title: A Question about Virtual Work related to Newton's Third Law

text: In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as, $$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$ where $F_l$ is the sum of the applied forces on $l^{th}$ particle, $f_{ml}$ being the internal force on $l^{th}$ particle due to an $m^{th}$ particle, and $C_l$ denoting the constraint forces. However considering the law of action and reaction it further states that $f_{ml}+f_{lm}=0$, which I have no trouble understanding. But considering a virtual displacement $\delta \bf{r}_l$ on $l^{th}$ particle, in the next line it concludes that the virtual work $\delta W$ done should be, $$\delta W~=~\sum_{l=1}^N(F_l+C_l)\centerdot\delta\textbf{r}_l,$$ ignoring $f_{ml}$ terms. But if we take those terms into account shouldn't it be $$\delta W~=~\sum_{l=1}^N(F_l+C_l)\centerdot\delta\textbf{r}_l+\sum_l\sum_mf_{ml}\centerdot\delta\textbf{r}_l.$$ > In other words, I don't see how comes > $$\sum_l\sum_mf_{ml}\centerdot\delta\textbf{r}_l~\stackrel?=~0.$$ For the purpose of illustrating my problem, consider a system of two particles for which one can expand the above double summation and write $$f_{11}\centerdot\delta\textbf{r}_1+f_{21}\centerdot\delta\textbf{r}_1+f_{12}\centerdot\delta\textbf{r}_2+f_{22}\centerdot\delta\textbf{r}_2.$$ How can this add up to zero in general? Even if I assume $f_{11}=0$ and $f_{22}=0$, I am left with $$f_{21}\centerdot\delta\textbf{r}_1+f_{12}\centerdot\delta\textbf{r}_2.$$ Do I have to assume $\delta\textbf{r}_1=\delta\textbf{r}_2$ i.e. that the virtual displacements of the particles correspond to merely a displacement of the system? Or have I missed out on something?


title: Entropy and Stability

text: What happens when you increase entropy in a system? Does it increase the instability? Is disorder the same as entropy?


title: How would you explain the difference between these situations?

text: I have been thinking about this for quite a while, but I just can't seem to formulate a coherent answer. In problem which I will denote (a), we have a man sitting on some sort of chair and he is trying to pull himself up using a pulley (Man in elevator, holding it, on a scale) {ignore the question, all you need is the sketch}. It is clear that the force needed to accomplish this feat is half the net weight of the man+chair. In the second problem which I will denote (b), we have to weights (m1=1 Kg, m2=2 Kg) that are connected to each other via rope on a pulley. (http://www.tutorvista.com/physics/physics-pulley-problems) {ignore the question, all you need is the sketch}. It is obvious that (m2) will go down and pull up (m1). My question is why in problem (a) half of the weight was enough to prevent the man in the chair from falling, while in problem (b), half of the weight (m1) is not enough to prevent m2 from falling?


title: Jordan Wigner Transformation in 1d Majorana chain

text: So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for both the interacting and non- interacting cases. I understand why we had a $\mathbb{Z}$ classification in the non-interacting case. Basically, it measures the number of dangling majorana modes when one cuts a TR-invariant majorana chain (see section II of the paper). But then, for the interacting case, apparently, in order to define the interacting invariants, and construct the matrix product states, we need to do a Jordan- Wigner transformation to a bosonic spin chain, so that the original hamiltonian is written in terms of the new bosonic variables, $\sigma_j^x$, $\sigma_j^y$,$\sigma_j^z$ (end of section II). I don't understand why do we need to do this transformation and what good does it do to get the $\mathbb{Z}_8$ classification in the interacting case. I don't see the connection between this transformation and the way he constructed the matrix product states in section III.


title: Flux, intensity, etc in ferromagnetic core by integral

text: Cheng, Field and Wave Electromagnetics (newest ed.), ex 6-10, page 252 Some of the examples in this book are kind of sloppily described. Having trouble with this one. We have a toroidal core with permeability $\mu$, mean radius $r_0$, circular cross section radius $a << r_0$ with $N$ turns and a steady current $I_o$. Also there's an an air gap of length $l_g$. > Determine magnetic flux density $\textbf{B}_f$ in the core. Disregard flux > leakage and fringe effects near the air gap. Solution: $\textbf{B} = B_f \ \hat{\theta}$, $\oint \textbf{H} \cdot \textbf{dl} = NI_0$ $\textbf{H}_g = (B_f/\mu_0)\hat{\theta}$, integrated from $0$ to $l_g$ $\textbf{H}_f = (B_f/\mu)\hat{\theta}$, integrated from $l_g$ to $2\pi r_0$ So far so good. But here I want to say $\textbf{dl} = (r_0 d\theta)\hat{\theta}$, which would give the integral $\oint_c \textbf{H} \cdot \textbf{dl} = \oint_0 ^{l_g} \frac{r_0}{\mu_0} B_f d\theta + \oint_{l_g}^{2\pi r_0} \frac{r_0}{\mu} B_f d\theta = NI_0$ But the author kind of skips the $ dl = r_0 d\theta$ step and just proclaims $\frac{B_f}{\mu}(2\pi r_0 - l_g) + \frac{B_f}{\mu_0}l_g = NI_0$ So apparently $dl = r_0 d\theta$ is incorrect. How come?


title: What is the difference between a linear and non-linear solution in the bending of beams?

text: I have been working on a simulator for bending of beams and came now to a tricky doubt: What should be the difference between a linear and non linear solution in this case (graphic at bottom)? The solution of the following ODE gives us the non-linear curvature: ![enter image description here](http://i.stack.imgur.com/roqd6.png) and for very small angles (dy/dx)^2 will tend to zero, so we can linearize (Sorry, $dy=dv$ in this image): ![enter image description here](http://i.stack.imgur.com/UqfVe.jpg) So we integrate the following equation (I used the bvp4c function on Matlab) , that includes the curvature, to obtain the deflection of the beam: ![enter image description here](http://i.stack.imgur.com/Qdawq.jpg) In red is the non-linear solution and in Blue the linear solution. ![enter image description here](http://i.stack.imgur.com/7QEEp.jpg) My doubt is: at the middle of the $x$-axis $dy/dx=0$ in both curves, should I expect then also to have the same value of $y$, since at that precise point I could also cancel $dy/dx$ in the formula and both equations would look the same? In other words, what should I expect as a difference between a linear and a non-linear solution in such equations?


title: Is every QFT non-local in the U.V.?

text: As much as I understand the renormalization group transformation and the concept of relevant/irrelevant operators, I'd say that if we push the reasoning of only looking at relevant operators when we follow a "renormalization group orbit" (a curve in the space of the parameters) to the infra-red regime the other way, so going to the U.V., we should keep all the operators this time (as they are no more suppressed by huge factors), or there is a infinite number of them, depending on an infinite number of the derivatives, and we end up with a non-local theory. As this reasoning can be applied to every QFT, does this mean that they are all non-local in the U.V.? or the reasoning fooled somewhere (breakdown of the QFT framework somewhere for example...)?


title: What is behind recoherence?

text: I am quite familiar with the concept of decoherence, and I heard that a system that has decohered could recohere after that, I was wondering what could cause the the coherences that have leaked into the environment to come back to the system. I heard about thermal fluctuations for example, without being able to understand this argument. Thanks.


title: Time dilation at the speed of light

text: Does a massless particle travelling at the speed on light in a vacuum (c) experience an infinite time dilation effect? i.e. Would the time dilation extend from the perspective of the massless particle past the measurable existence of the particle by a observer outside the dilation.


Imagine the following gedankenexperiment. Observer Alice is right here on Earth. Observer Bob is at say Alpha Centauri. A pair of maximally entangled qubits is formed with one qubit handed over to Observer Alice and the other sealed in a box on Pluto shielded from decoherence. It takes a few years for any signal to reach Observer Bob. Observer Alice measures her qubit and "collapses" it. To her, the pair of qubits is no longer entangled. However, for a few years, will Observer Bob think there is an entanglement? We have to be careful here. Once the qubit on Earth has been measured by Earthlings, the entanglement is "shared". By the well known monogamy of entanglement theorem, any correlation between both qubits by themselves will have become classical. However, Observer Bob, who has a very good working knowledge of quantum mechanics and relativity, knows that for a few hours, until light signals can reach Pluto, the system consisting of an expanding bubble around Earth and the Pluto qubit are in an entangled state. Alice, who lives inside this bubble, will beg to differ. Is quantum entanglement an objective or subjective property? Closely related to Wigner's friend and intersubjectivity in quantum mechanics and Given entanglement, why is it permissible to consider the quantum state of subsystems?.


title: Does magnetism only work on ferrous objects?

text: Ordinary magnets can attract only ferrous objects (correct if i am wrong), but celestial bodies attract almost anything (including light). Are these different types of magnetism?


title: Significance of letters in Hertzsprung-Russell (H-R) diagram

text: The Hertzsprung-Russell diagram that categorizes star types uses the letter codes O, B, A, F, G, K, and M to indicate a star's temperature/color. Hottest (blue) is O and coolest (red) is M. What do the letters themselves relate to? For example, why is "O" used for hottest/blue?


title: What is the name of the equation which led to the Schrödinger one?

text: What is the name of this equation: $$\frac {d^2\psi}{dr^2}+k^2\psi=0?$$ (I want a Wikipedia link for this equation, but I don't know what its name is.) * * * Point: In this equation, the wave function is only function of space $\psi (r)$, for example $\psi=Ae^{\mathrm i kr}$.


title: Matter and anti-matter collision energy problem

text: From _Beyond Einstein_ , by Michio Kaku and Jennifer Thompson, Chapter 13, _Antimatter_ : > Dirac, also focused on the fact that Einstein's equation $E=mc^2$ wasn't > totally true. (Einstein was aware that the true equation was $E=\pm mc^2$ > but as his theory was based on forces, he ignored the minus sign.) As Dirac > was creating a new kind of equation (known as Dirac's equation) for the > electron, he shouldn't ignore the minus sign. This sign was intriguing > people, because this, seems like he predicted a whole new type of matter. So if I understand correctly, matter (as always known) follow the equation $E=mc^2$ whereas anti-matter follows the equation $E=-mc^2$. But a few paragraph later, it says : > When matter and anti-matter collide, they annihilate each other and an > enormous amount of energy comes out. When I thought a little bit, if a particle carrying $mc^2$ amount of energy, and it's anti-patricle carrying $-mc^2$ amount of energy then the total energy carried by both particles should be $E_{total}=E_{particle}+E_{anti- particle}=mc^2-mc^2=0$. I think that everyone can see here the problem : there are not any energy in total with both particles. So where this huge energy comes from when the particle and anti-particle collide? Or is it me misinterpreting what I've read?


title: Do days and months on the Moon have names?

text: On Earth we have various calendars, for example, Days: Monday, Tuesday, Wednesday, etc., etc. Months: January, February, March Does the Moon have names for its "daily" rotations, etc.? It sounds like a silly question, and I am not sure if I've asked it using the correct terminology. I suppose what I'm trying to ask is; from a viewpoint of someone living on the Moon - does it have "day" names?


