetry and linearized gravity

When working with linearized gravity, is Poincaré symetry assumed to be the symmetry of space-time?


Why do ferromagnetic thermocouples change their behaviour near the Curie point?

Having read Wikipedia's current explanation that certain types of thermocouple (specifically Type K thermocouples) experience a step-change in their potential when they reach the Curie point of the constituent iron, it does not explain the reason, nor why the Curie point is important for the thermoelectric properties of ferromagnets. There seems to be no more of a hint on Wikipedia's page about the thermoelectric effect, nor for the page on the Curie temperature. Searching the internet more generally for information only seems to find similar assertions without explanations, or abstracts from academic papers hidden behind paywalls. Can anyone explain this connection in terms a layman reading Wikipedia would understand, or failing that an undergraduate physics student?


Optics, What is a Foil?

I was using a program for simulating optical elements; mirrors, gratings, foils, crystals, slits, and zoneplates are what it mentioned. Does anyone know what a foil is in terms of optics? I know what aluminum foil is, obviously, but in terms of an optical system I don't know what a foil, or foil, would be, and what it's unique optical properties are.


What age to start kids with astronomy

My son is getting interested in astronomy. I was thinking about getting him a scope when he gets a bit older, he is 4 now. At what age have other people gotten their kids telescopes? (I have a 10" dob that we use together) Also at what age is taking a kid on a dark sky camping trip start to make sense.


What prevents this magnetic perpetuum mobile from working?

As a child, I imagined this device, which may seem to rotate indefinitely. I have two questions.   1. **Is this perpetual motion machine already known?** If it is, could you please give some references?   2. **What is the exact mechanism that makes it stop?** By this, I mean an explanation, not simply "because it would break energy conservation". Of course energy conservation is true, and of course cannot rotate indefinitely. But for any known (presumed) perpetuum mobile, there was an explanation, usually based on showing that the force generating the motion is balanced by another force. For example, Stevin obtained the laws of the inclined plane from the perpetual motion device ![Magnetic perpetuum mobile](http://i.stack.imgur.com/rhhTv.gif) I would like to make some observations.   * I don't try to convince anyone that it will move forever, because **I don't believe in breaking the energy conservation**. It is true that great physicists like Bohr, Kramers and Slater admitted the possibility, and nowadays some who think that there may be energy exchanges between parallel worlds in MWI believe, but I don't.   * But I don't consider an enough explanation simply to refer to the energy conservation. I am interested in an explanation showing exactly how the magnetic forces making it rotate, are balanced.    * If the forces are balanced, **only then** friction will make it slow down and stop. I don't think that we can explain only by referring to friction, which in principle can be made as small as needed. There has to be a balance of forces.   * Why spending time trying to understand or explain something that admittedly can't work? Well, even though perpetual motion machines cannot actually work, I think they may be interesting as puzzles.


Why is there water coming out of a car’s tail pipe?

I notice yesterday that my neighbor’s car had water coming out of the exhaust pipe in the morning. My first response was since the hot exhaust is hotter than the cold tail pipe, heat is transferred from the hot exhaust through the pipe, and with enough moisture in the exhaust, enough heat leaves such that the humid air is condensed forming water drops.   1. Is my thinking correct here? If not, please correct me.   2. Does the water come from (i) a chemical byproduct of gasoline combustion or (ii) the humidity of the air (i.e. I've only noticed this in the winter, not the summer so I assume that during the summer this would not happen since the tail pipe is “already warm enough.”)


How is electric flux related to permittivity?

How is Gauss' law related to permittivity? I know that it equals $1/\epsilon_0$ times the magnitude of the charge enclosed. But, I'm unable to understand what it actually means. Can someone intuitively explain it to me?