I am trying to sort out my understanding regarding Photons. I understand the Electromagnetic Spectrum contains all radiation (energy) in all wavelengths (light) consisting of Photons. This can be broken down into Photons involving Waves and Packets- Electric and magnetic fields. The more Packets of Photons = the faster the wavelength which is the same as higher frequency/closer wave peaks. My best at understanding Electrons involves Electrons not creating energy themselves, but rather, simply act as a way to transport energy and have different levels of energy. I can also go as far as to say that when a level of energy is shifted closer or away from the Electron, the result is a shift in Light emitted as a reaction from the shift- and therefore; Photons are at work to represent the change in light within the Spectrum. I am looking at the EM Spectrum and focusing on the part I work with, Radio Waves, I can go into the endless theory which modern technology is based off of to create electronics, circuitry, and wireless propagation of radio waves transmitted and received from antennas. THIS IS MY QUESTIONS: If the EM Spectrum makes up all frequencies what is the connection between Photons and Electrons in electronics, mainly in RF Propagation? I am still new to quantum mechanics and my goal is to understand Photons. My cup is empty and any help would be greatly received. I admit I am still learning and I apologize ahead of time for trying to get a grasp on my lack of understanding. If Photos have no mass, yet make up the EM Spectrum through wavelength and packets and if Electrons have atomic mass and make up Radio Theory as we know it today then how is any of this even possible? How does the Electron have a relationship with the Photons to make RF Propagation work? The only theory I can ponder are Photons are the waves which Electrons ride on to pass/conduct energy which circuits use...? I won’t even ask what is Energy (at its start in the raw birth), rather just simply accept something provides a form of power/current/charged atoms which are manipulated to interact with electrical devices. Or, am I simply at trying to understand an ongoing division between normal physics with Hertz and Electrons vs Photons and QM? Any help would be amazing.


Intuitively, I would expect the change in kinetic energy of a particle to be frame independent. It just doesn't "feel" right that between two points in time-space, one frame should measure a change in kinetic energy of a particle different to another frame. Is my intuition right? Is the change in kinetic energy of a particle frame independent?


title: What is a ket of a vector with a bra of another one?

text: Suppose we have an orthonormal basis $\\{ \psi \\}$ in finite dimension of a hilbert space; What is the butterfly operator of a sum of the $ \psi $, say $\psi_i +\psi_j$ ? Since by linearity of "taking the dual" we must have, writing $B$ for but..terfly, $\lvert\psi_i +\psi_j\rangle\langle\psi_i +\psi_j \rvert = B_i + B_j + \lvert \psi_i\rangle\langle\psi_j\rvert + \lvert\psi_j\rangle\langle\psi_i\rvert$ but I have never seen expressions like $\lvert\psi_i\rangle\langle\psi_j\rvert$ when $i$ and $j$ are different. What happens when we relax orthonormality to get a general basis ? When we impose $\langle\psi_j\lvert\psi_i\rangle = \delta_{ij}$, what is the subspace the operator? How to visualize it? What if we have another basis $\\{ \phi \\}$ and impose $\langle\phi_j\lvert \psi_i \rangle = \delta_{ij}$; what does it mean to look at $\lvert\phi_j\rangle\langle\psi_i\rangle$? Can we say more than "it is projecting on $\lvert\phi_j\rangle$ when we take a ket and projection on $\langle\psi_i\rangle$ when we take a bra?


title: Conceptual way of understanding what material 'does' at critical angle

text: I would like a conceptual way of understanding how a material behaves at the critical angle. So why does it all reflect? I can see why it reflects in the maths, but conceptually, it doesn't make that much sense to me, especially the evanescent wave. Is the material basically acting like a lossless dielectric that absorbs the energy and then spits it back out? Thanks.


title: Uncertainty principle and multiple observers

text: My understanding is that an observer can measure the precise location of a particle so long as the corresponding uncertainty in momentum measurement is not an issue and vice-versa. Say there is such an observer, interested in the precise position of a particular particle. Now, consider a second, independent observer, unbeknownst to the first, who is trying to measure the exact momentum of the same particle without caring about the position. As a thought experiment, we assume that the two observers are somehow able to access the same particle at the same time in some way without being aware of each other. Can both observers get their desired results?


title: A quicker way to verify that a function is a constant of motion?

text: I have three particles that we can indicate with $\alpha$ ($\alpha$=0,1,2), they are identified by the $r^i_\alpha$ coordinates and $p^\beta_j$ conjugata momenta ($\beta=0,1,2$ and $i,j=1,2,3$). I have this transformation: $$\overrightarrow r_\alpha -> \overrightarrow R_\alpha + \overrightarrow \epsilon$$ $$\overrightarrow p_\alpha -> \overrightarrow P_\alpha=\overrightarrow p_\alpha$$ and I have to demonstrate that $\sum p_\alpha$ is a constant of motion. The Hamiltonian of the system is $$H=\frac{\overrightarrow p_0^2}{2m}+\frac{\overrightarrow p_1^2}{2m}+\frac{\overrightarrow p_2^2}{2m}-2V(\overrightarrow r_1- \overrightarrow r_0)+V(\overrightarrow r_2-\overrightarrow r_1)$$ I have calculated Poisson bracket $[H, \sum p_\alpha]$ that is equal to zero. Considering that the generating function of the transformation is: $$F_2=\sum \overrightarrow r_\alpha \cdot \overrightarrow P_\alpha+\overrightarrow \epsilon \cdot \sum \overrightarrow P_\alpha$$ I'd like to know if there is a quicker way to answer to the question.


title: Varying the amplitude of a driven wave

text: I' d like to know whether varying the amplitude of a system at resonance is possible or not and if it is, how? I've calculated the resonance frequency of a material and I'd like to know the displacement this frequency generates and whether I can control it by knowing the relationship between the amplitude (displacement) at resonance and power of driving source. It would make sense that increasing the power would increase the amplitude/displacement of the oscillation, but I need formulas. Thanks!


title: Why is the Ritchey–Chrétien telescope preferred in professional astronomy?

text: Hubble, as well as numerous other professional telescopes, use the Ritchey–Chrétien design. What optical and instrumental advantages does this kind of telescope have for professional astronomy?


title: What identifies an action-reaction pair of forces?

text: This is a question on a worksheet that I'm giving to a student: > A man of weight $W$ steps into the lift. The lift then moves downwards with > a constant speed. If $R$ represents the force acting on the man by the lift > floor, which of the following statements is/are correct? > > (3) $R$ and $W$ are an action-reaction pair according to Newton's third law Why isn't this statement (3) correct?


title: Why is this classical system seemingly non-deterministic?

text: So, I was solving this problem1 and I realized that the system given seems to be non-deterministic when analysed with classical laws. ![enter image description here](http://i.stack.imgur.com/Soh81.png) The situation is thus: A rope is wrapped for an angle of $\theta_0$ (symmetrically) around a fixed pole (the circle in the diagram). One end is fixed to a wall, and the other end is pulled with a force/tension $T_0$. The string is massless, but there is friction between the string and the pole of coefficient $\mu$. There is no slipping (the system is static). If necessary, the length of the string and the radius/relative position of the pole can be considered as givens. Now, if you follow my solution2, one gets that the tension in the string along the cylinder, where $\theta=0$ is the part on the right where the string just touches follows the following constraints: $T_0e^{-\mu\theta}\leq T(\theta)\leq T_0e^{\mu\theta}$. Extending this, you get that the tension $T_1$ in the part of the string connected to the wall follows the constraint $T_0e^{-\mu\theta_0}\leq T_1\leq T_0e^{\mu\theta_0}$. However, I don't see any way to _further_ analyse this system with Newtonian mechanics. Note that the inequality appears even if I consider a string with mass, though the calculation becomes more complicated. So basically I'm stuck with an inequality. Note that the system, as given, should be completely defined -- $\theta_0$, $T$,$\mu$, the radius of pole, and the relative positioning of pole are all that should be required to physically create this system. However, the macroscopic, measurable quantity $T_1$ somehow cannot be determined from Newton's laws. How does one resolve this apparent paradox? Classical systems are supposed to be able to fully determine measurable quantities. This doesn't seem to be the case here. 1\. This was a course on Classical (Lagrangian & Hamiltonian) mechanics, and our professor gave us some Newtonian mechanics problems as a first assignment. We're not yet sure why, but most were quite interesting. The assignment due date is over, this is just a residual point that has been bothering me. 2\. This is lifted from the document I submitted. I have blurred out the end because it contains my own resolution of the apparent paradox here. I may post it as an answer later, but for now I'd prefer to give everyone a fresh look at this situation.


title: The statistical nature of the 2nd Law of Thermodynamics

text: Ok, so entropy increases... This is supposed to be an absolute statement about entropy. But then someone imagines a box with a 10 particle gas, and finds that every now and then all particles are in the left. Conclusion, the 2nd law holds only in a statistical sense. But then Szilard comes up with a thought experiment with only one particle and a piston that can compress to the left or right. The apparent loss of entropy when finding the particle in the left half is compensated by the very bit of information that indicates where the particle is. So perhaps the 2nd Law actually holds in an absolute sense, except that... > Is there consensus on the absolute vs statistical nature of the 2nd Law, or > is it subject to interpretation? Can the issue be settled within a classical > setting, or does one have to go quantum? **Addendum:** (as per Ben Crowell's request) Here is the paper Szilard, L., 1929, _“On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings”_ , Zeitschrift fur Physik 53: 840–856. English translation in The Collected Works of Leo Szilard: Scientific Papers, B.T. Feld and G. Weiss Szilard (eds.), Cambridge, Massachusetts: MIT Press, 1972, pp. 103–129.


title: While finding the equivalent resistance why isn't the 8 ohm in series with the 5 ohm?

text: Is there any way to tell if the circuit below is in series or parallel quickly and accurately? ![refer to image](http://i.stack.imgur.com/uzR8Q.jpg)


title: What Did Nobelaureate Smoot Mean by "Modern Efforts to Find Violations of Special Relativity"

text: In George Smoot's 2006 Nobel Lecture, having won the Nobel prize for his research on the Cosmic Microwave Background (CMB), he refers to the possibility of the CMB being a special frame stating "modern efforts to find violations of Special Relativity look to this reference frame as the natural frame that would be special". Smoot's "aether.lbl.gov" website also states of the CMB: "This would seem to violate the postulates of Galilean and Special Relativity but there is a preferred frame". Are there peer reviewed articles that consider the possibility that the CMB represents a special frame?


title: Is there a systematic way to derive constraint equations?

text: There's this problem in Goldstein's (Classical Mechanics) derivations section: > 5\. Two wheels of radius $a$ are mounted on the ends of a common axle of > length $b$ such that the wheels rotate independently. The whole combination > rolls without slipping on a plane. Show that there are two nonholonomic > equations of constraint, > > $$\begin{align} \cos\theta dx + \sin\theta dy &= 0 \\\ \sin\theta dx - > \cos\theta dy &= \frac{1}{2}a(d\phi + d\phi'), \end{align}$$ > > (where $\theta$, $\phi$, and $\phi'$ have meanings similar to those in the > problem of a single vertical disk, and $(x,y)$ are the coordinates of a > point on the axle midway between the two wheels) and one holonomic equation > of constraint, > > $$\theta = C - \frac{a}{b}(\phi - \phi'),$$ > > where $C$ is a constant. And here's the image from the problem with a single vertical disk: ![image from problem with vertical disk](http://i.stack.imgur.com/jLhG2.png) Now, I believe I have successfully derived the equations for two of those constraints, but I'll write it anyway, in case my reasoning is somehow wrong or too sloppy. (I use the labels $1$ and $2$ for the wheels, instead of unprimed and primed.) $$\dot{x} = v \sin{\theta}$$ $$\dot{y} = -v \cos{\theta}$$ $$\implies \color{red}{\cos{\theta} \, dx + \sin{\theta} \, dy = 0}$$ And the second one: By rotating the wheels about the midpoint $(x,y)$, the angle $\theta$ changes such that $$d \theta = \frac{2}{b} \, dl$$ where $dl$ is the length of the arc swept by both wheels, satisfying $$dl = v_1 \, dt = - v_2 \, dt$$ because the wheels turn with anti- parallel velocities. $$ dl = v_1 \, dt = a \frac{d \phi_1}{dt} \, dt = a \, d\phi_1$$ $$ dl = -v_2 \, dt = -a \frac{d \phi_2}{dt} \, dt = -a \, d\phi_2$$ $$\implies \color{red}{d\theta = -\frac{a}{b} (d \phi_1 - d \phi_2) },$$ which implies the holonomic constraint equation, with flipped signs. (I guess I just picked different labels, right?) How can I get the last one? I don't have much experience with these sorts of problems, so I was wondering, **is there a systematic way to approach them** or is it always just hacking at the problem, hoping to pull out the constraint equations? _P.S. My question got edited because of policy reasons according to which I cannot ask some questions, so I would like to say that I don't want to know if my reasoning is correct for the derivation of first two constraints._ :) **EDIT, PLEASE READ:** Although I answered my own question regarding the specific problem mentioned here, if anyone provides a good answer regarding a systematic way to derive constraint equations, I will accept that answer instead.


title: Modeling viscous fluid flow in the curved pipe

text: I want to know if there is any simple (results don't have to be very precise) way of modeling viscous fluid flow throught the curved pipe with elastic walls. I need to gather preassure values and compare them to preassure of the fluid in the straight pipe with the same wall parameters.


title: Eigenvalue $a_n$

text: **Q1:** In Zetilli's book page 166 _(ch. "Postulates of QM", eq. 3.1)_ i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another form written like this $\hat{A}|\psi\rangle = a_n|\psi\rangle$ so now i am a bit confused. Is $|\psi\rangle$ equal to $|\psi_n\rangle$ or not? Whats the difference? **Q2:** Below the same expression there is a statement $a_n=\langle \psi_n | \psi(t)\rangle$. I interpret this like $a_n$ is an inner product between an $|\psi_n\rangle$ and $| \psi(t)\rangle$, but what is a physicall meaning of this? Maybee i would know this if i knew anwser to **Q1**. In the book it says that $a_n$ is _a component of $|\psi(t)\rangle$ projected on an $|\psi_n\rangle$_ and it states that this can even be seen if _we expand $|\psi(t)\rangle$ in terms of eigenvectors of $\hat{A}$ which form a complete basis_. Does this mean we calculate matrix multiplication $\hat{A}|\psi\rangle$? Author solves it like this: $$|\psi(t)\rangle = \sum_n |\psi_{n}\rangle\underbrace{\langle \psi_n| \psi(t)\rangle}_{a_n}=\sum_n a_n |\psi_n\rangle$$. Does this mean that the state of a system $|\psi(t) \rangle$ is a linear combination (if we have a finite possible states) of the eigenvectors multiplied by eigenvalues $a_n$? So the point of the QM is that the state of a system or a vector of a system $|\psi(t)$ remains the same but we use operators to change the eigenvectors and eigenvalues which then correspond to the operator (for example $\hat{x}$) associated with an observable (for example position). Enough for 1 question.


title: Do eigenvectors of quantum operators span the whole Hilbert Space?

text: I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of another operator $L_z$. I understand that I have to "expand" the state vector in the $L_z$ basis, which in this case is just the appropriate eigenvector, and then change the base to that of $L_x$, i.e to write the $L_z$ eigenstate in the $L_x$ base. The problem is that there is just no way of doing it, since the eigenvectors of $L_x$ don't span the whole Hilbert Space. I thought all quantum operators had a set of eigenvectors that spanned the whole space, I am wrong on that or am I wrong on my algebra?


title: Energy dispersion in graphene

text: I have two questions, in fact, both involving 2D graphene: (1) How may I determine the number of nearest neighbours? (2) Given that graphene has linear energy dispersion near the fermi level and the dispersion is given by E=$\hbar$v$_F$|$\vec{K}$|, I would like to determine the density of states. I think it is equal to g(E)=E/2$\pi$$\hbar^2$v$_F^2$, but how may I show that? I'd appreciate your help.


title: How to calculate Standing Waves in Electrical Cables?

text: I have a 20 metre Coaxial Cable. I send digital signals down the cable ranging from 5 KHz to 50 KHz. I have noticed a pattern in the noise ratio, an oscillating wave. I predict this is to do with standing waves in the cable. How can I calculate at what Frequencies these standing waves would occur in the cable?


title: Why is visible light used in Optical fibers (instead of other EM waves)?

text: Why aren't other electromagnetic waves used in optical fibres instead of visible light? Is it because the wavelength of light fits the internal reflection/refractive index of the material used for the fibre? e.g. Is the material that refracts light cheaper compared to material that refracts EM waves with other wavelengths?


title: Cylindrical wave

text: I know that a wave dependent of the radius (cylindrical symmetry), has a good a approximations as $$u(r,t)=\frac{a}{\sqrt{r}}[f(x-vt)+f(x+vt)]$$ when $r$ is big. I would like to know how to deduce that approximation from the wave equation, which is this (after making symmetry simplifications): $$u_{tt}-v^2\left(u_{rr}+\frac{1}{r}u_r\right)=0$$ Proving that's a good approximation is easy (just plug it in the the equation), I want to know how to deduce that from the above equation. I've been searching and I found this: http://vixra.org/abs/0908.0045, which actually solved me a couple of problems, but the way they do it looks a bit clumsy to me, saying for example that "assuming the function $g$ depends on $r$ so some terms just go away..." Thanks in advance.


title: Current vs Voltage Drive for Loudspeakers

text: Please see this article: here I don't know enough EMFT to comment on this but I am working on other tasks for a wireless speaker system prototype called "Busker's Friend". Completing my Schaum's EMFT book is on my list. So please comment on the truth or invalidity of this article and give an explanation. Thank you! This is physics related and I have a good reason for not posting on electronics.stackexchange. So please be kind. Thanks. :)


title: total noise power of a resistor (all frequencies)

text: Let's calculate the power generated by Johnson-Nyquist noise (and then immediately dissipated as heat) in a short-circuited resistor. I mean the total power at _all frequencies_ , zero to infinity... $$(\text{Noise power at frequency }f) = \frac{V_{rms}^2}{R} = \frac{4hf}{e^{hf/k_BT}-1}df$$ $$(\text{Total noise power}) = \int_0^\infty \frac{4hf}{e^{hf/k_BT}-1}df $$ $$=\frac{4(k_BT)^2}{h}\int_0^\infty \frac{\frac{hf}{k_BT}}{e^{hf/k_BT}-1}d(\frac{hf}{k_BT})$$ $$=\frac{4(k_BT)^2}{h}\int_0^\infty \frac{x}{e^x-1}dx=\frac{4(k_BT)^2}{h}\frac{\pi^2}{6}$$ $$=\frac{\pi k_B^2}{3\hbar}T^2$$ i.e. temperature squared times a certain constant, 1.893E-12 W/K2. Is there a name for this constant? Or any literature discussing its significance or meaning? Is there any intuitive way to understand why total blackbody radiation goes as temperature to the fourth power, but total Johnson noise goes only as temperature squared?


title: Is there a chance for Earth to have an additional satellite someday?

text: I am just an space enthusiast and not a physics professional so pardon me if this is not the right place to ask.   As title says, is there a chance for our Earth to have additional satellite(s) like our Moon? I tried to google it, without luck, however I know here there are very smart people that can perhaps give me some references to read or can give me an answer :)   Thanks


title: Gravitational force between two masses

text: I get it that there will be a gravitational force between objects attracted towards gravity but can there be a gravitational force between two objects resting on horizontal plane? In other words, does an object experience gravitational force in all directions?


title: Why is a suspended spinning body of liquid prolate?

text: Instead of a flat disc one would expect the centrifugal forces to push it in.. The body is not in a container but suspended in air spinning horizontally (i.e. left to right or vice-versa) - how to do this could be challenging - but that is not the issue. **Answer:** It is not prolate: http://www.youtube.com/watch?v=BxyfiBGCwhQ.


title: Non abelian gauge theory with charged scalar field

text: Suppose we have an SU(N) non abelian gauge theory coupled with a multiplet of complex scalar fields $\Phi$. The lagrangian would be $$ L= - \frac 12 \text{Tr } F_{\mu\nu}F^{\mu\nu} + |D_\mu \Phi|^2 - V(\Phi)$$ where $D_\mu$ is the covariant derivative: $D_\mu \Phi = \partial_\mu \Phi +ig A_\mu^at_a \Phi$, $A_\mu$ is the gauge field and $t_a$ are the generators of $SU(N)$. The kinetic term gives two interaction terms between $\Phi$ $$ \partial_\mu \Phi^* \partial^\mu \Phi + ig A^{\mu a} (\partial_\mu \Phi^* t_a \Phi - \Phi^* t_a \partial_\mu \Phi) + g^2 A_\mu^a A^{\mu b} \Phi^* t_a t_b \Phi$$ I would like to know if I extracted the vertices correctly from the lagrangian. Suppose all the momenta are ingoing. For the $\Phi^*\Phi A_\mu$ vertex I have $$ i g (q+q')^\mu (t^a)_{ij}$$ where $q$,$q'$ are the momenta of $\Phi^*$ and $\Phi$, but I'm not sure about the relative sign of the momenta. For the $\Phi^*\Phi A_\mu^a A_\nu^b$ vertex, I have $$-i g^2 g^{\mu\nu}\\{t^a,t^b\\}_{ij}$$ ($i,j$ are the component indices for the scalar multiplet). I would like to underline that this is not homework. Even if the question arises from the preparation of an exam, I'm not asking the solution to an exercise. This is just some speculation following a digression of the lecturer about gauge theories with scalar fields.