Reference paper to support information -- energy relation ($kT \ln2 \rm\frac{J}{bit}$)

In answer to Maxwell's Demon Constant (Information-Energy equivalence) there is stated that one bit of information allows to perform $kT \cdot \ln2$ Joules of work. Which paper supports the thesis? (there are many publications on Maxwell daemon, Szilard engine, Landauer's principle).


static friction problem without mass

I am having trouble with a physics problem. The problem is as follows: A car is traveling at 54.0 mi/h on a horizontal highway. A) If the coefficient of static friction between road and tires on a rainy day is 0.103, what is the minimum distance in which the car will stop? B) What is the stopping distance when the surface is dry and µs = 0.595? This is what I have so far for A)               fs = µs * Fn          Fn = Mg     fs = Ma          Ma = µs * Mg     a = µs * g          V = 54 - a * t     0 = 54 - a * t     t = 53.497 seconds          dist = 54*t - (a/2)*t^2     = 1444.42243 miles     = 2324572.57 meters      This is not the correct answer. Any help would me much appreciated.


Where can I find the full derivation of Helfrich's shape equation for closed membranes?

I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of variational calculus" to derive the following: $$\Delta p - 2\lambda H + k(2H+c_0)(2H^2-2K-c_0H)+2k\nabla^2H=0$$ But I'm having a lot of trouble tracking down the original derivation. The guy who did it first was Helfrich, and here's his and Ou-yang's paper deriving it: http://prl.aps.org/abstract/PRL/v59/i21/p2486_1 . However, they don't show an actual derivation, instead saying "the derivation will appear in a full paper by the authors" or something like that. Yet everybody cites the paper I just linked for a derivation. Does anybody know a source that can derive this, or can give me some hints to figure it out myself? To be honest I can't even figure out how to find the first variation. Edit: So, after some careful thought and hours and hours of work and learning, I realized that the answer that got the bounty was wrong. The author stopped replying to my messages after I gave him bounty.... thanks guys. That said, I've almost got it all figured out (in intense detail) and will post a pdf of my own notes once I'm done!


General definition of an event horizon?

Horizons are in general observer-dependent. For example, in Minkowski space, an observer who experiences constant proper acceleration has a horizon. Black hole horizons are usually defined as boundaries of regions from which no lightlike curve can reach null infinity $\mathscr{I}^+$. But how can this be interpreted in terms of an event horizon for an observer? Immortal material observers end up at timelike infinity $i^+$, not $\mathscr{I}^+$. Is there some nice way of unifying both cases? In other words, is there a general definition of an event horizon that has both these types of horizons as special cases? [Edited to clarify the question and remove a mistake about the dimensionality of $i^+$ versus $\mathscr{I}^+$.]


In a turning plane, will the vector of combined centrifugal plus gravity force experienced by passengers be perpendicular to the floor? In other words, will the passengers experience the feeling of inclination without looking into windows? I think that the reaction of air is perpendicular to the wings and as such, there will be no feeling of inclination. Am I right?


Does the positive mass conjecture indicate a necessity of interactions in our universe?

The positive mass conjecture was proved by Schoen and Yau and later reproved by Witten. Total mass in a gravitating system must be positive except in the case of flat Minkowski space, where energy is zero. Since QG is intended to be a theory of interaction with force particles called gravitons, one may begin to wonder if the interactions are in fact the important defining features of the space in question. So does a theory with interactions also require that space be curved?


Spooky action appears to contradict Relativity of time order of multiple events

It is well known that in special relativity observers can disagree on the time ordering of two events. It is also well known that entangled particles exhibit so called spooky action at a distance. Today I read in the New Scientist and on the arXiv that although the order of two events can be arbitrary, this is not so for multiple events. As the number of events increase the number of combinations of possible orders increase but not every one of these combinations will actually be possible to always observe. According to the NS article it is possible to entangle three particles a,b and c and then have them in such a way that the collaspe of c cannot be observed to precede the collapse of both a and b by any observer even though c spookily causes the collapse of a and b. Is this an outright contradiction between relativity and quantum mechanics or just a paradox ? What is the explantion ?


A good example of a nonlinear symplectomorphism?