title: If an object moved 5 meters in a second, how can its velocity be 10 m/s?

text: I am reading the tutorial on physics classroom.com, and am at Meaning of Slope for a v-t graph. It gives data for an object experiencing an acceleration: ![enter image description here](http://i.stack.imgur.com/sjs2K.png) This is the corresponding chart for the object's velocity: ![enter image description here](http://i.stack.imgur.com/jXFr1.png) At 0 seconds the position of the object is 0 meters. And the velocity is 0 m/s which makes sense. After 1 second, the object travels 5 meters. The chart says that its velocity is 10 m/s at time equals 1 second. How is this possible if the object only travelled 5 meters within that second? Shouldn't the velocity be 5 m/s and not 10 m/s?


title: Hydrogen atom: potential well and orbit radii

text: I happened to open up an old solid-state electronics book by Sah, and in it he says: "it is evident that the electron orbit radius is half the well radius at the energy level En" The orbit radius is $r_n=\frac{4\pi\epsilon_0 ℏ^2 n^2}{mq^2}$ and the potential well $V(r_n)=\frac{−q^4m}{(4\pi\epsilon_0)^2ℏ^2n^2}$ Of course the orbit has to be confined in the well, but it's not obvious to me why it should be exactly half the well radius? This isn't something I recall seeing before either in any other text. Thanks


title: Modeling a stool with three legs pushed on one leg

text: If a stool with three legs is pushed (horizontally) on one leg in the direction of the axis passing through the other two legs, it starts both moving and rotating. Is it correct to assume 1/3 of force on each of the legs? This comes a bit in contrast with intuition. If we imagine the legs flexible, then the pushed leg should deform more (i.e. stronger friction force). What is the correct modeling of the friction forces? p.s. this seems a simplified version of Push a box in a plane with friction. How to deal with the rotation? but there's no clear answer in that thread.


title: How is CMB related to the temperature of the universe

text: As I understand it, CMB (cosmic microwave background) is the radiation emitted when matter decoupled at the early stages of the big bang. The thing I don't understand is do all stars emit this kind of radiation or just the ones formed at the early stages of the big bang and are those stars travelling at the speed of light away from us also how is the fact that light is redshift related to the temperature of empty space.


title: Consider relativity. If a stick accelerates from a stationary state, how will it look like during the acceleration?

text: I mean how it will be shortened in the sight of an observer in a stationary coordinate system. It seems I was not clear enough. If the stick becomes shorter, then while this is happening, the observer should see the two ends have different velocities. Then, how big are they?


title: Physical interpretation related to a non-linear partial differential equation

text: I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting from a simple case with known applications, and then generalising it to a nonlinear (actually, quasilinear) case, and then finally explain the inverse problem. I admit to not having studied physics at university level. Hence, the notation and language I use might be exotic, for which I apologise. **Conductivity equation** Suppose we have a conducting body, $\Omega$, with conductivity (or conductance) $\sigma$. I will be using terminology from electricity, but this might also be heat conductance. Conductivity is the inverse of resistance, which I will not write more about. We take the material to be isotropic, which corresponds to conductivity with scalar (real) values. By (differential) Ohm's law the current J is proportional to conductivity and electric field, which we write as the gradient of potential v: $$ J = -\sigma \nabla v$$ By Kirchhoff's law the current is solenoidal, or divergence-free: $\nabla \cdot J = 0$. All in all, we have the conductivity equation $$-\nabla \cdot \sigma \nabla v = 0$$ If the conductivity is constant, say one, this is simply the Laplace equation $-\nabla^2 v = -\Delta v = 0$. **p-Laplace equation** We consider a nonlinear Ohm's law: $$ J = -\sigma |\nabla v|^{p-2} \nabla v $$ with finite parameter p > 1. The case p = 2 is the standard Ohm's law, and also the standard conductivity equation. Combining this with Kirchhoff's law we get the p-Laplace equation (with weight $\sigma$): $$-\nabla \cdot (\sigma |\nabla v|^{p-2} \nabla v) = 0$$ **Calderón's problem** The inverse problem of Calderón asks one to reconstruct the conductivity within the body by making boundary measurements: One prescribes either the current or the voltage on the boundary and measures the other one. These correspond to Neumann and Dirichlet boundary values for the equation - one sets the other and measures the other. This is a steady-state measurement, so one should wait for the current pattern to stabilise before making measurements. For the conductivity equation (p = 2 case) this problem has been studied, there are theoretical and computation results, and several applications, such as electrical impedance tomography which can be used to detect breast cancer, discover oil or image cracks in concrete. **Questions** Are there applications for the Calderón problem in the more general case where we need not have p = 2? That is, is there some medium on which the Ohm's law has the extra term $|\nabla v|^{m}$ for some non-zero m, and where finding out the conductivity by non-invasive boundary measurements would be useful? Is there some other nonlinear Ohm's law that is more realistic in situations where electrical impedance tomography or similar imaging methods would be useful?


title: How does the construction of a hydrophone differ from a microphone?

text: Both devices serve the same function, but a hydrophone must do it in a denser medium. What's the major difference in design? (besides being water-proof!) I'm asking about the relevant physics driving design in this case. I think a physicist would be able to explain this so that a good mechanical engineer would be able to design microphones for use alternately in air, underwater, submerged in honey, or at 20,000 feet.


title: Electric Potential Change

text: Imagine we have a conductor in the shape of a sphere with charge $Q$ on it. The conductor is not grounded. There is an associated potential $V=\frac{Q}{4\pi\epsilon_0 R}$ at and in the sphere, where $R$ is the radius of the sphere. If we now deform the spherical conductor into some other shape, e.g. an ellipsoid (or different shape) with the same volume, does it affect the potential (I don't mean the formula for $V$, but the amount of it)? How and why? Why it bothers me is that if the potential didn't change by deforming the conductor (and the total charge on it stayed the same) we would then have a relatively easy way of measuring the capacitance of such a conductor. We would just measure its volume, and calculate the capacitance of a sphere with the same volume.


title: Maximum temperature of an elliptical thermodynamic process

text: Given a heat engine made from the curve $$V(t)=V_0 +A \cos(\omega t); \qquad P(t)=P_0-B \sin(\omega t).$$ What are the hottest and coldest points of this cycle, and what are the temperatures at those points? I've tried setting $dT/dt=0$ with $T=PV/(Nk)$ many different ways, including stuff like having $P$ in terms of $V$ or using Lagrange multipliers, but the equations were always too messy. Any ideas?


title: Generalized Unitarity cut of Scalar One-loop Box integral

text: How does one perform the integrals in four particle cuts in generalized unitarity? It would be helpful how one finds solutions to the simplest case, the fully determined box integral given by: $\int_{\infty}^{\infty} \frac{d^4k}{(2 \pi)^4} \delta(k^2) \delta((k+p1)^2)\delta((k+p1+p2)^2)\delta((k+p1+p2+p3)^2)$ Where the k and $p_i$ are 4-dimensional complex momenta, and the external $p_i$ are massless and on shell i.e., $p_i^2=0$. I know there exist two solutions to the problem, and it would be helpful to be expressed in terms of mandelstam variables s,t,u I'm sure you can take a jacobian of each delta argument expanded explicitly in terms of (k0,k1,...) but I feel there should be an easier solution


title: Can the photoelectric effect be explained without photons?

text: Lamb 1969 states, > A misconception which most physicists acquire in their formative years is > that the photoelectric effect requires the quantization of the > electromagnetic field for its explanation. [...] In fact we shall see that > the photoelectric effect may be completely explained _without_ invoking the > concept of "light quanta." The paper gives a description in which an atom is ionized by light, with the atom being treated quantum-mechanically but the light being treated as a classical wave. Is it true that all the standard treatments in textbooks are getting this wrong? Lamb and Scully "The photoelectric effect without photons," in "Polarization, Matière et Rayonnement," Volume in Honour of A. Kastler (Presses Universitaires de France, Paris, 1969) -- can be found online by googling


title: Would it be consistent with QED to have leptons of different charges?

text: A recent question, Equality of electric charges of all leptons, made me wonder about a specific aspect of why exactly the charges of the different (free) fundamental particles are all the same. Specifically, why are the charges of the different (charged) leptons all equal? I understand this is a requirement in any treatment that includes the weak force, since the existence of $W^-\to e^-\bar{\nu}_e$, $W^-\to \mu^-\bar{\nu}_\mu$ and $W^-\to \tau^-\bar{\nu}_\tau$ vertices requires all the charges present to be equal. These vertices enable decays of the form $\mu^-\to e^-\bar{\nu}_e\nu_\mu$, which can only conserve charge if there's a single lepton charge unit. If you take away this interaction, though, and you simply consider a multi- species QED theory, this mechanism goes away, and it would seem that you could in principle have different charges for the different species. **Is this consistent with QED?** Or is there something else going on, within QED, which requires the charges to be equal?


title: Thermo-couple Eelectroomotive force

text: I am studying thermocouples. In a text book, the author said that, the electromotive force can be written as $$ E= \alpha \theta + \beta \theta^2 \tag{1}$$ where $\alpha$ and $\beta$ are constants and the thermocouple is connected at $0 \;\mathrm{°C}$ on one end and at $100 \;\mathrm{°C}$ at the other end. My problem is that I didn't find a good logic for the equation (1). Where does it arise from?


title: Determining The Coefficient of Viscosity and Diffusivity in an Estuary?

text: I am curious and eager to discuss about the experience of determining viscosity and diffusivity coefficient for our coastal or estuary model. I myself, just start to use the range of:               Vh = 5.d-5 to 5.d-3    # Uniform horizontal eddy viscosity coefficient (m2/s)     Dh = 0.05 to 0.2       # Uniform horizontal eddy diffusivity coefficient (m2/s)     Vv = 5.d-3 to 5.d-2    # Uniform vertical eddy viscosity coefficient (m2/s)     Dv = 0.05 to 0.2       # Uniform vertical eddy diffusivity coefficient (m2/s)      Are these values plausible? Are those ranges logical in representing a common estuary?


title: What is the Quantum equivalent of chaos on a classical system? (if there's any)

text: This is a question that bugging me around for some time now. It is not clear to me what is the meaning of a chaos if we consider a quantum system. What is the mathematical formalism (or the quantum analogy) to a orbit that is ergodic in the phase space? Can we still talk of a wave-function describing the particle? I mean, is it still well defined?


title: Would a Cup of Tea Be Hotter If you Add the Milk Before or After Boiling Water?

text: This is a bit of dispute between work colleagues. An answer would be greatly appreciated. **My argument is as follows:** If you add X amount of milk at a temperature of M to a mug at room temperature R before adding X amount of water at temp W, the result would be a cooler cup of tea than if you'd added the hot water first. This would be due to the milk reducing the overall temperature of the mug in the time that has been in it resulting in the addition of boiling water having less of an effect and resulting in a slightly cooler cup of tea/coffee. Conversely, if you add the hot water first, due to the mug being heated, the effect of adding the cool milk will be less and therefore the end result will be a hotter cup of tea. **My Colleagues arguement is simply that it would not make a difference but without any form of justification.** If anyone can give real scientific light to this issue it would be greatly appreciated :)


title: "Correlation energy" using the pair correlation function

text: In this paper on the Quantum Hall effect the authors refer to something called the _correlation energy_ of electrons. It is defined at the top of page 5 as $E=\frac{n}{2}\int (g(r)-1)V(r)dA\ ,$ where $n$ is electron density, $g(r)$ is the pair correlation function, $V(r)$ is the potential and $dA$ is the area element (area because the system is 2D). I'm trying to understand where this expression comes from, but searching around for "correlation energy" all I can find is connected to the Hartee-Fock approximation, which is unrelated I think. Testing the expression out it gives the same results as other ways of computing the energy per particle, with background charge included. Does anyone have an explanation for this expression? EDIT: prompted by an answer below I'll expand a bit on the mentioned "other ways of computing the energy." First we have the interaction between the electrons, which gives the following energy per particle: $E_{el}=\displaystyle\frac{\langle V\rangle}{N}=\frac{1}{N}\sum_{i<j}^N\int\prod_{k=1}^NdA_k\psi^*V(r_{ij})\psi=\frac{N-1}{2}\int\prod_{k=1}^NdA_k|\psi|^2V(r)\ , \ \ \ \ \ \ \ \ \ \ (1)$ where $\psi(r)=\langle r|\psi\rangle$ is the wavefunction and we've used the fact that it is an eigenfunction of $r$ and antisymmetric in electron exchange. At this point it is convenient to mention that all this takes place on a sphere; that is the electrons live on a spherical shell at the center of which we have a magnetic Dirac monopole. I didn't include this in the original post because I don't think it's relevant to the question of the expression for the correlation energy, but it explains the next term: To also include the effect of the positive background charge we can place it all in the center. The energy per electron resulting from the interaction between the background chance and the electrons, and the background charge with itself, will then be: $E_{bg}=-\frac{N}{2R}\ ,\ \ \ \ \ \ \ \ \ (2)$ where $R$ is the radius of the sphere (as explained better in e.g. Jain's book "Composite Fermions"). Returning to the expression for the correlation energy in the original post, we note that the definition of the pair correlation function for an isotropic system (so that we only need the relative coordinate, e.g. $r=r_1-r_2$), is: $g(r)=\frac{N(N-1)}{n^2}\displaystyle\int\prod_{k=3}^NdA_k|\psi|^2$. The term including $g(r)$ is therefore: $\frac{n}{2}\displaystyle\int dA\ g(r)V(r)=\frac{N-1}{2}\int\prod_{k=1}^NdA_k|\psi|^2V(r)\ ,\ \ \ \ \ \ \ \ \ (3)$ where we've used the fact that $g(r)$ refers to _any_ of the electrons to introduce an additional integration, over e.g. electron $1$, and cancel it's contribution with $1/A$, and the fact that $g(r)$ is isotropic to change an integration over a relative coordinate (the one in the original integral) to e.g. over electron $2$, bringing the integration to all electrons instead of just number $3$ to $N$. Lastly we look at the last term of the original expression: $-\frac{n}{2}\displaystyle\int dA\ V(r)=-\frac{n}{2}\int_0^R\frac{4\pi r\ dr}{r}=-\frac{N}{2R}.\ \ \ \ \ \ \ (4)$ We see that (1) corresponds to (3) and (2) to (4), so that this way of calculating the energy gives the same answer as the "correlation energy." The reason I'm interested in finding out about the latter is that this expression seems to be valid in more general cases, or at least easier to calculate, than the first method above. One example of this is when in the second Landau level and using an effective interaction in the first Landau level, so that $V(r)\neq \frac{1}{r}$ and it is less clear how to handle the background charge.


title: Does Clifford algebra depend on the topology of manifold?

text: We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or geometric algebra is believed to a reinterpretation of differential geometry suggested mainly by Hestenes and Doran. But as far as I know, many manifold-related theorem depends on the topology of the manifold such as connectedness, compactness, boundaryless or not. I want to know how Clifford algebra behave in different topologies?


title: Difference between a central potential that is a point and one that is a sphere?

text: In quantum physics there is a special case known as a particle in a spherically symmetric potential. I have a problem which is similar to the case of a hydrogen atom in that there is one free electron, but the "nucleus" is actually a group of several atoms. So I'm not sure how to find the radial wavefunction for this situation? If I could say that it's a point charge I could use the same radial wavefunction as for the hydrogen atom (I think). But since the group of atoms takes up space, how does that affect the potential/wavefunction?


title: Moving the plates of a charged capacitor to calculate energy density - where's the flaw in my argument?

text: For a charged air-dielectric capacitor, let the plates be parallel to the $xy$ plane, with the top carrying a positive charge $+Q$, the bottom a charge $-Q$. The force on an infinitesimal charge contained in a volume $dv$ inside the top plate is $$-\vec E_z\rho(x,y,z)dv$$ Integrating this over the volume of the plate gives the total force on its center of mass as $$-\vec E_z Q$$ which becomes using Gauss's law $$-\hat n_z E^2A\epsilon_0$$ If the plates are moved apart a distance $dz$ by an opposing force $\vec F$, the infinitesimal work done is $E^2\epsilon_0Adz$, and integrating this from zero to their final separation $d$ gives the energy density as $E^2\epsilon_0$. But this is different to the correct value of $1/2E^2\epsilon_0$, so where's the flaw in the above argument?


title: Can decompactification explain the inflation of the early universe?

text: I've just reread chapter 11 of this book where it is explained among other things, that our four dimensional universe could be unstable concerning a decompactification transition, since potential energy is needed to keep the extra dimensions rolled up. Are there some cosmological models that make use of this mechanism to explain the inflation of the early universe?


title: Force on Earth due to Sun's radiation pressure

text: I have been asked by my Classical Electrodynamics professor to calculate the force that the Sun exerts in the Earth's surface due to its radiation pressure supposing that all radiation is absorbed and a flat Earth, and knowing only that the magnitude of the Poynting vector in the surface is $\left\langle {\bar S} \right\rangle = 13000{\rm{[W/}}{{\rm{m}}^{\rm{2}}}{\rm{]}}$ using:   1. Maxwell's stress tensor.   2. The absorbed momentum. Using Maxwell's stress tensor I get ${\rm{35.6}} \cdot {\rm{1}}{{\rm{0}}^8}{\rm{[N]}}$, which seems plausible since we consider a flat Earth and no radiation reflection. But I'm lost on how to obtain an answer using the variation of electromagnetic momentum. I think I should start by writing $$\vec F = \frac{d}{{dt}}{\vec p_{EM}} = \frac{d}{{dt}}\int\limits_V {{\varepsilon _0}{\mu _0}\left( {\vec E \times \vec H} \right)dV}$$ But, how do I take it from here?


title: Conservation of momentum with MOND

text: Assuming as true the phenomenological MOND law for low accelerations ($< 10^{-10} m s^{-2} = a_0$), and considering a small mass $m$ attached to a larger mass $M$ by a faint spring (let's think of it as one of those ping pong tabs with a ball attached to the center by a springy thread). Now, the movement has two phases, when the ball is in the air and the ball acceleration points in the positive $X$ axis direction, the acceleration is always below $a_0$, so the MOND law says that the inertia (of $m$? or $M$ as well?) will be less than $m$, let's say it will be $\frac{m}{2}$. But when the ball hits the pad, it is bounced. The impulse during the impact lapse is presumably much higher than $a_0$, so the full inertia of the system applies to this phase. Now, linear momentum is conserved thanks to the third law of Motion (action and reaction), but I seek clarification for exactly how MOND affects linear momentum conservation? Does it violate it? Is Linear momentum conserved independently of it? Can MOND be fixed to conserve it? How does it look the change?


title: What's the differences among the concepts: binding energy, cohesive energy and formation energy?

text: In the papers about first principles (or ab initio) calculations, there are three energies which are often calculated: "binding energy", "cohesive energy" and "formation energy". Their meanings are alike, therefore they confuse me a lot. Can someone konw the correct definitions of them and show their differences in tail?


title: Compressed Spring Dissolving in Acid

text: This is just an interesting question a friend's uncle asked me that I was somewhat annoyed I couldn't answer. When a material dissolves in acid there is a chemical process that causes the changes in state/composition of the materials in question. There will be a change in energy due to chemical bonds forming/breaking and potentially a change in temperature. So we take a spring and compress it in a clamp. Obviously now there is a potential energy based on the compression of the spring as well as the chemical energy within the spring itself before compression. This is now placed into a bath of acid that will dissolve the spring but _not_ the clamp. There is a change of energy based on the dissolving of the spring, but **what happens to the potential energy due to it being compressed?** My thoughts were that in a realistic situation one (weak) point on the spring would be attacked, break, it would then probably just ping out of the clamp and lose the energy that way. But in the ideal scenario where is dissolves at a uniform rate, this wouldn't happen, so what happens at the instant that it is completely dissolved? This is obviously a massive assumption, so would be unlikely to be true. I thought maybe some kind of change of temperature would occur, due to the assumption that the compression would cause the atoms to be more tightly compacted causing a repulsion. As you remove layers this would gradually decrease, but there is still an inherent increase in energy for each atom, so this would most likely be expelled as something (like) heat. What are people's thoughts on this? **EDIT:** I'm mainly interested in the detail of the mechanisms for the energy tranfers on an atomic level as opposed to the broad picture. Specifically the transfer from coulomb repulsion to heat in the acid. The more low-level detail the better.


title: The conservation of energy when a spring is placed in acid

text: Stretch a spring out and give it a bunch of potential energy. Lower the spring into a vat of acid, disolving the spring into nothingness. Where did the potential energy go?


title: How does this shell hit the aircraft?