What is a good example of a simple, physically useful _nonlinear_ symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked have been purely mathematical and not connected to any physical problems. Obviously linear examples are well known and useful. Actually, it would suffice if I could see some nice example of a nonlinear diffeomorphism (i.e. change of coordinates) $\varphi: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ given by $x \mapsto y$, as one can then lift this diffeomorphism to a symplectomorphism by transforming the momenta via $p_y := (d\varphi^{-1})^{*}p_x$. Thanks in advance.


When one thinks of a field of operators in QFT, is it reasonable to think of a matrix being associated with each point in space time?

Is it correct to visualize operators existing as matrices parameterized by spacetime coordinates in the context of QFT?


Is it a goal of modern physical theory to avoid big bang singularities and how do they approach the problem?

During discussion on another question, the question of singularties in modern physical theories arose. The big-bang is an obvious singularity in modern conceptions of the cosmos. Is it a goal of modern theories to find a way to remove the big bang singularity from theory, and how is that accomplished in theories being proposed today?


Depletion region

In a semiconductor diode. There is a depletion region formed, it is formed when electrons from n type side migrate to p type side, now which electrons transfer? Valence band ones present at the boundary or the conduction band ones? If the conduction band ones only migrate, then are there enough present on the boundary such that they can create a field at the boundary that they can stop this diffusion process? By boundary i mean the border of n type and p type material.


Energy from the Feynman-Kikuchi Partition Function

The Feynman-Kikuchi Partition function is given as $$Z_{FK}=K_\beta \int dx \eta(x) \exp \left(-\frac{x}{\beta}\right) $$ where $K_\beta$ is a normalization constant and $$\eta(x)=\frac{1}{N!}\sum_P \delta\left(x-(R_0-PR_0)^2/4\lambda^* \right)$$ where $R_0$ is a set of particle coordinates for a perfect cubic lattice, the sum over P means over all perturbations, and $\lambda^*$ is defined as $$\lambda^*=\frac{\hbar}{2m^*}$$ where $m^*$ is the reduced mass. My questions is this: in Ceperley's notes "Path Integrals in the theory of condensed helium", he states that, if we differentiate the partition function above with respect to $\beta$, we obtain the energy, in the form $$\epsilon=\epsilon_0+\frac{3T}{2}-\frac{T^2 \langle x\rangle}{N}$$ I can't see how he would get that--I know that I have to calculate $$\epsilon=-\frac{1}{Z}\frac{\partial Z}{\partial \beta}$$ But wouldn't that just give me $$\epsilon=-\frac{1}{\beta^2}\frac{\sum_P (R-PR)^2\exp\left(-\frac{1}{\beta}(R-PR)^2 \right)}{\sum_P \exp\left(-\frac{1}{\beta}(R-PR)^2 \right)}$$ I just don't see how that would give me Ceperley's results. Thank you in advance.


Pauli principle for particles very far apart from each other

Can two electrons be in the _same state,_ when they belong to two different atoms, which are "far enough" (whatever that means) apart from each other? With "same state" I mean that (as far as specifiable) the states are really identical, except for the position of the electrons. More specifically:   I am still not clear of how the separation of particles is taken into account for the Pauli exclusion principle. E.g. in a crystal the electrons seem close enough for the exclusion principle to become meaningful, for particles "a universe apart" from each other, it seems that it is kind of redundant (is there specific maths to that?), but what about everything in between? (This is closely related to the talk Brian Cox gave, see this question: physics.stackexchange.com/q/18527/16689 )


Why black body radiation is all over the frequency range

I was studying black body radiation and how quantization of energy solves the problem of ultraviolet catastrophe. But I have a very fundamental doubt. A black body can be assumed as a cavity with a small hole with radiation leaking out of it. As the temperature of the black body is increased we can assume the the charge particles, electrons, on the metal surface will behave as harmonic oscillators and the energy of the harmonic oscillation will be equal to the energy density of the radiation inside the cavity at thermal equilibrium. My doubt is that at thermal equilibrium, the cavity(assuming it to be uniform) will have a uniform temperature. Since the oscillations of the charged particles is due to the thermal agitation, how can the charge particles radiate all over the frequency range? The temperature is uniform over the cavity, wont they all be experiencing same thermal agitation and oscillate at same frequency