text: > A fighter aircraft is flying horizontally at an altitude of 1500m with speed > of 200m/s. The aircraft passes directly overhead an anti-aircraft gun. The > muzzle speed of the gun is 600m/s. > > We are required to find the angle to the horizontal at which the gun should > fire the shell to hit the air-craft. Here is how I did it: I assumed that the shell will hit the aircraft at time $t$ secnods. During which the horizontal distance traveled by the former will be $600\cos(\theta)t$ meters and that by the later will be $200t$ meters. Now, since shell will hit the aircraft, both the horizontal distances will be equal $\implies 600 \cos(\theta) t = 200t$ $\implies \cos(\theta) = \dfrac{1}{3}$ $\implies \theta = \cos^{-1}({{1}\over{3}})$ Which is the right answer! According to my book. But I think something is wrong with my calculation. I did not consider the altitude at which the aircraft was flying. My calculation was independent of the altitude. Would the altitude make no difference to the angle at which the shell should be fired. What if the altitude was 1000m? Or what if the altitude was more than the maximum height the shell could reach, my calculation still won't be affected. Is this the right way of doing this?


title: Why must SUSY be broken?

text: **Background** One usually claims that supersymmetry must be spontaneously broken. The reasoning is roughly the following: Since $M^2=P^{\mu}P_{\mu}$ is a casimir operator of the supersymmetry algebra, all the particles in a supermultiplet will have the same mass. Therefore the electron and the selectron will have same mass, and we would be able to produce selectrons at the accelerators, which nowdays operate at an energy scale of $1 \ Tev >> m_e\approx 0.5 \ MeV$. But clearly no selectrons are ever seen at the $MeV$ scale. The standard way to avoid this is to introduce some kind of supersymmetry breaking mechanism, of a similar kind of the Higgs mechanism in the standard model. **Question** Is it possible that susy is exact and not broken, but still supersymmetric particles can not produced in reaction of ordinary particles, basically because these reactions would violate the conservation of an extra (not yet known) quantum number? Why is such a scenario discarded a priori, an so much effort is put into the study of the susy breaking mechanism?


title: Test for Many Worlds Hypothesis

text: If one actually ran an experiment similar to the Schrodinger's cat thought experiment, and ran it enough times (with the same cat) to get the confidence interval good enough, could one say that suggests the many worlds hypothesis is true? For example, with 50% probability of living, after 5 tests, there would only be a ~3% chance of the cat still being alive. If it is alive, then can we say with 95% confidence that it is true? If it is true, then is my intuition correct that this experiment would fail in most worlds, but succeed in a few/one?


title: How long does it take a block to slide up and down a plane with friction?

text: So I am studying for my exam on monday for analytical mechanics. I am kind of stuck on a question, so wondering if someone can give me a pointer. I am actually looking for some pointers and some techniques (hints and tips) to get me through this. the topic is just simple velocity as a function of time, velocity as a function of displacement, quadratic air resistance, oscillations. So here is the question: > A block of wood is projected up an inclined plane with initial speed $v_0$. > If the inclination of the plane is $30^\circ$ and the coefficient of sliding > friction is $\mu_k = 0.1$, find the total time for the block to return to > the point of projection. Heres my work: $$F_{\text{net}} = -F_f = -\mu_k \cdot m \cdot g \cdot sin(30) = m \cdot \frac{dv_x}{dt}$$ so the m's cancel and solving the differential equation you get the following: $$ \int_{v_0}^v dv = \int_0^t -\mu_k \cdot g \cdot sin(30) dt$$ $$v-v_0 = -\mu_k \cdot sin(30) \cdot g\cdot t$$ so $$ v(t) = v_0 - 0.98\cdot sin(30) \cdot t $$ so if i set that to 0 i know the time when it reaches the top and starts to slide backwards. Now how would I know when it reaches the point of projection? Would it just be double the time?


title: Graph of Lorentz force in function of angle between v and B-field?

text: I need to draw a graph of the magnitude of the Lorentz force in function of the angle between the velocity and B. Am i right in supposing this will be a sine wave?


title: How is a standard unit divided into equally smaller or fractional units physically/experimentally?

text: Consider the standard unit of length: meter. How was it divided into decimeter, centimeter, millimeter, etc. when there were no shorter lengths than the standard? What is the physical/experimental process involved?


title: Circuit with inductor and lightbulb when switch is turned off?

text: I want to know what happens with a lightbulb immediately after a switch is turned off. We're talking about a circuit with a battery (AC), an inductor, a lightbulb and a switch. I need to know it in case the inductor and lightbulb are in serie and in parallel. I know you can see the lightbulb as a resistor but i'm kinda lost on how to calculate this. Help would be much appreciated! I also need to draw the current in function of the time. Will this be a wave?


title: Heisenberg uncertainty principle and minimum energy

text: In an exercise, given the **average** lifetime $\tau$ of a particle, the author estimates the **minimum energy** using the uncertainty principle formula : $\Delta E \Delta t \geq \hbar/2$, assuming $\Delta t = \tau$ and interpreting $\Delta E$ as the minimum energy. Shouldn't $\Delta E$ and $\Delta t$ be **uncertainties** and not minimum or average values ?


title: What is the difference between the words transparent and translucent?

text: Merriam Webster defines **_transparent_** as: > Having the property of transmitting light without appreciable scattering so > that bodies lying beyond are seen clearly. And **_translucent_** as: > Transmitting and diffusing light so that objects beyond cannot be seen > clearly. Now if you see any object through a lens or a bottle filled with water then most of the times whatever you see is not at all clear. What does this signify? Are these two transparent or translucent? Or is their behaviour conditional to how we see through them?


title: Can Hubble red shift be interpreted as time dilation?

text: Can we interpret the de Sitter universe as a spherical cosmic horizon null surface of finite radius, centered at Earth, and containing the Hubble volume of space where time is dilated and radial dimensions contract closer to the edge in such a way that objects closer to the edge do not recognize that they are radially contracted? Everything is attracted to the edge, but the total radius remains more or less constant and emits de Sitter radiation at finite temperature. ![enter image description here](http://i.stack.imgur.com/xgtGt.png)


title: Yang-Mills Lagrangian invariant under BRST

text: In equation 16.47 in Peskin & Schroeder, it is claimed that $$ -\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}\,^{b}c^{d}c^{e}+A_{\mu}\,^{d}c^{e}c^{b}+A_{\mu}\,^{e}c^{b}c^{d}\right) ~=~ 0 \tag{16.47}$$ using the Jacobi identity $$ f^{ade}f^{bcd}+f^{bde}f^{cad}+f^{cde}f^{abd}~=~0, \tag{15.70}$$ where $A$ is the gauge field and $c$ is the ghost Grassmann field. I was trying to prove this assertion and failed. This is what I've tried: 1) Write $ f^{abc}f^{cde}=-f^{dbc}f^{cea}-f^{ebc}f^{cad} $ using the given Jacobi identity. 2) Relabeling some indices and using the anti-commutation of the ghost fields I am able to rewrite the expression as $$ +g^2f^{dbc}f^{cea}\left(A_{\mu}\,^{b}c^{d}c^{e}+A_{\mu}\,^{d}c^{e}c^{b}+A_{\mu}\,^{e}c^{b}c^{d}\right) $$ Now I am stuck.


title: What happens when you cut a biconvex lens in half?

text: Specifically, does the focal length change? How can this be rationalized?


title: One question about renormalization

text: The idea of renormalization of "naked" perturbation theory is in principal possibility of addition counterterms which reduce infinity when calculating matrix elements. But I have met such concepts as exact propagator and the exact vertex part. So, the question: are exact propagator and vertex part collected all countreterms of our theory (i.e. all countreterms are convoluted into expressions which are called exact propagators and vertex)? I.e., do they (exact propagators and vertes) contain all information about renormalization of the given theory?


title: Block on an inclined plane

text: If you take moments about the centre of mass of a block positioned on an inclined plane so that the gravitational force can be drawn from the centre of mass of the block to one corner of the block, why is there no turning effect due to the reaction?


title: Why is the universe so organized?

text: If you think about the Big Bang and the flow of matter in all directions, you would think, how unorganized would this universe be? No matter how long it would take. The idea that matter or most of it will be organized in such a beautiful way is unthinkable. Check this for example. The Moon orbits around our planet in a perfect fashion, not one but millions of system are like this. Our planet circles the Sun, again there are millions of examples like it. Our star circle the galaxy center in a predefined path. Again billions of stars do this! Now if you look at the matter that is organized (has found fixed orbit) vs that that is not, almost 90% of the matter is organized. They have fixed trajectories and have a stable path. In fact, if you want to look at matter that is not unorganized, you will find very little (we can think of our own solar system). Now there is not one galaxy but billions of them, all of them similar shape, like or spiral, which has stars. Outside this galaxy, there are few stars or maybe a supernova somewhere. Why and how is our universe so organized? Is the law incredibly simple, since the law has been repeated billions of times in every galaxy, or is it the work of some phenomenon of which the details are not yet known?


title: Deuterons can share quantum states, but why it is fine for the protons and neutrons inside?

text: Deuteron (2H) is composed of a neutron (spin-1/2) and a proton (spin-1/2), with a total spin of 1, which is a boson. Therefore, it is possible for two deuterons to occupy the same quantum state. However, the protons and neutrons inside are fermions, so they proton/neutron shouldn't be able the share states due to the exclusion principle. Why it does not happen?


title: Speed of neutrinos

text: Everyone knows it is close to $c$, but how close? What are the recent results?


title: On Bell inequality and bound entangled states

text: I have recently seen some presentation slides of Michał Horodecki (slide number 77) in which he discussed the following conjecture. > Bound entangled states satisfy all Bell inequalities The conjecture is not true for multi-partite systems, and the references are given in the talk itself. I believe by now, the bipartite case is also settled, one way or the other. Bell theorem has some applications in computer science areas, like cryptography (non-local games). Is there some similar application of the (falsification of) the above conjecture? I mean, what are the interesting applications which can take place by disproving the above conjecture? Advanced thanks for any help/suggestions. **To the Moderators:** I do not know, whether I should have asked this question in Physics or in Cstheory. Considering the fact that there can be physical applications also, I am posted the question here. Please suggest and/or take the required actions.


title: Organs & Oscillations: An Analysis on the Temperature Dynamics of Solids

text: Does temperature have an influence on the frequency of an oscillating organ pipe?


title: How do you calculate vortex shedding frequency?

text: I am attempting to try to find out if there is any effect of Von Karman vortices on a group of wind speed readings where it is presumed that due to a mountain nearby the data collection spot Von Karman vortices were formed which would register in the data. Based on what I have read the vortex shedding frequency which should be present in the data is simply calculated from the formula $F= \mathbb{St}\cdot \frac{V}{L}$ with $\mathbb{St}$ as the Strouhal number, $V$ as the velocity and $L$ as the characteristic length. From what I understand the Strouhal number is essentially constant for a given flow, but when I looked up how to find the Strouhal number it was defined as $\mathbb{St} = F\cdot \frac{L}V$, which is the exact same relationship as given in the first equation for the computation of the shedding frequency. My question then is in two parts: 1) How do I calculate the Strouhal number for a flow so that I can calculate the shedding frequency? 2) What should I use as my characteristic length for a landscape feature such as a mountain or cliff etc as everything online tells only in the case of uniform cylinders and using their diameter instead of something like a cone which can be used as an approximation of a mountain or other landmass. thank you for any help.