How to explain polarization in Zeeman effect

In the Zeeman effect, when we observe along the $B$ field, the polarization of light should be circular polarized. It can be understood by the conservation of angular momentum. $\Delta m=0$ can not be observed and $\Delta m=\pm1$ can be observed and thus circular polarization. However, I can't understand what we observe perpendicular to $B$ field. It should be linear polarized light. So the state of the photon should be something like $|\mathrm{linear~polarized}\rangle=\frac{1}{\sqrt{2}}|x_+\rangle+\frac{1}{\sqrt{2}}|x_-\rangle$($|x_\pm\rangle$ are circular polarized states in $x$ direction. We assume $B$ is parallel to $z$ axis) ,depends on $\Delta m$. How could it be like this? Why the light is linear polarized and can't be circular polarized? I am really confused.


Spreading UV light? Fresnel lenses?

I'm making a little UV exposure box and I'm looking for a way to evenly spread out the UV light. I've used Fresnel lenses to do just this in the past, but that was for visible light. But from what I've read, not all things work at all wavelengths and some materials will block UV. So my first question, would a Fresnel lens still work and evenly spread out UV light? What material should I look for? The lens I've seen are PVC and from what I've seen elsewhere, PVC has good UV resistance, but I don't know if that means it will just hold up to UV or if it will block UV all together. Any tips or suggestions would be great thank you


What is the simplest way to prove that the Earth orbits the Sun?

Assume you're talking to someone ignorant of the basic facts of astronomy. How would you prove to them that the Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits the Earth?


What does the complex electric field show?

We have a complex electric field. Is there any definition for absolute and imaginary part of a complex electric field? What do they stand for?


Coherent State in 2 dimensions

I am looking at a 2D harmonic oscillator $$H=\frac{1}{2m}(p_x^2+p_y^2)+\frac 12m(\omega_x^2x^2+\omega_y^2y^2)$$ Where $\omega_x=5\omega_y$. I am told that the oscillator is prepared in a coherent state with the following qualities: $$\langle x(0)\rangle=x_0$$ $$\langle p_x(0)\rangle=0$$ $$\langle y(0)\rangle=0$$ $$\langle p_y(0)\rangle=p_0$$ I am looking for the time dependent state of the system. My approach: Seeing as this is a coherent state, the expectation values act as classical variables, this means that the position and momentum will be given as $$q(t)=q(0)\cos(\omega t)+\frac{1}{m\omega}p(0)sin(\omega t)$$ $$p(t)=p(0)\cos(\omega t)-m\omega x(0)\sin(\omega t)$$ Additionally, the coherent state is given as: $$\alpha (t)=\frac{1}{\sqrt{2m\omega\hbar}}(ip(t)+m\omega x(t))$$ So if I want to find the state as a function of time, do I find the alpha for both x and y and add them together?


How would an electron bunch/beam look different in the rest and lab frames?

With respect to special relativity, I was wondering how the spatial dimensions would differ between the rest and LAB frame of an electron beam. System: Electron bunch/beam traveling in linear motion. How large/long would the bunch appear in the point of view of an electron (rest frame). How large/long would the bunch look in the lab frame? How would the answers change if the electron bunch/beam was accelerating?


Proving $RTln(a_i)$ equal to non PV work done by system,where $a_i$ is activity

First of all many here will might say that it is Chemistry question.If you think so then read this article and then answer question. Question is --- We know chemical potential is defined as $\mu_i =\mu_i^{standard}+RTln(a_i)$ .Here $a_i$ is activity of $i^{th}$ component of solution.In case of gases instead of $a_i$ it is $f_i$ that is fugacity of $i^{th}$ gas out of mixture of gases.Activity is here nothing but activity coefficient multiplied by concentration or in case of fugacity it is product of fugacity coefficient multiplied by partial pressure of gas. But now this is not the only definition of Chemical potential.Chemical potential also means **partial molar gibbs free energy** when pressure and temperature of system are constant.We know that Gibbs free energy of system at constant temperature and pressure also represents Non-PV work done by system.Now how can the term $RTln(a_i)$ can be correlated to Non-PV work done by system in order both two definitions will be equivalent.