title: Rotational inertia of a ball

text: This question refers to the solution of problem 12 here. It involves a spherical shell of mass $M$ filled with frictionless fluid of mass $M$ rolling down an inclined plane. (This is problem 12 of the 2013 F=Ma exam, a high school competitive physics exam). I understand the solution, but I don't understand how the given rotational inertia is obtained. We have a spherical shell of mass $M$ filled with frictionless fluid of mass $M$. The rotational inertia is apparently $$I_{contact} = \frac{2}{3}MR^2 + MR^2 + MR^2$$ Why? In particular, why is the $MR^2$ term added twice?


title: Is the concept of rest mass correct?

text: Is the concept of rest mass correct? All these years, we (me, and my classmates of Undergraduate 1st Year) have been accustomed to the concept of rest-mass, and the relativistic transformation of mass at speeds of the order of _c_. We had been introduced to this concept during our school-years, we were comfortable with it for quite a long period of time, until yesterday, when our teacher declared with a convincing degree of confidence, that it is not mass that changes relativistically, but rather, it is the momentum possessed by the body, that does so. Mass, as he says, is an invariant. P.S., he even declared that the conception that we had so long "cherished" in our hearts, is a wrong-one. Naturally, the whole class was wonder-struck. I am looking here for views on the subject from other, **reliable** sources, because I am (rather, _we_ all are) in rather a big-dilemma whether to accept our teacher's view, or to reject it. You should know, that our undergrad. classes have started just 3 days ago, and at the very onset, we have faced a heavy-weight _punch_.


title: In general, could any ad-hoc relationship of constants be useful?

text: In general; if one creates an ad-hoc relationship of constants, can we use it to solve equations OR is it just an abstract/artificial math construct? I'm a grad student and as we all know, these ad-hoc constants are very easy to make, but are they useful? Sometimes I think that I'm chasing my own tail... ![R0^73*w0^73*e0^2/\(c0^68*h0^74*A0^2*G0*N0^73\)=23/10390](http://www.sendspace.com/pro/dl/rbvvix) Granted that the example is an artificial relationship, and has an error of a few parts per billion; still this type of ratio seems both unit and Prius invariant. The constants aren't going to change, hence the ratio should stay the same in all units. ![maple sheet](http://www.sendspace.com/pro/dl/7chckl) In summary, if one creates an artificial ad-hoc relationship; then it is abusing dimensional analysis by definition?


title: How accurately can I expect to measure the gravitational constant with a club of college students?

text: I am a math instructor with almost no experimental physics background, but I run a math and engineering club that is interested in doing some experiments. I have read up a bit and see some obvious plans for calculating g with pendulums or falling objects, and a more complicated plan (Cavendish experiment). Maybe there are others out there. If I implement these methods with a group of college students, **how accurate is my result likely to be?** I'm particularly interested in getting a low-budget experiment that would be able to tell the difference between daytime gravity and nighttime gravity ([Our gravitational attraction toward the sun is approximately $0.006 \frac{m}{s^2}$; please excuse my issues with frames of reference]) but maybe that is completely out of reach. Thanks for your help or references.


title: How camera flash can send information

text: Nikon have CLS a system that allow to trigger one flash with other flash. I don't know how exactly it work from user point of view, I think that one flash (in command mode) can send information like flash compensation (+ or -) or metering using TTL (I think that when that option is turn on light meter from camera change how external flash function). How this is possible from physics point of view? How flash which is just light can send information? I know that you can send information with fiber optics but I don't think that this is the same.


title: Why are Navier-Stokes equations needed?

text: Can't we picture air or water molecules individually? Then, why are Navier- Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or inefficient to generate the picture of whole flow?


title: How to deduce E=(3/2)kT?

text: It says in my course notes that a particle has so-called "kinetic energy" $E=\frac{3}{2}kT=\frac{1}{2}mv^²$ Where does this formula come from? What is k?


title: Statistics of bound states of anyons with order pq

text: Anyons with fractional statistics are possible in 2 spatial dimensions, as shown by Wilczek. Suppose we have two identical anyons of spin 1/pq, where p and q are integers more than 1. Then, interchanging both of them will pick up a phase factor of $e^{-2\pi i/pq}$, right? Suppose there is a bound state of p such anyons. Then, they've got to have a spin of 1/q. However, interchanging two such identical bound states will pick up a phase factor of $e^{-2\pi i p/q}$ instead of $e^{-2\pi i/q}$?


title: Physical meaning of non-trivial solutions of vacuum Einstein's field equations

text: According to Einstein, the space-time is curved and the origin of the curvature is the presence of matter i.e. the presence of the energy-momentum tensor $T_{ab}$ in Einstein's field equations. If our universe were empty (i.e. $T_{ab}=0$ and the cosmological constant $\Lambda$ is setted to be $0$) then I would expect only the flat solution to the vacuum field equations $$R_{ab}=0$$ Surprisingly there are non flat (or non trivial) solutions to the above equations, for example the Schwarzschild solution. This conflicts with the fact that the matter curves the spacetime, so what is the origin of the curvature for these non trivial solutions? **I understand that mathematically $R_{ab}=0$ (Ricci-flatness) doesn't imply that the metric is flat** , i.e. non trivial solutions are formally admissible, but I don't understand how this is explained physically.


Do glass beads show any piezoelectric property? Since glass beads is mostly made out of SiO2 which is piezoelectric, will it show any piezoelectric property. Thanks. If anything wrong with my question kindly please tell me. Thanks.


title: How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)

text: This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it also correct to think of an even or odd Chern number in terms of an even or odd number of band crossings across the Fermi energy? (If it's odd, there must be a band connecting the valence to the conduction band and therefore provide a topological protected surface state.) * * * Edit: These lecture notes* (under Point H) state: "The formula (49) was not the first definition of the two-dimensional Z2 invariant, as the original Kane-Mele paper gave a definition based on counting of zeros of the “Pfaffian bundle” of wavefunctions. However, (49) is both easier to connect to the IQHE and easier to implement numerically." and furthermore: "...and the Chern numbers of the two spheres are equal so that the total Chern number is zero. The above argument establishes that the two values of the Z2invariant are related to even or odd Chern number of a band pair on half the Brillouin zone."   * http://socrates.berkeley.edu/~jemoore/Moore_group,_UC_Berkeley/Welcome_to_the_Moore_group,_Department_of_Physics,_UC_Berkeley_files/mitcourse.pdf


title: Force as gradient of scalar potential energy

text: My text book reads > _If a particle is acted upon by the forces which are conservative; that is, > if the forces are derivable from a scalar potential energy function in > manner $ F=-\nabla V $._ I was just wondering what may be the criteria for force to be expressed as negative gradient of scalar potential energy and HOW DO WE PROVE IT?


title: almost same mass, same distance, different energy needs

text: Imagine a car, driving 10 km at constant speed in 6th gear. There will be a measurable fuel consumption. Now imagine the same car driving 10 km again at a constant speed (same as above) in the 1st gear. The fuel consumption will be MUCH higher. I would expect a higher consumption because the internal organs of the engine make more rotations and therefor "travel" a longer distance. But the real consumption will be really much higher then the one i would expect since the moving organs of an engine are RELATIVELY lightweight compared to a car. Now, $W = F \cdot d$, there is no relation to speed, just the distance. $P = W / t$ but since the speed is the same in both cases also $t$ should be the same in both cases. So why does the car in the first gear use up so much more fuel?


title: What electric field vector should I use for modeling unpolarized light?

text: Regardless of computational cost, light is a kind of electromagnetic wave, so it can be simulated with Maxwell's equations. If we want to simulate light with Maxwell's equations, we need to express the electric field vector of light source with a formula. If the light source is polarized, this task won't be hard, but what if I want to simulate natural light: to be precise, unpolarized light? Is there any approximate formula for it? If the final result I want to get is just light intensity, can I simply replace the unpolarized light source with a polarized one? Well, to be honest, this question came to my mind when I read a paper that simulate light in a nanometric optic probe with Maxwell's equations and the incident light in that paper is polarized, I just want to know if unpolarized light is also available for Maxwell's equations.


title: Why aren't betavoltaics and alphavoltaics batteries widely used?

text: Betavoltaic batteries are devices which creates electricity from beta radiation of a radioactive material. Alphavoltaics operate similarly, using alpha radiation. The concept was invented roughly 50 years ago and they are safe enough to be used, for example, in pacemakers. However, the Wikipedia article on them states that they were "phased out as cheaper lithium-ion batteries were developed." I feel, though, that lithium ion batteries are hardly up to the task that consumers would want them to perform: for instance, iPhones hold their charge for about a day and notebooks can sometimes manage no more than four hours. Betavoltaics, on the other hand, can hold their charge for years. Why, then, are they not used in commercial applications? What are their relative advantages and disadvantages with respect to the current solutions, and in particular to lithium ion batteries? **UPDATE** Amount of electricity is tied with half-life. For example, if Ni-63 has half- life of 100 years this means, that mole of Nickel will produce Avogadro/2 electrons during that 100 years. This means 10^21 electrons per year and 10^14 electrons per second. This means up to 0.1 mA or electric current. The energy of electrons from Nickel is 67 keV. This means that each electron has 67 kilovolts of electric tension. So, the power of electricity from one mole of Nickel-63 is 67000*0.0001 = up to 6 watts. Other way to calculate. If Nickel-64 produces 10^14 electrons per second, each of 67 keV of energy, then the power is 7 * 10^4 * 10^14 ev/s = 7 * 10^4 * 10^14 * 10^(-19) = 0.7 Watts. So, the numbers are consistent to the order of magnitude. Approximating, one mole of Nickel-63 provides 1 watt of electricity approx. This looks sufficient for many cases including iPhone power consumption. 1 mole of Nickel-63 weights 63 grams. iPhones accumulator can weight more than 100. So atomic batteries can supersede conventional batteries, and serve for years. So why we don't use them?


title: Limits of superfluidity

text: Superconductivity has limits of currents and magnetic fields they can endure before dropping the superconductivity phase What are the corresponding limits of superfluidity? what are the limit velocities of superfluid substances relative to the container, above which the fluid stops being superfluid?


title: Harmonic Oscillator Energy to Momentum Expectation Value

text: If we are given a wave function written in terms of harmonic oscillator energy eigenfunctions how can we determine the maximum possible momentum expectation value? It's a combination of the first two energy states, weighted with equal probability - why is there even a range of possible momentum expectation values?


title: Perturbations in arbitrary dimensions

text: In general is it acceptable to say that if a perturbation is in only one spatial direction then the energy eigenvalue to second order is only changed in that spatial direction? For example 3D isotropic oscillator: $$ H = e \,E_0 \,x$$ for $e$ and $E_0$ constants. Energy is then changed second order by $\frac{e^2 E_0^2}{2 m \omega^2}$


title: Is Newton's Law of Gravity consistent with General Relativity?