How to find solutions to the gravitational potential metric h

I'm working on a problem in which a star of mass M1, radius R1 is surrounded by a thin shell of mass M2, , radius R2. I want to find the solutions to the gravitational potential h in the region in between the star and the shell and region outside both. Can we assume the mass is stationary, and not too large, and thus use the analogous Poisson's equation (Laplacian of potential = 4*pi*G*rho)? Or is there some physical reason that we should use the potential equation in the Hilbert gauge (d'Alembertian of tensor potential = 4*pi*G*(energy momentum tensor)/(c^2)? My thinking is that Poisson's is totally appropriate for this simple situation of a stationary star with a shell mass around it. Now, I don't see how to use Poisson's equation. I'm getting tripped up with how to consider the two masses. What if, for example, it's a very dense shell which is in fact more massive than the star. That would give a potential radially outward. How do I find a solution to the Laplacian?


Mixing of quarks, neutrinos... and leptons?

This is a quite simple question: quarks do mix (through the CKM matrix), neutrinos do mix (through the PMNS matrix). Then **why do charged leptons not mix?**


Does supercavitation create vacuum?

While reading the amazing things that the mantis shrimp can do, such as moving two of their limbs so quickly that the water around them 'supercavitates', a friend of mine told me that those limbs were creating vacuum, in the water. Does this creation of vacuum happen? Or is it something else? If so, am I creating vacuum when boiling water?


Energy of a string

What is the correct definition of the energy of a string ? I suddenly get confused with the definition of the energy of a string. Considering, for instance, a bosonic open string in the light-cone gauge, We have $H = p^-=p_+$, the hamiltonian, and we have $p^0$, which appears in the expansion of the string in light-cone gauge $X^o(\sigma, \tau) = x^0 + \frac{p^0}{p^+}\tau + ...$. a) In Zwiebach's First course in string theory (1), there is a calculus of the entropy of a string, which begins (Chapter $16.2$ p $354$) by a function partition for a non relativistic string with fixed points (the quantum violin string). For a given energy $E = N \hbar w_0$, where $N$ is the level, the entropy is computed (this is direcly linked to the number of partitions p(n)). More precisely, an expression for the entropy, $\sim T$, and the energy ,$\sim T^2$, are obtained, se we get a $S \sim \sqrt{E}$, and plugin the $E \sim N$ expression, we get a $S \sim \sqrt{N}$. "Suddenly", in chapter $16.3$ page $361$, it is said that "we now return to relativist strings that carry no spatial momentum, and this happens if the open string endpoints end on a D-0 brane, so the energy levels are given by the rest mass of its quantum states". Of course, we have $m^2 \sim N$ (for large $N$), so $m\sim \sqrt{N}$, so if the energy ($p^0 ?$) is identified with $m$, we have $S \sim E$ Now, if we take the hamiltonian $H$, for large $N$, we have $H \sim N$, so $S \sim \sqrt{H}$ So, this confused me, because I have $2$ law expressions for the entropy, depending on the definition of the energy. b) An other problem, is saying that the energy of the string is proportionnal to its length. If we look at $H$, we have $H=p^- \sim \frac{1}{l}$ (with zero trans verse momenta $p^i$), while $p^+ \sim \frac{l}{\alpha'}$. So, may I write that the correct energy is $p^o \sim (p^+ + p^-)?$


Scaling of non-gravitational energy in a black hole

When looking at a Schwarzschild black hole, for instance, we know that we may apply black hole thermodynamics. We may define a entropy of the black hole which scales like the area of the horizon : $$S \sim R_s^2$$. It is understood in the more general context of the holographic principle which states that " the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon" Now, the non-gravitationnal energy $E_{ng}$, so the mass $M$ for the Schwarzschild black hole, has a different scaling : $$E_{ng} \sim R_s$$ So, does that mean that the energy is encoded in a one-dimensional object (perimeter, loop, string, radius), and is it a different "holographic" principle ?