text: By 'Newton's Law of Gravity', I am referring to > _The magnitude of the force of gravity is proportional to the product of the > mass of the two objects and inversely proportional to their distance > squared._ Does this law of attraction still hold under General Relativity's Tensor Equations? I don't really know enough about mathematics to be able to solve any of Einstein's field equations, but does Newton's basic law of the magnitude of attraction still hold? If they are only approximations, what causes them to differ?


title: Can Lee-Yang zeros theorem account for triple point phase transition?

text: Now the prominent Lee-Yang theorem (or Physical Review 87, 410, 1952) has almost become a standard ingredient of any comprehensive statistical mechanics textbook. If the volume tends to infinity, some complex roots of the macro partition function may converge to some points $z_0,z_1,z_2,\dots$ on the real axis. Thus these $\\{ z_n \\}$ divide the complex plane to some isolated phases. According to the singularity near the $\\{z_n\\}$ every two neighboring phases may have phase transition phenomena occurr. Here comes my question. Considering three phases surrounding a triple point in a phase diagram, they can transit to each other (just think about water). _Since neighborhood along the real axis consists of only two possibilities, I wonder if this theory could account for a description of the triple point._ And what is the connection between neighborhood of patches on the complex plane and the neighborhood of phases in a phase diagram?


title: Does the Time Evolution Operator Commute with any Other Operators?

text: Does the time evolution operator in quantum mechanics commute with any other operators, with a commutator of zero? Also, what exactly is the utility of the time evolution operator, is it more convenient than simply writing the time dependent portion explicitly?


title: How is squeezed light produced?

text: Ordinary laser light has equal uncertainty in phase and amplitude. When an otherwise perfect laser beam is incident onto a photodetector, the uncertainty in photon number will produce shot noise with poisson statistics. However, laser light may be transformed into a 'squeezed state', where the uncertainty is no longer equally divided between the two quadratures, resulting in a reduction of shot noise. How is this done?


title: determining Induced emf/current through v x B . l

text: I am working on a question from my book. I am wondering how they conclude emf is negative in this case. looking at $$ \mathcal{E} = \vec{v} \times \vec{B} \cdot \vec{dl}$$ If there's some implied direction of the rod's direction a->b ($\hat{down}$) then the dot product would result in a negative (since $\vec{v} \times \vec{B}$ is $\hat{up}$). But is that really suppose to be implied? Am I missing something trivial? ![enter image description here](http://i.stack.imgur.com/NINjj.png)


title: What's with the very slightly larger mass of the neutron compared to the proton?

text: Neutron mass: 1.008664 u Proton mass: 1.007276 u Why the discrepancy? On a related note, how does one go about measuring the mass of a neutron or proton, anyway?


title: Expectation of a commutation relation

text: Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.


title: What was meant by the 'ponderomotive force' as understood by Minkowski?

text: Skimming through Minkowski's famous 1907 paper, he uses the term _ponderomotive force_. What does he mean by this?


title: Maximum electric field in a circle

text: I have a question as follows: "Show that $E_x $ on the axis of a ring charge of radius $a$ has its maximum value at $x = \pm a /\sqrt{2} $. Sketch $E_x$ versus x for both positivve and negative values of $x$" This is a bit more abstract than my other questions "Given some givens, find the electric field at this point". As I understand I need to find a function $E_x (x)$ along all x from $-a$ to $a$, and confirm the maxes by taking its derivative that $E_x '(x) = 0$ at the given maxes. I take a point on $x$ axis, I call it $x$. Considering all $dQ$ along the circle, all of the electric field should be along the x axis due to symmetry. That's nice I guess. The line towards $dQ$ forms an unknown length I'll call $w$. The line projected down from this point on $dQ$ I'll call $u$, and the point from the end of the projection back to $x$ is $v$. I don't have the rep to draw a pretty picture and share it here unfortunately. The direction I'm obviously going is some trigonometry trickery, but I have no idea where to go. I have $u,v,w$ as triangle sides and are unknowns and $w$ is needed for the distance part of the coulumb's equation. there is the triangle with $a,x,w$ but isn't a right triangle and I'm unsure of any useful properties to work with there. Perhaps there's some actual physics I have overlooked. Thanks for any help.


title: Human Kinematics:Does velocity or acceleration become zero at the moment of changing human limbs' direction of movement?

text: I need to know the answers to following questions about human kinematics. Can anyone kindly help me with it? When human moves hand in one direction then suddenly or not changes direction of hand movement, at the exact time of direction change does velocity or acceleration of hand become zero? Can human move hand or feet from one direction to other without velocity/acceleration being zero at the very moment of direction change?


title: Scattering problem: Expression for angular momentum of particle

text: I'm reading Goldstein's Classical Mechanics, the part on "Scattering" in the "Central Force" chapter. In relation to the figure below, he says that angular momentum, $l$, is given by $$l=mv_0s$$ where $v_0$ is the velocity of the particle and $s$ is the distance from the line of the center of force(as shown in figure). ![enter image description here](http://i.stack.imgur.com/JbPjP.jpg) However, given that $l=\overrightarrow r \times \overrightarrow p=r.mv_0. sin (\theta)$, and $l$ is measured about the center of the sphere shown, it seems as if he has concluded that perpendicular drawn from the center of the sphere to the point of closest approach of the particle is of length $s$, which generally need not be true. To put it simply, how did he get the above expression?


title: Characteristic of photons for constant speed

text: What characteristic of photons causes them to travel with constant speed in all frames of reference? Till the topics I have studied, we always assume this but never got any ideas what can be in the nature of photons (the quanta of electromagnetic field) that causes this. Maybe something is not clear to me. Please explain.


title: Can hair be used as an dielectric?

text: I was letting my mind freewheel, and it occurred to me that hair, containing keratin, is a polymer. It is also capable of acquiring and maintaining a charge as happens, on occasion, when brushed. So ...   1. Can hair be used as a dielectric in a capacitor?   2. Do any capacitors/manufacturers use hair as an dielectric already ?


title: The meaning of Goldstone boson equivalence theorem

text: The Goldstone boson equivalence theorem tells us that the amplitude for emission/absorption of a longitudinally polarized gauge boson is equal to the amplitude for emission/absorption of the corresponding Goldstone boson at high energy. I'm wondering what's the physical meaning of this theorem. Is there any relation between equivalence theorem and Higgs mechanism ?


title: Non-localities in Wilsonian effective action

text: Why terms non-analytical dependent on momenta in the effective action (in momentum space) are non-local? How to see this directly?


title: Do open strings with both ends attached to the same D-brane really exist?

text: Perturbatively, open strings with both ends attached to the same D-brane definitely exist. But D-branes have nonzero p-form charges. These charges give rise to a nonzero flux with nonzero stress-energy tensor. This backreaction leads to curved spacetime. Being BPS, the metric becomes that of an extremal black brane with the event horizon at an infinite distance (logarithmic divergence) away. I wish to apologize for asking bad questions. Please put up with a newbie like me. The energy of a string due to its tension is proportional to the length of the string and also the time factor aka the gravitational redshift factor. A string is a worldsheet after all. An open string extending radially from the event horizon has an infinite length, but the gravitational redshift factor cancels that, and the total energy of the open string remains finite. I have no question after all.


title: Finding force vectors

text: A tugboat tows a ship at a constant velocity. The tow harness consists of a single tow cable attached to the tugboat at point $A$ that splits at point $B$ and attaches to the ship at points $C$ and $D$. The two rope segments $BC$ and $BD$ angle away from the center of the ship at angles of $\varphi = 25.0^\circ$ and $\theta = 25.0^\circ$, respectively. The tugboat pulls with a force of $21,000\text{ N}$ . What are the tensions $T_{BC}$ and $T_{BD}$ in the rope segments $BC$ and $BD$? ![Ship](http://i.stack.imgur.com/lfvUh.png) I attempted to do this by finding $BD$ and $BC$ by $$ BD = 2100/\cos 25^\circ = 2317.1\text{ N} $$ But this is not right and I thought it was a bit too easy. Does anyone know anyway I would solve this?


title: Can magnetic flux be negative

text: I am studying magnetic flux linkage in an ac generator and it appears to be that magnetic flux linkage is negative half the time, how can this be?? Also with lenz's law why is emf defined as negative when magnetic flux is increasing and how does this relate to the direction of the current?


title: How to prove dp/dt = -dV/dx? Quantum mechanics

text: ![enter image description here](http://i.stack.imgur.com/74A2h.png) I got this problem from a book called Introduction to quantum mechanics, griffin 2nd edition. and I did not get why the solution says first term integrates to zero, integration by parts twice?! Please see the solution below! Thanks everyone for helping!


title: Why does the pion half-life differ between the charged and uncharged species?

text: Why does the uncharged pion have much shorter half-life than the charged pion despite the fact that the uncharged pion has a little bit less mass than the charged one, so that according to the uncertainty principle we should expect the uncharged pion to decay at a little bit slower rate when compared to the charged pion?


title: Earth as a conductor and emf produced?

text: Since about 73% of total surface of Earth is occupied by water, its internal is also occupied largely by hot molten mass.Let,the whole earth is considered a conductor.Sun is a source of gravitational force,lets consider that gravitational field of sun as magnetic field by a magnet(sun).Then,what the emf produced by the motion of conductor(earth) under a magnetic field(gravitational field) called?


title: Does change of coordinate system require acceleration?

text: This question came about from a side discussion that arose on this: Does GR provide a maximum electric field limit? Can we change our choice of coordinate system completely independent of physical motion, and still refer to coordinate dependent components as observations? For example if we have an inertial coordinate system in which we measure the electric field to be E, and now I want to boost the coordinate system to a different inertial coordinate system and state what the electric field E' is in this new coordinate system, do I need to physically accelerate to justify this change in coordinate system (and therefore need to worry about effect such as Unruh, etc. during "changing" the coordinate system)?


title: Chemical potential interpretation

text: Something that has bothered me for a while regards the interpretation of chemical potential for different statistics. While I understand its meaning in metals (and its relation with the Fermi surface), I cannot quite relate this definition with the thermodynamic chemical potential, defined as the change in energy of the system when one particle is added to it (or according to Wolfram Demonstrations, " It can be interpreted, for example, as the ability of the system to perform phase transitions or chemical reactions, or its tendency to diffuse"). 1) Are those concepts (thermodynamic vs Fermi-Dirac chemical potential) related or they should be thought as different things? 2) Am I missing something trivial or this cited demonstration is misleading? It is mentioned there that the x coordinate corresponds to (E- _u_ ). Shouldn't the function blow up for x = 0 independent of temperature, then? I do not understand the shift for negative values as temperature increases.